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TECHNICAL PAPERS149 Predictive Modeling of the Nonuniform Deformation of the Aluminum

Alloy 5182M. E. Bange, A. J. Beaudoin, M. Stout, S. R. Chen, andS. R. MacEwen

157 Multiaxial Cyclic Ratcheting in Coiled Tubing—Part I: TheoreticalModeling

Radovan Rolovic and Steven M. Tipton

162 Multiaxial Cyclic Ratcheting in Coiled Tubing—Part II: ExperimentalProgram and Model Evaluation

Radovan Rolovic and Steven M. Tipton

168 Thermomechanical Behavior and Modeling Between 350°C and 400°C ofZircaloy-4 Cladding Tubes From an Unirradiated State to High Fluence„0 to 85"1024 nmÀ2, EÌ1 MeV…

I. Scha ffler, P. Geyer, P. Bouffioux, and P. Delobelle

177 The Influence of Intrinsic Strain Softening on Strain Localization inPolycarbonate: Modeling and Experimental Validation

L. E. Govaert, P. H. M. Timmermans, and W. A. M. Brekelmans

186 Low-Cycle Fatigue of TiNi Shape Memory Alloy and Formulation ofFatigue Life

Hisaaki Tobushi, Takafumi Nakahara, Yoshirou Shimeno, andTakahiro Hashimoto

192 On the Nonuniform Deformation of the Cylinder Compression TestFuh-Kuo Chen and Cheng-Jun Chen

198 Fracture Static Mechanisms on Fatigue Crack Propagation in MicroalloyedForging Steels

M. A. Linaza and J. M. Rodriguez-Ibabe

203 Damage of Short-Fiber-Reinforced Metal Matrix Composites ConsideringCooling and Thermal Cycling

Chuwei Zhou, Wei Yang, and Daining Fang

209 Probabilistic Mesomechanics for High Cycle Fatigue Life PredictionRobert G. Tryon and Thomas A. Cruse

215 Study of Surface Residual Stress by Three-Dimensional DisplacementData at a Single Point in Hole Drilling Method

Z. Wu and J. Lu

221 A Study of Burr Formation Processes Using the Finite Element Method:Part I

I. W. Park and D. A. Dornfeld

229 A Study of Burr Formation Processes Using the Finite Element Method:Part II—The Influences of Exit Angle, Rake Angle, and Backup Materialon Burr Formation Processes

I. W. Park and D. A. Dornfeld

238 Densification Behavior of Ceramic Powder Under Cold CompactionK. T. Kim, S. W. Choi, and H. Park

Journal ofEngineering Materialsand TechnologyPublished Quarterly by The American Society of Mechanical Engineers

VOLUME 122 • NUMBER 2 • APRIL 2000

M. E. BangeA. J. Beaudoin

Department of Industrial andMechanical Engineering,

University of Illinois at Urbana-Champaign,Urbana, IL 61801

M. StoutS. R. Chen

Los Alamos National Laboratory,Los Alamos, NM 87545

S. R. MacEwenAlcan International Ltd.,

Kingston, Ontario, Canada

Predictive Modeling of theNonuniform Deformation of theAluminum Alloy 5182We pose an experimental model for hot deformation that, in complexity, falls between ahomogeneous laboratory test and an industrial process. Our objective is to document atransition between two distinct modes of thermally-activated deformation: diffusion con-trolled solute drag and hardening with concurrent recovery of dislocations. We demon-strate that constitutive equations for plasticity, describing different regimes of dislocationkinetics and calibrated with a minimum of adjustable constants, can be incorporated intofinite element analysis and used reliably to predict the mechanical response of a nonuni-form body.@S0094-4289~00!00302-9#

1 IntroductionThere exists considerable difference between the laboratory set-

ting used for characterization of metals and the industrial plantwhere the resulting material models are applied. In the laboratory,it is possible to pose a thermomechanical history that elicits aparticular microstructural response. In contrast, the typical indus-trial process will span a considerable range of conditions in de-formation, time, and temperature space to achieve desirable prop-erties. A thermomechanical history encompassing recovery,recrystallization, and work hardening will likely be encounteredduring hot rolling, for example. Coordination throughout the pro-cess is critical to achieve optimum properties. Taking aluminumalloys used in automotive and packaging applications as an ex-ample, critical demands on formability, strength, and surface fin-ish are achieved through careful manipulation of casting, hot roll-ing, and heat treatment. It is only through understanding of theinteraction between metallurgical actions~at the scale of the mi-crostructure! and thermomechanical conditions~dictated by themacroscale process pathway! that target properties can beachieved.

In the following, we pose a deformation model that, in com-plexity, falls somewhere between a homogeneous laboratory testand an industrial process. Our objective is to document a transi-tion between two distinct modes of thermally-activated deforma-tion: diffusion controlled solute drag and hardening with concur-rent recovery of dislocations. A further objective is the validationof the constitutive model and finite element implementation. TheAl-Mg alloy AA 5182 is studied, thereby providing a materialwith both microstructural complexity and industrial relevance.The end result is a test procedure that serves as a rigorous valida-tion of the constitutive model and finite element implementationat a level of complexity that lies between homogeneous testingand industrial processing.

The approach to the design and experimental evaluation of theinhomogeneous compression test~as well as a general outline ofthis manuscript! follows.

• The constitutive response of AA 5182 is detailed and twodistinct regimes of high temperature deformation are identified.Description of the constitutive response is achieved through twophysically-motivated relationships for plastic flow.

• A finite element formulation for the simulation of a visco-

plastic deformation is cast and simulations are conducted to pre-dict shape change for an inhomogeneous deformation. The result-ing design offers a specimen shape in which the activedeformation mechanism, by virtue of the rate sensitivity, createsvariations in the deformed geometry.

• Experimental testing is undertaken using the design devel-oped from the finite element simulations. Detail is also given toprocedures necessary to minimize both friction effects and ther-mal gradients.

• The deformation is characterized by the shape of the sampleafter compression and by optical metallography. Comparison isdrawn between the experimental and simulation results.

Finally, the implications of this work are discussed with regardto the significance of inhomogeneity inherent in the experimentaldesign and the predictive capability of the numerical simulations.

2 Constitutive Response of AA 5182The aluminum alloy AA 5182~4.5 Mg, 0.35 Mn! ~Hatch @1#!

has been studied using compression testing at constant strain rate~Chen et al.@2#, Stout et al.@3#!. It was found that the strain ratesensitivity ranges from a value of 1/3~a relatively high rate sen-sitivity! at high temperatures and low strain rates to a small nega-tive value at room temperature. As the experimental observationsand resulting constitutive formulation are key to the present re-search, this prior work will be briefly reviewed.

At low stresses, plastic flow is diffusion controlled through sol-ute drag~SD!. In this regime, the stresss is characterized by therelationship

«5AS s

m D 3 mb3

kTexpS 2

QD

RTD (1)

whereT is the absolute temperature,b is the Burger’s vector,k isBoltzman’s constant, andR is the universal gas constant.QD isthe activation energy of 131 kJ/mol, a handbook value for self-diffusion of magnesium in aluminum, andA is an adjustable con-stant. The temperature dependent shear modulus,m, is given bythe empirical relation~Varshni @4#!

m5m02D

expS T0

T D21

(2)

wherem0 , D, andT0 are fitted parameters.Shown in Fig. 1 are yield stress data taken from uniaxial com-

pression tests. The stress-strain rate relationship embodied inEq. ~1! is shown as a dashed line. Below a stress level of

Contributed by the Materials Division for publication in the JOURNAL OF ENGI-NEERING MATERIALS AND TECHNOLOGY. Manuscript received by the MaterialsDivision April 13, 1999; revised manuscript received November 13, 1999. AssociateTechnical Editor: H. M. Zbib.

Journal of Engineering Materials and Technology APRIL 2000, Vol. 122 Õ 149Copyright © 2000 by ASME

(s/m)'431023, the trend in the data demonstrates a similarpower-law behavior with a stress exponent close to 3~solid line!.Beyond this threshold stress level, however, there is an apparenttransition away from solute drag.

Increasing the stress, through reducing the temperature and/orincreasing the strain rate, leads to plastic flow with thermally-activated hardening and recovery~HR!. There is a correspondingreduction in the strain rate sensitivity. Such a regime is accuratelyrepresented by the Mechanical Threshold Stress~MTS! model~Follansbee and Kocks@5#!. The evolution of material state in theHR regime is described through a single scalar variable. Only avery brief outline of the MTS model will be provided here, moredetail may be found in references~Follansbee and Kocks@5#,Chen and Gray@6#, Chen et al.@2#!.

The flow stress,s, at constant structure is a function of tem-peratureT, strain rate«, and an internal variable which character-izes the structure,s« . The flow stress is given by

s

m5

sa

m1Si~ «,T!

s i

m01S«~ «,T!

s«

m0(3)

where

Si~ «,T!5H 12F kT

mb3g0ilnS «0i

« D G1/qiJ 1/pi

(4)

and

S«~ «,T!5H 12F kT

mb3g0«lnS «0«

« D G1/q«J 1/p«

(5)

In Eq. ~3!, sa represents an athermal contribution and the follow-ing two terms,Si andS« , provide thermal components of the flowstress arising from dislocation interactions with obstacles.Si andS« contribute to yield and hardening response of the stress, respec-tively. The evolution of the stateu5ds« /d« takes the form~Bronkhorst et al.@7#!

u

m5

u0

m0F12

s«

s«sGk

s«,s«s

u

m50 s«>s«s (6)

wheres«s is a function of temperature and strain rate,

s«s

s«s05S «

«0«sD ~kT/~mb3g0«s!!

(7)

The parameters for the model were fitted for AA 5182 over anappropriate range of strain rates and temperatures~Chen et al.@2#!. These parameters are listed in Table 1. While the number ofparameters is~at first sight! a bit disturbing, it must be noted thatmany of the parameters are relatively fixed—as dictated by thephysical motivation behind the MTS model. As an example, theparameterspi , p« , qi , andq« follow from a phenomenologicaldescription of the dislocation glide resistance profile and are lim-ited to the range~Kocks et al.@8#!.

0,pi ,p«<1; 1<qi ,q«<2 (8)

The terms «0i , «0« , and ««s0 are typically in the range of106– 107 s21. Parameterssa , s i /m0 , and g0i are readily deter-mined from simple linear regression applied to a so-called‘‘Fisher’’ plot of yield stress data~MacGregor and Fisher@9#!.

With regard to the forthcoming simulation procedures, an ap-proach must be adopted to transition between constitutive rela-tions for solute drag and HR. For a given temperature and strainrate the active mode of deformation is determined using Eqs.~1!and ~3!. Because of the high rate sensitivity, application of thepower law outside of the regime of solute drag quickly leads tovery high stress levels~dashed line of Fig. 1!. Hence, the mecha-nism presenting the lower stress is adopted—an algorithmicallyeasy choice. The state variable,s« is only evolved when HR is theactive deformation mechanism.

Finally, it should be noted that dynamic strain aging is observedfor conditions

F kT

mb3 lnS 1027 s21

« D G2/3

,0.32 (9)

Fig. 1 Yield stress data for AA 5182 plotted in the appropriatemanner for solute drag. The line, nÄ3, represents a strain rateexponent of 1 Õ3. Deviation from the dashed line provides anindication of an active mechanism other than solute drag.

Table 1 Parameters for mechanical threshold stress model

mm02

3440 MPa

expS215K

T D21m0 2.88153104 MPa

k

b3 0.5899 MPa/K

sa 10 MPas i

m00.010315 s«s0 1996 MPa

«0i 13107 s21 «0« 13107 s21 «0«s 13107 s21

g0i 1.196 g0« 1.6 g0«s 0.1058

pi 1/2 p« 2/3 u0 6800 MPa

qi 3/2 q« 1 k 2

150 Õ Vol. 122, APRIL 2000 Transactions of the ASME

Constitutive response is characterized by negative rate sensitivityand jerky flow. This deformation mechanism is not addressed inthe present work.

3 Design of the Test SpecimenInitially, an axisymmetric test specimen was postulated wherein

cross sections along the compression axis would have differentdiameters~Fig. 2!. This difference would lead to a gradient instrain rate through the specimen necessary to maintain equilib-rium. The notion is that a balance of axial force between the end~larger! and center cross sections would reflect the rate sensitivityassociated with the underlying mode of deformation. For example,the high strain rate sensitivity associated with solute drag wouldlead to a relatively smaller difference in the compressive defor-mation rate between cross sections as compared to the HR regime.A similar approach has been used by Lalli and DeArdo@10# toinvestigate the effects of hardening evolution in the deformationof pure aluminum.

Prior homogeneous compression tests demonstrated a transitionin the dominant kinetics at a temperature of roughly 400°C~Chenet al.@2#!. At this temperature, a strain rate of 1023 s21 resulted ina stress and strain-rate sensitivity indicative of solute drag~Fig.1!. Similarly, a strain rate of unity placed material response firmlywithin the HR regime~characterized by the MTS model!. Theconstitutive models described above were introduced into a hybridfinite element formulation with a viscoplastic description of ma-terial constitutive response~an outline of the formulation is given

in Appendix A!. Numerous simulations were carried out in thecourse of designing the test specimen. Geometries with both linearand curved transitions from the end~larger! diameter to the centerdiameter were considered. All specimen designs left some mate-rial at the ends with a constant cross section. The temperature inthe simulations was specified as 400°C and velocity boundaryconditions were manipulated to place constitutive response in ei-ther the realm of solute drag or HR.

The final choice for specimen geometry consisted of a constantradius transition between the end and center diameters~Fig. 3~A!!.Four dimensions were parametrically varied to arrive at the speci-men design: the end diameter, the diameter of the reduced~middle! section, the radius of curvature between these two sec-tions, and the total height of the specimen. For the configurationadopted, the overall height of the specimen was 12.5 mm and theend diameter was 10 mm~Fig. 4!. A reduced section, centeredaxially along the specimen, had a diameter of 4 mm. The radius ofcurvature between the end diameter and reduced section was 4mm. Associated with this geometry are cross-head velocities thatrender deformation within range of the desired model kinetics—solute drag~Fig. 3~B!! or the MTS model~Fig. 3~C!!. In bothfigures the stress invariant

s II 5A3

2s8:s8 (10)

is shown, wheres8 is the deviatoric stress tensor and : indicatesthe inner product. Constant crosshead velocities of 0.025 mm/sand 25.0 mm/s lead to specimen stresses in the domain of Eqs.~1!and ~3!, respectively.

4 Inhomogeneous Compression Tests

4.1 Test Procedure. An experimental procedure was devel-oped for the elevated temperature inhomogeneous compressiontests. This first series of tests was performed on an Instron model1331 servo-hydraulic machine equipped with a resistive furnace~AMTEL Laboratory at the University of Illinois at Urbana-Champaign!. A second series of tests was performed on a MTSmodel 880 servo-hydraulic machine~Los Alamos National Labo-ratory, Materials Science and Technology Division!. The secondset of experiments used cartridge heaters located in the ends ofmetal platens in order to achieve testing temperatures. The AA5182 specimens were turned from the center of a 25 mm thickplate and were oriented such that the compression axis was par-allel to the rolling direction. Load rings were also machined inboth planar ends of this specimen~Detail A of Fig. 4!. The load

Fig. 2 Simple notion that the deformed geometry at two dis-tinct cross sections 1 and 2 will reflect the strain rate sensitiv-ity m , following from force balance

Fig. 3 Effective stress from finite element simulations plotted in units of ln „sII Õm…:„A… undeformed specimen; „B… solute drag regime „SD…; „C… thermally activated hard-ening and recovery „HR…

Journal of Engineering Materials and Technology APRIL 2000, Vol. 122 Õ 151

ring creates a recess on the ends of each specimen which is filledwith lubricant ~Rastegaev@11#!. The lubricant used in these testswas a mixture of vacuum grease and colloidal graphite~2:1,respectively!.

For consistency between experiment and simulation procedure,the presence of a thermal gradient in the specimen must beavoided. An assessment of thermal gradients in the specimen wasundertaken during the first series of tests. Chromel-alumel ther-mocouples were placed along the compression axis in the centerof the specimen at three locations~Fig. 5!. In order to reduce heatconduction from the specimen to load frame, alumina platenswere placed on either side of the test specimen. The furnace con-trol thermocouple was spot welded to a metal washer which waslocated on the upper surface of the bottom alumina platen. Atemperature controller was programmed to ramp the resistive fur-nace to a test temperature of 400°C. The program consisted of aramp to 200°C with a short soak followed by a ramp to 400°C.The plot of the specimen temperature is shown in Fig. 6.

It can be see from the plot that no overshoot of the testingtemperature was observed. Also, the specimen remained within62°C of the test temperature for over six minutes. After each test,the specimens were quenched in water. The specimen entered thequench bath in under three seconds.

Corresponding to the simulations, all inhomogeneous compres-sion tests were performed using constant crosshead velocity.Crosshead velocities ofvz50.025 mm/s andvz525.0 mm/splaced specimen stress in the SD and HR regimes, respectively.Load and crosshead displacement data were collected for all tests.

4.2 Experimental and Numerical Results. The numericalsimulations resulted in two distinct deformed shapes for the SDand HR regimes~Figs. 3~B! and 3~C!, respectively!. The diam-eters of the center cross section from the two simulations arerecorded in Table 2. For the experimental tests, the stock materialis textured~Stout et al.@2#! so some ovaling of the compressionsample is expected~Beaudoin et al.@12#, Kalidindi and Anand@13#!. Major and minor diameters of the center cross section arelisted for the experimental samples. Finally, the ratio of diametersassociated with HR and solute drag is given for each pair of mea-surements. The simulation—for the most part—underpredicts themeasured dimensions. Yet, the ratio of the measured diameter forHR to that for solute drag shows a common trend between thesimulations and experimental tests~Table 2!.

Shadow graphs of the outline of each specimen were con-structed in order to provide rigorous comparison between defor-mation in the two constitutive regimes. This process involvesback-lighting each specimen, focusing the image on a screen, anddigitally scanning the image. The difference in deformation in thecenter region is clearly rendered in Figs. 7~A! and 7~B! for boththe simulation and experimental results, respectively. This trendwas repeated in all shadow graphs taken with the major and minordiameters, as well as the 45 deg bisector. The shadow graphs ofthe experimental specimens were overlayed with the side profilesfrom the simulations~Fig. 8!. Some fine details in the experimen-tal tests are not fully rendered by the simulations. The modelunderpredicts the deformation of the curved transition between themajor and minor diameters. This is possibly due to neglecting theload ring geometry in the finite element simulations. It is likelythat deformation of the load ring at the beginning of the test en-hanced deformation near the platen/sample interface~upper andlower portions of the sample in Fig. 8!. This may be a

Fig. 4 Final specimen geometry based on simulations

Fig. 5 Thermocouple positions and general experimentalsetup used for temperature verification „drawing not to scale …

Fig. 6 Specimen temperature recorded at the three thermo-couple positions shown in Fig. 5

Table 2 Measured diameter „mm … of the center „bulged … crosssection from simulation and experiment. Measurements on theexperimental specimens were performed using a machinistmicroscope.

Simulation

Experiment

Major Minor Average

Hardening and recovery (DHR) 7.619 8.133 7.608 7.871Solute drag (DSD) 7.271 7.744 7.287 7.516Ratio (DHR /DSD) 1.048 1.050 1.044 1.047


result of over-compensation for the effects of friction. That is, thehydrostatic pressure of the lubricant increases the radial force onthe load ring.

The load-displacement response is shown in Fig. 9. Displace-ment data collected from experimental tests was shifted 0.508 mmin order to correct for the deformation of the load ring geometry.This value is equal to the total height of the load ring. The simu-lation result slightly underpredicts the load developed in SD~Fig.9!. A slightly greater rate of hardening appears for the simulation,as compared to experiment, when HR is the active mechanism.However, the overall comparison is favorable. Control of friction

Fig. 7 Comparison of the deformed geometry: „A… for thesimulations and „B… for the experiments. Shadow graphsshown in „B… contain the minor axis of the center section.

Fig. 8 Comparison of the deformed geometry from the experi-ment and simulation for solute drag and HR

Fig. 9 Comparison of load-displacement curves between thesimulation and the experiment for both HR and SD regimes

Fig. 10 Micrograph of specimen deformed in the solute dragregime with ram velocity of v zÄ0.025 mm Õs: „A… unrecrystal-lized structure with compression axis oriented vertically; „B…

magnification of bulged section with compression axis ori-ented horizontally; „C… further magnification of the bulged re-gion, with compression axis oriented horizontally, showingserrated grain boundaries


effects in the experiment and accurate representation of the con-stitutive response in the finite element model are indicated by theload-displacement results.

4.3 Metallographic Analysis. Metallographic sampleswere prepared by making a longitudinal cut through the specimen,at roughly 45 deg to the major and minor axes. A Barker’s reagentwas used as an etchant. Optical metallographs are shown in Figs.10 and 11.

In the solute drag regime, grains take on a general aspect ratiothat reflects the deformation history corresponding to location inthe sample. Grains in the central section have a more elongated

appearance than grains in the upper section, reflecting a greaterdegree of plastic strain in the specimen center. A rather atypicalfeature lies in the appearance of the grain boundaries~Fig. 10~C!!.A coarse subgrain structure with planar arrays impinging on thegrain boundaries creates serrations.

In marked contrast, deformation atvz525.0 mm/s led to a re-crystallized microstructure~Figs. 11~A! and 11~B!!. The stressstate predicted by the simulation is reflected in the recrystallizedgrain structure. Grains in the center region are finer and reflect thehistory of locally elevated stress. The microstructure in this regionis equiaxed~Fig. 11~C!!. The experimental conditions chosen fortesting in the HR regime encourage recrystallization. For defor-mation taking place at 411°C and a strain rate of 10.2 s21, thetime needed for 50 percent recrystallization is 2.5 seconds~Wellset al.@14#!. While every attempt was made to quench quickly afterthe completion of the inhomogeneous compression tests, the timeto quench was greater than the rapid recrystallization times notedin the referenced work. We surmise that the microstructure shownin Fig. 11 follows from static recrystallization after completion ofthe test.

5 DiscussionThe key result of this work is the development of an experi-

mental test that highlights the constitutive response of a solution-strengthed aluminum alloy. As a further—and equallyimportant—objective, the test regime exercised both a constitutivemodel for viscoplastic response and implementation of that modelin a finite element code. Predictive capability of the computercode was demonstrated both in the use of the implementation as adesign tool and subsequent validation through experiment.Clearly, many repeated experimental iterations would be requiredto settle on a specimen design that gave a clear indication of thedominant mechanism of deformation. Use of the finite elementmodel as a design tool was indispensable in development of thetest procedure.

Uniaxial compression tests were used to develop the constitu-tive response~Chen et al.@2#!. The original concept of the presentexperiment followed a line of thought based on these homoge-neous tests. That is, to maintain equilibrium, the ‘‘compressive’’strain rate would be different in specimen cross sections of differ-ent diameter. This logic assumes that the shearing in the specimenis relatively small, and is reflected to some degree in the deformedmesh of Fig. 3~B!. However, a similar situation does not carryover to the case of deformation in the HR regime. From closeinspection of Fig. 3~C!, distortion in the mesh indicates that shear-ing is prevalent in proximity to the specimen center. The variationin recrystallized grain size, within the bulged region of Fig. 11~B!,emphasizes the non-uniformity of deformation. Simply put, thedeformation detailed by the simulation was rather more complexthan the simple stacking of homogeneous deformations that posedthe original conception of the test.

Finally, the approach to capturing the constitutive responsethrough distinct relations in accordance with distinct mechanismswarrants some comment. As noted above, for solute drag Eq.~1!involves a single free parameter,A. For the single temperatureconsidered—and when HR is the active mechanism—evolution ofstate,s« , is relatively modest. Stress response is dominated bythe first two terms of Eq.~3! and not the last term involvingS« . Inthe HR regime, the ‘‘free’’ parameterssa , g0i , and s i followfrom use of a Fisher plot~MacGregor and Fisher@9#, Kocks et al.@8#!. Other parameters are fixed for the most part. Accurate mod-eling of the constitutive response follows from a physically-motivated description, not a general curve fit.

Looking toward a more general application, one needs to con-sider the handling of the state variable with excursions of stressbetween HR and solute drag. Jump tests for a sequence of strainrates from 1 s21 to 1023 s21 and a return to 1 s21 show that asteady-state value of the flow stress is achieved after a slight tran-sient~Fig. 12!. This behavior, rapid establishment of a flow stress

Fig. 11 Micrograph of specimen deformed in the hardeningand recovery regime with a ram velocity of v zÄ25 mm Õs: „A…

recrystallized structure with variation in grain size along verti-cal compression axis; „B… magnification of bulged section withcompression axis oriented horizontally and revealing finegrains along regions of shear; „C… further magnification of thebulged region, with compression axis oriented horizontally


with little hardening, is central to the selection of 400°C as a testtemperature. Yet, recovery of prior hardening is indicated by thetransient present in the second jump from 1023 s21 to 1 s21. TheHR model is dependent on the concept of a state variable thatrepresents the resistance to further deformation; the SD model isnot, depending primarily on the amount of mobile solute in solidsolution. A general application of the constitutive formulationadopted herein must consider evolution of the state variable,s« ,for excursions into the regime of solute drag. The current workalso does not model plastic flow which occurs in the regime ofdynamic strain aging~high stress developed at lower testtemperatures!.

Future efforts will be aimed at the incorporation of a staticrecovery term into the evolution Eq.~6!. Such a step will furtherthe applicability of the constitutive model to a broad base of in-dustrially relevant problems.

AcknowledgmentsThis work was sponsored by NSF Career Award DMII-75154,

by a grant from Alcan International, Ltd., and with support fromU.S. Department of Energy, Basic Energy Sciences, Division ofMaterials Sciences, Center for Synthesis and Processing—MetalForming Project. Conversations with Prof. Paul R. Dawsonprompted consideration of thermal aspects of the test. Finally, thecareful attention paid by Dr. Peter Kurath, Mr. Manuel Lovato,and Dr. George Kaschner to the experimental tests is sincerelyappreciated.

Appendix A: Finite Element FormulationNeglecting inertial effects and body forces, the motion of the

body is described through

¹•s50 in Bs•n5 t on ]Bs

u5u on ]Bu (11)

Partitioning the deviatoric,s8, and hydrostatic,p, components ofthe Cauchy stress as

s5s82pI (12)

then forming a weighted residual in Eq.~11! and applying thedivergence theorem gives the equilibrium statement

EBs8:¹fdV2E

Bp¹•fdV5E

]Bs

t•f dS (13)

where f are weighting functions. The constitutive response iswritten as

s852meff~ «,T!D8 (14)

whereT is the temperature,D8 is the deviatoric deformation ratetensor, and« is taken as the second invariant of the deformationrate,

«5A2

3D8:D8 (15)

The effective viscosity,meff , is evaluated according to the appro-priate regime of behavior dictated by the stress, as describedabove. A weighted residual is formed on Eq.~14! giving

EBS D82

1

2meffs8D :YdV50 (16)

To complete the formulation, the constraint

2l¹•ui 115pi 112pi (17)

is employed where convergence of the pressure in thei 11 itera-tion provides near incompressible response~Zienkewicz et al.@15#!.

Trial functions are introduced into the residuals for the interpo-lated field variables. An eight node brick was employed with in-terpolation of the continuous linear velocity field, piecewise lineardeviatoric stresses, and piecewise constant pressure. Subse-quently, the discretized form of Eq.~16! may be inverted and theresult used to solve for the deviatoric stress in Eq.~13!.

A parallel implementation was used to conduct the simulations.All communication is effected through the Message Passing Inter-face~MPI! ~Gropp et al.@16#!. Global data structures from assem-bly of the finite element system of equations are contained inparallel objects. The maintenance of these objects through timestepping of the solution is enabled through the Portable ExtensibleToolkit for Scientific Computing~PETSc! ~Balay et al.@17#!. Theobject-oriented approach enables access to a variety of Krylovsubspace methods and preconditioners. The conjugate gradientmethod combined with Jacobi preconditioner was applied to thesesimulations.

References@1# Hatch, J. E., 1984,Aluminum: Properties and Physical Metallurgy, American

Society For Metals, Metals Park, OH, p. 353.@2# Chen, S. R., Kocks, U., MacEwen, S., Beaudoin, A. J., and Stout, M. G., 1998,

‘‘Constitutive Modeling of a 5182 Aluminum as a Function of Strain Rate andTemperature,’’Hot Deformation of Aluminum Alloys II. T. R. Bieler et al.,eds., The Minerals, Metals & Materials Society, Warrendale, PA, pp. 205–216.

@3# Stout, M. G., Chen, S. R., Kocks, U. F., Schwartz, A. J., MacEwen, S. R., andBeaudoin, A. J., 1998, ‘‘Mechanisms Responsible for Texture Development ina 5182 Aluminum Alloy Deformed at Elevated Temperature,’’Hot Deforma-tion of Aluminum Alloys II. T. R. Bieler et al., eds., The Minerals, Metals &Materials Society, Warrendale, PA, pp. 243–254.

@4# Varshni, Y. P., 1970, ‘‘Temperature Dependence of the Elastic Constants,’’Phys. Rev. B,2, pp. 3952–3958.

@5# Follansbee, P. S., and Kocks, U. F., 1988, ‘‘A Constitutive Description of theDeformation of Copper Based on the Use of the Mechanical Threshold Stressas an Internal State Variable,’’ Acta Metall.,36, pp. 81–93.

@6# Chen, S. R., and Gray, G. T., 1996, ‘‘Constitutive Behavior of Tantalum andTantalum-Tungsten Alloys,’’ Metall. Trans. A,27A, pp. 2994–3006.

@7# Bronkhorst, C. A., Kalidindi, S. R., and Anand, L., 1992, ‘‘Polycrystal Plas-ticity and the Evolution of Crystallographic Texture in Face Centered CubicMetals,’’ Philos. Trans. R. Soc. London, Ser. A,341, pp. 443–477.

@8# Kocks, U. F., Argon, A. S., and Ashby, M. F., 1975, ‘‘Thermodynamics andKinetics of Slip,’’ B. Chalmers, J. W. Christian, and T. B. Massalski, eds.,Progress in Materials Science, Vol. 19, Pergamon Press Ltd., Oxford, En-gland, p. 142.

@9# MacGregor, C. W., and Fisher, J. C., 1946, ‘‘A Velocity-Modified Tempera-ture for the Plastic Flow of Metals,’’ ASME J. Appl. Mech.,13, pp. 11–16.

@10# Lalli, L., and DeArdo, A., 1990, ‘‘Experimental Assessment of Structure andProperty Predictions During Hot Working,’’ Metall. Trans. A,21A, pp. 3101–3114.

Fig. 12 Strain rate jump test at 400°C with strain rates of 1,10À3, and 1 s À1 on homogeneous compression specimens


@11# Rastegaev, M. V., 1940, ‘‘A New Method of Homogeneous Compression ofSpecimens for Determining Flow Stress and the Coefficient of Internal Fric-tion,’’ ZADOVSK Lab., Vol. 6, p. 345.

@12# Beaudoin, A. J., Mathur, K. K., Dawson, P. R., and Johnson, G. C., 1993,‘‘Three-Dimensional Deformation Process Simulation with Explicit Use ofPolycrystal Plasticity Models,’’ Int. J. Plast.,11, pp. 501–521.

@13# Kalidindi, S. R., and Anand, L., 1994, ‘‘Macroscopic Shape Change and Evo-lution of Crystallographic Texture in Pre-Textured fcc Metals,’’ J. Mech. Phys.Solids,42, pp. 459–490.

@14# Wells, M. A., Lloyd, D. J., Brimacombe, I. V. S. J. K., and Hawbolt, E. B.,1998, ‘‘Modeling the Microstructural Changes During Hot Tandem Rolling of

aa5xxx Aluminum Alloys: Part 1. Microstructural Evolution,’’ Metall. Mater.Trans. B,29B, No. 3, pp. 611–620.

@15# Zienkewicz, O. C., Vilotte, J. P., and Toyoshima, S., 1985, ‘‘Iterative Methodfor Constrained and Mixed Approximation and Inexpensive Improvement offem Performance,’’ Comput. Appl. Mech. Engrg.,53, pp. 3–29.

@16# Gropp, W., Lusk, E., and Skjellum, A., 1994,USING MPI: Portable ParallelProgramming with the Message-Passing Interface, The MIT Press, Cam-bridge, MA.

@17# Balay, S., Curfman-McInnes, L., Gropp, W. D., and Smith, B. F., 1995,PETSc2.0 Users Manual~ANL Report ANL-95/11 ed.!, Argonne National Labora-tory, Argonne, IL.


Radovan Rolovic1

Research Associate

Steven M. TiptonProfessor

Department of Mechanical Engineering,University of Tulsa,

Tulsa, OK 74104

Multiaxial Cyclic Ratchetingin Coiled Tubing—Part I:Theoretical ModelingCoiled tubing is a long, continuous string of steel tubing that is used in the oil welldrilling and servicing industry. Bending strains imposed on coiled tubing as it is deployedand retrieved from a well are considerably into the plastic regime and can be as high as3 percent. Progressive growth of tubing diameter occurs when tubing is cyclically bent-straightened under constant internal pressure, regardless of the fact that the hoop stressimposed by typical pressure levels is well below the material’s yield strength. A newincremental plasticity model is proposed in this study that can predict multiaxial cyclicratcheting in coiled tubing more accurately than the conventional plasticity models. Anew hardening rule is presented based on published experimental observations. Themodel also implements a new plastic modulus function. The predictions based on the newtheory correlate well with experimental results presented in Part II of this paper. Somepreviously unexpected trends in coiled tubing deformation behavior were observed andcorrectly predicted using the proposed model.@S0094-4289~00!00402-3#

IntroductionWhen structural components are cyclically loaded in the plastic

regime, progressive plastic deformation can occur under a combi-nation of a primary~steady! loading and a secondary~cyclic!loading. In cases where maximum load levels induce plastic de-formation, small amounts of plastic strain are sometimes not re-covered during each cycle, which can lead to large accumulatedplastic strains. This can occur even when the primary load levelsare well within the elastic regime. This phenomenon is calledcyclic ratcheting. Coiled tubing undergoes such loading condi-tions in routine service. A schematic layout drawing of the coiledtubing deployment equipment is shown in Fig. 1.

The operational concept of the coiled tubing system involvesrunning a continuous string of steel tubing into a well to performa specific operation. When the operation is complete, the tubing isretrieved from the well and spooled onto its reel for storage andtransport. Throughout its life, coiled tubing endures severe cyclicdeformation imposed by the above-surface deployment hardware,high internal pressure, and axial forces. Due to tube dimensionsand curvatures of the spool and guide arch, the tubing is plasti-cally deformed every time it is deployed and retrieved from awell, with bending strains as high as 3 percent. The imposed in-ternal pressure can generate hoop stresses on the order of 50 per-cent of the material’s monotonic yield stress. When the tubing iscyclically bent under imposed internal pressure, a progressivegrowth of tubing diameter occurs. In some cases the tubing diam-eter can increase as much as 30 percent before a tubing failureoccurs. Severe diametral growth can indicate the onset of fatiguefailure and can cause other problems, such as tubing incompatibil-ity with implementation hardware.

A sophisticated multiaxial cyclic plasticity algorithm is neededto model the elastic-plastic behavior of coiled tubing. Investiga-tion of plastic deformation of engineering materials under multi-axial states of stress has mainly been confined to situations wherethe load history is relatively short and simple. In these cases, asmall inaccuracy in the theoretical representation of observed

physical phenomena usually results in satisfactory predictions.However, when the load history is more complex or repeatedmany times, inaccuracies can accumulate and final results candiffer significantly from physical observations. This inconsistencybetween theoretical predictions based on the existing plasticitymodels and physical observations is strongly manifested in thecase of multiaxial cyclic ratcheting in coiled tubing.

Conventional plasticity models based on kinematic hardeningrules proposed by Prager@1#, Ziegler@2#, Armstrong and Frederic@3#, Mroz @4#, Phillips and Lee@5#, Garud@6#, and Tseng and Lee@7#, cannot accurately describe multiaxial cyclic ratcheting. Tipton@8# showed that the Mroz hardening rule significantly overesti-mates the diametral growth of coiled tubing, especially for lowand moderate hoop to yield stress ratios. Bower@9#, Chaboche@10#, Hassan et al.@11#, Ohno and Wang@12–14#, Moreton@15#,and Jiang and Sehitoglu@16–18# also reported that conventionalplasticity models tend to overestimate cyclic ratcheting rates. Re-cently, Bower@9#, Chaboche@10#, Ohno and Wang@12–14#, andJiang and Sehitoglu@17,18# modified the nonlinear kinematichardening model by introducing new terms to the basicArmstrong-Frederic model. The ability of these new versions ofthe nonlinear kinematic model to describe multiaxial cyclic ratch-eting improved, but the complexity and the number of experimen-tal parameters increased significantly. Consideration of these fac-tors led to a new incremental plasticity model for multiaxial cyclicdeformation of coiled tubing.

New Plasticity ModelA yielding criterion is needed to define the onset of plastic flow

during an increment of loading. When the von Mises yield crite-rion is assumed, the kinematic yield function can be expressed asF53/2(S2a8):(S2a8)2sY

250, whereS is the deviatoric stresstensor,a8 represents the deviatoric quantity of the back stresstensor~center of the yield surface!, andsY is the yield strength inuniaxial tension. A bold letter denotes a Cartesian tensor. A colonbetween two second-order tensors denotes their scalar product.

A flow rule relates the plastic strain increment to the currentstate of stress and to the stress increment. The generally acceptedflow rule, the normality flow rule, proposed by Prandtl@19# andlater generalized by Reuss@20#, can be stated asde p

51/H^n:ds&n, wherede p andds are the tensors of incrementalplastic strain and stress,H is a proportionality factor called the

1Currently at Schlumberger Technology Corporation, Sugar Land, Texas, 77478~e-mail: [email protected]!.

Contributed by the Materials Division for publication in the JOURNAL OF ENGI-NEERING MATERIALS AND TECHNOLOGY. Manuscript received by the MaterialsDivision November 19, 1998; revised manuscript received September 21, 1999. As-sociate Technical Editor: H. Sehitoglu.


plastic modulus, andn is the unit normal to the yield surface. TheMacCauley bracket, &, is defined by x&5(x1uxu)/2.

A general stress-strain relationship for plastic deformation mustbe in the incremental form because plastic deformation is history~path! dependent. Assuming the additive decomposition of thestrain tensor,de5de e1de p, wherede e is the elastic strain ten-sor andde p is the plastic strain tensor, and the normality flowrule, the general stress-strain relation can be written asde5@De

11/Hnn#:ds, whereDe is the elastic compliance tensor, andnnis the dyadic product of two vectors, which is a special form ofsecond-order tensor whose components are given as@nn#5$n%$n%T.

New Hardening Rule. A hardening rule describes the evolu-tion of subsequent yield surfaces. Based on experimental observa-tions, a new hardening rule is proposed in this study. Phillips andTang @21#, Phillips and Lee@5#, and Phillips and Lu@22# con-ducted experiments on commercially pure aluminum, under bothproportional and nonproportional, axial-torsional loading condi-tions. They found that the center of the yield surface tends totranslate in a direction that lies between the stress incrementdsand the plastic strain incrementde p. They also found that in mostcases the motion of the yield surface is predominated by the stressincrementds; i.e., the center of the yield surface moves in thedirection that is closer tods than tode p. Similar observationswere reported by McDowell@23,24# for type 304 stainless steelunder proportional and nonproportional, axial-torsional loadingconditions. Based on these observations, a new hardening rule isproposed herein. The new hardening rule states that the motion ofthe center of the yield surface,a, is given by

da5dmS ds

idsi1q

de p

ide pi D , (1)

wheredm is a scalar which can be determined from the consis-tency conditionF(s1ds,a1da,sY1dsY)50, q is a hardeningparameter which, in general, is a function of material characteris-tics and loading, the norm of the second order tensords is de-noted byidsi5Ads:ds, andide pi5Ade p:de p. The graphicalrepresentation of the new hardening rule is shown in Fig. 2. Whenthe normality flow rule is assumed, it follows thatde p/ide pi5n, wheren is the normal unit vector to the yield surface, and thenew hardening rule can be written asda5dm(ds/idsi1qn).From the experimental observations mentioned above, it can beconcluded that in most cases 0<q<1.

Anisotropic distortion of yield surface, which manifests in anincreased curvature of the yield surface in the direction of plasticloading, has been observed by many researchers, for instance byPhillips and Tang@21#, Phillips and Lee@5#, Phillips and Lu@22#,and Eisenberg and Yen@25#. This anisotropic distortion of yieldsurface appears to cause the normal to the yield surface,n, at thecurrent stress point and the plastic strain increment,de p, to movecloser to the direction of loading~see Fig. 3!. The direction ofyield surface motion during plastic loading also becomes closer tothe loading direction as the anisotropic distortion progresses.

Thus, the hardening parameterq in Eq. ~1! should change accord-ingly if an ideal, undistorted yield surface is assumed. Otherwise,small inaccuracies in the calculation of yield surface motion canaccumulate and lead to large errors in the calculation of plasticstrain.

It is very difficult to examine and model the anisotropic distor-tion of yield surface, especially for long load histories. The influ-ence of anisotropic distortion of yield surface on the shift of yieldsurface motion toward the loading direction can be approximatedby using an undistorted yield surface and decreasing the harden-ing parameterq in Eq. ~1! as plastic deformation progresses. Thechange of the hardening parameterq can be modeled as a decreas-ing function of, for instance, the accumulation of equivalent plas-tic strain. If during the subsequent loading the stress incrementimpinges on a different, less distorted or undistorted part of theyield surface, some or all memory of previous anisotropic distor-tion can be lost, depending on how far the new loading point isfrom the center of the previously distorted region of the yieldsurface. This memory of anisotropic distortion of yield surfacecan be modeled by updating the accumulated equivalent plasticstrain, ee,acc

p , ~if ee,accp is used to describeq! when the stress

increment impinges on a different part of the yield surface.This study primarily deals with tubular structures submitted to a

combination of steady or quasi-steady internal/external pressure~primary loading! and a cyclic bending/axial loading~secondaryloading!. In this case the hardening parameterq can be describedas a decreasing function of the accumulation of equivalent plasticstrain, ee,acc

p , and the memory of anisotropic distortion of yieldsurface can be modeled by updatingee,acc

p as the steady~primary!stress level,sP , is changed. For coiled tubing applications theprimary stress is the hoop stress, i.e.,sP5sh . Based on experi-mental observations of multiaxial cyclic ratcheting in coiled tub-ing, the general form of the hardening parameterq should be ableto describe the behavior exhibited in Fig. 4.

Fig. 1 Schematic layout drawing of coiled tubing deploymentequipment

Fig. 2 Graphical representation of the new hardening rule

Fig. 3 Schematic illustration of anisotropic distortion of yieldsurface and the corresponding shift of the associated normalto the yield surface towards the loading direction


In a mathematical form, a hardening parameterq that satisfiesthe above considerations can be expressed as

q5Cq01Cq1

11Cq2@11Cq3usP /sYuCq4#~ ee,accp !2 , (2)

whereCq0 , Cq1 , Cq2 , Cq3 , andCq4 are material constants, andee,acc

p is equivalent plastic strain accumulated at a primary~steadyor quasi-steady! stress levelsP . The sum ofCq0 andCq1 is usedto define the initial value of the hardening parameterq ~see Fig.4!. The constantCq0 is also used to define the lowest value ofq—when anisotropic distortion of yield surface has been saturatedat the current primary stress level. The other three constants,Cq2 ,Cq3 , andCq4 , are used to define the nonlinear reduction ofq asthe distortion of yield surface progresses. An explanation of theempirical determination of the material constants in Eq.~2! isgiven in Part II of this paper. When the primary stress level ischanged byDsP relative to the previous load increment in theplastic regime,ee,acc

p is recalculated to account for previous an-isotropic distortion of yield surface as

ee,accp 5~ ee,acc

p ! i 21 [email protected]~E/sY!~DsP /sY!2#, (3)

where (ee,accp ) i 21 is accumulated equivalent plastic strain up to

the current load increment. The function given in Eq.~3!, which isused to update accumulated equivalent plastic strain between twoplastic loading steps, is shown schematically in Fig. 5. If there isno steady or quasi-steady stress, the primary stresssP in Eq. ~3!changes considerably with each load step, resulting inee,acc

p '0and q'Cq01Cq1 . Thus, when plastic loading is distributedaround the yield surface, it may be sufficient to assume that thehardening parameterq is constant.

It should be pointed out that the new hardening rule, given byEq. ~1!, would not ensure nesting of the yield surface with thenext inactive yield surface or limit surface in a multiple-surface ortwo-surface model. The new model uses a single yield surface todetermine if plastic flow will occur for an increment of loadingand the direction of plastic strain increment. The plastic modulusand material memory effects are modeled based on the back stressa. This way, the motion of the active yield surface can be speci-

fied according to experimental observations, which may not bepossible if auxiliary yield surfaces are placed around the activeyield surface.

Plastic Modulus. In this study, the plastic modulus,H, isdefined as a function of back stressa. The plastic modulus,H, isrelated to the uniaxial plastic modulus,Ep, as H52/3Ep. TheRamberg-Osgood equation for the uniaxial stress-plastic strain

curve,s5K8(ep)n8, can be used to calculate the uniaxial plasticmodulus asEp5ds/dep, whereK8 is the cyclic strength coeffi-cient andn8 is the cyclic strain-hardening exponent. Thus, theplastic modulus,H, can be obtained as

H52

3K8n8S sa

K8 D~n821!/n8

, (4)

wheresa is used instead ofs to account for material memoryeffects. In this study,sa is calculated based on the cyclic stress-strain curve wheneveria8i5ia8imax andse5se,max as

sa5sY1A32ia8i (5)

and in all other cases based on the hysteresis curve of the materialas

sa5sY10.5A32ia82ain8 i , (6)

wheresY denotes the size of the yield surface,a8 is the deviatoricback stress tensor~the deviatoric location of center of the currentyield surface!, se is the effective~von Mises! stress,ain8 is thedeviatoric back stress at the beginning of load reversal~see Fig.6!. The ‘‘max’’ subscript is used to indicate the maximum valueof the corresponding quantity during previous loading.ain8 is up-dated (ain8 5a8) every time the loading is changed from elastic toplastic, i.e., whenF(s),0 is followed with F(s)50 and]F/]s:ds.0, whereF(s) is the yield function. A new reversalof loading begins when elastic unloading occurs, i.e., whenF(s)50 and]F/]s:ds<0. Thus, once plastic deformation hasbeen initiated, a load reversal extends as long as the deformationstays plastic.

Analysis of Service Loading Applied to Coiled TubingDeformation of coiled tubing caused by bending loading is

strain-controlled because the tubing is bent over a cylinder or anarch with a constant radius of curvature. Therefore, the bendingstrain history in the axial direction of tubing,ex

b , is known, basedon the assumption that plane transverse sections of coiled tubingremain plane. This assumption has been supported experimentallyby Newburn@26#. Applied internal pressure will produce stresses

Fig. 4 Hardening parameter q as a function of the primarystress sP and ee,acc

p

Fig. 5 Accumulated equivalent plastic strain as a function ofthe primary stress level change

Fig. 6 Parameters used to determine the plastic modulus inthis study


in the hoop and radial direction of tubing. Thus, hoop stress,sh ,and radial stress,s r , histories are assumed to be known.

In this study, in order to decrease the computational time,coiled tubing deformation behavior is modeled based on the com-ponents of stress and strain calculated only as their average quan-tities at the midthickness radius of tubing. When tubing is con-strained onto a circular arch, the magnitude of bending strain inthe axial direction at the midthickness radius of tubing is given by

exb5

D2t

2Rbcosu, (7)

whereD is the outer diameter of tubing,t is the wall thickness,Rbis the radius of curvature of the bending neutral plane of tubing,and u denotes the angular position around the circumference oftubing relative to the top midthickness point on the convex side oftubing ~see Fig. 7!.

When tubing is subjected to internal pressure,p, hoop and ra-dial stresses are developed. Hoop stress at the midthickness oftubing can be estimated fairly accurately as average stress ob-tained from static equilibrium as

sh5pD22t

2t. (8)

Since Eq.~8! is derived from equilibrium consideration, it canbe used for both elastic and plastic deformation. The stress in theradial direction of tubing is caused by internal pressure and variesfrom 2p on the inside wall of tubing to zero on the outside wall.The radial stress distribution through the wall thickness is nonlin-ear and different for elastic and plastic deformation. The influenceof radial stress on plastic deformation of coiled tubing is muchsmaller than the influence of hoop and axial stress. Hence, radialstress at the midthickness radius of tubing for both elastic andplastic deformation can be approximated as

s r520.5p. (9)

When the ratio of the wall thickness to the tubing diameter issmall ~thin-walled tubing!, Eq. ~9! is fairly accurate. Based on thetubing geometry, bending radius, and applied internal pressure,the input loadingex

b , sh , and s r can be calculated from Eqs.~7!–~9!. The three remaining stress and strain components (sx ,eh , and e r) can be obtained from the general stress-strain rela-tionship given in the introductory part of the previous section.

The axial force resulting from the pressure on the closed endsof coiled tubing is computed by

Fx,p5p~D22t !2p

4. (10)

This axial force, together with any externally applied axial force,will cause additional uniform strain,ex

a , in the axial direction oftubing. In general, axial strainex

a cannot be determined directlyfrom the applied axial load. It has to be calculated iteratively from

the equilibrium condition in the axial direction. In practice, anidentical axial strainex

a is applied to every section around thecircumference in order to establish axial equilibrium.

DiscussionThe plastic modulus function, as described in Eq.~4!, accu-

rately describes cyclic material responses for the load historiesexamined in this study. However, the characterization of the ma-terial memory by only using Eqs.~5! and ~6! can result in over-estimating the plastic modulus for certain load histories. For theseload histories, a more robust description of the plastic modulusmay be employed. For instance, a more rigorous set of rules withadditional memory factors can replace Eqs.~5! and ~6!.

The introduction of the functionq in the new hardening rule isbased on observations of data from multiaxial yield surface prob-ing experiments conducted on aluminum and steel tubular samplesin axial-torsional and axial-bending-internal pressure loading.Specific multiaxial experiments are required to identify the param-eters defining the hardening functionq, but these can be standard-ized for particular applications. With coiled tubing, for example,diametral growth data from simple constant pressure, cyclic bend-ing tests were used to identify parameters in Eq.~2! empirically.The new hardening rule and plastic modulus function did an ex-cellent job correlating with diametral growth measurements, aswill be discussed in Part II of this paper.

The approach just described, with its parameters as identifiedfrom simple testing, was used by Rolovic@27# to make predictionsof tubing behavior under complex combinations of bending, axialforce, and internal pressure. Again, good correlation was demon-strated with experimental results. The routine made predictions ofcoiled tubing behavior that were previously unobserved and un-expected. These include diametral contraction during bendingcycles with positive internal pressure and axial contraction due tobending cycles following axial force applications. This mechani-cal behavior was subsequently substantiated experimentally. Theroutine also predicted accurately that slightly greater wall thinningoccurs at the compressive wall relative to the tensile wall, a trendpreviously noted in literature. Diametral shrinkage and wall thin-ning data and predictions are presented and discussed in Part II.

SummaryA single surface incremental plasticity model has been devel-

oped. The model is specialized for the prediction of multiaxialcyclic ratcheting of tubular structures submitted to pressure andbending/axial loading. A new hardening rule and plastic modulusfunction are implemented in the model. The hardening rule isbased on observations of experimental data available in the litera-ture. The model was used to successfully describe the mechanicalbehavior of coiled tubing under complex loading. The modelmade more accurate predictions of coiled tubing diametral growththan conventional plasticity theory and predicted a diametralshrinkage phenomenon that had never before been observed. Thisobservation was subsequently validated experimentally and is pre-sented in Part II of this paper.

AcknowledgmentsThe authors express appreciation for support of this work to

The University of Tulsa, The United States Department of En-ergy, and members of the University of Tulsa Coiled Tubing Me-chanics Research Project: Elongation and Diametral Growth.

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Fig. 7 Tubing geometry and the orthogonal coordinate systemfor an element of coiled tubing „x—axial, h—hoop, andr—radial direction …


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@21# Phillips, A., and Tang, J. L., 1972, ‘‘The Effect of Loading Path on the YieldSurface at Elevated Temperature,’’ Int. J. Solids Struct.,8, pp. 463–474.

@22# Phillips, A., and Lu, W., 1984, ‘‘An Experimental Investigation of Yield Sur-faces and Loading Surfaces of Pure Aluminum with Stress-Controlled andStrain-Controlled Paths of Loading,’’ ASME J. Eng. Mater. Technol.,106, pp.349–354.

@23# McDowell, D. L., 1985, ‘‘An Experimental Study of the Structure of Consti-tutive Equations for Nonproportional Cyclic Plasticity,’’ ASME J. Eng. Mater.Technol.,107, pp. 307–315.

@24# McDowell, D. L., 1987, ‘‘An Evaluation of Recent Developments in Harden-ing and Flow Rules for Rate-Independent, Nonproportional Cyclic Plasticity,’’ASME J. Appl. Mech.,54, pp. 323–334.

@25# Eisenberg, M. A., and Yen, C.-F., 1984, ‘‘The Anisotropic Deformation ofYield Surfaces,’’ ASME J. Eng. Mater. Technol.,106, pp. 355–360.

@26# Newburn, D. A., 1990, ‘‘Post Yield Cyclic Strain Response of PressurizedTubes,’’ Master of Science Thesis, The University of Tulsa, Tulsa, Oklahoma.

@27# Rolovic, R., 1997,Plasticity Modeling of Multiaxial Cyclic Ratcheting inCoiled Tubing,Ph.D. dissertation, The University of Tulsa, Tulsa, Oklahoma.


Radovan Rolovic1

Research Associate

Steven M. TiptonProfessor

Department of Mechanical Engineering,University of Tulsa,

Tulsa, OK 74104

Multiaxial Cyclic Ratcheting inCoiled Tubing—Part II:Experimental Program and ModelEvaluationAn experimental program was conducted to evaluate the plasticity model proposed in aseparate paper (Part I). Constant pressure, cyclic bend-straighten tests were performed toidentify material parameters required by the analytical model. Block pressure, bend-straighten tests were conducted to evaluate the proposed model. Experiments were per-formed on full-size coiled tubing samples using a specialized test machine. Two commonlyused coiled tubing materials and four specimen sizes were subjected to load historiesconsisting of bending-straightening cycles with varying levels of internal pressure. It wasobserved that cyclic ratcheting rates can be reversed without reversing the mean stress,i.e., diametral growth of coiled tubing can be followed by diametral shrinkage even whenthe internal pressure is kept positive, depending on the loading history. This materialbehavior is explained in the context of the new theory. The correlation between thepredictions and the test data is very good.@S0094-4289~00!00502-8#

IntroductionIn Part I of this paper, a single surface plasticity model was

presented which predicts multiaxial cyclic ratcheting in tubularstructures under a combination of pressure and bending/axialloading conditions. In Part II, experimental results from constantpressure cyclic bending tests are presented that were used to iden-tify material parameters for two coiled tubing materials. Resultsfrom more complex experiments are also presented which demon-strate the excellent predictive capabilities of the model. Coiledtubing tends to increase in diameter as it is cycled in bending withconstant internal pressure. This occurs in spite of the fact thattypical pressure levels cause hoop stress below 50 percent of theyield strength. However, the shape of the curve has not beenclosely studied and is usually considered linear during a constantpressure test. The data presented in this paper show nonlineardiametral growth, which varies with pressure level and bendingstrain range. Predictions from the model correlate well with theexperimental results. The model predicted another previously un-observed phenomenon: incremental diametral shrinkage duringcycling with positive internal pressure. This is predicted to occurwhen cycling follows previous cycling at significantly higher in-ternal pressure. Experimental results are presented from blockpressure experiments to validate this prediction. It should be notedthat additional complex experiments were conducted involvingbend-straighten cycling at various pressures with intermittentaxial loading, which are not included in this paper. Strain gagesmonitored axial elongation behavior during these tests and predic-tions from the analytical model correlated extremely well withthose data~Rolovic @1#!.

Experimental ProgramTwo coiled tubing materials, designated as QT-800 and QT-

1000, were used in this study. QT-800 is modified ASTM A-606Type 4 steel, normalized to a minimum nominal monotonic yield

strength of 552 MPa and minimum nominal tensile strength of620 MPa. QT-800 has a minimum elongation of 30 percent andthe maximum hardness of Rockwell C22. QT-1000 is modifiedASTM A-607 micro-alloyed steel, quenched and tempered tominimum yield strength of 690 MPa and minimum tensilestrength of 758 MPa. Low-cycle fatigue data were available forthese materials from a previous study~Tipton @2#!.

Diametral growth~ratcheting! tests were performed using full-size coiled tubing specimens. Straight specimens were cut at theend of the production line from the continuous tubing that wouldhave otherwise been spooled onto a take-up reel. Specimen geom-etry is shown schematically in Fig. 1. Tubing samples had nomi-nal outside diameters,D, of 31.75 mm~nominal wall thickness,t,of 2.41 and 3.96 mm! and 60.33 mm~nominal wall thickness,t, of3.40 and 3.96 mm!. The specimen length,L, was 1300 mm for the31.75 mm diameter tubing and 1400 mm for the 60.33 mm diam-eter tubing. The gage length,Lgage, was approximately 500 mmfor all tubing samples. This section received cyclic bending overthe full strain range associated with the curvature of the bendingmandrel.

Tubing samples were tested as received from the manufacturerwithout any modification. In some cases, the ends of specimenswere machined to fit into the pressure caps. The actual diameterand the wall thickness of the specimens were slightly differentthan their nominal values. Experiments were conducted using aspecialized test machine. Schematic layout drawings of the testmachine with a specimen in the straight and bent positions areshown in Fig. 2. The specimen is fixed at its upper end. Thebending mandrel has varying curvature along its length to providea transition into the gage section curvature. The form is straight atthe end, blending tangentially into a curvature of twice the gageradius for 50 mm and tangentially to its gage radius of curvature.Two forms were used for testing with gage radii of 1200 and 1800mm. Samples were bent and straightened using a hydraulic actua-tor with two V-profile rollers. Two mechanical limit switcheswere positioned to reverse the travel of the actuator, thus definingthe maximum bent and straight positions of the specimen. A pres-sure transducer was used to monitor pressure inside the specimen.Internal water pressure was controlled manually using propor-tional valves. A digital meter was used to display the instanta-neous value of internal pressure.

1Currently at Schlumberger Technology Corporation, Sugar Land, Texas, 77478~e-mail: [email protected]!.

Contributed by the Materials Division for publication in the JOURNAL OF ENGI-NEERING MATERIALS AND TECHNOLOGY. Manuscript received by the MaterialsDivision November 19, 1998; revised manuscript received September 21, 1999. As-sociate Technical Editor: H. Sehitoglu.

162 Õ Vol. 122, APRIL 2000 Copyright © 2000 by ASME Transactions of the ASME

Constant pressure diametral growth test data were collected atregular intervals throughout the life of each test. Measurementswere taken manually at the middle of the gage section using digi-tal calipers. The accuracy of the calipers was60.001 mm. Sincetubing becomes approximately 1 to 2 percent oval during bendingcycles, orthogonal measurements were taken across the tube andtheir average values are reported herein. One diameter,Dna , wasmeasured across the bending neutral axis and the other,Dtc , wasmeasured 90 degrees from the bending neutral axis from the con-vex tensile side of the tubing to the concave compressive side~seeFig. 3!. The average outside diameter was calculated asD50.5(Dna1Dtc).

After the specimens were taken out of the test machine, it wasobserved that the diametral growth varied along the gage length asshown in Fig. 4. This effect~referred to as ‘‘ballooning’’! causedthe location of the maximum diameter to vary randomly from testto test along approximately 200 mm of the gage length. Measure-ments could not be taken at the location of the tubing where themaximum growth occurred due to restricted physical access to thetubing and the random nature of the phenomenon. However, thegrowth trend demonstrated by the measurements was useful forvalidating the trend predicted by the analytical model. Diametermeasurements were taken at the maximum growth location after

the specimen was removed from the machine. These points wereused as targets for defining parameters in the incremental plastic-ity model.

Block-pressure diametral growth tests were conducted on tub-ing samples by applying a finite number of bend-straighten cyclesat constant internal pressure. The pressure level was changed sev-eral times during a test as shown in Fig. 5. Pressure levels werechanged to study the load sequence influence. High-low, low-high, and random pressure level sequences were investigated inthis study. Again, for these tests, diameter measurements weretaken at the middle of the gage section and the maximum diameterwas measured at the end of each test after the specimen wasremoved from the test machine.

Test Results and Model EvaluationExperimental results were used to examine coiled tubing defor-

mation behavior and evaluate the proposed analytical model.Based on the proposed model, a computer program was developedto simulate coiled tubing deformation behavior under any appliedloading history. Diametral growth data from relatively simple,constant pressure fatigue tests were used to determine parametersneeded for the model. Diametral growth data from more complexblock load histories were used to evaluate the model.

Computational Procedure. Based on the analytical modeldescribed in Part I, a computer program was developed to de-scribe deformation of coiled tubing. The plasticity model was ap-plied to discrete regions around the cross-section. Several loadhistories were simulated with the cross-section divided into 20,30, 40, 60, 100, and 200 segments. The results based on 40 seg-ments were negligibly different from the solutions obtained with200 segments. Based on this observation and the advantage ofshorter run times, the cross-section with 40 segments was used inthis study. Because of symmetry, only one half of the cross-section was modeled as shown in Fig. 6. Stress and strain quanti-ties were calculated as their average values at the center of eachsegment. All calculations assume that stress and strain are uniformalong the length of tubing.

Since the tubing has closed ends, there is an axial force actingon the ends of tubing due to internal pressure. The influence ofinternal pressure acting on the ends of tubing is modeled usingequilibrium of forces in the axial direction of tubing. This canonly be achieved iteratively, which increases the execution time ofthe computer program considerably compared to a closed formsolution. In order to reduce the number of iterations and the ex-ecution time, the axial equilibrium was satisfied by maintainingthe calculated axial force to within60.5 percent of the axial forcewhenever the axial force was not equal to zero. When the axialforce was equal to zero, the calculated axial force was maintainedwithin 6222 N. Based on the input data~material characteristics,tubing dimensions, and load history!, the program computed

Fig. 1 Specimen geometry

Fig. 2 Schematic layout drawing of bending Õtensile test ma-chine

Fig. 3 Tubing diameters measured during testing program

Fig. 4 Variation of tubing diametral growth along the gagelength

Fig. 5 Typical pressure variation during block-pressure tests


changes to the state of stress and strain for each segment. Thegeometry of each segment, as determined from its updated strainstate, was used to compute updated cross-sectional dimensions.These, in turn, were used to compute hoop and axial stressescaused by pressure for the next load application.

Determination of Material Constants and Model Param-eters. Uniaxial cyclic constants used with the proposed modelare given in Table 1. The model requires five material constantsfor multiaxial loading conditions to define the hardening param-eter q, Eq. ~2! of Part I. To determine these five material con-stants, three constant pressure, diametral growth tests were con-ducted for each material used in this study:~i! one test at a lowhoop stress level (sh /sY'0.1), ~ii ! one test at an intermediatehoop stress level (sh /sY'0.3), and~iii ! one test at a high hoopstress level (sh /sY'0.5). Through trial and error comparisons ofmodel predictions with these diametral growth data, the materialconstants required by the model are obtained as:Cq050.002,Cq150.218, Cq3575, andCq452.4 for both QT-800 and QT-1000;Cq259 for QT-800 andCq254 for QT-1000.

A total of six tests~three tests for each material! were used toestablish all material parameters required by the model. Addi-tional constant pressure tests were conducted on tubing specimenswith different diameter to wall thickness ratios and with differentmaximum bending strains to verify that the above material param-eters are not affected by the tubing size and loading conditions.Two sets of diametral growth measurements~two tests! are shownhere in Figs. 7 and 8. Additional test results are given by Rolovic

@1#!. Tests were not replicated in most cases because diametralgrowth data generated by tubing manufacturers generally showexcellent repeatability.

As mentioned earlier, diametral growth measurements weretaken at the center of the gage section (Lgage/2) because of thelimited physical access to the specimen. Since the maximum di-ameter in all tests occurred at a different~and varying! location, itcould only be measured at the end of each test, after the specimenwas removed from the test machine. The maximum diameter mea-surements are shown as diamond symbols in Figs. 7 and 8. Allpredictions in this study are based on the maximum diameter,Dmax.

Model Evaluation. Internal pressure during service loadingof coiled tubing is usually not constant and, in fact, can vary from

Fig. 6 Cross-section of tubing divided into equal segments

Fig. 7 Diametral growth test data and predictions for QT-800under low constant pressure, cyclic bending loading

Fig. 8 Diametral growth test data and predictions for QT-800under high constant pressure, cyclic bending loading

Fig. 9 Diametral growth test data and predictions for QT-1000under block pressure, cyclic bending loading

Fig. 10 Diametral growth test data and predictions for QT-1000 under block pressure, cyclic bending loading

Table 1 Uniaxial cyclic constants

Material E ~MPa! n sY ~MPa! K8 ~MPa! n8

QT-800 200000 0.29 379 785 0.10QT-1000 200000 0.29 517 806 0.055


cycle to cycle. In order to evaluate the proposed model for morerealistic loading, a number of block pressure tests were conducted.All material constants used by the model were determined fromconstant pressure, diametral growth tests. The model was thenused to predict coiled tubing behavior under more complex pres-sure histories. Some diametral growth measurements and predic-tions from the proposed model are shown here in Figs. 9–13.Additional test results and model predictions are given by Rolovic@1#.

DiscussionCoiled tubing routinely exhibits a significant amount of ratch-

eting when utilized with internal pressure. The tendency of the

tubing to substantially grow in diameter when subjected to bend-straighten cycles at internal pressure can be explained schemati-cally using Fig. 14 which shows a von Mises yield surface. Atzero internal pressure, the stress state in a highly strained regionof tubing impinges on the yield surface at points A and B. Sincethe plastic strain increment is normal to the yield surface at theloading point~normality flow rule!, the components of the hoopstrain increments at A and B are equal and opposite, which meansthat negative hoop strain increments at B are offset by positivehoop strain increments at A. Therefore, no diametral growth oc-curs at zero internal pressure. However, if internal pressure isapplied, resulting hoop stress brings cyclic axial loading at the CDlevel. At this level, the negative hoop component of the plasticstrain increment at D is smaller than the positive hoop componentat C. This leads to accumulation of plastic strain in the positivehoop direction during each cycle and, therefore, to diametralgrowth.

Block pressure, cyclic bending tests~Figs. 9–13! clearly showthe importance of loading sequence on diametral growth rates. Insome cases when bending cycles under internal pressure are fol-lowed by bending cycles under lower or zero internal pressure,diametral growth~ratcheting! rates can be reversed, which meansthat diametral growth can be followed by incremental diametralshrinkage. The amount of diametral shrinkage depends on thehoop stress difference between two load steps and the number ofbending cycles imposed at the higher hoop stress level. It in-creases when the hoop stress difference is increased and whenmore bending cycles are applied at the higher hoop stress level.Diametral shrinkage tapers off with cycling at a given hoop stresslevel and cannot cause the tubing to shrink below the originaldiameter~so long as hoop stress is greater than or equal to zero!.If the hoop stress level for subsequent bending cycles is not low-ered enough, diametral growth can continue, but this time at alower rate ~see Fig. 11!. Low-high, high-low, and mixed hoopstress sequences were used with different materials and tubingsizes to examine coiled tubing behavior and evaluate the proposedmodel. The predictions based on the proposed analytical modelare very good. In one case the proposed model predicted smalldiametral shrinkage, while the measurements showed slight dia-metral growth for one part of the test~second hoop stress level inFig. 13!. However, notice that the growth rate for this segment,conducted at a hoop stress of 21 percent of the yield strength,slowed considerably following the 50 cycles imposed with a hoopstress of 34 percent of the yield strength. Note in Fig. 12 thatinitial cycles at about the same hoop stress~20 percent of the yieldstrength! caused a considerably higher diametral growth rate thanthe 21 percent block in Fig. 13. The predicted final diameter wasvery close to the measured value for all tests.

The physics underlying diametral shrinkage under positivehoop stress is explained by considering the behavior of the kine-matic yield surface, as dictated by the new hardening rule. Cyclic




Fig. 14 Graphical representation of diametral growth mecha-nism when axial Õbending cycling is applied at CD hoop stresslevel


axial stress~due to bending! in the presence of a constant hoopstress~due to internal pressure! tends to ‘‘cam’’ the center of theyield surface,a, upward progressively~ahi in Fig. 15a!. Thiseffect correctly causes a lower predicted strain ratcheting rate inthe hoop direction than conventional hardening rules which tendto overpredict transverse ratcheting. The other consequence of theprogressive upward shift in the center of the yield surface is thefact that when the hoop stress level,sh , is dropped, the stressstate can impinge on the yield surface below its center~sh j,ah j in Fig. 15b!. The outward normal during axial stress excur-sions in the tension direction now has the negative hoop compo-nent that is greater than the positive hoop component during axialstress excursion in the compression direction. This results in nega-tive hoop ratcheting and diametral shrinkage. As the loading atthis hoop stress level proceeds, the yield surface is progressivelymoved downward, showing the predicted decreasing rate of dia-metral shrinkage.

The characterization of coiled tubing materials using the hard-ening parameterq described in Part I of this paper is promising inthat all but one of the parameters definingq were identical for thetwo materials examined. Furthermore, this parameter,Cq2 , tendsto decrease with the higher yield strength of the material. Havingexamined only two materials, it would be premature to suggestthat this trend would hold up for an entire class of coiled tubingmaterials. However, further research to identify a robust formula-tion for q is definitely warranted.

Variation of tubing diametral growth along the gage length ofthe specimens and localized occurrence of the maximum diameteris not a result of the specific specimen gage length or the testmachine design. The localization of tubing diametral growth hasbeen observed in field applications when long tubing sectionswere exposed to the same loading conditions. Similar tubing be-havior has been observed in different test machines that use tubingspecimens with different gage lengths. In all of these cases theproposed model accurately predicted the maximum tubing diam-

eter without any adjustments to the model. The random localiza-tion of the maximum tubing diameter in the case of tubing bal-looning is analogous to the random localization of specimennecking during a simple tensile test. Subtle differences in the ma-terial microstructure result in the existence of a tubing section thatis less resistant to plastic deformation than the surrounding sec-tions, which leads to locally higher plastic deformation and bal-looning.

Overall diametral growth~ratcheting! predictions in this studyare very good considering the extremely complex nature of plasticdeformation in coiled tubing. In order to improve predictions be-yond the current level, more fundamental research is needed. Thenormality flow rule, which is widely accepted in the theory ofplasticity, provides a good estimate for the direction of plasticflow, but yield surfaces are not simple to characterize. Surfacescan distort during plastic deformation, making it more difficult topredict plastic flow with certainty. Material anisotropy and tran-sient behavior~hardening, softening, or both! further complicateanalytical modeling. Even though these phenomena may not bepronounced, they can cause computational errors to accumulatewhen long load histories are considered. One or two degrees de-viation from the normality flow rule, or a small distortion of theyield surface can give negligible errors for a few load cycles, butit can cause large errors to accumulate over a long load history.

Another subtle characteristic of coiled tubing deformation be-havior was accurately described by the proposed model. Severalfatigue test specimens for constant pressure tests were cut trans-versely after the failure, and the wall thickness was measured attwo locations as shown in Fig. 16. The measurements and thepredictions from the proposed model are presented in Table 2. Itcan be seen that the predictions are very good, within 10 percentfrom the corresponding measured values. The model also predictsaccurately that for constant pressure cyclic bending tests, the wallon the concave side of the tubing becomes slightly thinner thanthe wall on the convex side of tubing~t1,t2 in Fig. 16!. This canbe explained by considering that an element att1 feels a combinedeffect of radial compressive stress from pressure and a mean com-pressive axial strain. This slightly accelerates the driving forcesfor plastic flow in the circumferential direction, relative to thecombination of radial compression and mean tension att2 .

ConclusionsMultiaxial cyclic ratcheting in coiled tubing was investigated in

this study. Diametral growth~ratcheting! of coiled tubing wasmeasured during cyclic bending tests under internal pressure. Thenew incremental plasticity model developed in Part I was used topredict coiled tubing deformation behavior. Based on the experi-mentally observed coiled tubing behavior and predictions from theproposed analytical model, the following conclusions can bemade:

1 When coiled tubing is cyclically bent-straightened in theplastic regime under imposed internal pressure, its diameter in-creases and wall thickness decreases. The maximum increase in

Fig. 15 Graphical representation of diametral shrinkagemechanism under positive hoop stress. Axial Õbending cyclingat shi hoop stress „a… is followed by axial Õbending cycling at shjhoop stress „b….

Fig. 16 Wall thinning during cyclic bending Õinternal pressureloading

Table 2 Wall thickness measurements and predictions

Original wall Measured wall thickness Predicted wall thicknessthickness~mm! t1 ~mm! t2 ~mm! t1 ~mm! t2 ~mm!

2.46 1.73 1.98 1.75 1.804.11 2.79 3.12 2.82 2.922.49 1.98 2.13 1.93 1.964.16 3.18 3.40 3.30 3.334.01 3.23 3.28 3.20 3.283.51 2.97 3.00 2.95 2.974.09 3.23 3.30 3.15 3.203.53 2.92 2.97 2.72 2.822.49 1.96 2.01 1.93 1.96


diameter observed in this study was approximately 15 percent andthe maximum reduction in wall thickness was about 30 percent.

2 Cyclic ratcheting rates can be reversed without reversing theprimary ~mean! stress, depending on the load history. Tubing di-ametral growth caused by cycling at high internal pressure can befollowed by incremental diametral shrinkage caused by bend cy-cling at a lower~positive or zero! internal pressure.

3 Plastic deformation of coiled tubing is highly dependent onload history; i.e., diametral growth of coiled tubing differs con-siderably when load sequence is changed.

4 The proposed analytical model demonstrates excellent capa-bilities in terms of predicting the complex nature of coiled tubingmechanical behavior.

5 More research would be useful in the following areas:~a! thin-walled tubes subjected to cyclic axial loading, internal

pressure, and torsion would provide additional important valida-tion and clues for refinement of the proposed model;

~b! transient cyclic behavior of materials~softening, hardening,or both! needs to be examined under both uniaxial and multiaxialloading conditions;

~c! a standard procedure needs to be identified for obtainingempirical parameters required by the plasticity model. Additionaldata sets for new materials would be useful towards this end.

AcknowledgmentsThe authors wish to express their appreciation for the support

from The University of Tulsa, The United States Department ofEnergy, and members of the University of Tulsa Coiled TubingMechanics Research Project: Elongation and Diametral Growth.

References@1# Rolovic, R., 1997,Plasticity Modeling of Multiaxial Cyclic Ratcheting in

Coiled Tubing, Ph.D. dissertation, The University of Tulsa, Tulsa, Oklahoma.@2# Tipton, S. M., 1998, ‘‘Low-Cycle Fatigue Testing of Tubular Materials using

Non-Standard Specimens,’’Effects of Product Quality and Design Criteria onStructural Integrity, ASTM STP 1337, T. L. Panontin and S. D. Sheppard,eds., American Society for Testing and Materials.


I. SchafflerP. Geyer

P. BouffiouxElectricite de France,

Direction des Etudes et Recherches,Departement MTC Route de Sens,

77250 Moret/Loing, France

P. DelobelleLaboratoire de Mecanique Appliquee R. Chaleat,

UMR 6604, 24 chemin de l’Epitaphe,25030 Besancon Cedex, France

Thermomechanical Behavior andModeling Between 350°C and400°C of Zircaloy-4 CladdingTubes From an UnirradiatedState to High Fluence(0 to 85"1024nm22, EÌ1 MeV…This paper first describes the effect of neutron irradiation on the thermomechanical be-havior of stress-relieved Zircaloy-4 fuel tubes that have been analyzed after exposure tofive different fluences ranging from nonirradiated material to high burnup. In the secondpart, a viscoplastic model is proposed to simulate, for different isotherms,350°C,T,400°C, out-of-flux anisotropic mechanical behavior of the cladding tubes overthe fluence range 0,f,100•1024 nm22 (E.1 MeV). The model, identified for tests con-ducted at 350°C, has been validated from tests made at 380°C and 400°C. The model iscapable of simulating strain hardening under internal pressure followed by a stress re-laxation period, the loading producing an interaction between the pellet and cladding.Introduction of a state variable characterizing the damage caused by a bombardment withneutrons into the model has allowed us to simulate the irradiation-induced hardening andcreep rate decrease, as well as the saturation noticed after two cycles of irradiation(>45•1024 nm22 (E.1 MeV)) in a pressurized water reactor (PWR). Finally, the numeri-cal simulations show the model is able to reproduce the totality of the thermomechanicalexperiments.@S0094-4289~00!00202-4#

1 IntroductionToday, over 70 percent of French electric power has nuclear

origin and is generated by 58 pressurized water reactor~PWR!units. This capacity makes it necessary for reactors to adapt theirproduction to network demand by operating under load followingconditions. Moreover, due to economic reasons, Electricity ofFrance~EDF! seeks to extend the fuel assembly life to ultimatelyreach a 60 GWd/tU~Giga-Watt-day/ton Uranium! burnup. Thesenew conditions of use bring about more severe mechanical load-ings of the cladding, owing to the greater number of power ramps.

Behavior laws currently used for predicting the mechanical be-havior of fuel rod cladding do not consider the loading modesinherent under operating conditions~Baron and Bouffioux@1#!.This requires developing a behavior model that incorporates theanisotropic behavior of the material more specifically, the stressrelaxation in cladding following loading from pellet-cladding in-teraction~PCI! resulting from a power ramp. Such a model shouldalso be able to take into account the irradiation effectsupon the mechanical properties at fluences, which may reach100•1024 nm22 (E.1 MeV), and at the cladding operating tem-peratures ranging from 350°C to 400°C. Conventionally, in reac-tor deformations have been classified as swelling, thermal or irra-diation enhanced. Thermal deformation models have been used tosimulate PCI~Baron and Bouffioux@1#!.

The literature gives many results of the mechanical testing ofirradiated claddings~Higgy and Hammad@2#, Northwood @3#,Franklin @4#, Petterson@5#, Baty et al. @6#, Yasuda et al.@7#!.

However, the diversity of alloys tested, as well as the irradiationand testing conditions examined were insufficient to provide thedata needed to construct a model. EDF has consequentlylaunched, with three industrial partners, new testing programs onstress-relieved Zircaloy-4 claddings used in EDF-PWR. This da-tabase is now available and has allowed us to develop a behaviormodel which takes into account the anisotropy, the temperature,and the irradiation effects upon the cladding mechanicalproperties.

A mechanical behavior model was developed during the firstphase of the project from uni- and biaxial tests conducted onrecrystallized and stress-relieved Zircaloy-4 tubes~Delobelleet al. @8,9#!. This paper presents the second phase of the projectduring which the model has been applied to unirradiated tubesbefore taking into account the temperature and the irradiation ef-fects upon the mechanical behavior. The model will first be ap-plied to tests performed at 350°C on nonirradiated tubes, whichare representative of those in service in PWRs. The analysis willthen be expanded to describe the effect of variable temperature~between 350°C and 400°C! on the model parameters. The modelwill be shown to describe the anisotropic thermomechanical be-havior of unirradiated claddings. The irradiation effects will beintroduced into the model through a damage variable which willaffect the viscoplastic state equation and the static recovery of thekinematic strain hardening variables.

2 Tested Materials and Experimental Results

2.1 Tested Materials. The cladding tubes are stress-relieved, low-tin content, Zircaloy-4. Their chemical compositionis given in Table 1. The thermal treatment consists of a 5 hranneal at 460°C. The content of alloying elements is consistentwith ASTM B 353 specification. The crystallographic texture of

Contributed by the Materials Division for publication in the JOURNAL OF ENGI-NEERING MATERIALS AND TECHNOLOGY. Manuscript received by the MaterialsDivision November 12, 1998; revised manuscript received July 6, 1999. AssociateTechnical Editor: Kwai S. Chan.


the tubes has been analyzed through x-ray diffraction. The analy-sis shows two preferred orientations of theC axis of the hexago-nal crystallites at640 deg in plane (r ,u) and 615 deg in plane(r ,z), wherer is the radial direction,u the tangential direction andz the longitudinal direction. The Kearns factor values which rep-resent the average of the squares of the cosines of the anglescbetween the poles$0002% and the direction~i! ~Van Swam et al.@10#!, are given in Table 2 for the three directionsr ,z,u: f r , f a , f l .

The grain structure is fine and elongated in the rolling direction.The microstructure is strain hardened similarly to that of a stress-relieved material~Bouffioux @11#!. In summary, the materialshows a strong texture due to the preferred crystallite orientationsand the shape of polycrystal grains. This structure will produceanisotropic mechanical behavior. In order to characterize this an-isotropy, tests on tubing have been conducted in the longitudinaldirection and under internal pressure. The nominal dimensions oftest specimens are an outside diameterBext of 9.5 mm for a wallthickness,e, of 0.57 mm.

2.2 Mechanical Tests Performed on Nonirradiated Clad-ding Tubes. Many tests have been performed on nonirradiatedcladding tubes so as to describe the reference state of the materialbefore irradiation.

2.2.1 The Longitudinal Tests.Longitudinal deformation wasmeasured using LVDT type sensor. The effective specimen lengthis 50 mm and the required deformation rate is controlled by thesensors. The stress state, which is homogeneous in the test area, isobtained directly by the ratio of the axial force divided by thesection of the sample.

Monotonic tests were performed at 350°C, 380°C and 400°Cwith strain rates between 2•1026 and 2•1023 s21 to obtain thematerial’s inelastic properties. Figure 1 shows the stress-strain

curves for different strain rates. The data show the strain-ratedependence of the material at these temperatures~Delobelle et al.@8#!. Creep tests were also conducted at stress levels ranging from140 to 400 MPa for these three temperatures. An example is givenin Fig. 2.

2.2.2 Biaxial Tests Under Internal Pressure.Since the clad-ding is anisotropic, strain hardening tests were performed at vari-ous biaxial stress ratiosa5szz/suu ~the longitudinal stress di-vided by the tangential stress! at 350°C. The diametraldeformation rates which were investigated, correspond to thoseimposed by the pellet on the cladding. Deformation rates rangedfrom 2.1025 to 2.1024 s21. Additional tests were performed at380°C and 400°C to determine temperature dependence. Manycreep and relaxation tests were also performed.

The test specimen is fitted with four diametral point sensors ofthe LVDT type that randomly contact four points along the tube.

During the strain-hardening tests, the testing machine exerts thediametral deformation rate while keeping the biaxial stress ratio,a, constant. During testing under internal pressure, the stress stateis not entirely uniform throughout the tube thickness. One uses themean variables of rational stresses as in Eq.~1! ~Bouffioux @12#!:

s rr 52p/2>0, suu5pS r ext2e

e Dexp~12Cu«uu!,

szz5asuu

(1)

s rr , suu , and szz are, respectively, radial, hoop, and axialstresses,p is the internal pressure,r ext5Bext/2 and,e, the wallthickness of the tubes.

In order to calculatesuu , knowing the deformation«uu , aconstantCu which varies with the biaxiality ratioa, is introduced.Figure 3 shows an example of biaxial results obtained witha520.5, 20.25, 0, 0.25, 0.5, 0.75 at 350°C.

At 350°C, creep tests were also performed witha50.5 andtangential stress levels ranging from 100 to 275 MPa~Fig. 4!.Repeated tests at 380°C and 400°C were conducted to better es-timate the temperature effects.

Lastly, tensile tests under internal pressure (a50.5) followedby a relaxation phase were performed for a future validation of themodel on loadings representative of a pellet-cladding interactionat two required diametral rates:«uu52•1024 and 2•1025 s21 at350°C, 380°C, and 400°C. Figure 5 gives an example of dataobtained with«uu52•1025 s21.

Fig. 1 Longitudinal tensile tests performed at 350°C for differ-ent strain rates «zz

T Ä2"10À3, 2"10À4, 2"10À5 and 2 "10À6 sÀ1.Experimental results and simulations.

Fig. 2 Longitudinal creep tests performed at 350°C for differ-ent stress levels: szzÄ275, 350, 380, and 400 MPa. Experimen-tal results and simulations.

Table 1 Weight composition of low tin content Zircaloy-4 clad-ding tubes

Alloying elements Impurities (pp)

Cr Fe Sn O Zr C N H0.10 0.21 1.25 0.109 bal. 120 31 ,2

Table 2 Kearn’s factors

f r f a f t

0.57 0.35 0.08

such that:f a1 f r1 f t51


2.3 Testing Program Performed on Irradiated CladdingTubes. This database consists of three programs respectivelydesignated by A, B, and C~CEA, Framatome and EDF@13#!.These programs will allow us to quantify the effects of irradiationupon a fuel rod cladding throughout the life span of assemblies upto high burnup.

2.3.1 Program A. Tests specimens taken from tubes similarto those described earlier are introduced into capsules and irradi-ated in an experimental reactor. Some specimens have been re-moved after 500, 1000, and 3000 hours so as to obtain claddingsirradiated at different fluence levels, say about 4•1024, 8•1024,and 21•1024 nm22 (E.1 MeV). The upper fluence level corre-sponds to that of one cycle of irradiation in PWR reactor. As thetest samples had been irradiated in a NaK medium~the samplesare corrosion resistant to this medium!, they have undergone nohydriding or corrosion and have not been exposed to any me-chanical loading.

Strain-hardening tests under internal pressure (a5szz/suu50.5) were conducted at two loading rates, namely:«uu

52.1024 and 5.1026 s21, at 350°C and 380°C. Figure 6 gives anexample of the influence of the irradiation on the stress-straincurves at 350°C and«uu52•1024 s21. These tests have beencomplemented by twelve creep tests performed at 350°C and threetests at 380°C. Some results of creep experiments are given in Fig.7 (f58.1024 nm22 andT5350°C). Thus, this program allows usto estimate the evolution of the mechanical behavior of the mate-rial during the first cycle (21•1024 nm22 (E.1 MeV)) in PWRreactor.

2.3.2 Program B. Tests are conducted on specimens fromfuel rods taken from a single assembly following two cycles(>45•1024 nm22 (E.1 MeV)) in a PWR reactor. Samples are

Fig. 3 Biaxial tensile tests performed at 350°C for differentstress-biaxiality ratio aÄsuu Õszz , aÄÀ0.25, À0.5, 0, 0.25, 0.5,and 0.75. Experimental results and simulations.

Fig. 4 Biaxial creep tests performed at 350°C with aÄ0.5 andfor different stress levels: suuÄ100, 140, 170, 200, and 275MPa. Experimental results and simulations.

Fig. 5 Biaxial tensile tests performed at 350, 380, and 400°Cwith aÄ0.5 and «uuÄ2"10À5 sÀ1 followed by a relaxation phase.These loadings are representative of a PCI transient. Experi-mental results and simulations.

Fig. 6 Biaxial tensile tests performed at 350°C with aÄ0.5 foran imposed hoop strain rate of 2 "10À4 sÀ1 and for different flu-ences: fÄ0, 5.1"1024, 7.21"1024, and 19 "1024 nmÀ2

„EÌ1 MeV…. Experimental results and simulations „program A ….

Fig. 7 Biaxial creep tests performed at 350°C with aÄ0.5 onirradiated cladding tubes „1000 h of irradiation … for differentstress levels: suuÄ359, 397, 445, and 517 MPa. Experimentalresults and simulations „program A ….


then cut and a chemical decladding of the fuel is made. Theseoperations therefore allow us to obtain specimens that have beenirradiated upto a fluence of 45•1024 nm22 (E.1 MeV). It isworth noting that claddings are corroded and hydrided as theyhave been in contact with the primary coolant. However, the hy-drogen content of the irradiated fuel cladding has not been mea-sured.

Irradiated test specimens are subjected to biaxial tensile tests(a5szz/suu50.5) at three deformation rates:«uu52•1024,2•1025, and 5•1026 s21 at 350°C, 380°C~Fig. 8! and 400°C, andto creep tests at 350°C for five stress levels ranging from 300 to520 MPa. Other creep tests were conducted at 380°C.

Therefore, this program describes the mechanical properties oftubes after two irradiation cycles (>45•1024 nm22 (E.1 MeV)) in PWR reactor.

2.3.3 Program C. Like program B, tests are performed onspecimens cut from a fuel rod irradiated during four cycles(>85•1024 nm22 (E.1 MeV)) in a PWR reactor. These clad-dings are consequently corroded and hydrided also. Biaxial tensiletests (a5szz/suu50.5) were performed at 350°C, 380°C, and400°C at deformation rates ranging from 1.4•1027 s21 to

1.4•1024 s21. An example of the stress strain curve at differentrates is given in Fig. 9. Creep tests were also performed at thesetemperatures fora5szz/suu50.5 and various tangential stresslevels ranging from 300 to 550 MPa. Figure 10 gives some creepcurves obtained atT5350°C forsuu5415 MPa and for four flu-ences corresponding to programs A, B, and C. This program al-lows us to evaluate the mechanical properties of a cladding tubeirradiated during four cycles (>85•1024 nm22 (E.1 MeV)) inPWR reactor.

One will note that specimens obtained within the framework ofprograms B and C have been oxidized, hydrided, and exposed toin-reactor loadings which remain unknown. This is why we mustbe careful with regard to a precise analysis of these programs.This question will be considered in detail in the next paragraphsand we will try to check the data base consistency as regards theevolution of irradiation-induced mechanical properties.

3 Analysis of the Results

3.1 Case of Unirradiated Tubes. As shown in Fig. 1 thematerial presents a strong strain-rate dependence at the tempera-tures of the tests. Uni- and biaxial tests show the isotropy of theelastic properties of the material and a linear decrease of Young’smodulus between 350°C and 400°C while Poisson’s ratio remainsconstant. The Rp ratio of the plastic hoop strain to the axial one,Rp5«uu

p /«zzp , is found to be, as a first approximation, independent

of temperature, plastic strain, and strain rate and equal toRp520.660.02. The knowledge of the Kearn’s factors allows us toevaluate ratioRp and then, the material incompressibility permitsto write the contractile strain ratio, CSR,~Van Swam et al.@10#!as:

CSR5f r

12 f r2 f a52

Rp

11Rp(2)

Kearn’s factors in radialf r and axialf a directions are recalled inTable 2. The calculation gives a value ofRp520.62 which is inaccordance with the value reported above. To simplify the de-scription of anisotropy in the model, the set of anisotropy coeffi-cients will be considered as independent of temperature and strainrate, which is corroborated by the results of Beauregard et al.@14#and Murty @15#.

In fact, this alloy is quite anisotropic for the observed texture,but the failure to detect the elastic anisotropy is an issue of ex-perimental sensitivity. For modeling, the elastic properties will beconsidered as isotropic, but for the opposite case, the generalizedHooke’s law can be used:

Fig. 8 Biaxial tensile tests performed at 350°C with aÄ0.5on irradiated cladding tubes until two cycles in PWR„fÄ45"1024 nmÀ2, „EÌ1 MeV…… and for different hoop strainrates: «uuÄ2"10À4, 3"10À4, and 5 "10À6 sÀ1. Experimental re-sults and simulations „Program B ….

Fig. 9 Biaxial tensile tests performed at 380°C with aÄ0.5on irradiated cladding tubes until four cycles in PWR„fÄ85"1024 nmÀ2, „EÌ1 MeV…… and for different hoop strainrates: «uuÄ1.4"10À6, 1.4"10À7 and 1.4 "10À8 sÀ1. Experimentalresults and simulations „program C….

Fig. 10 Biaxial creep tests performed at 350°C with aÄ0.5 andsuuÄ415 MPa on irradiated cladding tubes at different flu-ences: fÄ4.4"1024, 20.8"1024, 45"1024, and 85 "1024 nmÀ2

„EÌ1 MeV…, programs A, B , and C. Experimental results andsimulations.


@s#5@C#@«e#,

where@C# is a fourth rank tensor.The use of uniaxial creep tests conducted at 350°C, 380°C, and

400°C allows us to determine the apparent activation energyDHand the stress exponentn, according to the relations~3!:

DH5@] ln «zz/]~21/kT!#szz, n5@] ln «zz/] ln szz#T (3)

«zz is the steady creep rate andk the Boltzmann constant. We findan energy value of 2.060.1 eV between 350°C and 400°C andn>3.4 for the lowest stresses (85,szz,174 MPa). Then,n in-creases with increasing stress~Fig. 12! and can be represented asa hyperbolic sine dependence of«zz versusszz. TheDH value isclose to the one obtained in internal friction for the Snoek peak inimpure Zr and caused by the reorientation of the oxygen-impuritypairs in the hexagonal matrix; 0-Sn is a possible pair~Yi et al.@16#, Prioul @17#!. This value suggests that the dislocation sub-structure recovery is controlled either by the self-diffusion or bythe diffusion of a species within the matrix. This value is used inthe model formulation. Note that Murty et al.@18# and Matsuo@19# reported a value of 2.68 eV for the same temperature range,which is in close agreement with the value obtained for self-diffusion ~Lyashenko et al.@20#!.

3.2 Case of Irradiated Claddings. The elastic behavior ofthe material does not seem to be significantly modified by a neu-tron irradiation. However, the inelastic behavior of the claddingsis strongly affected by the irradiation. We have plotted on Fig. 11the variations of the flow stress required to obtain a given defor-mation versus the fluence. To do so, we have used biaxial tensiletests performed at 350°C and obtained through programs A and B

at diametral loading rates of 2•1024 and 5•1026 s21. The graphsprove that the irradiation makes the material harder at a high rateat fluences ranging from 10 to 20•1024 nm22 (E.1 MeV) andthat it tends to saturate at higher fluences. This saturation in theevolution of mechanical properties during tensile tests relative tothe fluence confirms the conclusions obtained on other types ofZircaloy ~EPRI @21#, Yasuda et al.@7#!.

In addition, as in the case of nonirradiated stress-relieved clad-dings, the material shows a stabilized flow stress during strainhardening tests even at high loading rates. The comparison of testsperformed at 2•1024 and at 5•1026 s21 ~Fig. 11! proves that wemainly obtain a strain hardening effect when the total hoop strainsare not important («uu,0.6 percent), whereas the recovery pre-vails when the strains are greater. The results of tests within theframework of program C also display a hardening and a consid-erable loss in material ductility.

We have also seen that the irradiation temperature affects thehardening amplitude through the original position of the testedspecimen along assembly of cladding tubes~Higgy and Hammad@2#, Petterson@5#, Franklin @14#!. The saturation is due to a bal-ance of the emergence of defects and their recombination. As thedefect mobility is high, the higher the irradiation temperature, thesmaller is the number of irradiation-induced defects present incladdings. These defects will thus promote a quick growth of ‘‘a’’dislocation loops. Per contra, at low temperatures, a greater den-sity of the smallest loops, which will slowly grow, can be ex-pected~Northwood @3#!. This might explain why the hardeningdue to irradiation is slower for claddings irradiated at low tem-peratures and this confirms the importance of the original state ofthe test specimen, depending on its position along the cladding inthe assembly, with regard to the mechanical behavior.

In these programs~no reported data obtained on recrystallizedsamples!, one also checks that the variations in the behavior ofstress-relieved and recrystallized claddings become smaller as ir-radiation progresses and that the overall hardening due to irradia-tion is much more important in the case of recrystallized materialthan for stress-relieved material when irradiation starts. These ob-servations confirm the conclusions obtained by Higgy and Ham-mad @2#, Franklin et al.@22#.

During the secondary creep phase, strain hardening could beregarded as being constant and therefore it is possible to analyzethe recovery phenomenon by plotting the diametral steady-statecreep rate versus the diametral constant stress for different flu-ences~Fig. 12!. We also take into consideration the steady flowstress obtained during tensile tests as a function of the requiredstrain rate; thus the points lie in continuation of those derived

Fig. 11 Evolution at 350°C of the flow stress for different totalstrain levels versus the fluence „a… «uuÄ2"10À4 sÀ1, „b… «uu

Ä5"10À6 sÀ1

Fig. 12 Representation of the equivalent Mises steady creeprates versus the equivalent Mises stress at 350°C and for dif-ferent fluence levels, 0 ÏfÏ85"1024 nmÀ2


from creep tests. Except for the case of nonirradiated material(a50), all the tests for irradiated material were performed witha5szz/suu50.5, thus the translation of the curves Ln«Mises5 f (LnsMises) ~Fig. 12!, is due to the effect of irradiation and notto the anisotropy. As in the case of nonirradiated tubes,n in-creases with increasing stress (3.8<n<14). This figure showsthat the state recovery of the material during creep tests is a de-creasing function of the fluence. Some points derived from tensileor creep test results do not lie in continuation of the other testsperformed at the same fluence. This may be due to experimentaluncertainties regarding the temperature and stress which exist dur-ing creep tests, the improper knowledge of corrosion and hydrid-ing effects~Chen et al.@23#! upon the mechanical behavior, andgenerally speaking, the difficulties met when conducting hot-labtests on specimens which have been weakened by irradiation. Inaddition, as for strain hardening tests, the cladding irradiationtemperature in reactors substantially affects the creep rate~Higgyand Hammad@2#, Petterson@3#, Franklin et al.@21#!.

Figure 13 shows the decrease of the creep rate, at 350°C, as afunction of the fluence. The points derived from tests performedwithin the framework of programs A, B, and C have been plotted.We note that the creep becomes much lower before stabilizing forhigh fluences according to the results given by Franklin et al.@21#.In fact, the creep rate decreases during two irradiation cycles toabout 45•1024 nm22 (E.1 MeV), and then decreases moreslowly. Baty et al.@6# also report a saturation of the hoop creeprate at 350°C between 30•1024 and 50•1024 nm22.

In conclusion, an analysis of the test results has allowed us todetermine, at fluence ranging from 0 to 85•1024 nm22 (E.1 MeV), consistent evolutions with regard to the mechanicalproperties of claddings made of Zircaloy-4. The model will bespecified, after a formulation has been proposed, on the bi-axialmonotonic tensile and creep tests conducted between 350°C and400°C for programs A, B, and C.

4 Modeling of the Temperature and Iradiation EffectsWe worked with a behavior model developed on other types of

materials and adapted to the Zircaloy case~Delobelle et al.@9#!.This model has proven its fitness to simulate the mechanical be-havior of tubes made of Zircaloy-4 in two different metallurgicalstates.

4.1 Initial Model. The equations of the initial model aregiven in Table 3. Initially in this model, the different parametersof the equations are constant and therefore, independent of thetemperature and of the irradiation, i.e.,T andD.

In the case of small deformations, the total strain@«T# can bedivided into two terms, an elastic strain@«e# considered isotropic

Fig. 13 Evolution of the equivalent steady creep rates versusthe fluence „programs A, B , and C… at 350°C and for differentstress levels; suuÄ313, 334, 368, and 415 MPa

Table 3 Equations of the model: Voigt matrix notations

• Equations of the strains@«T#5@«e#1@«#

@«e#511n

E~T!@s#2

n

E~T!@D#t@s#@D#

@«#53

2«@M #@s82a8#

s2a, «5 «0~T,D !F sinhS s2a

N~T,D!DGn

In these equations:@s8#5@s#21

3@D# t@s#@D#,

@a8#5@a#21

3@D# t@a#@D#,

s2a5S 3

2@s82a8# t@M #@s82a8# D 1/2

• Equations of the kinematical hardening variables@a#, @a (1)#, and@a (2)#

@a#5pS23

Y* ~T!@N#@ «#2@Q#@a2a~1!# « D 2r m~T,D !sinhS a

a0~T,D!Dm0

3@N#@R#@a#

a

@a~1!#5p1S 2

3Y* ~T!@N#@ «#2@Q#@a~1!2a~2!# « D

@a~2!#5p2S 2

3Y* ~T!@N#@ «#2@Q#@a~2!# « D with @a~k!#0

t 5@0,0,0,0,0,0#

and

a5S 3

2@a# t@R#@a# D 1/2

• Equation of the scalar variableY*Y*5Y0~T!1Y

Y5b~Ysat2Y!«, Y~0!50

The Voigt notation is adopted, so@s# t5@s115s1 ,s225s2 ,s335s3 ,s125s4,s135s5 ,s235s6#

@«# t5@«115«1 ,«225«2 ,«335«3,2«125«4,2«135«5,2«235«6#

@D# t5@1,1,1,0,0,0#

The matrices@M #, @N#, @Q#, and@R# have an orthotropic symmetry. Forexample, for@M #

@M #53M 11 M 12 M 13

M12 M 22 M 23

M13 M23 M 33 0

M44

0 M 55

M 66

4with the incompressibility relations

M111M 121M 1350

M121M 221M 2350

M131M 231M 3350


and a plastic strain@«# which is anisotropic. Experimental studiesincluding cyclic tests clearly indicated the kinematical nature ofthe hardening. So, the stress is the sum of two components, astrain rate dependent stress,s2a, and an internal stress@a#.Three kinematical variables are necessary to simulate the strongnonlinearity of the stress-strain curves. The evolution laws of thekinematical hardening variables@a#, @a (1)#, and@a (2)# define themechanical behavior completely. The first term of these equationsis the linear part of the hardening, the second term represents thedynamic recovery and the last one represents the time dependentstatic recovery. The introduction of the anisotropy in this model~Delobelle et al.@9#! is made via the four fourth rank tensorsaffecting the flow directions@M #, the linear part of the kinematicalhardening@N#, and the dynamic and static recoveries,@Q# and@R#, of these same hardening variables. The material has a mostlyradial-tangential texture, but the symmetry of the crystallographictexture in the tubing reference system~r, u, z! imparts the ortho-tropic properties, which explains the form of the anisotropy ma-trix. Y is a scalar variable which describes the cyclic softening ofthe material. In this study, only monotonic loadings are consid-ered and thereforeY is equal to zero andY* 5Y0 .

4.2 Modeling of the Temperature Effect. A detailed de-scription of the general method to identify the parameters of themodel is beyond the scope of this paper but is given by Delobelleet al. @9#.

Only the parametersE, «0 , N, r m , a0, andY0 have been con-sidered as temperature dependent~Table 4!.

Tensile tests show a linear decrease of Young’s modulus be-tween 350°C and 400°C while Poisson’s ratio remains constant.As mentioned earlier, in a first approximation, it has been as-

sumed that the temperature does not affectRP between 350°C and400°C. For an easier acknowledgment of the temperature in themodel, we consider that all the anisotropy coefficients are notlinked to the temperature. During the steady creep phase, the useof uniaxial tests performed at 350°C, 380°C, and 400°C allow usto determine the value ofDH ~Eq. ~3!! and consequently theevolution of r m versus the temperature. This is based on the hy-pothesis that during steady state first, the strain hardening may beconsidered as constant and does not vary much with the tempera-ture and secondly the strain-rate dependent stress level is low(a>s).

Dip-tests conducted at different temperatures~Delobelle et al.@9#! during creep tests allow us to determine the evolution of« asa function of sv5us2au for each isotherm, and consequently,the determination ofDH* . To take into account the decrease ofthe yield stress with the temperature, we have considered thatY0is a linearly decreasing function of the temperature. The strain-rate dependent stress increases with the temperature and thusN isconsidered as an increasing function of the temperature.

The five temperature dependent parameters are identified nu-merically with the SIDOLO software~Pilvin @24#! over all theuni- and biaxial tensile and creep tests performed at 350°C and400°C. Then, knowing the values of these parameters, we deter-mine the coefficients of their evolution laws~Table 4!. The iden-tification is then validated by comparing the calculation resultsand the test performed at 380°C. The other parameters which arenot temperature dependent are identified at 350°C. Numericalsimulations of all tests have been performed, but only a few fig-ures are presented~Figs. 1–5!. The model is in good agreementwith the results of the tensile tests at different strain rates~Fig. 1!,of the creep tests at different stress levels~Fig. 2!, of the biaxialtests at different biaxiality ratios~Fig. 3!, and of the biaxial creeptests witha50.5 and different stress levels~Fig. 4!. At this stageof the model development and of its identification on non irradi-ated claddings, the formulation is validated by comparing themodel predictions and the tensile tests performed at different tem-peratures under internal pressure (a50.5) and followed by a re-laxation phase~Fig. 5!. These tests simulate a pellet-cladding in-teraction type loading. The results can be considered assatisfactory.

It should be noted that the whole terms of the model are im-portant and activated for the simulation of the data base, as it wasshown in a previous paper~Delobelle et al.@9#!.

4.3 Modeling of the Irradiation Effects. As seen earlier,irradiation causes a marked hardening while reducing the creep-rate or strain of the material. These effects are particularly appar-ent during two cycles in PWR (>45•1024 nm22 (E.1 MeV))and become less perceptible when higher fluences are reached. Totake account of the saturation observed in the evolution of themechanical properties, we do not use the fluencef as a damagevariable~creation of random or linear defects linked to bombard-ment with neutrons, Franklin et al.@22# and Limback and Ander-son @25#! but a state variableD ~Table 4!, such that

D5D0~12exp2D1f!, with D~0!50 (4)

A number of model parameters will depend upon this variable~Table 4!. We have seen that the irradiation temperatureT* alsoaffects the cladding hardening amplitude, so the two parametersD0 andD1 must depend on this temperature. In this studyT* isnearly constant, (T* >330°C), therefore these two parameters areconsidered as constant. Such a formulation ofD variable corre-sponds to an isotropic representation. It expresses the fact that thenumber of defects existing in the material is equal to the numberof defects created less the number of defects which recombine~Matzke @26#!. A possible anisotropy of the damage might beeasily integrated into the model by introducing a second ranktensor@A# such that@D#5@A#D. This construction is not neces-

Table 4 Modifications introduced into the model to take intoaccount the temperature and the irradiation effects

• Temperature dependence of parametersE~T!5E01E1T

«0~T!5«1Sexp2DH*

kT D~N~T!!n

N~T!5N01N1T

Y0~T!5Y12Y2T

a0~T!5a12a2T

rm~T!5rm0Sexp2DH

kT D~a0~T!!m0

• Equation of the irradiation damage variableD

D5D1~T* !~D0~T* !2D!w with D~0!50

The integrated form is given byD5Do(T* )(12exp2D1(T* )f)

In these equations,w is the neutron flux,f the fluence andT* theirradiation temperature which may be different from the temperatureTof the test. The evolution ofD0(T* ) andD1(T* ) is not specified, thesetwo parameters are considered as constant in the present study.

• Irradiation dependence of the parameters

«0~T,D!5«0~T!~N~T,D!!n exp2~j1Dj2!

N~T,D!5N~T!1N2~12exp2N3D!

rm~T,D!5rm~T!exp2~x1Dx2!

a0~T,D!5a0~T!exp2~a3Da4!


sary to simulate the data base described above as only tensile andcreep tests under internal pressure (a50.5) are available.

Figure 12 has allowed us to demonstrate that the recovery dras-tically decreases as irradiation progresses. To take this observa-tion into account, as well as the hardening noticed~Fig. 11!, wepropose to rewrite the evolution law of the kinematical strainhardening variable@a# by making the parameters,a0 and r m ofthe static recovery term depend onD ~Table 4!. To take accountof the strain-rate dependent stress evolution with irradiation, thetwo parameters«0 andN also depend on the variableD ~Table 4!.One should note that the four parameters,a0 , r m , «0, and N,which depend on variableD, also depend on the temperature. Theformulation of the evolution of these four parameters, given inTable 4, is fairly nonlinear like the phenomena to be modeled.

The model parameters linked to irradiation through variableD,as well as those which control the evolution law ofD, are identi-fied numerically on biaxial tensile and creep tests made at 350°C.

Figure 6 compares biaxial strain hardening tests performed at350°C and their simulation at a specified hoop strain rate,«uu

52•1024 s21, on a nonirradiated tube and on irradiated tubeswithin the framework of programs A and B. Figure 7 presentssimulations of biaxial creep tests derived from program A. Fromthese two figures, the correlation of the simulation with the testsused to identify the model is satisfactory.

Figure 8 shows the simulation of the influence of the hoopstrain rate on an irradiated cladding up to two cycles(>45•1024 nm22 (E.1 MeV) in PWR at 350°C. The test has notbeen used for identification purposes and it allows us to check thatthe model and the parameter are satisfactory.

Temperature-controlled evolution laws and irradiation beingdetermined, one checks that the parameters calculated at 380°Cand 400°C allow the simulation of the irradiation-controlled evo-lution laws with the biaxial tensile and creep tests conducted atthese temperatures. Figure 9 compares the simulation of biaxialtensile tests performed at 380°C on tubes that have been irradiatedduring four cycles (85•1024 nm22 (E.1 MeV), Program C!. Onesimilarly plots on Fig. 10 the diametral strains versus time forcreep tests performed at 350°C, forsuu5415 MPa on claddingsirradiated within the framework of programs A, B, and C. Thepredictions made for these figures are realistic.

Figure 14 shows the simulations of biaxial tensile tests per-formed at 350°C and 380°C with«uu52•1025 s21 followed by arelaxation period on irradiated cladding tubes until four cycles(85•1024 nm22 (E.1 MeV)) in PWR, which are representativeof a PCI transient on irradiated tubes. There is a fairly good agree-ment between the test results and the simulations.

Lastly, the set of these results confirms the choice made todescribe the temperature and irradiation effects, as well as theidentification of evolution law parameters withT andf. Note thatthere are no systematic deviations in the simulations, the scatter-ing is due to the experimental uncertainties regarding the tempera-ture ~65°C! and the stress~65 MPa! which exist during creeptests on irradiated tubes, the improper knowledge of corrosion andhydridation effects upon the mechanical behavior and, generallyspeaking, the difficulties met when conducting hot-lab tests onspecimens which have been weakened by the irradiation.

5 ConclusionsThe mechanical behavior of Zircaloy-4 cladding tubes was de-

termined experimentally. The effect of the temperature on themechanical properties of nonirradiated and irradiated claddingswere reported. The analysis of the irradiated cladding tubes data,composed of biaxial tensile, creep, and relaxation tests, exhibitedradiation hardening and a decrease in creep strain.

A unified viscoplastic model was constructed to describe theexperimental observations. The model is able to simulate uni- andbiaxial tensile, creep, and relaxation tests for temperatures be-tween 350°C and 400°C. The effects of irradiation are integratedinto the model by means of an internal state variable representingneutron damage. This variable simulates a greater strain hardeningand a smaller static recovery.

Finally, we demonstrate the ability of the model to evaluate theanisotropic behavior of cladding tubes during tensile, high hoopstress creep and relaxation tests in the temperature range of 350–400°C and fluences from 0 to 85•1024 nm22 (E.1 MeV). Themodel allows the simulation of a pellet-cladding interaction typeloading.

The model has already been incorporated at EDF in the finiteelement code ASTER and implemented in the cladding tube cal-culation code CYRANO 3. This code is used to predict PCItransients.

Future developments of this model will incorporate ‘‘irradiationcreep’’ due to fast neutron flux.

AcknowledgmentsThe authors want to thank their industrial partners for financial

and technical supports in the research and development programsused to develop this study, i.e., Framatome Nuclear Fuel, Com-missariat a` l’Energie Atomique, EDF/SEPTEN.

A condensed version of this article has been presented atSMIRT XIV at Lyon ~France! in 1997.

Nomenclature

Mathematical Functions

A 5 scalar variableAA 5 derivative of the scalar variableA

@A# 5 vectorial representation of the second rank ten-sor A

@A# t 5 transpose of the vector@A#@A# 5 derivative of the vector@A# with respect to time@D# 5 vectorial representation of the Kronecker symbol@A# 5 matrix representation of the fourth rank tensorA

@A#21 5 inverse of the matrix@A#sinh 5 hyperbolic sine functionexp 5 exponential function

A(0), @A#0t

5 initial values of the variablesA and @A# t

Multiaxial Strains

@ «T#, @ «e#, @ «# 5 total, elastic, and inelastic strain rate vectors« 5 norm of the strain rate;

«5$2/3@ «# t@M #21@ «#%1/2

«zz, « rr , «uu 5 axial, radial, and tangential strains

Fig. 14 Biaxial tensile tests performed at 350 and 380°C withaÄ0.5, «uuÄ2"10À5 sÀ1, followed by a relaxation period onirradiated cladding tubes until four cycles in PWR„fÄ85"1024 nmÀ2, „EÌ1 MeV……. These loadings are represen-tative of a PCI transient on irradiated tubes. Experimental re-sults and simulations.


Multiaxial Stresses

@s# 5 stress vector@s8# 5 deviatoric stress vector

@a#, @a (1)#, @a (2)# 5 vectorial notation of the three kinematicalhardening variables

@a8# 5 deviatoric kinematical hardening variablea 5 norm of the kinematical hardening vari-

able @a#; a5$3/2@a# t@R#@a#%1/2

s2a 5 norm of the overstress~viscous stress! @s2a#;s2a5$3/2@s2a8# t@M #@s82a8#%1/2

@M #, @N#, @Q#, @R# 5 matrix representation of the four fourthrank tensors describing the material an-isotropy, i.e., orthotropic symmetry

szz, s rr , suu 5 axial, radial, and tangential stressesa5szz/suu 5 biaxiality ratio

Physical Parameters

T 5 temperature of the testT* 5 temperature of the irradiation

w 5 neutron fluxf 5 fluence,f5*w dtE 5 Young modulusn 5 Poisson ratio

DH, DH* 5 apparent creep activation energies respectively ats andsv constant

Y* , Y 5 scalar parameters describing the evolution of theamplitude of the kinematical hardening variablesduring cycling

D 5 irradiation damage variable~scalar!k 5 Boltzmann constant

References@1# Baron, D., and Bouffioux, P., 1989, ‘‘Le Crayon Combustible des Reácteurs a`

Eau Pressuriseé de Grande Puissance,’’ Rapport EDF, HT M2/88-27.@2# Higgy, R., and Hammad, F. H., 1972, ‘‘Effect of Neutron Irradiation on the

Tensile Properties of Zircaloy-2 and Zircaloy-4,’’ J. Nucl. Mater.,44, pp.215–277.

@3# Northwood, D. O., 1977, ‘‘Irradiation Damage in Zirconium and its Alloys,’’AT. Energy Review, p. 154.

@4# Franklin, D. G., 1982, ‘‘Zircaloy-4 Cladding Deformation During Power Re-actor Irradiation,’’ ASTM-STP 754, pp. 235–267.

@5# Petterson, K., 1982, ‘‘An Evaluation of Irradiation Temperature on the Irra-diation Hardening of Zircaloy,’’ Studvik Super-Ramp Project, SR 82/3.

@6# Baty, D. L., Pavinick, W. A., Dietrich, M. R., Clevinger, G. S., and Papazo-glou, T. P., 1984, ‘‘Deformation Characteristics of Cold-Worked and Recrys-tallized Zircaloy-4 Cladding in Zirconium in the Nuclear Industry,’’ 6th Inter-national Symposium, ASTM-STP,824, 306–339.

@7# Yasuda, T., Nakatsuka, M., and Yamashita, K., 1987, ‘‘Deformation and Frac-ture Properties of Neutron-Irradiated Recrystallized Zircaloy-2 Cladding Un-der Uniaxial Tension,’’ Zirconium in the Nuclear Industry, VIIth Int. ASTM939, pp. 734–747.

@8# Delobelle, P., Robinet, P., Bouffioux, P., Geyer, P., and Le Pichon, I., 1996,‘‘A Unified Model to Describe the Anisotropic Viscoplastic Behavior ofZircaloy-4 Cladding Tubes,’’ Zirconium in the Nuclear Industry, 11th Interna-tional Symposium, ASTM-STP 1295, pp. 373–393.

@9# Delobelle, P., Robinet, P., Geyer, P., and Bouffioux, P., 1996, ‘‘A Model toDescribe the Anisotropic Viscoplastic Behavior of Zircaloy-4 Tubes,’’ J. Nucl.Mater.,238, pp. 135–162.

@10# Van Swam, L. F., Knorr, D. B., Pelloux, R. M., and Shewbridge, J. F., 1979,‘‘Relationship between Contractile Strain RatioR and Texture in ZirconiumAlloy Tubing,’’ Metall. Trans. A,10, p. 483.

@11# Bouffioux. P., 1995, ‘‘An Experimental Method to Investigate the AnisotropicViscoplastic Behavior of Zircaloy Cladding Tubes,’’ 11th International Sym-posium, ASTM-STP.

@12# Bouffioux, P., 1994, ‘‘Etude du Comportement en Plasticite´ des Tubes deGainage en Zircaloy-4 Sous Sollicitations Biaxeés,’’ Report DER-EDF A4/94/012A.

@13# CEA, Framatome, and EDF Cooperative program, 1995, Proprietary data.@14# Beauregard, R., Clevinger, G. S., and Murty, K. L., 1977, ‘‘Effect of Anneal-

ing Temperature on the Mechanical Properties of Zircaloy-4 Cladding,’’Pro-ceedings of the SMIRT IV, paper C3/5.

@15# Murty, K. L., 1989, ‘‘Applications of Crystallographic Textures of ZirconiumAlloys in Nuclear Industry,’’ Zirconium in the Nuclear Industry, VIIIth Inter-national Symposium, ASTM-STP 1023, p. 570.

@16# Yi, J. K., Park, H. B., Park, G. S., and Lee, B. W., 1992, ‘‘Yielding andDynamic Strain Aging Behavior of Zircaloy-4 Tube,’’ J. Nucl. Mater.,189, pp.353.

@17# Prioul, C., 1995, ‘‘Le Vieillissement Dynamique Dans les Alliages de Zirco-nium: Consequences Sur Les Propriete´s Mecaniques,’’ SF2M, Journeéd’Etudes Proprie´tes-Microstructures-Les Edit, Physique, pp. 25–34.

@18# Murty, K. L., Clevinger, G. S., and Papazoglou, T. P., 1977, ‘‘Thermal Creepof Zircaloy-4 Cladding,’’ SMIRT IV, San Francisco, Aug. 15–19, paper C3/4.

@19# Matsuo, Y., 1987, ‘‘Thermal Creep of Zircaloy-4 Cladding Under InternalPressure,’’ J. Nucl. Sci. Technol.,24, No. 2, pp. 111–119.

@20# Lyashenko, V. S., Bykov, V. N., and Paulinov, L. B., 1959, Fiz. Met. Metall-oved,8, p. 362.

@21# EPRI, B and W, 1983, ‘‘Cooperative Program on PWR Fuel Rod Perfor-mance,’’ NP 2848, Project 711-1.

@22# Franklin, D. G., Lucas, G. E., and Bement, A. L., 1983, ‘‘Creep of ZirconiumAlloys in Nuclear Reactors,’’ ASTM STP 815, p. 35.

@23# Huang, P., Mahmood, T., and Adamson, R., 1996, ‘‘Effects of Thermome-chanical Processing on In-Reactor Corrosion and Post-Irradiation MechanicalProperties of Zircaloy-2,’’ 11th International Symposium, ASTM-STP 1295,pp. 726–755.

@24# Pilvin, P., 1988, ‘‘Identification Des Parametres de Mode`les de Comporte-ment,’’ Proceedings of Me`camat, International Seminar on Inelastic Behaviorof Solids, Models and Utilization, pp. 155–164.

@25# Limback, M., and Anderson, T., 1996, ‘‘A Model for Analysis of the Effect ofFinal Annealing on the in- and out-of- Reactor Creep Behavior of ZircaloyCladding,’’ ASTM-STP 1295, pp. 448–468.

@26# Matzke, H., 1993, ‘‘Radiation Damage in Nuclear Fuel Materials,’’ Solid StatePhenomena,30–31, pp. 355–366.


L. E. GovaertEindhoven University of Technology,Faculty of Mechanical Engineering,P.O. Box 512, 5600 MB Eindhoven,

The Netherlands

P. H. M. TimmermansPhilips Center for Manufacturing Technology,

Eindhoven, The Netherlands

W. A. M. BrekelmansEindhoven University of Technology,Faculty of Mechanical Engineering,P.O. Box 512, 5600 MB Eindhoven,

The Netherlands

The Influence of Intrinsic StrainSoftening on Strain Localizationin Polycarbonate: Modeling andExperimental ValidationIntrinsic strain softening appears to be the main cause for the occurrence of plasticlocalization phenomena in deformation of glassy polymers. This is supported by thehomogeneous plastic deformation behavior that is observed in polycarbonate samplesthat have been mechanically pretreated to remove (saturate) the strain softening effect. Inthis study, some experimental results are presented and a numerical analysis is performedsimulating the effect of mechanical conditioning by cyclic torsion on the subsequentdeformation of polycarbonate. To facilitate the numerical analysis of the ‘‘mechanicalrejuvenation’’ effect, a previously developed model, the ‘‘compressible Leonov model,’’ isextended to describe the phenomenological aspects of the large strain mechanical behav-ior of glassy polymers. The model covers common observable features, like strain rate,temperature and pressure dependent yield, and the subsequent strain softening and strain-hardening phenomena. The model, as presented in this study, is purely ‘‘single mode’’(i.e., only one relaxation time is involved), and therefore it is not possible to capture thenonlinear viscoelastic pre-yield behavior accurately. The attention is particularly focusedon the large strain phenomena. From the simulations it becomes clear that the precon-ditioning treatment removes the intrinsic softening effect, which leads to a more stablemode of deformation.@S0094-4289~00!01002-1#

1 IntroductionThe deformation behavior of glassy polymers is generally

strongly dominated by localization phenomena like necking, shearband formation, or crazing. This susceptibility to localization isdirectly related to the intrinsic large strain behavior of glassypolymers visualized in Fig. 1. These true stress-strain curves canbe obtained in uniaxial extension using a video-controlled tensiletest@1# or in uniaxial compression@2,3#. Typically, the yield stressdepends on strain rate, temperature, and pressure@4#. The post-yield behavior of glassy polymers is governed by two character-istic phenomena@1,5#. Immediately after the yield point the~true!stress tends to decrease with increasing deformation, an effect thatis usually referred to as intrinsic strain softening. At large defor-mations the softening effect is saturated and the true stress startsto rise again with increasing deformation. This strain hardeningeffect has been subject of a number of studies in the past~e.g.,@3,6,7#!, and is generally interpreted as a rubber elastic contribu-tion by the molecular entanglement network.

Although the origin of the intrinsic softening effect is not yetcompletely clear, it seems to be closely related to the physicalaging process~volume relaxation! that occurs in the glassy state@8#. With physical aging the specific volume decreases leading toan increase of the elastic modulus, a decrease of the time depen-dence~age-shift!, and an increase of the yield stress@8#. The in-crease of the yield stress seems to develop simultaneously withthe enthalpy overshoot that is observed around the glass transitiontemperature in DSC experiments on aged amorphous polymers@9,10#. The effect of aging on the deformation behavior of aglassy polymer is schematically represented in Fig. 2. During ag-ing, the yield stress increases and the intrinsic softening effectappears. As a result of intrinsic softening the large strain behavior

of the different samples is exactly the same: the effect of physicalaging has been removed and the material is rejuvenated@10–12#.The same effect can be achieved by heating the sample above theglass transition temperature and cooling it rapidly to the glassystate~quenching!. In some glassy polymers, as, for instance, PVC,intrinsic softening completely disappears after this quenchingfrom the rubbery into the glassy state@13#.

Although these experimental observations clearly connect theintrinsic softening effect to the physical aging process, the effectcannot be rationalized completely in terms of an increase of freevolume as a result of the imposed strain. The inability to explainintrinsic softening in experiments with a negative dilatationalstrain~compression! is probably the strongest argument. Xie et al.@14# measured a decrease of the actual free volume in polycarbon-ate under compression by means of positron annihilation lifetimespectroscopy~PALS!, whereas polycarbonate is known to displayintrinsic softening in compression@3#. In PALS measurementsduring compression tests on polymethylmethacrylate, however,Hasan et al.@15# observed an increase of the number of areas oflocal free volume evolving to a steady value. Based upon theseobservations they postulated a phenomenological law for the evo-lution of the densityD of these areas in a glassy polymer duringdeformation. During elastic deformation,D is constant~the mate-rial state does not change!. During plastic deformation,D evolvesto a saturation valueD` , indicating a maximum amount of re-gions with elevated levels of free volume, which is independent ofstrain rate or thermal history. Inclusion ofD in an originally non-intrinsic softening model resulted in a constitutive model that ex-hibited intrinsic strain softening@15#.

Intrinsic strain softening is an important factor in the initiationof strain localization. As during softening the deformation is al-lowed to proceed at a decreasing level of the~true! stress, smallstress variations will inherently lead to large differences in thelocal strain rate, nuclei for localized plastic deformation zones. Inthe absence of intrinsic softening, the deformation will be homo-geneous if the strain-hardening behavior is large enough to com-

Contributed by the Materials Division for publication in the JOURNAL OF ENGI-NEERING MATERIALS AND TECHNOLOGY. Manuscript received by the MaterialsDivision May 5, 1999; revised manuscript received November 23, 1999. AssociateTechnical Editor: H. M. Zbib.


pensate for the geometrical softening during a tensile test@4,7#.An extensive numerical study on the influence of strain softeningand strain hardening on neck formation in plane strain extensionwas performed by Wu and van der Giessen@16#. They showedthat intrinsic strain softening always leads to strain localization,whereas in the absence of softening strain localization can besuppressed if the amount of strain hardening is sufficient.

There is also some experimental evidence concerning the influ-ence of strain softening on neck formation. Cross and Haward@13# used samples of quenched PVC that display no intrinsic soft-ening and observed uniform deformation in a tensile test whereasslowly cooled samples necked. An alternative method to preventinhomogeneous behavior in glassy polymers is based on the initialelimination of intrinsic softening by raising the value of the soft-ening parameterD to its saturation valueD` by application ofplastic deformation~mechanical preconditioning!. A good ex-ample of the effect of mechanical preconditioning is the alternatedbending of PVC samples by Bauwens@17#, which suppressednecking in a subsequent tensile test. G’Sell@11# achieved thesame effect after plastic cycling in simple shear on polycarbonate.Recent experimental research@18# also shows the effect of theelimination of intrinsic softening by mechanical preconditioning:axisymmetrical samples were plastically cycled in torsion; tensiletests on these rejuvenated samples resulted in homogeneous de-formations and allowed for the characterization of the strain hard-ening behavior of polycarbonate.

The present study addresses the influence of intrinsic strainsoftening on the macroscopic deformation behavior of axisym-metric polycarbonate bars. To facilitate a numerical analysis, aconstitutive model which was derived in a previous study, theso-called compressible Leonov model@19#, is extended to incor-

porate the phenomena of intrinsic strain softening and strain hard-ening. The material characterization, including the determinationof the necessary parameters for the extended model, will be dis-cussed. With the parameters for polycarbonate known, the modelis employed to simulate neck formation and to predict the defor-mation behavior of a mechanically preconditioned sample~torsioncycling! in a subsequent tensile or torsion test.

2 Constitutive Modeling

2.1 The Compressible Leonov Model. For an arbitrarymaterial element of a loaded configuration the local actual defor-mation with respect to a predefined reference state is determinedby the deformation gradient tensorF ~e.g., Hunter@20#!. ThistensorF is multiplicatively decomposed into an elastic partFe anda plastic partFp , according to:

F5Fe•Fp (1)

The plastic contributionFp indicates the deformation~with re-spect to the reference state! of the relaxed stress-free configura-tion, which is defined as the state that would instantaneously berecovered when the stress is suddenly removed from the elementconsidered. The decomposition in Eq.~1! is not unique becauserotational effects can be assigned toFp as well as toFe . Unique-ness is achieved by the extra requirement that the plastic defor-mation occurs spin-free@21#.

The Cauchy stress tensors is elastically expressed in the leftCauchy Green tensorBe associated with the tensorFe which isdefined by

Be5Fe•Fec (2)

whereFec denotes the conjugate ofFe ~which is equivalent to the

transpose of the matrix representation of the tensor!. In this equa-tion it is presupposed that the elastic behavior is isotropic. In thatcase the application of expressions of the types5s(Be) guaran-tees the conservation of objectivity~if indeed the total spin andconsequently superimposed rigid body rotations are completelyattributed to the elastic part of the deformation!.

To specify the dependence of the stress on the deformation, aneo-Hookean relationship is chosen@19#:

s5K~Je21!I1GBed (3)

where the superscriptd indicates the deviatoric part.In this equation,K and G are the bulk modulus and the shear

modulus, respectively. The elastic volume change factorJe is de-fined by

Je5det~Fe!5Adet~Be! (4)

The tensorBe denotes the isochoric fraction of the elastic leftCauchy Green tensorBe according to

Be5Je22/3Be (5)

Based on purely kinematical considerations@19# the followingdifferential equation can be derived to calculate the evolution ofBe :

Be5~Dd2Dpd!•Be1Be•~Dd2Dp

d! (6)

The left-hand side of this equation represents the~objective! Jau-mann derivative of the isochoric elastic left Cauchy Green tensor.The tensorDp denotes the plastic deformation rate tensor. Theinitial condition necessary for the solution of the differential equa-tion ~6! reads:Be5I .

To complete the constitutive description the plastic deformationrate is expressed in the Cauchy stress by a generalized non-Newtonian flow rule@22#

Dp5s d

2h~teq!with teq5A1

2 tr~s d•s d! (7)

Fig. 1 Schematic representation of the effect of strain rate onthe true stress-strain curve of a glassy polymer

Fig. 2 Schematic representation of the effect of physical agingon the true stress-strain curve of a glassy polymer


The viscosityh depends on the equivalent stressteq according toan Eyring relationship@19#:

h~teq!5At0

teq/t0

sinh~teq/t0!(8)

In this equationA is a time constant andt0 a characteristicstress, respectively related to the activation energyDH and theshear activation volumeV according to@4,20#

A5A0 expFDH

RTG ; t05RT

V(9)

with R the gas constant,A0 a constant preexponential factor in-volving the fundamental vibration energy, andT the absolute tem-perature. It is emphasized that Eq.~7! implies that plastic defor-mation occurs at constant volume: tr(Dp)50 as tr(s d)50.ConsequentlyDp

d in Eq. ~6! may be replaced byDp . For the samereason,Je5det(Fe) in Eq. ~3! may be replaced byJ5det(F).

The model derived above was referred to as the compressibleLeonov model in the original paper by Tervoort et al.@19#. Todemonstrate the typical behavior of this compressible Leonovmodel, an application to uniaxial extension is performed. Thisleads, for constant strain rate, to the response schematically visu-alized in Fig. 3. The response of this Leonov model shows asudden transition from elastic-to-plastic behavior, which is verysimilar to that of an elastic-perfectly plastic material with a rate-dependent yield stress.

2.2 Extension to Intrinsic Strain Softening and StrainHardening. This section describes the extension of the com-pressible Leonov model to include both the intrinsic strain soften-ing and the strain hardening effect. Complementary to the outlinein Section 2.1 the Cauchy stress tensors is now redefined to becomposed of two distinguishable parts~in a parallel assemblage!,the driving stress tensors and the hardening stress tensorr , ac-cording to

s5s1r (10)

The expression for the driving stresss is adopted from the com-pressible Leonov model described above, see Eq.~3!:

s5K~J21!I1GBed (11)

The expression for the hardening stressr is obtained in the fol-lowing. In studies on the deformation behavior of glassy poly-mers, it is common practice to model the hardening behavior as ageneralized rubber elastic spring with finite extensibility, like theso-called three-chain and eight-chain models of Arruda and Boyce@3#, or the full chain model of Wu and van der Giessen@6#. On theother hand, Haward@7# applied network models employing

Gaussian chain statistics~leading to a neo-Hookean strain harden-ing response! to experimental uniaxial stress-strain curves, andconcluded that some amorphous and most semicrystalline poly-mers obeyed this formulation. The large amount of softening, ob-served in some glassy polymers, prevented the successful appli-cation of the Gaussian model to these polymers. However, asmentioned in the Introduction, it can be shown by means of me-chanical preconditioning@18# that this approach is also valid forpolycarbonate. Gaussian statistics leads to a neo-Hookean relationbetween stresses and strains. Generalization to three dimensions,in the assumption that the network is incompressible, this neo-Hookean relationship for the hardening stress tensorr can be writ-ten as

r5HBd (12)

with H the strain hardening modulus~assumed to be temperatureindependent!. Contrary to Boyce et al.@21# the hardening stress isnot related to the plastic deformation but to the total deformation.This adaptation is introduced because in the present approach bothelastic and plastic deformations are assumed to decrease the con-figurational entropy of the polymer.

To complete the constitutive description the plastic deformationrate is still expressed in the Cauchy stress tensor by a generalizednon-Newtonian flow rule

Dp5sd

2h~teq,D,p!(13)

whereteq, D, andp are state variables to be defined in the fol-lowing.

Particularly the driving stress tensors is relevant for the incor-poration of softening in the model. As suggested by Hasan et al.@15# a history variableD is specified, the softening parameter,which influences the viscosityh. During plastic deformationDevolves to a saturation levelD` , which is independent of thestrain history. The result forh reads

h~teq,D,p!5Am~D,p!t0

teq/t0

sinh~teq/t0!(14)

where the equivalent stressteq is redefined by

teq5A12 tr~sd

•sd! (15)

and with

Am~D,p!5A expS mp

t02D D (16)

p5213 tr~s!52

13 tr~s! (17)

wherep is the pressure~positive in compression!. The parameterm is a pressure coefficient, related to the shear activation volumeV and the pressure activation volumeV according to

m5V

V(18)

The evolution of the softening parameterD is specified by@15#

D5hF12D

D`G gp (19)

with initially D50;h is a material constant describing the relativesoftening rate andgp is the equivalent plastic strain rate, accord-ing to

gp5Atr~Dp•Dp!5teq

h&(20)

Fig. 3 Schematic representation of the response in uniaxialextension from the Leonov model


3 Experimental

3.1 Materials and Sample Preparation. The material usedin this study was polycarbonate, purchased as extruded rods~10.4mm in diameter! from Eriks BV ~Alkmaar, The Netherlands!. Ad-ditional to the mechanical parametersK andG, the values of thedensityr, the thermal conductivityk, the thermal expansion coef-ficient a, the specific heatc, and the glass-transition temperatureTg are given in Table 1.

The material properties, with the exception ofTg , G, and K,were provided by the supplier and are in good agreement withvalues reported in literature@23–26#. Tg was determined by dy-namic mechanical thermal analysis~DMTA ! and G and K weredetermined from the Young’s modulusE and the Poisson’s ratiovmeasured in the initial stages of a tensile test@19#. The thermalmaterial parameters will be used to perform a thermomechanicalanalysis in the subsequent sections.

For the uniaxial extension and torsion experiments, the speci-mens were designed as dog-bone shaped axisymmetric bars, de-picted in Fig. 4~a!. For the uniaxial compression experiments,cylindrical test specimens were used, the geometry shown in Fig.4~b!.

3.2 Uniaxial Extension. Uniaxial tensile tests were per-formed on a FRANK 81656 tensile tester at strain rates varyingfrom 1024 to 1022 @s21# and at temperatures of 22, 32, and 40@°C# ~295, 305, and 313@K#, respectively!. The true stress at theyield point, required for the determination of the yield parameters,was determined by assuming incompressibility in the viscoelasticarea, which introduces a small error~approx. 2 percent! comparedto a compressible approach. Neck formation and propagation wasrecorded by means of a video camera. From the images, the elon-gation factor in the neck is calculated from the diameter reductionin combination with the assumption of incompressibility. At theend of the test, the neck diameter was measured with the specimenstill in the load frame as an assessment of the video images.

3.3 Uniaxial Compression. Uniaxial compression testswere performed at room temperature at strain rates in the rangefrom 1024 to 1022@s21#, also on a FRANK 81656 tensile tester.A high performance lubricant~Hasco Z260! between the sampleand the polished stainless-steel shaft of the compression devicecould not prevent the samples from barreling at compressivestrains of approximately 0.20@-#. Since this phenomenon occurredat compressive strains beyond the yield point, it does not affectthe measured value of the true stress at the yield point. To avoidinfluence on the determination of the softening and hardening pa-rameters, data measured at compressive strains beyond 0.20@-#are omitted.

3.4 Torsion. Torsion experiments were performed on a test-ing machine consisting of an adjustable rigid support and a rotat-ing clamp. The sample is installed in the machine in a way thatinitial axial forces in the sample are avoided. In the testing devicethe length of the sample is fixed during deformation and torqueand axial load on the sample are measured during deformation bytwo independent load cells in the support. To determine the rota-tion, the angular displacement of the clamp is monitored. As areference, an axial line was drawn on the specimen, and it ap-peared that the torsion was restricted to the gauge section of thesample. During the torsion of polycarbonate initially narrow cir-cumferential shear bands were observed that broadened with on-going rotation, a phenomenon that has also been reported by Wuand Turner@27#. In order to obtain isothermal conditions, the ro-tation speed was limited to 360 degrees per minute resulting in anominal shear rate of 0.56@s21# at the outer surface of the bar.

In the case of the mechanical pretreatment~rejuvenation!, thetorsion experiments were performed by twisting polycarbonatespecimens to and fro over 720 degrees. After reversing the direc-tion of the twist, the rotation rate was the same as during loading.Heating of the rejuvenated samples above the glass transition tem-perature did not induce any residual motion, from which it wasconcluded that the specimens rejuvenated in this way regain isot-ropy. After mechanical conditioning, the rejuvenated sampleswere allowed to relax unconstrained for 3 hrs. Subsequently, theywere subjected to either uniaxial extension or torsion.

4 Material Characterization

4.1 Yield Parameters. The yield ~or Eyring! parameterscan be determined by measuring the true stress at the yield pointduring tension and compression experiments as a function ofstrain rate at different temperatures@28,29#. The strategy is basedon the application of the incompressible non-Newtonian viscousflow rule, Eq.~13!, which can be reformulated in axial direction~Fig. 4~a!! by

lp,zz

lp,zz5

1

3h~szz2srr ! (21)

with lp,zz the axial plastic elongation factor and withszz andsrrthe axial and radial components of the driving stress tensors,respectively. The expression for the viscosity, Eq.~14!, can thenbe replaced by:

h5h~szz,srr ,p,D,T!5At0

expF2mp

t01DG

F uszz2srr u

t0)G

sinhF uszz2srr ut0)

G(22)

To facilitate a straightforward analysis of the yield data, the fol-lowing considerations are made:

• At the yield point, the contribution of hardening is negligible,and therefore the components of the driving stress are equalto the components of the Cauchy stress.

• At the yield point, the plastic strain rate is equal to the nomi-nal strain rate«zz

0 applied.• At the yield point the value of the softening parameterD

equals 0.• The argument of the hyperbolic sine in the viscosity function

is large, and therefore the hyperbolic sine may be approxi-mated by an exponential function.

• During uniaxial tension and compression, the pressure isgiven byp521/3szz.

The incorporation of these considerations into the non-Newtonianflow rule leads to

Fig. 4 Geometry of axisymmetrical specimens for „a… uniaxialextension and torsion experiments and „b… uniaxial compres-sion experiments. Dimensions in †mm ‡.

Table 1 Material properties of polycarbonate at roomtemperature

r@kg m23#

k@W m21 K21#

a@K21#

c@J kg21 K21#

G@MPa#

K@MPa#

Tg@°C#

1200 0.21 65•1026 1200 860 4000 150


uszzuT

53R

)V1aVS ln@A0u«zz

0 u#1DH

RT2 lnF1

6)G D (23)

with a5sign(szz). This expression suggests that plots ofuszzu/Tagainst the logarithm of the strain rate for a series of temperaturesshould give a set of parallel lines. The result for polycarbonate isshown in Fig. 5, where the measured values ofuszzu/T of both theuniaxial tension as the uniaxial compression tests are plottedagainst the logarithm ofu«zz

0 u. The solid lines represent the best fitof the experimental results using a single set of the yield param-etersA0 , V, V, andDH, and seem to represent the actual yieldbehavior well over the entire range of strain rates experimentallycovered. It should be noted however, that the yield parametersshould be used with care outside the experimentally covered re-gion, as it has been shown in experiments by Bauwens-Crowetet al. @29# that the yield stress of polycarbonate tends to have amore substantial strain rate and temperature dependence at lowtemperatures and high strain rates than observed at high tempera-tures and low strain rates. This is related to secondary glass-transitions and implies that actually more than one Eyring flowprocess should be taken into account. For the range of strain ratesconsidered in the present work a single flow process seems tosuffice.

The values of the yield parameters obtained from the fit aregiven in Table 2. The values are in good agreement with valuesreported by Bauwens-Crowet et al.@29# and Duckett et al.@28#

4.2 Hardening Parameter. As mentioned before in the In-troduction, the hardening parameterH is, in the present study,determined from a uniaxial tensile test on a rejuvenated polycar-bonate sample. To rejuvenate the material, axisymmetric sampleswere subjected to a 720 deg to and fro fixed-end torsion treatment.After this mechanical pretreatment the intrinsic strain softeningbehavior has disappeared~saturated! with the astonishing resultthat the polycarbonate bars deform homogeneously~without neck-ing! in a subsequent tensile test. Figure 6 shows the result of suchan experiment, whereszz is plotted as a function of the strainmeasurelzz

2 2lzz21. This strain measure is, in a uniaxial tensile~or

compression! test, the component of the deviatoric isochoric leftCauchy Green deformation tensorBd in the load directioneW z . Theconstant slope of the tensile curve at large strain levels conse-

quently supports the neo-Hookean approach and directly reflectsthe value of the hardening modulus:H529@MPa#.

4.3 Strain Softening Parameters. The softening param-eters of polycarbonate are provisionally determined from auniaxial compression experiment. As was mentioned before, theuniaxial compression experiments showed barreling of the speci-men at compressive strains over 0.2@-# and therefore the data atlarger compressive strains were omitted. The values of the soften-ing parameters were determined by a fitting procedure on the postyield behavior of a compression test at a rate of 1023@s21#:h5200@-# and D`528@-# ~see Fig. 7~a!!. To facilitate the com-parison with the experimental data, the predictions by the Leonovmodel were shifted along the strain axis in order to overlap thepredicted and the measured yield points. The actual comparisonbetween the experimental data and the prediction using the com-pressible Leonov model is shown in Fig. 7~b!. The simulation wasperformed using the yield and hardening parameters obtained inthe previous sections and the softening parameters mentionedabove. Note the difference between the strains at yield in Fig.7~b!. Since the single mode compressible Leonov model displayselastic~rate independent! behavior up to the yield point, it is notable to describe the~multirelaxation time! viscoelastic behaviordisplayed by the material.

There are several possibilities to correct for this deficiency. Oneis the extension of the Leonov model to a spectrum of relaxationtimes ~multimode! as was suggested by Tervoort et al.@30#. An-other possibility was demonstrated by Hasan and Boyce@31# whoconsidered a distribution of activation energies. As the use ofeither of these extensions would dramatically increase the compu-tation time for the finite element analysis in the next sections thedeficiency at low strain levels will not be addressed.

Fig. 5 Yield stress over absolute temperature zszzzÕT as afunction of strain rate «zz . The solid lines are a best fit using asingle set of yielding parameters for each polymer.

Fig. 6 True stress szz versus lzz2 Àlzz

À1 during a tensile test at«Ä2.2"10À3

†sÀ1‡ of a polycarbonate tensile bar, precondi-

tioned in torsion

Fig. 7 Determination of the softening parameters in polycar-bonate at a compressive strain rate of 10 À3

†sÀ1‡ at room tem-

perature. „a… fitting the post-yield softening behavior; „b… simu-lation using the compressible Leonov model.

Table 2 Yield parameters for polycarbonate

A0@s#

V@m3 mol21#

V@m3 mol21#

DH@kJ mol21#

3.6•10225 3.4•1023 2.4•1024 290


5 Numerical Simulations

5.1 Uniaxial Extension of Untreated Polycarbonate. Forthe analysis an axisymmetric specimen, as used in the experiments~Fig. 4!, is considered. In the center of the bar, a Cartesian coor-dinate system is defined, in whichz refers to the axial directionandr to the radial direction. Because of symmetry of the materialgeometry and loading conditions, in Fig. 8 only one quarter of thelongitudinal cross-section is considered up toz50.5 L0 ~initialgeometry!. At the end face on this position the displacements inzdirection are prescribed~constant velocity!. In the simulations,nearz50, a geometric imperfection is introduced to initiate neck-ing. This imperfection is cosine shaped, defined by

Ri5R0F121

2~12j!cosS pz

zR0D G 0<z<zR0 (24)

with Ri the outer radius of the imperfection,R0 the outer radius ofthe gauge section of the perfect bar,j the measure of the imper-fection,j5(2Ri(z50)2R0)/R0 , andz controls the length of theimperfection. In the present calculationsj50.9925 andz50.85which is equivalent to an area reduction of 1.5 percent atz50.The nominal or engineering stress is defined byFz /(pR0

2), whereFz is the applied tensile force.

The result of a numerical simulation at a nominal strain rate inthe gauge section of 7.5•1023 @s21# is given in Fig. 9. The nomi-nal stress and the elongation factor in the neck are depicted as afunction of the nominal strain, together with the deformed meshesat different stages of the deformation. The nominal stress is de-fined as the tensile force divided by the original cross-sectionalarea and the draw ratio in the necklN is calculated by division ofthe original cross-sectional area of the bar by the actual cross-sectional area in the middle of the specimen. During the simula-tion, the specimen initially deforms homogeneously, both in theelastic region and in the first part of the viscoplastic region(a–b). At some stage, the deformation localizes and a neck isformed (b–d), which propagates along the specimen as deforma-tion continues (d–e). The neck propagation takes place underapproximately steady-state conditions. The steady-state value of

the nominal stress during propagation of the neck proved to beindependent of the geometry of the initial imperfection. The levelof the draw ratio, on the other hand, was slightly influenced. Thesimulated levels of the nominal stress during neck propagation arecompared to experimental data taken at room temperature in Fig.10. Using the parameter set determined in the previous section, itproved impossible to predict the right stress level~see Fig. 10,D`528@-#!. If it is assumed that the constitutive model is ad-equate, the quantification of the parameter set is indicated as thesource of this discrepancy. Apparently, the friction between thecompression platens and the sample also influenced the results ofthe compression test at low strain levels. To improve the descrip-tion of the material behavior, the value of the softening parameterD` was varied~see Fig. 10!. A good description of the experi-mental data was obtained with a value ofD`536@-#. To checkthis value, the simulations were repeated at temperatures of 305and 312 @K# and compared to the experimental values of thenominal stress level during neck propagation at these tempera-tures. As can be observed in Fig. 11, the valueD`536@-# yieldsa reasonable description for all temperatures. This value wastherefore adopted for the numerical simulations in the nextsections.

5.2 The Mechanical Pretreatment: Mechanical Rejuvena-tion. The geometry of the axisymmetric polycarbonate samplesthat were preconditioned by one cycle of fixed-end torsion wasshown in Fig. 4. The cylindrical surface of the bar is traction free,

Fig. 8 Definition of the geometry of the longitudinal cross-section of the axisymmetrical tensile bar

Fig. 9 Simulated tensile response of polycarbonate in termsof the nominal stress szz

0 and the draw ratio in the neck lN

versus the nominal strain «zz0 at a nominal strain rate «zz

0 Ä7.5"10À3

†sÀ1‡. The deformed meshes at different stages of the

simulation „a – e… are also shown.

Fig. 10 Comparison of simulated and experimental values ofthe nominal stress during neck propagation at room tempera-ture as a function of the nominal strain rate using different val-ues of D` . The lines are fitted through the results of the simu-lations.

Fig. 11 Simulated and experimental values of the nominalstress during neck propagation versus nominal strain rate atdifferent temperatures using D`Ä36. The lines are fittedthrough the results of the simulations.


and it is assumed that axial displacements are negligible. Thetwist w(t), dependent on the timet and defined per unit length ofthe bar, is applied at a low, constant, angular velocityw. Conse-quently, temperature effects have not been taken into account andthe simulations were performed under isothermal conditions. Theapplied twistw(t) leads to a torqueM (t), and an axial forceF(t)resulting from the axially constrained ends.

To simulate the mechanical rejuvenation treatment a dedicatedfinite difference scheme was developed, assuming the relevantstress and strain quantities to be only a function of the radius andthe loading time~homogeneous deformation over the length of thebar!. The axisymmetric specimens were subjected to one fullcycle of large strain torsion. First, the specimen was deformed upto a maximum twist ~defined per unit length! of w50.25@rad mm21#, which was applied at a constant angular ve-locity equal tow59.1•1024 @rad s21 mm21#. After this, the direc-tion of the twist was reversed, deforming the sample to a smallnegative twist value. Upon unloading of the sample, the cylindri-cal outer surface of the specimen approximately regains its initialstate, which was verified during the experiments by monitoring areference line on the sample.

Figure 12~a! shows the variation of the torqueM as a functionof the applied twistw during the applied process history. PathABC corresponds to the loading stage of the rejuvenation experi-ment and shows clearly the effect of intrinsic softening as thetorque decreases after the yield point~B! has been reached. Aftera subsequent twist of about 0.1@rad mm21# the torque level in-creases again as strain hardening sets in. The reversed twistingstage is represented by path CDE. The elastic unloading is shownby path EF. During this stage the cylindrical surface of the speci-men regains its original geometry.

Figure 12~b! shows the distribution of the softening parameterD over the ~dimensionless! radius r /Ra ~Ra is the actual outerradius! at stages C and F in Fig. 12~a!. It is clear that after theloading path ABC the softening parameterD in the outer layer ofthe specimen has almost reached its saturation level ofD`536.After the return twist, path CDEF, the softening parameter reachesits saturation level over 0.4,r /Ra<1 indicating that the intrinsicsoftening effect has been removed over approximately 84 percentof the specimen volume.

The effect of this rejuvenation on a subsequent twist is shownin Fig. 12~a!, path FGH, where it is clear that~a! the onset ofyield begins at a considerably smaller torque than for the originalmaterial and~b! no intrinsic softening is observed.

The distribution of the stress components over the~dimension-less! radius at stage C and of the residual stress components at theend of the rejuvenation by fixed-end torsion at stage F are shownin Fig. 13. It is clear that at the end of the mechanical pretreatmentthe residual stress level of most components is negligible, whereasthe componentsuz has a relatively high value in the core of the

specimen. Although these residual stresses may effect the subse-quent deformation behavior of the rejuvenated sample through thestress dependence of the viscosity~Eq. ~14!!, the elevated levelsof suz only influence a small part of the specimen~approx. 10percent of the volume!. In order to minimize the influence ofresidual stresses in the experiments performed, the samples wereallowed to relax unconstrained for a period of 1032104 @s# beforesubsequent mechanical experiments were performed, thus allow-ing the residual stress levels to decrease.

Therefore, residual stresses will not be taken into account insimulations of tension and torsion tests on rejuvenated samples.Only the distribution of the softening parameterD over the radiusafter the rejuvenation, depicted by stage F in Fig. 12~b!, will beused to characterize the ‘‘state’’ of the rejuvenated material.

5.3 Fixed-End Torsion of Rejuvenated Polycarbonate.The fixed-end torsion simulations on rejuvenated samples wererealized with a constant twist rate of 9•1024 @rad s21 mm21# atroom temperature to a maximum twist of 0.4@rad mm21#. Thesimulations were performed employing the same method as in theprevious section, whereas the profile of the softening factorD,depicted in Fig. 12~b! at stage F, was taken as the initial situation.It should be noted that the residual stresses were not taken intoaccount.

In Fig. 14~a! the variation of torque as a function of the twistwis given for both the simulation and the experiment. Both comparewell, which is a support for the values ofD`536@-# and H529@MPa# determined previously. The small deviations at lowertwist levels, w,0.05@rad mm21#, are attributed to viscoelasticeffects, which, as already stated before@30# are not adequatelyaddressed in the single mode compressible Leonov model.

Fig. 12 Results of simulations of the mechanical precondi-tioning by torsion at an applied twist rate wÄ9.1"10À4

†rad s À1 mmÀ1‡ at room temperature. „a… Torque versus

twist per unit length and „b… distribution of the softening pa-rameter D over the „dimensionless … radius at stages C and F in„a….

Fig. 13 Distribution of the relevant components of the Cauchystress tensor over the „dimensionless … radius at stage C andthe residual stress distribution after mechanical precondition-ing at stage F in Fig. 12 „a…

Fig. 14 Simulated and experimental curves of „a… torque ver-sus twist per unit length and „b… compressive normal force ver-sus twist per unit length for rejuvenated polycarbonate


Figure 14~b! shows a comparison of the experimental and nu-merical results of the normal force versus the twistw, which is amore critical assessment of the constitutive model. It is clear thatthere are strong deviations between simulation and experiment.Again it should be emphasized that the viscoelastic behavior atlow twist levels is not accurately described by the single modecompressible Leonov model. Also, at higher twist levels themodel is not able to capture these secondary effects accurately,although the description improves and the deviations are,strangely enough, observed to be constant with a value of approxi-mately 100@N# for 0.2,w<0.4.

5.4 Uniaxial Extension of Rejuvenated Polycarbonate.For the simulation of uniaxial extension of a rejuvenated sampleof polycarbonate, again the finite element method is employed. Inthe finite element model, the distribution of the softening param-eterD over the radius, depicted in Fig. 12~b! at stage F is specifiedas the initial condition. The distribution has been restricted to thegauge section of the specimen, since torsional deformations wereonly observed in this part. In the experiment and in the numericalsimulation, the deformation rate imposed at the free end of thespecimen corresponded with a nominal strain rate in the gaugesection of 2.25•1023 @s21#. To possibly trigger necking, exactlythe same imperfection was used as in the tensile simulations of theuntreated samples. For the present problem it was found that onlymeshes composed of ten elements or more in radial direction fa-cilitated an accurate description of the initial distribution ofDover the radius. Since during these experiments deformationswere observed over the entire sample length, the mesh is equallydistributed over the specimen.

Figure 15 shows the simulated and experimental true stressszz

versuslzz2 2lzz

21 for the mechanically rejuvenated polycarbonatetensile bar at a nominal strain rate of 2.25•1023 @s21#. Smalldeviations between simulations and experiment can be observed,

which at small values oflzz2 2lzz

21 may be attributed to inaccura-cies in the calculated value ofD after rejuvenation by torsion.Other causes for the differences between simulated and experi-mental results may be viscoelastic effects, where the mechanicalpretreatment might still influence the deformation behaviorthrough a memory effect. As mentioned before, the single modemodel used here will not capture these effects adequately.

Figure 15 also includes the deformed meshes at different stagesof the deformation, which confirms the absence of necking, de-spite the presence of the imperfection. In contrast to the findingsof Lu and Ravi-Chandar@32#, this result strongly suggests that, inpolycarbonate, strain softening is the main reason for localizationphenomena.

6 ConclusionIn this study an extension of the compressible Leonov model

has been presented, that captures the typical characteristics of thepost-yield behavior of glassy polymers: intrinsic strain softeningand strain hardening. Regarding the experimental assessment ofthe parameters needed for the model, it was found that the post-yield behavior during a compression test was too strongly influ-enced by barreling.

Mechanical rejuvenation by cyclic fixed-end torsion has beensimulated, and the results have been used as initial conditions fornumerical simulations of fixed-end torsion and uniaxial extensionof rejuvenated polycarbonate. Comparison of these simulationswith the experiments shows that although the post-yield behavioris described correctly by the compressible Leonov model, the de-formation behavior at small strains is not captured. This is espe-cially observed when comparing the axial forces during fixed-endtorsion on rejuvenated samples. Although at higher strain levelsthere appeared to be a qualitative agreement, the behavior at lowstrain levels deviated strongly. It can therefore be concluded thatthe single mode compressible Leonov model is not valid withrespect to second order effects during strain hardening.

On the other hand, the compressible Leonov model, extendedwith intrinsic strain softening and strain hardening, seems to beable to predict the transition from inhomogeneous~necking! tohomogeneous deformation as a result of a mechanical pretreat-ment. It is therefore concluded that the intrinsic strain softeningeffect is the main cause for localization phenomena inpolycarbonate.

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trolled Tensile Testing of Polymers and Metals Beyond the Necking Point,’’ J.Mater. Sci.,27, p. 5031.

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@3# Arruda, E. M., and Boyce, M. C., 1993, ‘‘Evolution of Plastic Anisotropy inAmorphous Polymers During Finite Straining,’’ Int. J. Plast.,9, pp. 697–720.

@4# Ward, I. M., 1983,Mechanical Properties of Solid Polymers, 2nd ed., Wiley,Chichester.

@5# Haward, R. N., and Young, R. J., 1997,The Physics of Glassy Polymers, 2nded., Chapman & Hall, London.

@6# Wu, P. D., and van der Giessen, E., 1993, ‘‘On Improved Network Models forRubber Elasticity and Their Applications to Orientation Hardening in GlassyPolymers,’’ J. Mech. Phys. Solids,41, pp. 427–456.

@7# Haward, R. N., 1993, ‘‘Strain Hardening of Thermoplastics,’’ Macromol-ecules,26, pp. 5860–5869.

@8# Struik, L. C. E., 1978,Physical Aging in Amorphous Polymers and OtherMaterials, Elsevier, Amsterdam.

@9# Struik, L. C. E., 1986, ‘‘Physical Aging: Influence on the Deformation Behav-ior of Glassy Polymers,’’Failure of Plastics, Brostow, W., and Corneliussen,R. D., eds., Hanser Publishers, Munich.

@10# Hasan, O. A., and Boyce, M. C., 1993, ‘‘Energy Storage During Deformationin Glasses,’’ Polymer,34, pp. 5085–5092.

@11# G’Sell, C., 1986, ‘‘Plastic Deformation of Glassy Polymers: ConstitutiveEquations and Macromolecular Mechanisms,’’ H. J. Queen et al., eds.,Strength of Metals and Alloys, Pergamon Press, Oxford, pp. 1943–1982.

@12# Aboulfaraj, M., G’Sell, C., Mangelinck, D., and McKenna, G. B., 1994,

Fig. 15 Simulated and experimental true stress szz versuslzz

2 ÀlzzÀ1 during a tensile test at a strain rate of 2.25

"10À3†sÀ1

‡ of a polycarbonate tensile bar, preconditioned intorsion. The indications „a – d … are related to the deformedmeshes at different stages of the deformation.


‘‘Physical Aging of Epoxy Networks After Quenching and/or Plastic Cy-cling,’’ J. Non-Cryst. Solids,172–174, pp. 615–621.

@13# Cross, A., and Haward, R. N., 1978, ‘‘Orientation Hardening of PVC,’’ Poly-mer,19, pp. 677–682.

@14# Xie, L., Gidley, D. W., Hristov, A., and Yee, A. F., 1995, ‘‘Evolution ofNanometer Voids in Polycarbonate Under Mechanical Stress and Thermal Ex-pansion Using Positron Spectroscopy,’’ J. Polym. Sci.: Part B: Polym. Phys.,33, pp. 77–84.

@15# Hasan, O. A., Boyce, M. C., Li, X. S., Berko, S., 1993, ‘‘An Investigation ofthe Yield and Postyield Behavior and Corresponding Structure of Poly~MethylMethacrylate!,’’ J. Polym. Sci., Part B: Polym. Phys.,31, pp. 185–197.

@16# Wu, P. D., and van der Giessen, E., 1995, ‘‘On Neck Propagation in Amor-phous Glassy Polymers Under Plane Strain Tension,’’ Int. J. Plast.,11, pp.211–235.

@17# Bauwens, J. C., 1978, ‘‘A New Approach to Describe the Tensile Stress-StrainCurve of a Glassy Polymer,’’ J. Mater. Sci.,13, pp. 1443–1448.

@18# Govaert, L. E., van Aert, C. A. C., Boekholt, J., 1997, ‘‘Temperature andMolecular Weight Dependence of the Strain Hardening Behavior of Polycar-bonate,’’ Proc. 10th Int. Conf. On Deformation, Yield and Fracture of Poly-mers, The Institute of Materials, pp. 424–426.

@19# Tervoort, T. A., Smit, R. J. M., Brekelmans, W. A. M., and Govaert, L. E.,1998, ‘‘A Constitutive Equation for the Elasto-Viscoplastic Deformation ofGlassy Polymers,’’ Mech. Time-Dep. Mat,1, pp. 269–291.

@20# Hunter, S. C., 1983,Mechanics of Continuous Media, 2nd ed., Ellis HorwoodLtd., Chichester, U.K.

@21# Boyce, M. C., Parks, D. M., and Argon, A. S., 1988, ‘‘Large Inelastic Defor-mation of Glassy Polymers. Part I: Rate Dependent Constitutive Model,’’Mech. Mater.,7, pp. 15–33.

@22# Bird, R., Armstrong, R., and Hassager, O., 1987,Dynamics of Polymer Liq-uids, Vol. 1: Fluid Mechanics, Wiley, New York.

@23# Moginger, B., and Fritz, U., 1991, ‘‘Thermal Properties of Strained Thermo-plastic Polymers,’’ Polym. Int.,26, pp. 121–128.

@24# Koenen, J. A., 1992, ‘‘Observation of the Heat Exchange During DeformationUsing an Infrared Camera,’’ Polymer,33, pp. 4732–4736.

@25# Boyce, M. C., Montagut, E. L., and Argon, A. S., 1992, ‘‘The Effects ofThermomechanical Coupling on the Cold Drawing Process of Glassy Poly-mers,’’ Polym. Eng. Sci.,32, pp. 1073–1085.

@26# Van Krevelen, D. W., 1990,Properties of Polymers: Their Correlation withChemical Structure, Their Numerical Estimation and Prediction from AdditiveGroup Contributions, 3rd ed., Elsevier, Amsterdam.

@27# Wu, W., and Turner, A. P. L., 1973, ‘‘Shear Bands in Polycarbonate,’’ J.Polym. Sci., Polym. Phys. Ed.,11, pp. 2199–2208.

@28# Duckett, R. A., Goswami, B. C., Smith, L. S. A., Ward, I. M., and Zihlif, A.M., 1978, ‘‘The Yielding and Crazing Behavior of Polycarbonate in TorsionUnder Superimposed Hydrostatic Pressure,’’ Br. Polym J.,10, pp. 11–16.

@29# Bauwens-Crowet, C., Bauwens, J. C., and Home`s, G., 1969, ‘‘Tensile YieldStress Behavior of Glassy Polymers,’’ J. Polym. Sci.: Part A-2,7, pp. 735–742.

@30# Tervoort, T. A., Klompen, E. T. J., and Govaert, L. E., 1996, ‘‘A Multi-ModeApproach to Finite, Three-Dimensional, Nonlinear Viscoelastic Behavior ofPolymer Glasses,’’ J. Rheol.,40, pp. 779–797.

@31# Hasan, O. A., and Boyce, M. C., 1995, ‘‘A Constitutive Model for the Non-linear Viscoelastic Viscoplastic Behavior of Glassy Polymers,’’ Polym. Eng.Sci., 35, pp. 331–344.

@32# Lu, J., and Ravi-Chandar, K., 1999, ‘‘Inelastic Deformation and Localizationin polycarbonate Under Tension,’’ Int. J. Solids Struct.,36, pp. 391–425.


Hisaaki Tobushie-mail: [email protected]

Takafumi Nakahara

Yoshirou Shimeno

Department of Mechanical Engineering,Aichi Institute of Technology,

1247 Yachigusa, Yagusa-cho,Toyota 470-0392 Japan

Takahiro HashimotoTakiron Co., Ltd.,

2-3-13 Azuchi-cho, Chuo-ku,Osaka 541-0052 Japan

Low-Cycle Fatigue of TiNi ShapeMemory Alloy and Formulation ofFatigue LifeThe low-cycle fatigue of a TiNi shape memory alloy was investigated by the rotating-bending fatigue tests in air, in water and in silicone oil. (1) The influence of corrosionfatigue in water does not appear in the region of low-cycle fatigue. (2) The temperaturerise measured through an infrared thermograph during the fatigue test in air is four timesas large as that measured through a thermocouple. (3) The fatigue life at an elevatedtemperature in air coincides with the fatigue life at the same elevated temperature inwater. (4) The shape memory processing temperature does not affect the fatigue life. (5)The fatigue equation is proposed to describe the fatigue life depending on strain ampli-tude, temperature and frequency. The fatigue life is estimated well by the proposedequation.@S0094-4289~00!01102-6#

Keywords: Shape Memory Alloy, Fatigue, Titanium-Nickel Alloy, Martensitic Transfor-mation, Rotating-Bending, Low-Cycle Fatigue, Frequency, Fatigue Equation, Atmosphere

1 Introduction

The shape memory effect and superelasticity appear in a shapememory alloy~SMA! ~Perkins@1#, Funakubo@2#, Doyama et al.@3#, Chu and Tu@4#!. The development of intelligent materials andmachine systems using these properties has attracted interest~Pel-ton et al.@5#!. In applications of SMAs, we use not only shaperecovery but also recovery force, energy storage, and dissipationof work. In applications to a robot, an actuator, and a solid-stateheat engine, a SMA element performs cyclic motions. In order toevaluate the reliability of the SMA element, fatigue properties ofthe material are important. SMAs are subjected to both mechani-cal cycle with cyclic loading and thermal cycle due to heating andcooling. The thermomechanical cycle affects fatigue propertiessignificantly ~McNichols and Brookes@6#, Melton and Mercier@7#, Miyazaki @8#!.

From the viewpoint of thermal response of SMA elements, thinwires are widely used in practical applications. The SMA ele-ments working under a certain constant stroke are more widelyused than those subjected to stress control. Considering thesepoints, the authors carried out the rotating-bending fatigue tests ofTiNi SMA wires subjected to strain control and ascertained thatthe fatigue life is longer than 107 cycles in the rhombohedral-phase transformation~RPT! region~Otsuka@9#, Miyazaki and Ot-suka@10#, Tobushi et al.@11#, Tobushi et al.@12#, Tobushi et al.@13#!. They confirmed also that the low-cycle fatigue propertiesassociated with the martensitic transformation~MT! are differentbetween in air and in water. The fatigue life in air shortens due totemperature rise. In the previous paper~Tobushi et al.@12#!, tem-perature was measured through a thermocouple. Therefore tem-perature was not measured with high accuracy.

In this work, by carrying out rotating-bending fatigue tests ofTiNi SMA wires, the relationship between strain amplitude andfatigue life is investigated. The fatigue life between in water andin silicone oil is compared and the influence of corrosion fatiguein water is discussed. By measuring temperature rise of the wires

in air through an infrared thermograph, the main cause of thetemperature rise that markedly affects the fatigue life is discussed.Influence of frequency and shape-memory processing temperatureon fatigue life is also investigated. Based on the results, the low-cycle fatigue life is formulated.

2 Experimental Method

2.1 Materials and Specimen. The material was a Ti-55.4wt percent Ni SMA wire, 0.75 mm in diameter. The specimenswere given shape memory of a straight line through shapememory processing~SMP!. This was done by holding the wires inthe straight line at a certain temperature for 60 min and cooling ina furnace. The specimens were straight lines with uniform crosssection. The reverse-transformation completion temperatureAfwas about 323 K at a SMP temperature of 673 K.

2.2 Experimental Apparatus. The SMA properties testingmachine composed of a tensile machine and a heating-coolingdevice ~Tobushi et al.@14#! was used for the tensile test. Thespecimen was heated by hot air or cooled by liquefied carbondioxide. Temperature was measured through a thermocouple, 0.1mm in diameter, put on the surface of the specimen. The gaugelength of the extensometer was 20 mm.

The rotating-bending fatigue machine~Tobushi et al.@12#, To-bushi et al.@13#! was used for testing fatigue. One end of thespecimen bent to a prescribed curvature was mounted to a motorand the other end, at which the number of cycles to failure wasmeasured, rotated freely. By keeping the bent form during theexperiment, the specimen was rotated. The surface element of thespecimen was subjected to tension and compression during onerevolution. Maximum bending strain of the surface of the speci-men was prescribed by using the radius of curvature in the centerbetween the supports. The rupture occurred in the central part ofthe specimens between the supports. Maximum bending strain ofthe ruptured specimens was obtained by using the radius of cur-vature of a ruptured part.

The infrared thermograph was used to measure temperature ofthe specimen during the fatigue test in air. By receiving the infra-red energy radiated from the specimen through the thermograph,temperature at each part was measured without contact.

Contributed by the Materials Division for publication in the JOURNAL OF ENGI-NEERING MATERIALS AND TECHNOLOGY. Manuscript received by the MaterialsDivision July 27, 1998; revised manuscript received November 8, 1999. AssociateTechnical Editor: C. Brinson.


2.3 Experimental Procedure. To investigate the rotating-bending fatigue properties of a TiNi SMA wire and temperaturerise during the fatigue test, the following four experiments werecarried out.

(1) Fatigue Test in Liquid. The fatigue properties were in-vestigated in water and in silicone oil. Test temperature was 303K, frequency 1000 cpm and SMP temperature 673 K.

(2) Test for Temperature Rise During Rotating-Bending.The temperature rise of the specimen in air was measured duringthe rotating-bending test. The test was carried out at various val-ues of frequency and strain amplitude at room temperature. Thetemperature was measured at every 10 s for 3 min from the startof the test.

(3) Fatigue Test at Various Frequencies.The fatigue prop-erties were investigated at various frequencies in air and in water.The value of frequencyf was 100–1000 cpm. The test tempera-tures were room temperature (RT5303 K), 333 K and 353 K.The test in air was carried out in a state of natural radiation ofheat. The value of strain amplitude«a was 0.5– 2.5 percent. TheSMP temperature of the wire was 673 K.

(4) Fatigue Test at Various SMP Temperatures.The fatigueproperties were investigated in water for the specimens at variousSMP temperatures. The SMP temperaturesTp were 623 K, 673 K,and 723 K. Test temperatureT was 333 K and frequencyf5500 cpm.

3 Experimental Results and Discussion

3.1 Influence of Liquid Atmosphere on Fatigue Life. Therelationship between strain amplitude«a and the number of cyclesto failure Nf obtained by the fatigue test in water and in siliconeoil is shown in Fig. 1. As seen in Fig. 1, the distinct difference infatigue life between in water and in silicone oil does not appear. Itmay be assumed that influence of corrosion on fatigue appears inwater and therefore fatigue life shortens in water. However, in therange of low-cycle fatigue, the time to failure is short and failuremay occur before the influence of corrosion fatigue appears. Be-cause coefficient of heat transfer in liquid is larger than that ingas, temperature rise in water and in silicone oil is small com-pared with that in air. Therefore the difference in fatigue life doesnot clearly appear between in water and in silicone oil.

As observed in the previous paper, because the low-cycle fa-tigue properties of SMA differs markedly between in air and inwater, the influence of test atmosphere on fatigue life is importantin applications. In the case of using shape memory effect in ap-plications, because SMAs are subjected to loading-unloading andheating-cooling, it is also necessary to take account of the influ-ence of these thermomechanical paths. In the case of high-cyclefatigue, it is necessary to take account of the influence of fatiguelimit and corrosion fatigue which depends on atmosphere. Thesepoints are future problems.

3.2 Temperature Rise of Specimen in Air. The tempera-ture distribution in the specimen obtained through the infraredthermograph is shown in Fig. 2. As seen in Fig. 2, the temperaturerise is distributed almost uniformly in the whole specimen.

The relationship between temperature rise of the specimen andlapse of time during the fatigue test at various frequenciesf mea-sured through the infrared thermograph is shown in Fig. 3. Asseen in Fig. 3, temperature increases rapidly until 20–30 s and issaturated in a certain value thereafter. Therefore the specimen iskept at an elevated temperature during almost the whole time.Temperature rise is large asf is large. In the case off51000 cpm, temperature rise is about 25 K. The relationship be-tween the MT stresssM and temperatureT is expressed by thefollowing equation called the transformation line~Tanaka et al.@15#!

sM5CM~T2Ms! (1)

whereCM and Ms denote a slope of the transformation line andthe MT starting temperature under no stress, respectively. Asfound from Eq.~1!, sM increases in proportion toT. For TiNiSMA, CM is 6 MPa/K ~Tobushi et al.@16#!. If temperature in-creases by 25 K, the MT stress increases by about 150 MPa.Therefore temperature rise markedly affects the fatigue properties.

The relationship between the saturated temperature riseDTRTand frequency is shown in Fig. 4. In the case of«a52.04 percent andf 51000 cpm, because the specimen was rup-tured at several ten seconds, the saturated temperature rise is notplotted. As seen in Fig. 4, temperature rise increases with increas-ing both frequency and strain amplitude. The relationship is not

Fig. 1 Strain amplitude versus fatigue life in water and in sili-cone oil

Fig. 2 Temperature distribution in the specimen obtainedthrough the thermograph „«aÄ1.54 percent, fÄ1000 cpm,tÄ60 s…

Fig. 3 Temperature rise with lapse of time during fatigue test„«aÄ1.54 percent …


proportional. As found from the relationship, the rate of tempera-ture rise decreases with increasing frequency. The temperaturerise shown in Fig. 4 is almost four times as large as that measuredthrough a thermocouple in the previous paper.

The reason why temperature increases in air can be explainedas follows. The surface elements of the wires under rotating-bending are subjected to tension and compression repeatedly. Thestress-strain curve of SMA shows a hysteresis loop with widewidth. The area surrounded by the hysteresis loop of the stress-strain curve represents dissipated work per unit volume. The dis-sipated work is lost as heat, resulting in temperature rise of thespecimen. In air, because coefficient of heat transfer is small andheat is hardly diffused, temperature increases due to the generatedheat. The temperature rise can be well estimated by consideringthe heat transfer characteristics around the specimen~Mikuriyaet al. @17#!.

3.3 Fatigue Properties at Various Frequencies in Air.The relationship between strain amplitude«a and the number ofcycles to failureNf obtained by the fatigue test at various frequen-cies f in air is shown in Fig. 5. As seen in Fig. 5, fatigue lifedepends markedly on frequency and shortens with increasingf.This is different from both the fatigue property of SMA in waterthat fatigue life does not depend on frequency~Tobushi et al.@12#! and that of stainless steel that fatigue life degrades withdecreasing frequency~Coffin @18#!. The slope of the curves be-comes gentle with increasingf. The difference inNf due to thedifference in f is large in the region of large«a . For «a52 percent,Nf at f 5100 cpm is more than ten times larger thanthat at f 51000 cpm. These properties appear due to temperaturerise of the specimen. On the other hand, the influence off on Nfdecreases in the region of small«a . Therefore the strain ampli-tude at fatigue limit may depend little on frequency. This is due tothe fact that strain amplitude at fatigue limit is in the region of theRPT ~Tobushi et al.@12#! and that temperature increases little atsmall «a which was shown in Section 3.2, resulting in slight in-fluence off on Nf . The fatigue life at an elevated temperature inair coincides with the fatigue life at the same elevated temperaturein water.

3.4 Influence of SMP Temperature on Fatigue Properties.The relationship between strain amplitude«a and the number ofcycles to failureNf obtained by the fatigue test at various SMPtemperaturesTp is shown in Fig. 6. As seen in Fig. 6, the strain-life relationships at eachTp are almost the same. The slope of theapproximate line in the low-cycle region is 0.49 atTp5723 K,0.47 atTp5673 K, and 0.48 atTp5623 K. The strain-life rela-tionship at eachTp has a knee in the region of«a50.5–0.7 per-cent andNf5104– 105. The «a2Nf relationship approaches thehorizontal line in the region of«a smaller than the knee.

The stress-strain curves obtained by the tensile test for thespecimens at variousTp are shown in Fig. 7. The tensile test wascarried out at the same temperatureT5333 K as the fatigue test.As seen in Fig. 7, the MT stress increases with decreasingTp .Because dislocations with high density induced through colddrawing to produce a SMA wire construct the stable internal

Fig. 4 Relationship between saturated temperature rise DTRTand frequency

Fig. 5 Strain amplitude versus fatigue life at various frequen-cies f in air

Fig. 6 Strain amplitude versus fatigue life for various shape-memory processing temperatures Tp

Fig. 7 Stress-strain curves for various shape-memory pro-cessing temperatures Tp


structure at lowTp , the MT stress increases. Therefore it may beassumed that the fatigue life at lowTp shortens because the MTstress is high. However, as observed in Fig. 6, the difference infatigue life does not clearly appear.

The reason whyTp does not affect the fatigue life may beexplained as follows. The dissipated workEd per unit volumeunder tension is obtained from the stress-strain curves shown inFig. 7. The value ofEd is 6.9 MJ/m3 at Tp5723 K, 5.4 MJ/m3 atTp5673 K, and 4.2 MJ/m3 at Tp5623 K. Ed decreases with de-creasingTp in inverse proportion to increase in the MT stress. IfEd is large, fatigue damage is large, resulting in short fatigue life~Sakuma et al.@19#!. Therefore, because the MT stress is low butEd is large at highTp , the fatigue properties do not clearly de-pend onTp due to both effects.

4 Formulation of Low-Cycle Fatigue LifeLet us discuss the formulation of the rotating-bending low cycle

fatigue life of TiNi SMA wires for constant SMP temperature of673 K.

4.1 Dependence on Strain Amplitude. As found from thestrain-life curves in Section 3, the relationship between strain am-plitude«a and the number of cycles to failureNf in the region oflow-cycle fatigue on a logarithmic graph is almost expressed by astraight line. Therefore, as similar as the Manson-Coffin relation-ship for normal metals in low-cycle fatigue, the relationship be-tween«a andNf is express as follows

«a•Nfb5a (2)

wherea andb represent«a in Nf51 and the slope of the log«a2log Nf curve, respectively. The value ofb is about 0.5 at eachtemperature which is valid for normal metals.

4.2 Dependence on Temperature. The dependence of thefatigue life on temperature will be discussed in water where tem-perature of the specimen increases little. The exponentb in Eq.~2! is about 0.5 at each temperature. The value ofa decreases withincreasing temperatureT. As expressed by Eq.~1!, the MT stressincreases in proportion toT. If the MT stress is high, fatiguedamage is large and the fatigue life is short, resulting in smalla.Based on these considerations, if the relationship betweena andTis plotted on a semilogarithmic graph, it is found that the relation-ship is expressed by a straight line. Therefore it becomes asfollows

a5as•102a~T2Ms! (3)

whereMs is 253 K which was obtained by the DSC test. Based onthe results of the fatigue test in water, the coefficients in Eq.~3!are determined asas58.56 anda50.012 K21. Therefore, fromEq. ~2!, the relationship between«a and Nf is expressed by thefollowing equation

«a•Nf0.558.5631020.012~T2Ms! (4)

Thus the dependence of the fatigue life on«a andT is describedby Eq. ~4!.

4.3 Dependence of Temperature Rise in Air on Frequency.Because temperature increases little in water due to high heattransfer, temperature rise is ignored. On the contrary, temperatureincreases depending on strain amplitude and frequency in air, re-sulting in short fatigue life. The temperature rise appears due todissipated workEd . In order to consider the fatigue properties inair, it is important to take account of the temperature rise.

Based on tensile deformation properties of SMA, dissipatedwork Ed depends on strain rate« and maximum strain«m ~To-bushi et al.@20#!. ThereforeEd is expressed as follows

Ed5Ed~ «,«m! (5)

where«m corresponds to strain amplitude«a and« is proportionalto frequencyf in rotating-bending fatigue test. With respect to the

surface element under rotating-bending, strain which varies in onecycle for strain amplitude«a is 4«a . Therefore average strain rateat frequencyf is

«54«a• f 54 f «a (6)

From these relationships,Ed becomes a function of«a and f asfollows

Ed5Ed~«a , f ! (7)

Because temperature increases based onEd , temperature riseDTis also a function of«a and f

DT5DT~«a , f ! (8)

Let us obtain the concrete form of Eq.~8!. The relationshipbetween temperature rise at room temperatureDTRT and fre-quencyf is shown in Fig. 8 which is rearranged in a logarithmicscale from Fig. 4. As seen in Fig. 8, the relationship between logDTRT and log f is expressed by a straight line. ThereforeDTRTbecomes

DTRT5S f

f 0D b

(9)

wheref 0 represents a value on the frequency axis. As seen in Fig.8, f 0520 cpm. In Fig. 8,b represents the slope of the straight lineand changes depending on strain amplitude. The slopeb increasesin proportion to strain amplitude in a logarithmic scale. Thereforeb is expressed as follows

b5c~ log «a2 log « l ! (10)

wherec52.13 and« l50.006. Because the RPT strain is 0.6 per-cent, Eq.~10! reveals the fact that temperature increases markedlyin the MT region of strain amplitude above« l50.6 percent.

From Eqs.~9! and~10!, temperature rise at room temperature isexpressed as follows

DTRT5S f

f 0D c log «a /« l

(11)

4.4 Dependence on Temperature in Air. In order to obtaintemperature rise at an arbitrary temperature, dissipated workEd ateach temperatureT will be discussed.Ed is obtained from tensiletest results for maximum strain of 2 percent. Assuming that stress-strain curves are symmetrical with respect to tension and com-pression,Ed in one cycle is determined. The relationship betweenEd andT is shown in Fig. 9. As seen in Fig. 9, logEd decreases inproportion toT at temperatures aboveTl5320 K. This means thatEd decreases in the superelastic region becauseTl5320 K is closeto the reverse-transformation completion temperatureAf5323 K. TakingEd as unit in the region of temperature below

Fig. 8 Relationship between saturated temperature rise DTRTand frequency in a logarithmic scale


320 K at which the shape memory effect is observed and roomtemperature exists,Ed decreases at the rate ofr at temperaturesabove 320 K. From Fig. 9,r is

r 5102h~T2Tl ! (12)

whereh50.0054 K21 andTl5320 K.Temperature riseDT appears based onEd . Therefore, com-

pared with temperature rise at room temperatureDTRT , tempera-ture riseDT at temperatures in the superelastic region decreases atthe rate ofr as follows

DT5DTRT•102h~T2Tl ! (13)

From Eqs.~11! and ~13!, DT becomes

DT5S f

f 0D c log~«a /« l !

3102h~T2Tl ! (14)

Considering temperature rise, the fatigue equation at an arbitrarytemperatureT in air becomes from Eq.~4! as follows

«a•Nf0.558.5631020.012~T1DT2Ms! (15)

The fatigue life of SMA wires in the low-cycle region in air isdetermined from Eq.~15!.

4.5 Calculated Results and Discussion.The calculated re-sults between strain amplitude and fatigue life in water obtainedfrom Eq. ~4! are shown in Fig. 10. In Fig. 10, solid curves repre-sent the calculated results and plotted points the experimental re-sults, respectively. As seen in Fig. 10, the low-cycle fatigue life inwater is well expressed by Eq.~4!.

The calculated results between strain amplitude and fatigue lifeat various temperatures in air obtained from Eq.~15! are shown inFig. 11. The calculated results at various frequencies in air areshown in Fig. 5. In both figures, the calculated results are shownby solid curves. As seen in Figs. 11 and 5, the overall fatigue lifein the low-cycle region in air is expressed by the equation inwhich temperature rise is considered. As seen in Fig. 5, the cal-culated fatigue life is shorter than the experimental one in the

region of low frequency and small strain amplitude. This occursdue to the assumption of Eq.~15! in which Eq.~4! in water is usedas the condition of no temperature rise. As observed in the previ-ous paper~Tobushi et al.@12#!, in the high-cycle region, the fa-tigue life in water is shorter than that in air.

If the state of air flow changes, coefficient of heat transferchanges and accordingly temperature rise changes. Therefore it isimportant to decide the fatigue life by considering carefully theatmosphere. Furthermore, it is necessary to formulate the fatiguelimit for high-cycle applications. It is necessary to consider cor-rosion for the fatigue limit in water. These are future problems.

5 ConclusionsThe low-cycle fatigue properties of TiNi SMA wires subjected

to rotating-bending have been investigated in water and in air. Themain results obtained are summarized as follows.

1 The influence of corrosion fatigue in water does not appear inthe region of low-cycle fatigue.

2 The temperature rise measured through an infrared thermo-graph during the fatigue test in air is four times as large as thatmeasured through a thermocouple.

3 The fatigue life at an elevated temperature in air coincideswith the fatigue life at the same elevated temperature in water.

4 Although stress-strain curves are different at various SMPtemperatures, fatigue lives are almost the same.

5 The fatigue equation is proposed to describe the fatigue lifedepending on strain amplitude, temperature and frequency in wa-ter and in air. The fatigue life is estimated well by the proposedequation. The proposed fatigue equation is useful for the design ofSMA elements.

AcknowledgmentsThe experimental work of this study was carried out with the

assistance of the students of Aichi Institute of Technology, towhom the authors wish to express their gratitude. The authors alsowish to extend thanks to the Scientific Foundation of the JapaneseMinistry of Education, Science and Culture for financial support.

References@1# Perkins, J., 1975,Shape Memory Effects in Alloys, Plenum Press, New York.@2# Funakubo, H., 1987,Shape Memory Alloys, Gordon and Breach Science, New

York.@3# Doyama, M., Somiya, S., and Chang, R. P. H., 1989, ‘‘Shape Memory Mate-

rials,’’ MRS Int’l. Mtg. on Adv. Mats., Vol. 9, Materials Research Society,Pittsburgh.

@4# Chu, Y. Y., and Tu, H. L., 1994, ‘‘Shape Memory Materials ’94,’’Proc. ofInter. Sym. on Shape Memory Materials, International Academic, Beijing.

@5# Pelton, A. R., Hodgson, D., and Duerig, T., 1995, ‘‘SMST-94,’’Proc. of FirstInter. Conf. on Shape Memory & Superelastic Technologies, MIAS, Monterey.

@6# McNichols, J. L., Jr., and Brookes, P. C., 1981, ‘‘NiTi Fatigue Behavior,’’ J.Appl. Phys.,52, No. 12, pp. 7442–7444.

@7# Melton, K. N., and Mercier, O., 1979, ‘‘Fatigue of NiTi Thermoelastic Mar-tensites,’’ Acta Metallurgica,27, pp. 137–144.

@8# Miyazaki, S., 1996, ‘‘Development and Characterization of Shape MemoryAlloys,’’ Shape Memory Alloys, M. Fremond and S. Miyazaki, eds., SpringerWien, New York, pp. 69–147.

Fig. 9 Relationship between dissipated work and temperature

Fig. 10 Strain-life curves at various temperatures in water

Fig. 11 Strain-life curves at various temperatures in air


@9# Otsuka, K., 1990, ‘‘Introduction to the R-Phase Transition,’’Engineering As-pects of Shape Memory Alloys, T. W. Duerig, K. N. Melton, D. Stockel, and C.M. Wagman, eds., Butterworth-Heinemann, London, pp. 36–45.

@10# Miyazaki, S., and Otsuka, K., 1986, ‘‘Deformation and Transition BehaviorAssociated with the R-Phase in Ti-Ni Alloys,’’ Metall. Trans. A,17, pp. 53–63.

@11# Tobushi, H., Yamada, S., Hachisuka, T., Ikai, A., and Tanaka, K., 1996,‘‘Thermomechanical Properties Due to Martensitic and R-Phase Transforma-tions of TiNi Shape Memory Alloy Subjected to Cyclic Loadings,’’ SmartMater. Struct.,5, pp. 788–795.

@12# Tobushi, H., Hachisuka, T., Yamada, S., and Lin, P. H., 1997, ‘‘Rotating-Bending Fatigue of a TiNi Shape-Memory Alloy Wire,’’ Mech. Mater.,26, pp.35–42.

@13# Tobushi, H., Hachisuka, T., Hashimoto, T., and Yamada, S., 1998, ‘‘CyclicDeformation and Fatigue of a TiNi Shape-Memory Alloy Wire Subjected toRotating Bending,’’ ASME J. Eng. Mater. Technol.,120, pp. 64–70.

@14# Tobushi, H., Tanaka, K., Kimura, K., Hori, T., and Sawada, T., 1992, ‘‘Stress-Strain-Temperature Relationship Associated with the R-Phase Transformationin TiNi Shape Memory Alloy,’’ JSME Int. J. Ser. I,35, No. 3, pp. 278–284.

@15# Tanaka, K., Kobayashi, S., and Sato, Y., 1986, ‘‘Thermomechanics of Trans-formation Pseudoelasticity and Shape Memory Effect in Alloys,’’ Int. J. Plast.,2, pp. 59–72.

@16# Tobushi, H., Lin, P. H., Tanaka, K., Lexcellent, C., and Ikai, A., 1995, ‘‘De-formation Properties of TiNi Shape Memory Alloy,’’ J. Phys. IV,C2, No. 5,pp. 409–413.

@17# Mikuriya, S., Nakahara, T., Tobushi, H., and Watanabe, H., 1999, ‘‘The Esti-mation of Temperature Rise on Low Cycle Fatigue of TiNi Shape MemoryAlloy,’’ Trans. Jpn. Soc. Mech. Eng., Ser. A,65, No. 633, pp. 1099–1104.

@18# Coffin, L. F., 1978, ‘‘Fatigue in Machines and Structures-Power Generation,’’Materials Science Seminar, Fatigue and Microstructures, St. Louis, AmericanSociety for Metals, pp. 1–28.

@19# Sakuma, T., Iwada, U., Kariya, N., and Ochi, Y., 1998, ‘‘Fatigue Life ofTiNiCu Shape Memory Alloy under Thermo-mechanical Conditions,’’Proc.of 11th Inter. Conf. on Exp. Mech., Oxford, Vol. 2, pp. 1121–1126.

@20# Tobushi, H., Shimeno, Y., Hachisuka, T., and Tanaka, K., 1998, ‘‘Influence ofStrain Rate on Superelastic Properties of TiNi Shape Memory Alloy,’’ Mech.Mater.,30, pp. 141–150.


Fuh-Kuo ChenProfessor

Cheng-Jun ChenGraduate Student

Department of Mechanical Engineering,National Taiwan University,

Taipei, Taiwan

On the Nonuniform Deformationof the Cylinder Compression TestA theoretical model based on Hill’s general method is developed in the present study tocalculate the flow stress in a cylindrical specimen under axial compression in the pres-ence of friction at the die-specimen interface. Unlike most of the published methods whichstudied the incipient barreling only, the proposed theoretical model takes the barreledshape of the deforming specimen into account. In order to construct the stress-straincurve, the mean effective strain of the barreled specimen was also calculated on the basisof an assumed velocity field. As the present study shows, the proposed theoretical modelprovides good results, both in magnitude and in trend, for the prediction of flow stressesin the barreled specimen during the compression test.@S0094-4289~00!00602-2#

1 IntroductionThe cylinder compression test has been widely adopted for de-

termining a material’s flow stress as a function of the strain duringthe compression of cylindrical specimens between two flat dies. Ifthe friction is absent at the die-specimen interface, the deforma-tion in the cylinder is uniform and the free surface of the cylinderremains straight during the compression. Thus, the flow stress ofthe specimen is simply the value obtained by dividing the appliedforce ~P! by the contact area of die-specimen interface~A!, andthe effective strain is the same as the axial compressive straingiven by ln(2Ho/2H), where 2H0 and 2H are the initial heightand the instantaneous height of the deforming specimen, respec-tively. In reality, due to the presence of friction at the die-specimen interface, the radial displacement of the material nearthe interface is restrained while the remaining portion of the speci-men bulges out changing the free surface into the form of a barrel.The barreled shape leads to an inhomogeneity of deformation, andtherefore, the method used to calculate the flow stress and theeffective strain for the uniformly deformed specimen is notapplicable.

Many theoretical methods have been proposed to derive theflow stress for a barreled specimen. Siebel@1# assumed no barrel-ing and a low coefficient of friction, and derived a simple well-known formula for the relationship between the mean axial pres-sure acting at the die-specimen interface and the flow stress of thematerial under frictionless uniaxial compression. With the use ofthe upper-bound method, Avitzur@2# has determined a relation-ship between the applied load and flow stress of the material for acylinder under uniaxial compression with friction at the die-specimen interface. Later, Lee and Altan@3# also proposed anupper-bound velocity field that considers bulging of the cylinderduring the upsetting. They developed computer programs to de-termine the strain, strain-rate, velocity, and flow-stress distribu-tions. Although the previous works mentioned above have takenthe effect of friction into consideration, the efforts were mainlyfocused on the incipient barreling of the free surface. In otherwords, the specimen with a straight free surface was used for theanalysis, and no attempt was made to consider a barrel-shapedspecimen for the analysis.

Since the compression load acting on any cross-sectional areaof the barreled specimen has the same value, another way to cal-culate the flow stress of the specimen may rest on the analysis forthe mean cross sectional area. Following this concept, a simpleexpression for the flow stress~Y! is therefore suggested as

Y5P

A, (1)

whereA is the mean cross-sectional area obtained by dividing thevolume of the specimen (V) by the instantaneous height 2H ofthe specimen. This method is termed the ‘‘mean flow stressmethod’’ thereafter in this paper.

In 1983, Ettouney and Hardt@4# extended the concept, whichBridgman@5# had developed for calculating the stress distributionat the neck of tensile specimen, to determine the flow stress of thebarreled specimen. Considering the barreled shape in the analysis,they derived a formula of the following nature:

Y5~sz!mH F122R

R2G• lnF12

R2

2RG J 21

, (2)

where

~sz!m5P

pR22 ,

Y is the flow stress of the material under frictionless uniaxialcompression,P is the compression load,R is the radius of thebulge curvature, andR2 is the maximum radius of the barreledspecimen, as shown in Fig. 1.

In the present study, a theoretical model based on Hill’s generalmethod was developed to calculate the flow stress of the barrel-shaped specimen, which is compressed under frictional condition.In the theoretical model, the longitudinal nonuniformity was un-der investigation, and an expression for the flow stress in terms ofthe compression load and the geometric parameters defining thebarreled specimen was obtained. Finite element simulations werealso performed in the present study in order to validate the theo-retical calculation. The flow stresses calculated according to theproposed theoretical model are compared with the true flow stressand with those predicted by the Ettouney and Hardt’s formula~termed the ‘‘E-H formula’’ for short!, and the mean flow stressmethod, based on the compression loads and the geometric param-eters obtained from the finite element simulations.

It should be noted that the strain distribution of the barreledspecimen is also not uniform and the effective strain cannot there-fore be determined explicitly. Nevertheless, the overall axial com-pressive strain has been commonly used by researchers, withoutproof, to approximately represent the effective strain. In thepresent study, an admissible velocity field was proposed to calcu-late the mean effective strain, and the detailed derivation is pre-sented following the calculation of the flow stress.

Contributed by the Materials Division for publication in the JOURNAL OF ENGI-NEERING MATERIALS AND TECHNOLOGY. Manuscript received by the MaterialsDivision October 6, 1998; revised manuscript received February 13, 1999. AssociateTechnical Editor: G. Ravichandran.


2 Calculation of the Flow StressIn 1963, Hill @6# proposed a new method of analysis for metal-

forming processes based on a variational principle. Although Hillbelieved that the proposed method comes close to the ideal, veryfew investigations on the method have appeared in the publishedliterature. Among them, Lahoti and Kobayashi@7# carried out theanalysis of ring compression with barreling, spread in Steckelrolling, and thickness change in tube sinking.

Hill’s method begins by selecting a class of velocity fieldv ifrom which the best approximation will eventually be taken. Theselected velocity field must satisfy all the kinematic conditions.However, the associated stress field in the deformation zone,which is determined according to the material constitutive law,will not generally satisfy all the static requirements. Hence, theselection criterion for the best approximating stress fields i j maybe considered as

EVs i j

]wj

]xidV5E

SI

t jwj dS1ESC

@~nit i !nj1mklj #wj dS (3)

for a sufficiently wide subclass of virtual orthogonalizing motionwj , wheret i denotes the surface traction computed from the con-sidered approximating stress fields i j , ni , and nj are the localunit outward normals,l j is a unit tangent vector opposite in senseto the relative velocity of sliding in the approximating field, andmk represents the constant frictional stress with 0<m<1, k beingthe shear yield stress. For metal-forming process, the surfaceSofthe deforming zone usually consists of three distinct parts:SC isthe interface between the die and the workpiece;SF is the uncon-strained surface;SI is the interface between the deforming zoneand the rigid zone. It is to be noted that use of the reverse of thevirtual work-rate principle has been made to derive Eq.~3!, andalthough the method is applicable to all types of friction, constantfriction is adopted in Eq.~3! for simplicity.

In applying Eq. ~3!, the orthogonalizing familywj must besufficiently wide and extensive to identify a single approximatingvelocity field in the particular class constructed for satisfying thekinematic conditions. Once the orthogonalizing family is chosen,the calculus of variations technique is applied to Eq.~3!, treatingwj as a variation parameter. This furnishes a system of equilib-rium equations and boundary conditions, suited to the particularapproximating class, allowing us to determine its best member~Lahoti and Kobayashi@7#!. The detailed description of themethod can be found in~Hill @6#!.

To facilitate the analysis, the following assumption are made:

1 The deforming material is isotropic and incompressible.2 The elastic deformation is negligible, i.e., the material is con-

sidered as rigid-plastic.

Now, consider a barrel-shaped specimen under compression be-tween two flat dies which move toward each other with a unitspeed at any stage of compression. The specimen has an instanta-neous height of 2H and a radius ofR1 at the die-specimen inter-face, as shown in Fig. 1. Since the nonuniform deformation in thez-direction is our main concern, the selected velocity field must bea function ofz. A simple velocity field representing the incipientbarreling of the specimen has been shown by Kobayashi et al.@8#to be in the form

v r512rF8~z!, vu50, and vz52F~z!, (4)

where v r ,vu ,vz are the velocities in ther, u, and z directions,respectively, the prime denotes differentiation,F is a function thatis sufficiently differentiable. The velocity field given in Eq.~4! implies the incompressibility of material, whileF(z)52F(2z), F(H)52F(2H)51 due to the symmetry with re-spect to the mid-plane and the velocity boundary conditions, re-spectively. It is reasonable to suppose that the velocity field givenin Eq. ~4! continues to hold during a finite compression of thebarreled cylinder, and it is therefore adopted in the present studyto calculate the flow stress for the barreled cylinder.

One way to choose the virtual orthogonalizing velocities is justto take the similar form as that given in Eq.~4!, such as

wr512rf8~z!, wu50, and wz52f~z!, (5)

with f(z)52f(2z).Making use of the following conditions:SI50, $nr ,nu ,nz%

5$0,0,1%, and$ l r ,l u ,l z%5$21,0,0%, Eq. ~3! can be written as

2E0

HH EAFs r

]wr

]r1su

wr

r1sz

]wz

]z1t rzS ]wr

]z1

]wz

]r D GdAJ dz

52EA@szwz2mkwr #z5H dA, (6)

noting thatnj and l i are the unit normal vectors in thez and rdirections, respectively.

Substituting Eq.~5! into Eq. ~6!, yields

2E0

HH EAFs r

2f8 1

su

2f82 sz f81

t rzr

2f9GdAJ dz

522EAFszf1

mkrf8

2 Gz5H

dA. (7)

Applying integration by parts and rearranging terms, Eq.~7!becomes

E0

HEAF ~s r1su22sz!f82r

]t rz

]zf8GdA dz

1E0

HEA

]~t rzrf8!

]zdA dz

52Pf~H !2mkf8~H !EAH

r dA, (8)

whereP is the compression load given byP52*AHsz dA, and

AH is the contact area at the die-specimen interface.Since the area of cross section of the barreled cylinder is itself

a function ofz, it is convenient to consider Eq.~8! in relation to anequivalent specimen having a uniform cross section with a meanareaA5V/(2H), whereV is the volume of the specimen whichremains constant during compression. The configuration of theequivalent specimen is also shown in Fig. 1 by the dashed-line,the equivalent radius being denoted byR. With the above as-

Fig. 1 Configurations of barrel-shaped and equivalentspecimens


sumption in mind, the integrations with respect todA and dz inEq. ~8! can then be interchanged. Consequently, the second inte-gral on the left-hand side of Eq.~8! can be written as

E0

HEA

]~t rzrf8!

]zdA dz5E

AF E

0

H ]~t rzrf8!

]zdzGdA

5EAt rz~H !rf8~H !dA, (9)

with t rz50 at z50.In the cylinder compression with friction at the die-specimen

interface, Hill@6# has shown that the von Mises yield criterion canbe taken in the form

sz212~s r1su!52Y, (10)

to a close approximation.Substituting Eq.~9! and Eq.~10! into Eq. ~8! and noting that

Pf(H)5P*0Hf8(z)dz, results in

E0

HH EAS 2Y2r

]t rz

]z DdA22PJ f8dz

2f8~H !H mkEAH

r dA1EAt rz~H !•r dAJ 50,

(11)

where the first and the third integrals are evaluated over the meanarea of the barreled cylinder.

Sincef8 is arbitrary, it follows from Eq.~11! that

EAS 2Y2r

]t rz

]z DdA22P50, (12)

and

mkE0

R1

2pr 2 dr1E0

R~t rz!z5H2pr 2 dr50, (13)

whereR is the radius corresponding to the mean areaA, andR1 isthe radius of the surface of contact, as shown in Fig. 1.

Integrating out Eq.~13!, yields

~t rz!z5H52S R1

RD 3

mk. (14)

Integrating Eq.~12! with respect toz between the limits 0 toH,and noting the fact thatt rz50 at z50, gives

EA@2YH2r ~t rz!z5H#dA22PH50. (15)

Evaluating Eq.~15! at z5H and substituting from Eq.~14!, weobtain the expression for the compression load as

P5YA1pR1

3mk

3H. (16)

When the barreling is disregarded, i.e.,R15R, andmk is replacedwith mY, Eq. ~16! reduces to the well known Siebel formula.

For the von Mises yield criterion,k5Y/), Eq. ~16! becomes

P5YS A1pR1

3m

3)HD . (17)

The expression of the flow stress therefore becomes

Y5PS A1pR1

3m

3)HD 21

. (18)

3 Calculation of Effective StrainSince the barreling of the cylinder leads to a nonuniform defor-

mation, the stress-strain relation may be best represented byadopting the mean flow stress and the associated mean effectivestrain of the deforming cylinder. In order to determine the meaneffective strain, the upper half of the barreled specimen, as shownin Fig. 1, was considered, with a velocity field suggested byChakrabarty@9# being adopted, which has the form

v r52HH r

2H1mc8S z

H D J , vu50,

and

vz5HH z

H1

mH

rcS z

H D J , (19)

where H is the rate of change of the specimen height,m is anarbitrary small coefficient representing the frictional condition atthe die-specimen interface, andc is an approximating function ofz/H. Also, c(0)5c(1)5c(21)50. The arbitrariness of thefunction c, which represents the barreling effect, makes the ve-locity field flexible enough to include the whole range of possibleshapes of the barreled specimen. The velocity field given in Eq.~19! characterizes the feature of longitudinal nonuniformity in thebarreled specimen, and the associated strain rates are

e r5]v r

]r52

H

2H,

eu5v r

r52

H

H H 1

21

mH

rc8S z

H D J ,

ez5]vz

]z5

H

H H 11mH

rc8S z

H D J ,

g rz51

2 S ]v r

]z1

]vz

]r D52mH

2H H c9S z

H D1H2

r 2 cS z

H D J . (20)

The corresponding expression for the effective strain ratee isgiven by

e252

3~ e r

21 eu21 ez

212g rz2 !5H 11

2mH

rc8S z

H D J S H

HD 2

,

(21)

the terms of orderm2 being omitted since they are small comparedto unity. This gives the effective strain rate as

e52H

H F11mH

rc8S z

H D G , (22)

with the terms of orderm2 being omitted. The negative sign on theleft-hand side of Eq.~22! is used to ensure a positive value for theeffective strain rate.

A mean effective strain rateeG may be defined with sufficientaccuracy as

eG 51

pR2HE

0

HE0

Re2pr dr dz

52H

H H 112m

R2E

0

RdrE

0

H

c8S z

HD dzJ

52H

H, (23)


since c vanishes at both limits of integration, i.e.,c(0)5c(1)50, R being the radius of the mean area. The total mean effectivestrain e at any stage is therefore given by

e5E eG dt5E 21

H

dH

dtdt52E

H0

H dH

H5 lnS H0

H D , (24)

where 2H0 is the initial height of the underformed specimen.Equation~24! indicates that the mean effective strain for a bar-reled specimen is the same as in the case of compression withoutbarreling to a first order of approximation. This result provides thetheoretical justification for using the axial compressive strain,ln(Ho /H), or ln(2Ho/2H), as the effective strain to construct thestress-strain curve for a barreled specimen in the compression test.

4 Finite Element ModelIn an actual compression test, it is very difficult to determine

the exact frictional condition at the interface between die face andspecimen; while the coefficient of friction can be easily incorpo-rated in the finite element method which has been considered as awell-developed technology for the analysis of metal forming pro-cesses. It is therefore convenient to perform the finite elementsimulations, instead of conducting experiments, to validate theproposed theoretical model. The procedure begins with the con-struction of the finite element model, as shown in Fig. 2. Becauseof symmetry, only a quarter of the specimen is analyzed. Both thetop and bottom dies are considered as rigid and the specimen ismodeled by 4-node axisymmetric elements, as shown in Fig. 2. Itis seen in this figure that the mesh is very dense so that the profileof the deformed specimen can be determined with sufficient ac-curacy. The material constitutive relation of the specimen used inthe simulations isY5K en with K5516 MPa andn50.23, wheree is the effective plastic strain. Since the barreling of the specimendepends also on the ratio of the initial height (2Ho) to the initialdiameter (do), known as the aspect ratio, in addition to the fric-tion coefficient, specimens of 10 mm in diameter and of two dif-ferent heights of 15 mm and 10 mm were used in the simulations,the corresponding aspect ratios being 1.5 and 1.0, respectively.

The simulations were performed for both the specimen sizesunder the frictional conditions specified bym50, m50.3, andm50.5, respectively. The compression load and the geometricparameters of the deforming specimen, such as the radius of thecontact area between die and specimen (R1), the maximum radiuscorresponding to the central section of the specimen (R2), and theinstantaneous height of the specimen (2H), as shown in Fig. 1,

obtained from the finite element simulations are used to calculatethe flow stressesY according to the proposed theoretical model,the E-H formula, and the mean flow stress method. The predictedflow stresses are then compared with each other and with thegiven material stress-strain relation for the required validation.

The finite element program DEFORM was adopted to performthe simulations, and the program was run on a SGI R4000workstation.

5 Results and DiscussionsIn order to validate the finite element simulation itself as an

effective method for providing the necessary information for thecompression loads and the geometric parameters of the barreledspecimen, the simulation was first performed for the specimen thatis 10 mm in height (2Ho /do51.0) under frictionless condition, sothat the simulation results can be compared with theoretical pre-dictions. The free surface of the deforming cylinder, obtainedfrom the finite element simulation, remains straight even under alarge reduction in height to 4 mm. The flow stresses calculatedaccording to the relationP/A at various effective strains~or axialcompressive strains! during the compression are almost the sameas those given by the stress-strain curve, as shown in Fig. 3, whereP is the compression load, andA is the cross-sectional area of thespecimen obtained from the finite element simulation. Althoughthis good agreement between the simulation results and the theo-retical predictions only validates the finite element simulationsunder the frictionless condition, it also implies the feasibility ofusing this method to simulate the cylinder compression in thepresence of friction. It is therefore meaningful to compare the trueflow stress given in the finite element method as input data withthe flow stresses calculated according to the present theoreticalmodel, the E-H formula, and the mean flow stress method, usingthe compression loads and the geometric parameters of the de-forming specimen furnished by the finite element simulationresults.

The compression load at various height reduction ratios (Ho2H)/Ho for both the specimens under different frictional condi-

Fig. 2 Initial mesh for the finite element simulation „unit: mm …

Fig. 3 Comparison of the simulated flow stresses „PÕA … withthe true flow stresses

Fig. 4 Compression load at various height reduction ratios„2Ho Õd oÄ1.0…


tions were obtained from the finite element simulations. The re-sults for the specimen of aspect ratio of 1.0 are plotted in Fig. 4.As seen in the figure, the compression load increases as the coef-ficient of friction increases.

In the E-H formula, the bulge profile is treated as a circular arcand the radius of the bulge curvatureR is determined from anempirical formula~Horton et al.@10#! which has the form

R5~2H !21~d22d1!2

4~d22d1!, (25)

whered1 , d2 are the minimum and maximum diameters of thebarreled specimen, respectively, and 2H is the instantaneousheight of the specimen. The finite element simulation results wereutilized to calculate the corresponding radius of the bulge curva-ture for the specimens under different frictional condition accord-ing to Eq. ~25!. Figure 5 shows the calculated radii of the bulgeprofile versus the mean effective strain of the cylinder with anaspect ratio of 1.0 under constant shear friction given bym50.1, 0.3, and 0.5, respectively. As seen in Fig. 5, the radius ofthe bulge profile decreases almost exponentially with the currentheight, and the specimen under higher friction coefficients hassmaller radii of curvature as expected.

With the compression loads and the geometric parameters gen-erated from the finite element simulation results, the flow stressescalculated according to the proposed theoretical model, the E-Hformula, and the mean flow stress method under various frictionalconditions were evaluated. To limit the length of this paper, onlythe results of higher friction coefficients are discussed. The flowstress calculated from the three methods and the true flow stressare plotted against the mean effective strain in Figs. 6 and 7,respectively, for the specimen of aspect ratio 1.5 under the fric-tional conditions corresponding tom50.3 andm50.5. It is seenin Fig. 6 and Fig. 7 that the mean flow stress method shows goodresults over a range of low compressive strains in which the bar-reling is not significant. However, when the cylinder is deformed

to large effective strains, saye>0.8, the calculated mean flowstress deviates from the true flow stress and the difference be-comes larger as the reduction increases, and the overestimatedvalue of the flow stress at higher compressive strains becomesmore significant as the coefficient of friction increases. It followsthat the calculation based on the mean cross sectional area (A) isnot sufficiently accurate, and the flow stress calculated from thismethod requires correction at higher reductions in height.

It is also seen in Fig. 6 and Fig. 7 that the flow stresses pre-dicted by the E-H formula agree reasonably well with the trueflow stresses at lower compressive strains for both the frictionalconditions ofm50.3 andm50.5, but significant differences arenoted at compressive strains higher than 0.4. As noted in both thefigures, the calculated flow stress according to the E-H formulaincreases very rapidly when the compressive strain is greater than1.0 for m50.3, and greater than 0.8 form50.5. The differencebetween the predicted flow stresses and the true flow stresses maybe attributed to the magnification of error in the approximate for-mula ~Eq. ~25!! for the radius of curvatureR of the bulge curva-ture. The rapid increase in the calculated flow stress is due to thefact thatR2 approaches the value of 2R as the strain increases,which makes the term@12R2 /(2R)# appearing in Eq.~2! tend tozero. WhenR2.2R the flow stress is not obtainable from Eq.~2!.This suggests that the E-H formula is applicable only to cylindersof relatively large aspect ratios.

The flow stresses calculated according to the proposed theoret-ical model form50.3 are also plotted in Fig. 6. It is seen in thisfigure that the theoretical model underestimates the flow stressesover a range of axial compressive strains lower than 1.2, andoverestimates the flow stresses at higher axial compressive strains.However, no marked difference is observed almost throughout thecompression process. Although the difference would tend to in-crease with the increasing compressive strains, the variation issmall and the flow stress still agrees reasonably well with the trueflow stress. When the friction coefficient is increased tom50.5,the flow stress calculated by the proposed theoretical modelagrees very well with the true flow stress, even for large compres-sive strains, as indicated in Fig. 7, while the other two theoriesresult in significant deviations from the true flow stress. Figure 7also suggests that the proposed theoretical model would providebetter results for specimens with significant barreling. On thewhole, the proposed theoretical model is an improvement over theE-H formula and the mean flow stress method, both of whichpredict significantly higher flow stresses at higher compressivestrains, and therefore increasingly deviate from the true flow stressof the material.

The comparison of the flow stresses calculated by the threemethods with the true flow stress is also made in Fig. 8 for thespecimen with an aspect ratio of 1.0 and a friction coefficient of0.5. The lower the aspect ratio, the less is the radius of the barrelcurvature, implying that an infinitely long cylinder would not bar-rel at all, as suggested by Johnson and Mellor@11#. Hence, de-creasing the aspect ratio from 1.5 to 1.0 significantly decreases the

Fig. 5 Radius of bulge profile at various effective strains„2Ho Õd oÄ1.0…

Fig. 6 Comparison of flow stresses at various effective strains„2Ho Õd oÄ1.5;mÄ0.3…



barrel radius when subjected to the same height reduction ratio,thereby significantly increasing the inhomogeneity of deforma-tion. It is seen in Fig. 8 that the proposed theoretical model stillpredicts the flow stress consistently in close agreement with thetrue flow stress, and is superior to those given by the other twotheories. This confirms the validity of the proposed theoreticalmodel for calculating the flow stress for a barreled specimen un-der axial compression over the whole range of reductions inheight.

6 Summary and Concluding RemarksThe purpose of this paper is to calculate the flow stresses for a

barreled specimen, resulting from the presence of frictional forceat the die-specimen interface during the compression test. Thefinite element simulations, instead of experimental results, wereemployed to generate the compression loads and the geometricparameters of the barreled specimen to validate the theoreticalmodel developed in the present study. Unlike most of the pub-lished methods, which studied the incipient barreling only, theproposed theoretical model takes the continued barreling of thedeforming specimen into account. As the present study shows, theanalysis which is based on Hill’s general method for metal form-ing problems has yielded better results, both in magnitude and intrend, for the prediction of flow stresses in the barreled specimenin a compression test. It is also noted from the comparison madebetween the three methods that the proposed theoretical modelcan be used more effectively to the cylinder compression test withsignificant degrees of barreling.

In spite of extensive derivations and complex integrations re-quired in its application, Hill’s general method is certainly basedon sound mathematical principles, and is flexible enough for in-

troducing various simplifying assumptions. The satisfactory esti-mate of the flow stress for the barreled specimen under inhomo-geneous compression indicates that the assumptions made in theproposed theoretical model are sufficiently realistic.

The velocity field adopted in the present study to calculate themean effective strain for the barreled specimen also proves to beeffective. The fact that the mean effective strain of the barreledspecimen is approximately equal to the overall axial compressivestrain to the first order simplifies the construction of the stress-strain relation in the compression test.

Furthermore, although the analysis correctly predicts the maintrend for the flow stresses in a barreled specimen during the com-pression test, the proposed theoretical model would tend to under-estimate the flow stress of specimen at lower compressive strains.However, the difference between the calculated flow stresses andthe true flow stresses is not significant. It is suggested that theinclusion of a shape factor in Eq.~18!, which can allow for thechange in the barrel geometry during the compression, might im-prove the accuracy of the proposed theoretical model.

AcknowledgmentThe authors wish to thank the National Science Council of the

Republic of China for their grant under the project NSC85-2216-E-002-027, which makes this research possible. They also wouldlike to thank Professor J. Chakrabarty for his helpful discussions.

References@1# Siebel, E., 1923,Stahl und Eisen, Duesseldorf,43, p. 1295.@2# Avitzur, B., 1968,Metal Forming: Process and Analyses, McGraw-Hill, New

York, pp. 102–111.@3# Lee, C. H., and Altan, T., 1972, ‘‘Influence of Flow Stress and Friction Upon

Metal Flow in Upset Forging of Rings and Cylinders,’’ ASME J. Eng. Ind.,94,Aug., pp. 775–782.

@4# Ettouney, O., and Hardt, D. E., 1983, ‘‘A Method for In-Process Failure Pre-diction in Cold Upset Forging,’’ ASME J. Eng. Ind.,105, pp. 161–167.

@5# Bridgman, P. W., 1952,Studies in Large Plastic Flow and Fracture, McGraw-Hill, New York, pp. 9–86.

@6# Hill, R., 1963, ‘‘A General Method of Analysis for Metal-Working Pro-cesses,’’ J. Mech. Phys. Solids,11, pp. 305–326.

@7# Lahoti, G. D., and Kobayashi, S., 1974, ‘‘On Hill’s General Method of Analy-sis for Metal-Working Processes,’’ Int. J. Mech. Sci.,16, pp. 521–540.

@8# Kobayashi, S., Oh, S. I., and Altan, T., 1989,Metal Forming and the Finite-Element Method, Oxford University Press, Oxford, pp. 78–83.

@9# Chakrabarty, J., 1996, private communications.@10# Horton, H. L., Ryffel, H. H., and Schubert, P. B., 1959,Machinist’s Hand-

book, 16th Edition, The Industrial Press, New York, p. 152.@11# Johnson, W., and Mellor, P. B., 1975,Engineering Plasticity, Van Nostrand

Reinhold, London, pp. 110–114.



M. A. LinazaJ. M. Rodriguez-Ibabe

CEIT and ESII de San Sebastian,Pde Manuel de Lardizabal, 15, 20018,San Sebastian, Basque Country, Spain

Fracture Static Mechanisms onFatigue Crack Propagation inMicroalloyed Forging SteelsThe influence of static mechanisms on fatigue crack propagation in Ti and Ti-V microal-loyed steels is considered. Small inclusions originate void nucleation. In contrast, TiNcoarse particles contribute to the formation of bursts of cleavage in the fatigue zone.Taking into account the microstructural characteristics of the matrix that surrounds theparticle, the microcrack can be confined within the particle or propagate along the matrixforming a cleavage burst. The influence on macroscopic crack propagation of both typesof static micromechanisms is considered.@S0094-4289~00!00902-6#

1 IntroductionIn the fatigue crack propagation corresponding to the Paris re-

gime, together with the conventional fatigue mechanisms, otherprocesses of static nature can take place in some materials. Thepresence of these static mechanisms can contribute in a relevantway to the crack behavior, giving place to higher macroscopicpropagation rates. For example, it is well known that in lowtoughness materials the local cleavage mechanisms contribute inincreasing them exponent value~Ritchie and Knott@1#, Beeverset al. @2#!. On the other hand, ductile voids are related to nonme-tallic particle decohesion and their influence on fatigue is not sowell defined as in the case of brittle fracture mechanisms. This canbe as a consequence of the different behavior of the nonmetallicinclusions depending on their nature~Nicholson and Gladman@3#!and on testing orientation~Wilson @4#!.

In the case of forging steels, microalloying addition has becomeone of the main procedures in achieving ferrite-pearlite micro-structures, with strengths similar to the more expensive quenchedand tempered steels. Among these types of steel, Ti, V, and Ti-Vmicroalloyed steels must be considered. Although equivalent me-chanical strength has been achieved with these steels, usually frac-ture toughness behavior remains lower~Naylor @5,6#!. In thesetypes of steels, the cleavage process has been attributed to thenucleation of microcracks in some nonmetallic particles and theirdynamic brittle propagation across a coarse ferrite-pearlite micro-structure~Linaza et al.@7#!. This type of brittle mechanism, actingunder static conditions, can play a relevant role during fatiguecrack propagation in the as-hot worked microstructures. On theother hand, in order to improve machinability or toughness prop-erties ~by promoting intragranular ferrite nucleation~LaGrecaet al.@8#!!, a significant volume fraction of nonmetallic inclusions~mainly MnS inclusions! can be present in forging steels. Theseinclusions can give rise to the nucleation of ductile voids, whichcan increase crack growth rates.

In this work, the influence of both ductile and brittle staticmechanisms, on the fatigue crack propagation, is considered forthe case of two microalloyed medium carbon steels~Ti and Ti-V!with different microstructures.

2 Material and Experimental ProcedureTwo hot-rolled bar steels~50 mm square! with compositions

listed in Table 1 were used in this investigation. Both materialsare medium carbon steels with Ti additions and, in the case of theTi-V steel, an addition of V is done in order to obtain a higher

strength. In this latter case, a higher sulfur content improves themachinability of the steel. The as-rolled ferrite-pearlite conditionwas analyzed in both steels and, for the Ti-V microalloyed steel,three additional microstructures were considered. These micro-structures were obtained by laboratory thermomechanical simula-tions performing plane strain compression tests (50350310 mm3) at 850, 950, and 1000°C, developing different fineferrite-pearlite and acicular ferrite microstructures. The main mi-crostructural parameters~da ferrite grain size andf a ferrite vol-ume fraction! and mechanical properties~sy yield stress, UTSultimate tensile strength andJ integral fracture toughness param-eter ~ASTM E813 standard, three point bending specimen withB56 mm!! are listed in Table 2. A more detailed description ofthe microstructures and their corresponding toughness behaviorhave been reported elsewhere~Linaza et al.@9,10,7#!

Specimens for fatigue crack growth analysis were machinedinto CT geometries~B56 mm and W536 mm! from the as-rolledbars and plane strain compression samples. With this geometry,both the threshold and the Paris equation were determined at roomtemperature, in a 20 kN resonant testing machine at a load ratio ofR50.03. In the case of as-rolled microstructures, in order to mini-mize the crack closure effect on crack propagation, some testswere performed at R50.5. Tests for the threshold measurementwere carried out applying a manual load shedding technique. Forthe determination of the Paris equation, the tests were carried outunder load control mode, using a sinusoidal load signal, in agree-ment with the ASTM E647 standard. During the test, the fatiguecrack length was measured periodically and the correspondingnumber of cycles recorded. The fracture surface of the testedspecimens was extensively studied by scanning electronmicroscopy.

3 ResultsFigures 1 and 2 show theDK2da/dN curves corresponding to

the Ti steel and the Ti-V steel, respectively. In Fig. 1, the resultshave been obtained with the same ferrite-pearlite microstructure,considering the influence of the load ratio R. As can be observed,in the analyzedDK range, the crack propagation rate is smallerfor R50.03 than for R50.5. In Fig. 2 the influence of four dif-ferent microstructures is considered. It is worth emphasizing that

Contributed by the Materials Division for publication in the JOURNAL OF ENGI-NEERING MATERIALS AND TECHNOLOGY. Manuscript received by the MaterialsDivision April 13, 1999; revised manuscript received October 14, 1999. AssociateTechnical Editor: S. Mall.

Table 1 Chemical compositions of the studied microalloyedmedium C steels „wt. percent …

Steel C Mn Si P S V Al TiN

~ppm!

Ti 0.35 1.56 0.33 0.004 0.007 - 0.027 0.028 89Ti-V 0.37 1.45 0.56 0.010 0.043 0.11 0.024 0.015 162


the fine ferrite-pearlite microstructure obtained by hot working at850°C shows a lower crack growth rate for a given cyclic stressintensity value. In a similar way, the higher crack growth ratecorresponds to the coarse as-rolled ferrite-pearlite microstructure.The acicular ferrite and the ferrite-pearlite microstructures ob-tained by hot working at 950°C lie between these two extremes.

A summary of the results~C andm Paris coefficient values andDKth threshold data! is shown in Table 3. As can be seen in thetable,m exponent values are between 3.4 and 4.7 for R50.03; incontrast, when R50.5, them exponent value ranges between 2.7and 3. In relation to the fatigue threshold, the influence of loadratio is more relevant than the microstructure effect.

The fractographic analysis shows the classical flat and quasi-amorphous aspect of the fatigue fracture surfaces. Apart from thepresence of striations, more visible for highDK values, ductilevoids nucleated at inclusions and brittle islands, formed by several

cleavage facets, have been identified. In relation to the ductilevoids, they are more frequently observed in the Ti-V steel, whichexhibits a higher density of inclusions due to the higher sulfurconcentration~see Fig. 3 corresponding to voids nucleated at MnSinclusions!. On the other hand, the void volume fraction remainsconstant during the entireDK range corresponding to the Parisregime, although there are some local heterogeneities. In the caseof the Ti-V microalloyed steel, an area fraction of around 1 per-cent of cavities in the fracture surfaces has been estimated bypoint counting technique in theKmax range from 20 to 50 MPaAm.

Brittle cleavage islands can be observed in both steels withferrite-pearlite microstructures~they have not been identified inthe case of an acicular ferrite microstructure!, although more inthe Ti-V steel, and usually they appear forKmax values higher than

Fig. 1 da ÕdNÀDK fatigue crack propagation curves of as-rolled ferrite-pearlite Ti steel

Fig. 2 da ÕdNÀDK fatigue crack propagation curves of Ti-Vmicroalloyed steel for different ferrite-pearlite and acicularferrite microstructures developed by thermomechanicaltreatments

Table 2 Microstructural parameters and mechanical behavior of analyzed microstructures

Steel Processda

~mm!f a

~%!sy

~MPa!UTS

~MPa!J

~kJ/m2!KJ

~MPaAm!

Ti As-rolled 5.3 26.1 440 740 58.1* 116.3As-rolled 5.5 38.0 590 875 23.7* 74.3

Ti-V HW 950°C1SC 5.1 26.0 655 927 - -HW 850°C1SC 5.6 39.0 607 907 78.51 135.2HW 1000°C1FC - - 560 920 1031 154.9

HW: hot worked; SC: slow cooled; FC: fast cooled~* ! J5Jc ~1! J5JIc

Table 3 C and m coefficient values of Paris equation and fatigue threshold values

Steel Process Microstruct. R C mDKth

~MPaAm!

Ti-V As-rolled ferrite-perl. 0.03 4.74310213 3.7 -As-rolled ferrite-perl. 0.03 1.17310212 3.4 8.8

HW 1000°C1FC acicular ferrite 0.03 5.15310213 3.6 8.5HW 8501SC ferrite-perl. 0.03 1.02310214 4.7 11.6HW 9501SC ferrite-perl. 0.03 8.05310214 4.2 -

As-rolled ferrite-perl. 0.5 1.31310211 2.8 -As-rolled ferrite-perl. 0.5 8.26310212 3.0 4.7

Ti As-rolled ferrite-perl. 0.03 4.14310214 4.3 10.5As-rolled ferrite-perl. 0.5 6.84310212 3.0 5.6As-rolled ferrite-perl. 0.5 1.59310211 2.7 -


50 MPaAm. In all the cases, they are originated by the rupture ofTiN particles coarser than 3.5mm ~see Fig. 4! and consist ofseveral facets. In order to analyze the characteristics of thesebrittle islands, some geometrical measurements have been per-formed for the case of the Ti-V steel with a ferrite-pearlite micro-structure. The measured parameters, listed in Table 4, are theKmaxvalue at which the island has been formed, the dimensions of theTiN particle responsible for the cleavage initiation, the number ofcleavage facets constituting the island and the maximum andminimum dimensions of the island. Another important feature,

common to all the identified islands, is the small misorientationbetween the facets and the macroscopic fatigue crack plane. Fi-nally, some coarse TiN broken particles have also been observedin the fatigue fracture surface, which have not promoted the for-mation of cleavage facets. This behavior is more common in thecase of acicular ferrite and fine ferrite-pearlite microstructures~hot-worked samples at 850 and 950°C!.

4 DiscussionReferring to the results obtained in the Paris region, the most

relevant parameter influencing crack propagation is the load ratio,as can be seen in Figs. 1 and 2. This behavior has also beenobserved in other ferrite-pearlite steels and it can be correlatedwith the presence of a crack closure effect. Although crack clo-sure measurements have not been carried out, taking into accountthe results obtained by Tanaka and Soya@11# with different Vmicroalloyed structural steels, the slight differences observed inthe Paris zone between the different microstructures can be con-sidered through the crack closure. Similarly, Costa and Ferreira@12# have shown that in the case of thin specimens (B56 mm)there is a stronger influence of the crack closure effect in the Parisregime. These authors have proposed the following empirical re-lationships to take into account the crack closure effect:

U50.43310.716R10.012DK10.144~B/W! if b,0.567

U51 if b.0.567(1)

whereb50.716R10.0121DK10.144(B/W) andDKeff5UDKTaking into account these expressions, it can be concluded that

for R50.5, in the Paris regime, there is not a closure effect. Onthe other hand, ifDKeff is considered instead ofDK, the datacorresponding toR50.03 overlap the data obtained for R50.5 forboth steels, showing that the R effect in the Paris zone corre-sponds to the crack closure.

All the data are within the scatter band of fatigue crack growthrates published in the bibliography corresponding to ferrite-pearlite microstructures obtained with different hypoeutectoidsteels~Rodriguez-Ibabe and Gil-Sevillano@13#!. Similarly, thresh-old data exhibit fair agreement with reported data for ferrite-pearlite steels~Bulloch @14#!.

Figure 5 shows the Paris equations corresponding to the meanvalues of the different tests of both steels. In the figure the resultsobtained with a ferrite-pearlite C-Mn steel without static mecha-nisms are also included for comparison~Linaza et al.@15#!. Toavoid the possible effect of crack closure, only the results mea-sured for R50.5 have been considered. The Paris straight linescorresponding to the three materials are parallel and the propaga-tion rate is fastest for the Ti-V steel, followed by the Ti steel andby the C-Mn steel, respectively. In consequence, while them ex-ponent is very similar in all cases, there are some differences in Ccoefficient values that can be correlated to the contribution ofstatic mechanisms in crack propagation.

Fig. 3 Fracture surface in the Paris zone of Ti-V steel withas-rolled ferrite-pearlite microstructure showing ductile voidsnucleated at inclusions „RÄ0.5, da ÕdNÄ1.2Ã10À7 mÕc…

Fig. 4 Brittle island in the Paris region originated by the rup-ture of a TiN coarse particle „Ti steel, R Ä0.03…

Table 4 Characteristics of the brittle cleavage islands on the fatigue surface of Ti-V steel with ferrite-pearlite microstructure

RKmax

~MPa Am!

TiN size

facetnumber

Island size

amin~mm!

amax~mm!

dmin~mm!

dmax~mm!

0.03 58 6.1 6.9 8 45 8960 4.4 6.5 9 50 9080 5.4 6.8 1 22 3086 8.0 8.0 7 44 90

0.5 53 3.3 8.3 7 90 14259 4.2 8.3 10 68 9760 2.2 5.8 8 50 8051 2.2 10.4 10 40 12853 1.6 8.3 2 40 52


4.1 Brittle Static Mechanisms. In relation to the staticfracture mechanisms, the number of brittle features identified ineach specimen is very small and, as a result, they did not influencethe macroscopic crack propagation, as can be observed in theresults of Figs. 1 and 2 and in them exponent values of Table 3.Nevertheless, the characteristics of these islands can contribute tothe study of the micromechanisms taking place in the static brittlefracture of microalloyed forging steels.

The brittle islands are nucleated at coarse TiN particles and, inconsequence, the presence of these cleavage facets, completelysurrounded by fatigue mechanisms, must be related to the natureof these TiN particles. Coarse TiN particles are formed during thesolidification process of the steel~Herman et al.@16#! and, due totheir size, they are ineffective in controlling austenite grain coars-ening during forging. On the other hand, as a consequence of thethermal expansion coefficient mismatch~Brooksbank and An-drews @17#!, TiN particles have a very strong bonding with thesteel matrix. Brittle fracture requires the nucleation of a micro-crack in some microstructural feature, usually as a consequence ofa plasticity induced process~Bowen et al.@18#!, and its propaga-tion across the matrix. In the case of TiN particles, once the par-ticle is broken, its good inclusion/matrix bonding facilitates thebrittle propagation of the microcrack across the interface. In aprevious study, Linaza et al.@9,20# identified that the static brittlefracture toughness of both steels was related to the nucleation ofmicrocracks at coarse TiN particles and their propagation acrossdifferent microstructural barriers~particle-matrix and high anglematrix-matrix barriers!. Once one of these microcracks has tra-versed several grains~sometimes one is enough!, its length to-gether with the local stress state is sufficient for catastrophicpropagation to failure. This behavior takes place in the completelybrittle regime and in the ductile-brittle transition, and has beenidentified as one of the main microstructural features promotingpoor toughness values in the as-forged ferrite-pearlite microstruc-tures~Linaza et al.@20#!.

Figure 6 shows the histograms of the measured minimum andmaximum dimensions of coarse TiN particles present in the Ti-Vsteel. In the static fracture toughness tests performed with thesame steel in a previous work~Linaza et al.@10#!, the interval ofTiN particles identified as responsible for cleavage nucleationranged from 1 to 4mm. Comparing these previous results with thedata of Table 4, it is worth emphasizing that the particles observedin the fatigue surfaces promoting brittle islands are coarser thanthose identified in the static tests, and that they belong to the

group of largest particles present in the steel. In order to analyzethis size difference of the TiN particles acting as brittle fracturenucleators, a probability effect must be considered originating as aconsequence of testing samples of different volume in the fatigueand fracture toughness experiments. Taking into account the ge-ometries tested, in the static fracture toughness tests~three pointbending specimen with B56 mm and W511 mm! the materialvolume sampled by the peak stress is at least three times smallerthan in the case of fatigue specimens~considering the materialvolume corresponding to the propagation of the crack forKmax.50 MPaAm, minimum value observed for the appearance ofbrittle islands in the fatigue tests!. In consequence, the probabilityof having coarse TiN particles able to nucleate brittle fracturebursts is higher in the fatigue samples, leading to the differencesobserved between the static fracture toughness results and the val-ues of Table 4.

Another relevant characteristic is that there are broken TiN par-ticles which have not promoted the formation of cleavage facets.This means that the nucleation of a sharp microcrack by the rup-ture of a TiN particle is not a sufficient criterion to cause theformation of brittle facets. In order to explain this behavior, itmust be considered that, as a consequence of a Weibull volumeeffect ~Knott and King @21#!, large TiN particles can break atlower stresses than small particles. If cracks are nucleated in largeparticles at stress values too low to be able to promote their propa-gation across particle-matrix interfaces, microcracks will bluntwithout forming cleavage facets. This behavior can explain why aminimum Kmax value is required for the appearance of brittleislands.

In relation to the characteristics of the islands, independent ofthe number of facets, in all the cases they are well oriented inrelation to the macroscopic plane. This means that the first facetformation only takes place in those cases in which there is a goodcrystallographic orientation of the~100! planes surrounding thebroken TiN particle. In consequence, in those cases without anappropriate crystallographic orientation~for example in the caseof an acicular ferrite microstructure!, the local stress value actingnormally to the~100! plane would be smaller, increasing the dif-ficulty for propagation of the microcrack from the broken TiNparticle to the matrix and across the different microstructural bar-riers. This is the reason why after some brittle propagation, cracksare arrested at matrix-matrix boundaries and the cleavage processahead of the fatigue crack is completely stopped. Similarly, brittleislands nucleated at TiN broken particles and arrested after cross-ing some matrix-matrix boundaries have been identified in statictests in the ductile-brittle regime~San Martin and Rodriguez-Ibabe@22#!.

Finally, although these brittle fracture mechanisms have notmodified them slope value of the Paris curve, it is worth empha-

Fig. 5 Paris equations of Ti-V and Ti steels for R Ä0.5. Resultsobtained with a ferrite-pearlite C-Mn steel are included „Linazaet al. †15‡….

Fig. 6 Histograms of the minimum and maximum dimensionsof coarse TiN particles in the Ti-V steel


sizing the deleterious effect that they can have in industrial appli-cations, promoting premature brittle fracture of components. It iswell documented that coarse TiN particles are deleterious fromthe point of view of fracture toughness when they are combinedwith coarse microstructures~Linaza et al.@9,20#!. This refers tothe situation corresponding to the lower part of the ductile-brittleregime of microstructures at room temperature. In this zone, thereis a large dispersion of toughness values, dependent upon theprobability of activating the brittle process after some ductilepropagation~Landes et al.@19#!. As a consequence of the fatiguecrack propagation process, during the crack extension a highermaterial volume could be sampled by the peak stress than is thecase for a stationary crack, thus increasing the probability of hav-ing adequate conditions to promote cleavage fracture~a combina-tion of the presence of a TiN particle of adequate size surroundedby a coarse ferrite-pearlite matrix!.

4.2 Ductile Static Mechanisms. Ductile voids nucleated atinclusions appear in the entire range of measuredDK values. Theorder of Paris straight lines of Fig. 5 can be correlated to thecleanness of the steels: while the lower growth rate for a givenDK value corresponds to C-Mn steel, the higher rate propagationis achieved with the Ti-V steel. The Ti-V steel has a higher vol-ume fraction of MnS and oxides, while the C-Mn steel exhibits avery low volume fraction of inclusions~;0.020 percent!. Thedifferent volume fraction of inclusions results in the appearance ofdifferent percentages of ductile cavities on the fatigue surface.These cavities have been nucleated ahead of the crack tip in theplastic zone. In consequence, fatigue crack growth will be easier,due to the fact that the crack must propagate across a lower effec-tive surface. If the effect produced by a cavity nucleated from aninclusion is assimilated to a pore in a sintered material, the propa-gation rate can be obtained by the equation proposed by Bompardand François @23# for those types of materials:

da

dN5

C~DK !m

Dm11 (2)

whereD is the ratio between the effective area and the total area.In this equation, it is supposed that the ductile static fracture

events are independent of stress intensity and, as a result, theycontribute only to increasing the coefficient of the power-law, butnot its exponent. If the area fraction of cavities of around 1 per-cent, estimated for Ti-V steel, is substituted in Eq.~2! the differ-ences between Ti-V steel and C-Mn can be explained~the C co-efficient value for the Ti-V steel changes from 1.0310211 ~meanvalue of tested specimens! to 5.9310212, very close to the7.5310212 value obtained for the C-Mn steel!. A similar behavioris observed for the Ti steel.

Summarizing, the existence of ductile mechanisms in the fa-tigue crack propagation in the microalloyed forging steels consid-ered in this study has a very limited effect, introducing only aslight increase in the C coefficient value of Paris equation.

5 ConclusionsThe influence of static mechanisms on fatigue crack propaga-

tion in the considered microalloyed steels leads to the followingconclusions:

1 These steels exhibit fatigue threshold values and Paris equa-tions close to those found in plain C-Mn steels with ferrite-pearlitemicrostructures. The brittle islands nucleated at TiN coarse par-ticles which appear in both microalloyed steels have no influenceon the fatigue crack propagation rate.

2 The formation of these brittle islands agrees with the brittlefracture process, which takes place in Ti microalloyed steels withcoarse TiN particles. During fatigue crack propagation, due to a

probability effect, the toughness of these steels, in the ductile-brittle regime, can correspond to the lower bound of the valuesmeasured by static fracture toughness tests.

3 The formation of ductile cavities at non-metallic inclusionsincreases the crack propagation rates slightly by increasing the Ccoefficient of the Paris equation. In the Paris regime, the forma-tion of the ductile cavities is independent of the appliedDK value.

AcknowledgmentsPart of this work has been carried out under an ECSC Research

Project. Financial support by CICYT is also acknowledged.

References@1# Ritchie, R. O., and Knott, J. F., 1973, ‘‘Mechanisms of Fatigue Crack Growth

in Low Alloy Steels,’’ Acta Met.,21, p. 639.@2# Beevers, C. J., Cooke, R. J., Knott, J. F., and Ritchie, R. O., 1975, ‘‘Some

Considerations of the Influence of Subcritical Cleavage Growth DuringFatigue-Crack Propagation in Steels,’’ Met. Sci.,9, p. 119.

@3# Nicholson, A., and Gladman, T., 1986, ‘‘Non-Metallic Inclusions and Devel-opment in Secondary Steelmaking’’ Ironmaking and Steelmaking,13, p. 53.

@4# Wilson, A. D., 1984, ‘‘Fatigue Crack Propagation in Steels: The Role of In-clusions,’’ Fracture: Interactions of Microstructure, Mechanisms and Me-chanics, J. M. Wells and J. D. Landes, eds., AIME, p. 235.

@5# Naylor, D. J., 1989, ‘‘Review of International Activity on Microalloyed Engi-neering Steels’’ Ironmaking and Steelmaking,16, p. 246.

@6# Naylor, D. J., 1998, ‘‘Microalloyed Forging Steels’’ Mater. Sci. Forum,284–286, p. 83.

@7# Linaza, M. A., Rodriguez-Ibabe, J. M., and Urcola, J. J., 1997, ‘‘Determinationof the Energetic Parameters Controlling Cleavage Fracture Initiation inSteels,’’ Fatigue Fract. Eng. Mater. Struct.,20, p. 619.

@8# LaGreca, P. D., Matlock, D. K., and Krauss, G., 1996, ‘‘Short-rod FractureToughness Testing of Microalloyed Steels as a Function of Sulfur and Intra-granular Ferrite Content,’’Fundamentals and Applications of MicroalloyedForging Steels, C. J. Van Tyne et al., eds., TMS, Warrendale, p. 357.

@9# Linaza, M. A., Romero, J. L., Rodriguez-Ibabe, J. M., and Urcola, J. J., 1993,‘‘Influence of the Microstructure on the Fracture Toughness and FractureMechanisms of Forging Steels Microalloyed with Ti with Ferrite-PearliteStructures,’’ Scr. Metall. Mater.,29, p. 451.

@10# Linaza, M. A., Romero, J. L., Rodriguez-Ibabe, J. M., and Urcola, J. J., 1995,‘‘Cleavage Fracture of Microalloyed Forging Steels,’’ Scr. Metall. Mater.,32,p. 395.

@11# Tanaka, Y., and Soya, Y., 1990, ‘‘Metallurgical and Mechanical Factors Af-fecting Fatigue Crack Propagation and Crack Closure in Various StructuralSteels,’’Fatigue 90, H. Kitagawa and T. Tanaka, eds., MCE Publications, Vol.2, p. 1143.

@12# Costa, J. D. M., and Ferreira, J. A. M., 1998, ‘‘Effect of Stress Ratio andSpecimen Thickness on Fatigue Crack Growth of CK45 Steel,’’ Theor. Appl.Fract. Mech.,30, p. 65.

@13# Rodriguez-Ibabe, J. M., and Gil-Sevillano, J., 1984, ‘‘Fatigue Crack Path inMedium-high Carbon Ferrite-Pearlite Structures,’’Advances in Fracture Re-search, S. R. Valluri et al., eds., Pergamon Press,3, p. 2073.

@14# Bulloch, J. H., 1992, ‘‘Effects of Mean Stress on the Threshold Fatigue CrackExtension Rates of Two Spherical Graphite Cast Irons,’’ Theor. Appl. Fract.Mech.,18, p. 15.

@15# Linaza, M. A., Rodriguez-Ibabe, J. M., and Fuentes, M., 1992, ‘‘Fatigue CrackGrowth and Closure Behavior of Pressure Vessel C-Mn Welded Steels,’’Re-liability and Structural Integrity of Advanced Materials, S. Sedmak et al., eds.,EMAS, Vol. 1, p. 397.

@16# Herman, J. C., Messien, P., and Greday, T., 1982, ‘‘HSLA Ti ContainingSteels,’’ Thermomechanical Processing of Microalloyed Austenite, A. J.DeArdo et al., eds., AIME, Warrendale, p. 655.

@17# Brooksbank, D., and Andrews, K. W., 1968, ‘‘Thermal Expansion of SomeInclusions Found in Steels and Relation to Tessellated Stresses,’’ JISI,206, p.595.

@18# Bowen, P., Druce, S. G., and Knott, J. F., 1987, ‘‘Micromechanical Modellingof Fracture Toughness,’’ Acta Metall.,35, p. 1735.

@19# Landes, J. D., Heerens, J., Schwalbe, K. H., and Petrovski, B., 1993, ‘‘Size,Thickness and Geometry Effects on Transition Fracture,’’ Fatigue Fract. Eng.Mater. Struct.,16, p. 1135.

@20# Linaza, M. A., Romero, J. L., Rodriguez-Ibabe, J. M., and Urcola, J. J., 1997,‘‘Influence of Thermomechanical Treatments on the Microstructure andToughness of Microalloyed Engineering Steels,’’Thermomechanical Process-ing in Theory, Modelling and Practice, B. Hutchinson et al., eds., SFMC, p.351.

@21# Knott, J. F., and King, J. E., 1990, ‘‘Fatigue in Metallic Alloys ContainingNon-Metallic Particles,’’Fatigue 90, H. Kitagawa and T. Tanaka, eds., MCEPublications, Vol.4, p. 2557.

@22# San Martin, I., and Rodriguez-Ibabe, J. M., 1999, ‘‘Determination of the En-ergetic Parameters Controlling Cleavage Fracture in a Ti-V MicroalloyedFerrite-Pearlite Steel,’’ Script. Mat.,40, p. 459.

@23# Bompard, P. H., and François, D., 1984, ‘‘Effect of Porosity on Fatigue CrackPropagation in Sintered Nickel,’’ Advances in Fracture Research, S. R. Valluriet al., eds., Vol.3, p. 2049.


Chuwei ZhouWei Yang

Daining FangFML, Department of Engineering Mechanics,

Tsinghua University,Beijing 100084, China

Damage of Short-Fiber-Reinforced Metal MatrixComposites Considering Coolingand Thermal CyclingMechanical properties and damage evolution of short-fiber-reinforced metal matrix com-posites (MMC) are studied under a micromechanics model accounting for the history ofcooling and thermal cycling. A cohesive interface is formulated in conjunction with theGurson-Tvergaard matrix damage model. Attention is focused on the residual stressesand damages by the thermal mismatch. Substantial stress drop in the uniaxial tensileresponse is found for a computational cell that experienced a cooling process. The stressdrop is caused by debonding along the fiber ends. Subsequent thermal cycling lowers thedebonding stress and the debonding strain. Micromechanics analysis reveals three failuremodes. When the thermal histories are ignored, the cell fails by matrix damage outsidethe fiber ends. With the incorporation of cooling, the cell fails by fiber end debonding andthe subsequent transverse matrix damage. When thermal cycling is also included, the cellfails by jagged debonding around the fiber tops followed by necking instability of matrixligaments.@S0094-4289~00!01202-0#

1 IntroductionShort-fiber-reinforced Metal-Matrix-Composite~MMC! has ad-

vantages in both formability and enhanced mechanical properties,such as high specific stiffness and high specific strength. Onecritical requirement for short-fiber-reinforced MMC is the surviv-ability under a severe history of thermal cycling. For example, aspace vehicle may encounter thermal cycling ranging from anelevated temperature to a cryogenic temperature. The mismatch inthermal expansions between the fiber and the matrix producesresidual stresses in MMC. Damage accumulation during a severethermal cycling history may degrade the mechanical properties,such as the strength and the toughness, of MMC. Multiple damagesources exist in the MMC: the interfaces may debond, voids mayoccur in the metal matrix, and fibers may break under unbearabletensile stress. The present work is focused on the interface andmatrix damages, since fiber breakage does not dominate the dam-age pattern for the MMC reinforced byshort-fibers.

The evolution of damage is dictated by the local stress andstrain fields in MMC, thus raising the issue of sensitivity to theFEM scheme in the simulation. A viable FEM scheme shouldhave the resolution on stress, strain, and damage fields finer thanthe diameter of a fiber, and have the ability to account for thehistory of thermal-mechanical processing. FEM studies of MMCwere initiated by Christman et al.@1#, by Levy and Papazian@2#,and by Tvergaard@3#. These pioneer works did not discuss theissue of residual stresses and strains in MMC when cooled fromthe solution-treatment temperature to the room temperature. Theimportance of the residual stresses and the thermal cycling wassoon recognized. Levy and Papazian@4#, and Davis and Allison@5# discussed the residual stresses in MMC after cooling, andLevy and Papazian@6# considered the effect of thermal cycling.Unfortunately, those works neither referred to the damage evolu-tion within the material, nor accounted explicitly the damages ininterface and matrix. That provides a strong impetus for thepresent work to investigate the effects of cooling and thermalcycling on the strength and damage evolution of MMC.

2 Formulation on Interface and Matrix DamageThe fiber/matrix interface is usually a thin layer of different

material properties, or a bimaterial interlayer~Yang and Shih@7#!.The present work adopts elements based on the cohesive interfacemodel~Needleman@8#, Tvergaard@9#, Chaboche et al.@10#! withminor modifications, that enables numerical simulation of inter-face damage and debonding.

The cohesive interface is formulated under a phenomenologicalmodel. It relates to the interface properties and the damage ofMMC by choosing the interface parameters properly. Damages inthe normal and the tangential directions are coupled. The interfacetraction components,Tn and Tt , are correlated to the displace-ment jumps across the interface,Dn andD t , by

Tn5~12lmax!2EnDnH~Dn!1KnDnH~2Dn! (1)

Tt5~12lmax!2EtD t (2)

where

lmax5AS Dnmax

dnD 2

H~Dnmax!1S D t

max

d tD 2

(3)

The quantitiesDnmax andD t

max denote the maximum displacementjumps ofDn andD t in the history. In Eqs.~1! and~2!, subscriptsn andt label the normal and the tangential components. SymbolHdenotes the Heaviside function that is utilized to distinguish inter-face responses under tension and compression. SymbolsEn andEt , dn andd t , are two sets of interface parameters that representthe interface moduli and the critical interface separations, respec-tively. The quantityKn denotes the normal compressive stiffnessand is assigned a large value to minimize the fiber/matrix penetra-tion. The scalarlmax is a monotonically increasing parameter in-dicative to the interface damage:lmax50 refers to a perfect inter-face andlmax>1 a complete separation. Figure 1 depicts thetraction-displacement jump curve for the case of normal separa-tion. As the interface displacement jump increases from zero, theinterface traction rises at first, then declines after the traction peak,and eventually diminishes whenDn>dn . The descending regimein the cohesive curve describes the interface weakening. The situ-ation thatlmax ceases to evolve refers to a pseudo-elastic state,where the interface traction and interface displacement jump are

Contributed by the Materials Division for publication in the JOURNAL OF ENGI-NEERING MATERIALS AND TECHNOLOGY. Manuscript received by the MaterialsDivision March 24, 1999; revised manuscript received November 4, 1999. AssociateTechnical Editor: E. Busso.


linked by linear unloading or linear reloading, as marked in Fig. 1.The shaded area represents the energy dissipated during normalinterface separation. Oncelmax reaches unity, the interface isseparated freely except blocked by contact, then Eqs.~1! and ~2!are replaced by:

Tn5KnDnH~2Dn! (4)

Tt5mKnDn sgn~D t!H~2Dn! (5)

The scalarm denotes the friction coefficient. If a fully detachedinterface is recompressed, the interface will cease to slide whenuTtu<muTnu.

The maximum strengths that a cohesive interface can sustain inpure normal and tangent directions can be derived as

sn54

27Endn , s t5

4

27Etd t (6)

One can use the interface strengthssn and s t , instead of theinterface moduli En and Et , as the independent interfaceparameters.

Nucleation, growth and coalescence of voids characterize thematrix damage. The matrix damage is treated in a distributedmanner in the present work. Within metallic matrix, the void evo-lution is described by Gurson-Tvergaard model~Gurson @11#,Tvergaard@12,13#, Needleman and Tvergaard@14#!. The onlydamage indicator is the volume fraction of voids. This averagingtreatment of damage makes sense if the initial void size is muchsmaller than the fiber diameter~in the order of one micron!. Oneshould bear in mind that when coupling with the finite elementmethod, the Gurson-Tvergaard model can also handle large, dis-crete cavities formed due to void coalescence. Void nucleationand growth in the matrix of MMC result in macroscopic dilatancy,which should be incorporated into the constitutive relation of ma-trix materials. The yield condition of the Gurson-Tvergaard modelis expressed as

F5seq

2

sM2 12 f * q1 coshS q2skk

2sMD2~11~q3f * !2!50 (7)

f * 5H f if f < f C

f C1~q12Aq1

22q32!/q3

22 f C

f F2 f C~ f 2 f C! if f . f C

(8)

wheres i j is the macroscopic stress tensor,seq the macroscopicequivalent stress, andsM the actual yield stress of the matrixmaterial. Symbolsf and f * represent the actual and the effectivevoid volume fractions,f C the critical volume fraction for acceler-

ated void growth, andf F the void volume fraction to lose thestress carrying capacity. Numbersq1 , q2 , andq3 are the adjust-able parameters introduced by Tvergaard@12#. The ratef consistsof the contributions from the growth of the existing voids and thenucleation of the new ones. That is

f 5~12 f !«kkp 1A«M

p (9)

where«kkp is the plastic part of the macroscopic volume deforma-

tion rate and«Mp the equivalent plastic strain rate of the matrix

material. The latter is obtained by equating the macroscopic andthe microscopic plastic powers:

«Mp 5

s i j « i jp

~12 f !sM(10)

The first term in Eq.~9! is derived from the condition of plasticincompressibility of the matrix material. For the strain-controllednucleation, the second term in Eq.~9! takes the Gaussian distri-bution form proposed by Chu and Needleman@15#

A5f N

SNhA2pexpF2

1

2 S «Mp 2«N

SND 2G (11)

where f N is the volume fraction of void nucleating particles,«Nthe profusion strain for nucleation,SN the standard deviation ofthe distribution, andh the matrix hardening modulus. The plasticflow within the matrix is modeled by an isotropic hardening law:

sM5h~«Mp !5s0S 11

EM«Mp

s0D N

(12)

with the strain-hardening exponentN, the Young’s modulusEMand the initial yield stresss0 . The ductile matrix may withstandlarge plastic flow before failure, so a finite deformation frame-work is adopted~Simo and Ortiz@16#, Moran et al.@17#!.

3 Computational CellA periodic distribution of fibers is assumed for simplicity. A

longitudinal section is shown in Fig. 2~a!, where short fibers of

Fig. 1 Cohesive force curve during normal separation

Fig. 2 Aligned short fibers arranged in a transverse hexagonarray. „a… A longitudinal section shows the fiber alignments; „b…a transverse section shows the hexagon array; „c… finite ele-ment mesh with indication of boundary conditions.


radiusr f and half-lengthl f are aligned in a rectangular array of ahorizontal spacing 2l c . In the transverse section, the fiber centersare arranged in a hexagonal array as shown in Fig. 2~b!, with acenter-to-center spacingas . This configuration is further simpli-fied by the axisymmetric cells, namely, to replace the hexagons bycircles of the same area as shaded in Fig. 2. The radius of thecircle in nondeformed state isr c5(31/4/A2p)as . The volumefraction of fibers is

Vf5r f2l f /r c

2l c (13)

The aspect ratios of the fibers and the initial cells are

b f5 l f /r f , bc5 l c /r c (14)

Subjected to macroscopically uniform deformation, the periodic-ity dictates that each cell remains cylindrical though its aspectratio varies with the deformation. Consequently, the local defor-mation and the damage are also periodic. In a finite deformationstate, the present cell model cannot be used to the place where themacroscopic strain localizes.

4 Material CharacterizationWe assume the MMC is stress free at 525°C, the annealing

temperature of the matrix material. It first undergoes a coolingphase, from 525°C to the room temperature~25°C!. Then it issubjected to a specified thermal cycling phase that terminates atthe room temperature. After cooling and thermal cycling, the me-chanical behavior of the MMC is evaluated by a tensile test.

The simulation is conducted for SiC whisker-2124 aluminumalloy, whose material constants at the room temperature are sum-marized by Tvergaard@18#. The SiC whisker has a Poisson’s ration f of 0.2 and a coefficient of thermal expansion~CTE! a f of4.931026/°C. The whisker has a diameter of 0.5mm and anaverage length of about 2.5mm, resulted from the breakage in theextruding process. Accordingly, the fiber has an aspect ratio ofb f55 ~Tvergaard@18#!. The matrix of 2124 aluminum alloy has aEm /s0 ~Young’s modulus to the initial yield stress! ratio of 200,a Poisson’s rationm of 0.30, a work-hardening exponentN of 0.13and a CTEam of 23.231026/°C. The Young’s modulus of thefiber is 5.7 times of that of the matrix.

A value of 20 percent is selected for the volume fraction offibers. A cell aspect ratio ofbc54.3 is chosen to give uniformlongitudinal and transverse spacings among fibers. The finite ele-ment mesh and the boundary conditions are depicted in Fig. 2~c!.The periodicity and the predominately tensile loading leave thegeometry of the computation cell undistorted. The horizontal cellboundary remains horizontal and the vertical boundary remainsvertical during the deformation. The elements within the fiber arehighlighted by gray color. Between the fiber and the matrix, alayer of interface elements~with zero initial thickness! is inserted.

Determination of the parameters in Eqs.~7!, ~8!, and ~11! re-quires detailed experiments on the matrix material. Several mea-surable material parameters~such as the yield strength, the ulti-mate strength, the ductility and the void volume fraction atfailure! of the Chinese equivalent of 2124 aluminum alloy wassummarized by Du et al.@19#. Specifically, thef F value is re-ported to be 0.20. From those experimental data, and the numeri-cal simulation for the uniaxial tensile response of a bar made of2124 aluminum alloy, Zhou et al.@20# correlated the followingmatrix damage parameters: the Tvergaard’s parameters for modi-fied Gurson yield surface are found to beq15q351.25 andq251.0; the void nucleation parameters are found to bef N50.04,SN50.1 and«N50.3, identical to those used by Biner@21# forMMC; and the critical void volume fraction for accelerated voidgrowth is found to bef C50.15.

Following Tvergaard@3,18#, the interface parameters are cho-sen asdn5d t[d50.02r f , sn5s t52.5s0 andm50.3.

5 Results and DiscussionWe now present the simulation results on the computational

cell made of SiC whisker-2124 aluminum alloy. When cooledfrom the annealing temperature~525°C! to the room temperature~25°C!, residual stresses develop by the CTE mismatch betweenthe fiber and the matrix. The residual shear stress concentratesnear the corner of the fiber end and induces the initial damagealong the interface and within the matrix. The fiber/matrix inter-face is compressed at this stage. We next consider a thermal cy-cling about the room temperature. The MMC is first cooled fromthe room temperature to a cryogenic temperature. The residualstress and the damage grow as the temperature declines. Then theprocedure is reversed, the MMC is heated from the cryogenictemperature, through the room temperature, to an elevated tem-perature. Tensile stress develops along the fiber/matrix interface.The residual stresses in matrix relaxes first and then changes toopposite signs. Figure 3 shows the most severe thermal stressdistributions during a thermal cycling of6200°C. Graphs~a!, ~b!,and ~c! plot the contours of radial, longitudinal, and shear stresscomponents. The results indicate a considerable axial tensilestress ahead of the fiber end, and a large shear stress around thecorner of the fiber. The effective stress near the fiber corner sur-passes the yield stress, and promotes void nucleation and growthin the matrix surrounding the fiber corner. Graph~d! delineatesthe distribution oflmax along the interface. The suppression ofmatrix damage would raise the interface damage slightly, asshown in Graph~e!. Graphs~f ! and ~g! plot the contours of thematrix damage with and without the consideration of the interfacedamage. The occurrence of interface damage slightly reduces thelevel of matrix damage. Both the interface and the matrix dam-ages concentrate near the fiber end, especially near the fiber cor-ner. The highest value of interface damage is 0.178 against thevalue of lmax51 for the interface debonding; while the highestvalue of matrix damage is 0.00105 against the value off F50.20for the matrix failure. Accordingly, the matrix damage is smallcompared with the interface damage and consequently secondaryfor the subsequent failure of MMC.

Figure 4 plots the curves of the average tensile stresssx versusthe average logarithmic tensile strain«x under different historiesof thermal cycling. CurveA refers to the case without cooling andthermal cycling, and consequently free of the initial damage. Theabsence of initial damage suppresses the interface damage, anddebonding never occurs when the tensile loading is applied to thecomputation cell. The stress strain curve attains the highest ulti-mate strength and the gradual declination of tensile stress iscaused by the matrix damage. The MMC exhibits good strengthand good toughness~measured by the energy dissipation in thefailure process! in this case. CurvesB, C, andD in Fig. 4 are allpreceded by a cooling process from 525°C to 25°C. Interfacedamage takes place during cooling, and leads to debonding in thesubsequent tensile loading. As the result, the average tensile curvewill suffer a sudden drop in the average tensile stress. Since thefirst stress drop is triggered by debonding along the fiber end, weterm the average tensile stress and strain corresponding to thisevent the debonding stresssd and the debonding strain«d . Afterthe first drop and a plateau of sustained strength, the second dropin the stress strain curve appears by damage coalescence in thematrix, which destroys the loading capacity of the composite.

The effect of thermal cycling further reduces the strength andthe toughness of MMC. CurveB refers to the case without thermalcycling, while curvesC and D correspond to thermal cyclingswith ranges of6100°C and6200°C, respectively. Figure 4 indi-cates that thermal cycling reduces the ultimate strength of MMC,as well as the average tensile strain at the first stress drop. Thewider the thermal cycling range, the lower the ultimate strengthand the debonding strain. Figure 5 plots thesx versus«x curvesafter zero, one, two and twenty thermal cycles of6200°C. Thesimulations indicate that the effect of thermal cycling stabilizesafter two complete temperature cycles. Fatigue effects are not


included in this research since neither the cohesive interfacemodel nor the Gurson-Tvergaard matrix damage model accommo-dates fatigue damage. In the present formulation, phenomena suchas interface separation, matrix plastic deformation, residual stressand residual strain are dominated by the thermal cycling rangerather than the number of thermal cycles. Under a given thermalcycling range, interface separation and matrix plastic deformationshake down after about two thermal cycles.

The qualitative changes in the uniaxial tension curve of MMCby cooling and thermal cycling become transparent when theirrespective failure modes are examined. Figure 6 delineates thematrix damage distribution at the failure stage when the compu-tation cell undergoes different thermal processes. If the effects ofcooling and thermal cycling were ignored, the failure mode would

assemble the damage pattern in Fig. 6~a!. It is featured by matrixdamage that circumvents a cap ahead of the fiber end, without anytrace of interface debonding. The incorporation of a cooling phasealters the failure mode, as shown in Fig. 6~b!. The fiber end deb-onds completely, and the matrix damage progresses across theligament to form a flat failure surface. Thermal cycling of a suf-ficient temperature range further twists the failure mode, as shownin Fig. 6~c!. Debonding occurs not only along the fiber end, butalso along the side surface of the fiber. The cell eventually fails bynecking instability of the matrix ligaments, leading to a jaggedfractography.

The effect of thermal cycling can be measured by a dimension-less ratioj[d/L f(am2a f)DT, where the numerator representsthe largest separation the interface can sustain without debonding,

Fig. 3 Stress and damage fields during two thermal cycles of Á200°C. „a… Maximum radial stress distribution inmatrix; „b… maximum longitudinal stress distribution; „c… maximum shear stress distribution; „d… distribution of theinterface damage lmax with matrix damage; „e… distribution of lmax without matrix damage; „f … distribution of thematrix damage f with interface damage; „g… distribution of f without interface damage.


and the denominator measures the mismatch at the fiber/matrixinterface due to thermal cycling of a temperature rangeDT. Thecharacteristic lengthL f should be assigned asl f for debondingalong the fiber end, andr f for debonding along the fiber side.When j is large, the effect of thermal cycling is insignificant;when j is small, the effect of thermal cycling may change thefailure mode. In the present simulations,d/r f50.02,d/ l f50.004andam2a f518.331026/°C. A thermal cycling of6100°C canaffect the debonding at the fiber end (j51.09), but is insufficientto alter the debonding status along the fiber side (j55.46). Athermal cycling of6200°C, on the other hand, affects not only thedebonding status at the fiber end (j50.55), but also the debond-ing status along the fiber side (j52.73).

The dimensionless groupj depends critically on the relativeseparationd/r f that governs the debonding along the side surfaceof fibers. For a given MMC after a prescribed manufacturing pro-cess~namely, a fixed history of cooling and thermal cycling!, theratio d/r f becomes the critical parameter. Since the value ofd canbe altered by interface coating, the ratiod/r f is an adjustableparameter. The influences ofd/r f on sd and «d are plotted inFigs. 7~a! and 7~b!, respectively. From Fig. 7~a!, one observesthat sd declines asd decreases. The declination becomes faster

Fig. 4 Average tensile stress sx versus average logarithmicstrain «x curves „f fÄ20 percent, snÄs tÄ2.5s0 , b fÄ5 and bRÄ4.3… after various thermal histories. A, without cooling andthermal cycling; B, with cooling from 525°C to 25°C but notthermal cycling; C, cooling followed by thermal cycling ofÁ100°C; D, cooling followed by thermal cycling of Á200°C.

Fig. 5 Average tensile stress sx versus average logarithmicstrain «x curves after zero, one, two, and twenty thermal cycles

Fig. 6 The interface and matrix damage distributions at thefailure stage after „a… thermal history A, „b… thermal history B ,„c… thermal history D, preceding the uniaxial tension at theroom temperature.

Fig. 7 Average debonding tensile stress and strain versus therelative interface thickness dÕr f curves under thermal historiesB ,C, and D. „a… sd Õs0 versus dÕr f curves; „b… «d versus dÕr fcurves.


under larger thermal cycling, as predicted by the form of the di-mensionless ratioj. The same trend is observed in Fig. 7~b! forthe debonding strain. Under a large thermal cycling range, Figs.7~a! and 7~b! indicate that the ratiod/r f has large influence on thedebonding stress and strain.

6 ConclusionsIncorporating interface and matrix damage, the present analysis

reveals the damage evolution in cooling, thermal cycling, and thesubsequent tension of MMC. During cooling, the CTE mismatchplays a central role in provoking a high residual stress distribution~tensile in front of the fiber end, and shear around the fiber cor-ner!. Interface damage occurs along the fiber end and matrix dam-age initiates near the fiber corner.

The initial damage results in substantial stress drop~nearly ahalf of the ultimate strength! in the overall flow curve of thecomputation cell when subjected to mechanical loading at theroom temperature. The drop is caused by the interface debondingalong the fiber ends. The stress drop reduces the toughness of theMMC. The ultimate strength corresponds to the debonding stress,the critical strain for the stress drop corresponds to the debondingstrain. Both debonding stress and debonding strain decrease as therange of thermal cycling increases, and increase as the relativeinterface thicknessd/r f , adjustable by interface coating,increases.

Cooling and thermal cycling change the failure modes ofMMC. Under uniaxial tension, the SiC whisker-2124 aluminumalloy fails by matrix damage in front of the fiber end when coolingand thermal cycling are absent. With cooling but not thermal cy-cling, it fails by debonding along the fiber ends and transversedamage across the matrix. With the further participation of ther-mal cycling, the MMC fails by debonding along the fiber ends andfiber sides, followed by necking instability of the matrix liga-ments. Though this conclusion is specified to the MMC of SiCwhisker-2124 aluminum alloy, and the results might change if thedamage parameters of matrix material are altered, we still expectthe occurrence of three failure modes mentioned above in otherMMC when cooling and thermal cycling are involved.

The effect of thermal cycling can be measured by a dimension-less ratio ofd/L f(am2a f)DT. The numerator indicates the larg-est separation the interface can sustain without debonding, and thedenominator measures the mismatch due to a thermal cycling oftemperature rangeDT.

References@1# Christman, T., Needleman, A., and Suresh, S., 1989, ‘‘An Experimental and

Numerical Study of Deformation in Metal-Ceramic Composites,’’ Acta Met-all., 37, pp. 3029–3050.

@2# Levy, A., and Papazian, J. M., 1990, ‘‘Tensile Properties of Short Fiber-Reinforced SiC/Al Composites: Finite Element Analysis,’’ Metall. Trans. A,2,pp. 411–420.

@3# Tvergaard, V., 1990, ‘‘Effect of Fiber Debonding in a Whisker-ReinforcedMetal,’’ Mater. Sci. Eng., A,125, pp. 202–213.

@4# Levy, A., and Papazian, J. M., 1991, ‘‘Elastoplastic Finite Element Analysis ofShort-Fiber-Reinforced SiC/Al Composites: Effect of Thermal Treatment,’’Acta Metall. Mater.,39, pp. 2255–2266.

@5# Davis, L. C., and Allison, J. E., 1993, ‘‘Residual Stresses and Their Effects onDeformation in Particle-Reinforced Metal-Matrix Composites,’’ Metall. Trans.A, 24, pp. 2487–2496.

@6# Levy, A., and Papazian, J. M., 1993, ‘‘Finite Element Analysis of Whisker-Reinforced SiC/Al Composites Subjected to Cryogenic Temperature ThermalCycling,’’ ASME J. Eng. Mater. Technol.,115, pp. 129–133.

@7# Yang, W., and Shih, C. F., 1994, ‘‘Fracture along an Interlayer,’’ Int. J. SolidsStruct.,31, pp. 985–1002.

@8# Needleman, A., 1987, ‘‘A Continuum Model for Void Nucleation by InclusionDebonding,’’ ASME J. Appl. Mech.,54, pp. 525–531.

@9# Tvergaard, V., 1990, ‘‘Analysis of Tensile Properties for a Whisker-Reinforced Metal Matrix Composite,’’ Acta Metall. Mater.,38, pp. 185–194.

@10# Chaboche, J., Girard, R., and Levasseur, P., 1997, ‘‘On the Interface Debond-ing Models,’’ Int. J. Damage Mech.,6, pp. 220–257.

@11# Gurson, A., 1977, ‘‘Continuum Theory of Ductile Rupture by Void Nucleationand Growth: Part I-Yield Criteria and Flow Rules for Porous Ductile Media,’’ASME J. Eng. Mater. Technol.,99, pp. 2–15.

@12# Tvergaard, V., 1982, ‘‘Ductile Fracture Cavity Nucleation Between LargeVoids,’’ J. Mech. Phys. Solids,30, pp. 265–286.

@13# Tvergaard, V., and Needleman, A., 1984, ‘‘Analysis of the Cup-cone Fracturein a Round Tensile Bar,’’ Acta Metall.,32, pp. 157–169.

@14# Needleman, A., and Tvergaard, V., 1984, ‘‘An Analysis of Ductile Rupture inNotched Bars,’’ J. Mech. Phys. Solids,32, No. 6, pp. 461–490.

@15# Chu, C., and Needleman, A., 1980, ‘‘Void Nucleation Effects in BiaxiallyStretched Sheets,’’ ASME J. Eng. Mater. Technol.,102, pp. 249–256.

@16# Simo, J., and Ortiz, M., 1985, ‘‘A Unified Approach to Finite DeformationPlasticity Based on the Use of Hyperelastic Constitutive Equations,’’ Comput.Methods Appl. Mech. Eng.,49, pp. 221–245.

@17# Moran, B., Ortiz, M., and Shih, C. F., 1990, ‘‘Formulation of Implicit FiniteElement Methods for Multiplicative Finite Deformation Plasticity,’’ Comput.Methods Appl. Mech. Eng.,29, pp. 483–514.

@18# Tvergaard, V., 1993, ‘‘Model Studies of Fiber Breakage and Debonding in aMetal Reinforced by Short Fibers,’’ J. Mech. Phys. Solids,41, pp. 1309–1326.

@19# Du, Z. Y., Liu, J. Y., Zhang, S. L., and Zhu, X. W., eds., 1995, ‘‘A ConciseHandbook of Engineering Materials,’’ Electronical Industry Press~in Chi-nese!.

@20# Zhou, C., Yang, W., and Fang, D., 1999, ‘‘Strength and Damage of MetalMatrix Composites,’’ Acta Mech. Solida Sinica,31, pp. 372–377~in Chinese!.

@21# Biner, S. B., 1994, ‘‘The Role of Interfaces and Matrix Void NucleationMechanism on the Ductile Fracture Process of Discontinuous Fiber-ReinforcedComposite,’’ J. Mater. Sci.,29, pp. 2893–2902.


Robert G. TryonThomas A. Cruse

Fellow ASMEBrentwood Technologies, Inc.,

Brentwood, TN 37027

Probabilistic Mesomechanics forHigh Cycle Fatigue Life PredictionThis paper presents an analytical modeling approach to characterize and understandhigh cycle fatigue life in gas turbine alloys. It is recognized that the design of structuressubjected to fatigue cannot be based on average material behavior but that designs mustconsider23s or some other appropriate extreme value (tail of the distribution) loadingand/or material properties. Thus, a life prediction capability useful in a design applica-tion must address the scatter inherent in material response to fatigue loading. Further,the life prediction capability should identify the key micromechanical variables that arecritical in the tail of the materials durability distribution. The proposed method addressesthe scatter in fatigue by investigating the microstructural variables responsible for thescatter and developing analytical and semi-analytical models to quantitatively relate thevariables to the response. The model is general and considers the entire range of damageaccumulation sequences; from crack nucleation of the initially unflawed structure to finalfast fracture.@S0094-4289~00!01302-5#

Introduction

Many material and structural design factors influence compo-nent reliability in terms of the defined durability problems. From amaterial performance standpoint, many of these factors are atwork in the durability ‘‘size effect.’’ Two important aspects of thesize effect influence high cycle fatigue~HCF! in mechanical com-ponents: the relative size of the stressed area compared to the sizeof the component and the relative size of the damage~crack!compared to the size of the microstructure.

The size effect was first reported by Peterson@1# when he no-ticed that the mean fatigue life and variation in fatigue life were afunction of the stressed volume. The size effect must be carefullyconsidered in regards to HCF of such components as aeroengineairfoils. The stresses that cause HCF are often mode shapes bend-ing stresses induced by vibratory excitation. Only a very smallportion of the total airfoil area is subjected to the high stresses.

The size effect has another fundamental role in controlling HCFbecause damage accumulation often starts on a small scale. HCFfailures are not usually initiated by the large microstructural de-fects associated with low cycle fatigue failures but often nucleate‘‘naturally’’ at local regions of high stress. The local regions ofhigh stress may be caused from vibratory resonance or foreignobject damage. The damage grows through various mechanisms,including crack nucleation, microstructurally small crack growth,and linear elastic long crack growth. Each mechanism is associ-ated with a characteristic size and each characteristic size has itsown geometric complexities, constitutive laws and heterogene-ities. Fatigue behavior cannot be fully understood and predictedwithout obtaining information about each of the characteristicsizes, or what can be called mesodomains. Nested models can linkeach of the mesodomains to determine the response of the mac-rodomain.

The overall fatigue response of a component is predicted bynesting the individual mesoscale models. The lowest level modeluses the appropriate mesoscale parameters to determine the initialstate of the next level. This level uses the results from the previ-ous level along with the appropriate parameters to determine theinitial state of the next level and so on. Using nested models, fleetreliability can be linked to the heterogeneities at each meso-

domain. Additionally, by modeling each level of the fatigue pro-cess individually, and rigorously linking the levels, various sizeeffects are included.

Three Level Fatigue ModelFigure 1 shows the three levels of damage accumulation that

are assumed in the present study. First, the crack nucleates on asmall scale on the order of the grain size. Then the crack grows asa microstructurally small crack in which the crack front lies inrelatively few grains. The material properties, averaged along thecrack front, approach bulk material properties as the crack growand the number of grains interrogated by the crack front increase.At this point, linear elasticity can be assumed and the crack growsas a typical long crack until final failure.

The models used to predict the behavior for each of the threelevels of damage accumulation have been discussed elsewhere@2#and only those aspect of the models related to HCF of gas turbinealloys will be discussed.

Crack Nucleation Model. Models used in the research musthave two attributes. They must be quantitative with regards to thenumber of cycles needed to produce a crack to a specific size ifthey are to be used for lifetime predictions. The models must alsobe able to address the microstructural parameters in order to pro-vide a physical link between the microstructure and the fatiguebehavior. The crack nucleation model used in the current researchaddresses slip band cracking within a grain that is a preferredmode of damage accumulation for HCF in gas turbine alloys@3#.The model used in this effort is based on a model proposed byTanaka and Mura@4# and extended to account for grain orienta-tion by Tryon and Cruse@3# as

Contributed by the Materials Division for publication in the JOURNAL OF ENGI-NEERING MATERIALS AND TECHNOLOGY. Manuscript received by the MaterialsDivision June 23, 1998; revised manuscript received November 1, 1999. AssociateTechnical Editor: H. Sehitoglu. Fig. 1 Three-stage mesomechanical fatigue model


Nn54GWs

S 1

MSDs22kD 2

p~12n!d

(1)

whereNn is the number of cycles needed to grow a crack to thesize of the grain,G is the shear modulus,Ws is the specific frac-ture energy per unit area,s is the local applied normal stress,MSis grain orientation factor~reciprocal Schmid factor!, k is the fric-tional stress which must be overcome to move dislocations,y isPoissons ratio, andd is the grain diameter.

Small Crack Growth Model. The behavior of small cracksdiffers from the behavior of long cracks. Long crack behavior canbe predicted using conventional continuum based LEFM tech-niques. Small crack growth rates vary widely, from several ordersof magnitude greater than that predicted by continuum basedDKto complete arrest. A small crack can be thought of as a crack witha size on the order of the microstructure. The anomalous growthof small cracks has been attributed to two competing factors: highgrowth rates due to lack of closure and growth retardation due tomicrostructural obstacles.

Crack Tip Opening Displacement. The experimentallyobservable parameter that has been correlated to the varyingsmall crack growth rate is the crack tip opening displacement~CTOD! @5#

da

dN5C8Df t (2)

wherea is the crack length,N is cycles,f t is the CTOD, andC8is a material constants derived from test data. The CTOD is ameasure of the amount of damage associated with the crack tip.The larger the CTOD, the higher the crack growth rate. This phe-nomenon was first observed by Laird and Smith@6# and has beenwell established in long crack growth behavior@7#. The directproportionality of Eq.~2! has been observed in small crack growthof aluminum, nickel and titanium alloys@8#. Nisitani and Takao@9# showed that small crack arrest could be associated with zeroCTOD.

In the current research, the CTOD is modeled as a function ofthe random microstructural variables based on the approach usedby Tanaka et al.@10# and extended by Tryon@2#. Consider a crackof lengtha with the crack tip in thej th grain as shown in Fig. 2.The slip band has a length ofw with the slip band tip in thenthgrain. The total length of the damage,c, is the crack length plusthe slip band length. If the slip band is propagating~not blockedby the grain boundary!, the size of the slip band zone can be foundfrom

05pt j

22kj arcsin

a

c

2 (i 5 j 11

n

@~t i 212ki 21!2~t i2ki !#arcsinS Li 21

c D (3)

where,t i is the applied resolved shear stress in thei th grain,ki isthe frictional stress of thei th grain,a is the crack length,c is thecrack length plus slip band length,Li is the distance from the freesurface to grain boundary of thei th grain preceding the slip bandtip as shown in Fig. 2.

The CTOD is given by

f t52kja

p2Aln

c

a1 (

i 5 j 11

n~t i 212ki 21!2~t i2ki !

p2Ag~a;c,Li 21!

g~a;c,L !5L lnUAc22L21Ac22a2

Ac22L22Ac22a2U2a lnUaAc22L21LAc22a2

aAc22L22LAc22a2U (4)

A5G/2p~12n! for edge dislocations

A5G/2p for screw dislocations

For the slip band blocked by the grain boundary, the size of theslip band zone is

v5Ln2a (5)

The CTOD is given by

f t5bt

pAAc22a21

2kja

p2Aln

c

a

1 (i 5 j 11

n~t i 212ki 21!2~t i2ki !

p2Ag~a;c,Li 21!

b5122kj

pt jarccos

a

c

2 (i 5 j 11

n2@~t i 212ki 21!2~t i2ki !#

pt jarccosS Li 21

c D (6)

The microscopic stress intensity factor at the slip band tip is

Km5btApc (7)

To account for the crystallographic orientation of the individualgrains, the applied resolved shear stresst in Eqs.~3! through~7!can be replaced with

t5s

MSfor surface grains

t5s

MTfor interior grains

wheres is the local normal applied stress,MS is the reciprocalSchmid factor andMT is the Taylor factor@12#.

Modeling the Physical MicrostructureConsider a random array of grains as shown in Fig. 3. A crack

nucleates in the surface grainX0 and then grows along thex axisas a semi-circle through zones in which the effective materialproperties are uniform. The boundaries of the zones are repre-sented by the concentric half circles. The zones are composed ofgrains represented by the semi-circular arc segments. The arclength of the semi-circular segments is a random variable equal tothe grain diameter. The surface grains are represented by the in-Fig. 2 Crack tip slip band in multiple grains


tersection of the zones and the surface. Note that because thecrack grows as a semi-circle, the surface crack growth is the sameas the in-depth crack growth through the zones represented bySection A-A.

After successful crack nucleation, the crack grows from grainX0 into zone 1. In the example shown in Fig. 3, zone 1 containsthree grains. The effective material propertyP1 eff , of zone 1 isthe average of the properties of the individual grains.

P1i weighted with the area of thei th grain.~In the current studyP1 eff represents the local frictional strengthk or the local appliedstresst.!

P1 eff5P11d1

21P12d122 1P13d13

2

d121d12

2 1d132 (8)

In thenth zone composed ofj grains, the effective material prop-erty is

Pn eff5( i 51

j Pnidni2

( i 51j dni

2 (9)

As the crack becomes long,Pn eff approaches the bulk properties.Using the concepts of effective material properties, crack

growth is modeled as one-dimensional. Consider a cut along thexaxis ~Section A-A in Fig. 3!. The fatigue damage is modeled as aone-dimensional crack growing through zones of varying sizel nand varying effective material propertiesPn eff .

The above microstructural modeling technique is approximateand does not capture some of the nuances of crack/microstructureinteractions. In particular, the model does not allow for spatialvariation of properties along the crack front that can cause a non-smooth or ragged crack front shape. If the crack front encountersstrong grains~due to unfavorable orientation or high frictionalstress! in a matrix of weaker grains, the crack front will retard inthe region near the strong grains and tunnel in the region of theweak grains. However, crack growth mechanisms tends to have asmoothing effect on the crack front shape. The crack front willtunnel around the blockage until the shape of the crack front at theblockage is such that the stress intensity overcomes the blockageand the crack front resumes its smooth shape@11#. If the blockageis not overcome, the crack front will not continue to tunnel. Thecrack growth will arrest.

Long Crack Growth Model. The long crack growth is mod-eled using the Paris law representation of a surface crack in asemi-infinite body subjected to a constant stress cycle. If the finalcrack size is much greater than the initial crack size, Tryon andCruse@12# showed that

Ng5ai

12n/2

CDsnbnS n

221D (10)

where,Ng is the number of cycles needed for the crack to grow tofailure, ai is the initial crack size at the start of the long crackgrowth phase,Ds is the global stress range,b is the geometryconstant~1.12p!, andC andn are based on material properties.

Monte Carlo Simulation ModelThe statistical characteristics of variables used in the Monte

Carlo simulation have been discussed in detail in Tryon and Cruse@13#. The random variables and the associated statistical distribu-tions are shown in Table 1. Normalized distributions are used ford, a, k, andC. This allows the average values to be easily changedwithout having to re-evaluate the distribution parameters. The dis-tribution parameters only need to be re-evaluated if a change incoefficient of variation~COV! is desired. The orientation factorsare not normalized. A change in the average value of the orienta-tion factor would require texturing the microstructure. The values

Fig. 3 Array of random grains

Table 1 Values used in the Monte Carlo simulation

Variable DescriptionDistribution

typeDistributionParameters Average COV

C Paris Law Coefficient Lognormal l50.034 z50.30 4.431029 MPaAm 0.30C8 CTOD Law Coefficient Deterministic N/A 0.10 N/Ad Grain diameter Lognormal l520.076 z50.39 55.8mm 0.40da Small Crack Growth Interval Deterministic N/A 0.5 N/AG Bulk Shear Modulus Deterministic N/A 7631023 MPa N/Ak Frictional Strength Weibull h51.12 z53.7 69 MPa 0.30

KcritM Critical Microstructural Stress

Intensity FactorDeterministic N/A 769 MPaAm N/A

MS Schmid Orientation Factor Curve Fit~See Ref.@12#! 2.21 0.08MT Taylor Orientation Factor Curve Fit~See Ref.@12#! 3.07 0.13n Paris Law Exponent Deterministic N/A 3 N/A

WS Specific Fracture Energy Deterministic N/A 440 kN/m N/An Poisson’s Ratio Deterministic N/A 0.3 N/As Applied Micro-stress Normal m51 s50.3 Variable* 0.30

*Note: Several different stress levels are modeled as discussed in the Section on Model Results.


in Table 1 would no longer be valid and new representationswould be required. The deterministic input variables are alsoshown in Table 1.

The basic flow of the Monte Carlo simulation is outlined asfollows. A crack is nucleated in each surface grain of a compo-nent. Each crack goes through the small crack growth phase andlong crack growth phase. The total life associated with each crackis the summation of the cycles in the crack nucleation, small crackgrowth and long crack growth phases. The life of the componentis equal to the minimum total life of all of the cracks. The detailsof the simulation are described in Tryon@2#.

Model ResultsThe predictions of the individual crack nucleation, small crack

growth, and long crack growth simulations have been shown tocorrelate very favorably with experimental observations and arediscussed elsewhere@3,12#. In this section, we will discuss theprediction of the total fatigue life for a simple test specimen.

Predicted Total Fatigue Life of a Test Specimen. It is dif-ficult to compare the probabilistic model predictions for total fa-tigue life directly with experimental data because the parametersused in the model are usually not reported. However, the predictedscatter in fatigue data is compared with trends in the experimentaldata and the predicted mean life for different size specimens iscompared with size effect observations.

The distribution of fatigue life for test specimens was predictedby assuming the parameters in Table 1 in the Monte Carlo analy-sis. These values are characteristic of a stainless steel. The speci-men has a circular cross section with radius 7.62 mm, and a shal-low notch with a gauge surface area of 1.61 mm2. This gaugesurface area results in about 4000 grains per specimen. Differentspecimens will have a different number of surface grains andtherefore the number of surface grains is a random variable. Thepredicted CDF of fatigue life for the specimens is shown in Fig. 4.The mean life of the specimens is 60,000 cycles with a COV of0.17.

Figure 4 shows that fitting the model results to a lognormaldistribution give a correlation coefficient of 0.993. Fitting themodel results to a normal distribution~not shown! gives a corre-lation coefficient of 0.999. Both the normal and lognormal distri-butions provide an adequate representation of the model resultsand both have been used to represent experimental data@14,15#, p.380#.

A thorough investigation of the scatter in fatigue life is notavailable in the literature for most alloys. Many manufacturers,particularly in the aerospace industry, have the large compilation

of data used for statistical characterization. However, the cost as-sociated with such test is considerable and the data is tightly held.However, Bastenaire@14# performed a thorough investigation ofthe scatter in fatigue life for five different grades of low alloysteel.

Steels may nucleate cracks by mechanisms other than slip bandcracking depending on the alloy composition and the impurities.However, the trend in the scatter in steel data has been observedin other metallic alloys@16#. Bastenaire performed rotating bend-ing fatigue experiments for many stress levels for each grade ofsteel with several hundred specimens for each stress level.

Figure 6 shows the trends in the scatter exhibited in Bas-tenaire’s data plotted on lognormal paper.~The curves are replot-ted from the data in Fig. 7 of@14#.! If data plots as a straight linein Fig. 6, the lognormal distribution is valid. If data plots as anonstraight line in Fig. 6, the lognormal distribution is no longervalid. The general trend is that the COV~indicated by the slope ofthe curves! is fairy constant for applied stresses well above thefatigue limit ~363–324 MPa!. As the applied stress decreases, theCOV starts to increase~304–285 MPa!. As the applied stressapproaches the fatigue limit, the fatigue life increases and run-outsstart to occur. The right tail of the distribution becomes moreheavily populated than a lognormal distribution, which causes aline through the data to bend to the right~265–245 MPa!. The 363MPa data curves slightly to the left indicating the right tail ofdistribution is less populated than a lognormal distribution and thedata can also be fitted to the normal distribution. As the appliedstress decreases, the curvature shifts to the right.

Comparison of Fig. 5 with the results in Fig. 6 shows that themodel predicts all of the above trends observed in the experimen-tal data. Figure 7 presents the same data in the familiar form of aSN diagram. The runouts~suspensions! are the percentage ofspecimens that did not fail at 106 cycles.

The Monte Carlo simulation showed that most of the failureswere caused by the largest grain in the specimen and almost allthe failures were initiated in one of the five largest grains. Thelower the stress the more failures initiated in the largest grain.This indicates that the ‘‘weak links’’ in crack nucleation are thelargest grains. Experimental evidence shows that failures can beassociated with the largest grains@11#.

The distribution of the largest defects~or the largest grains inthe present model! lead to the size effect model developed byWeibull @17#. Size effect is the phenomenon that small compo-nents have a higher fatigue life than larger geometrically similarcomponents. Weibull assumed that the larger component is more

Fig. 4 Fatigue life distribution of the specimens plotted onlognormal probability paper

Fig. 5 Predicted fatigue life distribution plotted on lognormalpaper


likely to have a larger life-controlling defect. This approach iswidely used in the design of ceramics and it has also been appliedto ductile materials@18#.

The reliability of different size~defined by the mean number ofsurface grains! specimens was determined and the mean fatiguestrength at an arbitrary life is plotted against size in Fig. 8. Themodel indicates that very large structures have zero life. This isbecause a lognormal distribution of grains allows an infinitelylarge grain in an infinite population. In reality, the grain size can-not be infinite and the true distribution of grain size is truncated ata size no larger than the component. Maximum truncated grainsize would control the fatigue life of a very large structure.

The predicted size effect on fatigue strength is linear in logspace as shown in Fig. 8. Trantina@18# predicted the same rela-tionship using a weakest link theory. Trantina’s experimental ob-servations on smooth, bolt hole and sharp notched specimens havebeen scaled with respect to fatigue strength for comparison withthe model predictions in Fig. 8. A direct comparison cannot bemade because the data exhibited is for a different material thanthat modeled. The important point demonstrated by Fig. 8 is thatthe model predicts that the fatigue life decreases linearly with anincrease in the log of volume~or surface area!. The intercept ofthe line depends on the specified fatigue life. The slope of the line,which represents the sensitivity of the material to size effect, de-

pends on the scatter of the fatigue strength controlling variablesand can vary with material processing and material alloy@18#.

Sensitivities. The sensitivities shown in Figure 9 representthe sensitivity of the total fatigue life COV to the random variableCOV.

To change in total life COV a Monte Carlo simulation wasperformed using the nominal variations in Table 1. Then a sepa-rate Monte Carlo simulation was performed for each of the ran-dom variables in which the COV the random variable was de-crease by 5 percent. The sensitivities have been normalized suchthat the summation of sensitivities is one.

Figure 9 shows that at low stress~high cycle fatigue!, the varia-tion in fatigue life is most sensitive to the variation in the grainorientation. It is well known that texturing can greatly effect highcycle fatigue life. The variation in high cycle fatigue life is shownto be least sensitive to the variation in grain size. The Monte Carlosimulation showed that at low stress, the largest grains were re-sponsible for the failure-causing crack. It would seem that thefatigue life would be sensitive to the grain size distribution. How-ever, the distribution of the largest grains in each specimen is anextreme-value distribution and will only change slightly with a 5percent decrease in the COV of grain size for all of the grains inthe specimen.

Figure 9 shows that at high stress~low cycle fatigue!, the varia-tion in fatigue life is most sensitive to the variation in the applied

Fig. 6 Fatigue life test data plotted on lognormal paper „datafrom Bastenaire …

Fig. 7 Predicted mean fatigue life for various size specimens

Fig. 8 Predicted stress versus life curve

Fig. 9 Importance of the random variable variation on the fa-tigue life variation


microstress. In low cycle fatigue, the crack tip plastic zone is largeand not as sensitive to the local material property variations. Thescatter in fatigue life is more sensitive to grain size variations inlow cycle fatigue than in high cycle. This is because the failure-causing crack is less likely to be associated with the largest grainin the specimen. The distribution for the entire population of grainsize in the specimen will effect the failure causing cracks in lowcycle fatigue. High cycle fatigue is a function of the extreme valuedistribution of grain size.

SummaryThis study develops a probabilistic mesomechanical fatigue

model to relate the variation in the material microstructure to thevariation in the fatigue life of macrostructural components. Onlythe microstructural effects were investigated. Variations in theapplied loading, stress concentrations, residual stresses, and globalgeometry are not considered.

Single-phase polycrystalline components are modeled. Grainshape is assumed equiaxial and the grain orientation is untextured.Loading and material properties within a grain are homogeneousalthough not isotropic and vary from grain to grain. Componentgeometries are simple smooth test specimens.

The fatigue process is divided into three phases. The first phaseis the crack nucleation phase. The second phase is the small crackgrowth phase. Local microstructural variables considered randomare grain size, grain orientation, micro-stress and frictional stress.The variables are common to both the crack nucleation and smallcrack growth models. The third phase is the long crack growthphase. Long crack growth rate is modeled using Paris law andmicrostructural variations are not explicitly considered. All varia-tion in long crack growth is model by allowing the Paris lawcoefficient to be a random variable.

The model predicted many aspects of fatigue observed in theexperimental data. These include:

• The shape of the total fatigue life distribution.• The applied global stress effects on the shape of the total

fatigue life distribution.• The knee in the SN curve and run-outs.• The size effect.

This study demonstrates the feasibility of developing probabi-listic mesomechanical material models that can link the variationin the material microstructure to the scatter in fatigue life.

AcknowledgmentsThis work was supported by NASA SBIR Contract NAS

399040, Dr. P. L. N. Murthy, Technical Representative, andGSRP Project No. NGT-51053, Glenn Research Center, Dr. C. C.Chamis, Technical Advisor.

References@1# Peterson, R. E., 1939, ‘‘Methods of Correlating Data from Fatigue Test of

Stress Concentration Specimens,’’Contributions to the Mechanics of Solids,Macmillan, pp. 179–183.

@2# Tryon, R. G., 1996, ‘‘Probabilistic Mesomechanical Fatigue Model,’’ Ph.D.thesis, Vanderbilt University.

@3# Tryon, R. G., Cruse, T. A., 1998, ‘‘A Reliability-Based Model to PredictScatter in Fatigue Crack Nucleation Life,’’ Fatigue Fract. Eng. Mater. Struct.,21, pp. 257–267.

@4# Tanaka, K., Mura, T., 1981, ‘‘A Dislocation Model for Fatigue Crack Initia-tion,’’ ASME J. Appl. Mech.,48, pp. 97–103.

@5# Chan, K. S., Lankford, J., 1983, ‘‘A Crack Tip Strain Model for the Growth ofSmall Fatigue Cracks,’’ Scr. Metall.,17, pp. 529–532.

@6# Laird, D., Smith, G. C., 1962, ‘‘Crack Propagation in High Stress Fatigue,’’Philos. Mag.,7, pp. 847–857.

@7# Weertman, J., 1979, ‘‘Fatigue Crack Propagation Theories,’’ Fatigue and Mi-crostructure, ASM, Metals Park, Ohio, pp. 279–206.

@8# Hicks, M. A., Brown, C. W., 1984, ‘‘A Comparison of Short Crack GrowthBehavior in Engineering Alloys,’’ Fatigue 84, Engineering Materials AdvisoryServices Ltd., England, pp. 1337–1347.

@9# Nisitani, H., and Takao, K-I., 1981, ‘‘Significance of Initiation, Propagationand Closure of Microcracks in High Cycle Fatigue of Ductile Metals,’’ Eng.Fract. Mech.,15, No. 3, pp. 445–456.

@10# Tanaka, K., Kinefuchi, M., and Yokomaku, T., 1992, ‘‘Modelling of StatisticalCharacteristics of the Propagation of Small Fatigue Cracks,’’ Short FatigueCracks, Miller, K. J., and de los Rios, E. R., eds., ESIS 13, Mechanical Engi-neering Publications, London, pp. 351–368.

@11# Forsyth, P., 1969, The Physical Basis of Metal Fatigue, American ElsevierPubl., New York.

@12# Tryon, R. G., Cruse, T. A., 1997, ‘‘Probabilistic Mesomechanical FatigueCrack Nucleation Model,’’ ASME J. Eng. Mater. Technol.,19, No. 1, pp.65–70.

@13# Tryon, R. G., Cruse, T. A., 1995, ‘‘Probabilistic Mesomechanical FatigueCrack Initiation Model, Phase 1: Crack Nucleation,’’ ASME/JSME PressureVessel and Piping Conference, Honolulu, HI, Published in PVP-95-MF2.

@14# Bastenaire, F. A., 1972, ‘‘New Method for the Statistical Evaluation of Con-stant Stress Amplitude Fatigue-Test Results,’’ Probabilistic Aspects of Fa-tigue, Ed., Heller, R. A., ASTM STP 511, pp. 3–28.

@15# Dieter, G. E., 1986,Mechanical Metallurgy, McGraw-Hill, Third Edition.@16# Sasaki, S., Ochi, Y., Ishii, A., Hirofumi, A., 1989, ‘‘Effects of Material Struc-

tures on Statistical Scatter in Initiation and Growth Lives of Surface Cracksand Failure Life in Fatigue,’’ JSME Inter. J., Series I,32, No. 1, pp. 155–161.

@17# Weibull, W., 1961,Fatigue Testing and Analysis of Results, Pergamon Press.@18# Trantina, G., 1981, ‘‘Statistical Fatigue Failure Analysis,’’ J. of Test. Eval.,9,

No. 1, DD. 44–49.


Z. Wu

J. LuPh.D. Student. Professor. Mem. ASME

LASMIS,University Technologie de Troyes,

12 Rue Marie Curie-BP2060-10010,Troyes, Cedex, France

e-mail: [email protected]

Study of Surface Residual Stressby Three-DimensionalDisplacement Data at a SinglePoint in Hole Drilling MethodA method combining moire´ interferometry, Twyman–Green interferometry, and blind holedrilling method is proposed for simple and accurate determination of residual stress. Therelationship between the three-dimensional surface displacements produced by introduc-ing a blind hole and the corresponding residual stress is established by employing theFourier expansion solution containing a set of undetermined coefficients. The coefficientsare calibrated by 3D finite element method. The surface in-plane displacements Ux , Uy ,and the out-of-plane displacement Uz produced by the relaxation of residual stress aremeasured by moire´ interferometry and Twyman–Green interferometry, respectively, afterthe hole-drilling procedure. The complete three-dimensional displacement data at anysingle point around the hole can be used for residual stress determination. The accuracyof the method is analyzed and the experimental procedure is described to determine thesign of residual stresses. As an implementation of the method, a shot peening residualstress problem is studied.@S0094-4289~00!00802-1#

IntroductionStrain gage hole-drilling method was first proposed by Soete

@1# and continuously developed by many contributors@2–8#.Strain gage hole drilling method is the most widely used residualstress measurement method. For planar uniform residual stresses,it is believed reliable, simple, and fast to implement. It requiresrelatively simple equipment and modest operator skills. However,in the method, strain data are only obtained at three specifieddirections and random measurement errors are inevitable. More-over, for blind-hole problems, the radial distribution of surfacedeformations due to the relaxation of residual stress is very non-uniform and highly localized to the boundary of the hole. Thenonlinear distributions of the deformations depend on both thehole depth-to-diameter ratio and the residual stress which is un-known previously. As the size of a standard strain gage rosette istypically two to four times the hole diameter,r 0 and 30 deg ofcircumferential angle,u ~ASTM E837-92@9#!, the region coveredby a strain gage is too large compared to the released strain field.Thus, the average strain measured within the covered area, whichwill be used to determine residual stress, might not be equivalentto the possible strain at the calibration point. Another commonlyrecognized drawback of the method is the eccentric hole-drillingerror.

Recently, some laser interferometry methods such as moire´ in-terferometry and holographic interferometry have been used inconjunction with hole-drilling technique as an alternative meansfor residual stress studies@10–16#. These optical methods canprovide whole-field displacements around the hole to cope withthe drawbacks of strain gage method. However, they often requirethree measurement points with large angle intervals, which is timeconsuming and easy to introduce measurement errors.

In this study, moire´ interferometry and Twyman–Green inter-ferometry are to be used together to provide complete three-dimensional displacement fields when a blind-hole is drilled. Thethree-dimensional displacement data at any single point is to be

used for residual stress determination. This approach can savemeasurement time and improve the measurement accuracy. Byusing multipoint analysis, random measurement errors can be re-duced. The proposed method is to be implemented for a shotpeening residual stress problem.

Background: Moire Interferometry and Twyman –Green Interferometry

Moire interferometry is an optical method, providing real timeand whole field contour maps of in-plane displacements. The highdisplacement measurement sensitivity and high spatial resolutionmade it suitable for a broad range of problems in solid mechanics.A typical setup of an optical system for moire´ interferometry isdepicted schematically in Fig. 1. The specimen grating of fre-quencyf s is replicated on the surface of the specimen. Accordingto grating diffraction equation, Eq.~1!, two beams of symmetri-cally incident light are mutually coherent and are diffracted by thespecimen grating to different diffraction orders,

sinbm5sina1m fsl (1)

wherea is the incident angle,m is diffraction orders,bm is thedirection of themth order diffraction beam,l is the wavelength ofthe laser, andf s is the frequency of the specimen grating as shownin Fig. 1.

When the relation of Eq.~1a! is satisfied, the11 and21 dif-fraction orders emerge as normal to the specimen,

sina5 f sl (1a)

If the specimen grating is perfect, the two diffracted beamscoexist in space but their angle of intersection is zero. Their mu-tual interference produces a uniform intensity throughout the field,which is called the null field. When the specimen deforms, thespecimen grating deforms correspondingly and the resulting inter-ference patterns represent the contours of constantUx and Uydisplacements. The displacements can then be determined fromthe fringe orders by the following relationships~@17# et al.!,

Ux~x,y!51

2 f sNx~x,y!, Uy~x,y!5

1

2 f sNy~x,y! (2)

Contributed by the Materials Division for publication in the JOURNAL OF ENGI-NEERING MATERIALS AND TECHNOLOGY. Manuscript received by the MaterialsDivision September 29, 1997; revised manuscript received October 14, 1999. Asso-ciate Technical Editor: Kwai S. Chan.


whereNx and Ny are fringe orders inUx and Uy fields, respec-tively. In routine practice, a frequency of 1200 lines/mm is used,providing a contour interval of 0.417mm/fringe order.

Twyman–Green interferometry was originally introduced totest optical elements@18#. A combined setup of Twyman–Greeninterferometry and moire´ interferometry is also depicted in Fig. 1.The collimating beam of laser light is separated by a half mirror.One beam of light, traversing the half mirror, is intersected by ahigh quality reference mirror. The returning light from the mirroris reflected by the half mirror to the CCD camera. Another beamof light reflected from the half mirror is intersected by the speci-men grating at the normal of the grating plane. A high qualitygrating with a metal coating is similar to a flat mirror since thegrating frequency is very high. The returning light from the grat-ing traverses from the half mirror to the CCD camera. The halfmirror is at 45 deg to the incident light. Both the normal of thespecimen grating and the reference mirror coincide to the incidentand emergent light, thus, their mutual interference produces a uni-form intensity throughout the field, too. When the specimen hasan out-of-plane deformation, the change of the light path willproduce a interfering fringe pattern representing contours ofUzdisplacement by the following relationship@18#,

Uz~x,y!5l

2Nz~x,y! (3)

wherel is the wavelength of the laser andNz is the fringe ordersrepresenting the out-of-plane displacement contours.

The combination of Eqs.~2! and ~3! provides complete three-dimensional displacements in the whole field with the same orderof sensitivity.

3D Displacement Field for Blind Hole Drilling MethodFor blind hole problems, a Fourier expansion solution was pro-

posed by Schajer@3# to account for the effect of out-of-planedisplacement and was expressed as

ur~r ,u,z!5(n50

`

urn~r ,z!cosnu

uu~r ,u,z!5(n50

`

uun~r ,z!sinnu (4)

uz~r ,u,z!5(n50

`

uzn~r ,z!cosnu

whereurn , uun , anduzn are the contributions of thenth displace-ment components to the total displacement componentsur , uu ,

and uz , respectively, in the cylindrical coordinate system. Thecorresponding load distribution for each value ofn has the sameharmonic form, and thusn50,62,64, . . .

For biaxial planar residual stresses, as the first order approxi-mation,n50, 2, the displacement field near a blind hole can beexpressed as@12#

ur~r ,u!5A~sxx1syy!1B@~sxx2syy!cos 2u12txy sin 2u#

uu~r ,u!5C@~sxx2syy!sin 2u22txy cos 2u# (5)

uz~r ,u!5F~sxx1syy!1G@~sxx2syy!cos 2u12txy sin 2u#

where, A, B, C, F, and G are undetermined coefficients whichdepend on the material constants and the geometrical parametersof the blind hole;sxx , syy , and txy are residual stress compo-nents in a Cartesian coordinate system;ur , uu , anduz are surfacedisplacements in a cylindrical coordinate system.

Determination of Calibration CoefficientsThe coefficientsA, B, C, F, andG in Eq. ~5! can be determined

by three-dimensional finite element analyses by using two specificloading cases:

1 sxx5syy5s, txy50, an equibiaxial residual stress field,which is equivalent to a uniform pressurep5s acting on the holeboundary. By using Eq.~5!, the coefficientsA and F are deter-mined as,

AS E,n,r 0 ,r ,h

r 0D5

ur~r ,u!

2sFS E,n,r 0 ,r ,

h

r 0D5

uz~r ,u!

2s(6)

2 sxx52syy5s, txy50, a pure shear residual stress field,which is equivalent to the harmonic distributions of the normalstresss rr 52s cos 2u and the shear stresst ru5s sin 2u acting onthe hole boundary. By using Eq.~5! again, the coefficientsB, C,andG are determined as,

BS E,n,r 0 ,r ,h

r 0D5

ur~r ,u!

2s cos 2u

CS E,n,r 0 ,r ,h

r 0D5

uu~r ,u!

2s sin 2u(7)

GS E,n,r 0 ,r ,h

r 0D5

uz~r ,u!

2s cos 2u

In Eqs.~6! and~7!, ur , uu , anduz are the surface displacementscalculated from the finite element analysis. It is to be noted thatthe calibration coefficients are independent of angleu. The coef-ficients depend on the material constants of Young’s modulusEand Poisson ration, the geometrical parameters of the hole radiusr 0 , hole depthh, and the radial coordinater at the calibrationpoint.

Determination of Residual StressMoire interferometry provides displacement data with the Car-

tesian coordinates. The radial displacementUr , the circumferen-tial displacementUu can be expressed in terms ofUx andUy as,

Ur~r c ,u!5Ux~x,y!cosu1Uy~x,y!sinu(8)

Uu~r c ,u!52Ux~x,y!sinu1Uy~x,y!cosu

wherex21y25r c2 andu5tan21(y/x).

By combining Eqs.~2!, ~3!, ~5!, and ~8!, the relationship be-tween the residual stress and the fringe orders at a surface point(r c ,u) around the hole can be rewritten in a matrix form as,

Fig. 1 Schematic setup of an optical system combining moire ínterferometry and Twyman–Green interferometry


@Nx~x,y! Ny~x,y!#Fcosusinu G

52 f s@A1B cos 2u A2B cos 2u 2B sin 2u#F sxx

syy

txy

G@Nx~x,y!Ny~x,y!#F2sinu

cosu G52 f s@C sin 2u 2C sin 2u 22C cos 2u#F sxx

syy

txy

Gl

2Nz~x,y!5@F1G cos 2u F2G cos 2u 2G sin 2u#F sxx

syy

txy

G(9)

The three unknown residual stress componentssxx ,syy ,txycan be determined by solving the above three homogeneous linearequations.

Determination the Sign of Residual StressFringe patterns of moire´ interferometry and Twyman–Green

interferometry, produced by residual stresses with the same distri-bution and the opposite signs, are identical to each other. The signof the fringe orders should be determined during the experimentssince the sign of residual stresses is previously unknown. Thereare many methods available for sign determination in moire´ inter-ferometry and Twyman–Green interferometry, but the simplestand the most convenient way is the carrier fringe method. Inmoire interferometry, if the specimen grating and the incidentbeams have a relative rigid body rotation around each other with-out the change of incident angle, a series of uniformly spacedfringes perpendicular to the grating lines will be created, whichare called the carrier fringes of rotation. Similarly, if the angle ofthe incident beams is changed, a series of uniformly spacedfringes parallel to the grating lines will be created, which arecalled the carrier fringes of extension or compression. When car-rier fringes are added to a load induced fringe pattern, the result-ant fringe gradient vector at every point of the fringe pattern willbe the superposition of the original and the carrier fringe gradientvectors. This is clearly illustrated in Fig. 2, where the moire´fringes which represent the releasedUx displacement field under

the equibiaxial residual stress state are modified in different waysby two known carrier fringes of rotation with opposite signs. Attwo typical regions of Fig. 2, the original displacement gradientF0 is changed in both direction and magnitude due to the effect ofcarrier fringe of rotationFc as illustrated in the inserted parts. InTwyman–Green interferometry, similar to those in moire´ interfer-ometry, the carrier fringes can be created by rotating the referencemirror and the fringe lines are perpendicular to the direction ofrotation. The carrier fringes of ration can be superposed to theload induced out-of-plane displacement as illustrated in Fig. 3 forthe same residual stress problem, where the originally circularUzfringes are altered in different ways when the orthogonal carrierfringes are introduced. For residual stresses have identical magni-tude but opposite signs, the fringe gradient at any point of the fieldshould have opposite directions. Thus, the carry fringes of rotationor extension can alter the residual stress induced fringes in oppo-site ways. Therefore, the sign of residual stress is easy to deter-mine by this method.

Accuracy AnalysisFrom Eq.~9! we know that the accuracy of residual stress result

depends on the accuracy of calibration coefficients and fringe or-ders. Generally, calibration coefficients can be determined accu-rately for given material geometrical parameters. In moire´ inter-ferometry and Twyman–Green interferometry, the integral andfractional fringe orders can be determined accurately through theintensity distribution analysis of the fringe patterns or phase shift-ing technique. The reasonable accuracy in fringe order determina-tion will be 60.05 fractional fringe orders which represents60.02 mm of displacement in moire´ interferometry (f s51200 lines/mm), and60.016 mm in Twyman–Green interfer-ometry (l5632.8 nm). The accuracy would be sufficient formany residual stress problems. As the fringe counting uncertain-ties are random measurement errors, the average value of the ran-dom displacement errors should tend to zero if a large amount ofmeasurement points are considered. Therefore, the residual stressmeasurement accuracy can be further improved through multi-point analysis.

It is important to notice that, in the previous hole-drilling meth-ods, the three measurement points should be chosen with rela-tively large angle intervals so that the homogeneous linear equa-tions are not ill-conditioned. However, in the proposed method,the homogeneous linear equations, Eq.~9!, composed by the com-plete three-dimensional displacements at a single point are alwayswell-conditioned.

Fig. 2 Ux field moire´ pattern modified by carrier fringes of ro-tation with opposite fringe vectors

Fig. 3 Uz field fringe pattern modified by carrier fringes of ro-tation with orthogonal fringe vectors


Implementation of the Method for Shot-Peened Speci-men

In order to implement this method, a shot peening residualstress problem was studied. Shot peening processing is widelyused in engineering to increase the working life under fatigue loador to enhance the resistance to surface corrosion of metallic ormetal based composite components by introducing moderate com-pressive residual stresses in the near surface layer of the material@19#.

The specimen was cut from a large plate of AS10U3NG alumi-num alloy, one side of which was shot peened. The material prop-erties were:E579.5 GPa, n50.3. The holographic specimengrating with a frequency of 1200 lines/mm was replicated on theshot peened surface of the specimen at room temperature. Basedon a previous study, compressive residual stress existed within a0.3 mm surface layer, thus a blind hole of 2.0 mm in diameter and0.3 mm in depth was carefully drilled at the grating side of thespecimen. It was drawn schematically in Fig. 4. The drill bit wascontrolled precisely at a speed of 1.0mm/s during the hole drillingprocedure and the machining stress proved experimentally to bevery small. The influence of grating delamination and the plasticzone was diminished within 0.1r 0 . The bottom of the hole wasflat.

After the hole was drilled, the specimen was put into the com-bined optical system of moire´ interferometry and Twyman–Greeninterferometry as shown in Fig. 1. The fringe patterns representingthe contour maps ofUx , Uy , and Uz displacements were cap-tured by a high resolution CCD camera (131731035) as shownin Figs. 5–7. The sign of the fringes was determined by carrierfringe method. Generally, in order to improve the accuracy infringe counting, the displacements should be measured as close aspossible to the boundary of the hole. By considering the influenceof grating delamination, the displacement data in this experimentwere obtained atr c51.2r 0 .

Shot peening was a random process and thus, the in-plane re-sidual stress produced by the process is regarded to be a uniformequibiaxial residual stress state~sxx5syy52s, txy50!. Basedon the above analysis, in order to improve the measurement accu-racy, residual stress was determined separately from 16 pointsaround the hole with an identical angle interval ofp/8. The fringeorders of Ux , Uy , and Uz displacements at the points(1.2r 0 ,np/8) are listed in Table 1. The calibration coefficients

Fig. 4 Schematic drawing of the shot peened specimen con-taining compressive residual stress

Fig. 5 Ux displacement field obtained from shot peenedAS10U3NG aluminum alloy specimen with the hole radius of1.0 mm and the hole depth of 0.3 mm

Fig. 6 Uy displacement field obtained from shot peenedAS10U3NG aluminum alloy specimen with the hole radius of1.0 mm and the hole depth of 0.3 mm

Fig. 7 Uz displacement field obtained from shot peenedAS10U3NG aluminum alloy specimen with the hole radius of1.0 mm and the hole depth of 0.3 mm


determined atr c51.2r 0 by Eqs.~6! and~7! are listed in Table 2.By using Eq.~9!, residual stress is determined, respectively. Thedetermined residual stress results at these points are plotted to-gether in Fig. 8. It is obvious that the maximum compressiveresidual stresses are2140.0 MPa forsxx and2148 MPa forsyy ,respectively. The minimum compressive residual stresses are2113.7 MPa forsxx and2109.7 MPa forsyy , respectively. Theaverage values ofsxx and syy at the points around the hole are2129.5 MPa and2128.6 MPa, respectively. The determinedshearing stresses from these points are within630.0 MPa and theaverage value of the shear stresses at the points around the hole is22.2 MPa. Therefore, the average results can reflect the equibi-axial compressive residual stress induced by a shot peeningprocedure.

DiscussionThe proposed method in this study could be used in the follow-

ing cases:

„1… Nonuniform Planar Residual Stress Evaluation. Forplanar nonuniform residual stresses, which exist in some localregions, when a hole is drilled through these regions, the relax-ation of residual stress at different parts has no effect on eachother as shown in Fig. 9. The relationship between the local re-sidual stress and the surface displacements can be proposed as,

ur~r ,u!5A8~r !s rr 2B8~r !suu

uu~r ,u!5C8~r !t ru (10)

uz~r ,u!5F8~r !s rr 2G8~r !suu

whereA8, B8, C8, F8, andG8 are calibration coefficients whichcan be determined by using the similar finite element analysis~FEA!.

The three-dimensional displacement data at the coefficientcalibration point can be used for the local residual stressdetermination.

Fig. 8 Residual stress determined at the radial coordinate r cÄ1.2r 0 and the angle interval uÄpÕ8 for AS10U3NG aluminumalloy specimen

Fig. 9 Potential application of the proposed method for planarnonuniform residual stresses

Table 1 Fringe orders obtained from the Ux , Uy , and Uz fringe patterns at the correspondingpoints

uN 0 p/8 p/4 3p/8 p/2 5p/8 3p/4 7p/8

Nx 22.1 22.05 21.8 21.05 20.1 0.9 1.95 2.4Ny 0 20.75 21.7 22.2 22.4 22.2 21.95 21.0Nz 20.7 20.7 20.7 20.7 20.7 20.7 20.7 20.7

uN p 9p/8 5p/4 11p/8 3p/2 13p/8 7p/4 15p/8

Nx 2.45 2.3 1.7 1.1 0 20.6 21.5 21.95Ny 0 0.9 1.6 2.0 2.2 2.0 1.55 0.8Nz 20.7 20.7 20.7 20.65 20.65 20.65 20.7 20.7

Table 2 Calibration coefficients determined by 3D finite element analyses when r 0Ä1.0 mmand hÄ0.3 mm at r cÄ1.2 r 0

A B C F G

0.456631025 0.64431025 20.337631025 0.107531025 0.485231025


„2… Determination of Residual Stress Distribution inDepth. As the accuracy of the proposed method is sufficientlyhigh, by combining the method with incremental hole-drillingtechnique, the distribution of residual stress in depth can be deter-mined by using the similar procedure as above. The accuracy canalso be improved by multipoint measurements.

ConclusionsA method is proposed to determine residual stress by three-

dimensional displacement data at a single point. The relationshipbetween the three-dimensional surface displacements produced byintroducing a blind hole and the corresponding residual stress isestablished by using a Fourier expansion solution with a set ofcalibration coefficients. The three-dimensional surface displace-ments are provided by whole-field and high sensitivity moire´ in-terferometry and Twyman–Green interferometry. The method issufficiently accurate. Through multipoint analysis and averageprocessing, accuracy of residual stress result is further improved.As an implementation, a shot peening residual stress problem isstudied. The equibiaxial compressive residual stress is found.More applications of the method are anticipated.

References@1# Soete, W., 1949, ‘‘Measurement and Relaxation of Residual Stress,’’ Sheet

Met. Ind.,26, No. 266, pp. 1269–1281.@2# Kelsey, R. A., 1956, ‘‘Measuring Non-Uniform Residual Stresses by the Hole

Drilling Method,’’ Proc. SESA, No. 1, pp. 181–194.@3# Schajer, G. S., 1981, ‘‘Application of Finite Element Calculations to Residual

Stress Measurements,’’ ASME J. Eng. Mater. Technol.,103, No. 4, pp. 157–163.

@4# Lu, J., Niku-Lari, A., and Flavenot, J. F., 1985, ‘‘Mesure de la Distribution desContraintes Residuelles en Profondeur par la Me´thode du Trou Incre´mentale,’’Memoire et Etudes Scientifiques, Revue de la Me´tallurgie, Feb., pp. 69–81.

@5# Lu, J., and Flavenot, J. F., 1989, ‘‘Applications of the Incremental Hole-

Drilling Method for Measurements of Residual Stress Distribution,’’ Exp.Tech.,13, No. 11, pp. 18–24.

@6# Flaman, M. T., and Manning, B. H., 1985, ‘‘Determination of Residual StressVariation with Depth by Hole-drilling Method,’’ Exp. Mech.,25, No. 3, pp.205–207.

@7# Schajer, G. S., 1988, ‘‘Measurement of Non-Uniform Residual Stresses Usingthe Hole Drilling Method, Part I—Stresses Calculation Procedures,’’ ASME J.Eng. Mater. Technol.,110, No. 4, pp. 338–343.

@8# Schajer, G. S., 1988, ‘‘Measurement of Non-Uniform Residual Stresses Usingthe Hole Drilling Method, Part II—Practical Applications of the IntegralMethod,’’ ASME J. Eng. Mater. Technol.,110, No. 4, pp. 344–349.

@9# ASTM E837-92, 1992, Standard Test Method for Determining ResidualStresses by the Hole Drilling Strain-Gage Method, Annual Book of ASTMStandards, Section 3,03.01, pp. 747–753, American Society for Testing andMaterials, Philadelphia, PA.

@10# Antonov, A., 1983, ‘‘Inspecting the Level of Residual Stresses in WeldedJoints by Laser Interferometry,’’ Weld Prod.,30, No. 9, pp. 29–31.

@11# Nicoletto, G., 1991, ‘‘MoireInterferometry Determination of Residual Stressesin the Presence of Gradients,’’ Exp. Mech.,31, No. 3, pp. 252–256.

@12# Makino, A., and Nelson, D., 1994, ‘‘Residual Stress Determination by SingleAxis Holographic Interferometry and Hole Drilling, Part I: Theory,’’ Exp.Mech.,3, pp. 66–78.

@13# Makino, A., and Nelson, D., 1997, ‘‘Determination of Sub-surface Distribu-tions of Residual Stresses by Holographic Hole-Drilling Technique,’’ ASMEJ. Eng. Mater. Technol.,119, No. 1, pp. 95–103.

@14# Nelson, D., Fuchs, E., Makino, A., and Williams, D., 1994, ‘‘Residual StressDetermination by Single Axis Holographic Interferometry and Hole Drilling,Part II: Experiments,’’ Exp. Mech.,3, pp. 79–88.

@15# Wu, Z., Lu, J., and Han, B., 1998, ‘‘Study of Residual Stress Distribution by aCombined Method of Moire´ Interferometry and Incremental Hole-Drilling,Part I: Theory,’’ ASME J. Appl. Mech.,65, No. 4, pp. 837–843.

@16# Wu, Z., Lu, J., and Han, B., 1998, ‘‘Study of Residual Stress Distribution by aCombined Method of Moire´ Interferometry and Incremental Hole-Drilling,Part II: Implementation,’’ ASME J. Appl. Mech.,65, No. 4, pp. 844–850.

@17# Post, D., Han, B. and Ifju, P., 1994,High Sensitivity Moire´ ~ExperimentalAnalysis for Mechanics and Materials!, Springer-Verlag.

@18# Brown, M., and Wolf, E., eds., 1985,Principles of Optics, Sixth Edition,Pergamon Press, pp. 302.

@19# Metal Improvement Company, Inc., 1981, Applications du Shot Peening.


I. W. ParkIntegrated Surgical Systems, Inc.,

1850 Research Park Dr.,Davis, CA 95616

D. A. DornfeldFellow ASME

Department of Mechanical Engineering,University of California, Berkeley,

Berkeley, CA 94270-1740

A Study of Burr FormationProcesses Using the FiniteElement Method: Part IA finite element model of orthogonal metal cutting including burr formation is presented.A metal-cutting simulation procedure based on a ductile failure criterion is proposed forthe purpose of better understanding the burr formation mechanism and obtaining a quan-titative analysis of burrs using the finite element method. In this study, the four stages ofburr formation, i.e., initiation, initial development, pivoting point, and final developmentstages, are investigated based on the stress and strain contours with the progressivechange of geometry at the edge of the workpiece. Also, the characteristics of thick andthin burrs are clarified along with the negative deformation zone formed in front of thetool edge in the final development stage.@S0094-4289~00!00702-7#

1 IntroductionBurrs are defined as undesirable projections of the material

formed as the result of the plastic flow from cutting and shearingoperations. Manufacturing of precision components often requiresa deburring or finishing operation. Studies have shown that formanual deburring of parts, the time required to deburr a part in-creases exponentially with an increase of burr thickness. Henceautomation of the deburring process has become a prime objectiveas part of efforts to automate the entire production system. How-ever, deburring automation is quite difficult, especially force-based control technique for precision components due to burr pro-file variation. Thus, under any circumstance, any attempts toautomate deburring requires a reliable model of burr formation.More important, understanding of the burr formation mechanismmay allow prevention or minimization of burrs.

In most machining operations, rollover burrs are formed at theend of cut as a result of a chip pushed out of the cutter’s pathrather than shearing. There have been many studies to predict thegeneral burr formation characteristics in both ductile and brittlematerials. The first quantitative analysis of burr formation wasperformed by Gillespie@1# assuming that the burr is formed bybending deformation of the end portion of the workpiece. Hecharacterized the burr by the distance of the tool from the end ofthe workpiece at which bending deformation starts. Despite theunsuccessful quantitative prediction of the burr size, the qualita-tive effect of each parameter is quite often observable. Iwata et al.@2# observed the burr formation during machining inside a scan-ning electron microscope~SEM!, and experimentally determinedthe effect of exit angles of the workpiece on the burrs. Theyexplained the fracture of the workpiece at the tool edge using thestrain obtained by the finite element method~FEM! analysis. Koand Domfeld@3# developed a model for burr formation in mate-rials exhibiting ductile and brittle behavior, and evaluated themodel at slow machining speeds. Later, Chern and Dornfeld@4,5#continued the development of burr formation models, extendedKo and Dornfeld’s model with more realistic machining opera-tions and conditions. They also included the study of edge break-out phenomenon as well as burr formation. Pekelharing@6# triedto explain poor tool life and described the formation of ‘‘foot’’ ininterrupted cutting with sharp tools at the exit of cut.

In general, closed-form analytical solution for general problemsof elastic-plastic with large deformation such as chip and burr

formation processes are very difficult to derive. Thus, simulationanalysis using the finite element method technique is a reasonableapproach to model the metal-cutting and burr formation processes.In the last two decades, a great deal of research has gone into theestablishment of FEM techniques for orthogonal cutting. In 1973,Klamecki @7# introduced the study of metal cutting using FEM bypresenting finite element models in his Ph.D. dissertation. In1984, Usui and Shirakashi@8# made a significant contribution tothe development of FEM as the best predictive theory because itcan predict chip formation, cutting force, and distributions ofstress and strain without any input from cutting experiments.Later, Iwata et al.@2# presented a rigid-plastic finite elementmodel for steady-state plane strain orthogonal machining. Stren-kowski and Caroll@9# also presented finite element models basedon general purposed two-dimensional finite element codeNIKED2D. Recently, Park et al.@10# conducted a finite elementanalysis of burr formation in ductile material. In this study, ageneral purpose finite element software package, ABAQUS/Standard, was used to simulate the chip and burr formation pro-cesses in orthogonal cutting. Based on the geometrical change atthe edge of the workpiece and a series of stress and strain con-tours, the fundamental burr formation mechanism was found anddivided into four stages as the tool approaches the edge of theworkpiece. The stages are initiation, initial development, pivotingpoint, and final development stages. The results from the finiteelement analysis were also qualitatively verified with the experi-mental data.

Although the finite element analysis provided physical insightinto the fundamental burr formation mechanism, a final burr con-figuration could not be simulated due to assumptions such as anisothermal condition and ignorance of strain-rate influence. Theisothermal condition is assumed so that the temperature of theworkpiece remains constant at ambient throughout the simulation.However, during the cutting of metal, high temperatures are gen-erated in the region of the tool cutting edge, and these tempera-tures have a controlling influence on the friction between the chipand tool. Also, considering the geometry of the process, plasticflow at these high temperatures implies that the strain rate is quitehigh for the chip to form because of the geometry of the process.The most significant problem in the previous model was the con-cept of the parting line and a separation criterion. During the burrformation process, the distorted parting line was constantly re-zoned to a predefined location associated with the metal-cuttingprocess involving extensive computing. Further, the distanceseparation criterion@11# was adopted to simulate the metal-cuttingand burr formation processes under the assumption that the tooltip radius is quite small compared to the depth of the cut. This

Contributed by the Materials Division for publication in the JOURNAL OF ENGI-NEERING MATERIALS AND TECHNOLOGY. Manuscript received by the MaterialsDivision June 26, 1998; revised manuscript received December 31, 1999. AssociateTechnical Editor: Kwai S. Chan.


approach was sufficient to find the fundamental burr formationmechanism but is inappropriate to give a quantitative estimate ofburr dimensions.

The objective of this study is to develop a finite element modelto further investigate the burr formation process including theanalysis of a burr or edge breakout. In this study, the burr forma-tion process in each stage is reexamined. An explicit dynamicsFEM software package, ABAQUS/Explicit, is used for the simu-lation of the burr formation. The ABAQUS/Explicit code has sev-eral important features, which make it more suitable for studyingburr formation problems. First, instead of the parting line crite-rion, a ductile failure model is adopted to separate the chip fromthe workpiece. Second, an adiabatic heating model is adopted tosimulate the heat generation effects due to plastic work of theworkpiece and chip. Thus, the temperature-dependent materialproperties~in this case, 304 L stainless steel~SS304L!! and strain-rate property are incorporated into the finite element analysis.

2 Finite Element ModelingThe two-dimensional finite element model is generated under

the plane strain assumption that the width of the cut is at least fivetimes greater than the depth of the cut~0.5 mm!, i.e.,w>5t. Themesh configuration and tool geometry is shown in Fig. 1. Theelement type used in this model is a four-node bilinear and re-duced integration with hourglass control to deal with the largedeformation caused by the chip and burr formation processes.Also, the tool is assumed to be perfectly rigid. Total number ofelements and nodes are 1054 and 1180, respectively. In thismodel, the boundaryA–B–C isconstrained against displacements(DU50), and the boundaryC–D–A is unconstrained (DF50)to allow a burr or edge breakout to form. The finite elementmodel with an incorporated initial chip geometry is largeenough to quickly reach steady-state cutting conditions beforethe burr initiation occurs. Also, total five element groups forthe workpiece part of the model are assigned and tied togetherwith *CONTACT PAIR definition and*TIED option usingABAQUS/Explicit @12#.

The perfectly rigid tool with a rake angle of 5 degrees containsa slide-line capability along the rake face to describe the chip flowas the tool advances through the workpiece. Also, a slide-line onthe rake face allows friction to form a secondary shear zone due tointeraction on the tool-chip interface. Coulomb friction ofm50.3 is used in this model. However, it should be pointed out thathigher Coulomb friction would be more appropriate to describethe interaction on the tool-chip interface in actual metal cutting.The rigid tool advances through the stationary workpiece with auniform velocity ofv55.0 m/s.

The heat generated during metal cutting, especially in highspeed cutting, is of importance in the chip formation process. The

thermal model assumes adiabatic conditions within each elementso that no heat transfer occurs within the workpiece and on theworkpiece free surface. Adiabatic assumption is typically used inthe simulation of high speed manufacturing processes involvinglarge amounts of inelastic strain, where the heating of the materialdue to its deformation is an important effect because oftemperature-dependence of the material properties, i.e., thermal-elastic-plastic material behavior. The fraction of inelastic dissipa-tion rate that appears as heat flux per volume, i.e., the work-to-heat conversion factor, is 0.8 in the finite element model. Foradiabatic heating, the temperature change can be written as

DT5Q

rCp, (1)

whereDT is the temperature change,Q is the heat generated perunit volume, andr and Cp are the material density and specificheat, respectively. The temperature increase is calculated directlyat the material integration points, and temperature is not a degreeof freedom in this model.

As strain rates increase during metal cutting, many metals showan increase in their yield strength. In primary shear zone, materialis usually deformed at a bulk shear strain rate on the order ofgp5103 to 105 s21 @13#. This effect is very important as it com-monly occurs in high energy dynamic events or in manufacturingprocesses. The model for this purpose is

«G pl5DS s

so21D p

for s>so , (2)

wheres is the effective yield stress at a nonzero strain rate,so isthe static yield stress,«G pl is the uniaxial equivalent plastic strainrate, andD andp are material parameters that may be functions oftemperature and, possibly, of other predefined state variables. InFig. 1 A finite element model of burr formation

Table 1 Material properties of SS304L and cutting conditions


this study, constant values ofD andp are used under the assump-tion that they are temperature-independent and not influenced byany other predefined state variables.

Due to the adiabatic assumption, the workpiece boundary of thefinite element model is insulated as shown in Fig. 1. The thermal-elastic-plastic material properties of SS304L along with the cut-ting conditions are given in Table 1. Finally, no build-up-edge~BUE! formation or tool wear is assumed.

3 Solution ProcedureFor the simulation of the metal cutting and burr formation pro-

cesses, ABAQUS/Explicit, an explicit dynamics finite elementprogram is used@12#. The explicit dynamics analysis procedure isbased upon the implementation of an explicit integration rule to-gether with the use of diagonal or lumped element mass matrices.The equations governing motion of the body are integrated usingthe explicit central difference integration rule

uG ~ i 11/2!5uG ~ i 21/2!1Dt~ i 11!1Dt~ i !

2uJ ~ i ! (3)

and

u~ i 11!5u~ i !1Dt~ i 11!uG ~ i 11/2!, (4)

where the subscripti refers to the increment number,u representsthe displacement vector, andDt represents the time increment.The acceleration at the beginning of the increment is computed by

uJ5@M #21~ F2 I !, (5)

where@M# is the diagonal lumped mass matrix,F is the appliedload vector, andI is the internal force vector. The central differ-ence integration operator is explicit in that the kinematic state maybe advanced using known values ofuG ( i 21/2) and uJ ( i ) from theprevious increment. From Eqs.~3! and ~4!, it can be noticed thatthe explicit integration rule is quite simple; furthermore, the com-putational efficiency of the explicit procedure is obtained by theuse of a diagonal lumped matrix. The explicit procedure requiresno iterations and no tangent stiffness matrix.

The explicit procedure integrates through time by using manysmall increments. The central difference operator is conditionallystable, and ABAQUS/Explicit uses an adaptive algorithm to de-termine conservative bounds for the stable time increment. Thetime incrementation scheme in ABAQUS/Explicit is fully auto-matic and requires no user intervention. The use of small incre-ments allows the solution to proceed without iteration. Althoughthe explicit dynamics analysis procedure can apply to the analysisof slower processes, it is obvious that it is ideally suited for ana-lyzing high speed dynamic events. The analysis proceeds by inte-grating the equations of motion~3! and~4! with a time incrementsize determined by the time incrementation scheme. While theanalysis takes an extremely large number of increments, each in-crement is relatively inexpensive and often economical. For theanalysis of the metal cutting and burr formation presented here,approximately 400,000 iterations are performed.

4 Metal-Cutting Simulation ProcedureIn order to separate the chip from the workpiece in finite ele-

ment analysis of metal cutting, various material separation crite-ria, such as an energy approach method@8#, a total effective plas-tic strain approach@9#, a distance separation method@11#, etc.@14,15#, have been proposed. In steady-state cutting, the proposedmaterial separation criterion remains constant and ensures thesmooth chip flow on the rake surface of the tool in order to predictthe shear angle formed from the tool edge to the free undeformedsurface. However, to include the burr formation process, imple-menting a suitable separation criterion becomes the most difficultobstacle in the finite element simulation. Recently, Park et al.@10#used the distance separation criterion to simulate the burr forma-

tion under the assumption that the tool edge is relatively sharpcompared to the depth of cut, that is, the ratio of the tool tip radiusto the depth of cut is on order of 1022.

Considering the work in previous FEM studies, a new approachto the simulation of a metal cutting process including both steady-state cutting and burr formation is proposed here. The metal-cutting process developed in this study is based on the ductilefailure model using the*FAILURE option in ABAQUS/Explicit@12#. This simple failure criterion is designed to allow the stableremoval of elements from the mesh as a result of tearing or rip-ping of the structure. ABAQUS/Explicit automatically deletes el-ements from the mesh as they exceed the failure criterion. Thefailure model is based on the value of the effective plastic strain.As shown in Fig. 2, when any element of the mesh reaches«o

pl ,damage on the corresponding element initiates. And when thisdamage reaches« f

pl due to further deformation, the material pointfails, and the element loses its ability to resist any further load.The damage is calculated from the effective plastic strain as

D5«pl2 «o

pl

« fpl2 «o

pl , (6)

where «pl is the current effective plastic strain of the material.Also, the material’s elastic response is based on the damagedelasticity. The damaged elastic moduli are given by

GD5~12D !G (7)

and

KD5~12D !K, (8)

where GD and KD are the damaged shear modulus and bulkmodulus, respectively. Figure 2 shows the unloading path alongthe damaged modulus. The damaged plastic yield surface is de-fined as

syd5~12D !sy~ «pl!. (9)

This causes the yield surface to shrink to a single point in stressspace when the damage reaches a value of one. In this study, themetal-cutting procedure is performed based on the ductile failurecriterion by removing the elements in contact with the tool edgeas the tool advances through the workpiece. As a matter of con-venience, the ductile failure criterion is defined as

FCi5~ «opl ,« f

pl!, (10)

wherei represents the element number. It should be noted that theductile failure model approach does not necessarily reflect themechanism involving the actual material separation during themetal cutting operation. Instead, the ductile failure criterion valuesare assigned to delete the elements contacted with the tool tip andshould assure that the surrounding elements exhibit steady-statecutting and burr formation characteristics as shown in Fig. 3.

Fig. 2 Stress-strain curves of the ductile failure model


In order to determine the values of«opl and « f

pl in ~10!, themetal-cutting process in the finite element model is divided intothree parts: steady-state cutting, initiation to pivoting point stage,and pivoting point to final development stage.

4.1 Ductile Failure Criterion in Steady-State Cutting. In-stead of attempting to predict the shear angle or chip formation inorthogonal cutting, the ductile failure criterion values are deter-mined with shear angle information obtained from the metal-cutting handbook@16# according to the cutting conditions and toolgeometry. With the information of the shear angle,«o

pl is foundunder the assumption that the only shear occurs on the shear planeand can be represented by

g5cosa

sinf cos~f2a!(11)

and

d«pl5dg

), (12)

where g and g represent a shear strain and shear strain on theshear plane, respectively, anda andf are rake and shear angles,respectively. Assuming that the shear deformation on the shearplane begins withgo50, Eq. ~12! is integrated from 0 tog andgives

«opl5

g

). (13)

Thus, the ductile failure criterion on thei th element is determinedby

FCi5~ «opl ,«o

pl1D«!, (14)

whereD«.0. Hence« fpl5 «o

pl1D«. In this study,D« is chosennot only to assure the cutting characteristics of the surroundingelements, but also to control the plastic work rate for the chipseparation from the workpiece. It is obvious that with an increaseof D«, the total strain energy increases for the ductile failure, i.e.,chip separation from the workpiece. Hence, the energy rate re-quired for the chip separation in the finite element model alsoincreases. As a result, the numerical results indicate that the sizeof D« gives some effects on the amount of residual stress remain-

ing on the machined surface and the normal stress built up on therake surface of the tool. In steady-state cutting, two constraintsshould be enforced to insure the validity of the ductile failurecriterion as follows:

«opl, « f

pl,« f~T!, (15)

and

«opl.«d , (16)

where« f represents the effective plastic strain at the maximumfracture strain value~a function of the temperature,T!, and«d isthe effective plastic strain value of the elementd, shown in Fig. 3.Constraints~15! indicate that«o

pl should not exceed« f(T) whichis considered as the upper bound value of the ductile failure cri-terion. Also, constraint~16! indicates that during steady-state cut-ting, any contact mechanism characteristics should be avoided;otherwise, the elementd is removed before the elementh is re-moved in Fig. 3. This phenomenon takes place if«o

pl is overesti-mated so that the workpiece is largely dragged by the tool. There-fore, although Eq.~11! and~13! provide the initiation point for thesteady-state cutting simulation, the ductile failure criterion valuesshould be accordingly adjusted by constraints~15! and ~16!. Fig-ure 4 shows the mesh deformation in steady-state cutting andexhibits the characteristics of orthogonal cutting such as primaryshear zone, secondary shear zone, etc. The analysis shows that theductile failure criterion approach to chip separation from theworkpiece is as effective as the other separation criteria discussedealier.

4.2 Ductile Failure Criterion From the Initiation to Pivot-ing Point Stage. In this part, we utilize the information obtainedfrom the previous FEM study. The initiation stage represents thepoint where the plastically deformed region appears on the partedge. The main contributor to the initiation is the high compres-sion developing at the edge of the workpiece; whereas, high ten-sile stress beneath the tool edge develops. This equilibrium stateindicates that the burr initiation is similar to bending deformation.Hence, after the initiation stage, bending takes place at the edge ofthe workpiece. This stage is called the initial development stage.As the tool approaches the edge of the workpiece, it is found thatthe effective plastic strain of the element closest to the tooledge monotonically increases with an increase of the amount ofbending:

d«c

d«e.0, (17)

where«c represents the effective plastic strain closest to the tooltip and forms a part of the chip, and«e is the largest effectiveplastic strain at the edge of the workpiece, representing theamount of bending at the edge. It now follows that we introducethe interpolating function:

Fig. 3 Metal-cutting simulation procedure by ductile failuremodel

Fig. 4 Mesh deformation in steady-state cutting


«opl5F~«1 ,«2 , . . . ,«n!5(

i 51

n

F i« i , (18)

where F i is the weighting factor, andi represents the elementnumber surrounding the element contacted with the tool edge inFig. 3. The weighting factor should be subjectively chosen andshould be validated by constraints~15!, ~16!, and~17!. However,one more constraint should be enforced to assure the consistencyof cutting process in this part by

uR2Reu,z, (19)

whereR represents the distance from the reference point to pre-defined machining surface,Re is the distance from the referencepoint to the machining surface obtained from the finite elementsimulation, andz is the element size near the machining surface,shown in Fig. 5. Constraint~19! indicates that the allowable sur-face finish in the finite element simulation should be less than theelement size on the machined surface.

Therefore,«opl should be accordingly determined to assure the

burr formation environment specified by the constraints. Also,D«for the element damage procedure should be determined to assurea continuous burr formation process. In this part of the simulation,the four constraints~15!, ~16!, ~17!, and~19! are enforced to vali-date the ductile failure criterion values obtained from the interpo-lating function~18!. It should be pointed out that constraint~19!would be the most important one because the smaller the elementsize, the more difficult the choice of the ductile failure criterionvalues become despite obtaining more reliable results. In thisstudy, the element size is determined based on constraint~19!with consideration of the computation time.

4.3 Ductile Fracture Criterion From the Pivoting Point toFinal Development Stages. The pivoting point stage representsthe point where material instability at the edge of the workpiecetakes place due to high stress built-up. From this point, cata-strophically large deformation, known as the rollover process, oc-curs instead of bending deformation. In the pivoting stage, thelength from the pivoting point and cutting surface is known asburr thickness. After the pivoting point stage, the chip no longerforms, and material in front of the tool is pushed ahead by bothshearing and plowing processes. This stage is called the final de-velopment stage where the rollover process continues to give finalburr shape or edge breakout. For the simulation from the pivotingpoint to final development stages, constraints~15! and ~19! areenforced to determine the ductile failure criterion values from Eq.~18!. However, constraint~17! is not enforced since no informa-tion is available with respect to the relationship between the ef-fective plastic strain near the tool edge and the amount of rolloverat the edge. Constraint~16! is not appropriate in this case becausethe edge breakout can take place. Therefore, we introduce anotherconstraint instead of constraint~16! that

H « fpl,« f~T! burr

« fpl.« f~T! breakout.

(20)

Constraint~20! can be considered to be the criterion to predict theformation of a burr or edge breakout. It should be noted that theedge breakout phenomenon involves fracture, and a crack formsnear the tool edge and propagates into the workpiece. Althoughthe ductile failure criterion would be inappropriate as a burr/breakout criterion from the viewpoint of fracture mechanics, it isadopted in this study since the precise criterion is not known yet.

As shown in Fig. 6, the finite element model is divided intothree regions. Region I simulates the chip formation, and the duc-tile failure model is not applied by assuming that a continuouschip forms. Region II supports the metal-cutting process. Sincethis region primarily involves elastic deformation, the ductile fail-ure model is omitted. The ductile failure model is applied in re-gion III to simulate metal cutting and burr formation. Also, Fig. 6shows that a total of seven ductile failure criteria are determinedalong the cutting path for the simulation of a thick burr with theirlocations.

5 Simulation ProcedureThe flowchart of the finite element simulation of chip and burr

formation is shown in Fig. 7. First, the simulation starts withassigning the ductile failure criterion values for steady-state cut-ting. If the ductile failure criterion values are not properly as-signed, then they are adjusted according to the constraints. Other-wise, the ductile failure criterion values are kept and used todescribe the steady-state cutting process. When it reaches the ini-tiation stage, the ductile failure criterion values for steady-statecutting are no longer valid. From this point, the ductile failurecriterion values must be constantly adjusted based on the con-straints to describe the burr formation process. This procedure iscontinuously repeated until all ductile failure criterion values from

Fig. 5 Constraint for allowable machined surface finish

Fig. 6 Element groups of a FEM model and cutting path inthick burr simulation


the steady-state cutting to final development stages are found.Table 2 shows the ductile failure criterion values applied to thesimulation of the thick and thin burrs. However, it should be notedthat the complete temperature-dependent maximum fracture strainvalues of SS304L are not available for this study. Therefore, weassume that the ductile failure criterion values are validated for aburr formation process in Table 2. The computation is performedon an IBM RS6000 Model 590 workstation.

6 ResultsThe results from the finite element simulation are presented in

this section to enhance the understanding of the burr formationmechanism. With the proposed metal-cutting simulation proce-dure, the study is extended to investigate the physical phenom-enon occurring in the final burr development stage. The charac-teristics of thick and thin burrs in orthogonal cutting are clarifiedalong with the investigation of the development of the pivoting

point and the negative deformation zone. The simulated results arecompared and verified with SEM micrographs obtained from theactual metal-cutting data using the micro machining stage de-signed by Stiles@17# and modified by Ko@3#.

In order to explain the development of the pivoting point, asimple model, shown in Fig. 10, is introduced. In the initiationstage, yielding occurs at the edge of the workpiece, and the maincontributor to the yielding is the compressive stress developing atthe edge of the workpiece. This phenomenon is similar to thebending mechanism. Also, in this stage, the initial pivoting pointis established. In Fig. 10, the cutting force acting on the machinedsurface causes the bending or deflection of the workpiece. On theother hand, since the workpiece is elastically deformed below theyielding zone, it is assumed that the workpiece from the yieldingzone to the unmachined workpiece surface is attached and fixed atthe workpiece below the yielding zone. Therefore, in order tostatically balance forces and moments, i.e.,(F5(M50, thenegative moment must be introduced around the yielding zone. Inthe finite element model, this moment is displayed in terms of thepositive shear. The development of the positive shear is quiteimportant to determine the final pivoting point in burr formation.Figure 11 shows the development of the pivoting point whichinitially forms quite a bit below the predefined machined surfaceand gradually moves up to form at the edge of the workpiece inthe pivoting point stage. The initial formation of the pivotingpoint is quite important not only for determining the burr thick-ness but also for the development of burrs in the final develop-ment stage.

The prediction of the final location of the pivoting point is quiteimportant for the deburring process because the length of the burrthickness is measured from the machined surface to the final lo-cation of the pivoting point. As mentioned previously, the timerequired to deburr a part increase exponentially with an increaseof burr thickness. It is found that the location of the pivoting pointwould be largely determined by the plastic work rate required inthe steady-state cutting since the burr initiation is similar to bend-ing deformation. Hence, in order examine the burr thickness char-acteristics without any significant change of shear angle, two dif-

Fig. 7 Flowchart of metal cutting and burr formation simula-tion

Fig. 8 Progressive change of geometry in thick burr simula-tion

Fig. 9 Progressive change of geometry in thin burr simulation

Table 2 Ductile failure criterion for thick and thin burrs


ferent values ofD« in Eq. ~10! are assigned. However, it shouldbe noted thatD« is chosen to examine the influence of the totalstrain energy for ductile failure modes on the burr formation pro-cesses within the bounds of the constraints. Figures 8 and 9 showthe progressive changes of geometry at the edge of the workpieceof the thick and thin burr formation, respectively. With a smallvalue ofD«, the thin burr is formed; whereas, with a large valueof D«, the thick burr is formed. They clearly exhibit that a plastic

hinge, known as the pivoting point is formed at the edge of theworkpiece, as described in Fig. 10. Also, according to numericalresults, for a thick burr, the location of the pivoting point from thepredefined machined surface appears further below the depth ofcut (l c.t), and the edge of the workpiece exhibits a smoothcontour. On the other hand, in the thin burr case,l n,t, and theedge of the workpiece shows a sharp corner. Chern and Dornfeld@4# have developed a formula to predict the burr size in orthogonalcutting. One of the parameters in the equation is the angle of thenegative deformation zone. After the initiation stage, the shearangle tends to decrease as the edge of the workpiece deflects. Asa result, high negative shear stress in front of the tool edge devel-ops; whereas, positive shear is formed around the pivoting pointin Fig. 11. As the tool further approaches the edge of the work-piece, the negative shear formed near the tool edge increases andgreatly influences the rollover process. In a thin burr case, Fig. 12shows that the negative shear formed near the tool edge extends tothe edge of the workpiece as seen in the deformed mesh, and theresult is compared to a SEM micrograph of micro-burr formation.As a result, the rollover process is initiated by the shearing effect,which results in a sharp corner at the edge of the workpiece. Onthe other hand, in a thick burr case, the pivoting point is formedquite below the predefined machined surface; consequently, thenegative deformation zone does not extend to the edge of theworkpiece, as shown in Fig. 13. Since the shearing effect is notaccounted for during the rollover process, the edge shows asmooth contour. This is also compared to a micrograph from mi-cromachining stage inside the SEM.

Since a burr generally forms for machining conditions usedhere for SS304L, edge breakout phenomenon is not considered inthis study. However, it now becomes clear that if the edgebreakout occurs, a crack would form in the negative deformationzone due to excessive distortion of the material near the tool edge.As the tool further approaches the edge of the workpiece, thecrack would initially propagate through the negative deformationzone by mode II~shear rupture!. And later the edge of the work-piece would be separated from the main body by mode I openingtype of fracture@18# ~normal rupture!.

Fig. 10 Formation of positive shear on the workpiece edge

Fig. 11 Development of positive shear on the workpiece edgein burr formation

Fig. 12 Negative deformation zone and SEM micrograph of athin burr

Fig. 13 Negative deformation zone and SEM micrograph of athick burr


7 ConclusionA finite element method of a metal-cutting process including

burr formation has been presented. A new procedure for a chipseparation criterion has been proposed in this study. This proce-dure is based on the ductile failure criterion included inABAQUS/Explicit FEM package. The finite element simulation isdivided into three parts: steady-state cutting, initiation to pivotingpoint stage, and pivoting point to final development stage for burrformation. In each part, several constraints are enforced to deter-mine the ductile failure criterion values in order to assure that themetal cutting simulation is as close to reality as possible. And anadiabatic heating model is adopted to include the heat generatedby the element distortion consistent with the temperature depen-dent material properties of SS304L.

The physical phenomenon occurring in the final developmentstage is more focused than the other stages, i.e., initiation, initialdevelopment, and pivoting point stages. The characteristics of thethick and thin burrs are clarified along with the investigation ofthe negative deformation zone based on the numerical results. It isfound that the formation of thick and thin burrs are significantlyinfluenced by the size of the negative deformation zone. The thinburr forms when the development of high negative shear near thetool edge extends to the edge of the workpiece; consequently, asharp corner is formed at the edge due to the shearing effectduring the rollover process. On the other hand, a thick burr exhib-its a smooth contour on the edge of the workpiece because thenegative deformation zone does not extend to the edge of theworkpiece.

This work attempts to clarify the burr formation mechanism inorthogonal cutting, especially, for the final development stage,and also to introduce metal cutting simulation using the ductilefailure criterion in order to obtain a quantitative analysis of burr/breakout formation.

AcknowledgmentsThe authors would like to thank the members of the Consortium

on Deburring and Edge Finishing~CODEF! at the University ofCalifornia, Berkeley for their financial support. The authors wouldalso like to thank Dr. N. Rebelo~Hibbitt, Karlsson, and Sorenson,Inc.! for his helpful comments and suggestions.

References@1# Gillespie, L. K., and Blotter, P. T., 1976, ‘‘The Formation and Properties of

Machining Burrs,’’ ASME J. Eng. Ind.,98, Feb., pp. 64–74.@2# Iwata, K., Ueda, K., and Okuda, K., 1982, ‘‘Study of Mechanism of Burrs

Formation in Cutting Based on Direct SEM Observation,’’ J. Japan Society ofPrecision Engineering,48-4, pp. 510–515.

@3# Ko, S. L., and Dornfeld, D. A., 1991, ‘‘A study on Burr Formation Mecha-nism,’’ ASME J. Eng. Ind.,98, No. 1, pp. 66–74.

@4# Chern, G. L., 1993, ‘‘Analysis of Burr Formation and Breakout in MetalCutting,’’ Ph.D. Thesis, University of California, Mechanical Engineering De-partment, pp. 25–42.

@5# Chern, G. L., and Dornfeld, D. A., 1996, ‘‘Burr/Breakout Development andExperimental Verification,’’ ASME J. Eng. Mater. Technol.,118-2, pp. 201–206.

@6# Pekelharing, A. J., 1978, ‘‘The Exit Failure in Interrupted Cutting,’’ CIRPAnn., 27, pp. 5–10.

@7# Klamecki, B. E., 1973, ‘‘Incipient Chip Formation in Metal Cutting—ThreeDimension Finite Element Analysis,’’ Ph.D. dissertation, Univ. of Illinois,Urbana-Champaign.

@8# Usui, E., and Shirakashi, T., 1982, ‘‘Mechanics of Machining-From ‘Descrip-tive ’ to ‘Predictive’ Theory,’’ On the Art of Cutting Metals—75 Years Later,ASME Publication PED,7, pp. 13–35.

@9# Strenkowski, J. S., and Carroll, III, J. T., 1985, ‘‘A Finite Element Model ofOrthogonal Metal Cutting,’’ ASME J. Eng. Ind.,107, pp. 349–354.

@10# Park, I. W., Lee, S. H., and Dornfeld, D. A., 1994, ‘‘Modeling of Burr For-mation Processes in Orthogonal Cutting by the Finite Element Method,’’ESRC Report No. 93–34, Univ. of California, Berkeley, Dec.

@11# Komvopoulos, K., and Erpenbeck, S. A., 1991, ‘‘Finite Element Modeling ofOrthogonal Metal Cutting,’’ ASME J. Eng. Ind.,113, No. 3, pp. 253–273.

@12# Hibbitt, Karlsson, and Sorenson, Inc., 1988,ABAQUS/Explicit User’s Manu-als, Version 5.3, Providence, RI.

@13# Wright, P. K., 1982, ‘‘Predicting the Shear Plane Angle in Machining fromWorkmaterial Strain-Hardening Characteristics,’’ ASME J. Eng. Ind.,104, No.3, pp. 285–292.

@14# Lin, Z. C., and Lin, S. Y., 1992, ‘‘A Coupled Finite Element Model ofThermo-Elastic-Plastic Large Deformation for Orthogonal Cutting,’’ ASME J.Eng. Mater. Technol.,114, pp. 218–226.

@15# Cockcroft, M. G., and Latham, D. J., 1968, ‘‘Ductility and Workability ofMetals,’’ J. Inst. Met.,96, pp. 33–39.

@16# ASME Research Committee on Metal Cutting Data and Bibliography, 1952,Manual on Cutting of Metals with Single-Point Tools, The American Societyof Mechanical Engineering.

@17# Stiles, T. A., 1985, ‘‘A Scanning Electron Microscope~SEM! Machining Sub-stage for Metal Cutting Observation and Acoustic Emission Analysis,’’ MSthesis, Univ. of California, Berkeley, Department of Mechanical Engineering.

@18# Barsom, J. M., and Rolfe, S. T., 1987,Fracture & Fatigue Control in Struc-tures, Prentice-Hall Inc., New Jersey.


I. W. ParkIntegrated Surgical Systems, Inc.,

1850 Research Park Dr.,Davis, CA 95616

D. A. DornfeldFellow ASME

Department of Mechanical Engineering,University of California, Berkeley,

Berkeley, CA 94270-1740

A Study of Burr FormationProcesses Using the FiniteElement Method: Part II—TheInfluences of Exit Angle, RakeAngle, and Backup Material onBurr Formation ProcessesFinite element models in orthogonal cutting are presented in order to examine the influ-ences of exit angles of the workpiece, tool rake angles, and backup materials on burrformation processes in 304 L stainless steel in particular. Based on the metal-cuttingsimulation procedure proposed by the authors, a series of stress and strain contours andfinal burr/breakout configurations are obtained. The burr formation mechanisms withrespect to five different exit angles are found, and duration of the burr formation processincreases with an increase of exit angle, resulting in different burr/breakout configura-tions. Based on the development of negative shear stress in front of the tool tip, the tooltip damage, what is called ‘‘chipping,’’ is investigated. Also, with fixed cutting conditionsand workpiece exit geometry, the influence of the rake angle is found to be closely relatedto the rate of plastic work in steady-state cutting because the larger the rate of plasticwork in steady-state cutting, the earlier the burr initiation commences. Furthermore, inorder to effectively minimize the burr size, three cases of backup material influences onburr formation processes are examined. It is found that the burr size can be effectivelyminimized when the backup material supports the workpiece only up to the predefinedmachined surface.@S0094-4289~00!01402-X#

1 Introduction

Rollover burrs, the result of material behavior at the end of an‘‘orthogonal-like’’ cutting process, are formed in most machiningoperations. Some attempts have been made to analyze rolloverburrs by controlling the cutting conditions and material properties.The first quantitative analysis of burr formation was carried out byGillespie @1# assuming that the burr is formed by bending defor-mation of the end portion of the workpiece. Ko and Dornfeld@2#developed a model for burr formation in material exhibiting duc-tile or brittle behavior and evaluated it at slow machining speeds.Further, Chern and Dornfeld@3,4# continued the development ofburr formation models, extended Ko and Dornfeld’s model withmore realistic machining operations and conditions. Within theboundary of the model and cutting conditions, the predictionsfrom Chern and Dornfeld’s orthogonal cutting model correspondwell with the actual measured burr/breakout values from orthogo-nal impact machining tests. Pekelharing@5# tried to explain thepoor tool life often seen in interrupted cutting with sharp tools atthe exit of a cut. In his study, the interrupted cutting was carriedout on the workpieces with various exit angles, defined as theangle between the cutting velocity and the edge of the workpiece.Iwata et al. @6# observed the burr formation during machininginside a scanning electron microscope~SEM! and experimentallydetermined the effect of the exit angles of the workpiece on the

burr size. They explained the fracture of the workpiece at the tooledge using the strain obtained by a finite element method~FEM!analysis.

Park et al.@7# conducted a finite element simulation of the burrformation using the general purpose finite element software pack-age, ABAQUS/Standard, and attempted to model the process asclose to reality as possible. In this study, based on the progressivechanges of the edge of the workpiece along with a series of stressand strain contours, the fundamental burr formation mechanismwas observed and divided into four stages: initiation, initial devel-opment, pivoting point and final development. However, due toseveral assumptions used in the finite element model, the simula-tion study on the burr formation process could not be completedto provide the final burr/breakout configuration. Recently, Parkand Dornfeld@8# continued the development of the finite elementmodel and extended the previous model with explicit dynamicsfinite element software, ABAQUS/Explicit. In this study, themetal-cutting simulation procedure form the steady-state to finaldevelopment stage was developed not only to confirm the funda-mental burr formation mechanism obtained from the previousstudy @7#, but also to provide a quantitative analysis of the finalburr/breakout configuration. Furthermore, the physical phenom-enon occurring in the final development stage was intensivelystudied. As a result, the characteristics of thick and thin burr for-mation were clarified along with the investigation of the negativedeformation zone.

The objective of this study is to examine the influences of exitangles of the workpiece edge, tool rake angles, and backup mate-rials on the burr formation processes using the finite elementmethod. Finite element models with various exit angles at theworkpiece edge and tool rake angles are developed in this study.

Contributed by the Materials Division for Publication in the JOURNAL OF ENGI-NEERING MATERIALS AND TECHNOLOGY. Manuscript received by the MaterialsDivision June 26, 1998; revised manuscript received December 31, 1999. AssociateTechnical Editor: Kwai S. Chan.


In a drilling operation, backup material is often placed behind thedrill exit surface of the workpiece not only to minimize the burrsize but also to reduce the residual shear stress formed around thehole. Thus, in this study, the use of backup material, which comesinto perfect contact with the workpiece edge, is modeled to ob-serve the influence of the backup material in orthogonal cutting.In each simulation, the metal cutting simulation procedure devel-oped in the burr formation study@8# is applied, and thetemperature-dependent material properties of 304 L stainless steel~SS304L! are also incorporated into the finite element analysis.Furthermore, the finite element study includes the effect of dam-age on the tool edge occurring at the exit of cut, and the resultsare also compared with the experimental results obtained byPekelharing@5#.

2 Finite Element ModelingTwo-dimensional finite element models with plane strain as-

sumption to examine the effects of exit angles, rake angles, andbackup materials are developed and displayed in Figs. 1–3. Totalnumber of elements and nodes in each two-dimensional model are1080 and 1160 with plane strain assumption, respectively. In thefinite element model for the workpiece, total five element groupsare assigned and tied together with*CONTACT PAIR definitionand *TIED option @9#. The boundaryB–C–D is constrainedagainst displacements (DU50), and boundaryD–E–F–G–A isunconstrained (DF50) to allow a burr to form. For the backupmaterial simulation, the boundary Q–R is constrained against dis-placements, and the boundaryR–S–P–Q isunconstrained. Thedepth of cut is 0.5 mm, and the tool is assumed to be perfectlyrigid with a slide-line capability. Also, Coulomb friction ofm50.3 is used on the rake surface of the tool to describe the inter-action on the tool-chip interface. Furthermore, the heat model as-sumes adiabatic conditions within each element so that no heattransfer occurs within the workpiece and on the workpiece freesurface for moderate cutting speed processes (n55.0 m/s). Thecutting process including burr formation involves large amountsof inelastic strain, where the heating of the material caused by itsdeformation is an important effect because of temperature-dependence of the material properties, i.e., thermal-elastic-plasticmaterial behavior. The strain rate effect is also considered in themodel because many metals show an increase in their strength asthe strain rate increases during the metal-cutting process.

In order to investigate the exit angle influence on burr forma-tion, a total of six finite element models with exit angles of 60, 80,90, 100, and 120 degrees are introduced and displayed in Fig. 1.The figure shows that the exit angle of each model is referenced topoint E at the edge of the workpiece. The distance from the pre-defined machined surface B–F to point E is determined so as toassure that the significant part of burr formation processes, i.e.,burr initiation, pivoting point, etc., takes place above point E.

Hence the workpiece below point E is furnished simply to supportthe metal-cutting process. For the investigation of rake angle in-fluence, two finite element models are developed with rake anglesof 5 and 20 degrees, displayed in Fig. 2. In this part of the simu-lation, the exit angle of the workpiece and cutting speed are fixedto 90 degrees andn55.0 m/s in each case, respectively. In theinvestigation of the backup material influence, a total of threefinite element models for backup material influences are devel-oped, displayed in Fig. 3. The first model, Fig. 3~a!, examines theburr formation mechanism with a 10 mm thick backup materialwhich comes into perfect contact with the workpiece material.The second model, Fig. 3~b!, use a backup material is half as thickas that in Fig. 3~a! in order to examine the effect of the backupmaterial. Finally, the last model, Fig. 3~c!, examines the effect ofthe backup material supporting the workpiece up to the predefinedmachined surface under the condition of perfect contact.

The most important part of the finite element simulation is thedescription of the metal cutting process including burr formation.In this study, the metal-cutting simulation procedure developed in

Fig. 1 Finite element models of 60, 80, 90, 100, and 120 degreeexit angles

Fig. 2 Finite element models of 5 and 20 degree rake angles

Fig. 3 Finite element models of backup materials


the study done by Park and Dornfeld@8# is applied to each finiteelement simulation. The temperature dependent material proper-ties of SS304L and cutting conditions are displayed in Table 1.Table 2 shows the sequence of the ductile failure criterion deter-mined for the simulation of 80 degree exit angle and 100 degreeexit angle as the tool advances toward the edge of the workpiece.

3 Finite Element Simulation Results

3.1 Exit Angle Influences. The focus of the finite elementstudy in this section is to investigate the location of a plastichinge, known as the pivoting point, with respect to the exit angleof the workpiece edge. The exit angle is defined as the angle

between the cutting velocity vector in orthogonal cutting and theworkpiece edge, Fig. 1. It is important to observe the location ofthe pivoting point since it determines burr thickness. Also, it isimportant to observe the duration of the burr formation processand how the edge deflection and rollover develop with regard tothe exit angle variation. These play an important part in the de-velopment of negative shear near the tool edge. As Pekelharing@5# pointed out, the negative shear near the tool edge is the maincause of the tool tip damage, known as ‘‘chipping,’’ at the exit ofa cut.

In the 60 degree exit angle simulation, the pivoting point at theedge of the workpiece appears quite close to the machined sur-face, Fig. 4. Also, Fig. 5 shows that the shear stress contour infront of the tool edge reveals a similar pattern seen in the contactmechanisms@10#, which indicates that the workpiece material infront of the tool would no longer form a chip. Moreover, since thepivoting point is too close to the machined surface, this particularstress environment could not allow the workpiece edge to rollover. Hence since the tool moves with the uniform speed, a modeII shearing type of fracture@11# would most likely take place.Consequently, it leads to edge breakout phenomenon displayed inFig. 6. As a result, due to the edge breakout, the burr size, thick-ness or height, for example, can be effectively minimized with alow exit angle at the workpiece edge.

When the exit angle is 80 degrees, the pivoting point appears toform a little below the machined surface in Fig. 7 compared withthe 60 degree exit angle case. And this environment could allow

Fig. 4 Equivalent stress contour of 60 degree exit angle

Fig. 5 Shear stress contour of 60 degree exit angle

Table 1 Material properties of SS304L and cutting conditions

Table 2 Ductile failure criterion for 80 and 100 degree exitangles


the workpiece edge to roll over to certain degree. Also, with thedevelopment of positive shear as shown in Fig. 8, the negativedeformation could appear in this case. However, as the tool ap-proaches the edge of the workpiece, according to the shear stresscontour in front of the tool edge in Fig. 8, the contact mechanismcharacteristics still appears and dominate for 304 L stainless steel,which would lead to the edge breakout phenomenon. It is alsoimportant to note that a mode II shearing type of fracture initiallyoccurs; however, due to the edge rollover, the crack later propa-gates with a mode II opening type of fracture@11#, shown in Fig.9. As a result, the burr size can be also minimized due to the

Fig. 6 Edge breakout of 60 degree exit angle



Fig. 9 Edge breakout of 80 degree exit angle




breakout. However, there exists a small burr, a remnant from theedge rollover, as shown in Fig. 9.

When the workpiece with a 90 degree exit angle is machined,Fig. 10 shows that the pivoting point forms quite below the ma-chined surface compared to the 60 and 80 degree exit angle case.

Furthermore, the magnitude and size of the positive shear stressconsiderably increase, and the effect of the contact mechanismdecrease, shown in Fig. 11. Consequently, the mode II shearingtype of fracture would not take place, and the workpiece edgewould most likely roll over. Thus the burr formation would bemore likely to occur than edge breakout phenomenon around the90 degree exit angle under the given machining conditions andmaterial properties. During the rollover process, the magnitude ofthe negative shear stress near the tool edge further develops, lead-ing to the formation of the negative deformation zone, shown inFig. 12. This has been experimentally observed by Chern andDornfeld@3# and verified by Park and Dornfeld@8# using the finiteelement method.

As the exit angle increases up to 100 and 120 degrees, themagnitude and size of the positive shear stress further increase,Fig. 13, compared with the lower exit angle cases. Also, it can beobserved from Fig. 14 that the higher the exit angle becomes, thelower the location of the pivoting point becomes from the ma-chined surface. As a result, Fig. 15 shows that the burr size be-comes relatively thick and long compared with the 90 and lowerexit angle case although the amount of edge rotation during therollover process decreases due to the exit geometry of the work-piece. Also, it can be observed that with an increase of exit angle,the influence of negative deformation zone on the edge of theworkpiece decreases leading to a smooth edge contour. Therefore,from the viewpoint of burr minimization and economical deburr-ing, the workpiece with a large exit angle should be avoided.

Although the tool is assumed to be perfectly rigid in the finiteelement model, tool tip damage during the burr/breakout forma-

Fig. 12 Formation of negative deformation zone in burr forma-tion

Fig. 13 Shear stress contours of 100 and 120 degree exitangle

Fig. 14 Equivalent stress contours of 100 and 120 degree exitangle


tion can be indirectly examined through a series of shear stresscontours of the workpiece near the tool edge. Pekelharing@5#conducted an experiment with regard to the number of cuts beforetool tip damage, i.e., ‘‘chipping,’’ occurs with respect to the exitangle, Fig. 16. The figure shows that the most severe damage onthe tool edge occurs around an exit angle of 90 degrees. On theother hand, the tool tip damage is considerably reduced with bothincreasing and decreasing the exit angles around 90 degrees de-spite the different burr/breakout configurations. It is known thatthe tool tip damage occurring at the exit of cut is mainly associ-ated with the development of the shear stress near the tool edgeduring the rollover process. The development of the negativeshear stress of the element closest to the tool edge was investi-gated as the tool approaches the edge of the workpiece. The nu-merical results are displayed in Fig. 17. First of all, it is importantto observe that the duration of the burr formation process from theedge where burr formation commences, i.e., the burr initiation,increases with an increase of exit angle, which is also closelyrelated to the development of the shear stress closest to the tooledge.

With a 60 degree exit angle, the negative shear remains almostconstant prior to edge breakout. In other words, when the tooledge is approximately 1.5 mm behind the workpiece edge, thebreakout occurs immediately after the pivoting point stage fol-lowed by burr initiation, and its development. Second, any signifi-cant change in shear stress of the element closest to the tool edgeis not observed. Instead, due to the contact mechanism character-istics, the higher shear stress in front of the tool edge developsprior to the edge breakout. Thus there is no time for the negativeshear to develop and effectively damage the tool tip. With the 80

Fig. 15 Final burr configuration of 100 and 120 degree exitangles

Fig. 16 Pekelharing’s tool life experiment results in orthogo-nal cutting, from †5‡

Fig. 17 Development of negative shear stress closest to thetool edge


degree exit angle, it is observed that the edge rollover initiatesprior to the edge breakout. As shown in Fig. 17, after the rapidincrease of the magnitude of the negative shear stress caused bythe rollover process, the edge breakout immediately occurs. Thiscould lead to another sudden change in the stress field at the tooledge. Under the circumstances, the tool tip would undergo a me-chanical impact-shock at the exit of cut, which could be substan-tially harmful to the tool tip. From the investigation of the 60 and80 degree exit angles, it can be deduced that the magnitude of amechanical impact-shock would depend on the amount of theedge rollover prior to the breakout. Thus the lower the exit anglebecomes, the less edge rollover there is, which obviously leads tolonger tool life.

When the exit angle is 90 degrees, Fig. 17 shows that the burrinitiation occurs early compared with the 80 degree exit anglecase. The sudden decrease of the negative shear stress takes placeimmediately after the rapid increase of the negative shear stressdue to the rollover process. Hence the development of the nega-tive shear stress in the 90 degree exit angle would be similar tothat in the 80 degree exit angle although a burr forms instead ofbreakout. Thus the tool tip could be still vulnerable to fracture dueto a mechanical impact-shock.

On the other hand, as the exit angle increases to 100 and 120degrees, the burr initiation occurs quite early compared with thelower exit angle cases. Instead of the rapid change in the shearstress closest to the tool edge, the magnitude of the negative shearstress gradually increases during the initial development stage andgradually decreases during the edge rollover. This indicates that amechanical impact-shock could be considerably relieved by thegradual increase and decrease of the negative shear stress com-pared with the 80 and 90 degree exit angle cases. As a result, thedamage on the tool tip could be considerably reduced with anincrease of the exit angle.

3.2 Rake Angle Influences. In this section, the finite ele-ment simulation is conducted to examine the rake angle influenceson the burr formation. The rake is defined as the angle from theline perpendicular to the cutting velocity vector in orthogonal cut-ting to tool rake surface, Fig. 2. The rake angles of 5 and 20degrees are chosen with a fixed 90 degree exit angle in both cases.In steady-state cutting, the rate of plastic work is dependent on theshear angle forming from the cutting edge to workpiece free sur-face. According to Rowe and Spick’s simple model@12# based onthe minimum energy approach, i.e., the Hamiltonian principle be-ing used, Wright@13# showed that the minimum power of primaryshear for machining commercial steel with low friction on thetool-chip interface arises when the shear angle is around 48 de-grees for a rake angle of 6 degrees. It is important to consider therate of plastic work in steady-state cutting in the study of the burrformation because the mechanism of burr initiation is quite simi-lar to the bending mechanism. Thus, it can be deduced that thelarger the rate of plastic work in steady-state cutting, the earlierburr initiation begins. The simulation of steady-state cutting ofSS304L in this study yielded the shear angles of approximately 28and 40 degrees for rake angles of 5 and 20 degrees, respectively.Therefore, under the cutting conditions specified in this study,especially, with low friction on the tool-chip interface, the rate ofplastic work for a 5 degree rake angle tool would be greater thanthat for a 20 degree rake angle tool in steady-state cutting. Thiscertainly makes qualitative sense.

In case of the 20 degree rake angle tool, the numerical resultshows that the burr initiation takes place whenl 20>3.3t in Fig.18, wherel 20 represents the burr initiation distance from the tooltip to the edge of workpiece with a 20 degree rake angle tool, andt is the depth of cut. For a 5 degree rake angle tool, the burrinitiation takes place whenl 5>4.5t, Fig. 19. Also, it can be ob-served from Figs. 18 and 19 that the location of yielding at theedge of the workpiece with the 20 degree rake angle is closer tothe machined surface than that with the 5 degree rake angle. As aresult, a thinner burr would be obtained with the 20 degree rake

angle tool than that with the 5 degree rake angle tool. Gillespie@14# pointed out that for hand deburring of precision parts, thetime required to deburr a part increases exponentially with in-creased burr thickness and burr accessibility. Therefore, it wouldbe helpful to machine with a higher rake angle tool. However,from a viewpoint of tool wear, as the rake angle increases, the tooltip is mechanically more vulnerable to fracture. Furthermore, amechanical impact-shock at the exit of cut could further damagethe tool tip if the duration of the burr formation process is short.Hence it would be desirable to find the optimal selection of cut-ting tool geometry with the consideration of both deburring effortsand tool wear.

3.3 Back-Up Material Influences. The most effective wayto minimize the burr size would be to put a back-up materialbehind the edge of the workpiece. Gillespie@15# conducted ex-periments to examine the backup material influence for the pur-pose of burr minimization in drilling. Based on the Gillespie’sexperiments, finite element models are analyzed to investigate thebackup material influence on burr formation. In this study, abackup material, whose material properties are twice as stiff~Young’s modulus! and twice as strong~yielding and ultimatestrength! as those of the workpiece material, is used.

Backup materials with a thickness of 10 mm and 5 mm are

Fig. 18 Location of burr initiation of 20 degree rake angle

Fig. 19 Location of burr initiation of 5 degree rake angle


placed at the end surface of the workpiece with condition of per-fect contact between the two materials. As seen earlier, after theburr initiation, deflection at the edge of the workpiece occurs.With the 10 mm thick backup material, the deflection of the work-piece causes local deformation of the backup material at point P,Fig. 20, and the backup material significantly reduces the amountof deflection at the workpiece edge. As a result, continuous chipformation continues until the tool advances toward the very end ofthe workpiece, and fracture takes place at the last moment. Con-sequently, the burr can be effectively minimized. However, theslight deflection at the workpiece edge creates a small gap be-tween the two materials. Therefore, depending on the size of thegap, a small burr, a remnant from the fracture, would be expectedsince the size of the gap is an indirect measure of the amount ofdeflection or rollover at the workpiece edge.

With the 5 mm thick backup material, instead of local defor-mation, the whole backup material exhibits the bending character-istics. Around the fixed boundary Q–R, the normal stress contourin y-direction in Fig. 21, shows that the tensile and compressivestress develop at point Q and R, respectively. The bending of theback-up material results in a large gap between the two materialscompared with the 10 mm thick backup material case, which in-dicates that the workpiece edge with the 5 mm thick backup ma-terial deflects more than with the 10 mm thick backup materialprior to the end of cut. In this case, the burr size also can beeffectively minimized although a relatively large remnant, com-pared to the 10 mm backup material case, would be expected to beleft at the edge. As a result, it would be desirable to have backupmaterials thick enough to cause only local deformation near theedge of the workpiece by avoiding the bending of the backupmaterial.

The final simulation examines the case when the 5 mm backupmaterial contacted with the workpiece only up to the predefinedmachined surface. A similar case has been experimentally carriedout by Gillespie@15# to minimize the size of a drilling burr. Al-though orthogonal cutting is quite different from drilling, the char-acteristics of the rollover process in orthogonal cutting would besimilar to those seen in drilling. Initially, the deflection of theedge above the predefined machined surface takes place, whichbrings point P into contact with point F instead of forming a gapbetween the two materials. As the tool further approaches theedge, the burr size is effectively minimized by a mode II shearingtype of fracture before the tool edge reaches the edge of the work-piece, in Fig. 22, which quite resembles the mechanism in the caseof a 60 degree exit angle. As a result, this effectively reduces thesize in any resulting burr and would be also the mechanism be-hind minimizing the size of a drilling burr in Gillespie’s experi-ment.

4 ConclusionA finite element analysis has been performed to examine the

influences of exit angles at the edge of the workpiece, rake angles,and backup materials on the burr formation processes based on themetal cutting simulation procedure proposed in the study done byPark and Dornfeld@8#.

In the study of exit angle influences, when the exit angle at theedge is 60 degrees, the edge is the more susceptible to breakoutthan burr formation. It has been found that the pivoting pointforms close to the machined surface as the exit angle decreases.As a result, the burr size can be effectively minimized becausethis particular environment triggers breakout by a mode II shear-ing type of fracture at the edge. On the other hand, when the exitangle is 90 degrees, burr formation is more favorable than break-out. From this point, considerable deburring efforts would be re-quired. As the exit angle further increases, the amount of edgerotation during the rollover process is reduced due to the exitgeometry; however, a burr becomes relatively thick and long com-pared with the 90 degree exit angle case since the pivoting pointforms quite below the machined surface. From the viewpoint of

Fig. 20 Equivalent stress contour with 10 mm backup material

Fig. 21 Normal stress contour in y -direction with 5 mmbackup material

Fig. 22 Influence of backup material partially supportingworkpiece


tool wear, it has been found that 80 to 90 degree exit angle wouldbe the most damage to the tool tip because the sudden change ofshear stress in front of tool tip creates a mechanical impact-shockcharacteristic. On the other hand, as the exit angle increases, thetool tip damage can be substantially reduced because a mechani-cal impact-shock on the tool tip is relieved due to the long dura-tion time of the burr formation process. In other words, the shearstress field closest to the tool edge does not involve impact-shockcharacteristics. As the exit angle decreases, the tool tip damagecan be also reduced because a mechanical impact-shock hardlydevelops due to a decrease in the amount of edge rotation due tothe development of the contact mechanism characteristics in frontof the tool edge.

With a fixed exit angle of 90 degrees, the location of burrinitiation is dependent on the rake angle in terms of the rate ofplastic work, i.e., the larger the rate of plastic work in the steady-state, the earlier the burr initiation commences. Thus in order toobtain a thin burr for an economical deburring process, it wouldbe desirable to find cutting conditions which assure an optimalrate of plastic work while considering the tool wear. The use ofbackup material is the most effective way to minimize the burrsize. It has been found that the backup material should be thickenough to allow only local deformation without bending of thebackup. Also, an effective way to minimize burr size is to placethe backup material in contact with the workpiece at the pre-defined machined surface.

This study has shown that the finite element method is a valu-able approach to understand the burr formation mechanism in ma-chining and the influences of workpiece exit angles, tool rakeangles, and backup materials. Within the limits of the finite ele-ment models, the numerical results provide the physical insightinto the burr/breakout formation and qualitatively agree with theexperimental results.

AcknowledgmentsThe authors would like to thank the members of the Consortium

on Deburring and Edge Finishing~CODEF! for their financial

support at the University of California, Berkeley. The authorswould also like to thank Dr. N. Rebelo of Hibbitt, Karlsson, andSorenson, Inc. for his helpful comments and suggestion.

References@1# Gillespie, L. K., and Blotter, P. T., 1976, ‘‘The Formation and Properties of

Machining Burrs,’’ ASME J. Eng. Ind.,98, pp. 64–74.@2# Ko, S. L., and Dornfeld, D. A., 1991, ‘‘A study on Burr Formation Mecha-

nism,’’ ASME J. Eng. Ind.,112, No. 1, pp. 66–74.@3# Chern, G. L., 1993, ‘‘Analysis of Burr Formation and Breakout in Metal

Cutting,’’ Ph.D. Thesis, University of California, Mechanical Engineering De-partment, pp. 25–42.

@4# Chern, G. L., and Dornfeld, D. A., 1996, ‘‘Burr/Breakout Development andExperimental Verification,’’ ASME J. Eng. Mater. Technol.,118-2, pp. 201–206.

@5# Pekelharing, A. J., 1978, ‘‘The Exit Failure in Interrupted Cutting,’’ Ann.CIRP,27, pp. 5–10.

@6# Iwata, K., Ueda, K., and Okuda, K., 1982, ‘‘Study of Mechanism of BurrsFormation in Cutting Based on Direct SEM Observation,’’ JSPE,48-4, pp.510–515.

@7# Park, I. W., Lee, S. H., and Dornfeld, D. A., 1994, ‘‘Modeling of Burr For-mation Processes in Orthogonal Cutting by the Finite Element Method,’’ESRC Report No. 93-34, Univ. of California, Berkeley, Dec.

@8# Park, I. W. and Dornfeld, D. A., 1995, ‘‘A Study of Burr Formation ProcessesUsing the Finite Element Method Part I,’’ ESRC Report No. 95-32, Univ. ofCalifornia, Berkeley, Sept., 1995.

@9# Hibbitt, Karlsson, and Sorenson, Inc., 1988ABAQUS/Explicit User’s Manuals,Version 5.3, Providence, RI.

@10# Hills D. A., and Novell D., 1994,Mechanics of Fretting Fatigue, KluwerAcademic Publisher, London.

@11# Barsom, J. M., and Rolfe, S. T., 1987,Fracture & Fatigue Control in Struc-tures, Prentice-Hall, Englewood Cliffs, NJ.

@12# Rowe, G. W., and Spick, P. T., 1967, ‘‘A New Approach to Determination ofthe Shear Plane Angle in Machining,’’ ASME J. Eng. Ind.,89, pp. 530–538.

@13# Wright, P. K., 1982, ‘‘Predicting the Shear Plane Angle in Machining fromWorkmaterial Strain-Hardening Characteristics,’’ ASME J. Eng. Ind.,104, pp.285–292.

@14# Gillespie, L. K., 1975, ‘‘Hand Deburring Precision Miniature Parts,’’ Precis.Eng.,1, No. 4, pp. 189–198.

@15# Gillespie, L. K., 1975, ‘‘Burrs produced by Drilling,’’ Bendix Corporation,Unclassified Topical Report BDX-613-1248, Dec., 1975.


K. T. KimProfessor, Assoc. Mem. ASMEe-mail: [email protected]

S. W. Choi

H. ParkDepartment of Mechanical Engineering,

Pohang University of Science and Technology,Pohang 790-784, Korea

Densification Behavior of CeramicPowder Under Cold CompactionDensification behavior of ceramic powder under cold compaction was investigated. Ex-perimental data were obtained for zirconia powder under triaxial compression with vari-ous loading conditions. For densification of ceramic powder during cold compaction, anovel hyperbolic cap model was proposed from the iso-density curves based on experi-mental data of zirconia powder under triaxial compression. The proposed model wasimplemented into a finite element program (ABAQUS) to study densification behavior ofzirconia powder under die compaction. The modified Drucker–Prager/cap model wasalso employed to compare with experimental data and the finite element results from theproposed model in the present work. By including the effect of friction between thepowder and die wall, density distributions of a zirconia compact were measured andcompared with finite element results under die compaction.@S0094-4289~00!00102-X#

Keywords: Cap Model, Ceramic, Densification, Die Compaction, Drucker-Prager/CapModel, Finite Element Analysis, Triaxial Compression

IntroductionCeramics have become increasingly important in modern indus-

try because of their good mechanical and physical properties@1#.Ceramic parts are generally produced by cold die compaction withsubsequent sintering and finishing or by hot pressing or hot isos-tatic pressing and finishing. Among these procedures, sinteringhas been investigated theoretically and experimentally by manyresearchers. The study on densification behavior of ceramic pow-der under cold die compaction prior to sintering, however, hasbeen dependent mainly on experimental trial-and-error method.

The friction between the powder and die wall during die com-paction typically causes residual stress and inhomogeneous densi-fication. Inhomogeneous density distributions in a powder com-pact lead to nonuniform shrinkage or distortion during sinteringprocess making it difficult to control the shape of final ceramicparts. Residual stress causes cracks in a powder compact duringthe ejection of the compact from the die or during the sinteringprocess and thus affects the mechanical properties of the compact.The density that can be achieved under cold die compaction ofceramic powder is fairly low compared with that of metal powdercompacts, and the ceramic compact then undergoes a significantvolume change during sintering. Therefore, the effects of inhomo-geneous density distributions and residual stress on ceramic pow-der compacts are more serious than those on metal powdercompacts.

Process simulations by using a finite element analysis may beuseful for optimizing the mold design, minimizing the requiredcompaction pressure and density gradients, and controlling shapesof final ceramic parts. The numerical modeling of the powdercompaction processes requires appropriate constitutive modelsthat can describe densification behaviors of ceramic powder.

The densification behaviors of ceramic powder may be studiedby investigating interactions between powders, frictional behav-ior, and so on after characterizing the powder by size, hardness,and shape@2#. An alternative approach is to treat powders as ahomogeneous continuum. The numerical analysis from constitu-tive models based on continuum mechanics has been widely usedin soil mechanics to obtain stress distributions under the complexloading conditions@3#.

So far, the constitutive models for densification of ceramic

powder have been generally adopted from those of soil mechanicsand powder metallurgy. To study densification behaviors of ce-ramic powder, Strijbos et al.@4# used the double hardening modelfor sand@5# and Schwartz and Weinstein@6# and Broese@7# usedthe Mohr–Coulomb model. Shima and Mimura@8# adapted amodel from experimental data of iron powder and copper powderto ceramic powder.

Ceramic powder is densified by rearrangements including slid-ing and rolling of powders@9#. Ceramic powder also shows atendency to be in a state of agglomerates caused by van der Waalsattraction between them@10#. Thus, it is essential to obtain theconstitutive model from experiments to investigate densificationbehaviors of ceramic powders. From experimental data of zirconiapowder under triaxial compression, Bortzmeyer@11# proposed aroughly linear cap model and compared experimental data withfinite element calculations from the proposed model. Recently,Aydin et al. @12# investigated densification behavior of aluminapowder under die compaction by comparing finite element calcu-lations from the modified Drucker–Prager/cap model with experi-mental data.

The present paper reports on densification behaviors of zirconiapowder under cold compaction. To analyze densification behaviorof ceramic powder under cold compaction, a novel hyperbolic capmodel was proposed based on experimental data of zirconia pow-der under triaxial compression. The proposed model was imple-mented into a finite element program~ABAQUS! to compare withexperimental data of zirconia powder under die compaction. Themodified Drucker–Prager/cap model was also employed to com-pare with the proposed model and experimental data in the presentwork. By including the effect of friction between the powder anddie wall, finite element calculations were compared with experi-mental data for density distribution of a zirconia powder compactunder die compaction.

AnalysisThe strain rate in a ceramic powder compact during cold com-

paction may be decomposed into an elastic part and an inelasticpart @13–15#. Thus,

« i j 5 « i j~el!1 « i j

~in! (1)

where « i j~el! and « i j

~in! are elastic and inelastic strain rate tensors,respectively.

By using the generalized Hooke’s law in the elastic part, thestress tensor can be written

Contributed by the Materials Division for publication in the JOURNAL OF ENGI-NEERING MATERIALS AND TECHNOLOGY. Manuscript received by the MaterialsDivision March 12, 1998; revised manuscript received September 13, 1998. Associ-ate Technical Editor: G. Ravichandran.


s i j 5Di jkl~el! «kl

~el! (2)

whereDi jkl~el! is a fourth-order elastic modulus tensor. The inelastic

strain rate can also be written

« i j~in!5l

] f

]s i j(3)

wheref is the yield function for ceramic powder under cold com-paction andl is a positive scalar. From the mass conservation, thedensification rate can be written

D52D «kk~in! (4)

whereD is the relative density of a powder compact.Considering the relative density as a hardening parameter@8,9#,

the yield function for densification of ceramic powder can be writ-ten

f ~s i j ,D !50. (5)

Assuming that ceramic powder is isotropic and neglecting thethird invariant of stress tensor@8#, Eq. ~5! can be written

f ~p,q,D !50 (6)

wherep andq, respectively, are the first and second invariant ofstress tensor. When the direction 1 is regarded as the axial direc-tion and the directions 2, 3 are regarded as the lateral directions,respectively, thens25s3 and «25«3 in the standard triaxialcompression test. Hence, the hydrostatic pressurep in Eq. ~6! canbe written

p5213 ~s11s21s3!52

13 ~s112s2!. (7)

The effective stressq in Eq. ~6! can also be written:

q5~32 s i j8 s i j8 !1/252~s12s2! (8)

wheres i j8 is the deviatoric components of stress, i.e.,

s i j8 5s i j 213 skkd i j . (9)

Finite Element Analysis. The constitutive equations~1!–~9!were implemented into the user subroutine UMAT of ABAQUS@16# to analyze densification behavior of ceramic powders undercold compaction. The numerical scheme used in this paper can befound elsewhere@17–19#. Finite element calculations were alsoobtained by using the modified Drucker–Prager/cap model in theconstitutive library provided in ABAQUS@16#. For the modifiedDrucker–Prager model, we determined the parameters in themodel from experimental data of zirconia powder under triaxialtest, for instance,a50.03, b554.3°, d51.5 MPa, and R50.835. The definitions of these parameters can be found in Fig.4.4.4-1 in ABAQUS@16#.

To study densification behavior of ceramic powder under diecompaction by single action pressing, we used 200 four-node axi-symmetric elements in the finite element analysis. The axisym-metry for they axis allowed a half model of the powder compact.The die wall was assumed to be a rigid surface. The relativedensity of the powder compact in the finite element analysis wasobtained from the volume average of relative density at each ele-ment. Thus,

Davg5( j 51

m D jVj

( j 51m Vj

(10)

wherem is the number of total element andD j and Vj , respec-tively, are relative density and the volume of thej th element.

ExperimentsZirconia powder~HSY-3.0, Daiichi-Kigenso Kagaku Kogyo

Co. Ltd., Japan! stabilized by 3 mol percent Y2O3 was used in thiswork. Table 1 shows physical properties and chemical composi-

tion of zirconia powder. Table 2 shows mechanical properties ofzirconia powder.

Triaxial Compression. Ideal triaxial compression test shouldpermit independent control of the three principal stresses(s1 ,s2 ,s3) or strains («1 ,«2 ,«3). However, it was not feasibleto fabricate such an apparatus. Thus, a standard triaxial compres-sion apparatus that applies an axial load to the compact underconstant confining pressure was used in this work. Figure 1 showsa schematic drawing of the triaxial compression apparatus used inthis work.

Samples for triaxial compression test were produced as follows:6.4 g of zirconia powder was poured into a die, then the ceramicpreforms were produced by cold die compaction. Samples havethe average initial relative density ofDo50.31. The sample is13.1 mm in diameter and 25.3 mm in height with the aspect ratioabout 2 to reduce the friction between the specimen and platens@20#.

A rubber mold of 0.7 mm in thickness was used to prevent thepenetration of water into a sample during triaxial test. Teflonsheets were also placed between the sample and platens to reducethe friction. Then, the sample assembly was placed in the triaxialcompression apparatus after vacuum sealed. The confining pres-sure of 20–200 MPa and the axial force of 0–250 kN were ap-

Fig. 1 A schematic drawing of the triaxial compressionapparatus.

Table 1 Physical properties and chemical composition ofHSY-3.0 zirconia powder

Specific surface area~m3•g21! 6.4

Average particle size~mm! 0.53Chemical composition~wt%!

ZrO2 94.06Y2O3 5.41CaO 0.02Na2O 0.01

Loss of ignition 0.13

Table 2 Mechanical properties of HSY-3.0 zirconiapowder †32‡

Theoretical density~g•cm23! 6.08Young’s modulus~GPa! 206

Poisson’s ratio 0.31


plied to the sample simultaneously. The stress path during triaxialcompression test was a straight line with the slopeq53p in the~p, q! plane.

The frictional force caused by the O-ring between the pressurechamber and the moving ram to prevent the leakage of watervaried with the confining pressure. Thus, the frictional force wasconsidered for calculating the axial force on the sample. The ramspeed was set at 1.2 mm mn21 during the test.

Die Compaction. The cold die compaction of zirconia pow-der was carried out by single action pressing. 4 g of zirconiapowder was poured into the die, then the compaction pressure of10–350 MPa was applied to the powder. The relative density ofthe powder compact was measured by Archimedes’ method. Thesample was coated by alkyd lacquer to prevent from water pen-etration. The mass and volume of the coating material were com-pensated in calculating the density of the sample.

Density Distributions. The density distribution in a ceramiccompact may be obtained indirectly from the relationship betweenhardness and relative density. Abe et al.@21# obtained densitydistribution in a ceramic compact by using Vickers hardness. Ra-jab and Coleman@22# also used Vickers hardness to show thedensity distribution in an iron powder compact with a complicatedshape.

The samples for hardness test were prepared as follows: A ce-ramic compact was sintered for 40 min at 1000°C in a vacuumfurnace without changing its relative density. The sample was cutvertically by a diamond wheel. Then, the sample was coated byalkyd lacquer~0.005 mm thick!, because the marks by a diamondindenter on the cut plane was very obscure. The sample has gridlines with 1 mm distance along the radial direction. A total of 11points was measured for a sample by applying 25 g force on thesample for 15 s. The average value of measured hardness wasused to obtain the relationship between hardness and relative den-sity. To obtain the density distribution in a ceramic compact undercold die compaction, a total of 84 points was measured in eachsample. The cut sample has grid lines with 0.5 mm distance nearthe edge and 1 mm distance in the central portion of the sample.

Results and Discussion

Triaxial Compression. The tensile direction was regarded asthe positive direction in this paper. Figure 2 shows the variation ofYoung’s modulus with relative density for zirconia powder com-pacts. Young’s modulus of the powder compact was obtainedfrom measuring the variation of the axial strain with the axialstress in infinitesimal unloading during triaxial compression test,because unsintered ceramic compacts show very low uniaxialcompressive strengths. The solid curve in Fig. 2 was obtainedfrom the curve fitting@23#. Thus,

E5E0 exp@2~bf1cf2!# (11)

where

E05206 GPa, b512.6, c526.99.

In Eq. ~11!, E0 is Young’s modulus of the matrix material,b andc are constants, andf(512D) is the porosity of the powdercompact. The Poisson’s ration for a porous ceramic compact wasobtained from the literature@24#. Thus,

n51

4 S 4n013f27n0f

112f23n0f D (12)

wheren0 is the Poisson’s ratio of the matrix material.Figure 3 shows the variation of relative density with hydrostatic

pressure for zirconia compacts at various confining pressures un-der triaxial compression. It is observed in Fig. 3 that the densifi-cation is enhanced by adding shear stress. The effect of shearstress on densification is more pronounced under low confining

pressure than high confining pressure. The variations of relativedensity under various confining pressures were found to be almostlinear as shown in Fig. 3.

During triaxial compression of ceramic powder, the criticalstate was observed as shown in Fig. 4~a!. In the critical state, thecompact deforms continuously without changing its volume@25#.The critical state was observed at several confining pressures inFig. 3. The critical state in this experiment can be represented bythe following equation in the~p, q! plane~see: Fig. 4~b!!:

q5Mp (13)

where M51.41. In the ideal critical state, the volumetric strainmust be zero as in Fig. 4~a!. However, experimental data showthat the relative density increased slightly while the stress remainsalmost constant. This may be due to the friction between the pow-der compact and platens although teflon sheets were placed be-tween them.

Figure 5 shows zirconia compacts~a! before deformation,~b!after isostatic compression under 200 MPa, and~c! after triaxialcompression under confining pressure of 60 MPa withp5117 MPa andq5171 MPa. The friction between the specimenand teflon sheets prevented the ideal shrinkage of the sample un-der isostatic pressing~see: Fig. 5~b!! and also caused barreling~see: Fig. 5~c!!.

Figure 6 shows the variation of the effective stressq with axialstrain «1 and lateral strain«2 for zirconia compacts at variousconfining pressuresPc during triaxial compression. Here the av-erage lateral strain was used because of barreling as shown in Fig.

Fig. 2 Variation of Young’s modulus with relative density for azirconia powder compact obtained from triaxial compressiontest

Fig. 3 Variation of relative density with hydrostatic stress forzirconia powder at various confining pressures Pc


5~c!. As confining pressure increases, the effective stress increasesrapidly as strains increase. As strains further increase, the increasein the effective stress slows down and then plateaus eventually.

Figure 7 shows the iso-density curves of zirconia compacts inthe range of relative densityD50.42– 0.51. Data points were ob-tained from the relationship between relative density and hydro-static stress at various confining pressures under triaxial compres-sion. The iso-density curves~solid curves! at various relativedensities in Fig. 7 were obtained from the least-square curve fit ofa hyperbolic function. Thus,

q1cosh$A~D !•p%5B~D ! (14)

with

A~D !51.9231024•D27.008, B~D !53.1963105

•D7.915

(15)

The dashed curves were obtained from the modified Drucker–Prager/cap model with parametersa50.03, b554.3°, d51.5 MPa, andR50.835 in ABAQUS@16#.

Bortzmeyer@9# showed that relative density of ceramic powdercompacts behaves as the hardening parameter. The iso-densitycurve in Eq.~14! can be regarded as the yield function for densi-fication of zirconia powder under cold compaction. The yieldfunction~14! has a special form of the cap model, which is widelyused in soil mechanics and powder metallurgy@26,27#. In particu-lar, Eq. ~14! represents hyperbolic caps instead of roughly linearcaps as observed by Bortzmeyer@11# for zirconia powder undertriaxial compression.

The iso-density curves in Fig. 7 were obtained for zirconiapowder with relative density greater thanD50.42. Thus, the ex-trapolation of Eq.~14! is necessary for ceramic compacts withinitial relative density lower than those in Fig. 7. To show thevalidity of the extrapolation for ceramic compacts with low rela-tive density, we compared the extrapolation of Eq.~14! with ex-perimental data under hydrostatic compression.

Figure 8 shows comparisons between the iso-density curvesextrapolated from Eq.~14! with Eq. ~15! and from the modifiedDrucker–Prager/cap model for experimental data of zirconia com-pacts under hydrostatic pressing. The extrapolations of both Eq.

Fig. 4 „a… Description of the critical state in the stress andstrain plane and „b… stress path in the „p ,q … plane during tri-axial compression

Fig. 5 Zirconia powder compacts „a… before deformation, „b…after isostatic compression under 200 MPa, and „c… after triaxialcompression under confining pressure of 60 MPa.

Fig. 6 Variation of the effective stress q with axial and lateralstrains for zirconia powder compacts at various confining pres-sures Pc

Fig. 7 Comparison between the proposed hyperbolic capmodel „solid … and the modified Drucker–Prager Õcap model„dashed … for iso-density curves of zirconia powder compactsunder triaxial compression


~14! and the modified Drucker–Prager/cap model agree reason-ably well with experimental data for zirconia compacts with lowrelative density under hydrostatic compression.

Cold Isostatic Pressing. The yield function in Fig. 7 showvertices in thep axis. Plastic strain rate under hydrostatic loadingmay have a deviatoric component as well as a volumetric compo-nent, since the direction of plastic strain rate is not unique at thevertex. There also may be discontinuity when the stress movesfrom hydrostatic pressure to triaxial stress. To solve this problem,the vertex on the yield surface was replaced by a small sphericalcap ~see Fig. 9! as suggested by Govindarajan and Aravas@28#.

Figure 10 shows comparisons between experimental data andfinite element calculations for the variation of relative densitywith hydrostatic pressure for zirconia compacts with initial rela-tive densityD050.31 during cold isostatic pressing. The solidcurve was obtained from finite element calculations by using theproposed cap model, i.e., Eq.~14! with Eqs. ~11! and ~12!. Thedashed curve was obtained from finite element calculations byusing the modified Drucker–Prager/cap model. The proposedmodel agrees well with experimental data for zirconia compactsunder hydrostatic compression, while the modified Drucker–Prager/cap model slightly underestimates experimental data.

Cold Die Compaction. Many researchers investigated densi-fication behaviors of ceramic powders under cold die compaction.For instance, Thompson@29# proposed a theoretical model basedon the model by Janssen@30# and experimental data by Unckel@31#. Thus,

sz~r ,z!5BS r 2

R2DexpS 24ma

RzD1CS 12

r 2

R2D (16)

wheresz is the axial stress andr and z are the radial and axialcoordinates, respectively. Also,R is the radius of the sample,mand a are material constants, andB and C are constants. In Eq.~16!, the stress distributions in the radial and axial direction havethe parabolic and exponential forms, respectively. At the centerline (r 50), the axial stresssz is constant along thez axis.

To compare finite element calculations by using the proposedcap model, Eq.~14! with experimental data of a zirconia compactunder die compaction, we determined the friction coefficient be-tween the powder and die walls during die compaction by follow-ing the approach of Kwon et al.@19#. Thus, the friction coefficientwas obtained indirectly from the relationship between the com-paction pressure and ejection pressure by comparing experimentaldata with finite element results.

Figure 11 shows comparisons between experimental data andcalculated results for the variation of ejection stress with axialstress of zirconia powder during die compaction. The ejectionstress was measured by pushing out the compact out of the die.The data points in Fig. 11 were obtained for zirconia powderunder single action die pressing. The dotted curve, solid curve,and dash–dotted curve were obtained from finite element calcula-tions by using Eq.~14! with various friction coefficients. Here, thesolid curve with the friction coefficientm50.2 shows a good

Fig. 8 Comparison between the extrapolated curves from theproposed hyperbolic cap model „solid … and the modifiedDrucker–Prager Õcap model „dashed … for experimental data ofzirconia powder under cold isostatic pressing

Fig. 9 The modified yield surface in the vicinity of hydrostaticstress axis

Fig. 10 Comparison between finite element calculations fromthe proposed hyperbolic cap model „solid … and the modifiedDrucker–Prager Õcap model „dashed … for the variation of relativedensity with hydrostatic pressure for zirconia powder duringcold isostatic compression

Fig. 11 Comparison between experimental data and calcu-lated results with various friction coefficients for the variationof ejection stress with axial stress of zirconia powder under diecompaction


agreement with experimental data for zirconia powder in Fig. 11.The friction coefficients were found by a trial-and-error method.

Figure 12 shows comparisons between experimental data andfinite element calculations for the variation of relative densitywith axial stress of zirconia powder during die compaction. Thefriction coefficient m50.2 was used for zirconia powder. Thesolid curve was obtained from finite element calculations by usingEq. ~14! and the dashed curve from the modified Drucker–Prager/cap model. The proposed model slightly overestimates experimen-tal data, while the modified Drucker–Prager/cap modelunderestimates.

Figure 13 shows the variation of relative density with Vickershardness (Hv) for zirconia powder compacts. The relationship ofrelative density and Vickers hardness can be represented by thefollowing equation:

D50.3134910.01781•Hv (17)

Figure 14 shows a comparison between experimental data andfinite element calculations for relative density contour plots of azirconia powder compact ejected from a die after compacted bysingle action pressing under axial stress of 100 MPa. It is under-stood only right halves of the compact are shown in Figs. 14~a!,14~b!, and 14~c!. Figure 14~a! was obtained from Eq.~17! withVickers hardness. Figures 14~b! and ~c!, respectively, were ob-tained from finite element calculations by using Eq.~14! and themodified Drucker–Prager/cap model withm50.2. In Fig. 14~a!,relative density is the highest at the corner of contact surfacebetween the upper punch and the die wall and the lowest at the

corner of contact surface between the lower punch and the diewall. A similar trend of the density distribution was also observedfor a stainless steel powder compact by single action pressing@19#. Thompson@29# assumed that the relative density is constantalong the center line of the sample during die compaction, how-ever, it varies as shown in Fig. 14~a!. The finite element results inFigs. 14~b! and 14~c! show some of the same trends with experi-mental data in Fig. 14~a!. The overall density distribution in Fig.14~b! is slightly overestimated experimental data in Fig. 14~a!,while that in Fig. 14~c! is underestimated experimental data.

Figure 15 shows finite element calculations for Figs. 15~a! nor-mal stress and 15~b! shear stress distributions on a zirconia com-pact under axial stress of 100 MPa by single action pressing.Figure 15 was obtained from finite element calculations by usingEq. ~14! with m50.2. In Fig. 15~a!, the normal stress was repre-sented by the ratio to the axial stress ofs5100 MPa. The normalstress on the die wall was the highest in the top corner and thelowest in the bottom corner due to the effect of friction. The shear

Fig. 12 Comparison between finite element calculations fromthe proposed hyperbolic cap model „solid … and the modifiedDrucker–Prager Õcap model „dashed … for the variation of relativedensity with axial stress of zirconia powder under diecompaction

Fig. 13 Variation of relative density with Vickers hardness„Hv… for zirconia powder compacts

Fig. 14 Comparison between „a… experimental data and finiteelement calculations from „b… the proposed hyperbolic capmodel, and „c… the modified Drucker–Prager Õcap model for rela-tive density contour plots of a zirconia powder compact ejectedfrom a die after compacted by single action pressing underaxial stress of 100 MPa

Fig. 15 Finite element calculations from the proposed modelfor „a… normal stress and „b… shear stress distributions on azirconia powder compact under axial stress of 100 MPa bysingle action pressing


stress which affects the life of the die by wear is zero at thebottom corner and the highest at the top corner. The shear stress iszero at the center line due to the axisymmetric condition.

Figure 16 shows finite element calculations for distributions ofFigs. 16~a! residual Mises stress and 16~b! residual hydrostaticstress of a zirconia powder compact ejected from a die afterpressed under 100 MPa. Figure 16 was obtained from finite ele-ment calculations by using Eq.~14! with m50.2. In Fig. 16~a!,the highest residual Mises stress is observed at the upper corner ofthe compact. In Fig. 16~b!, tensile residual hydrostatic stress isobserved at the top and bottom of the compact, although it issmall. Almost no residual hydrostatic stress was observed at thecentral part of the compact. It is also observed that the residualstress in the compact are much smaller than the compactionpressure.

ConclusionsDensification behaviors of zirconia powder during cold com-

paction were investigated. Based on experimental data of zirconiapowder under triaxial compression, a hyperbolic cap model wasproposed. The proposed model was implemented into a finite el-ement program to compare with experimental data of zirconiapowder under die compaction. The modified Drucker–Prager/capmodel was also employed to compare with the proposed model.From the relationship between the compaction and ejection pres-sures, the friction coefficientm50.2 was obtained for zirconiapowder between the powder and die walls.

Finite element calculations from the proposed model agreedreasonably well with experimental data of zirconia powder undercold compaction for both global densification and local densitydistribution. Finite element calculations from the modifiedDrucker–Prager model showed some of the same trends with ex-perimental data, however, underestimated.

AcknowledgmentsThis work was financially supported by a grant from the Korean

Science and Engineering Foundation~KOSEF! under grant no.971-1007-042-2. We are grateful for this support.

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Fig. 16 Finite element calculations from the proposed modelfor distributions of „a… residual Mises stress and „b… residualhydrostatic stress of a zirconia powder compact ejected from adie after compacted under 100 MPa


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