A numerical method for elasto-plastic notch-root stressâ€“strain analysis

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2013 48: 229 originally published online 5 April 2013The Journal of Strain Analysis for Engineering DesignAyhan Ince and Grzegorz Glinka

strain analysis−A numerical method for elasto-plastic notch-root stress

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Original Article

J Strain Analysis48(4) 229–244� IMechE 2013Reprints and permissions:sagepub.co.uk/journalsPermissions.navDOI: 10.1177/0309324713477638sdj.sagepub.com

A numerical method for elasto-plasticnotch-root stress–strain analysis

Ayhan Ince and Grzegorz Glinka

AbstractIn this article, a computational modeling method of the multiaxial stress–strain notch analysis has been developed tocompute elasto-plastic notch-tip stress–strain responses using linear elastic finite element results of notched compo-nents. Application and validation of the multiaxial stress–strain notch analysis model were presented by comparing com-puted results of the model to the experimental data of SAE 1070 steel notched shaft subjected to severalnonproportional load paths. Based on the comparison between the experimental and computed strain histories, theelasto-plastic stress–strain model predicted notch strains with reasonable accuracy using linear elastic finite elementstress histories. The elasto-plastic stress–strain notch analysis model provides an efficient and simple analysis methodpreferable to expensive experimental component tests and more complex and time-consuming incremental nonlinearfinite element analysis. The elasto-plastic stress–strain model can thus be employed to perform fatigue life and fatiguedamage estimates associated with the local material deformation.

KeywordsMultiaxial, elasto-plastic, stress–strain, notch analysis, nonproportional loading, cyclic plasticity, fatigue

Date received: 19 November 2012; accepted: 16 January 2013

Introduction

Fatigue failure is considered to be the most commontype of failure mode experienced by most engineeringcomponents. Many mechanical components containnotches and geometrical irregularities, which cannot beavoided in practice. In addition, most of the mechanicalnotched components are subjected to biaxial/multiaxialloadings in services. Multiaxial loading paths producecomplex stress and strain states near notches and cancause a fatigue failure at or near notch roots even withoutany evident large-scale plastic deformation. Therefore,notches have been considered as one of the most impor-tant problems in the design of machine components.

Local strain-based fatigue life prediction of notchedcomponents subjected to multiaxial loading paths requiresdetail knowledge of stresses and strains in such regions.Early researches focused primarily on determining theore-tical stress concentration factors using either elasticity the-ory or photoelastic analysis. Peterson1 has compiledtheoretical stress concentration factors for various geome-tries into one book. Stowell2 and Hardrath and Ohman3

investigated stress concentrations in the plastic range. Themost well-known approximation formula that relates thetheoretical stress concentration factor to the product ofthe elasto-plastic stress and strain concentration factors

was originally proposed by Neuber.4 Neuber studied asemi-infinite prismatic notched body obeying a nonlinearstress and strain law. He proposed that the product of thestress and strain at the notch tip in any arbitrary notchedgeometry in a prismatic body is not dependent on materi-al’s nonlinear parameters but can be related to material’selastic parameters and the far field boundary conditions.Neuber’s rule for uniaxial loading is

saea =K2t Se ð1Þ

where sa and ea are the elasto-plastic notch-tip stressand strain, S and e are the nominal section stress andstrain and Kt is the elastic stress concentration factor.Alternatively, Neuber’s rule can be written in the form

sa ea =seee ð2Þ

Department of Mechanical and Mechatronics Engineering, University of

Waterloo, Waterloo, ON, Canada

Corresponding author:

Ayhan Ince, Department of Mechanical and Mechatronics Engineering,

University of Waterloo, 200 University Avenue West, Waterloo, ON,

Canada N2L 3G1.

Email: [email protected]



where se =KtS and ee =Kte are the fictitious elasticstrain and stress. Equation (2) states that the total strainenergy density and complementary energy density atthe notch tip are equal to the fictitious strain energydensity and complementary energy density as if a mate-rial hypothetically remained elastic.

Topper et al.5 extended validation of Neuber’s ruleto several notch geometries subjected to uniaxial cyclicloading. Their results showed that Neuber’s rule forcyclic loading was in good agreement with experimentalresults for notched aluminum alloy sheets. Molski andGlinka6 proposed the equivalent strain energy density(ESED) method as an alternative to Neuber’s rule.They postulated that the strain energy density at anotch equals that if the body were to hypotheticallyremain elastic. The authors showed6 that the ESEDmethod provided good agreement with experiment forseveral notched specimens subjected to monotonicloading. The ESED method can be written in terms ofstresses and strains as

1

2seee =

ðea

0

sadea ð3Þ

Glinka7 later extended the ESED method to addressnotched bodies subjected to cyclic loading. As the localstrain-life approach was extended to multiaxial loadingusing multiaxial fatigue damage parameters, these dam-age parameters require multiaxial strains and stresses tobe determined at notch areas. Since nonlinear finite ele-ment analysis (FEA) is too costly to compute multiaxialstresses and strains for a long load history, simple uni-axial notch stress and strain approximation techniqueswere extended to states of multiaxial stress and strain.Hoffmann and Seeger8,9 proposed a method for multi-axial proportional loading by applying an equivalentform of Neuber’s rule as

saeqe

aeq =se

eqeeeq ð4Þ

where saeq and eaeq are the notch-tip elastic–plastic equiv-

alent stress and strain, respectively, and seeq and eeeq are

those that would be obtained if the material remainedelastic. They assumed that the ratio of minimum princi-pal strain components at the notch tip remains constantduring loading.

Ellyin and Kujawski10 proposed a method that themaximum stress and strain at the notch roots can becalculated for monotonic and cyclic loadings from theknowledge of the theoretical stress concentration factorKt. This method is based on an averaged similaritymeasure of the stress and strain energy densities alonga smooth notch boundary. The method can also beused in the case of multiaxial states of stress. Ellyin andKujawski reported that the predicted stresses andstrains at the notch root were in good agreement withthe available experimental data and FE results. Thegeneralization of both the ESED method and Neuber’srule for proportional multiaxial loading for notched

bodies was suggested by Moftakhar.11 Numerical andexperimental studies conducted by Moftakhar et al.12

showed that the generalized ESED method andNeuber’s rule provide upper and lower bounds of theactual strains. Their study concluded that Neuber’s ruletends to overestimate the notch-tip elastic–plasticstrains and stresses and the ESED method tends tounderestimate the notch-tip inelastic strains and stres-ses. Hoffman et al.13 presented a method to estimatenotch-root stresses and strains for bodies subjected tononproportional loading. In their method, the multiax-ial loads are first separated and notch-root strain his-tories are calculated for the loads independently byfollowing the same solution procedure as for propor-tional loading. Compatibility iteration is then used toaccount for interaction between strain components thatresult from nonproportional loading. Their calculationscompared well with FEA. Barkey14 and Barkey et al.15

later proposed a method to estimate multiaxial notchstrains in notched bars subject to cyclic proportionaland nonproportional loading, using the concept of astructural yield surface. The structural yield surfacedescribes the relationship between nominal stresses andnotch strains. The hardening parameter is found byusing the uniaxial form of the ESED method as thebasis of nominal load to notch plastic strain curve.Their results showed good agreement with experimen-tal nonproportional tension–torsion tests for a notchedsteel bar. Kottgen et al.16 extended Barkey’s approachby incorporating the notch effect into the constitutiverelation. They first obtained pseudo stress history byassuming elastic material behavior. The yield criterionand the flow rule were determined using elastic stresshistory, and the hardening parameter was determinedusing pseudo stress and local plastic strain curveobtained from a uniaxial simplified rule. The resultingelastic–plastic strain increments were then fed back intothe flow rule to calculate notch stresses. Kottgen et al.reported correlation of their method with elasto-plasticFEA for various geometries and applied loads. Singh17

extended the ESED and Neuber’s methods to estimatethe notch-root stresses and strains for monotonic non-proportional loading. The approach is an incrementalgeneralization of the ESED method and Neuber’s rule.In generalizing the ESED method, Singh suggested thatfor a given increment of external load, the correspond-ing increment of the strain energy density at the notchtip in an elastic–plastic body is equal to the incrementof the strain energy density if the body were to remainelastic.

Recent research studies18–20 have shown that thenotch correction method can be combined with the cyc-lic plasticity model to compute the local stress andstrain history from the pseudo-elastic stress and strainat the notch area. Coupling the notch correctionmethod and the cyclic plasticity to compute the localstresses and strains at critical locations in componentsprovides a great advantage over experiments and incre-mental elastic–plastic FEAs due to its simplicity,

230 Journal of Strain Analysis 48(4)



computational efficiency, low cost and reasonable accu-racy. The multiaxial stress–strain notch analysismethod originally proposed by Buczynski and Glinka19

is adapted to develop the elastic–plastic stress–strainmodel to compute local stress–strain responses usinglinear elastic FE results of notched components. A gen-eral numerical algorithm is developed on the basis ofincremental relationships, which relate the elastic andelastic–plastic strain energy densities at the notch rootand the material stress–strain behavior, simulated bythe Garud21 cyclic plasticity model. The algorithm com-putes multiaxial elasto-plastic stress–strain responsesfor many nodal points that define critical notch areasof the FE model. A flowchart for the algorithm imple-menting the stress and strain analyses for notched com-ponents subjected to the multiaxial loadings is shown inFigure 1.

Linear elastic stress–strain histories

FE models are used often to analyze engineering com-ponents. A linear elastic FEA can be used to calculatelinear elastic stresses/strains for a notched component.Once the elastic stresses/strains are known, the elasto-plastic stress–strain analysis (combined with the multi-axial notch correction and the cyclic plasticity) can beused to compute the actual elastic–plastic stress andstrain responses at notch areas. The linear elastic FEAassumes that there is a linear relationship between theapplied external load and stress–strain results. Axle

and shaft components often experience combined bend-ing and torsion loads. Let us consider a notched shaftshown in Figure 2, which has two applied load his-tories, namely, bending and torque. The FEA is per-formed to calculate unit-load linear elastic stresstensors for each applied load. The elastic stress tensorat each node from the linear elastic FEA is multipliedby the corresponding load history to compute a timehistory of elastically calculated stress tensor. If the elas-tic stress tensor at a node is s

epij for a unit load of p, the

time history of the elastic stress tensor seP(t)ij for the

load history P(t) can be calculated as

Figure 1. Algorithm for notch stress and strain analyses.FEA: finite element analysis.

Figure 2. A notched shaft with applied load histories P(t) andQ(t).

Ince and Glinka 231



seP(t)ij =P(t)s

epij ð5Þ

If the elastic stress tensor at the same node is seqij for

a unit load of q, the time history of the elastic stress ten-sor s

eQ(t)ij for the load history Q(t) can be calculated as

seQ(t)ij =Q(t)s

eqij ð6Þ

Time histories of elastic stress tensors seP(t)ij and

seQ(t)ij for the same node are then superimposed to

obtain the resultant time history of the elastic stresstensor for both load histories P(t) and Q(t), appliedsimultaneously

seij =seP

ij +seQij ð7Þ

Figure 3 shows a schematic representation of theimplementation procedure for superimposing elasticstress tensors. Since fatigue crack initiation is formedon the surface of a component, elastic stress tensors forsurface nodes at critical notch areas are used for esti-mating actual elastic–plastic stress–strain responses. Amacro routine written in ANSYS Parametric DesignLanguage (APDL) is used to compute elastic stressresults for surface nodes for each unit load. Unit-loadelastic stress results from the FEA and correspondingtime histories for each unit load are used as input to acomputer program to calculate combined time historiesof elastic stress tensor at critical notch areas.

If components of a stress tensor change proportion-ally during the loading, the loading is called propor-tional. When the applied load results in the change ofthe principal stress directions and the ratio of the prin-cipal stresses, the loading is called nonproportional.When plastic yielding takes place at the notch tip, thenthe stress path at the notch-tip region is usually non-proportional regardless of whether the external loadingis proportional or not. Nonproportional loading/stresspaths are usually defined by increments of load/stresscomponents, and therefore, stress–strain calculationshave to be carried out in incremental form. Time his-tories of elastic stresses are transformed to incrementsof elastic stresses. A schematic representation of elasticstress increments used as input to the stress–strainnotch analysis model is shown in Figure 4.

For the case of general multiaxial loading applied toa notched body, the state of stress near the notch tip istriaxial. However, the stress state at the notch tip is

Figure 3. Superimposing FEA elastic stress results from two load histories.

Figure 4. The input elastic stress increments of the timehistory of the stress.




biaxial because of the notch-tip stress for a free surfaceas shown in Figure 5. Since equilibrium of the infinite-simal element at the notch tip must be maintained, thatis, se

23 =se32 and ee23 = ee32, there are three nonzero

stress components and four nonzero strain components(Figure 5)

seij =

0 0 00 se

22 se23

0 se32 se

33

24

35 and eeij =

ee11 0 00 ee22 ee230 ee32 ee33

24

35ð8Þ

Seven fictitious linear elastic stress and strain com-ponents (se

ij, eeij) are obtained from the linear elastic FE

solution; however, the actual elastic–plastic stress andstrain components (sa

ij, eaij) at the notch tip are

unknown. Therefore, a set of seven independent equa-tions are required for the calculation of actual elastic–plastic stress and strain components at the notch tip.The multiaxial Neuber notch correction rule and theGarud cyclic plasticity model are integrated to providethe required seven equations to solve for all unknownstress and strain components. The cyclic plasticity pro-vides four equations, and the notch correction rule pro-vides the remaining three equations.

Constitutive governing equations

Under applied cyclic loadings, stress–strain responsesat critical regions (notches) often show a transientresponse but stabilize over a number of cycles. Fornonproportional load histories, an incremental plasti-city analysis is required to estimate the material’sstress–strain response. A number of incremental plasti-city models21–27 have been developed to estimate con-stitutive material behavior, and some of those modelsare sophisticated to include the transient hardeningresponses. However, these complex models require sig-nificant material testing to characterize model

parameters and are not appropriate for practical engi-neering use. Furthermore, for the multiaxial fatigueanalysis, the transient nature of deformation behavioris not as critical as the behavior of cyclically stabilizedmaterial behavior. Therefore, a relatively simple Garudhardening model is used to deal with proportional andnonproportional multiaxial loadings.

The cyclic plasticity model provides a set of govern-ing equations to relate stress and strain components.Since the fatigue cracks most often initiate on the sur-face of a component, governing equations are providedfor stress–strain state on the free component surface.Governing equations are presented in deviatoric spacebased on a rate-independent, homogenous and isotro-pic material because it is easier to formulate the equa-tions using deviatoric stress and strain quantities.

In elastic loadings, the deformation process is rever-sible, and when the applied load is removed from thebody, the deformed body returns to its original state.There is a direct relationship between stress and strain,and relations between stress and strain tensors aredetermined by Hooke’s law (equation (9)).

The generalized Hooke’s law in an incremental formis represented as

deeij =1+ v

Edsij �

v

Edsijdij ð9Þ

When the body is deformed beyond the material’syield limit, permanent plastic deformation occurs, andthe deformation process becomes irreversible. Whenthe applied load is removed from the body, the perma-nent deformation remains in the body. Based on theplasticity theory, the stress and strain states are depen-dent on the loading path. As well known, there arethree main elements required in order to model consti-tutive behavior of the material: a yield criterion, whichdefines a boundary between elastic and elastic–plasticstress states; a flow rule, which describes the relation-ship between stress and strain increments; and a hard-ening rule, which describes how yield function changesduring the plastic deformation. The von Mises yield cri-terion has been the most popular criterion for modelingthe material constitutive behavior. Since it is widelyaccepted that hydrostatic stresses do not influenceyielding, the yield function, equation (10), for the iso-tropic hardening can be described as a uniform andsymmetrical expansion of the yield surface in all direc-tions during plastic loading

F=3

2SijSij � s2

y kð Þ=0 ð10Þ

where Sij is the deviatoric stress tensor and sy kð Þ is thecurrent size of the yield surface as a function of k.

The yield function, equation (11), for the kinematichardening can be represented as a translation of theyield surface without any expansion during plasticloading. The yield surface for the kinematic hardeningmaintains its shape and size

Figure 5. Stress and strain states at a notch tip.

Ince and Glinka 233



F=3

2Sij � aij� �

Sij � aij� �

� s2yo=0 ð11Þ

The flow rule, equation (12), defines the relationshipbetween stress and plastic strain increments. The flowrule, based on the normality postulate by Drucker,22

implies that the increment of plastic strain is in the nor-mal direction to the yield surface during plasticdeformation

depij = dl∂F

∂sijð12Þ

where deij is the plastic strain increment tensor, sij is thestress tensor, F is the yield function, and dl is a scalar-valued function.

For elastic–plastic loading, total strain tensor is thesum of elastic strain determined by Hooke’s law andplastic strain determined by the flow rule

eij = eeij + epij ð13Þ

Similarly, the elastic and plastic strain increment canbe added to obtain the total strain increment

deij = deeij + depij ð14Þ

Substituting equations (9) and (12) into equation(14) yields a general form of the total elastic–plasticstrain increment. The generalized constitutive elasto-plastic stress–strain relationships are derived from theuniaxial stress–strain curve by using principles of thetheory of elasticity and plasticity

deij =1+ v

Edsij �

v

Edsijdij + dl

∂F

∂sijð15Þ

In the case of proportional stress path, the Henckytotal deformation of plasticity equations can be usedfor stress–strain analysis

eaij =1+ v

Esaij �

v

Esakkdij +

3

2

epaeqsaeq

� Saij ð16Þ

The normality flow rule, also called the Prandtl–Reuss relation, is considered one of the most frequentlyused model in the incremental plasticity. The totalstrain increment, equation (15), can be expressed in theform of the Prandtl–Reuss strain–stress relationship

Deaij =1+ v

EDsa

ij �v

EDsa

kkdij +3

2

Depaeqsaeq

� Saij ð17Þ

The notch-tip deviatoric stresses of the hypotheticallinear elastic body are determined as

Seij =se

ij �1

3sekk dij ð18Þ

The elastic deviatoric strain and stress incrementscan be calculated from Hooke’s law

Deeij =DSe

ij

2Gð19Þ

The actual deviatoric stress components in the notchtip can analogously be defined as

Saij = sa

ij �1

3sakkdij ð20Þ

The incremental deviatoric stress–strain relationsbased on the associated Prandtl–Reuss flow rule can besubsequently written as

Deaij =DSa

ij

2G+ dl � Sa

ij ð21Þ

where

1

2G=

1+ n

E, dl=

3

2

Depeqsaeq

, sa2eq =

3

2SaijS

aij,

Depeq =df(sa

eq)

dsaeq

Dsaeq

This relation assumes that the plastic strain incrementsat any instant of loading are proportional to the devia-toric stress components. The relation between theequivalent plastic strain increment and the equivalentstress increment in the uniaxial stress–strain curve canbe used to determine the multiaxial incremental stress–strain relation

Depaeq =Dsa

eq

EpT

ð22Þ

where Depaeq is the equivalent plastic strain increment,Dsa

eq is the equivalent stress increment and EpT is the

current value of the generalized plastic modules.The function Depaeq = f Dsa

eq

� �is identical to the plas-

tic strain–stress relationship obtained experimentallyfrom uniaxial tension test. The plastic strain–stress rela-tionship can be expressed according to the uniaxialRamberg–Osgood equation

s =K9 epð Þn9 ð23Þ

where the constants K# and n# are determined by uniax-ial tensile test.

The analytical expression of the generalized plasticmodules Ep

T can be derived using the Ramberg–Osgoodequation

EpT =

2

3

ds

dep=

2

3Ep =

2

3K9n9

s

K9

� �n9�1n9 ð24Þ

where Ep is the slope of the uniaxial stress–plastic straincurve (or the uniaxial plastic modulus).

The uniaxial stress–strain curve is divided into anumber of stress fields (Figure 6). Each stress surfacedefines regions with constant plastic modulus in thestress space. Figure 6 shows a graphic interpretation ofgeneralizations of the seq � epeq curve in the stress spaceand stress fields of constant plastic modules.

In the case of stress and strain states on the free sur-face of the notch, material constitutive equations aregiven in terms of deviatoric stresses as




Dea11 =DSa

11

2G+ dl � Sa

11

Dea22 =DSa

22

2G+ dl � Sa

22

Dea33 =DSa

33

2G+ dl � Sa

33

Dea23 =DSa

23

2G+ dl � Sa

23

ð25Þ

Four notch strain increments (Dea11, Dea22, Dea33, Dea23)and three notch stress increments (Dsa

22, Dsa33, Dsa

23) inequation (25) form seven unknowns to be solved. Fourdeviatoric stress increments (DSa

11, DSa22, DSa

33, DSa23) in

equation (25) are functions of three notch stress incre-ments (Dsa

22, Dsa33, Dsa

23).

Coupling constitutive equations andNeuber’s notch correction relation

The main goal of this section is to show how to com-bine a cyclic plasticity model and notch correctionmethod to determine a set of governing equations tocompute the notch stress and strain components.

The load is usually represented by the nominal stressbeing proportional to the remote loading for notchedbodies. In the case of notched bodies in a plane stressor plane strain state, the relationship between the elas-tic notch-tip stresses and strains and the actual elastic–plastic notch-tip stresses and strains in the localizedplastic zone is often approximated by the Neuber ruleor the ESED method. It was shown17,19 that bothmethods can also be extended for multiaxial propor-tional and nonproportional modes of loading. Similarapproaches were proposed by Hoffman and Seeger13

and Barkey.14 All methods consist of two parts,namely, the constitutive equations and the notch cor-rection relating the pseudo-linear elastic stress–strainstate (se

22, ee22) at the notch tip with the actual elastic–plastic stress–strain response (sa

22, ea22).The Neuber rule4 for proportional loading, where

the Hencky stress–strain relationships are applicable,can be written for the uniaxial and multiaxial stress

states in the form of equations (26) and (27),respectively

se22 e

e22 =sa

22 ea22 ð26Þ

seij e

eij =sa

ij eaij ð27Þ

The ESED method6 represents the equality betweenthe strain energy density at the notch tip of a linear elas-tic body and the notch-tip strain energy density of ageometrically identical elastic–plastic body subjected tothe same load. The strain energy density equations forthe ESED method for the uniaxial and multiaxial stressstates can be written in the form of equations (28) and(29), respectively

ðee220

se22de

e22 =

ðea220

sa22de

a22 ð28Þ

ðeeij

0

seijde

eij =

ðeaij2

0

saijde

aij ð29Þ

The strain energy density equality, equations (27)and (29), relating the pseudo-elastic and the actualelastic–plastic notch strains and stresses at the notchtip, has been widely used as a good approximationmethod, but additional conditions are required for thecomplete solution of a multiaxial stress state problem.However, those conditions have been the subject ofcontroversy. Hoffman and Seeger13 assumed that theratio of the actual principal strains at the notch tip isequal to the ratio of the fictitious elastic principal straincomponents, while Barkey14 suggested using the ratioof principal stresses. Barkey’s approach was based onassumptions that if the applied loads are proportional,the notch-tip deviatoric stress components remain infixed proportion, and if the applied loads are nonpro-portional, the ratio of the normal stresses remains infixed proportional during the loading history. Theseassumptions are sufficiently accurate in the case of cir-cumferential notches for cylindrical bodies but are nottrue in general multiaxial loading. Moftakhar11 foundthat the accuracy of the stress or strain ratio–basedanalysis depended on the degree of constraint at thenotch tip. Therefore, Moftakhar et al. proposed12 touse the ratios of strain energy density contributed byeach pair of corresponding stress and strain compo-nents. Singh17 confirmed later that the additionalenergy equations provide a good accuracy when usedin an incremental form. Because the ratios of strainenergy density increments seem to be less dependent onthe geometry and constraint conditions at the notch tipthan the ratios of stresses or strains, the analyst is notforced to make any arbitrary decision about the con-straint while using these equations. It was also foundthat the conflict between the plasticity model (normal-ity rule) and strain energy density equations at somespecific ratios of stress components may cause singular-ity for a set of seven equations. Such a conflict can be

Figure 6. A graphical representation of fields of constantplastic modules and the multilinear material curve.

Ince and Glinka 235



avoided if the principal idea of Neuber is implementedin the incremental form. It should be noted that theoriginal Neuber rule was derived for bodies in pureshear stress state. It means that the Neuber equationstates the equivalence of only distortional strain ener-gies. In order to formulate the set of necessary equa-tions for a multiaxial analysis of elastic–plastic stressesand strains at the notch tip, the equality of incrementsof the total distortional strain energy density should beused. Therefore, a set of seven equations from the nor-mality rule and strain energy density should be formu-lated in a deviatoric stress space to be consistent withthe original Neuber rule. However, Moftakhar et al.12

and Singh17 suggested these equations in the normalstress space. Thus, the equations were overconstraineddue to inclusion of hydrostatic stress components inthe solution.

Buczynski and Glinka19 proposed, analogous to theoriginal Neuber rule, to use the equivalence of incre-ments of the total distortional strain energy densitycontributed by each pair of associated stress and straincomponents, that is

Se22Dee22 + ee22DSe22 =Sa

22Dea22 + ea22DS

a22

Se33De

e33 + ee33DS

e33 =Sa

33Dea33 + ea33DSa

33

Se23De

e23 + ee23DS

e23 =Sa

23Dea23 + ea23DSa

23

ð30Þ

The equalities of strain energy increments for eachset of corresponding hypothetical elastic and actualelastic–plastic strains and stress increments at the notchtip can be shown graphically in Figure 7. The area ofdotted rectangles represents the total strain energyincrement of the hypothetical elastic notch-tip inputstresses, while the area of the hatched rectangles repre-sents the total strain energy density of the actualelastic–plastic material response at the notch tip.

Consequently, a combination of four equations fromthe elastic–plastic constitutive equation (25) and threeequations from the equivalence of increments of thetotal distortional strain energy density, equation (30),yields the required set of seven independent equationsnecessary to completely define elastic–plastic notch-tipstrain and stress responses for a notched componentsubjected to multiaxial nonproportional cyclic loads.The final set of equations written as a set of sevensimultaneous equations, equation (31), from which allunknown deviatoric strain, Deaij, and stress, DSa

ij, incre-ments can be calculated, based on the linear hypotheti-cal elastic notch-tip stress history, that is, incrementsDse

ij and Deeij, are known from the linear elastic FEA

Dea11 =DSa

11

2G+

3

2

Dsaeq

EpTsa

eq

� Sa11

Dea22 =DSa

22

2G+

3

2

Dsaeq

EpTsa

eq

� Sa22

Dea33 =DSa

33

2G+

3

2

Dsaeq

EpTsa

eq

� Sa33

Dea23 =DSa

23

2G+

3

2

Dsaeq

EpTsa

eq

� Sa23

Se22Dee22 + ee22DSe

22 =Sa22Dea22 + ea22DS

a22

Se33De

e33 + ee33DS

e33 =Sa

33Dea33 + ea33DS

a33

Se23De

e23 + ee23DS

e23 =Sa

23Dea23 + ea23DS

a23 ð31Þ

Since the equation set (31) is linear, the solution of theequations requires the Gaussian elimination approach.For each increment of the external load, represented bythe increments of pseudo-elastic deviatoric stresses DSe

ij,the deviatoric elastic–plastic notch-tip strain and stressincrements Deaij and DSa

ij are computed from equation(31). The calculated deviatoric stress increments DSa

ij

can subsequently be converted into the actual stressincrements Dsa

ij using equation (32)

DSa22 =Dsa

22 �1

3Dsa

22 +Dsa33

� �

DSa33 =Dsa

33 �1

3Dsa

22 +Dsa33

� �DSa

23 =Dsa23

ð32Þ

The deviatoric and the actual stress components Saij

and saij at the end of given load increment are deter-

mined from equations (33) and (34)

Sanij =Sao

ij +Xn�1k=1

DSakij +DSan

ij ð33Þ

sanij =sao

ij +Xn�1k=1

Dsakij +Dsan

ij ð34Þ

where n denotes the load increment number.The actual strain increments Deaij can finally be deter-

mined from equation (25). Before the solution of plasti-city equations, loading/unloading conditions must be

Figure 7. Graphical representation of the incremental Neuberrule.




checked. At the beginning of initial loading cycle, it isassumed that all strains/stresses are elastic, and thenotch-tip stress–strain response is equal to the knownelastic solution, equation (9)

saij =se

ij

eaij = eeijð35Þ

When the elastic loading reached the initial yield sur-face, subsequent load increment may result in elasticunloading, tangential (neutral) loading and elastic–plastic (active) loading. The mode of loading is deter-mined based on the loading criterion. When the stressincrement moves inward from the yield surface, elasticunloading will take place, that is, the inner product oftensors dS, ∂F=∂S, satisfies the condition

∂F

∂SijdSij \ 0 ð36Þ

If the stress increment is tangential to the yield sur-face, the neutral loading will occur, that is, the innerproduct of tensors dS, ∂F=∂S, satisfies the condition

∂F

∂SijdSij =0 ð37Þ

Since the computer implementation of the neutralloading is virtually impossible to determine, this load-ing criterion is regarded as an elastic unloading. Therelevant constitutive relations for the elastic unloadingand tangential loading are described based on Hooke’slaw, equation (9). When the current state of stress incre-ments moves out from the yield surface, the elastic–plastic loading will take place, that is, the inner productof tensors dS, ∂F=∂S, satisfies the condition

∂F

∂SijdSij . 0 ð38Þ

The elastic–plastic loading condition states that theprojection of the stress increment onto the normal ofthe yield surface must be greater than zero. The rele-vant constitutive relations for the elastic–plastic loadingare described based on Hooke’s law and Prandtl–Reussequation (17).

Substituting the yield surface equation (11) for kine-matic hardening into these loading criteria, the loadingmode criterion LC, in terms of finite increments can bewritten for the purpose of numerical implementation asfollows

LC=Xj=2, 3

i=2, 3

(Saij � an

ij)Dseij ð39Þ

The loading mode criterion LC, for the incrementalNeuber method states that elastic unloading or theelastic–plastic loading takes place as

LC40 Elastic Unloading

LC. 0 Elastic�Plastic Loadingð40Þ

The coupled constitutive governing and Neuber’sincremental equations discussed before are related withthe Garud cyclic plasticity model to compute the actualnotch-tip stress–strain response of a notched compo-nent subjected to proportional and nonproportionalmultiaxial cyclic loading. After the notch stressincrements are determined, the translation of the yieldsurface is updated by employing the multisurface hard-ening model proposed by Garud.21

Cyclic plasticity model

In order to estimate the elasto-plastic stress and strainresponses at critical notch location for a notched com-ponent subjected to the multiaxial cyclic loading, a cyc-lic plasticity model has to be related with the equationset, equation (31). In several past decades, many plasti-city models have been developed to model the materialbehavior using different levels of complexity from sim-ple to complicated solutions. Modeling complex mate-rial behavior such as cyclic hardening/softening,ratcheting and nonproportional hardening requiresextensive material testing to determine material con-stants required for the modeling of complex materialbehaviors. It is intended in this article to focus on aplasticity model for its simplicity and efficiency to char-acterize the material stress–strain responses with a rea-sonable accuracy. Such a plasticity model is consideredto be suitable for the multiaxial stress–strain notchanalysis. The differences in the plasticity models aregenerally based on the translation rule that governs themovement of the yield surface. One of the most popu-lar cyclic plasticity models was proposed by Mroz.23

Mroz defined a field of plastic moduli in stress spacefor better approximation of the stress–strain curve andgeneralization of the plastic modulus in multiaxial case.Each surface is represented by its center coordinates akij,a yield stress sk

yo and a plastic modulus Ep(k)t . If the von

Mises criterion is used to represent the surface, theyield surface is defined as

Fk Sij, aij� �

=3

2Sij � akij

� �Sij � akij

� �� sk

yo2=0

ð41Þ

Garud21 showed a possibility of intersecting yieldsurfaces for Mroz model under certain loading; there-fore, Garud proposed an improved translation rule thatprevents any intersections of plasticity surfaces. Garudsuggested that the movement of the stress surfacedepends not only on the current state of stress but alsoon the direction of stress increment. Garud postulatedthat increases in stress induce the evolution of plasticdeformation, and the surfaces are subject to transla-tion, keeping constant shape and size without rotationin the stress space. The assumed direction of movementprovides locations of the current state of stress on thesurface at the new location and eliminates the possibil-ity of intersection between adjacent surfaces. The

Ince and Glinka 237



principle idea of the Garud translation rule is demon-strated in the following steps.

It is assumed that an applied load results in the cur-rent stress state settled at the point S and the two yieldsurfaces F1 and F2 with the corresponding yield limitsRF1 and RF2 have been moved in the stress space, sothat their centers have been located to the points aF1ijand aF2ij as it has been arbitrarily assumed and illu-strated in Figure 8.

Based on the applied load path, the current stressincrement induces the plastic strain increment, and it isforwarded outside the surface F1. According to the con-sistency condition, the yield surface must follow thestress state evolution, and the updated stress stateSij+DSij must satisfy the updated yield functionF1(Sij+DSij � aF1ij � DaF1ij )� RF1 =0. During the plas-tic strain evolution the yield surface with a fixed sizeRF1 translates without rotation in the stress space.Taking into account this assumption that the consis-tency condition is satisfied by the condition as the sur-face centered at the point Sij+DSij corresponding tothe updated stress state

F1½(aF1ij +DaF1ij )� (Sij+DSij)� � RF1 =0 ð42Þ

Thus, the translation rule of the yield surface mustbe defined. The details of the yield surface translationrule according to Garud’s proposition are discussed inthe following. The translation is associated with theapplied load path that generates a plastic strainevolution.

In order to translate the yield surface according tothe Garud rule, the following steps should beperformed.

(a) Extend the current stress increment to intersectthe next inactive yield surface F2 at the point A2

as it is presented in Figure 8.

In the index notation, the following equationdescribes the present step

Sij + x � DSij � aF2ij

�� =RF2 ð43Þ

where the unknown coordinates of the point A2 areexpressed as

SA2ij =Sij + x � DSij ð44Þ

Find a normal vector on the next surface at the pointof intersection

nA2ij =

ffiffiffi3

2

rSA2ij � aF2ijRF2

ð45Þ

(b) Determine a point A1 on the active surface F1

with the same normal vector.

The two operations are described in the index notationin the following form

nA2ij = nA1ij =

ffiffiffi3

2

rSA1ij � aF1ijRF1

SA1ij =

RF1

RF2SA2ij � aF2ij

� �+ aF1ij

ð46Þ

(c) Connect the conjugate points A1 and A2 to findthe direction of translation of the yield surfaceF1, as shown in Figure 8 (current yield surfacemoves in the direction of the connection betweentwo normal vectors)

(A1A2)ij =SA2ij � SA1

ij ð47Þ

(d) Translate the surface F1 from point aF1ij to (aF1ij )9in the direction of the vector (A1 A2) till the stressincrement DSij is found on the translated surface,as presented in Figure 8.

In terms of the index notation, the following equa-tion can be defined

Sij +DSij � aF1ij � DaF1ij

�� =RF1 ð48Þ

in which the components DaF1ij of the translation vectorare expressed

DaF1ij = y SA2ij � SA1

ij

� �ð49Þ

Thus, the updated coordinates of the center of theyield surface F1 are given as

(aF1ij )9= aF1ij +DaF1ij ð50Þ

The yield surface movement is determined by calcu-lating the scalar parameter y defined by the consistencycondition represented by equation (48). Barkey14 and

Figure 8. A graphical representation of the Garud cyclicplasticity model.




Singh17 utilized Mroz’s multisurface plasticity modelfor the notch-tip stress and strain calculations.However, it is found that Mroz and Garud’s plasticitymodel produces identical stress–strain predictions whena number of plasticity surfaces exceeded a certain valueand a loading increment is infinitesimal. Otherwise, forthe finite loading increment, the intersection of plasti-city surfaces occurs in the Mroz model, and the Garudmodel generates more accurate stress–strain resultsthan the Mroz model.

Numerical implementation

The elastic–plastic stress–strain model consists of twoparts, namely, the Garud cyclic plasticity model andthe multiaxial Neuber notch correction rule to computethe actual elastic–plastic stress–strain response atnotches using the FE linear elastic stress data. In orderto implement equations defined in this article for anotched component subjected to multiaxial cyclicloads, a general numerical algorithm has been devel-oped. The flowchart of the algorithm is shown inFigure 1. The algorithm starts with the linear elasticFEA results of a notch component for unit loads (amacro written in APDL is used to output linear elasticsolution for each unit load). In the next step, two sepa-rate computer programs written in Fortran 90 are usedperforming the elastic–plastic stress–strain analysis; thefirst computer program is used to superimpose linearelastic stress histories at notch region of the FE modelsubjected to multiaxial loads and the second computerprogram is used to implement the notch stress andstrain analysis using increments of linear elastic stresshistories (output from the first program).

For the implementation of the first program, the lin-ear elastic FE results for each unit load are multipliedby the corresponding load history to compute elasticstress history for that applied load history, and then theelastic stress histories for all applied load histories aresuperimposed to obtain the combined time histories oflinear elastic stresses in accordance with equations (5)–(7). The resultant elastic stress history for each node atthe critical notch region is divided into small incrementsof stresses for numerical implementation of elastic–plastic stress–strain analysis (Figure 4).

The second program computes the actual elastic–plastic stress and strain responses at notch areas. In thebeginning of the program, loading criterion conditionsmust be checked. The yield surfaces are initially cen-tered at the origin (no loading) and all stresses areassumed to be elastic, the stress–strain solution beingdetermined by equation (9). When the elastic loadingreached the initial yield surface, unloading criterion,equation (40), is used to determine elastic unloading/tangential loading and elastic–plastic loading duringstate of stress increment. If the elastic–plastic loadingtakes place, the actual elastic–plastic strain and stressincrements are calculated using equation (31). A set of

seven equations (three notch correction equations fromNeuber’s rule and four constitutive equations) is solvedsimultaneously to determine the actual elastic–plasticstrain and stress increments. Then, the active surfacesare translated according to equations (42)–(50). If stres-ses exceed the outer yield surface as governed by equa-tion (43), the stress increment is bisectioned, and thenstresses are updated to the point where new stress statelies on the yield surface. The current state of the activesurface is also updated. The elastic–plastic stress–straincalculation is repeated for the remaining portion ofstress increments. If the stresses after the load incre-ment remain on the current active yield surface, thestresses are updated and the active surface and any inte-rior surfaces are translated. The procedure is repeateduntil the last elastic stress increment is reached.

One critical part of the actual stress–strain calcula-tion is based on the cyclic plasticity model. The cyclicplasticity model is used to determine the active yieldsurface, which is necessary to define the parameter dl

in the governing equation (25). In other words, theplasticity model determines which piece of the stress–strain curve (the slope of the actual stress–strain curve)has to be utilized for given stress/load increment Dsi todefine the plastic modulus Dseq=Depeq. The plasticitymodels are generally used for calculating stress or strainincrements that result from given stress–strain elements.In the case of the notch analysis, neither stresses norstrains are direct inputs to the plasticity model. Theinput is provided in the form of the total deviatoricstrain energy density increments, and both the actualdeviatoric stress and strain increments DSa

ij and Deaij areunknowns, and these unknown stress and strain incre-ments are simultaneously calculated by solving theequation set, equation (31). After calculating the devia-toric stress increments DSa

ij, the plasticity surfaces aretranslated as shown in Figure 8. The calculated deviato-ric stress increments DSa

ij are subsequently convertedinto the actual stress increments Dsa

ij using equation(32).

The process is repeated for each subsequent incre-ment of the ‘‘elastic’’ input Dse

ij. The cyclic plasticitymodel assumes a stable material response such that notransient hardening effects of the nonproportionalhardening are taken into account.

Results and discussion

In order to predict the fatigue failure of notched com-ponents under the multiaxial cyclic loadings, it is essen-tial to provide a good understanding of how externalloads relate to the state of stress and strain at the criti-cal location and material constitutive behavior. Toestablish the prediction capability of the notch stressand strain analysis model presented in this investiga-tion, calculated elastic–plastic notch strains and stressesobtained from the elasto-plastic stress–strain model arecompared to the experimental strain and stress data of

Ince and Glinka 239



SAE 1070 steel notched bar for six different nonpro-portional load paths.14 Pseudo-elastic stress historiesfor each load path were calculated using linear elasticFE stress results. Calculated elastic stress histories arethen used as input to the analytical elasto-plastic notchanalysis model to compute actual elastic–plastic strainsand stresses at the critical notch area.

Barkey14 performed experiments on circumferentialnotched shafts subjected to various nonproportionalload paths. The notched shafts were subjected to cyclictension and torsional load histories under conditions ofload controls by using Instron and MTS tension–torsion biaxial test frames. Strain gauges were mountedon the notch root for strain measurements. The experi-mented notch shafts were a cylindrical bar with a cir-cumferential notch similar to that shown in Figure 5.Each cylindrical specimen was machined from SAE1070 steel stack to the proper geometry, then heattreated to give uniform material properties. The actualradius of the cylindrical specimen was R=25.4mm,and notch dimensions of the cylindrical specimen werer/t=1 and R/t=2. The FEA and experimental stressconcentration factors are listed in Table 1. These stressconcentration factors are relatively mild and wouldexist on typical notched components such as thosefound in many ground vehicle applications. The ratioof the measured notch-tip hoop stress to the axial stressunder tensile axial loading was se

33=se22 =0:184.

The FE model for the analyzed notched shaft isshown in Figure 9. The geometry of the notched shaftwas modeled in ANSYS FE code and then meshedusing three-dimensional (3D) hexagonal (brick) solidelements, and the area near the notch root was care-fully refined as shown in Figure 9 to obtain accuratepseudo-elastic stress tensors at the notch location. Thepseudo-elastic stress se

ij tensors obtained from the FEmodel are based on the cylindrical coordinate systemthat is defined as: y axis is the primary axial axis, z axisis the tangent to the notch surface and x axis is perpen-dicular to the notch. The coordinate system x-y-zdefined for the FE model is interchangeably used as1-2-3 coordinate system for the stress and strain ten-sors. Linear elastic tensors (nodal stresses on the criti-cal notch area) for the applied load paths were readand converted (using the computer program) to a for-mat readable by the elastic–plastic stress–strain model.Linear elastic stress results from two unit load cases(axial and torsion) were combined with actual axialand torsion load paths using the principle of superposi-tion to obtain increments of pseudo-elastic stress his-tories. The increments of hypothetical ‘‘elastic’’ stresscomponents se

22, se33 and se

23 were used as input for theanalytical elastic–plastic stress–strain model.

The material for the notched bar was SAE 1070 steelwith a cyclic stress–strain curve (Figure 10) approxi-mated by the Ramberg–Osgood relation. The materialproperties were given as E=210MPa, n=0.3,SY=242MPa, n#=0.199 and K#=1736MPa.

Pseudo-elastic notch stresses, se22–se

23, for clockwiseand counterclockwise box-shaped cyclic stress pathsare shown in Figures 11 and 12, respectively. Theclockwise/counterclockwise box-shaped load pathswere repeated more than 100 cycles while recording thestrains at the notch tip. The box path indicates a highdegree of nonproportionality loading. This load pathwas designed to show regions that axial and shearresponses are uncoupled (elastic response) and wherethey are coupled (elastic–plastic response). Therefore,the box-shaped load path provides a critical test for theproposed stress–strain model for notch-tip strain andstress calculations. The maximum nominal tensile andtorsion stresses were sn=296MPa and tn=193MPa,respectively. The corresponding pseudo-elastic notchstresses were se

22 =417:3MPa and se23 =221:9MPa,

respectively. Comparison of the measured and calcu-lated notch strain responses for the clockwise andcounterclockwise box-shaped load paths is shown in

Figure 9. FEA model for SAE 1070 steel notched specimen.

Figure 10. SAE 1070 cyclic stress–strain curve.

Table 1. FEA and experimental stress concentration factors ofSAE 1070 notched bar.

se22=sn se

33=sn se23=sn

FEA 1.42 0.30 1.15Experiment 1.41 0.26 1.15

FEA: finite element analysis.




Figures 13 and 14, respectively. It can be noted that theagreement between the calculated and measured strainresponses are qualitatively and quantitatively good.The strain paths show deviation from the box-shapedload paths at the onset of local plasticity for both the

experiment and the model. It can also be seen fromFigures 13 and 14 that the proposed elastic–plasticstress–strain model predicts the elastic unloading ateach corner of the box (the axial and shear strains areuncoupled) and followed by the elastic–plastic responseto the next corner (the axial and shear strains arecoupled). Figures 13 and 14 also indicate that thestrains ranges are predicted as approximately 5%–15%smaller than experimental ones. These underestima-tions in strain ranges might result in nonconservativefatigue life predictions.

Several nonproportional cyclic loading paths duringwhich ratios of the frequency of applied loads wereunequal were applied to the notched-bar specimen. Themaximum nominal stresses were sn=296MPa andtn=193MPa. Nonproportional load paths fromunequal frequencies of applied loads are a commontype of loadings experienced by many machine compo-nents. Four of those load paths at unequal frequenciesof tensile to torsional load paths in the ratio of 3:1, 5:1,1:3 and 1:5 are analyzed here. Three cycles of tensileload were applied in the same time period as one cycleof torsional load (Figure 15). Five cycles of tensile loadwere applied in the same time period as one cycle oftorsional load (Figure 16). Three cycles of torsionalload were applied in the same time period as one cycleof tensile load (Figure 17). Five cycles of torsional loadwere applied in the same time period as one cycle oftensile load (Figure 18). Axial and shear strain historiesobtained from the model and experiments are plottedin Figures 19–22 for the tensile to torsional frequencyratios of 3:1, 5:1, 1:3 and 1:5, respectively. As seen fromthese figures that there are slight offsets between themeasured strain paths and the calculated strain paths,which were obtained from the stabilized cyclic stress–strain curve. The offsets might be attributed to the factthat the measured strain paths are not symmetric withrespect to the center of coordinates during the first

Figure 12. Box cyclic stress–load path—counterclockwise.

Figure 13. Experimental and calculated strain paths in thenotch tip induced by the box input loading path—clockwise.

Figure 14. Experimental and calculated strain paths in thenotch tip induced by the box input loading path—counterclockwise.

Figure 11. Box cyclic stress–load path—clockwise.

Ince and Glinka 241



cycle, which are always slightly different from the sub-sequent cyclically stabilized material response, mightcause the shift of the strain responses. In spite of thisoffset, strain responses computed by the elastic–plasticstress–strain model reasonably agree well with theexperiment results in terms of the general trend andnumerical strain values.

Conclusion

In this article, the simple analytical multiaxial notchanalysis model, which is based on the Garud cyclicplasticity model integrated with the multiaxial Neubercorrection rule, has been developed to estimate theelastic–plastic notch-tip material behavior of the notchcomponents subjected to the multiaxial nonpropor-tional loadings using linear elastic FE stress solution.

In order to determine the predictive accuracy of themultiaxial elasto-plastic notch analysis model, themodel was validated against the experimental results ofSAE 1070 steel notched shaft obtained by Barkey.Based on the comparison between the experimentaland computed strain histories for six different nonpro-portional load paths, the elastic–plastic stress–strainmodel predicted notch strains with reasonable accuracyusing linear elastic FE stress histories.

The implementation of the incremental strain energydensity equations in the deviatoric stress space and theutilization of the Garud plasticity model provided moreaccurate notch-tip stress–strain calculations than previ-ously suggested strain energy density approaches byMoftakhar et al.12 Barkey14 and Singh.17

Figure 16. Unequal frequency (ratio 5:1) tension–torsionstress/loading path.




Figure 19. Experimental and calculated strain paths in thenotch tip induced by the unequal frequency (ratio 3:1) tension–torsion input loading path.




The multiaxial elasto-plastic stress–strain modeldeveloped in this article provides a more efficient and

logical analytical approach to estimate notch-rootelastic–plastic stress and strain responses for the notchcomponent under the multiaxial nonproportional load-ing. For this reason, the multiaxial notch analysismodel may be employed to perform fatigue life estima-tion for the notched components subject to complexmultiaxial loadings. The effect of changes in material,geometry and loads on the fatigue life can then beassessed in a short time frame in practical engineeringapplications; thus, the design of notched componentscan be evaluated and optimized for the service life inthe early design phase.

Funding

This research received no specific grant from any fund-ing agency in the public, commercial, or not-for-profitsectors.

References

1. Peterson RE. Stress concentration factors. New York:John Wiley & Sons, 1977.

2. Stowell EZ. Stress and strain concentration at a circular

hole in an infinite plate. NACA Technical Report 2073,April 1950. Washington, DC: NACA.

3. Hardrath HF and Ohman H. A study of elastic plastic

stress concentration factors due to notches and fillets in

flat plate. NACA Technical Report 2566, December1951. Washington, DC: NACA.

4. Neuber H. Theory of stress concentration for shearstrained prismatic bodies with arbitrary stress–strain law.

J Appl Mech 1961; 28: 544–550.5. Topper TH, Wetzel RM and Morrow JD. Neuber’s rule

applied to fatigue of notched specimens. J Mater 1969; 1:

200–209.6. Molski K and Glinka G. A method of elastic-plastic

stress and strain calculation at a notch root. Mater Sci

Eng 1981; 50: 93–100.7. Glinka G. Energy density approach to calculation of

inelastic strain–stress near notches and cracks. Eng Fract

Mech 1985; 22: 485–508.8. Hoffmann M and Seeger T. A generalized method for

estimating multiaxial elastic–plastic notch stresses and

strains. Part I: theory. J Eng Mater: T ASME 1985; 107:250–254.

9. Hoffmann M and Seeger T. Stress–strain analysis and

life predictions of a notched shaft under multiaxial load-ing. In: Leese GE and Socie D (eds) Multiaxial fatigue:

analysis and experiments AE-14. Warrendale, PA: Society

of Automotive Engineers, 1989, pp.81–101.10. Ellyin F and Kujawski D. Generalization of notch analy-

sis and its extension to cyclic loading. Eng Fract Mech

1989; 32(5): 819–826.11. Moftakhar AA. Calculation of time independent and time-

dependent strains and stresses in notches. PhD Disserta-tion, Department of Mechanical Engineering, Universityof Waterloo, Waterloo, ON, Canada, 1994.

12. Moftakhar A, Buczynski A and Glinka G. Calculation of

elasto-plastic strains and stresses in notches under multi-axial loading. Int J Fracture 1995; 70: 357–373.

13. Hoffmann M, Amstutz H and Seeger T. Local strain

approach in non-proportional loadings. In: Kussmaul K,




Ince and Glinka 243



McDiarmid D and Socie D (eds) Fatigue under biaxial and

multiaxial loadings—ESIS 10. London: Mechanical Engi-

neering Publications, 1991, pp.357–376.14. Barkey ME. Calculation of notch strains under multiaxial

nominal loading. PhD Dissertation, University of Illinois

at Urbana-Champaign, Urbana, IL, 1993.15. Barkey ME, Socie DF and Hsia KJ. A yield surface

approach to the estimation of notch strains for propor-

tional and non-proportional cyclic loading. J Eng Mater:

T ASME 1994; 116: 173–180.16. Kottgen VB, Schoen M and Seeger T. Application of

multiaxial load-notch strain approximation procedure to

autofrettage of pressurized components. In: McDowell

DL and Ellis R (eds) Advances in multiaxial fatigue,

ASTM STP 1191. Philadelphia, PA: American Society

for Testing and Materials, 1993, pp.375–396.17. Singh MNK. Notch tip stress-strain analysis in bodies sub-

jected to non-proportional cyclic loads. PhD Dissertation,

Department of Mechanical Engineering, University of

Waterloo, Waterloo, ON, Canada, 1998.18. Kottgen VB, Barkey ME and Socie DF. Pseudo stress

and pseudo strain based approaches to multiaxial notch

analysis. Fatigue Eng Mater 2001; 34: 854–867.19. Buczynski A and Glinka G. Elastic-plastic stress-strain

analysis of notches under non-proportional loading

paths. In: F Ellyin and JW Provan (eds) Proceedings of

the international conference on progress in mechanical

behaviour of materials (ICM8), Victoria, BC, Canada,16–21 May 1999, vol. 3, pp.1124–1130.

20. Ye D, Hertel O and Vormwald M. A unified expressionof elastic-plastic notch stress-strain calculation in bodiessubjected to multiaxial cyclic loading. Int J Solids Struct

2008; 45: 6177–6189.21. Garud YS. A new approach to the evaluation of fatigue

under multiaxial loadings. J Eng Mater: T ASME 1981;103: 118–125.

22. Drucker DC. A more fundamental approach to plasticstress-strain relation. In: Proceedings of the first U.S. con-

gress of applied mechanics, Chicago, USA, June 1951,pp.487–491. New York: ASME.

23. Mroz Z. On the description of anisotropic workharden-ing. J Mech Phys Solids 1967; 15: 163–175.

24. Chu CC. A three-dimensional model of anisotropic hard-ening in metals and its application to the sheet metal

forming. J Mech Phys Solids 1984; 32(3): 197–212.25. Amstrong PJ and Frederick CO. A mathematical repre-

sentation of the multiaxial Bauschinger effect. CentralElectricity Generating Board. Technical Report RD/B/N731, 1966. UK: Berkeley Nuclear Laboratories, Researchand Development Department.

26. Chaboche JL. Time-independent constitutive theories forcyclic plasticity. Int J Plasticity 1986; 2(2): 149–188.

27. Jiang Y and Kurath P. A theoretical evaluation of plasti-city hardening algorithms for nonproportional loading.Acta Mech 1996; 118: 213–234.




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