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~~~~ ~~ONSTANT ~~)LUME ~~YPOTHES IS FOR THEF ~~NELASTIC pEFOPMPTION OF ~~~TALS IN THE

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~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~• Aeronautical Eng ineerin g & Mechanics

Rensselaer Polytechnic InstituteTroy, New York 12181

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1.

j ~~~~ CONSTA~~ VOLUME HYPOTHESIS FOR THE INELASTIC DEFOFJ~~TIONOF METALS IN THE SNMPL STRAIN RAI~~E

P. Hewelt and E. Xrempl

H rL In plasticity and creep theory it is generally assumed that

inelastic deformations are volume preserving. Available tensile

and creep tests were evaluated to see whethe r the experiments confirm

the constant volume assumption . None of the experiments which were

done on a variety of structural metals confirmed the constant volume

hypothesis.

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ITHE CONSTANT VOLUME HYPOTHESIS FOR ~~~~~ INELASTIC DEFORMA TION

OF METAlS IN ‘lEE SW~LL STRAIN RANGE

P. Hewelt and B. ]~~empiDeparthent of Mechanical Engineering,Aeronautical Engineering & Mechanics

• Rensselaer Polytechnic Institutepray, New york 12181

Introduction

F I In the plasticity and creep theories for infinitesimal deformation the

plastic strains and the creep strains , respectively , are usually introduced as

deviators* and the total strain is split add~ tively into elastic , plastic and

— creep partsci p1 cr

~ d€~~ 4 de~~ + dcii

(1)

• so that

• de~~ ~ ds~~ (2)

i.e., the classical theories predict that for every stress state the mean-

normal strain is elastic and that the mean normal strain and pressure are

relate d by the elastic bulk modulus. Usually the above results are motivated

by the statement , plastic and creep deformations are isochoric and this is

- referred to as the constan t volume hypothesis.

In the following we evaluate published experimental results to see

whether the constant volume hypothesis is verified . We mention that the

— results are not from the “classical” plasticity or creep literature .

* Other construction s are of course possible .

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11 2.

Check on the Constan t Volume Hypothesis

. a) Tensile Test*

A simple tensile test affords the possibility of checking on the

validity of (1) if the applied stress , the axial strain and the two perpendicu-

- lar transverse strains are measure d simultaneously. If the material is truly

isotropic the measurement of one transverse strain is sufficient. If (1) is

a realistic representation of metal behavior the gra ph of a~~ — a vs —~ll

+ € 22 + determined from a tensile test should be a straight line (1, 2 1.•

L A literature survey was made and the results are su~~~~r js ed in Table 1.

All tests are done at room temperature, In none of the cases was a linear

relationship of a~~ vs t~~~ observ ed, rather the two types of curves shown in

pigs. 1 and 2, respectively appeared as typical result s. In some of the in-

vestigations (3 , 4, 5,61 permanent volume changes were determined . with the

i i exception of (33 where a decrease in volume was found, a permanent volume

increase was reported . This permanent volume change depended on the maximum

strain and was generally small ~‘d 2 x ~~~~ for a true plastic strain of 4% ,

(5] Fig.9) but measurable.

Measurements of poisson ’ s ratio ~ during cyclic testing of tensile test

• specimens under completely reversed strain contro l (9, 101 show clearly that

the usually assumed value of sj — 1/2 is not attained at a total strain range

of 2% ((9 1 , Figs. 7, 8, 9) and at a plastic strain f ive times the elastic

strain ([10], Fig.8). Moreover, the graphs in (9] and (10] reveal a gradual

transition from the elastic value of Poisson ’ s ratio to ~ — 1/2. These tests

then do not support the usual assumption of ~~ = 1/2 for plastic strains. It

11 ________________

* We are only considering test data in small stra in range .

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should be noted that a total of four different steels were investigated in [93• I! and ilO!.

I b) Creep Tests

No elevated temperature creep tests were found during which the axial

and the transverse strains were measured simultaneously. However, room

temperature creep tests during which the three perpendicular strains were1 - measured by means of strain gages on 1100 Al ((11] , Figs.12 and 13) show

clearly that volume is not preserved during creep , rather increases con-

tinuously with time, see (121 , Fig. 3 . The increase is considerable and canno t

be neglected .

Discussion

The foregoing literature survey showed very clearly that the cons tan t

volume assumption usually employed in creep and plastici ty theory is not-

supported by the experiments . Supp orters of the constant volume hypothesis

could point to the fact that isotropy had to be invoked in most of the tests

listed in Table 1 to arriv e at our conclusion . While the assumption of iso-

• tropy was necessary in most cases, there are other instances listed in Table 1

1! where strain was measured in three perpendicular directions. In addition the

permanent volume changes were determined by othe r than strai n measurements.

Since they are shown to be consist ent with the strain measuremen ts [4, 51 the

• volume changes obtained from the strain measurements ar e reliable. Indeed,

no direc t experimental support for the constant-volume hypothesis could be

- - found and most plastici ty textbooks tak e the constant volume hypothesis for

I granted . An exception is (13] . The experimental evidence cited in (13] ,

= however, pertains to large prior deformations (16 % and higher reduction s in diamet er) .

The experiments show that volume increased permanently by an amount that is —

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4.

TMLE 1

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SURVEY OF p~ vious WORX ON a~~ vs €~~~ IN A TENSILE TEST

• Isotro py a~~ vs

~~~~ Material Assumed is Straight Remarks

Line

[7] Al 24ST sheet, flat Yes No All a~~ vs curve sL specimen have the shap e of Fig . 1.

Fully killed low carbon Yes NO Max. axial strain ~~10% . -

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steel , circular specimen

[8] 24S-T4 AL rolled CT~~~ vs curves have145-16 AL extruded NO No f 275S-T6 AL extruded $ 2ape 0 Fig.

- - (3] Six “h igh quality steels” Sharp yield point , aj~j~

Carbon steels Yes NO vs e~~ shape of pig. 2;Anneal 625°C, 1½ hrs in permanent volume decreaseprotective atmosphere .1 to 2% observed after

T testing. Maximum axialstrains ~~~ 3% .

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(1,21 AL 1100 as received Yes NO vs curves haveAL 1100, annealed the shap e of Fig. l except600 F, 2 hrs for carbon steel where

Cu as received both the shape of Figs. 1Cu, annealed 750 F, and 2 is observed .

2 hrsLow carbon steel ,annealed 12500 F, 1 hr

• (4] AISI 4330 ~ with special 110* No 3% maximum strain, Un-

AISI 4310 -, hea t treat- loading and reloading ex-ment periments ; permanent

volume increase; vs

• curves have the

• H _______ __________ shape of Fig.l

(51 NY 80 ~ special 110* NO ~~‘ ~axi~nu~ strai n, Un-

MaBAGfl~~ steel -‘ heat loading and reloading ax-treatment periments ; permanent

volume increase; a~~ vscurves have the

shap e of Fig.l

* Readings from four 90° strain gage rosettes ; we infer that isotropy was notassumed in (4] and (5] .

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small compared to the total deformation at the first measured point. From

I these measurements very little can be inferred about the behavior in the small

— strain range of interest here.

We further see that the detailed nonlinear behavior of the volume change

- with deformation is still in doubt (compare Figs. 1 and 2) and may depend on

• the material. To this end additional experiments are necessa ry.

Behavior of Metal s under Hydrostatic Pressure

1 Another experimental observation related to volume change is the behavior

of metals under hydrostatic pressure . Available experimental evidence [14]

and the results cited in (3] (specifically Ref . (1] in (3] ) supp ort a linear

elastic relation between hydrostatic pressure and volume change*. In many

h instances this linear relation under hydrostatic stress is thought to imply (2 ) .

However, this is not so. It rather requires

LI- - and ~ } for a~ — p6~~ (3)

crde ij~~~

0

Ii i.e., the first invariants of plastic and creep strain increments must be

• 1 zero if the stress state is hydrostatic , In the above c~~ denotes the

stress tensor , p the pressure and is the Xronecke r delta. Equation (2 )

1! implies that (3) holds for every stress state not just the one given in (3) .

It is therefore more restrictive than required by the test results under

hydrostatic stress.

* In this study we are only interested in pressure levels comparable in magnitudeto stresses encountered in stress analysis. The ultrahigh pressure s to which the

I ~ data of (141 extend are of no interest here.

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1 6.

It appears therefore that the constant volume hypothesis is not repro-

duced by the recent experiments quoted herein . Since they cover a wide

• U variety of metals there is reason to believe that the inelastic deformation

of metals may not be volume preserving . In addition the constant volume

H assumption requires special precautions in computational applications (15]

which could conceivably be avoided if an improved constitutive model would be

available. It is therefore urgent to develop faithful constit utive equation s

for metals reflecting the recent observations which are obtained with much

• better equipment than was availabl e at the time of the development of the

I “ classical theories ” of creep and plastici ty.

Acknowledgemen t

The support of the National Science Foundation and the Office of Naval

Research is gratefu lly acknowledged .

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REFERE1~~ES

‘ 1. Valanis, LC. and Han-thin Wu, Material Instabilities in the Expe rimentalStudy of the Plastic compressibility of Some Important Metals, Departmentof Mechanics and Hydraulics, Univ. of Iowa, Report No. M&H 1. 03 (August1971).

2. valanis, K. C., On the Foundations of the Endochronic Theory of Visco-.plasticity , Archives of Mechanics , iL 857—868 (1975).

3 . xafka , V. and R. Novotny, A contribution to the Solution of the Questionof Volume changes of Polycrysta lline Materials in plastic Deformation,ACTA Technica CSAV, !~

41 (1971).

4. Spitzig , W .A. , R. Y. Sober and 0. Richmond, Pressure Dependence of Yieldingand Associated Volume Expansion in Tempered Materials, Acts Met ., j~,885—893 (1975).

5. Spitzig , W A . , R. J. Sober and 0. Richmond, The Effect of HydrostaticPressure on the Deformation Behavior of Ma raging and gy-80 Steels andits Implication for Plastici ty Theory , Met. Trans. A, ~~~

1703—1710 (1976).

6. Man n, j. and L. W. Hu, On the validity of Assumptions Made in Theorj es ofPlastic Flow for Metals , Trans. of the AS~~ , Z2~

1181 (1953).

7. Stang, A.H. , M. Greenspan and S.B. Newman, Poisson ’ s Ratio of Some Struc-tural Alloys for Large Strains , Journal of Research of the National Bur eau

• of Standards, ~~~,

211 (1946).

8. Gerard, G. and S. Wildhorn, A study of Poisson ’s Ratio in the Yield Region ,MACA TN 256 1 (1952).

9. Krempl, E. , Evaluation of High Elongation Foil Strain Gages for Measuringcyclic Plast ic Strains , Experimental Mechanics , Al-A8 (1968).

10. Dowling, N.E . , Performanc e of Metal Foil Strain Gages During Large CyclicStrains , Westinghouse Research Labo ratories Report 75-1E7—PAL FA-R2,october 1975 , Experimental Mech anics, L~, 193—197 (1977).

11. Phillips , A. and M. Ricciuti , Fundamental Experiments in Plast icity andCreep of Aluminum-Extension of previous Results , m t . r . Solids andStructures, ~~~, 158—171 (1976).

12. Leis , B.N. , Nonlinear Aspects of Structural Fatigue Damag e Assessment andAccumulation , Pap er L3/3 , Proceeding s SMiRT IV, Coimnission of the EuropeanCommunities , Brussels , 1977 .

13. Hill , R. , The Mathematical Theory of Plasticity , Oxford at the Clare ndonPress , 1950.

14. Bell, i. F. , The Experimental Foundations of Solid Mechanics, Handbuch derPhysik , vIa/ l, Springer 1976 , specifically Fi g.4.46 .

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15. Nagtegaal, J. C ., D.M. Parke s and J. R. Rice, On Numerically. AccurateFinite Elemen t Solutions in the Fully plastic Range , Report NASA NGL 40-

• 002—080/14, Brown university , March 1974 .

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FIGURE CAPTIONS

Figure 1 Axial Stress a (or Mean Normal Stress a& vs. Measured

Vnlume Change €~~ as Determined from the Results of Tensile

- -~~ Tests in [71.-

• • Figure 2 Absolute Value of Axial Stress a (or Absolute Mean Normal

- Stress Ia1~~I ) vs. Measured Absolute Volume Change

as Determined from the Results of Tensile and Compression

Tests Reported in (8] .

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