MODELLING OF RATE-DEPENDENT ANISOTROPIC MATERIAL BEHAVIOR BY PLASTICITY THEORY

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MODELLING OF RATE-DEPENDENT ANISOTROPIC MATERIAL BEHAVIOR BY

PLASTICITY THEORY

W. Drysdale

To cite this version:

W. Drysdale. MODELLING OF RATE-DEPENDENT ANISOTROPIC MATERIAL BEHAV- IOR BY PLASTICITY THEORY. Journal de Physique Colloques, 1985, 46 (C5), pp.C5-73-C5-80.

�10.1051/jphyscol:1985510�. �jpa-00224740�

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MODELLING O F RATE-DEPENDENT ANISOTROPIC MATERIAL BEHAVIOR

BY PLASTICITY THEORY

W.H. Drysdale

U.S. Army B a Z Z i s t i c Research Laboratory, Aberdeen Proving Ground, MaryZand 21005, U.S.A.

Resume - Dans l e c a s de materiaux a n i s o t r o p e s , nous avons B t a b l i l e s Qquations c o n s t i t u t i v e s pour l a deformation p l a s t i q u e dependant de l a v i t e s s e . Lamethode employee c o n s i s t e 2 i n c l u r e l a v a r i a b l e t a u x de deformation a p p r o p r i e e dans l a d e f i n i t i o n f o n c t i o n n e l l e de l a s u r f a c e d'dcoulement e t 2 imposer que l e p o i n t r e p r e s e n t a n t 1 ' B t a t demeure s u r c e t t e s u r f a c e drQcoulement l o r s d'un a c c r o i s - sement de l a deformation p l a s t i q u e .

A b s t r a c t - C o n s t i t u t i v e e q u a t i o n s a p p r o p r i a t e t o t h e rate-dependent p l a s t i c deformation of a n i s o t r o p i c m a t e r i a l s a r e developed. The method used i n t h e d e r i v a t i o n c o n s i s t s of t h e i n c l u s i o n of t h e a p p r o p r i a t e r a t e of deformation v a r i a b l e i n t h e f u n c t i o n a l d e f i n i t i o n of t h e y i e l d s u r f a c e and the requirement t h a t t h e s t a t e p i n t remain on t h i s y i e l d s u r f a c e during an increment of p l a s t i c s t r a i n .

I - INTRODUCTION

The p r e s e n t work w i J l g e n e r a l i z e a p r e v i o u s paper p r e s e n t e d a t t h e Third Oxford Conference on Mechaqical P r o p e r t i e s a t High Rates of S t r a i n / I / , which was concerned with r a t e dependent p l a s t i c deformation of i s o t r o p i c m a t e r i a l . The a p p l i c a t i o n of t h e c u r r e n t e f f o r t is t o t h e p l a s t i c deformation of m a t e r i a l s which cannot be modelled a s i s o t r o p i c , e.g. h e a v i l y worked metals with d i r e c t i o n a l l y o r i e n t e d c r y s t a l s , p a r t i a l l y o r i e n t e d s h o r t f i b e r composite m a t e r i a l s , and u n i d i r e c t i o n a l f i b e r composites. B r i t t l e f i b e r composites which behave e l a s t i c a l l y u n t i l f a i l u r e and t h e v i s c o e l a s t i c behavior of some r e s i n matrix m a t e r i a l s with r a t e dependent moduldi a r e n o t included w i t h i n t h e intended scope of t h e theory.

The r a t e e f f e c t of primary i n t e r e s t i s t h e observed i n c r e a s e i n y i e l d s t r e s s with s t r a i n r a t e . This d a t a is obtained from a sequence of c o n s t a n t r a t e , u n i a x i a l s t r e s s t e s t s performed a t d i f f e r e n t s t r a i n r a t e s . Yield s t r e s s i n c r e a s e s of 10% t o 20% over a " s t a t i c " value may be measured f o r some m a t e r i a l s f o r even moderate r a t e s of load a p p l i c a t i o n /2/. This s o - c a l l e d " s t r a i n - r a t e s t r e n g t h e n i n g " should be included i n s t r u c t u r a l a n a l y s i s .

Another m a t e r i a l behavior t r a c e a b l e t o r a t e e f f e c t is t h e appearance of c r e e p - l i k e phenomena a t room temperature / 3 / . I f a m a t e r i a l i s loaded t o a s t r e s s above t h e y i e l d p i n t and t h i s load maintained, p l a s t i c s t r a i n w i l l c o n t i n u e t o grow u n t i l a l i m i t i n g v a l u e i s reached. Thus, t h e t o t a l p l a s t i c s t r a i n from a l o a d i n g p u l s e depends not only on t h e maximum s t r e s s achieved, b u t a l s o on t h e d u r a t i o n of t h e p u l s e .

These e x p e r i m e n t a l l y observed r a t e dependent phenomena a r e most commonly modelled a t t h i s time by one of two methods. The f i r s t method i n v o l v e s t h e use of t h e theory of rate-independent p l a s t i c i t y . However, t h e parameters used i n t h e model, e.g. t h e y i e l d s t r e s s , a r e o b t a i n e d from s t r e s s - s t r a i n experiments performed a t high s t r a i n r a t e s . The use of t h e s e "dynamic" p r o p e r t i e s i n t h e rate-independent theory has

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been shown t o g i v e a c c e p t a b l e p r e d i c t i v e c a p a b i l i t i e s f o r some problems where a l a r g e p o r t i o n of t h e deformation occurs near t h e "dynamic" r a t e , most n o t a b l y f o r u n i a x i a l p l a s t i c wave propagation /4/. This model cannot be a s a c c u r a t e f o r widely v a r i a b l e l o a d i n g r a t e s and does n o t g i v e i n s i g h t i n t o t h e m a t e r i a l behavior a t t h e peak of a load p u l s e , where t h e l o a d i n g r a t e i s zero.

The second common method is o v e r s t r e s s v i s c o p l a s t i c i t y /5/. This model assumes t h a t t h e p l a s t i c deformation i s a viscous flow, d r i v e n by t h e excess of t h e a p p l i e d s t r e s s component over t h e c u r r e n t s t a t i c y i e l d s t r e s s . The l i n e a r form of t h i s o v e r s t r e s s c o n s t i t u t i v e r e l a t i o n i s seldom used, r a t h e r an e x p o n e n t i a l o r power f u n c t i o n i s assumed i n o r d e r t o f i t c o n s t a n t r a t e d a t a . It is p r e f e r a b l e from a t h e o r e t i c a l view t o f i t a m a t e r i a l parameter t o t h e r a t e d a t a and then t o d e r i v e t h e form of t h e c o n s t i t u t i v e f u n c t i o n f o r v a r i a b l e l o a d i n g r a t e s .

There a r e o t h e r models of m a t e r i a l i n e l a s t i c behavior t h a t a l s o p r e d i c t r a t e e f f e c t s . (See t h e review by Bodner/6/.) However, t h e s e c o n s t i t u t i v e e q u a t i o n s a r e n o t widely used i n computations, such a s f i n i t e element s t r e s s a n a l y s i s codes.

The purpose of t h e p r e s e n t work then w i l l be t o d e r i v e a c o n s i s t e n t model of o r t h o t r o p i c p l a s t i c i t y with r a t e e f f e c t s . The p o i n t of view w i l l be t h a t of t h e s t r u c t u r a l a n a l y s t ; t h e r e s u l t i n g c o n s t i t u t i v e e q u a t i o n s must be computationally u s e f u l and compatible with numerical s t r e s s a n a l y s i s codes.

The approach w i l l be t o make t h e s i m p l e s t e x t e n s i o n of t h e c l a s s i c a l a n i s o t r o p i c t h e o r y which y i e l d s a c o n s i s t e n t t h e o r y of rate-dependent p l a s t i c i t y . 1t w i l l be shown t h a t it i s s u f f i c e n t t o add an a p p r o p r i a t e i n t e r n a l r a t e v a r i a b l e t o t h e formal d e f i n i t i o n of t h e y i e l d s u r f ace.

11 - EQUATIONS OF THE PLASTIC STATE

The c e n t r a l concept of most p l a s t i c i t y t h e o r i e s is t h e y i e l d s u r f a c e . This is a s u r f a c e i n s t r e s s space, where t h e s t r e s s components a r e a x e s , which e n c l o s e s t h e zone of e l a s t i c response /7/. A s t h e load i s i n c r e a s e d i n i t i a l l y from a s t r e s s f r e e s t a t e i n any d i r e c t i o n , t h e m a t e r i a l behaves e l a s t i c a l l y u n t i l some p o i n t , where p l a s t i c behavior i n i t i a t e s . The l o c u s of a l l such p o i n t s is t h e y i e l d s u r f a c e . The shape of t h i s s u r f a c e may be approximated a n a l y t i c a l l y a s a f u n c t i o n of t h e s t r e s s components. For i s o t r o p i c m a t e r i a l s , t h e a n a l y t i c r e p r e s e n t a t i o n i s simple; f o r a n i s o t r o p i c m a t e r i a l s t h e f u n c t i o n a l form may be very involved.

Most m a t e r i a l s harden due t o p l a s t i c deformation, t h a t i s , t h e y i e l d s t r e s s i n t h e d i r e c t i o n of p r i o r p l a s t i c s t r e s s i n g i s i n c r e a s e d , s o t h e y i e l d s u r f a c e must extend i n t h i s d i r e c t i o n . Thus, t h e f u n c t i o n a l d e f i n i t i o n of t h e y i e l d s u r f a c e must c o n t a i n some i n t e r n a l hardening v a r i a b l e . The most commonly used parameters i n t h e c l a s s i c a l f o r m u l a t i o n s a r e t h e kinematic "back s t r e s s , " o r t h e e f f e c t i v e p l a s t i c s t r a i n , both f u n c t i o n s of t h e p l a s t i c s t r a i n t e n s o r , o r t h e t o t a l p l a s t i c work.

Since t h e p l a s t i c work t o a p a r t i c u l a r p l a s t i c s t r a i n i s n o t unique f o r r a t e - dependent s t r e s s l e v e l s , t h e p l a s t i c s t r a i n t e n s o r is assumed t o be t h e i n t e r n a l hardening v a r i a b l e .

F i n a l l y , t h e y i e l d s t r e s s i s shown e x p e r i m e n t a l l y to vary with r a t e of loading. For a rate-dependent y i e l d s u r f a c e , some i n t e r n a l r a t e parameter must a l s o be included among t h e v a r i a b l e s s p e c i f y i n g t h e y i e l d f u n c t i o n /8/. It i s c r u c i a l f o r a c o h e r e n t t h e o r y t h a t the c o r r e c t r a t e v a r i a b l e be used f o r t h i s f o r m u l a t i o n . From a

c o n c e p t u a l p o i n t of view, it is no longer convenient t o view t h e y i e l d s u r f a c e a t high v a l u e s of t h e r a t e v a r i a b l e a s e n c l o s i n g only t h e r e g i o n of e l a s t i c response.

S t r i c t l y , only t h e " s t a t i c " y i e l d s u r f a c e , corresponding t o a z e r o value of t h e r a t e v a r i a b l e , e n c l o s e s o n l y e l a s t i c m a t e r i a l response.

I t i s assumed t h a t t h e increments of s t r a i n a r i s i n g from d i f f e r e n t p h y s i c a l mechanisms may be d i s t i n g u i s h e d . Thus a t o t a l s t r a i n increment i s t h e sum of e l a s t i c and p l a s t i c p a r t s .

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of an a p p l i e d f o r c e increment. This behavior i s e f f e c t i v e l y i n s t a n t a n e o u s and t h e r e f o r e cannot be a f a c t o r i n a r a t e / t i m e dependent phenomena.

The p l a s t i c p a r t of t h e s t r a i n increment r e f l e c t s change i n t h e c r y s t a l s t r u c t u r e due p r i m a r i l y t o t h e accumulated passage of d i s l o c a t i o n s . It i s a c t i v a t e d by t h e a p p l i e d f o r c e s , b u t must develop i n time due t o t h e d e f i n i t e average v e l o c i t y a t t a i n e d by t h e d i s l o c a t i o n s w i t h i n t h e c r y s t a l . Hence, t h e p l a s t i c s t r a i n increment, o r t h e r e l a t e d r a t e , i s d i r e c t l y dependent on t h e m i c r o s t r u c t u r a l r a t e - c o n t r o l l i n g phenomena and s o may b e s t s e r v e a s a macroscopic measure of t h i s q u a n t i t y , i.e. a s an i n t e r n a l r a t e v a r i a b l e .

The c l a s s i c a l q u a n t i t y , e f f e c t i v e p l a s t i c s t r a i n increment ( o r t h e r e l a t e d r a t e q u a n t i t y ) , d e f i n e d a s

sums t h e p l a s t i c s t r a i n increment t e n s o r and s o sums t h e t o t a l p l a s t i c " a c t i v i t y "

occuring i n t h e c r y s t a l . Therefore, it i s assumed t h a t t h i s e f f e c t i v e p l a s t i c s t r a i n r a t e is an i n v a r i a n t measure f o r t h e r a t e e f f e c t . Other measures of r a t e may be used i n t h e f o r m u l a t i o n of the r a t e dependent y i e l d s u r f a c e ; a r e c e n t example c o n t a i n s t h e t o t a l s t r a i n r a t e /9/. However, use of t h i s v a r i a b l e r e q u i r e s s e p a r a t e assumption of c o n s i s t e n t s t r e s s - s t r a i n behavior d u r i n g change of r a t e m a t e r i a l

t e s t s

.

Thus, t h e r e is a g e n e r a l y i e l d s u r f a c e i n t h e extended s t r e s s space, given by

where super d o t s r e p r e s e n t time d e r i v a t i v e s . Now, being on t h i s y i e l d s u r f a c e d e f i n e s t h e " p l a s t i c s t a t e , " s o t h a t throughout t h e p l a s t i c deformation e q u a t i o n (2) must hold. Changes i n t h e i n t e r n a l s t a t e v a r i a b l e s must be such t h a t

dF =- _aoijaF d o . . ₁₁+

_apj

d c P . _l1

+ = _a2'

^dZP⁼^0.

Also r e q u i r e d i s a s p e c i f i c a t i o n of t h e d i r e c t i o n of p l a s t i c s t r a i n , i.e. the r e l a t i v e s i z e of t h e components of t h e i n c r e m e n t a l p l a s t i c s t r a i n t e n s o r . The c l a s s i c a l a s s o c i a t e d flow r u l e is adopted without m o d i f i c a t i o n , thus

Using e q u a t i o n ( 4 ) i n t h e d e f i n i t i o n of e f f e c t i v e p l a s t i c s t r a i n increment g i v e s

m e n , using e q u a t i o n s ( 4 ) and ( 5 ) i n t h e d i f f e r e n t i a l e x p r e s s i o n dF = 0 g i v e s t h e incremental governing e q u a t i o n of t h e p l a s t i c s t a t e

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a € ~aa.

i j l j

dZP + ^dEP+ - aF

a

ZP

²^aF ^aF

a q j

" I j =

Dividing t h e e q u a t i o n ( 6 ) by d t , t h e time increment g i v e s

aF "

a€:. a o i j

.

- ^EP+ ^{c p =} - - ^{a F}

a

EP

²âF âF âa.11 ^.^'ij

Equation ( 7 ) i s a n e v o l u t i o n a r y e q u a t i o n f o r t h e i n t e r n a l r a t e v a r i a b l e ?. The e q u a t i o n was d e r i v e d simply from t h e d e f i n i t i o n of a r a t e dependent y i e l d s u r f a c e and d i d n o t need t o be s e p a r a t e l y assumed.

An approximate s o l u t i o n t o e q u a t i o n s ( 6 ) o r ( 7 ) i s o b t a i n e d by a method common i n n o n l i n e a r numerical s t r u c t u r a l a n a l y s e s , t h e use of s m a l l , f i n i t e time increments.

Thus t h e v a r i a b l e c o e f f i c i e n t s i n e i t h e r e q u a t i o n a r e assumed t o be c o n s t a n t , w i t h t h e value from t h e beginning of t h e time s t e p , and only t h e incremental o r r a t e q u a n t i t i e s evolve with time. A s t h e time s t e p i s made s m a l l e r , t h e s o l u t i o n o b t a i n e d from t h i s procedure w i l l approach t h e e x a c t one.

With t h i s s i m p l i f i c a t i o n e q u a t i o n (71, f o r example, becomes a l i n e a r f i r s t o r d e r d i f f e r e n t i a l e q u a t i o n f o r

ZP

w i t h i n t h e small time increment, with an i n i t i a l c o n d i t i o n

which s p e c i f i e s t h e c o n t i n u i t y of e f f e c t i v e p l a s t i c s t r a i n r a t e between time increments. For t h e f i r s t p l a s t i c s t r a i n increment, of c o u r s e

Solving e q u a t i o n ( 7 ) t h e n g i v e s

Applying t h e i n i t i a l c o n d i t i o n , e q u a t i o n ( 7 a ) , allows t h e c o n s t a n t of i n t e g r a t i o n , B l r t o be e v a l u a t e d a s

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2 aF aF

;p =

-

aF aF a a . . 11

-- _=I

aeEn aamn

For s m a l l time s t e p s , a s assumed, deP =

rpt

^,

and the a s s o c i a t e d flow r u l e , equation (4) i s

In equation (101, t h e f i r s t l i n e corresponds t o c l a s s i c a l , r a t e - independent p l a s t i c i t y t h e o r y . The remaining terms a r e c o r r e c t i o n s f o r r a t e e f f e c t . The form of t h e y i e l d f u n c t i o n has n o t been s p e c i f i e d and is f u l l y g e n e r a l , so t h a t any type of o r t h o t r o p i c m a t e r i a l behavior i s included i n t h e r e l a t i o n .

I11 - APPLICATION

The form of y i e l d f u n c t i o n used t o model a n i s o t r o p i c m a t e r i a l behavior may be a form of t h e H i l l ' s y i e l d c o n d i t i o n with kinematic hardening, t h a t i s

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where Y l l , YZ2 and Y33 a r e t h e y i e l d s t r e s s e s i n simple t e n s i o n i n t h e t h r e e o r t h o t r o p i c d ~ r e c t i z n s and Y12, Y23 and Y13 a r e t h e corresponding y i e l d s t r e s s e s i n simple s h e a r . The Yterms a r e f u n c t i o n s of t h e y i e l d s t r e s s e s . These f u n c t i o n s a r e

The y i e l d s u r f a c e is assumed t o t r a n s l a t e i n t h e d i r e c t i o n of p l a s t i c s t r a i n . Thus t h e t r a n s l a t i o n i s given by

da. . = C dEij P r (1 2 1

1 3

where C i s a unique m a t e r i a l parameter. For i s o t r o p i c m a t e r i a l s t h e r e i s no ambiguity i n t h e d e f i n i t i o n .of C. However, f o r a n i s o t r o p i c behavior, t h e d e t e r m i n a t i o n of C must be s p e c i f i e d

.

The d e t e r m i n a t i o n of t h e m a t e r i a l hardening parameter, C, w a s performed f o r a n i s o t r o p i c s t a t i c p l a s t i c i t y i n / l o / , where t h e following assumptions were made

i . The m a t e r i a l i s t r a n s v e r s e l y i s o t r o p i c ii

.

^The^{1 1}d i r e c t i o n i s predominant

iii. The s t r e s s - s t r a i n r e l a t i o n s f o r " s t a t i c " , u n i a x i a l s t r e s s m a t e r i a l t e s t s have a b i l i n e a r form.

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T

c = - 2 ( 1 3 )

3 E - E T '

11 11

where E~ i s t h e s t a t i c p l a s t i c modulus.

11

The r a t e e f f e c t i s a l s o presumed to be e q u a l i n t h e t h r e e o r t h o t r o p i c d i r e c t i o n s , with t h e y i e l d s u r f a c e expanding uniformly due t o i n c r e a s i n g r a t e of loading. With t h e same assumption used i n t h e d e r i v a t i o n of t h e hardening parameter, u n i a x i a l s t r e s s t e s t s i n t h e 11 d i r e c t i o n over a range of s t r a i n r a t e s a r e f i t t o t h e s t a n d a r d r e p r e s e n t a t i o n

where Y i s t h e " s t a t i c " y i e l d s t r e s s i n 11 d i r e c t i o n , b i s t h e s t r a i n - r a t e

" l o

hardening parameter, and E i s t h e t r a n s i t i o n s t r a i n r a t e .

The method of f i t t i n g t h i s equation t o experimental s t r e s s - s t r a i n curves r e q u i r e s some c a r e . The experimental u n i a x i a l s t r e s s t e s t s a r e g e n e r a l l y performed a t a c o n s t a n t t o t a l s t r a i n r a t e . However, t o t a l s t r a i n r a t e is not t h e r e q u i r e d r a t e v a r i a b l e f o r use i n e q u a t i o n ( 1 4 ) . For a b i l i n e a r model of t h e experimental curve, t h e p l a s t i c s t r a i n r a t e becomes c o n s t a n t a f t e r a very l i m i t e d amount of p l a s t i c s t r a i n . Back e x t r a p o l a t i o n t o t h e e l a s t i c l i n e along a l i n e a r p l a s t i c l i n e w i l l g e n e r a t e a p p a r e n t y i e l d s t r e s s v e r s u s c o n s t a n t p l a s t i c s t r a i n r a t e d a t a .

Since we a l s o assume, along with e q u a t i o n ( 1 4 ) , P

Y~~ = Y~~~ [ I + b gn ( I +

F)]

^,^{e t c .}

E

m u l t i p l y i n g e q u a t i o n (1 1 ) by the square of t h e q u a n t i t y i n b r a c k e t s i s o l a t e s t h e r a t e dependence i n a s i n g l e term of t h e y i e l d s u r f a c e e x p r e s s i o n . This s i m p l i f i e s t h e form of t h e r e l e v a n t d e r i v a t i v e s r e q u i r e d i n e q u a t i o n (1 0 ) .

IV - DISCUSSION

Equation ( 1 0 ) r e p r e s e n t s t h e increment of rate-dependent p l a s t i c s t r a i n f o r a small time i n t e r v a l i n a form u s e f u l i n i n c r e m e n t a l n o n l i n e a r f i n i t e element codes. The c a l c u l a t i o n of y i e l d s u r f a c e d e r i v a t i v e s from t h e e x p r e s s i o n s generated a s e q u a t i o n s

( 1 1 ) - ( 1 4 ) completes t h e s p e c i f i c a t i o n of t h e approximate i n c r e ~ ~ n t a l t h e o r y . A l l of t h e q u a n t i t i e s r e q u i r e d to e v a l u a t e t h e s e d e r i v a t i v e s , e.g. E = , a r e known a t t h e beginning of t h e time i n t e r v a l . Hence t h e use of t h e s e e x p r e s s i o n s i n i n c r e m e n t a l n o n l i n e a r s t r u c t u r a l codes does not r e q u i r e t h e m o d i f i c a t i o n of t h e i n c r e m e n t a l s o l u t i o n s t r a t e g y .

The s e l e c t i o n of t h e c o r r e c t v a r i a b l e t o r e p r e s e n t t h e r a t e e f f e c t can s i g n i f i c a n t l y s i m p l i f y t h e behavior of t h e m a t e r i a l model. The use of t h e p l a s t i c s t r a i n r a t e a s i n t h e c u r r e n t model, means t h a t t h e " y i e l d " of t h e m a t e r i a l always occurs a t t h e s t a t i c y i e l d s t r e s s , s i n c e t h e p l a s t i c s t r a i n r a t e is zero b e f o r e p l a s t i c

deformation begins. The p l a s t i c s t r a i n r a t e i n c r e a s e s c o n t i n u o u s l y a f t e r y i e l d , a s

(9)

JOURNAL DE PHYSIQUE

d e s c r i b e d by equation (10). The appearance of t h e u n i a x i a l s t r e s s - s t r a i n curve a f t e r y i e l d shows t h a t t h e i n i t i a l slow p l a s t i c s t r a i n r a t e does n o t v i s i b l y change t h e s l o p e of t h e e l a s t i c l i n e , u n t i l an a b r u p t t r a n s i t i o n occurs t o t h e p l a s E i c l i n e a t a s t r e s s l e v e l corresponding t o t h e f i n a l p l a s t i c s t r a i n r a t e /I/. During r a t e increment t e s t s , where t h e s t r a i n r a t e i s i n s t a n t a n e o u s l y i n c r e a s e d from a lower t o a h i g h e r value, t h e model with p l a s t i c s t r a i n r a t e a s v a r i a b l e r e t u r n s t o a n e a r l y e l a s t i c l i n e u n t i l t h e p l a s t i c l i n e corresponding t o t h e h i g h e r r a t e i s approached, where a r a p i d t r a n s i t i o n t o t h i s l i n e occurs.

In c o n t r a s t , t h e use of t o t a l s t r a i n r a t e a s r a t e v a r i a b l e r e q u i r e s t h a t t h e " y i e l d "

of t h e m a t e r i a l depends on t h e e x t e r n a l l y a p p l i e d s t r a i n r a t e . Thus continuous checking i s r e q u i r e d t o determine whether o r n o t t h i s m a t e r i a l is i n a p l a s t i c s t a t e . In a d d i t i o n , f o r r a t e increment t e s t s , t h e m a t e r i a l must be assumed t o i n s t a n t a n e o u s l y t r a n s i t i o n between d i f f e r e n t l e g s of t h e p l a s t i c l i n e corresponding t o t h e d i f f e r e n t r a t e s , i n o r d e r t h a t t h e uniqueness of t h e d e f i n i t i o n of p l a s t i c s t r a i n increment may be maintained /9/. This behavior d u r i n g changes of r a t e i s i n c o n s i s t e n t with t h e e x p e r i m e n t a l l y observed behavior of m a t e r i a l s /3/.

REFERENCES

1. Drysdale, W.H., Mechanical P r o p e r t i e s a t High Rates of S t r a i n , (1984), 143.

2. Lindholm, U.S., Yeakley, L.M., and Bessey, R.L., A i r Force M a t e r i a l Lab Tkchnical Report AFML-TR-68-194, ( 1 968)

.

3. P h i l l i p s , A., P l a s t i c i t y i n S t r u c t u r a l Engineering, Fundamentals and A p p l i c a t i o n s , (1979), 187.

4. Ripperger, E.A., and Watson, H., Mechanical Behavior of M a t e r i a l s Under Dynamic Load (1968), 294.

-

5. Perzyna, P., Advances i n ~ p p l i e d Mechanics, (1 966), 243.

6. Bodner, S.R., Mechanical P r o p e r t i e s a t High Rates of S t r a i n , (1 984), 451

.

7. Martin, J.B., P l a s t i c i t y : Fundamentals and General R e s u l t s , ( 1975 )

.

8. Olszak, W., P l a s t i c i t y i n S t r u c t u r a l Engineering, Fundamentals and A p p l i c a t i o n s , (19791, 273.

9. Naghdi, P.M., J. Appl. Mech., 2, (1 9841, 93.

10. Zak, A.R., and Craddock, J . N . , B a l l i s t i c Research Laboratory C o n t r a c t o r Report ARBRL-CR-00423, (1 980).