FURTHER RESULTS ON THE INITIATION AND GROWTH OF ADIABATIC SHEAR BANDS AT HIGH STRAIN RATES

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FURTHER RESULTS ON THE INITIATION AND GROWTH OF ADIABATIC SHEAR BANDS AT

HIGH STRAIN RATES

T. Wright, R. Batra

To cite this version:

T. Wright, R. Batra. FURTHER RESULTS ON THE INITIATION AND GROWTH OF ADIA-

BATIC SHEAR BANDS AT HIGH STRAIN RATES. Journal de Physique Colloques, 1985, 46 (C5),

pp.C5-323-C5-330. �10.1051/jphyscol:1985541�. �jpa-00224772�

J O U R N A L DE PHYSIQUE

Colloque C5, supplirnent au n08, Tome 46, aoOt 1985 page C5-323

FURTHER RESULTS ON THE INITIATION AND GROWTH OF ADIABATIC SHEAR BANDS AT HIGH STRAIN RATES

T.W. Wright and R.C. ~ a t r a *

~ a Z Z i s t i c Research Laboratory, Aberdeen Proving Ground, MD. 21005, U.S.A.

U n i v e r s i t y o f Missouri-RoZZa, RoZZa, MO. 65401, U.S.A.

Resume

-

Par un calcul aux elements f i n i s on a etudie l a croissance de per- turbations dans l a deformation par cisaillement simple d'un solide thermo- visco-plastique avec adoucissement. On a trouve que l e s t r e s p e t i t e s pertur- bations croissent lentement au debut mPme s i l e pic de contrainte de l a courbe contrainte-deformation homogene a

@te

franchi mais qu'une croissance explosive peut avoir l i e u par l a s u i t e . Durant l a phase de croissance l e n t e , l a contrainte s u i t de trPs prPs l a reponse homogene mais dans l a phase ex- plosive l a contrainte chute precipi tamment. Les grandes perturbations crois- sent

a

une vitesse i n i t i a l e plus importante e t a c d l P r e n t rapidement mGme avant l e pic de contrainte de l a dsformation homogene. L'introduction d'un e f f e t plastique dipolaire semble P t r e s t a b i l i s a n t e .

A b s t r a c t , - Growth of perturbations in the simple shearing deformation of a softening thermo-visco-plastic solid have been examined by means of a f i n i t e element technique. I t has been found t h a t very small perturbations grow slowly a t f i r s t , even though the underlying homogeneous s t r e s s - s t r a i n response has passed peak s t r e s s , but t h a t eventually explosive growth occurs. During the slow growth phase, the s t r e s s follows the homogeneous response q u i t e closely, but in the explosive phase, the s t r e s s drops precipitously. Large perturbations grow a t a larger i n i t i a l r a t e , and accelerate rapidly even before peak homogeneous s t r e s s . Inclusion of a dipolar p l a s t i c e f f e c t appears to be s t a b i l i z i n g .

I - INTRODUCTION AND BASIC EQUATIONS

Adiabatic shear banding i s a localization phenomenon t h a t occurs during high r a t e p l a s t i c deformation in many materials. Since t h e heat generated by p l a s t i c flow usually tends t o lower the flow s t r e s s , the response curve f o r adiabatic homogeneous deformation will f a l l below t h e isothermal response. If t h i s thermal softening i s stronger than both work and r a t e hardening together, then eventually a peak s t r e s s occurs a t a c r i t i c a l value of s t r a i n , followed by decreasing s t r e s s f o r f u r t h e r increments of s t r a i n . Such a s i t u a t i o n i s generally unstable and presents an opportunity f o r s t r a i n localization t o occur. In recent years there has been con- siderable i n t e r e s t in developing a quantitative, dynamical theory of adiabatic shear banding. Pluch of the recent l i t e r a t u r e , a s well as some of the pioneering works have been l i s t e d and discussed b r i e f l y by Clifton, e t a l . / I / . In t h i s paper we present f u r t h e r calculations of the kind given previously by Mright and Batra / 2 / . In partic- u l a r , we show the e f f e c t of changing the c o n s t i t u t i v e r a t e response and the magnitude of the perturbation. We also examine the e f f e c t of adding a gradient or dipolar response to the c o n s t i t u t i v e equations, a t l e a s t f o r early times.

In order t o concentrate on fundamentals, consider one dimensional shearing of a block of material t h a t l i e s between the boundaries y = + h . The deformation i s assumed to be given completely by horizontal shearing. Thus-the velocity f i e l d may be written

x

⁼ V ( j , f ) ;

y

^{= 0 ;}

z

^{= 0.} (1

As used in / 2 / , the balance and c o n s t i t u t i v e laws f o r t h i s case a r e given here in

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1985541

JOURNAL DE PHYSIQUE

nondimensional form.

blomen t u m : s = p v

.

^'Y

Energy:

e

= ke,yy + s i p

E l a s t i c Response:

i

= e(v,,

- up)

Reference P l a s t i c Response: K = ( 1 +

:ln

= O '

Work Hardening: K $ = S Y ~

Yield Surface: Is1 = ( 1

-

a e ) ( l + b l ; p l ) m ~

Boundary conditions a r e v ( + l , t ) =

5 , e,,

( + l , t ) = 0. In these equations the shear s t r e s s i s s , mass density i s p, p a r t i c l e velocity I s v , temperature r i s e i s e , thermal conductivity i s k , p l a s t i c s t r a i n r a t e i s y p , shear modulus i s p , work hard- ening parameter i s K , and p l a s t i c s t r a i n in a reference t e s t i s

+.

The comma denotes partial d i f f e r e n t i a t i o n with respect to the s p a t i a l coordinate y , and t h e superimposed dot denotes p a r t i a l d i f f e r e n t i a t i o n with respect to time t . In the energy equation p l a s t i c working a c t s a s a source term. A l i n e a r e l a s t i c response i s assumed as well as the additive decomposition of shear s t r a i n into e l a s t i c and p l a s t i c parts. The reference p l a s t i c response i s intended t o be simply a curve f i t t o a slow isothermal t e s t . The work hardening parameter i s assumed t o depend only on the p l a s t i c work no matter what the r a t e of the t e s t . Finally, the yield surface i s taken to depend o n p l a s t i c s t r a i n r a t e , a s well as s t r e s s and temperature. When ( 2 ) 6 i s solved f o r y p as a function of s , e , and K , t h i s l a s t condition i s seen t o be simply an overstress r u l e f o r p l a s t i c flow when the stress-temperature point l i e s outside the s t a t i c yield surface. The form chosen here i s similar to one due to Litonski / 3 / . Note t h a t the thermal softening depends only l i n e a r l y on temperature.

In ( 2 ) the nondimensional variables and parameters a r e related to t h e i r dimensional (barred) counterparts a s follows:

K = K / K ~

y = Y $ = ;L p = (ph2;;)/~, k = K/(PcViOh2)

u = ;/K, a = ( ~ K ~ ) / ( P c ~ ) b = 6+,

;,

⁼

^Yp/;,

where ;o = v(h,E)/h i s the average applied s t r a i n r a t e between

y

⁼

kh,

^{K~ i s the}

i n i t i a l y i e l d s t r e s s in the reference t e s t , cV i s the s p e c i f i c heat a t constant volume, and

a, 6 ,

$,, m and n a r e material constants.

I1

-

HOMOGENEOUS SOLUTIONS AND PERTURBATIONS

If we s e t v = y and assume t h a t s , e , and K a r e independent of y , then equations ( 2 ) reduce to a s e t of ordinary d i f f e r e n t i a l equations in time with s , e , and ^K as depen- dent variables. For i n i t i a l conditions we assume s ( 0 ) = 1, e ( 0 ) = 0, and ~ ( 0 ) = 1 , so t h a t yielding begins a t the i n i t i a l time. For the assumed v i s c o / p l a s t i c flow law, the s t r e s s / s t r a i n response i s a smooth curve, which r i s e s a t the e l a s t i c slope i n i t i a l l y and then, as p l a s t i c flow increases and the temperature r i s e s , the curve bends over, passes through a s i n g l e maximum, and f i n a l l y decreases with f u r t h e r increments of s t r a i n . This type of calculated behavior i s well known in the l i t e r - a t u r e , eg. / 4 , 5, 6 , 7/. Figure 1 shows typical r e s u l t s . For curve A the nondimen- sional parameters a r e

and f o r c u r v e B a l l t h e numbers a r e t h e same except b = 5 x

l o 5

and m = 0.02. A l s o shown i n t h e f i g u r e a r e t h e r e f e r e n c e s t r e s s / s t r a i n c u r v e (a = 0, b = 0 ) and t h e i s o t h e r m a l response (a = 0 ) c o r r e s p o n d i n g t o A (case B i s s i m i l a r ) .

CASE A

s so thermal

V) a = O )

(Reference a =

b =

0)

9 ^I

-0 0.1 0.2

Y

F i g . 1 - Homogeneous response curves. The r e f e r e n c e c u r v e , t h e i s o t h e r m a l c u r v e f o r case A, and t h e a d i a b a t i c c u r v e s f o r b o t h cases A and B a r e shown.

P e r t u r b a t i o n s t o t h e homogeneous response have been i n t r o d u c e d i n t h e f o l l o w i n g manner. J u s t p r i o r t o t h e occurance o f peak s t r e s s , t h e t e m p e r a t u r e was m o d i f i e d

by adding a symmetric bump a t t h e c e n t e r . Two cases have been c o n s i d e r e d as shown ir:

F i g u r e 2, t h e l a r g e r p e r t u r b a t i o n b e i n g f i v e t i m e s h i g h e r and b r o a d e r t h a n t h e s m a l l e r one, b u t s i m i l a r i n shape. W i t h t h e new t e m p e r a t u r e d i s t r i b u t i o n g i v e n , t h e s t r e s s was r e c a l c u l a t e d so t h a t (2)6 i s s t i l l s a t i s f i e d , a l l o t h e r v a r i a b l e s b e i n g h e l d f i x e d . Then t h e f u l l problem u s i n g a l l o f ( 2 ) and t h e s t a t e d boundary c o n d i t i o f i s was r e s t a r t e d w i t h t h e new f i e l d v a r i a b l e s a s i n i t i a l d a t a . A f t e r c a s t i n g t h e

e q u a t i o n s i n t o weak form, c a l c u l a t i o n s were made u s i n g t h e f i n i t e element method, eg. see Becker, e t a l . /8/.

Y

F i g 2.

-

Temperature p e r t u r b a t i o n s added t o t h e homogeneous response a t I .

Case B w i t h t h e s m a l l e r t e m p e r a t u r e bump as a p e r t u r b a t i o n was p r e v i o u s l y d e s c r i b e d i n some d e t a i l i n /2/. F i g u r e s 3 t h r o u g h 7 i l l u s t r a t e and compare t h e main f e a t u r e s o f t h e response f o r a l l cases c o n s i d e r e d t o date. Because o f t h e c o n s t a n t v e l o c i t y m a i n t a i n e d a t t h e boundaries, average s t r a i n r a t e i n t h e s c a l e d v a r i a b l e s f o r t h e s t r i p rave i s e x a c t l y equal t o one. T h i s f a c t a l l o w s d i r e c t comparisons between t h e homogeneous and inhomogeneous cases. F i g u r e 3 shows t h e s t r e s s a t t h e c e n t e r o f t h e s t r i p as a f u n c t i o n o f average s t r a i n ( o r t i m e ) . The p e r t u r b a t i o n was added a t t h e p o i n t marked I, and t h e peak o f t h e homogeneous response i s marked P. F o r t h e small t e m p e r a t u r e bump t h e s t r e s s f o l l o w s t h e homogeneous response u n t i l w e l l p a s t t h e p o i n t P, and t h e n drops r a p i d l y as t h e shear band a c c e l e r a t e s and l o c a l i z e s . The

C5-326 JOURNAL DE PHYSIQUE

s t r e s s a t o t h e r p o i n t s o f t h e s t r i p i s n e a r l y t h e same, b u t l a g s s l i g h t l y i n t i m e . Note t h a t i n r e a l t i m e ( f o r yo = 500 s - I , say) t h e s t r e s s c o l l a p s e i s delayed

Fig. 3 - S t r e s s vs. average s t r a i n ( o r t i m e , s i n c e Vave = 1). Homogeneous and p e r t u r b e d responses f o r cases A and B.

a p p r o x i m a t e l y 125 us p a s t peak s t r e s s P, and t h e r a p i d a c c e l e r a t i o n i t s e l f takes more than 20 us. T h i s b e h a v i o r i s e n t i r e l y s i m i l a r f o r b o t h cases A and B. The case o f t h e l a r g e r temperature p e r t u r b a t i o n c o n t r a s t s s h a r p l y . I n t h i s case t h e s t r e s s c o l l a p s e begins even before p o i n t P .

r\-

F i g u r e s 4 and 5 show t h e e v o l u t i o n o f temperature and p l a s t i c s t r a i n r a t e a t t h e c e n t e r and edge o f t h e band i n comparison t o t h e homogeneous case. As i s t o be expected t h e b e h a v i o r p a r a l l e l s t h e s t r e s s response. A t f i r s t t h e temperature and s t r a i n r a t e d e v i a t e s l o w l y f r o m t h e homogeneous response, b u t when t h e s t r e s s c o l l a p s e s , t h e y b o t h r i s e s t e e p l y i n t h e c e n t e r . A t t h e edge t h e temperature l e v e l s o f f t o a p l a t e a u w h i l e t h e ' p l a s t i c s t r a i n r a t e ( n o t shown f o r t h e edge p o s i t i o n ) drops towards zero. A t f i r s t i t appears t h a t t h e temperature and p l a s t i c s t r a i n r a t e change so as t o compensate each o t h e r w i t h o u t a p p r e c i a b l y a f f e c t i n g t h e s t r e s s response, b u t e v e n t u a l l y thermal s o f t e n i n g wins o u t i n t h e c e n t e r . A t t h e edge t h e s t r e s s drops due t o momentum t r a n s f e r f r o m t h e c e n t e r and e f f e c t i v e l y quenches b o t h t h e temperature r i s e and t h e p l a s t i c d e f o r m a t i o n .

-

V)

Y -

Fig. 4

-

Temperature r i s e f o r case A. Upper c u r v e s a r e f o r t h e c e n t e r o f t h e band, l o w e r c u r v e s f o r t h e edge. S o l i d c u r v e i s t h e homogeneous case.

1

- --

\

^P --- ^-

^{\ CASE A}

\\ large

>

^\small

l

A @ \ A @ I

e

CASE B

- 1 .

small

\

A 8

small A 0

Fig. 5

-

P l a s t i c s t r a i n r a t e i n t h e c e n t e r of t h e band f o r c a s e A. Reference r a t e (homogeneous deformation) is c l o s e t o 1.

Figures 6 and 7 show cro,ss s e c t i o n s of temperature and p l a s t i c s t r a i n r a t e f o r c a s e A (with t h e l a r g e r temperature p e r t u r b a t i o n ) a t several times during t h e rapid

v,",

0.0842

...

\ \ \

0.0892

\ . --

-

- - - . . -

-

_{0.0942 (P)}

0.0967

. .

Fig. 6

-

Cross s e c t i o n s of temperature r i s e a t times i n d i c a t e d .

Fig. 7

-

Cross s e c t i o n s of p l a s t i c s t r a i n r a t e a t times i n d i c a t e d . -p

0

\.

^{-.-.- 0.0842}

...

I \

^0.0892

- - -

-

- - -

-

- -

\.

^{0.0942 (P)}

" '..* \

^0.0967

C5-328 JOURNAL DE PHYSIQUE

t r a n s i t i o n . The most n o t a b l e f e a t u r e i s t h e n a r r o w i n g o f t h e most a c t i v e r e g i o n o f d e f o r m a t i o n . T h i s i s p a r t i c u l a r l y e v i d e n t i n t h e c r o s s s e c t i o n s o f p l a s t i c s t r a i n r a t e . Note t h a t f o r t h e l a t e s t t i m e t h e p l a s t i c s t r a i n r a t e i n t h e c e n t e r o f t h e band i s n e a r l y 8 0 t i m e s t h e average a p p l i e d s t r a i n r a t e .

I 1 1

-

MODIFICATIONS FOR A DIPOLAR EFFECT

As a shear band forms, i t i s e v i d e n t t h a t v e r y steep l o c a l s t r a i n g r a d i e n t s must develop. Therefore, i t has seemed w o r t h w h i l e t o r e f o r m u l a t e t h e p r o b l e m i n terms o f a d i p o l a r t h e o r y o f p l a s t i c i t y . T h i s has been accomplished i n a s t r a i g h t f o r w a r d manner by m o d i f y i n g t h e t h e o r y due t o Green, M c I n n i s , and Naghdi / 9 / t o i n c l u d e a r a t e e f f e c t . I n a d i p o l a r m a t e r i a l i t i s assumed t h a t , i n a d d i t i o n t o t h e u s u a l t r a c t i o n f o r c e s t h a t do work a g a i n s t v e l o c i t i e s , t h e r e a r e h y p e r t r a c t i o n s , denoted

o

and h a v i n g t h e dimensions o f s t r e s s t i m e s l e n g t h , t h a t do work a g a i n s t v e l o c i t y g r a d i e n t s . Decomposition o f s t r a i n gradient^ i n t o e l a s t i c and p l a s t i c p a r t s i n t r o d u c e s a new i n t e r n a l v a r i a b l e , denoted d, i n t h i s paper, w h i c h r e q u i r e s i t s own e v o l u t i o n a r y c o n s t i t u t i v e e q u a t i o n . As skown i n /9/, d i p o l a r p l a s t i c i t y i n t r o - duces l e n g t h s c a l e s t h a t a r e c h a r a c t e r i s t i c o f t h e m a t e r i a l i t s e l f . I n t h i s paper i t i s supposed t h a t t h e r e i s o n l y one i n t r i n s i c l e n g t h s c a l e f o r e l a s t i c , p l a s t i c , o r v i s c o p l a s t i c d e f o r m a t i o n , which i s denoted L. I n a d d i t i o n t o t h e nondimensional q u a n t i t i e s i n t r o d u c e d p r e v i o u s l y , we r e q u i r e

The f u l l s e t o f e q u a t i o n s w i t h a Von Mises t y p e f l o w c o n d i t i o n now t a k e s t h e f o r m Momentum: ( s

-

a u ) = p t

'Y 'Y

Energy:

6

= ke

'YY

+

s ; ~ + C o n s t i t u t i v e E q u a t i o n s :

Reference Response and Work Hardening:

Y i e l d C o n d i t i o n :

( s 2 +o2)' = ( 1

-

a e ) ~ l

+

b ~ ( s 2

+

0 2 ) 4 p K

I n t h e s e e q u a t i o n s h p l a y s t h e r o l e o f a p l a s t i c m u l t i p l i e r and i s t o be d e t e r m i n e d from y i e l d c o n d i t i o n when t h e l o a d i n g p o i n t (s, o,e) l i e s o u t s i d e t h e s t a t i c y i e l d s u r f a c e , a s determined b y t h e l a s t e q u a t i o n w i t h h s e t equal t o zero. E x t r a boundary c o n d i t i o n s a r e a l s o r e q u i r e d . These a r e chosen t o be a(+l,t) = 0, so t h a t t h e d i p o l a r e f f e c t w i l l e f f e c t i v e l y v a n i s h o u t s i d e t h e shear band.

Homogeneous s o l u t i o n s t o t h e s e e q u a t i o n s a r e e x a c t l y t h e same as b e f o r e , b u t now t h e response t o p e r t u r b a t i o n s , computed f o r case B w i t h t h e s m a l l e r t e m p e r a t u r e bump, appears t o be q u i t e d i f f e r e n t , a l t h o u g h l o n g c o m p u t a t i o n a l r u n t i m e s have n o t y e t been achieved. F i g u r e 8 shows t h e e v o l u t i o n o f t h e c e n t r a l p l a s t i c s t r a i n r a t e f o r a nondimensional i n t r i n s i c l e n g t h a = 10 2 . The r a t e decays r a p i d l y back toward t h e average a p p l i e d r a t e , a t l e a s t f o r s h o r t t i m e s a f t e r i n t r o d u c t i o n o f t h e p e r t u r b a t i o n , so t h a t t h e d i p o l a r e f f e c t appears t o s t a b i l i z e t h e d e f o r m a t i o n .

i SIMPLE MATERIAL

--- ----

CASE B

"I

DIPOLAR MATERIAL

s l '

^{P ,}

-

^{0.095 0.1 0.105}

rave

F i g . 8

-

P l a s t i c s t r a i n r a t e i n t h e c e n t e r of t h e band f o r case B a t e a r l y t i m e s . D i p o l a r m a t e r i a l compared t o s i m p l e m a t e r i a l .

I V

-

DISCUSSION AND CONCLUSIONS

The response t o t h e small temperature p e r t u r b a t i o n i s q u a l i t a t i v e l y v e r y s i m i l a r f o r b o t h cases A and B. F i g u r e 3 shows t h a t , a l t h o u g h t h e s t r e s s l e v e l s a r e d i f f e r e n t due t o t h e d i f f e r e n t v i s c o u s response, t h e shapes o f t h e homogeneous respone curves a r e s i m i l a r , and so a r e t h e shapes o f t h e response t o p e r t u r b a t i o n . For something on t h e o r d e r o f 5 o r 6% average s t r a i n p a s t t h e peak o f t h e homogeneous c u r v e t h e s t r e s s p a t h d e v i a t e s o n l y s l i g h t l y u n t i l s t r e s s c o l l a p s e s e t s i n a t t h e end o f t h e c a l c u l a t i o n . S i m i l a r b e h a v i o r was r e p o r t e d by Merzer /6/ who used a c o m p l e t e l y d i f f e r e n t f l o w l a w f o r v i s c o p l a s t i c i t y . Thus, i t appears t h a t s t r e s s c o l l a p s e i s i n h e r e n t i n t h e process o f band f o r m a t i o n w i t h t h e d e t a i l s depending on t h e s p e c i f i c m a t e r i a l model used.

The c a l c u l a t i o n w i t h t h e l a r g e r p e r t u r b a t i o n i n temperature f o r case A shows t h a t t h e d e t a i l s o f s t r e s s c o l l a p s e a l s o depend s t r o n g l y on t h e magnitude o f t h e p e r t u r - b a t i o n . A l t h o u g h case B has n o t been r u n w i t h t h e l a r g e r temperature d i s t u r b a n c e , t h e r e i s no reason t o expect t h a t t h e r e s u l t s would be q u a l i t a t i v e l y d i f f e r e n t f r o m case A.

The most s t r i k i n g a s p e c t o f t h e response p a t t e r n s i n F i g u r e 3 i s t h e i r s i m i l a r i t y t o t h o s e o b t a i n e d i n a n a l y z i n g i m p e r f e c t i o n s e n s i t i v i t y f o r a b i f u r c a t i o n problem, eg., see

/ l o / .

The o n l y t h i n g m i s s i n g i s t h e b i f u r c a t i o n branch i t s e l f . F i g u r e s 4 and 5 f o r temperature and p l a s t i c s t r a i n r a t e r e i n f o r c e t h i s idea.

A l t h o u g h c a l c u l a t i o n s f o r t h e d i p o l a r case have n o t y e t progressed f a r enough t o be d e f i n i t i v e , i t appears t h a t t h e e f f e c t , even f o r a v e r y small i n h e r e n t m a t e r i a l l e n g t h s c a l e , w i l l be h i g h l y s t a b i l i z i n g . T h i s would be i n accordance w i t h t h e experience t o d a t e f o r e l a s t i c systems where, as i n /11/, h i g h e r o r d e r e f f e c t s a l l o w smooth t r a n s i t i o n s i n s t e a d o f d i s c o n t i n u i t i e s .

U n f o r t u n a t e l y , because o f computational i n s t a b i l i t i e s t h a t s e t i n , i t has n o t y e t been p o s s i b l e t o c o n t i n u e c a l c u l a t i o n s t o t h e p o i n t where a f i n a l c o n f i g u r a t i o n f o r t h e shear band appears. The c a l c u l a t i o n s r e p o r t e d i n /2/ i n d i c a t e d t h a t a peak v a l u e f o r p l a s t i c s t r a i n r a t e may occur, b u t r e e x a m i n a t i o n has i n d i c a t e d numerical i n s t a b i l i t y . The d i f f i c u l t y seems t o be t h a t e v e n t u a l l y t h e extreme l o c a l i z a t i o n o f t h e band d e f e a t s t h e a b i l i t y o f t h e p r e s e n t numerical method t o r e s o l v e t h e d e t a i l s .

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