Flow Localization in Strain Hardening Crystalline Solids

8/18/2019 Flow Localization in Strain Hardening Crystalline Solids

1/7

S c r i p t a M E T A L L U R G I C A V o l . 1 8, p p . 4 2 9 - 4 3 5 , 1 9 8 4 P e r g a m o n P r e s s L t d .

P r i n t e d i n t h e U . S . A . A l l r i g h t s r e s e r v e d

V I E W P O I N T S E T N O . 6

F L O W L O C A L I Z A T I O N I N S T R A I N H A R D E N I N G C R Y S T A L L I N E S O L I DS

R . J . A s a r o a n d A . N e e d l e m a n

D i v i s i o n o f E n g i n e e r i n g

Brown University, Providence9 Rhode Island 02912

( R e c e i v e d F e b r u a r y 2 1 , 1 9 8 4 )

I. Introduction

P l a s t i c d e f o r m a t i o n i n m e t a l s i s a n i n h e r e n t l y n o n u n i f o r m p r o c e s s ; w h e n e v e r d i s l o c a t i o n

s l i p o r t w i n n i n g o c c u r s t h e r e i s a s i z e s c a l e b e l o w w h i c h t h e d e f o r m a t i o n i s h e t e r o g e n e o u s a n d

d i s c o n t i n u o u s . H e n c e , a n y r e f e r e n c e t o t h e u n i f o r m i t y o f p l a s t i c f l o w i m p l i e s s o m e s c a l e o f

o b s e r v a t i o n . H o w e v e r , o n a s i z e s c a l e o f t e n s o f m i c r o n s , e n c o m p a s s i n g s e v e r a l g r a i n s i n

p o l y c r y s t a l s , t h e d e f o r m a t i o n m a y a p p e a r u n i f o r m e v e n t h o u g h t h e m i c r o m e c h a n i c a l d e f o r m a t i o n

p r o c e s s e s a r e n o n u n i f o r m . L o c a l i z e d m i c r o s t r u c t u r a l m o d e s c a n b e o p e r a t i v e a t t h i s s i z e s c a l e

a s w e l l ; e x a m p l e s i n c l u d e m i c r o t w i n n i n g a n d t h e ' ~ i c r o b a n d s t h a t H a l i n a n d H a t h e r l y ( 1 ) h a v e

s h o w n t o g i v e r i s e t o t h e s l i p b a n d s c o m m o n l y o b s e r v e d i n h i g h s t a c k i n g f a u l t e n e r g y

m a t e r i a l s . E v e n w h e n t h e s e d e f o r m a t i o n m o d e s o c c u r , o n a m a c r o s c o p i c l e v e l o f a t e n t h o f a

m i l l i m e t e r o r m o r e , p l a s t i c d e f o r m a t i o n c a n a g a i n a p p e a r u n i f o r m . W h en s u f f i c i e n t p l a s t i c

d e f o r m a t i o n h a s o c c u r r e d , a l l s u c h m a c r o s c o p i c a l l y h o m o g e n e o u s d e f o r m a t i o n p a t t e r n s b r e a k d o w n,

p e r h a p s i n e v i t a b l y , a s A s a r o ( 2 ) h a s r e m a r k e d , i n t o o n e s i n v o l v i n g l o c a l i z e d d e f o r m a t i o n i n t h e

f o r m o f s h e a r b a n d s .

O n t h e g r a i n s i z e s c a l e i t h a s b e e n r e p o r t e d t h a t s h e a r b a n d s c a n b e c o m e t h e d o m i n a n t

d e f o r m a t i o n m e c h a n i s m ( 3 ) . A s s u c h t h e y d e t e r m i n e m a t e r i a l s t r e n g t h a n d d e f o r m a t i o n k i n e m a t i c s ,

t h e r e b y c o n t r i b u t i n g d i r e c t l y t o t h e d e v e l o p m e n t o f t e x t u r e a n d m a t e r i a l a n i s o t r o p y . I n s o m e

c a s e s t h e s e s h e a r b a n d s l e a d t o f r a c t u r e a n d t h u s s e r v e a s f a i l u r e m o d e s . O n a m o r e m a c r o s c o p i c

l e v e l , s h e a r b a n d s o f t e n p l a y a n i m p o r t a n t r o l e in l i m i t i n g d u c t i l i t y ( 4 , 5 ) . O n c e a m a c r o -

s c o p i c s h e a r b a n d f o r m s , v e r y l i t t l e a d d i t i o n a l o v e r a l l s t r a i n i n g t a k e s p l a c e w h i l e l a r g e

s t r a i n s a c c u m u l a t e i n t h e b a n ~ l e a d i n g t o f r a c t u r e . I n v i e w o f t h e d u a l s i g n i f i c a n c e o f s h e a r

b a n d s in d e t e r m i n i n g d e f o r m a t i o n p a t t e r n s a n d a s a p r e c u r s o r t o f r a c t u r e , i t i s i m p o r t a n t t o b e

a b l e t o p r e d i c t t h e i r i n i t i a t i o n a n d d e s c r i b e t h e i r d e v e l o p m e n t . H e r e , w e d e s c r i b e s o m e r e c e n t

p r o g r e s s i n d o c u m e n t i n g a n d a n a l y z i n g s h e a r b a n d s a n d i n d i c a t e , w h a t s e e m t o u s , s o m e f r u i t f u l

a r e a s f o r f u t u r e i n v e s t i g a t i o n .

2 . A F r a m e w o r k f o r A n a l y z i n g L o c a l i z a t i o n

A l t h o u g h t h e i n i t i a t i o n o f s h e a r b a n d s i s f r e q u e n t l y a s s o c i a t e d w i t h s o f t e n i n g , f o r i s o -

t h e r m a l d e f o r m a t i o n s t h e r e i s l i t t l e e v i d e n c e f o r m a t e r i a l s t r a i n s o f t e n i n g o c c u r r i n g p r i o r t o

l o c a l i z e d s h e a r i n g e x c e p t , p e r h a p s , i n s p e c i a l c a s e s , e . g . r a d i a t i o n d a m a g e d a l l o y s . I n d e e d ,

t h e r e a r e n u m e r o u s r e p o r t s i n d i c a t i n g t h a t m a t e r i a l w i t h i n s h e a r b a n d s c o n t i n u e s t o h a r d e n , a t

l e a s t u n t i l m i c r o f r a c t u r e o c c u r s . T h i s d o e s n o t r u l e o u t s o m e s o r t o f s o f t e n i n g m e c h a n i s m f r o m

p l a y i n g a r o l e i n p r e c i p i t a t i n g s h e a r b a n d s . A p a r t i c u l a r l y i n t e r e s t i n g s u g g e s t i o n , m a d e b y

D i l l a m o r e e t a l . ( 6 ) , C h a n g a n d A s a r o ( 7 ) , a n d M o r i i a n d N a k a y am a ( 3 , 8 ) , i s t h a t t e x t u r a l o r

g e o m e t r i c s o f t e n i n g d u e t o l a t t i c e r e o r l e n t a t i o n d u r i n g d e f o r m a t i o n l e a d s t o t e x t u r e s t h a t a r e

s o f t w i t h r e s p e c t t o s h e a r b a n d s . O n a m o r e m a c r o s c o p i c s c a l e t h e i n i t i a t i o n a n d g r o w t h o f

v o i d s m ay l e a d t o a n e f f e c t i v e s o f t e n i n g a l t h o u g h t h e s u r r o u n d i n g m a t r i x c o n t i n u e s t o s t r a i n -

h a r d e n .

T h e s e e f f e c t s , a l o n g w i t h t h e p a t h d e p e n d e n c e o f s t r a i n - h a r d e n i n g c a n b e a n a l y z e d w i t h i n a

framework that regards localizati on as a material instabi lity, where a material element is

consider ed to be subject to prescri bed all around displa cements that are consistent with a

h o m o g e n e o u s d e f o r m a t i o n . D e f o r m a t i o n s i n a lo c a l i z e d b a n d a re p e r m i t t e d p r o v i d e d t he v e l o c i t y

field remains continuous and continu ing eq uilibr ium at the band interface is satisfied (9,10).

Bifurcation and imperfect ion analyses with in this fra mework have proved useful in revealing the

influence of constitutiv e features and stress and strain state on localizat ion (I0~11).

4 2 9

0 0 3 6 - 9 7 4 8 / 8 4 $ 3 . 0 0 + . 00

C o p y r i g h t ( c) 1 9 8 4 P e r g a m o n P r e s s L td .


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4 3 0 F L O W L O C A L I Z A T I O N I N C R Y S T A L L I N E S O L I D S V o l . 1 8, N o. 5

M athemat ica l ly , a s hea r band b i fu rca t ion co inc i des w i th lo s s o f e l l ip t i c i t y o f the

governing increments1 equa t io ns , Hi l l (9) , Rice (10) ; the governing equat ions become ei t her

pa rabo l i c o r hyperbo l i c in cha rac t e r and ve ry d i f f e r en t behav io r s a r e a s s oc ia te d w i th thes e two

possibilities. Rice s (I0) three dimensiona l generaliza tion of the imperfect ion analysis of

Marcin iak and Kuczyns kl (12), is important since, for example, a localization bifur catio n is

ruled out for rate dependent mate rials but it can occur when there are very small

i nhomogene i t i e s . Q u i te genera l ly , loca l i za t ion invo lves a change in load ing d i r ec t ion s o tha t

in a s s es s in g the ro le o f mic ros t ruc tu r a l f ea tu r es , s uch as t ex tu r a l s o f t en i ng on loca l i za t io n ,

i t i s nece s s a ry to eva lua te e f f ec t s on the s t i f fn es s governing the r e s pons e to a change in

loading path .

There a r e however l imi ta t ions to th i s des c r ip t i on o f loca l i za t ion . F or example , loca l i -

za t ion occu r s s imu l taneous ly w i th in the band so tha t the ana lys i s i s incapab l e o f des c r i b ing the

p ropaga t ion o f a r eg ion o f loca l i za t i on f rom some loca l s t r a in concen t r a to r . A l s o , no l eng th

s ca le en te r s the ana lys i s un les s i t i s exp l i c i t ly pu t the re by be ing inco rpo r a ted in to the

cons t i tu t ive r e l a t ion . I t s hould be emphasized tha t thes e l imi ta t ions a r e no t inhe ren t to the

cont inuum desc r ip t io n but are consequences of i t s app l ic at i on to (p ie cewise ) homogeneous and

homogeneous ly deformed mat er i a l e lemen ts . When these res t r i c t i ons are re laxe d, by car r y ing out

a fu l l s o lu t i on to the r e l evan t boundary va lue p rob lem, r eg ions o f loca l i za t i on p ropaga te , qu i t e

typ ica l ly , f rom s t r a in concen t r a t ion s and the de fo rmat ion f i e ld g r ad ien t s in t roduce an imp l ic i t

l eng th . F u r the rmore , i t i s then pos s ib l e to exp l i c i t ly inc lud e s pa t i a l d i s t r i bu t ions o f

mate r i a l inhemogene i t l e s in to the ana lys i s . I n thes e more genera l c i r c , - , s t ances , the ons e t o f

loca l i za t i on and i t s in i t i a l d i r ec t ion o f p ropaga t ion a r e r a the r w e l l r ep res e n ted by a ma te r i a l

in s t ab i l i ty ana lys i s , a l though the s ubs equent developmen t i s g r ea t ly a f f ec t ed by g rad ien t s in

the s u r round ing f i e ld .

Not a l l c i rcum stance s where s t r a in conc ent ra t es in to a more or le ss wel l def i ned band

corr espo nd to whet is meant her e by loca li za ti on. Figur e 1, ta ken from LeMonds (13) shows two

s tages o f de fo rmat ion in a ca lcu l a t ion o f a p lana r b i - c rys ta l model s ub jec t to p lane s t r a in

tens io n . A cros s the cen te r o f the b i - c ry s ta l the re i s an in i t i a l l a t t i ce mis o r i e n ta t io n tha t

t r i gger s inhomogeneous deforma t ion . In F igure la , the or ien ta t ion of the band of inc reas ed

s t r a in i s s e t by the boundary cond i t ions . A t a s l igh t ly g r ea te r s t r a in , F igu re lb , the de fo rma-

t ion pa t t e rn has s h i f t ed to one w here the band o r i en ta t ion i s cha rac t e r i s t i c o f the ma te r i a l ,

w hich i s in exce l l en t ag reement w i th tha t g iven by a b i fu rca t ion ana lys i s . F igu re l a co r r e -

s ponds to a geomet r i ca l l y induced s t r a in concen t r a t io n ; F igu re lb i l lu s t r a t e s loca l i za t ion .

LeMonds ' ana lys i s (13) may also provide a s ta r t i ng poin t for ana lyzing the model proposed by

Mori i and Nakayama (3) for micro- shea r band in i t i a t i on a t twin c lus t ers as d iscuss ed again in

Section

3.

0 1 6

0 4~

o T

~//// ~ o ~6

~

/ ~ > . > / ×

o ~ 0 6

% '

l o . o l T : : :

o '

0 70

Figure 1 Contours of cons tan t

maximum princlple

logarithmic strain in

a b l - c ry s ta l s ub jec t to

p l a ne s t r a i n t e n s i o n .

There is an in i t ia l

l a t t i c e m l s o r l e n t a t i o n

of 10 ° ac ross the cen ter

o f t h e b i - c r y s t a l .

(a) at an extens ion of 0.12,

before shear bands

(b) at an extensi on of 0.15,

after shear bands.

From LeMonds (13).

o)

b)


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V o l . 1 8, N o . 5 F L O W L O C A L I Z A T I O N I N C R Y S T A L L I N E S O L I D S 4 3 1

2.1 Locali zation in Single Crystals

Within the continuum sllp framework for crystal

plasticity,

Asaro (2) carried out a

bifurca tion anal ysis for a two dimensional model crystal oriented for symmetric double slip,

drawing on Hill and Hutchins on's (14) general analysis of bifurca tion in plane strain tension.

Asaro's (2) analysis points out the vital role of geometrical softening in shear band initi ation

in single crystals, as is also brought out by the experimental studies of Saimoto et al. (6),

Chang and Asaro (7) and Morii and Nakayama (3,8). Three essentially different types of bifurc a-

tion mode occur: (i) a shear band bifurcation mode which involves continued double slipping in

the shear band and which is such as to induce a lattice rotation that causes the d o m i n a n t slip

system to undergo geom etrical softening; (ii) a localization mode where she aring occurs parallel

to the tensile axis and the shearing mode itself leads to unloading on one of the currently

active slip systems - as discussed by Peirce et al. (15,16) it is this mode that leads to

'patchy' slip; a nd (iii) a mo de that involves an excess of conjugate slip in a band that is most

closely aligned with the primary slip system. Although there is an initial geometrical

softening acco mpanying this latter mode, eventu ally the

c o n j u g a t e

s y s t e m g e o m e t r i c a l l y h a r d e n s

(as does the primary

system

so

that

very large strains do not

a c c u m u l a t e

in the band (16).

Which of these phenomeno logies develops d epends on the crystal geometry and material

harden ing properties. Latent harden ing plays a particul arly important role in this regard. For

a geometry r e p r e s e n t a t i v e of an f.c.c, crystal oriented for symmetric double slip, and when the

latent hardening ratio does not exceed Taylor's isotropic hardening, shear band formation is the

preferred l ocalizat ion mode. Pelrce et al. (15) carried out a full finite element analysis of

shear band development based on the planar double slipping model of Asaro (2) and using a small

geometric imperfection to initiate diffuse necking. When using materia l parameter values repre-

sentative of Chang and Asaro's (7) hard, li ghtly hardeni ng aluminum -copper alloy crystals, the

computed lattice rotations are in close corresponden ce with the

e x p e r i m e n t a l

m e a s u r e m e n t s .

Also, consist ent with the observations, the shear bands form abruptly after very little diffuse

necking and a r e sharp ly de l inea te d . I n the ca l cu la t ion s fo r so f t e r , h igher ha rdening c rys t a l s

diffuse nec king is found to induce lattice rotations which promote geometric softening, as

suggested by Saimoto et al. (17), and the bands form gradua lly and are rather broad (15). This

is most likely a case where the shear band thickness is set by gradients in the surrounding

field, altho ugh the possibility of some numerical mesh induced shear band broadening cannot be

ruled out. By way of contrast, there is nothing in the continuum analysis to set a minim um

width for shear bands so that the fineness of the calculated shear bands in the stronger, low

hardeni ng crystals is undoubt edly limited by the mesh.

Peirce et al. (16) also found that latent hardening rates suff iciently greater t han self

harden ing rates led to the development of patchy slip. However, for somewhat greater latent

hardenin g ratios, but still withi n the range 1.0 - 1.4 found by Kocks (18) to encompass much of

the available data, the incremental equilibri um equations become parabolic in character and the

numerical procedure of Peirce et al. (15) could not obtain a unique solution. Interestingly,

the critical conditi on for parabolic behavior occurs only sligh tly before failure of the condi-

tion for slip mode uniqueness unde r prescribed stress rates. This behavior is not an artifact

of the planar double slipping model as Peirce (19) found that this type of instability also

occurred in an actual three dimensiona l f.c.c, geometry. Indeed, it appears that this loss of

constit utive uniqueness, and the associated inability to obtain solutions to bounda ry value

problems, is inherent to the rate independent ideali zation of crystal plastihity for a signifi-

cant range of materia l properties. There is a related, long appreciated lack of uniqueness in

the choice of active slip systems in multi -slip modes. This behavior of the rate independent

idealiz ation apparently precludes a unique, consistent p redictio n of lattice rotations and hence

of finite strain polycrystalline response from single crystal properties. Rate dependent

plastici ty formulatio ns do, however, lead to uniquely determ inedsli pping rates on each system--

Pan and Rice (20) and Peirce et al. (16)--so that even for material s with no perceptible rate

dependence o ver the full range of quasi-stati c strain rates the inherent rate dependence of

plastic ity plays an essential role. It is worth noting in this regard that, although a rate

dependent pla sticity formul ation appears essential for the calculati on of macroscop ic proper-

ties, rate independent phen umenolo gical theories of plasticity, suitable for use in localizat ion

analyses, can be formulated which do not suffer the constitutive nonuni queness of rate indepen-

dent crystalline plas ticity form ulations (although the effect of materi al rate sensitivity on

localizat ion is itself an important issue (35,36)).


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4 3 2 F L O W L O C A L I Z A T I O N I N C R Y S T A L L I N E S O L I D S V o l . 1 8, N o. 5

The rate dependen t cons titutive formulati on of Peirce et al. (16) did permit a calculatio n

of all three modes of localizati on noted above. One partic ularly interesti ng phenomeno logy that

develop ed in these calculations was that the lattice rotations associated wit h patchy slip could

act as a kinemati c barr ier to shear band propagatio n and, hence, p atchy slip could tend to

increase the crystal s ductility. The role of lattice misorient ations, either within a grain or

at grain boundaries, in inhibiting shear band propagati on appears to merit further attention.

Figure I is from an initial stud y of shear bend propaga tion across such e boundary.

2.2 Phenome nologic al Models for Shear Band Develop ment in Polycr ystalli ne Solids

Much of the effort in theoretical anal yses of localizati on in polycry stallin e metals has

aimed at identifying the constituti ve features triggering the observed localization. For

example, observat ions of void nucleation and growth within shear bands are not uncommon. One

possibi lity is that an inherent ins tability o f the plastic flow process initi ates shear banding

and the large strains that accumul ate in the band then lead to the initiation of ductile

rupture. Anothe r possibility is that the weake ning induced by void nucleation end growth itself

triggers the observed localization; as remarked in (ll),which of these possibilities occurs can

be stress state dependent for a given material. Significant recent progress has been made, for

example by Tvergaar d (21-23), Fisher and Gurland (24), Hancock, MacK enzie and Brown (25,26) and

Saje et al. (27), in developing a quantitative descript ion of microvoi d induced shear band

localization. It has also been shown (28) how void induced shear band localizatio n coupled with

the kinematic constraint of axisymm etry natur ally gives rise to the familiar cup-cone fracture

in simple tension.

When microruptur e effects are absent and when the material response is appropr iately

idealized as rate independent, the key feature of plastic material response for localizat ion is

the verte x yield surface structure implied by the discrete nature of crystallo graphic slip. A

yield surface ver tex results in a much reduced stiffness to a change in loading path and, as

noted above, it is this stif fness that plays a primary role in the onset of locali zation (for

rate dependent solids there is also a verte x llke reduction in this overall st iffness but its

charact erizati on is more complex (20,42)). The phenomeno logical corner theory of plastic ity

introduced by Christoffers en and Hutchinson (29) embodies this feature and another aspect of

p o l y c r y s t a l l i n e p l a s t i c r e s p o n s e t h a t g r e a t l y a f f e c t s s u b s e q u e n t s h e a r b a n d d e v e l o p m e n t ; w i t h

i n c r e a s i n g s h e a r i n g i n th e s h e a r b a n d , th e s h e a r s t i f f n e s s i n c r e a s e s , m o d e l i n g t h e e f f e c t o f

some inltially active slip systems unload ing as the deforma tion pat h shifts from the pre-

localizatio n one. This mech anism stro ngly influences the spatial distribut ion of shear bands as

illustrated in the numerical calculatio ns of Tvergaar d et al. (30); a manif estat ion of this

vertex stiffening being that initially formed shear bands saturate, giving increased straining

in a nearby reg ion which resu lts in further shear band initiation.

S u c h p h e n o m e n o l o gi c a l c o n t i n u u m a n a l y s e s g i v e a p i c t u r e o f m a c r o s c o p i c s h e a r ba n d d e v e l o p -

ment at least qualitatively in accord with observ ation (30,31). However, quite typically local-

ization initiates locally around some local strain concentrator or in some favorably oriented

grains, Clausin g (32), Hahn and Rosenfi eld (33), Anand and Spitzig (34). These bands ultimately

can lead to a band extending across the entire specimen when some macro -bandi ng condition is

met. The details of this transition are not fully explained and are important in particu lar in

relation to the role of localizat ion in setting fracture toughness.

A q u a n t i ta t i v e c o m p a r i s o n b e t w e e n m a c r o s c o p i c s h e a r b a n d p r e d i c t i on s b a s e d o n p h en o m e n o -

logical corn er theory and experiment is contain ed in the study of Larsso n et al. (5) on inter-

nally pressurized copper and aluminum alloy tubes. The calculations gave good agreement for the

deformed shapes of the tubes and for

the

angle at which shea r failure ultimat ely occurred.

A l t h o u g h t h e r e p r e s e n t a t i o n o f

the

effective stress effective st rain hardeni ng properties was

based on measur ed un iaxial response for the tube

materials

the additional c orner properties

which quan tify the response to a loading path change were chose n arbitrarily. Larsson et al.

(5) attri bute, at least in part, quantitative discrepancies b etw een th eir calcu late d and mea sur ed

pressure-r adius values to different harden ing rates under plane strain and axisym~etr ic cond i-

tions, i.e., to texture effects. Also, the use of corner theory pr esumes that the stability

against flow localization is limited by the plastic flow process itself with the initiation of

ductile rupture ensuing in the highly strained shear band. This appeared an appropria te con-

clusio n for the aluminum a ll o~ but Larsson et al. (5) note that for the copper tubes the

weakeni ng of void nucleatio n and growth could also play a role in precip itating the

l o c a l i z a t i o n .


5/7


The examples of shear band analyses and observation s described above illus trate the need

fo r accu ra te ma te r i a l cha rac te r i za t i on and cons t i tu t ive des c r i p t ion in the p red ic t ion o f s hea r

bands . This is known to be par t icu lar ly t rue in the case of hea vi l y ro l le d metals and a l l oys

w here s hea r bands p lay a v i t a l r o le in de te rmin ing de fo rmat ion pa t t e rns , t ex tu re and u l t ima te ly

duct i l i ty . The phenomenology of shear band format io n in ro l l in g has been wel l documented

experi menta lly and it is known that prexisting rolling textures themselves are important in

causing shear bands which then lead to texture changes. In closing, a brief descr iption of the

reported phe nomenol ogy is given and we propose that an important topic for future study is the

theoretical analysis of shear bands in heavi ly rolled polycrystal line metals.

3. Shear Bands in Rolled Metals and the Development of Texture

The texture of heavily rolled f.c.c, metals is generally described as the pure metal type

or alloy type as typified by copper and a-brass, respectively. In the pure metal texture the

orientatio ns of the sheet nor mal and rolling dir ection are contained in a band extendin~__from

{110} to the region on the - symmetr y boundary lying between {211} and

{311} . This texture is predicted from simple slip theory assuming slip on the f.CoC, slip

systems {111} . The alloy texture is characte rized by a predomin ance of orientati ons of

the sheet plane near {110}; the texture component near the ideal texture {211} is absent

and this is the most fundamental difference in the two textures. The reasons for these differ-

ences is reported to lie in the different mechanisms and deformation patterns that develop in

these materlals, vlz. twinning versus ~lip, the operation of shear bands and the different

mlcrostructures that result.

E 7

o

~o

¢

o

6

> 5

p

~ 4

~ 3

2

D E F O R M A T I O N M O D E S I N C O P P E R A T R O O M

{ 1 11 } S L i P T E M P E R A T U R E

R O L L I N G R E D U C T I O N , %

3 0 5 0 6 0 7 0 8 0 9 0 9 5 9 8

i I I , , i I I

I = I

I D E F O R M A T I O N M O D E S

I S H E A R I B A N O S ~ I N ( 2 - B R A S S

~ , : I p ~ j . . . . , . . . . . . s . ~ , . / , . R o s

= I

T O A L L I G N E D

,..~1 TWNS I ~

( H O }

{1 1 1 )

I O 2 . 0 3 4

T R U E R O L L I N G S T R A I N

Figure 2

Inve r s e po le f igu re

da ta o f the dens i ty o f

c r y s t a l l o g r a p h i c p l a n e s

pa ra l l e l to the ro l l in g

plane of s -brass deformed

by p lane s t r a in ro l l in g

at room temp erat ure

(take n .from Vuggan et al .

(37)). The predominant

m i c r o - s t r u c t u r a l

deform at ion modes in the

a s s o c i a t e d s t r a i n

in te rva ls for c~-brass and

copper deformed at room

temperature , deduced f rom

the work of Morli and

Nakayama (3,8) and

Duggan et al. (37) ,

are indicated on the

figure.

Figure 2 contains data, taken from Duggan et al. (37), on the development of texture in 70-

30 brass defor med by rolling at room temperature. The character of the deforma tion at various

stages of rolllng, as deduced from the recent work of Morli and Nakay,ma (3,8) and Duggan et al.

(37), is brief ly discussed on the figure. During approxim ately the first 30 percent redu ction

in thickness the texture that develo ps in brass is very s imilar to the pure meta l texture, as

indicated by an increase in the {110} sheet plane component and rapid decrease in the {IIi}

component. The deformation mode in brass in this strain interval is octahedra l slip of extended

dislocat ions (37). In pure copper, on the other hand, Malin and Hathe rly (37) report that after

i0 percent reduction the primary deforma tion mode involves the formation of microban ds. In

brass the primary deform ation mechani sm in the strain interval between 30 and 50 percent r educ-

tion is twinning which occurs by the formation of fine microtwins, aligned at nearly 30 to the


6/7

4 3 4 F L O W L O C A L I Z A T I O N I N C R Y S T A L L I N E S O L I D S V o l . 1 8 , N o . 5

rolling plane preferential ly in grains havin g the {211} texture component. At 50 percent

reduction the rolling textures of copper and s-brass are quite similar except that the occur-

rence of twins contributes a {255} component in brass. Beyond 50 percent, as indicated in

Figure 2, the brass texture undergoes a transition char acterized by a buildup of a {III} compo-

nent of the sheet plane whic h is later transformed into a {II0} component. This transition is

characterized by the appearance of shear bands that become the dominant d eformation mode.

Shear bands form at approximatel y 35 to the rolling plane, although as discussed by

Hatherly and Malin (38) they are observed to form withi n a range of angles near 35 . They are

between 0.I and 1 ~m wide and propagate across grain boundaries with little change in trajec-

tory. The number of bands, and therefore their volume fraction, increases monoto nicall y with

strain. Eventually, at near 90 percent reduction, they occupy as much as 90 percent of the

volume whereup on further deformation can again be described as occurring by {Iii} sllp.

Morii and Nakayama (3) have shown that the phenomen ology in copper rolled at liquid nitrogen

temperatures is very similar to that found in e-brass. Shear bands develop very large strains;

in low SFE material s or in high SFE materials deformed at very l ow temperatures(the strains are

as large as 10),whereas in high SFE materials deformed at room temperature the strains are some-

what less. In low SFE materials, such as s-brass, shear bands account for nearly all the strain

between reductions of 50-90 percent; in high SFE materia ls defo rmation by {iii} sllp continues

to be important concomitant with the shear bands. In both high and low SFE materials the

observations suggest that one set of shear bands tend to dominate. In brass a conjugate set

forms at later strains.

Between 50 and 70 percent reduction in thickness the deformat ion in brass occurs essen-

tially by slip along those {III} planes parallel to the microtwins and by shearing along the

primary set of shear bands. This leads to what has been described as an extreme form of

overshooting that causes the twins to reorient parallel to the rolling plane. An effect of

this is the texture tran siti on {211} + {255} T ~ {iii} + {III} w here M

and T refer to matrix and twinned regions respectively. In copper, slip in the matrix betw een

shear bands offsets the rotation caused by the shear bands which stabilizes the {211} texture

component (3,41).

Until microfract ure begins shear bands are observed to harden and so material strain

softening does not appear to be a cause. In brass, or in copper deformed at low temperatures,

shear bands are observed to initiate only in twinned regions. Morii and Nakayama (3) have

observed that shear bands initiate within twin lamella and have propos ed that the critical

micromec hanical events involve local lattice rotatlon( s) which cause a geometrical softening

of the slip systems responsible for the concentrated str aining in the bands. In copper at

higher temperatures, they have proposed that similar local lattice curvatures can develop in

tangled dislocatio n wall(s). These ideas are a direct microscopi c analog to the experimen tal

and detailed theoretical findings of Chang and Asaro (7), Asaro (2) and Peirce et al. (15,16)

for macroscop ic shear bands. In these studies it has been shown that the kinematics of necking

in single crystals causes nonuni form lattice rotations and geometrical softening that, in turn,

promote localized shearing. Dillamore et al. (6) have suggested textural softening occurs in

heavil y rolled materia ls and is responsible for the overall softening which drives the localized

shearing mode. This idea has been pursued very recently by Canova et al. (39), among others,

who, following Dillamore et al. (6) , describe the effect in terms of the evolution of the

Taylor factor. This is defined as the ratio of the cumulative glide strains on all operative

slip systems (in a grain) to the maximum principal strain. If the Taylor factors decrease due

to the particular prevailing texture and strain state, this implies less strain hardening and a

less stiff response.

The concepts of geometrical and textural softening provide a unifying set of mechanisms for

explaining how shear bands develop in strain hardenin g materials. Lattice rotations withi n

individual grains as well as effects of texture and misorien tations across grain boundaries in

polycrysta ls can be rigorously analyzed within a comprehensi ve mechani cs framework, alth ough to

date only a few solutions to complete boundar y value problems for single crystals are available.

Theoretic al study of shear bands in plane strain compression, or rolling, would be of great

value and would require two separate analyses. The first involves the propagati on of shear

bands in textured polycrystals. This requires a constitutive model which accounts for material

anlsotropy. We are carrying out studies directed toward developing such models (42) whic h can

then be incorporated within the theoretical fr amework described in Section 2. The second

analysis involves a study of the initiation of shear bands within individual grains. A first


7/7


attempt at this could be pursued al ong the lines developed by Peirce et al. (15,16) for

analyzing macrosc opic shear bands in single crystals and used in the hi-crystal analysis of

LeMonds (13). Here it may be possible to include the mechanisms suggested by Morii and Nakayama

(3) for the initiation of shear bands from microtwi n clusters. Including such microst ructura l

features explicitl y in the modelli ng should lead to a specification, or prediction, of shear

band width from the analysis. Such an analysis would undoubt edly provide considerable new

insight into the mechanics of shear band formation as well as help delineate the range of

micros cale phen omena that can he described with continuum analysis.

A c k n o w l e d g m e n t s

R.J.A. gratefu lly acknowled ges support from the Metallu rgy Sect ion of the U.S. National

Science Foundati on under grant DMR26190. A.N. is grateful for the support provided by the U.S.

N a t i o n a l S c i e n c e F o u n d a t i o n S o l i d M e c h a n i cs P r o g r a m u n d e r g r a n t M E A - 8 1 0 1 9 4 8.

Re

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Flow Localization in Strain Hardening Crystalline Solids

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