rice the localization of plastic deformation

8/3/2019 Rice the Localization of Plastic Deformation

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Theoret ical and Applied Mechanics, W. T. Koiter, ed . NOrth-Holland Publishing Company (1976)

J. R. Rice, liThe Localization of Plastic Deformation", inTheoretical and Applied Mechanics (Proceedings of the14th International Congress on Theoretical and AppliedMechanics, Delft, 1976, ed. W.T. Koiter), Vol. 1, NorthHolland Publishing Co., 1976, 2.07-220.

- THE LOCALIZATION OF PLASTIC DEFORMATION

James R. Rice

Division o f EngineeringBrown University

P r o v i d e n c e ~Rhode Is land t U.S.A.

The loca l i za t ion o f p l a s t i c deformation in to a shear band i s discussed as an i n s t a b i l i t y o f- p l a s t i c flow and a precursor to rup ture . Experimental observat ions are reviewed, a g e ~ e r a l

theore t ica l framework i s presented, and spec i f ic calculat ions of c r i t i c a l condi t ions arecarr ied out fo r a variety o f mater ia l models. The interplay between features of ine las t iccons t i tu t ive d e s c r i p t i o n ~espec ia l ly deviations from normality and vertex-l ike yielding, andth e onset of loca l i za t ion is-emphasized.

1 . INTRODUCTIOl-r

I t i s remarkably common among duc t i l e so l idstha t when deformed -suff icient ly into the plas t icrange, an essential ly smooth and continuously

varying deformation pa t te rn gives way-to highlylocal ized deformation in th e form o f a tlsheal"bandt!. : Sometimes such _bands, once fbrmed, pers i s t and the subsequent deformation proceeds ina markedly non-uniform m a n n e r ~O f t e n ~however,such- intense loca l deformation leads di rec t ly toducti le f rac tu re , so tha t the onset of localiz-ation: i s synonymous with th e inception of rupture .

While observed in duc t i l e metals . polymers, an din rocks an d granular aggregates under compress i ve s t r e sses , -there i s l i t t l e ' in th e way -of acomprehensive understanding of th e phenomenon.Some basic theore t ica l pr inc ip les follow fromHadamard's [1 ] s tud ies o f e l a s t i c s t a b i l i t y, extended to the non-elast ic context by Thomas [ 2 ] ,Hil l [3] , an d Mandel [4-]. But i t i s only recently t h a t conclitions- fo r the onset o f loca-lizat-ion have been t-ied t o r e a l i s t - i c cons t i tu t ivedescr ipt ions of- ine las t ic r-esporise. Indeed-, aswil l be real ized from th e work to be reviewed(sect ion 3) , - thesecondi t ionsdepend c r i t i c a l l yon sub t le - fea tu res o f - these desc r ip t ions , specifi ca l ly on vertex-like yie ld ing ' e ffec ts and' departures from plasticnnormali yll, as well as onth e t ensor ia l nature o f th e pre - loca l i za t ion deformation f ie ld


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208 J.R. RICE

when cleavage i s precluded, such mater ia ls f a i lby th e nucleation o f holes from the b r i t t l ecracking o r decohering of inc lus ions , with subsequent plas t ic growth to coalescence o f th ecavit ies thus created. Cox and Low [9] show int h e i r f ig . 16a a magnified sect ion o f a p l a s t i -ca l ly deformed high-strength AISI-4340 s t e e lnear i t s point o f rupture . Cavi t ies have formed

around some b r i t t l e inclusions an d have enlargedwith th e plas t ic deformation. One might thenassume that rupture wil l enta i l th e large plast i c growth o f these holes unt i l the remainingligaments between them neck to zero thickness ,fo r many such cases have been observed ( e . g . ,McClintock [10]). But instead th e hole growthprocess in th e Cox and Low specimen ha s beenterminated by th e formation o f a band of loca lized shearing extending between the l a rgecav it i e s . Hithin the band. a number of very muchsmal ler cav i t i es have formed an d grown towardcoalescence. One cannot sa y whether these smaller rupture cavit ies f i r s t began to form, an dt h i s le d to loca l iza t ion , o r whether t ~ e plas t icflow f i r s t local ized an d nucleated th e smallcav i t i es . But cer ta in ly, there i s no detectableevidence of the nucleat ion o f th e smal ler cavit i e s a t points outs ide the shear band.

IndeeQ, such loca l i za t ions seem to be cruc ia l inse t t ing th e l imits to achievable -fracture duct i l i t y . Forexample, in sharply pre-crackedduc t i l e so l ids , th e onset of crack growth i s expected to occur when th e crack has been s u f f icient ly opened at i t s t ip so tha t the zone o flarge plas t ic s t ra in extends over an adequates i z e , by comparison to the spacings of void nucleat ion s i t e s , to grow a representat ive void tocoalescence with the crack t ip (Rice and Johnson[11]) . In studies by Green and Knott [12] on

s tee l s and Hahn and Rosenfield [13] on aluminuma l loys , i t i s shown tha t the predictiQns o f sucha model ar e often well followed exper imental ly,but that there ar e also cases in which the f racture duct i l i ty i s s t r ik ing ly less than predicted;these cases seem to i n v o l v ~th e termination o fthe hole- joining process by s t rong loca l i za t ionso f th e type revealed by Cox an d Low.

At a more macroscopic l eve l , Tanaka and Spretnak[14] subjected round bars o f a high st rengths t e e l to large to r s iona l s t ra ins and observedloca l i za t ions which, they suggest. correspondedto th e achievement ot" an "ideal ly p las t i c" s t a t eof stat ionary shear s t ress with ongoing deforma

t ion .Also,

Berg[15J

suggests th e poss ib i l i tyof macroscopic local izat ion in void-containingduc t i l e mate r ia l s , when the hardening o f theso l id matrix surrounding the voids , in an increment of deformation, i s outweighed by the sof tening due to porosi ty increase through increment a l void growth. The range of s i tua t ions towhich th i s model applies remains uncertain.Studies of duc t i l e rupture- in-progress in th enecks of Cu t ens i le specimens, nominally said tof a i l through void growth and coalescence, byRogers [16] (his f igs . 6,7,8 ,10) and Bluhm andMorrissey [17] ( the i r f ig . 42) reveal zones of

highly local ized deformation within which voidsare indeed coalescing. But conditions are highly nonuniform across t h e i r specimens and i t i sd i f f i c u l t to determine whether a Berg-like loc a l i z a t i o n , due to sof tening through porosi tyinc rease , ha s occurred o r whether loca l i za t iono f plas t ic flow ha s caused the extensive voidgrowth. Certainly, the f i n a l rupture surface

made up o f a "void shee t" , but th i s alone i s noind ica t ive o f the process leading to i t .

Further examples o f loca l i za t ion i n s t a b i l i t i e sare provided by the formation o f narrow neckedzones in duc t i l e metal sheets deformed, a t l easpr ior to loca l iza t ion , in plane s t r e s s . Judgedas 3-D problems, these are dis t inc t ly di ffe ren tfrom th e o ther cases c i ted and involve "geometr i c " as opposed to "material" i n s t a b i l i t y. Hoever, to the extent tha t such duc t i l e sheets aremodelled as 2-D cont inua, th e problem of loca li za t ion may be t r ea ted through ident ical m a t hmatical steps and i s well considered within th esame general framework.

Geological mater ia ls are r i ch a l so in examplesof local izat ion. For slope s t a b i l i t y fa i lu reso f overconsolidated clays an d clay sha les , i ta common observation tha t deformations concent r a t e in a narrow shear zone, perhaps only a femm across , on which large downslope mass movements take place. Laboratory deformation o fsuch clays reveal a corresponding concentrationo f deformation ( e . g . , Hvorslev [ 1 8 ] . f ig . 28),which seems to f i t th e concept o f bifurcat ioninto a loca l i zed mode. Field occurrences may,however, involve st rongly non-uniform conditionswith a crack-l ike mechanism for propagation o fthe shear band (Palmer and Rice [19]) a f t e r i t si n i t i a t i o n a t a s i t e o f loca l s t ra in concentra

t ion . Rowe [20] shows loca l i za t ions within sans p ~ c i m e n s(h i s f igs . l5b ,c ) , deformed in thet f t r iaxial" apparatus under ax ia l ly ' symmetriccompression. The effec t o f end conditions i sevident in tha t an arrangement intended to. provide shear-free ends of th e specimen leads tos ign i f ican t ly l a rger s t ra in a t loca l i za t ion thafo r an, effec t ive ly, f ixed end-piece specimen.Also, i t i s evident ' tha t there i s no necessaryassociat ion o f loca l i za t ion with th e "ideal lyp l a s t i c ll s t a t e , fo r the loca l i za t ions occur whthe mater ia l i s well beyond tha t s ta te and inthe s t ra in - sof ten ing range (s ignif icant ly negat ive slope of th e l o a d ~ d e f o r m a t i o ndiagram). Ideed th e t ens i le fracture t e s t s on metal speci

mens by Bluhm and Morrissey [17] were done ins t i f f t e s t ing apparatus so tha t a st rongly negat ive load-deformation slope could be accommodatewithout i n s t a b i l i t y, and these suggest to o tha ta s t ra in - sof ten ing s t a t e (i n terms o f t rues t ress ) may well have prevai led, a t the onsetmacroscopic rupture through hole coalescence athe cen te r of . the necked region.

Fina l ly, natural rock specimens t e s ted undercompressive pr inc ipa l s t resses also show examples of loca l iza t ion , usual ly referred to as"faul t ing lt Wawersik and Brace [21] study the


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THE LOCALIZATION OF PLASTIC D E F O ~ t A T I O N 20 9

post-"fai lure Tl behavior o f specimens through as t i f f t e s t ing apparatus , with provis ion for rap1dunloading, and show ( e . g . , t h e i r pla tes 4a,b fo ra Frederick diabase) examples of loca l i za t ion .Here th e ine las t i c deformation ar i ses from fric-:t iona l s l iding on closed microcracks' and p r o ~ sive enlargement of th e microcrack networkthrough local f issuring; th e f ina l macroscopic

fau l t l inks a large number of such microcracks,~ l t h o u g ht h e i r individual direct ions of growthdo no t c o i n c i d ~with tha t of th e f ina l fau l t .This case, l ike tha t o f granular mater ia ls , i sinterest ing because th e Coulomb f r i c t i o n a l nature of th e yieiding means tha t plas t ic normalitywil l no t apply (Mandel[4]) and t h i s ha s interestin g consequences for local izat ion i n s t a b i l i t i e s .

1.2. Mechanisms of local izat ion_The work to follow explores a par t icu la r ap

proach to explanation o f th e local izat ion of deformation, viewing th e process as an i n s t a b i l i t ytha t can be predicted in terms o f th e pre - loca lizat ion const i tut ive re la t ions o f th e mater ia l .The mater ia l i s modelled as rate-independent andc r i t i c a l conditions a re sought a t which i t s cons t i t u t i v e relat ions allow a bifurcat ion from homogeneous or smoothly varying deformation into ahighly concentrated shear band mode, o r , perhapsin'stead, a t which th e accelerated growth, withongoing deformation, of some i n i t i a l l y smallnon-uniformity of mater ia l propert ies can occurin such a manner tha t th e same sort o f shearband i s th e end resu l t .

Of course, no t a l l local izat ion phenomena can beexpected to f i t t h i s concept. An a l te rna t ivehypothesis would be tha t some essen t ia l ly newphysical deformation mechanism comes into play,abrupt ly, and rapidly degrades the s t rength of

the mater ia l . In such cases th e pre - loca l i zat ion const i tut ive relat ions cannot be continuedanalyt ical ly a t th e c r i t i c a l point , and theyprovide no basis fo r predict ion of loca l i za t ion .Indeed, to . the extent tha t upper yie ld pointsarise from the sudden breaking f ree o f dislocat ions from pinning obstacles , with only l igh t lyimpeded subsequent g l i d i n g , th e Luders band casemust be considered as one which i s dominated byonset of some new mechanism, and thus the bi furcation approach, explored here , does not apply toi t . ( I t i s , however, curious tha t us e o f a moresophis t icated rate-dependent plas t ic flow theorywould return t h i s case to one fo r which localizat ion could be understood in terms of c o n ~ t u t i v ere la t ions , although th e de ta i l s of th e analysiswould be very di ffe ren t from what i s to fol low.)

Further, th e approach to be explored here, beingessen t ia l ly a bi fu rca t ion approach, envis ions aprocess of simultaneous or nearly simultaneousoccurrence of concentrated deformation a t a l lpoints of th e (ul t imate) zone of loca l i za t ion .But in contrast , there may be s i tua t ions tha tare dominated by some s t rong loca l inhomogeneity,which concentrates deformation in i t s vic in i tyan d causes th e i n i t i a t i o n of a local ized zonewhich, subsequently, creates i t s own s t ra in con-

centrat ion and thereby t raverses the mater ia l a tnominal deformation ~ o n d i t i o n stha t are well r e ~moved from those for loca l i za t ion . This i s , ofcourse , the way in which a Gri ff i th flaw cancause a crack to t raverse a body a t averages t resses tha t are well below the s t rength levelfor an unflawed sol id (although s t resses a t ornear t h a t l eve l would be achieved loca l ly a t th e

propagating crack t i p ) .

Such c r a ~ k - l i k epropagation processes do no t inval idate th e present approach to - local izat ion,bu t do require tha t i t be general ized, in a waytha t i s not ye t ful ly c l e a r, to encompass th ehighly non-uniform condi t ions prevai l ing nearth e t ip o f the crack-l ike zone. An elementaryapproach i s to assume tha t once loca l deformat ions reach conditions fo r loca l iza t ion , th e cons t i tu t ive re la t ions fo r c o n t i n u u m ~ l i k edeformat ion ar e suspended in favor of a relat ion betweent r ac t ions and re la t ive displacements o f the surfaces o f th e zone of local izat ion (presumed t h i ~ .This approach i s embodied in the Palmer and Rice[19] model for shear band propagation in overcons o l i dated clay so i l s : fo r very small i n i t i a lflaws (o r fau l t s ) th e strength as so predictedapproaches tha t fo r th e unflawed body, whereasfo r large flaws th e response tends t o become i n ~dependent o f the s t rength fo r the unflawed body,and i s expressible instead in terms o f flaw sizeand parameters of th e t r a c t i o n v s . separat iondisplacement relat ion ( s p e c i f i c a l l y ~in terms ofth e net work of separat ion) for the local izedmater ia l .

Clear ly, the mere observation tha t a zone of local ized deformation ex i s t s within a deformedsol id i s an inadequate basis fo r choosing which,i f any, of the various mechanisms discussed i s

correct for i t s explanat ion.

2 . GENERAL THEORY

vle consider deformations which carry points ~of some reference s t a t e t o posi t ions x , whereboth ~ an d ~ ar e coordinate se t s ( ; . g . , Xl ,X2'X 3 ) referred to a fixed Cartes ian frame. Thedeformation gradient tensor i s r = a ~ / a ~ands t ress i s measured by th e nominal s t ress tensors , defined so tha t no s i s th e force act ing,per uni t reference a r ; a : on an element of sur..;.face having normal vector n in the references t a t e . I t s a t i s f i e s

s + f . = a ( s k :: as . .lax. )1J ,1 J 1J , 1 J - 1 0 a n d , as Berg [15] remarks, i f the mater ia l o f the so l id matr ix i smodelled loca l ly by continuum plas t ic i ty of akind fo r ~ h i c hth e normality ru le holds , then th erule app11es to th e aggregate also and thus ~ = 8 .Gurson [35] gives spec i f i c forms , for 8 and ubased on a r ig id-p las t i c model of a voided mater i a l , an a r e ~ r k stha t th e inclusion-in the cons t i t u t i v e re la t ions o f a hydrostat ic s t ress dependent cr i t e r ion fo r void nucleat ion (say, byth e bri t t le-cracking or decohering of inclusions)leads to deviat ions from normality with u > 8

Le t us f i r s t analyze th e mater ia l descr ibed by(26) according to th e r ig id-p las t ic model. In

tha t case loca l i za t ion i s possible only i f th eintermediate pr inc ipa l value of f -vanishes ,i . e . i f

ail = - (28/3) T ,an d then l o c a l i z ~ t i o ni s possible no matter howlarge or smal l th e value of h i f 8 1 u ,whereas loca l i za t ion i s possible only when h= 0i f B : : p . Rudnicki and Rice [22] obtained d irec t ly th e r e s u l t corresponding to (25) fo r thee l a s t i c - p l a s t i c model and, searching ou t theplane which f i r s t allows loca l i za t ion ( t h a t corresponding to the maximum h over a l l or ien tat ions n) , they showed tha t provided u and B~ r e

no t to o large (see p. 384 o f [22] ) , the cr i tl .cal value i s

h . /Gcr1.t

Here v

O f

:: l+ v 2 l + v I I ~ + 8 2 1:9 ( l _ v ) { U - B ) - 2 ( - 1 - + - 3 - ) + O(G')

i s th e e l a s t i c Poisson r a t i o . (27)

The r e s u l t i ~ in te res t ing in several respects .F i r s t note that to neglect of terms of O(T/G) ,ar i s ing from th e dis t inc t ion between ~ and aloca l i za t ion can never occur with pos i t ive h 'i f normality appl ies ( i . e . , i f U = 8). On th eothe r hand, i f normality does no t apply, i t i s


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21 6 J .R. RICE

possible fo r loca l i za t ion to occur with pos i t iveh ~ depending on th e value of GrI/T . Indeed treplacing t h i s quant i ty in favor of FIr ( reca l lt h a t according to the r ig id-p las t ic model, loca l i za t ion ca n occur only when Plr = 0) , th ec r i t i c a l condition i s

Hence, when PIr = 0 , loca l i za t ion ca n occur.with th e posi t ive hardening ra te

(1+v)2 2hcri t /G = 18(I-v) ( ~ - e )

bu t in contrast to th e r ig id-p las t ic case, loca l i za t ion ca n a l so occur fo r othe r values ofP rI ( the most c r i t i c a l case i s PIr = - ( ~ - e ) / 6 ) .I f th e in termediate plas t ic s t ra in ra te depar tsto o much from zero, t h e . c r i t i c a l h value turnsnegat ive. This i s consis tent with what appearsto be a grea te r tendency fo r loca l i za t ion inplane s t ra in than under axial ly symmetric condit ions , t ens i le duct i l i ty being l e s s in the fo rmer case (Clausing [ 3 6 ] ) .

Several numerical tabulat ions of these r e s u l t sare given in [22J, in which i t i s a l so pointedou t tha t ver tex- l ike y ie ld effec t s have a stronginf luence on th e predicted condi t ions fo r loca li za t ion (see sect ion 3. 5 to follow).

3.1.j.. Localization in a single crys ta lConsider.a duc t i l e single crys ta l undergoingpla!;>tic flow by s l i p on a s ing le system of planeshaving normal in th e x2 di rec t ion and s l i p d irect ion in the Xl di rec t ion . Fo r brev i ty, weanalyze the problem by neglect ing th e dis t inct ionbetween 2 and Q and otherwise neglect ingterms of order s t r e s s divided by e l a s t i c modulus;

. a f u l l analysis i s to be presented in the nearfuture by the author in col laborat ion with R. J .Asaro. I f th e Schmid law of resolved shears t ress governing s l i p i s followed, th e plas t icresponse i s given by

=where h i s th e hardening r a t e . This law corresponds t o r = 2. , hence normality ~ withP12 = P21 = 1/ 2 and a l l other P . = O . Anon-deforming surface always e x i s t ~in t h i s case(namely the x1 ,x 3 plane) so tha t when s tudied asa r i g i d - p l a s t ~ cproblem, the c r i t i c a l condi t ion i s h = o . Sometimes t h i s f i t s the experimentllfac t s a t onset of coarsened s l i p (Jackson andBasinski [8 ] ) . However, i t i s in te res t ing fo rother cases to study sources of deviat ions fromSchmid's law, because th e corresponding non-normality ca n be des tab i l i z ing .

One of th e most promising candidates i s crosss l i p . When screw segments of dis locat ions surmount local obstacles by th is mode of s l ip th eincremental plas t ic deformation shquld dependnot only on th e increment of 012 , bu t also ofth e s t r e s s resolved onto th e c r o s s ~ s l i pplaneand of tha t which serves to coalesce dislocation

stacking f a u l t s , so as to make th e loca l changof s l i p plane possible . Fo r effec t s o f n u m1y comparable s i z e , tha t due t o th e resolveds t ress on th e cross - s l ip plane i s by fa r the mimportant fo r loca l i za t ion ( i t corresponds t onOD-zero OaB in ( 1 7 , an d hence fo r simpliciwe rewri te th e plas t ic cons t i tu t ive relat ion tinclude only t h i s effec t :

= (2

Fo r t h i s r e l a t i o n , t i s as given above, hutQ12 = Q2 l = 1 /2 , Ql3 = Q31 = p/2 and otherQ = 0

~ J

Following th e general solut ion (25) fo r h a tloca l i za t ion on a plane of normal E , thesevalues of and 9 lead to

2 2 2 + ~ n/G = I + n l + n 2 + p n2n 3 - Xn

ln 2 (n 2

2 . 2 ~ s i n , cos, sinOSl.n 4> s ~ n 0 +

- X

. 2 28s ~ n 4> co s cos , ( cos , + ~ s i n , s in

where X = 4(A+G)/(A+2G) and where, in the l aform" i s th e angle between n an d the x 2axis and 8 i s th e angle made with th e Xl aby th e project ion of n .onto the x1 ,x 3 plane increases hy n /2 in a ro ta t ion From th et o x3 di rec t ion . r f ~ i s smal l as expecteds6 also wil l be the deviat ion of 4> for the mc r i t i c a l plane from zero. Thus, expanding ther e s u l t to quadrat ic order in ,

h/G ~ , ~ s i n e- , 2 ( s in 2e + X co s 2e)

This i s eas i ly shown to take on i t s maximum vue when a = n/2 and 4> = ~ / 2 , correspondingto

2h c r i t ~ ~ G/4 (3

This shows again the des tab i l i z ing influencedeviat ions from normality. A value of ~ onth e order of 1/10 would give loca l i za t ion according to (30) a t what would have to be judgas a s ign i f ican t ly st ra in-hardening s t a t e ~

As remarked e a r l i e r , i t remains an open questiofo r many dUctile metals as to whether th e present ucoarse .s l ipH type of loca l i za t ion servesto concentrate s t ra in an d lead to the nucleatioand growth to coalescence of voids in f rac tu re ,

or whether i t i s instead th e inc ip ien t nucleat ion and growth o f voids which leads to a locai za t ion , as described in th e previous sect ion.

3.5. Yield v ~ r t e xeffec t sWhen th e plas t ic port ion of the pre- local izeddeformation ra te f i e ld contains no non-deformingplane, th e cr i t e r ion fo r loca l i za t ion i s cont ro l led by those cons t i tu t ive parameters whichmark th e s t i ffness of response to abrupt , thouperhaps smal l , changes in th e Hdirection H of tdeformation r a t e . These changes correspond tothe superposi t ion of the local ized shear band


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THE LOCALIZATION OF PLASTIC DEFORMATION 21 7

mode on th e given homogeneous f ie ld . Forttsmoothyield surface and plas t ic potent ial" models, therelevant response moduli a ~ e of the same orderas e l a s t i c moduli, s ince superposed deformationincrements tha t are ~ t h o g o n a l t oth e plas t icflow direct ion f induce only an e l a s t i c r esponse. This feature i s a t root of the stronglynegative values predicted for h in (28) when

f depar ts great ly from conditions for exis tenceof a non-deforming plane.

However, physical models of ra te - insens i t ivep l a s t i c flow based on microstructural s l i p ,e i ther of a Schmid (metal p l a s t i c i t y ) or f r ict iona l (geOlogic mater ia ls) type, lead universally to th e predict ion of vertex development a t th ecurrent s t ress point on ftzero-offset" yield andplas t ic poten t ia l surfaces [22,24,26,27] . Suchmodels e n t a i l a considerably sof te r e l a s t i c - p l ~t i c response t o small superposed deformation increments, orthogonal t o th e prevai l ing plas t icflow direct ion (see, e . g . , f ig . 2 of Hutchinson[27] for crys ta l l ine s l i p ) , and t h i s ha s a

st rong, general ly destabi l iz ing effec t on predicted conditions for loca l i za t ion .

The matter ha s been s tudied a t length by Rudnickiand Rice [22] , who add vertex effec t s to th e cons ~ i t u t i v emodel of sect ion 3. 3 and demonstrateboth separate and combined destabi l iz ing effec t stha t a r i se from vertex yielding and nqn-normali t y. Also, Stren and Rice (29] show t h a t avertex yield model leads to predict ions o f 10 -calized'necking in thin sheets , under posi t ivein:"'plane pr inc ipa l extensions, a t condi t ionstha t compare favorably with experimental resu l t swhereas, in th e same circumstances, the smoothyield-surface r ig id p l a s t i c model (without imperfect ions [28]) predicts unlimited duct i l i ty.

Fo r brevi ty we consider here only the non-di lat a n t , pressure- insensi t ive version of the ver texconst i tut ive re la t ion studied in [22] , f i r s ts p e c i a l i ~ i n gi t as in [29] to a r ig id-p las t i cmodel. The re la t ion i s

2 H - 1 t 1 v t- a' - + - (a ' - 0 ' - )h ~ T h ~ N T (3l)where 2T2 = ~ I : ~ tand or i s the deviator icstresS. The' re la t ion i s ' intended to model response on deformation paths t h a t di ffe r onlymodestly from f ixed-principal-axis deformationwith ' ~ ' . When deformations comply precisely with t h i s , the l as t term of (31) vanishes

and we are l e f t only with the f i r s t , correspondin g t o c lass ica l Mises r ig id-p las t ic response a ta hardening modulus h (i n shear ) , with = 9 = 1/2T. The second term o f (31) represents th e vertex yie ld e f f e c t , and hI i s themodulus of vertex response (defined analogouslyto an e l a s t i c shear modulus) fo r small superposed ra tes d' tha t are no t coaxial with a ' .To th e extent- that approximately p a t h - i n d e p e ~ d 6 t tre la t ions between sui tably defined s t ress and deformation measures [29'1 resu l t for such onlymoderately non-proportional deformation paths,the "deformation theory" of p l a s t i c i t y appl ies

and hI can be iden t i f i ed as the secant modulust o the s t ress -s t ra in re la t ion in shear (see [27]fo r a comparison\with response moduli of " inc remental" models). In any event , one may assumetha t hI > h

By applying th e theory of sect ion 3.1 to th e rela t ion (31) , we se e tha t in the absence of avertex effec t (h I = ~ ) local izat ion can occuronly i f O il = 0 and then, s ince (31) as writteninvol'ves plas t ic normality, -only when th e tangentmodulus h = O . To se e how the ver tex effec tmodifies these r e s u l t s , f i r s t invert (31) to

2 = 01 + g - g-g + 2hl ~ - (hl-h)g ' (2 t :!2) /T 2 (32)where a = t r ( o ) / 3 . Now, s ince th e mater ia l i si n c o m p r e s s i b l e ~ t h ebifurcat ion vector g of ( 1 )must take th e form gm l-lhere m i s a uni t vecto r perpendicular t o - ~ n, i . e . ly ing in th e planeof local izat ion. We take the reference s t a t e ascoincident ins tantaneously with the current s t a t ~operate with n in (32) , and dot with ~ to

obtain= nno+g{- . !m.o - !{n-o .m)n+!(n .o -n )m- . 2'" '" 2 '" ~ "" ~ 2,-v,. , '" ",

+ h m - (h - h ) ( n ~ o t ) ( n . o ' . m1 - 1 ~ - - N ~

By dott ing t h i s equation successively with unitvectors m and ~ , the l a t t e r being perpendicular t o ' n a n d m and hence also in th e planeof l o c a l i ~ a t i o n ,we obtain two simultaneous condit ions for th e exis tence of a non-zero bifurcat ion amplitude g . These are

2 2(onn- omm)/2 + hI - (hl-h)omn/ T

a /2 + (hl-h)o /T2 =mz nz mn

o (33)

o (34)

where the re i s no summation implied and indicesdenote components of ~ on the axes n , ~ , ~ .

We may view (33) as an equation giving th e c r i ti c a l value of h fo r a given h I ' ~ , a n d n ,and view (34) as a constraint on th e correspondin g direct ion m . The most c r i t i c a l plane n ,fo r a given h l ~and ~ , i stha t which m a x i m ~the correspond1ng h value s ince , for appl icat i o n s , h i s general ly a non-increasing functionof th e amount of deformation imposed on th e mater i a l .

By se t t ing th e variat ion oh = 0 fo r 0E infuedirect ion of z , one f inds tha t (34) as well a s

th e equation2

a /2 - (hl-h}o 0 /Tnz . mz mn = o

must be sa t i s f ied simultaneously. This can bed o ~ e o n l yi f th e ~ direct ion i s pr inc ipa l ,0nz = 0mz = O . That i s , B a n d ID must l i e ina plane formed by two of th e principal direct ions . Next, se t t ing oh = 0 when on i s inthe d i rec t ion o f m , and recognizing tha t theassociated om ha ; direct ion - B , one obtains

220 mn + 2(hl-h)omn(onn-omm)/T = 0 (35)


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> 218 J .R . RICE

Assuming tha t th e cr i t i caL plane does no t correspond to one o f vanishing shear s t ress the c r i t i ~cal or ientat ion i s tha t fo r which

( 0 - 0 )/ Tmm nn = (36)

and i t can be shown tha t a plane ex is t s meetingt h i s condition i f

T ~ 2 ( h l _ h ) ( 1 _ 3 0 1 2 / 4 ~ 2 ) l f 2zz

Thus, observing tha t

(37)

T2 = ~ g Y 2= (Omm-Onn)2/4 + O ~ n+ 3 0 ~ ; / 4,and using (36) , one may subs t i tu te in to (33) toobtain th e c r i t i c a l condition

o (38)

H ~ n c e tto summarize, the c r i t i c a l condition i sgiven by (38) where ~ i s one o f th e pr inc ipa ld i rec t ions , provided tha t th e associated cr i t i ca l

condition allows th e inequal i ty (37) to be sat isf ied. I t i s easy to show tha t ~ must be chosenas th e intermediate pr inc ipa l d i rec t ion (denotedby I I ) to give the maximum of possible solut ionsfo r h. " and i t i s convenient a t t h i s point tointroduce the dimensionless s t ress s ta te parameter

u = (39)

noting tha t u = 0 fo r pure shear and t h a t utakes on i t s maximum value, 1 /4 , fo r axi-symmet r i c extension or compression.

The condition (38) can then be writ ten2

T = (40)and t h i s i s the c r i t i c a l condi t ion provided theinequal i ty (37) i s met. That inequal i ty ca n nowbe r ~ a r r a n g e dto th e requirement tha t the ra teof hardening s a t i s f i e s

(4l)

which does no t seem r e s t r i c t i v e in terms of the.physical in te rp re ta t ion o f h Equation (40)reduces to tha t given by Hil i land Hutchinson [3Dfor s ta tes of plane s t r a i n , u = 0 , and t h i ss ta te allows loca l i za t ion a t a smal ler equivalents t ress T than does any other.

I f Tfhl i s regarded as a small parameter, (38)may be solved for the c r i t i c a l hardening modulusat loca l i za t ion and to the order of the terms reta ined the resu l t i s

(42)

Thus for small T/hl unless u i s close tozero ( s ta tes approaching plane s t ra in conditions)th e loca l i za t ion condi t ion requires s t ra in s o f tening behavior, .h < 0

For th e corresponding e l a s t i c - p l a s t i c mater ia lwith a yie ld v e r t e ~ b u tt r ea ted as e l a s t i c a l l yimpressible as wel l , one need only add a termgiG to the r i g h t s ide of ( 3 l ) , where G i s thshear modulus, and then th e same analysis as gien here appl ies provided t h a t the replacements

h -+ hG/{G+h) ,

ar e made in a l l formulae. In p a r t i c u l a r, whenth e c r i t i c a l condi t ion analogous to (38) i ssolved fo r h , an d the r e s u l t expanded to theorder of terms as in (42) , on e obtains

= hlG { u + (l-u+u 2 )h l +(I-u)Gh c r i t hl+{I-u)G - hI + (l-u)G

(h l +G)2 T2 + . }

4 h 2 G 21Retention o f only th e f i r s t term in th e bracketsnamely -u , corresponds t o writ ing a fo r gin th e cons t i tu t ive re la t ion an d thenNthe r e smay be compared with (27) , se t t ing ~ = a = 0and v = 1 /2 in the l a t t e r . Evidently, when tvertex modulus h I i s ~ u c hl e ss than th e elas t ishear modulus, th e c r i t i c a l lfhardening U modulusfo r loca l i za t ion a t s ta tes othe r than pl ines t r a i n i s considerably l e ss negative than wouldbe t h ; case in absence of a vertex (h I = ~ ) .Numerl.cal resu l t s and comparisons are given byryhdnicki and Rice [22] .

Fo r example, in axi-symmetric extension or compression, u = 1/4 and th e r e s u l t i s

.hlG=

4h l +3G

when both T/G and Tl h l are smal l compared tu n ~ t y With hllG in the range . 1 to .0 1whl.ch ml.ght be regarded as representat ive forheavily deformed metals based on a secant modulin te rpre ta t ion , t h i s formula gives c r i t i c a l hlvalues of - .03 to - .003 (compared to - .25when vertex effec t s are neglected) . Strain sofening effec t s of t h i s magnitUde might well resultin the necked regions of some duc t i l e metal t ens i l e specimens pr ior to fracture [ l6 ,17] , withthe ~ o f t e n i n gresu l t ing from progressive cav i tat ion a t inclusions. Of course , imperfect ion efec t s could also be very important in such caseas could th e sources of non-normality discussede a r l i e r .

4. CONCLUSION

The loca l i za t ion of p l a s t i c flow i s a fascinatingand widely observed phenomenon, which seemsimportant in se t t ing a l imi t to the achievable dut i l i t y of a so l id . Ye t th e top ic ha s remainedouts ide th e mainstream of work on the mechanicsof i n e l a s t i c s o l i d s , save for the elucidat ion ofgeneral principles in th e s p i r i t of Hadamard byThomas, Hil l , and Mandel. The present studyshows tha t conditions fo r loca l i za t ion r e l a t eclosely to subt le and not well understood fea-


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THE LOCALIZATION OF PLASTIC DEFORMATION 21 9

tu res of the cons t i tu t ive descr ipt ion of p l a s t i cflow. Localization i s favored by a low p l a s t i chard-ening modulus.; bu t th e matters of how low arrlwhether s t r a i n sof tening i s required are de te rmined by th e nature of the pre- local ized deformat i o n , those s ta tes with non-deforming planes being highly suscep t ib le , an d by deviat ions fromp l a s t i c normal i ty. The l a t t e r may, fo r example,

ar i se from Coulomb f r i c t iona l e ffec t s in yieldingo r from non-Schmid effec t s in crys ta l s as bycross - s l ip or other triggeTed processes , wheres t resses other than th e resolved shear s t resscontr ibute to flow on a given s l i p system". Vertex yielding effec t s are predicted on physicalgrounds and these to o have a s t rong inf luence onlocal izat ion condi t ions, fo r example, in mitigatin g predict ions of s t rongly negat ive h a r d e n i n ~fo r loca l i za t ion in axisymmetrically deformedso l ids an d l ike cases.

While the cons t i tu t ive modelling of these featu res needs "to be improved in re la t ion to th edetai led mechanisms of deformation, so also i s

the reneed

for a f u l l e r assessment of the ro leof imperfections o r i n i t i a l non-uniformities inmater ia l propert ies in promoting loca l i za t ion .Indeed, th e l a t t e r approach seems mandatory fo rrate-dependent p l a s t i c flow models and these , aswell as th e range o f thermornechanically coupledloca l i za t ion phenomena would seem to meri t fu rthe r study.

The basic theory o f uniqueness in re la t ion t oloca l i za t ions and s ta t ionary waves i s a l so no tyet adequately developed for mater ia ls deviat ingfrom normal i ty, and ne i the r ca n the case of vertex yielding be handled in f u l l general i ty within th e ex i s t ing framework. Yet both a re featureswhich seem inherent to much of p l a s t i c cons t i tut ive behavior and th e examples and analysis ofthe present study suggest tha t both a re import an t destabi l iz ing features fo r th e process o floca l i za t ion .

AcknowledgementDiscussions with C. Dafermos, R. Hil l and A.Needleman on the general theory, and with R. J .Asaro a n d J . W. Rudnicki on par t icu la r c lassesof mater ia ls have been very helpful to th e writenThe study was supported by th e "U.S. Energy Research and Development Administration under cont r a c t (11-1)3084 with Brown Univers i ty.

REFERENCES

1. Hadamard, J . ; Lecons sur l a Propagation de sOndes e t l e s Equations de L'Hydrodynamique; Par i s , Chp. 6 , 1903.

2. Thomas, T.Y.; Plas t i c Flow and Fracture inSolids; Academic Press , 1961.

3. H i l l , R.; Accelera t ion waves in s o l i d s ; J .Mech. Phys. Sol ids , 10, p . I -16 , 1962.

4. Mandel, J . ; Conditions de s tab i1 i te e t postul a t de Drucker; in Rheology and Soil Me-chanics , eds . J . Kravtchenko an d P.M.Si r ieys , Springer-Verlag, p.5B-68, 1966.

5. Nadai, A., P l a s t i c i t y, McGraw-Hill, 1931.

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f r ac tu re of single crys ta l s ; in Fracture(Prec . Conf. a t Swampscott, Mass. ) , eds.B. L. Averbach e t a I . , Technology Press-MtT,p.474-497, 1959.

7. Pr ice , R.J . and Kelly, A. ; Deformation of ag ehardened aluminum a l loy c r y s t a l s - I I , f ractu re ; Acta Met. , l 2 , p . 9 7 9 - 9 9 2 , 1964.

8. Jackson, P.J . and Basinski , Z.S . ; Latenthardening and th e flow s t ress of coppersingle crys ta l s ; Can. J . Phys. , 45 ,p. 707-735, 1967. -

9. Cox, T.B. and Low, J . R . ; An inves t iga t ion ofth e p l a s t i c f r ac tu re o f AISI 4340 and 18Nickel-200 grade maraging s t e e l s " ; Met.,Trans . , ~ , p.1457-"1470, 1974.

10 . McClintock, F .A. ; A cr i t e r ion fo r duc t i l ef r ac tu re by the growth of holes; J . Appl.Mech., ~ , p.363-37l , 1968.

11 . R i c ~ - !J .R. and Johnson, M.A.; The ro le oflarge crack t i p geometry" changes in planes t ra in f rac tu re ; in Ine las t i c Behavior ofSo l ids , eds . M.F. Kanninen e t a l . , McGraw-H i l l , p.641-672, 1970.

12. Green, G. and Knott, J . F. ; The i n i t i a t i o n andpropagation o f duc t i l e fracture in lowst rength s t e e l s ; Trans. ASME, 98, Ser. H,J . Engr. Mat. Tech. , p ;37-46, 1976.

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eds . M.F. Kanninen e t a l . , McGraw-Hill,p.171-209, 1970.

16 . Rogers, H.C.; The t ens i le f r ac tu re of duct i lemetals ; Trans. AnfE, 218. p . 498-506, 1960.

17 . Bluhm, J . I . and Morrissey, R.J".; Fracture ina t ens i le specimen; in Proc. 1 st Int .ConFrac tu re , eds. T. Yokobori e t a l . , Japanese Soc. fo r Strength and Fracture ofMater ia ls , Vol.3, p.1739-l780, 1966.


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18 . Hvorslev, M.J. ; Physical components of theshear st rength o f saturated c lays ; in Research Conf. on Shear Strength of CohesiveS o i l s , ASCE, p.169-273, 1961.

19 . Palmer, A.C. and Rice, J . R . ; The growth o fs l i p surfaces in th e progressive fa i lu reof overconsolidated c lay ; Proc. Roy. Soc.

Lond., ~ , p.527-548, 1973.

20 . Rowe, P.W.; The s t ress -d i l a tancy re la t ion fo rs t a t i c equi l ibr ium of an assembly of part i c l e s in contact ; Proc. Roy. Soc. Lond.,A269, p.500-527, 1 9 6 2 ~

21 . Hawersik, W.R. and Brace, W.F.; Post-fai lurebehavior of a grani te and a diabase; RockMech., ~ , p . 6 l , 1971.

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23. H i l l , R.; Eigenmodal deformation in e l a s t i c /p l a s t i c continua; J . Mech. Phys. Sol ids ,15 , p.371-386, 1967; On cons t i tu t ive inequa l i t i e s f ~ simple m a t e r i a l s - I I ; J .Mech. Phys. Sol ids , 1 6 , p.315-322, 1968.

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30. Needleman, A.; Necking of pressur ized spheri c a l membranes; Brown Univ. Rep. MRL E-97,1976 (submitted t o J . Mech. Phys. Sol ids ) .

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32. van Hove, L .; Su r 1'extension de l a conditiode Legendre du c a l c u l de s var ia t ions auin tegra les mult iples a plus ieurs fonctioninconnues; Proc. Kon. Ned. Akad. Wetensch~ , p . 1 8 - 2 3 ~1947.

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cent Progress in Applied Mechanics - TheFolke Odqvist Volume, eds. B. Broberg ea l . , Almquist and Wiksel l , Stockholm,p.24l-249, 1967.

34. Spi tz ig , W.A., Sober, R . J . , an d Richmond,Pressure dependence of yielding and assciated volume expansion in tempered mart e n s i t e ; Acta Met., ~ , p.885, 1975.

35. Gurson, A.L.; Plas t ic flow and f r ac tu re beh a v i o r o f duc t i l e mater ia ls incorporat ingvoid nucleat ion, growth, and "interact ion;Ph.D. Thes i s , Brown Uuiv., 1975 (availablfrom Univ. Microfilms, Ann Arbor, Mich.)

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rice the localization of plastic deformation

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