A generalisation of J2.flow theory

8/7/2019 A generalisation of J2.flow theory

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C o m p u t e r M e t h o d s i n A p p l i ed M ech an i c s an d E n g i n ee r i n g 1 03 (1 9 93 ) 3 4 7 - 3 6 2N o r t h - H o l l a n dC M A 3 20

A genera l isat ion of JE-flow theory for polarcont inuaR e n 6 d e B o r s t

Delft University o f Technology, D epartment o f O vil Engineering~TNO Building and ConstructionResearch, P.O. Box 5048 , 260 0 G A Delft, The N etherlandsR ece i v ed 9 O c t o b e r 1 9 8 9Revised manuscr ip t r ece ived 16 Apr i l 1992

A p r e s s u r e - d ep en d en t ] 2 -f lo w t h eo r y i s p r o p o s ed f o r u s e w i t h in t h e f r amew o r k o f t h e C o s s e r a tcon t inuu m . To th i s end the d ef in i t ion of the s ec ond invar ian t o f the dev ia tor ic s tr es ses is genera l i s ed toinc lude couple- s t r es ses , and the s t r a in-hardening hypo thes i s o f p last i ci ty i s ex te nde d to t ak e acc ount o fmicro-curva tures . The t empora l in tegra t ion of the r esu l t ing se t o f d i f f e ren t ia l equat ions i s ach ievedus ing an impl ic i t Eule r backwa rd schem e. This r e turn-m appin g a lgor i thm resu l t s in an exac t s a t is f ac t iono f t h e y i e l d co n d it i o n a t t h e en d o f t h e l o ad i n g s tep . M o r eo v e r , t h e i n t eg r a ti o n s ch eme is amen ab l e t oex ac t l i n ead s a t i o n , s o t h a t a q u ad r a t i c r a te o f co n v e r g en ce is o b t a i n ed w h en N ew t o n ' s m e t h o d i s u s ed .An impor tan t charac ter i s t i c o f the model i s the incorpora t ion of an in te rna l l ength sca le . In f in i t ee lem ent s imu la t ions of Ioca l is a t ion , th i s p rope r ty war ran t s convergen ce of the load-d ef lec t ion curve toa ph ys ica lly rea l is t ic so lu t ion upo n m esh r e f inem ent and to a f ini te w id th of the | oca li s a t ion zone . Thisi s dem on s t ra te d for an in f in i t e ly long shear l ayer and for a b iaxial spec ime n com pos ed of as t ra i n - so f t en in g D r u ck e r - P r ag e r ma t e r ia l .

1 . I n t r o d u c t i o nW h e n t e s t in g s t r u c tu r e s c o m p o s e d o f m a t e r ia l s su c h a s c o n c r e t e , r o c k a n d s o i l , a m a r k e dp e a k i n t h e l o a d - d i s p l a c e m e n t c u r v e i s o f t e n f o u n d f o l l o w e d b y a d e s c e n d i n g b r a n c h .D e p e n d i n g o n t h e t y p e o f m a t e r i a l , t h e r e s i d u a l l o a d - c a r r y i n g c a p a c i t y o f t h e s t r u c t u r e m a yo n l y b e m a r g i n a l l y b e l o w t h e p e a k s t r e n g t h o r m a y v a n i s h a l t o g e t h e r .T h e c l as s ic a l a p p r o a c h t o w a r d s t h i s b e h a v i o u r i s t o s i m p l y e x t e n d t h e p r o c e d u r e s u s e d i n th ep r e - p e a k r e g i m e a n d t o c o n v e r t t h e l o a d - d e f l e c t i o n c u r v e a t a s t r u c t u r a l l e v e l i n t o a

s tre ss - s tra in curve a t a loca l l eve l . However , in do ing so , i t i s tac i t ly a ssumed that a l lh y p o t h e s e s w h i c h a re n o r m a l l y m a d e t o ar riv e a t a c o n t i n u u m m o d e l a n d w h i c h h a v e p r o v e nt o b e r e a s o n a b l e a s s u m p t i o n s i n t h e p r e - p e a k r e g i m e s t i l l h o l d b e y o n d p e a k s t r e s s l e v e l . O n eo f t h e s e a s s u m p t i o n s i s t h a t t h e t r a n s m i s s i o n o f f o r c e s b e t w e e n t h e m a t e r i a l o n b o t h s i d e s o fa n i n fi n i te s i m a l s u r f a ce e l e m e n t i s c o m p l e t e l y d e s c r ib e d b y a fo r c e v e c t o r a n d d o e s n o t r e q u i r et h e i n t r o d u c t i o n o f a c o u p l e v e c t o r [ 1 ] . R o t a t i o n a l e q u i l i b r i u m o f a n e l e m e n t a r y c u b e o fm a t e r i a l s h o w s t h a t t h i s a s s u m p t i o n i s t a n t a m o u n t t o t h e p o s t u l a t e t h a t t h e s t r e s s t e n s o r i sCorrespondence to- P r o f e ss o r R en 6 d e B o r s t, D e l f t U n i v e rs i ty o f T ech n o l o g y , D e p a r t m en t o f C i vil E n g i n ee r i n g /T N O B u i l d i n g an d C o n s t r u c ti o n R es ea r ch , p . o . B o x 5 04 8 , 2 60 0 G A D e l f t, T h e N e t h e r lan d s .

0045-7 825/93/$0 6 .00 1993 E lsev ier Sc ience Publ i sher s B .V . A l l r igh ts r eserved


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348 R. de Bo rst, A generalisation o f J:-flow theorysymmetr ic , which is somet imes refer red tO as Bo l t zmann ' s Ax iom. Never the le ss , con t inuumtheor ies do ex is t tha t a re no t roo ted in th is assumpt ion . Al though such theor ies a re usual lymuch more complicated than classical continuum theories, their use is just if ied if the classicalconcept o f a cont inuum fa i ls to produce meaningfu l answers .The t r ans fo rma t ion o f an expe r imen ta l ly ob ta ined load -d i sp lacem en t cu rve tha t exh ib i ts adescending branch a t the s t ruc tura l level in to an af fine s t ress-s t ra in curve a t the m ater ia l po in tlevel is usually associated with the terminology 'strain-softening ' . A consti tut ive law which hasa descending branch in the s t ress-s t ra in curve is then named a s t ra in-sof ten ing type const i tu -tive law.In the las t few years , the no t ion has t ransp ired tha t a s t ra igh tforward t ransla t ion ofload-d isp lacement curves in to s t ress-s t ra in curves en ta i l s some ser ious compl ica t ions , bo thf rom a mathemat ica l and a physica l po in t o f v iew [2-5] . From a mathemat ica l v iewpoin t ,which wi l l be the pr ime sub jec t o f concern in th is contr ibu t ion , we face the problem tha tin t roduct ion of s t ra in sof ten ing in a c lass ica l cont inuum model may conver t the boundaryvalue problem for s ta t ic load ing condi t ions f rom an e l l ip t ic p roblem in to a hyperbol icproblem. For dynam ic load ing condi t ions on the o the r han d , hyperbol ic ity i s los t and the w avespeeds of load ing waves become imaginary . In bo th cases , the ra te boundary va lue problem isno longer well-posed. Numerically , this i l l -posedness manifests i tself in pathological meshdependence, i .e . localisat ions which inevitably accompany fai lure processes in the classes ofmater ia ls d iscussed above tend to be de termined en t i re ly by the spacing of the f in i te e lementmesh in l ieu of being gove rned by the physics of the u nderlying p rob lem [6, 7] .For t ransien t p roblems, the loss of hyperbol ic i ty and the ensu ing pa thologica l meshsensit ivi ty can, at least par t ial ly , be remedied by the consideration of heat f low and theinc lusion of thermo-mechanica l coupl ing te rms and/or by resor t ing to the in t roduct ion ofviscosity in the consti tut ive descr iption [8-12] . These approaches implici t ly introduce aninternal length scale into the go vernin g set of equ ations , which causes the init ial value prob lemto remain well-posed.For s ta t ic load ing condi t ions , the above enhancements of the const i tu t ive model apparen t lycannot in t roduce an in terna l length sca le in to the problem. In the past , th ree d i f feren tapproaches have been fo l lowed to in t roduce such an in terna l length sca le . The f i r s t approachin t roduces h igher -order s t ra in grad ien ts in the const i tu t ive descr ip t ion [13-18] , whi le thesecond method employs an averag ing procedure wi th respect to the ine las t ic s ta te var iab les(non-local const i tu tive eq uat ions [19 , 20] ). The approac h pursued in th is pap er i s the use of aso-ca l led genera l ised or micro-polar cont inuum. Such a cont inuum model , in which th reero ta t ional d egrees-of - f reedom are in t roduced in addi t ion to th t~ convent ional th ree t ransla t ion-a l degrees-of - f reedom, was proposed as ear ly as in 1909 by E. a nd F . C ossera t [21] . Probab lybecause of i t s re la t ive complexi ty , i t rece ived l i t t le a t ten t ion . Never the less , renewed in teres tarose af te r a dormant per iod of some 50 years , p r imar i ly due to the works of Er ingen [22] ,G~inther [23], M indlin [24, 25] , Schae fer [26, 27] and To upin [28]. Th ese con tr ibutio ns hav econsiderab ly broadened the or ig ina l concept o f E . and F . Cossera t [21] and the te rminologym icro .po lar e las t ic i ty has becom e the vogue to descr ibe these ex tend ed or genera l ised e las t ic i tytheor ies . Yet , in teres t d ied in the la te 1960s, p robably because of the inheren t complexi ty ofthe theory , w hich resu lts in a govern ing se t o f d i f feren t ia l equat ions t ha t i s inso luble exce pt forthe m ost s imple cases [24, 26] . O the r a rgum ents against the use of micro-polar e las t ic ity wereput forward by Koi ter [1 ] who unfor tunate ly based h is conclusions on a ra ther specia l type ofmicro-polar e las t ic so l id , which may have b lur red a proper assessment .


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R. de Borst , A generalisa tion o f J , - f low the ory 349Recent years have again wi tnessed a pro l i fera t ion of in teres t in micro-polar so l ids , asev idenced in the contr ibu t ions of Bogd anova-Bontcheva and Lippm ann [29] and M iih lhausand Vardou lak is [30-32 , 37]. In these approache s the em phasis i s on micro-polar plasticityra the r th an on e las tici ty and i t has been s t ressed tha t micro-polar theor ies by the i r very na turein t roduce a charac ter is t ic length in the const i tu t ive descr ip t ion , thus render ing the govern ingse t o f equat ions to remain e l l ip t ic whi le a l lowing for loca l isa t ion of deformat ion in a nar row,

but f in i te band of mater ia l . In th is paper we shal l amongst o ther th ings show tha t thesefavo urab le prop ert ies a re prese rved also af ter discretisation of the continu um in to f initee lemen ts .Before enter ing the discussion how to implement Cosserat elasto-plast ici ty in a f ini tee lement contex t , inc lud ing aspects such as tempora l in tegra t ion of the const i tu t ive equat ionsand consis ten t l inear isa t ion of the so-der ived re turn-mapping a lgor i thm, a br ief d iscuss ion ofthe scope , the mer i ts and l imi ta t ions of the approach p ioneered here i s in order . F i r s t , thepr ime mot iva t ion of the presen t invest iga t ion was to const ruc t a model tha t does no t suf ferf rom pathologica l mesh dependence as encountered in convent ional s t ra in-sof ten ing modelsfor so i ls , concre te and rock , whi le be ing s t ra igh tforward ly ex tendib le to two and threed imensions . This goal has been reached comple te ly . A l imi ta t ion of the developed micro-polar , p ressure-dependent Jz- f low theory is tha t i t cannot rea l is t ica l ly model mode- I f rac turewhich is the prom inent fa i lu re m echanism in concre te and rock u nde r low confining pressures .For the la t te r c lass of p roblems, the inc lusion of h igher -order grad ien ts in the const i tu t ivem odel a ppe ars to be m ore ef fec tive [13] . M ore w ork is a lso needed on addi t ional exper im enta-t ion and ex t rac t ion of mater ia l p roper t ies f rom tes t da ta , an i ssue tha t i s on ly marg inal lyaddressed in th is invest iga t ion . For ins tance , in the examples a t the end of the paper thecharac ter is t ic length has been chosen ra ther a rb i t rar i ly .2. Cosserat elasticity

In the presen t t rea tment , we l imi t our a t ten t ion to two-d imensional , p lanar deformat ions .In tha t case , each mater ia l po in t in a micro-polar so l id has two t ransla t ional degrees-of -fre ed om , nam ely ux and uy and a rotat ional deg ree-of-freed om oJ~, the rotat ion axis of which isor thogonal to the x , y -p lane . As in a s tandard cont inuum the normal s t ra ins are def ined asOu~ 0uy (l a ,b )

e ~ = Ox a n d e ~ y = Oy "However , the shear strains are given a sl ightly modif ied form:

~ux Ouy (2a ,b)e ~ y = ~ + ~ o ~ a n d e y e = # x ~oz.I t i s observed tha t on ly for the choice

of the m acro- ro ta t ion , do ~xy and e . becom e equa l , thus ma in ta in ing sym m etry of the s t ra intensor . This rather special case of micro-polar elast ici ty has been the subject of much research,


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350 R. de Borst, A generalisation o f J2-flow theorybut has a lso given r ise to som e confusion . M indl in [24] has cons t ruc ted exact so lu t ions fors t ress concentra t ions around c i rcu lar ho les for th is theory , whi le Koi te r [1 ], on the basis o f th isspecial theory, has concluded that , for plate f lexure, micro-polar elast ici ty results in anincrease in r igid i ty ove r tha t p red ic ted by c lass ica l elas tic ity , tha t i s un l ike ly to have ' rem ain edunnot iced ' had micro-polar elas tic ity ef fec ts been s ign ifican t . Unfor tu nate ly , the im press ion issomet imes c rea ted tha t the theo ry o r ig ina ted by E . and F . C osse ra t embod ies a ssumpt ion (3 ) ,which is no t t rue , (cf . [22 , 27 , 28] ) . The or ig ina l theory o f E . and F . Cosse ra t does n o t req uire(3) , a l though the possib i l i ty of const ra in ing the theory by imposing (3) was no t iced by them.L a te r , E r ingen [22 ] nam ed th is spec ial ca se o f the Cosse rat theo ry the ' i nde te rmina tecouple-s t ress theory ' . In the remainder of th is t rea tment , we wi l l no t use (3) and we wi l ladhere to the or ig ina l fo rmula t ion of the Cossera ts for der iv ing the e las t ic par t o f the theory .In addi t ion to normal s t ra ins and shear s t ra ins , Cossera t theory a lso requires the in t ro-duct ion of m icro-curvatures

000: a~o~ (4 a ,b )K x ~ = O x and K~ y- - 0y "Ant ic ipa t ing the t rea tment for e las to-p las t ic i ty we wi l l ra ther use the genera l ised curvaturesg,xi and Kxyl, wh ere I is a ma ter ial p ara m ete r w ith the dimen sion of leng th. I t is this pa ram et erwhich ef fec tive ly se ts the in terna l length sca le in the cont inuum , an d the refore has the ro le ofa 'character ist ic length ' .The s t ra in components in t roduced so far may be assembled in a vec tor ,

e e , y , e + , + , , e , + , , , + ' ,,.+ + , < , , J , , < + , , l ] ' . ( S )Note tha t in addi t ion to the s t ra in components in t roduced in (1) , (2 ) and (4) , the normalstrain in the z.direction , e++, has also bee n include d in the strain vector e . T his has bee n d onebecause , a l though th is s t rain com pon ent remains zero und er p lane s t ra in condi tions dur ing theentire loading process, this is not necessar i ly the case for the elast ic and plast ic contr ibutionsof th is s t ra in com ponent . Also , the norma l s t ress o '~ ., which ac ts in the z-d i rec t ion , ma y benon-zero, which necessi tates inclusion of e~ and o-~ in the stress-strain relat ion. I t isfur thermo re no ted tha t by mul t ip ly ing the micro-curvatures ~++ and ~ by the len gthparameter l , a l l components o f the s t ra in vec tor have the same d imension .Let us now consider the s ta t ics o f a Cossera t cont inuum. While the s t ra in vec tor i scompr ised of seven components for p lanar deformat ions , so i s the s t ress vec tor o ' . As in aclassical continuum, we have the normal stresses o '~ , , , r~yan d o'~+, an d the sh ea r s tress es o-+y,%x (Fig . l ) . For the Cossera t cont inuum, we a lso have to in t roduce s t ress quant i t ies tha t a reconjuga te to the curva tures Kz+ and ~y . Figure 1 show s that the c o u p l e . s t r e s s e s m + ~ and m~yserve th is purpose . We observe tha t fo r th is cont inuum model , a couple vec tor ac ts on anelementary surface in addit ion to the familiar stress vector .Div id ing the couple-s t resses by the length param eter l , we obta in a s t ress vec tor z r in whichal l the en t r ies have the same d imension:

o . = [ o .,+ .,+ , # , , o .o .+ . , o . + , ,, % , + , m + + l + , m + + I t ] ' . ( 6 )Leaving aside body forces and body couples for the sake of simplici ty (see e.g . [33] for an


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R . de Borst, A generalisation o f 12-flow theory 35 1

%ymzy

o~

1

Fig. 1. Stress and couple-s t ress in a two-dimensional conf igurat ion.

ex ten s iv e t r ea tmen t ) , t r an s l a t i o n a l eq u i l i b r iu m in t h e x an d y - d i r ec t io n s r e su l t s i n0c% 0%~ O%yO'xx + = O , ' + = O , (7a ,b)Ox ay Ox i~y

r e sp ec t iv e ly , w h ich r ep l i ca t e s t h e r e su l t o b t a in ed f o r a c l a s s i ca l , n o n - p o la r co n t in u u m.H o w ev er , f o r r o t a t i o n a l eq u i l i b r iu m, w e f in d th a ti~m,,,O......~ 0m,,0y (o 'xy % x) = 0 , (8)

w h ich sh o w s th a t t h e s t r e s s t en so r i s in g en e r a l o n ly sy mm et r i c ( 0 % = % , , ) i f t h e co u p le -s t r e s ses m , x an d m z y v a n i s h ( B o l t z m a n n ' s A x i o m ) .A n t i c ip a t in g th e t r ea tmen t o f mic r o - p o la r p l a s t i c i t y i n t h e n ex t s ec t io n , w e d eco mp o se th es t r a in v ec to r i n to an e l a s t i c co n t r ib u t io n e ' an d a p l a s t i c p a r t eP : f f i e + e p , ( 9 )

w h i l e w e a s s u m e t h a t t h e e la s t i c s tr a i n v e c t o r i s l i n e a r l y r e l a t e d t o t h e s t r e ss v e c t o r ~r:= D e e ; ( t o )

D c i s t h e s t if f n ess m a t r ix t h a t co n ta in s t h e e l a s t i c mo d u l i :

D C __-2V , c t 2V, c 2 2/. c 0 0 0 0

2 / ~ C 2 2 ~ c t 2V, c 2 0 0 0 02V, c 2 2/~ c 2 2V, ct 0 0 0 00 0 0 / ~ + ~ / z - / z ~ 0 00 0 0 / ~ - ~ t ~ + / ~ 0 00 0 0 0 0 2V, 00 0 0 0 0 0 2 ~

g

( i t )


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3 5 2 R . d e B o r s t , g e n e ra l i sa t i on f J , - l l o w h e o r yw ith c t = ( 1 - v ) / ( 1 - 2 v ) a nd c 2 = u / ( 1 - 2 v ) . T h e e l a s t ic co n s t an t s / t an d v h av e th ec la ss ical m ean in g o f th e sh e a r m o d u lu s an d Po i s so n ' s r a t i o , r e s p e c t iv e ly . / t c i s an ad d i t i o n a lm a te r i a l co n s t an t , co m p le t in g th e t o t a l o f f o u r m a te r i a l co n s t an t s , v i z . / . t , v , I an d V.c t h a t a r en e e d e d t o d e s c r i b e t h e e l a s t i c b e h a v i o u r o f a n i s o t r o p i c C o s s e r a t c o n t i n u u m u n d e r p l a n a rd e f o r m a t io n s . T h e co e f f ic i en t 2 h as b een in t r o d u ced in t h e t e r m s D 6 6 an d D ~ 7 in o r d e r t oa r r iv e a t a co n v en ien t f o r m o f t h e e l a s to - p l a s t i c co n s t i t u t i v e eq u a t io n s . T h e to t a l ( b en d in g )s t if fn es s t h a t s e t s th e r e l a t i o n b e tw een th e m ic r o - cu r v a tu r es a n d th e co u p le - s t r e s ses i s b as i ca l lyd e t e r min ed b y th e v a lu e o f t h e i n t e r n a l l en g th s ca l e 1 .W e o b se r v e th a t a l l s t if fn es s mo d u l i t h a t en t e r t h e e l a st ic s t r e s s - s t r a in m a t r ix D ~ h av e th esame d imen s io n . T h i s is a t t r i b u t ab l e t o t h e f ac t t h a t a l l co m p o n e n t s o f t h e s t r a in v ec to r e an dth e s t r e ss v ec to r z r h av e th e s am e d im en s io n .3 . M i c r o - p o l a r e l a s to - p l a s ti c it y

I n t h e p r e s e n t t r e a t m e n t , w e e m p l o y a p r e s s u r e - d e p e n d e n t J 2 - f l o w t h e o r y ( D r u c k e r - P r a g e rmo d e l ) . A cco r d in g ly , t h e y i e ld f u n c t io n f can b e w r i t t en a s

f = (3 J2 ) 2 a p - # ( y ) , (12)w i t h ~ a f u n c t i o n o f t h e h a r d e n i n g p a r a m e t e r y and a a ( cons tan t ) f r ic t ion coef f ic ien t .p - ~ o'xx + ryy+ crzz) and ]2 is the sec ond inva r ian t o f the de v ia tor ic s t r esses , w hich , fo r amic r o - p o la r co n t in u u m , can b e g en e r ah sed a s [ 30 , 3 1]

, l 2 m a t sd /$ ~ / + a 2 5 o $ 1 ~ + a 3 r n ~ j r n J l 2 . ( 1 3 )I n ( 1 3 ) , t h e su mmat io n co n v en t io n w i th r e sp ec t t o r ep ea t ed in d i ces h as b een ad o p ted , s# i s t h ed ev ia to ri c s t re s s t en so r an d a t , a 2 an d a s a r e ma te r i a l p a r am ete r s . I n t h e ab sen c e o fcou ple-s tre sses , i .e . m 0 = 0, s~j = sj~ an d (13) red uc es to

J 2 - ( a t + a D s : o , ( 1 4 )w h i c h i m p l i e s t h a t t h e c o n s t r a i n t a~ + a 2 = m u s t b e e n f o r c e d s o t h a t t h e c l a s si c a l e x p r e s s i o nf o r J2 b e r e t r i ev ed p r o p e r ly .

F o r t h e c a s e o f p l a n a r d e f o rm a t i o n s , J 2 c a n b e e l a b o r a t e d a s2 2 2J2 -- t[s~x + syy 0 % + 2a 2 o ', ,y % x + a t o.y ,, + a s [ ( m v , l l ) 2 + ( m , y l l ) 2 ] .

Co mb in in g ( 1 2 ) an d ( 1 5 ) , an d in t r o d u c in g th e ma t r ix

p =

m 2 / 3 - I / 3 - 1 / 3 0 0 0 0 "- 1 / 3 2 / 3 - 1 / 3 0 0 0 0- 1 / 3 - 1 / 3 2 / 3 0 0 0 00 0 0 2 a t 2 % 0 00 0 0 2a 2a I 0 00 0 0 0 0 2~l3 0. 0 0 0 0 0 0 2 a 3 .

( 1 5 )

( 1 6 )


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R . d e Bors t , A gen eral isa t ion o f J . , -f low theory 35 3then leads to an appeal ing ly compact form of the y ie ld funct ion

f = [3 o " e e r ] u 2 + a z r ' m " - + ( 1 + ) , (17)wi th ~ r t= [1 /3 , 1 /3 , 1 /3 , 0 , 0 , 0 , 0] . A (non-associa ted) f low ru le i s now obta in ed in anident ica l fash ion to tha t in a non-polar cont inuum.

, p = , ( a ga t r ' ( 1 8 )g = [ + o . ' e o . ] + , s o - ' , . . -w h e r e (19)

is the p las t ic po ten t ia l funct ion and/3 is a d i la tancy fac tor . Not ing tha t dur ing p las t ic f lowf- - -0 , we obta in for the p las t ic s t ra in ra te[ 3Po" ]e P = A 2X/3"-~wtPw + f l , r . (20)

In (1 8 ) an d (20) , A is the plast ic mult iplier w hich, in analogy with classical plast icity , isde termined f rom the consis tency condi t ion f = 0 .I t now rema ins to iden t i fy the p las tic s t ra in m easure 3 ' ( the hardenin g param eter ) fo r aJ2-f low theory in a Cosserat medium. For this purpose we f irst recall the conventionals t ra in-hardening hypothesis ,3' = [ t - ,j- ,,j+ + P .+ + . P . I" 2 ( 2 1 )

wi th ~ j the p las t i c dev ia to r i c s t ra in - ra te tensor . Fo r "an iax ia l s t ress ing , q reduces to theun iax ia l p las t i c s t ra in ra te , ~ , = "p, ,, . S ince the re a re no coup le -s t ress e f fec ts i n un iax ia l l oad ing ,w e r e q u i r e t h a t a n y m o d i f i c a t i o n t o ( 2 1 ) f o r C o s s e r a t m e d i a d o e s n o t a f f e c t t h e r e s u l t f o r p u r eu n i a x i a l l o a d i n g . C o n s i d e r i n g t h i s p r e r e q u i s i te , a p o s s ib le g e n e r a l i s a t io n , a n a lo g o u s t o ( 1 3 ) , i st o p o s t u l a t e t h a t [ 3 0 , 3 1 ]

h , ~ P , ~ P + / ~ , : . P d . P 1 2 1 1 1 2 (22)~ , = [ b # ~ + ~ + " v ' , / ' J , " ' " O " ' r J 'with b t + b= = ~ in orde r th at def init ion (21) fo r the strain-hard ening hypothe sis in a non-pola rso l id can be re t r ieved .Fo r the case o f p lana r de fo rma t ions , 1 can be e labo ra ted a s

p . p b l ( ~ ; x ) 2~ , = [ t [ ( ~ x p x ) 2 + ( t ~ y p y ) 2 + ( ~ = p ) 2 ] + b t ( ~ x p y ) 2 + 2 b 2 f x y e y x + p 2 1 / 2+ b 3 [ ( ~ P z l ) 2 + (Ry= / ) ] ] .

In t roduc t ion o f the ma t r ixI 2 / 3 - 1 / 3 - 1 / 3 0 0 0 0- 1 / 3 2 / 3 - 1 / 3 0 0 0 0- 1 / 3 - 1 / 3 2 /3 0 0 0 0

0 0 0 3/ 2b 1 3 /2 b 2 0 00 0 0 3 /2 b 2 3/2 b 1 0 00 0 0 0 0 3 / 2b 3 0_ 0 0 0 0 0 0 3 /2b 3.

(23)

(24)


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354 R. de B orst , A generalisation o f J, .f low theorya l lo w s th e r a t e o f t h e h a r d en in g p a r amete r ~ t o b e w r i t t en in a s imi l a r f o r ma t a s t h e y i e ldf u n c t io n ,

. - - [ 2 ( ~ p ) t Q ~ p ] l / 2 , ( 2 5 )w i th t h e v ec to r ~ p a s semb l in g th e p l a s ti c s t r a in - ra t e co m p o n en t s .

W e n ex t i n t ro d u ce th e f lo w ru l e ( 2 0 ) i n ex p r es s io n ( 2 3 ) f o r t h e r a t e o f t h e h a r d e n in gp ar am ete r . S in ce ,~ an d 6 a r e a lw ay s n o n - n eg a t iv e , t h e r e su lt i s g iv en b y. /= A ( ~ t P Q P t F ~ 1 ,2

cr tpcr / ( 2 6 )s in ce Q ~ ' - O . I f t h e p a r am ete r s a , , a 2 , a 3 an d b ~ , b 2 , b 3 a r e ch o sen su ch th a t

P O P = P , ( 2 7 )( 2 5 ) r ed u ces t o ex ac t ly t h e s ame f o r ma t a s o b ta in ed in s t an d a r d J2 - f lo w th eo r y :

- A . ( 2 8 )I n t h e r ema in d e r o f t h i s p ap e r , w e a s su me th a t co n d i t i o n ( 2 7 ) i s s a t i s f i ed .Wi th t h e ab o v e d e f in i t i o n s , J~ - f lo w th eo r y can b e ca r r i ed o v e r t o a Co sse r a t med iu m in as t r a ig h t fo r w ar d an t e l eg an t f a sh io n . I n t eg r a t io n a lg o r i t h ms f o r th i s mo d e l w h ich a r e b ase d o nth e co n cep t o f r e tu r n map p in g a r e d i s cu s sed in t h e n ex t s ec t io n .

4 . A r e t u r n . m a p p i n g a l g o ri th mWi th th e g o v er n in g r a t e eq u a t io n s f o r t h e mic r o - p o la r e l a s to - p l a s t i c so l id a t h an d , w e canse t o u t t o d ev e lo p an a lg o r i t h m th a t d e t e r min es t h e s t r e s s i n c r emen t i n a f i n i t e l o ad in g s t ep .H e r e , a v a r i e ty o f a lg o ri t h ms ex i s t, b u t w e sh a ll o n ly co n s id e r tw o p o s s ib l e can d id a t e s , n am elyt h e E u l e r b a c k w a r d a l g o r it h m a n d a o n e -s t e p r e t u r n - m e p p i n g a l g o r it h m i n w h i c h th e g r a d i e n ti s evaluated a t the t r ia l s t r ess s ta te ,

o ' t = o ' 0 ~ " D e A e , ( 2 9 )O " b e in g th e s t r e s s a t t h e b eg in n in g o f t h e l o ad in g s t ep . Fo r t h e l a t t e r a lg o r i t h m, i t h as b eensh o w n in [3 4] t h a t , f o r ce r ta in c l a s ses o f t h e m a te r i a l p a r am ete r s , t h e r e ex i s ts a s e t o f v a lu esa t , o 2, a 3 and b t , b2 , b3 , w h ich en su r es a r e tu r n to t h e y i e ld su r f ace i n a s in g l e i t e r a t i o n . T h e sead v an tag eo u s p r o p e r t i e s a r e o b ta in e d f o r a t = a 2 = ~ , a 3 = , b t = b 2 ~, b 3 ~ . T h i s s e t o fco n s t an t s w i l l h en ce f o r th b e r e f e r r ed to a s t h e ' s t an d a r d ' s e t .

W h e n t h e s e c o n d i t i o n s a r e n o t s a t i s f i e d , a n E u l e r b a c k w a r d a l g o r i t h m a s d e l i n e a t e d b e l o wsh o u ld b e emp lo y ed . I n t h i s a lg o r i t h m, t b e t o t a l s t r a in i n c r emen t i n a f i n i t e l o ad in g s t ep A s i sd eco m p o sed in to an e l a st ic co n t r ib u t io n A e ~ an d a p l a s t i c p a r t A ~ P ,A e = A ~ ~ + A e r . ( 3 0 )


9/16

R. de Borst, A generalisalion of ~-flow theory 355Be twe en the s t ress increm ent Ao- and the e l as t i c s tra in increm ent Ae~, we have the b i j ect ivere la t ionship

Ao" = D e A e e . (31)Fur thermore , the express ion for the p las t i c s t ra in ra t e (20) i s in t egra ted us ing a s ingle -pointEu l e r ba c kwa r d r u l e . Th i s r e su l t s i n

[ 3 , ~ . ]A e p - - A A - ~ + ~ . , ( 3 2)2~/~ O'nPO""wh ere the su bscr ip t n re fe rs to the va lue a t the end of the load ing s t ep . B y def in it ion o- . i sg iven by

o ' . = o" + A o ' , (33)so t ha t c ombi na t i on o f ( 29 ) - ( 33) r e su l t s i n

o ' . f f i ( r , - A A 2 ~ / i w ' .e o ' . + f l D ' z r . (34)We next use the condi t ion tha t a t the end of the loading s t ep the y ie ld condi t ion must besa ti sf ie d : f ( ~ . , % ) ff i 0 . The n , ( 34 ) t r a ns f o r ms i n t o

. 3 D P o . no ' . -- o', - ~ A 2 [ ~ ( ~ - ~ _ ~ - r , cr . l + ~ D ~ r . ( 3 5 )A com pl ica t ion now ar i ses , s ince we wish to express n as a func t ion of o ', and AA, whi le in( 35) o ' , a l so oc c ur s i n t he de nomi na t o r o f t he se c ond t e r m on t he r i gh t - ha nd s i de . The r e f o r e ,we express ~ '*o ', as a func t ion of ~r*o" and AA by prem ul t ip ly ing (35) b y the pro jec t ion vec tor~ ' . Thi s g ives

lr*(r. = ~rto", - A A ~ K , ( 36)wi th K ffi zrtDezr. In the case o f i sot ropic e last ic i ty, K is the b ulk m od ulus . Sub st i tut ion of thisident i ty in to (35) resul t s in the des i red formula t ion:

o ' . = A - I [ o ", - A A f l D ' ~ ] , ( 3 7)w h e r e

A = i + 3AAD~P (38)2 [ ~ ( 7 . ) + A A a [ J K - a 'o ', ] "Subs t i t u t i on i n t he y i e l d c ond i t i on f ( o ' . , %) = 0 t hen resul t s in a non- l inear equa t ion in ~A,


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3 56 R . d e B o r s t , g e n er a l i sa t i o n f J z - f l o w h e o r yf ( A A ) - ( ~ ( ~ , , - A / 3 O e ~ )~ t - ' e A -' ( o - A A ~ O ~ ) ) ''2 , ~ - 1 ( o - , - A A ~ O = ) - ,~

= 0 , ( 3 9 )which is solved using a Regula-Falsi method. Normally, convergence is achieved within 4-5iterations.

$. Consistent l lnearisat ion modul iFo r the derivation of a prop erly linearised set of tangential mod uli, we w ill restrict ourselvesto the form at o f a pressure-depe ndent ,/z micro-polar plasticity theory that obeys the constraint(27). Differentiating (34) then yields

~ - u ~ - , ~ ~-~ ,where ag ~ Po"~ - ~ + O ~a n d

. /'~ c r tP~ rP- Po 'o : PH - ' = [ 0 " I ' + A ^ ~ j ~ ~ ~ ~ .

Since f = f (~r , y ) , the cons i s tency condi t ion f ,~ 0 can be e laborated as

I n t r o d u c i n g t h e h a r d e n i n g m o d u l u s0 5

h ( ~ , ) = 0~,a n d u s i n g t h e y i e l d c o n d i t i o n 1 2 ) , w e o b t a i n

Equationrelation

( 4 0 )

( 4 1 )

( 4 2 )

( 4 3 )

( 4 4 )

T ~ b - h , ~ = 0 . ( 4 S )(45) can be combined with (40) to give the explicit consistent tangential stiffness


11/16

R. de B orst, A generalisa tion o f J2"flow th eo l . 3576 . E x a m p l e s : s h e a r l a y e r a n d b i a x i a l t e st

Th e m ode l d i scus s ed i n the p reced i ng s ec ti on has been i m p l em en t ed i n a 6 -node d t r iangu l a rp l a n e s t r a i n e l e m e n t . T h i s e l e m e n t h a s 1 8 d e g r e e s -o f - f re e d o m d u e t o t h e f a c t t h a t e a c h n o d eh a s t h r e e d e g r e e s - o f -f r e e d o m , t w o t r a n sl a ti o n s a n d o n e r o t a t io n .To i l lu s t r a t e t he e f f ec ti venes s o f the C os s e ra t m ode l i n p red i c t i ng phys i ca ll y r ea li s ti cs o l u t i ons fo r m ode- I I f a i l u re p rob l em s , t he s hea r l aye r o f F i g . 2 has been ana l y s ed . I t i sa s s um ed t h a t t he s hea r l aye r i s i n fi n it e ly l ong i n bo t h t he nega t i ve an d t he pos i ti ve x -d i r ec t ion .Th e d i s c re t i s a ti on o f t he s hea r l aye r , wh i ch has a he i gh t H = 100 m m , i s s hown i n F i g . 2 fo rt h e c a s e o f 2 0 e l e m e n t s .B as i ca l l y , t he p rob l em i s one -d i m ens i ona l and cou l d a l s o have been ana l y s ed u s i ng l i nee l e m e n t s . U s e o f t w o - d i m e n s io n a l e le m e n t s r e q u i r e s t h a t l i n e a r c o n s tr a i n t e q u a t i o n s a r e a d d e dt o t he s e t o f a l geb ra i c equa t i ons wh i ch r e s u l t a f t e r d i s c re t i sa t ion o f t he con t i nuum . T hey havet o be app l i ed t o t he d i s p l acem en t s i n t he x -d i r ec t i on a s we l l a s t o t he ro t a t i ona l deg rees -o f -f r eedom , s i nce a l l d i s p l acem en t s i n t he y -d i r ec t i on a re p reven t ed ( i s ocho r i c m o t i on ) . Thebo t t om o f t he s hea r l aye r is fi xed (u x = 0 , uy = 0 ) an d t he up pe r bou nda ry i s s ub j ec t ed t o as hea r fo rce o r ,y (pe r un i t a r ea ) , w h i ch has bee n co n t ro l led u s i ng a s t anda rd a rc - leng t ht echn i que [6 , 7 , 35 ]. The add i t i ona l bound ary cond i t i on o~ = 0 has been e n fo rced a t t he l owera n d u p p e r b o u n d a r i e s .T h e s t a n d a r d e l as ti c m o d u l i h a v e b e e n c h o s e n a s s h e a r m o d u l u s / z = 4 0 0 0 M P a a n dPo i s s on ' s r a t i o v - 0 .25 . Th e i n it ia l y i e ld s t r eng t h has bee n t ak en equa l t o 6 = 100 M Pa , w i t hno f r i c ti on o r d i l a t ancy ( a = / 3 = 0 ) , wh i le a l i nea r s o f ten i ng d i ag ram has been u s ed w i th ah a r d e n i n g ( s o f t e n i n g ) m o d u l u s h - - - 0 . 1 2 5 / ~ . F o r t h e C o s s e r a t c o n t i n u u m t h e a d d i t i o n a lm a t e r i a l c ons t an t s /~c = 2000 M Pa and l - 12 m m have been i n s e r t ed . F i r s tl y , t he s t anda rdvalues for a l , a2 , a3 and b 1 , b 2 , b 3 have bee n u t i li sed .

I n c o n t r a s t t o a s t a n d a r d c o n t i n u u m , a h o m o g e n e o u s s t r a in s t a t e is n o t o b t a i n e d u n d e r p u r es hea r l oad i ng fo r a C os s e ra t con t i nuum , a t l ea s t no t w i t h t he e s s en t i a l boundary cond i t i onsl i st ed above . A l read y i n t he e l a s ti c reg i m e a bou nda ry l aye r w i t h a he i gh t t ha t i s p ropo r t i ona lt o I deve l ops a t t he u ppe r an d l ower pa r t s o f the s hea r l aye r . As a cons eq uence t he l oca l i sa t iond e v e l o p s s m o o t h l y a n d g r a d u a l l y i n t h e m i d d l e o f t h e s h e a r l a y e r w i t h o u t t h e n e e d f o ri n t roduc i ng i m per fec t i ons o r fo r add i ng a pa r t o f the e i genvec t o r t o t he hom ogen eous s o l u t i on[6, 7 , 36].W hen t he d i s c re t i s a ti on is re f i ned t o s uch an ex t en t t ha t m ore t h an one s e t o f two t r i angu l a re l em en t s a r e p l aced ove r t he l oca l i s a t i on zone , t he w i d t h o f t he num er i ca l l y p red i c t edl oca l is a t ion zone becom es cons t an t (F i g . 3 ) . F i gu re 4 (a ) s hows al s o t ha t t he l oad - d i s p l ac em en tcurv e co nve rges to a phy s ica l ly rea l i st i c so lu t ion .

FE-model

o~

Fig. 2. Infinitely long shear layer: applied loading and finite element lay-out.


12/16

358 R. de Bo rn, A general isation of ~- f lo w theory

/

( e ) ( b )

/

( a ) ( d )Fig. 3, Incre m ental displacem ent pa ttern s at a residual load lev el of ~r,y/# ffi 0.28. (a) tO lemnts; (b) 20 elem ent s;( ) 100 e lemen ts ; (d) 200 e lements.

As a l luded to in the in t roduct ion , a major ques t ion tha t needs to be add ressed whenin t roducing h igher -order cont inuum models i s the de terminat ion of the addi t ional mater ia lparameters . The ro le of the in terna l length sca le I i s ra ther obvious , s ince i t governs thebri t t leness and the width of the Iocalisat ion zone. A smaller value for I implies a smaller widthof the loca l isa t ion zone and a s teeper post -peak response as i s shown in F ig . 4 (b) . Theinf luence of the ra t io a I :a 2 i s shown in F ig . 4 (c). A pa r t f ~ m the s tanda rd se t (a I = a 2 = ~) ,the so-called kinematic model of Miihlhaus and Vardoulakls [30, 31, 37] (as f f i ~ , a2 = ~) andthe stat ic m ode l of M iihlhaus and Va rdoulakis [30, 31, 37] (a ! = ~, a 2 ffi - ~ ) hav e b ee n used .I t appears that the specif ic choice of these parameters has t i t t le inf luence on the results in thef ir st par t o f the post -peak reg ime , bu t tha t beyon d som e cri tica l po in t in the post -peak reg im e,the gradual evolu tion of the loca l isat ion zone as observed for the s tan dard m odel b reaks dow nfor the k inem at ic model and a t an ev en ear l ie r s tage for the s ta t ic m odel . Bey ond these po in ts ,the k inemat ic and s ta t ic models reac t in a very br i t t le manner . F ina l ly , F ig , 4 (d) shows theinf luence of the sof ten ing modulus on the s t ruc tura l hehaviour .


13/16

R. de Borst, A generalisationof .12-flowtheory 359

0.6

u,lH

O.b4 O.b8 0u,lH

(4

o.b4 O.b8 0.u,lH

i2

2

Fig. 4. L~arkiisplacement iagramor shear layer. Results for a Cosseratcontinuum with strain softening. (a) Theeffect of mesh refinement; (b) the role of the characteristic ength I; (c) influence of the parameter et (a,, a,, aJ);(d) influenceof the stPftening odulush.

To demonstrate the effectiveness of the Cosserat continuum also in two-dimensionalboundary value problems where frictional sliding is the prevailing failure mechanism, aplane-strain biaxial test has been simulated. The specimen that has been considered has awidth B = 60 mm and a height H = 80mm. Smooth boundary conditions (u,, = 0) have beenand lower boundaries and natural boundary conditions have beenssumed at the uppera -lFIB- 1 I

Fig. 5. Load-displacement curves for a plane-strain biaxial test (Drucker-Prager yield function).


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360 R. de Bo rst, A generalisation o f Jz-flow theory

(a ) (b )F i g . 6 . In c r e m e n t a l d is p la c e m e n t s f o r a p l a n e -s tr a in b i a x i a l t e st ( D r u c k e r - P r a g e r p l a s t i c it y ) . ( a ) M e d i u m m e s hwith 432 elemen ts; (b) Fine m esh with 1728 elements.assumed a t a l l s ides for the ro ta t ions (% f ree) . A Drucker -Prager y ie ld condi t ion wi th anon-associated f low rule was employed. The mater ial data were as follows: shear modulus

~ 1 0 0 0 M P a , P o is s on 's r a ti o v = 0 . 2 , ~ = 5 0 0 M P a , ! ~ 6 r a m , a ~ 1.2 , / ~ - 0 . 0 a n d # =1.2V'~(1- 25y) , For the coef f ic ien ts a~ , a z and a~ the ' s tandard ' va lues have been analysed .Three d i f feren t m eshes have been ad opted , wi th t08 , 432 and 1728 s ix-noded t r iangularelements, respectively. The load-displacement curves for al l three discretisat ions are indist ing-uishable as is shown in Fig. 5 . This f igure also shows the solution that is obtained under theassumpt ion of homogeneous deformat ions (no loca l isa t ion) .Because in pure compress ion ro ta t ional degrees-of - f reedom are no t ac t iva ted , an imperfec te lement (5% reduct ion in the in i t ia l y ie ld s t rength) has been inser ted in the model a t the le f tboundary nea r the hor izonta l cen t re l ine. Fro m th is po in t , two shear ba nds in it ia l ly prop aga te ,bu t la te r on ly one band persis ts . Th e increm enta l d isp lacements a t th is s tage are shown in F ig .6 for the medium (432 e lements) and the f ine mesh (1728 e lements) , respect ive ly .7 . C o nc lud ing rem a rks

Micro-polar plast ici ty does n o t share the d isadvantages tha t a d h e r e t o some o the r non -classical plast ici ty approaches. In part icular , the concept of return-mapping algori thms can beappl ied s t ra igh tforward ly and in a po in twise fash ion . There i s no need to res ta te theconsistency condit ion in a format that involves distr ibution functions, as is the case fornon- local approaches [38] . This observat ion grea t ly s impl i f ies the programming ef for t .


15/16

R. de Bo rst , A generalisation o f J. ,-f low tl leory 361A n o t h e r i m p o r t a n t a d v a n t a g e is t h e f a c t t h a t t h e m a j o r s y m m e t r y in t h e t a n g e n t i a l st if fn e s sm a t r i x r e m a i n s p r e s e r v e d w h e n a s s o c i a t e d p l a s ti c it y is u s e d . A g a i n , t h i s r e s u l t c o n t r a s t s w i t hn o n - l o c a l p l a s t i c i t y / d a m a g e t h e o r i e s w i t h a l o c a l s t r a in f ie l d , w h e r e t h is p r o p e r t y i s l o s t [ 1 9,20] .

P r e v i o u s r e s e a r c h o n c o n v e n t i o n a l s t r ai n - s o ft e n i n g m o d e l s [ 3 5 , 3 9 , 4 0 ] h a s s h o w n t h a t , w h e nf i n i t e e l e m e n t s a r e u s e d f o r s u c h m a t e r i a l m o d e l s , t h e s o l u t i o n t e n d s t o b e c o m e u n s t a b l e a ts o m e s t a g e o f t h e l o a d i n g p r o ce s s . S o m e e l e m e n t s o r g r o u p s o f e l e m e n t s t e n d t o sh o wd e f o r m a t i o n p a t t e r n s t h a t a r e t o t a l l y u n r e a l i s t i c a n d p h y s i c a l l y u n a c c e p t a b l e . T h e m i c r o - p o l a ra p p r o a c h , w h i c h b a s i c a l l y e n r i c h e s t h e e l e m e n t f o r m u l a t i o n w i t h h i g h e r - o r d e r g r a d i e n t s ,a p p e a r s t o s t a b il is e t h e e l e m e n t b e h a v i o u r w h e n s t r a i n - s o f t e n i n g t y p e c o n s t it u t iv e l a w s a r eu s e d .

A c k n o w l e d g m e n tT h e c a l c u la t io n s h a v e b e e n c a r r i e d o u t w i t h th e D I A N A f in it e e l e m e n t c o d e o f T N O

B u i l d i n g a n d C o n s t r u c ti o n R e s e a r c h . D i s c u s s io n s w it h D r . H a n s - B e r n d M i i h l h a u s o f t h eC S I R O D i v is io n o f G e o m e c h a n i c s a n d w it h P r o f e s so r H o w a r d L . S c h r e y e r o f T h e U n i v e rs i tyo f N e w M e x i c o h a v e p r o v e n t o b e v e r y v a lu a b l e t o o b t a i n m o r e i ns ig h t in n o n -c l as s ic a lc o n t i n u a .

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