Critical state plasticity. Part VI: Meso-scale finite element simulation of strain localization in discrete granular materials

Transcription

1 Comut. Method Al. Mech. Engrg. 195 (006) Critical tate laticity. Part VI: Meo-cale finite element imulation of train localization in dicrete granular material Ronaldo I. Borja *,1, Joé E. Andrade Deartment of Civil and Environmental Engineering, Stanford Univerity, Stanford, CA 9405, USA Received 1 March 005; received in revied form 0 Augut 005; acceted 1 Augut 005 Abtract Develoment of more accurate mathematical model of dicrete granular material behavior require a fundamental undertanding of deformation and train localization henomena. Thi aer utilize a meo-cale finite element modeling aroach to obtain an accurate and thorough cature of deformation and train localization rocee in dicrete granular material uch a and. We emloy critical tate theory and imlement an elatolatic contitutive model for granular material, a variant of a model called Nor-Sand, allowing for non-aociative latic flow and formulating it in the finite deformation regime. Unlike the reviou verion of critical tate laticity model reented in a erie of Cam-Clay aer, the reent model contain an additional tate arameter w that allow for a deviation or detachment of the yield urface from the critical tate line. Deending on the ign of thi tate arameter, the model can reroduce latic comaction a well a latic dilation in either looe or dene granular material. Through numerical examle we demontrate how a tructured atial denity variation affect the redicted train localization attern in dene and ecimen. Ó 005 Elevier B.V. All right reerved. Keyword: Granular material; Strain localization 1. Introduction Develoment of accurate mathematical model of dicrete granular material behavior require a fundamental undertanding of the localization henomena, uch a the formation of hear band in dene and. For thi reaon, much exerimental work ha been conducted to gain a better undertanding of the localization roce in thee material [1 11]. The ubject alo ha urred coniderable interet in the theoretical and comutational modeling field [1 9]. It i imortant to recognize that the material reone oberved in the laboratory i a reult of many different micro-mechanical rocee, uch a mineral article rolling and liding in granular oil, micro-cracking in brittle rock, and mineral article rotation and tranlation in the cement matrix of oft rock. Ideally, any localization model for geomaterial mut rereent all of thee rocee. However, current limitation of exerimental and mathematical modeling technique in caturing the evolution in the micro-cale throughout teting have inhibited the ue of a micro-mechanical decrition of the localized deformation behavior. To circumvent the roblem aociated with the micro-mechanical modeling aroach, a macro-mechanical aroach i often ued. For oil, thi aroach ertain to the ecimen being conidered a a macro-cale element from which the material reone may be inferred. The underlying aumtion i that the ecimen i reared uniformly and deformed * Correonding author. Fax: addre: borja@tanford.edu (R.I. Borja). 1 Suorted by US National Science Foundation, Grant No. CMS and CMS /$ - ee front matter Ó 005 Elevier B.V. All right reerved. doi: /j.cma

2 5116 R.I. Borja, J.E. Andrade / Comut. Method Al. Mech. Engrg. 195 (006) homogeneouly enough to allow extraction of the material reone from the ecimen reone. However, it i well known that each ecimen i unique, and that two identically reared amle could exhibit different mechanical reone in the regime of intability even if they had been ubjected to the ame initially homogeneou deformation field. Thi imlie that the ize of a ecimen i too large to accurately reolve the macro-cale field, and that it can only cature the train localization henomena in a very aroximate way. In thi aer, we adot a more refined aroach to invetigating train localization henomena baed on a meo-cale decrition of the granular material behavior. A a matter of terminology, the term meo-cale i ued in thi aer to refer to a cale larger than the grain cale (article cale) but maller than the element, or ecimen, cale (macro-cale). Thi aroach i motivated rimarily by the current advance in laboratory teting caabilitie that allow accurate meaurement of material imerfection in the ecimen, uch a X-ray comuted tomograhy (CT) and digital image roceing (DIP) in granular oil [1,4,8,9,9]. For examle, Fig. 1 how the reult of a CT can on a biaxial ecimen of ure ilica and having a mean grain diameter of 0.5 mm and reared via air luviation. The gray level variation in the image indicate difference in the meo-cale local denity, with lighter color indicating region of higher denity (the large white ot in the lower level of the ecimen i a iece of gravel). Thi advanced technology in laboratory teting, combined with DIP to quantitatively tranfer the CT reult a inut into a numerical model, enhance an accurate meo-cale decrition of granular material behavior and motivate the develoment of robut meo-cale modeling aroache for relicating the hear banding rocee in dicrete granular material. The modeling aroach urued in thi aer utilize nonlinear continuum mechanic and the finite element method, in combination with a contitutive model baed on critical tate laticity that cature both hardening and oftening reone deending on the tate of the material at yield. The firt laticity model exhibiting uch feature that come to mind i the claical modified Cam-Clay [4,0 6]. However, thi model may not be robut enough to reroduce the hear banding rocee, articularly in and, ince it wa originally develoed to reroduce the hardening reone of oil on the wet ide of the critical tate line, and not the dilative reone on the dry ide where thi model oorly relicate the oftening behavior neceary to trigger train localization. To model the train localization roce more accurately, we ue an alternative critical tate formulation that contain an additional contitutive variable, namely, the tate arameter w [7 9]. Thi arameter determine whether the tate oint lie below or above the critical tate line, a well a enable a comlete detachment of the yield urface from thi line. By detachment we mean that the initial oition of the critical tate line and the tate of tre alone do not determine the denity of the material. Intead, one need to ecify the atial variation of void ratio (or ecific volume) in addition to the tate arameter required by the claical Cam-Clay model. Through the tate arameter w we can now recribe quantitatively any meaured ecimen imerfection in the form of initial atial denity variation. Secifically, we ue claical laticity theory along with a variant of Nor-Sand model rooed by Jefferie [8] to decribe the contitutive law at the meo-cale level. The main difference between thi and the claical Cam-Clay model lie in the decrition of the evolution of the latic modulu. In claical Cam-Clay model the character of the latic modulu deend on the ign of the latic volumetric train increment (determined from the flow rule), i.e., it i oitive under comaction (hardening), negative under dilation (oftening), and i zero at critical tate (erfect laticity). In andy oil thi may not be an accurate rereentation of hardening/oftening reone ince a dene and could exhibit an initially contractive behavior, followed by a dilative behavior, when heared. Thi imortant feature, called hae tranformation in the literature [40,41], cannot be reroduced by claical Cam-Clay model. In the reent formulation the growth or collae of the yield urface i determined by the deviatoric comonent of the latic train increment and by the oition of Fig. 1. Cro-ection through a biaxial tet ecimen of ilica and analyzed by X-ray comuted tomograhy; white ot i a iece of gravel.

3 the tre oint relative to a o-called limit hardening dilatancy. Such decrition reroduce more accurately the oftening reone on the dry ide of the critical tate line. The theoretical and comutational aect of thi aer include the mathematical analye of the thermodynamic of contitutive model characterized by elatolatic couling [4,4]. We alo decribe the numerical imlementation of the finite deformation verion of the model, the imact of B-bar integration near the critical tate, and the localization of deformation on the dry ide of the critical tate line. We reent two numerical examle demontrating the localization of deformation in lane train and full D loading condition, highlighting in both cae the imortant role that the atial denity variation lay on the mechanical reone of dene granular material. Notation and ymbol ued in thi aer are a follow: bold-faced letter denote tenor and vector; the ymbol Æ denote an inner roduct of two vector (e.g. a Æ b = a i b i ), or a ingle contraction of adjacent indice of two tenor (e.g. c Æ d = c ij d jk ); the ymbol : denote an inner roduct of two econd-order tenor (e.g. c : d = c ij d ij ), or a double contraction of adjacent indice of tenor of rank two and higher (e.g. C : e ¼ C ijkl e kl ); the ymbol denote a juxtaoition, e.g., (a b) ij = a i b j. Finally, for any ymmetric econd-order tenor a and b, (a b) ijkl = a ij b kl, (a b) ijkl = a jl b ik, and (a b) ijkl = a il b jk.. Formulation of the infiniteimal model We begin by reenting the general feature of the meo-cale contitutive model in the infiniteimal regime. Extenion of the feature to the finite deformation regime i then reented in the next ection..1. Hyerelatic reone R.I. Borja, J.E. Andrade / Comut. Method Al. Mech. Engrg. 195 (006) We conider a tored energy denity function W e ( e ) in a granular aembly taken a a continuum; the macrocoic tre r i given by r ¼ owe o ; e where ð:1þ and W e ¼ ew e ð e v Þþ le e ; ewð e v Þ¼ 0 ~j ex x; x ð:þ ¼ e v e v0 ; l e ¼ l ~j 0 þ a 0 Wð ~j e e vþ. ð:þ The indeendent variable are the infiniteimal macrocoic volumetric and deviatoric train invariant rffiffi e v ¼ trðe Þ; e ¼ ke e k; e e ¼ e 1 e v1; ð:4þ where e i the elatic comonent of the infiniteimal macrocoic train tenor. The material arameter required for definition are the reference train e v0 and reference reure 0 of the elatic comreion curve, a well a the elatic comreibility index ~j. The above model roduce reure-deendent elatic bulk and hear moduli, in accord with a well-known oil behavioral feature. Eq. (.) reult in a contant elatic hear modulu l e = l 0 when a 0 = 0. Thi model i conervative in the ene that no energy i generated or lot in a cloed elatic loading loo [44]... Yield urface, latic otential function, and flow rule We conider the firt two tre invariant ¼ 1 rffiffi trr; q ¼ kk; ¼ r 1; ð:5þ where 6 0 in general. We define a yield function F of the form F ¼ q þ g; ð:6þ where ( g ¼ M½1 þ lnð i=þš if N ¼ 0; ð:7þ ðm=nþ½1 ð1 NÞð= i Þ N=ð1 NÞ Š if N > 0.

4 5118 R.I. Borja, J.E. Andrade / Comut. Method Al. Mech. Engrg. 195 (006) Fig.. Comarion of hae of critical tate yield urface. Here, i < 0 i called the image tre rereenting the ize of the yield urface, defined uch that the tre ratio g = q/ = M when = i. A cloed-form exreion for i i ( i exðg=m 1Þ if N ¼ 0; ¼ ð:8þ ½ð1 NÞ=ð1 gn=mþš ð1 NÞ=N if N > 0. The arameter N P 0 determine the curvature of the yield urface on the hydrotatic axi and tyically ha a value le than 0.4 for and [8]; an increae, the curvature increae. Fig. how yield urface for different value of N. For comarion, a lot of the conventional ellitical yield urface ued in modified Cam-Clay laticity theory i alo hown [0]. Next we conider a latic otential function of the form Q ¼ q þ g; ð:9þ where ( g ¼ M½1 þ lnð i=þš if N ¼ 0; ð:10þ ðm=nþ½1 ð1 NÞð= i Þ N=ð1 NÞ Š if N > 0. The latic flow i aociative if N ¼ N and i ¼ i, and non-aociative otherwie. For the latter cae, we aume that N 6 N reulting in a latic otential function that i flatter than the yield urface (if N < N), a hown in Fig.. Thi effectively yield a maller dilatancy angle than i redicted by the aumtion of aociative normality, imilar in idea to the thermodynamic retriction that the dilatancy angle mut be at mot equal to the friction angle in Mohr Coulomb or Drucker Prager material, ee [45,46] for further elaboration. The variable i i a free arameter that determine the final ize of the latic otential function. If we et Q = 0 whenever the tre oint (,q) lie on the yield urface, then i can be determined a ( i exðg=m 1Þ if N ¼ 0; ¼ ð:11þ ½ð1 NÞ=ð1 gn=mþš ð1 NÞ=N if N > 0. Fig.. Yield function and family of latic otential urface.

5 The flow rule then write _ ¼ kq; _ q :¼ oq or ; ð:1þ where k _ P 0 i a non-negative latic multilier, and q ¼ oq o þ g or or þ og rffiffi or ¼ 1 M g 1 þ ; 1 N kk N P 0. ð:1þ In thi cae, the variable i doe not have to enter into the formulation ince g can be determined directly from the relation g ¼ g. The firt two invariant of _ are _ v ¼ tr _ ¼ k _ M g rffiffi ; _ 1 N k_e k¼ k; _e ¼ _ 1 _ v1; ð:14þ where N P 0. Note that _ v > 0 (dilation) whenever g > M, and _ v < 0 (comaction) whenever g < M. Platic flow i urely iochoric when _ v ¼ 0, which occur when g = M. Furthermore, note that f :¼ of or ¼ oq or þ g o or þ og or ¼ 1 M g 1 N rffiffiffi 1 þ ; N P 0. ð:15þ kk Since trf = 0 whenever g = M, then latic flow i alway aociative at thi tre tate regardle of the value of N and N. Non-aociative latic flow i oible only in the volumetric ene for thi two-invariant model. For erfect laticity the reduced diiation inequality require the tree to erform non-negative latic incremental work [47], i.e., D ¼ r : _ ¼ kr _ : q P 0. ð:16þ Uing the tre tenor decomoition r = + 1 and ubtituting relation (.1) into (.16), we obtain D ¼ k _ M g M g gþ P 0 ) g þ 1 N 1 N P 0 ð:17þ ince 6 0. Now, if the tre oint i on the yield urface then (.7) determine the tre ratio g, and (.17) thu become " MN # N=ð1 NÞ N 1 ð1 NÞ þ M P 0. ð:18þ i However, M > 0 ince thi i a hyical arameter, and o we get " # N=ð1 NÞ 1 N 6 N 1 ð1 NÞ. ð:19þ i The exreion inide the air of bracket i equal to unity at the tre ace origin when = 0, reduce to N at the image tre oint when g = M and = i, and i zero on the hydrotatic axi when g = 0 and = i /(1 N) (1 N)/N. The correonding invere are equal to unity, 1/N > 1, and oitive infinity, reectively. Hence, for (.19) to remain true at all time, we mut have N 6 N; a otulated earlier... State arameter and latic dilatancy R.I. Borja, J.E. Andrade / Comut. Method Al. Mech. Engrg. 195 (006) In claical Cam-Clay model the image tre i coincide with a oint on the critical tate line (CSL), a locu of oint characterized by iochoric latic flow in the ace defined by the tre invariant and q and by the ecific volume v. The CSL i given by the air of equation q c ¼ M c ; v c ¼ v c0 ~ k lnð c Þ; ð:1þ where ubcrit c denote that the oint (v c, c,q c ) i on the CSL. The arameter are the comreibility index ~ k and the reference ecific volume v c0. Thu, any given oint on the yield urface ha an aociated ecific volume, and iochoric latic flow can only take lace on the CSL. ð:0þ

6 510 R.I. Borja, J.E. Andrade / Comut. Method Al. Mech. Engrg. 195 (006) To aly the model to and, which exhibit different tye of volumetric yielding deending on initial denity, the yield urface i detached from the critical tate line along the v-axi. Thu, the tate oint (v,,q) may now lie either above or below the critical ecific volume v c at the ame tre deending on whether the and i looer or dener than critical. Following the notation of [8], a tate arameter w i introduced to denote the relative ditance along the v-axi of the current tate oint to a oint v c on the CSL at the ame, w ¼ v v c. ð:þ Further, a tate arameter w i i introduced denoting the ditance of the ame current tate oint to v c,i on the CSL at = i, w i ¼ v v c;i ; v c;i ¼ v c0 ~ k lnð i Þ; ð:þ where v c,i i the value of v c at the tre i, and v c0 i the reference value of v c when i = 1, ee (.1). The relation between w and w i i (ee Fig. 4) w i ¼ w þ ~ k ln i. ð:4þ Hence, w i negative below the CSL and oitive above it. An uhot of diconnecting the yield urface from the CSL i that it i no longer oible to locate a tate oint on the yield urface by recribing and q alone; one alo need to ecify the tate arameter w to comletely decribe the tate of a oint. Furthermore, iochoric latic flow doe not anymore occur only on the CSL but could alo take lace at the image tre oint. Finally, the arameter w i dictate the amount of latic dilatancy in the cae of dene and. Formally, latic dilatancy i defined by the exreion D :¼ _ v =_ ¼ g M 1 N. ð:5þ Thi definition i valid for all oible value of g, even for g = 0 where Q i not a mooth function. However, exerimental evidence on a variety of and ugget that there exit a maximum oible latic dilatancy, D *, which limit a latic hardening reone. The value of D * deend on the tate arameter w i, increaing in value a the tate oint lie farther and farther away from the CSL on the dene ide. An emirical correlation ha been etablihed exerimentally in [8] between the latic dilatancy D * and the tate arameter w i, and take the form D ¼ aw i ; ð:6þ where a.5 tyically for mot and. The correonding value of tre ratio at thi limit hardening dilatancy i g ¼ M þ D ð1 NÞ ¼M þ aw i ð1 NÞ ¼M þ aw i ð1 NÞ; ð:7þ and the correonding ize of the yield urface i ( i ¼ exðaw i=mþ if N ¼ N ¼ 0; ð:8þ ð1 aw i N=MÞ ðn 1Þ=N if 0 6 N 6 N 6¼ 0; where ab ¼ a; b ¼ 1 N 1 N. ð:9þ In the above exreion we have introduced a non-aociativity arameter b 6 1, where b = 1 in the aociative cae. Fig. 4. Geometric rereentation of tate arameter w.

7 R.I. Borja, J.E. Andrade / Comut. Method Al. Mech. Engrg. 195 (006) Conitency condition and hardening law For elatolatic reone the tandard conitency condition on the yield function F read _F ¼ f : _r H k _ ¼ 0; k _ > 0; ð:0þ where H i the latic modulu given by the equation H ¼ 1 of _ k _ i ¼ 1 1=ð1 NÞ M _ o i k _ i. i ð:1þ Since / i > 0, the ign of the latic modulu deend on the ign of _ i : H > 0if_ i < 0 (hardening), H <0if _ i > 0 (oftening), and H =0if _ i ¼ 0 (erfect laticity). In claical Cam-Clay theory the ign of H deend on the ign of _ v, i.e., H i oitive for comaction and negative for exanion. However, a noted above, thi imle criterion doe not adequately cature the hardening/oftening reone of and, which are hown to be deendent on the limit hardening latic dilatancy D *, i.e., H i oitive if D < D * and negative if D > D *. Thu, any otulated hardening law mut atify the obviou relationhi gnh ¼ gnð _ i Þ¼gnðD DÞ ¼gnðg gþ ¼gnð i i Þ; ð:þ where gn i the ign oerator. Furthermore, in term of the cumulative latic hear train Z ¼ _ dt; ð:þ t we require that lim H ¼ lim ð _ i Þ¼ lim ðd DÞ ¼ lim ðg gþ ¼ lim ð!1!1!1!1 i i Þ¼0. ð:4þ!1 Thu, any otulated hardening law mut reflect a condition of erfect laticity a the latic hear train become very large. Note that the above retriction i tronger, e.g., than the weaker condition D * D = 0, without the limit, which could occur even if the tre and image oint do not coincide. The limiting condition!1inure that the tre and image oint aroach the CSL, and that thee two oint coincide in the limit. A general evolution for _ i atifying the requirement tated above may be given by an equation of the form _ i ¼ f ð i i Þ_ ¼ f ð i i Þ_ k; ð:5þ where f i a imle odd calar function of it argument, i.e., f( x)= f(x) and gnf = gnx. (Alternately, one can ue either D or g in the argument for f.) In thi cae, the exreion for the latic modulu become H ¼ M 1=ð1 NÞ f ð i i Þ. i ð:6þ Taking f(x) = hx, where h i a oitive dimenionle contant, we arrive at a henomenological exreion of the form imilar to that reented in [8], f ð i i Þ¼hð i i Þ. ð:7þ Thi reult in a latic modulu given by the equation H ¼ Mh 1=ð1 NÞ ð i i Þ. i ð:8þ To ummarize, the tranition oint between hardening and oftening reone i rereented by the limit hardening dilatancy D *, which aroache zero on the CSL..5. Imlication to entroy roduction Conider the Helmholtz free energy denity function W ¼ Wð e ; Þ. For now, we avoid making the uual additive decomoition of the free energy into W ¼ W e ð e ÞþW ð Þ; in fact, we hall demontrate that uch decomoition i not oible in the reent model. Ignoring the non-mechanical ower, the local Clauiu Duhem inequality yield r : _ dw dt P 0. ð:9þ

8 51 R.I. Borja, J.E. Andrade / Comut. Method Al. Mech. Engrg. 195 (006) Alying the chain rule for dw/dt and invoking the tandard Coleman relation reult in the contitutive relation r ¼ owðe ; Þ ; ð:40þ o e lu the reduced diiation inequality r : _ i _ P 0; i ¼ owðe ; Þ o. ð:41þ Here, we have choen i to be the tre-like latic internal variable conjugate to. With an aroriate et of material arameter we have enured before that r : _ P 0 (ee Section.). Since _ ¼ k _ P 0 and i < 0, the reduced diiation inequality a written above hold rovided that owð e ; Þ o 6 0. ð:4þ Now, conider the evolution for i a otulated in (.5). Integrating in time yield owð e ; Þ Z o i ¼ i0 f ð i i Þd ; ð:4þ 0 where i0 i the reference value of i when ¼ 0. Recalling that f i a imle odd function, the right-hand ide of the above equation i negative rovided that gn ð i i Þ¼gnH ¼ oitive. Thi imlie that the reduced diiation inequality i identically atified in the hardening regime. Integrating once more give a more definitive form of the Helmholtz free energy, Z Z Wð e ; Þ¼ Z i0 d f ð i i Þd d þ We ð e ÞþW 0 ; ð:44þ where W e ( e ) i the uual elatic tored energy function. The firt two integral rereent the latic comonent of the free energy, Z Z W ¼ W ð e ; Þ¼ Z i0 d f ð i i Þd d. ð:45þ Note in thi cae that W deend not only on but alo on e through the variable i. The Cauchy tre tenor then become Z r ¼ owe ð e Þ Z þ f 0 ð o e i i Þ o i o e d d ; ð:46þ 0 0 fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} where D ¼ 0, and OðD Þ o i o ¼ð1 NÞ a i =M ow i e 1 aw i N=M o þ i e o ; N P 0. ð:47þ oe Strictly, then, the Cauchy tre tenor deend not only on e but alo on. Attemt have been made in the at to cature thi deendence of r on ; for examle, a nonlinear elaticity model in which the elatic hear modulu varie with a tre-like latic internal variable imilar to i ha been rooed in [4,48]. However, thee develoment have not gained much accetance in the literature due, rimarily, to the lack of exerimental data and to the difficulty with obtaining uch tet data. It mut be noted that the oberved deendence of W on the elatic train e occur only rior to reaching the critical tate where D remain relatively mall, and thu, the econd-order term in (.46) may be ignored (uch a done in Section.1). Mot of the intene hearing (i.e., large D ) in fact occur at the critical tate where f ð i i Þ¼0, at which condition the additive decomoition of the free energy into W e ( e ) and W ð Þ hold, ee (.44)..6. Numerical imlementation Even though the latic internal variable i deend on the tate arameter w i, and that thi variable i deely embedded in the latic modulu H, the model i till amenable to fully imlicit numerical integration. Box 1 ummarize the relevant rate equation ued in the contitutive theory. Box ummarize the algorithmic counterart utilizing the claical return maing cheme. For imroved efficiency, the return maing algorithm in Box i erformed in the train invariant ace,

9 R.I. Borja, J.E. Andrade / Comut. Method Al. Mech. Engrg. 195 (006) Box 1. Summary of rate equation for laticity model for and, infiniteimal deformation verion 1. Strain rate: _ ¼ _ e þ _.. Hyerelatic rate equation: _r ¼ c e : _ e ; c e = o W e /o e o e.. Flow rule: _ ¼ _ kq. 4. State arameter: _ w i ¼ _v þ ~ k_ i = i from (.). 5. Hardening law: _ i ¼ f ð i i Þ_ k from (.5). 6. Conitency condition: f : _r H _ k ¼ Kuhn Tucker condition: _ k P 0, F 6 0, _ kf ¼ 0. a demontrated below, leading to a ytem of nonlinear equation with three unknown. A uual, it i aumed that D i given, which imlie that both the elatic trial train etr and the total train are known. Note that v 0 i the initial value of the ecific volume at the beginning of the calculation when = 0, and hould not be confued with v c0. A uual, the main goal i to find the final tree r and the dicrete latic multilier Dk for a given train increment D. Following [4,46], conider the following local reidual equation generated by the alied train increment D 8 >< r ¼ rðxþ ¼ >: e v e etr v etr þ Dkbo F þ Dko q F F 9 >= >; ; x ¼ 8 9 >< e v >= e >: >; ; ð:48þ Dk where b i the non-aociativity arameter defined in (.9). The goal i to diiate the reidual vector r by finding the olution vector x * uing a local Newton iteration. In develoing the local Jacobian matrix r 0 (x) ued for Newton iteration, it i convenient to define the following maing induced by the numerical algorithm 8 >< y ¼ q >: i 9 >= >; ¼ 8 ð e v ;e Þ 9 >< >= qð e v ;e Þ ) y ¼ yðxþ. ð:49þ >: i ð e v ;e ; DkÞ >; The tangent y 0 (x) =D define the loe, given by D 11 D 1 D 1 o e v o e D ¼ 4 D 1 D D 5 ¼ 4 o e v q o e q ð:50þ D 1 D D o e v i o e i o Dk i Box. Return maing algorithm for laticity model for and, infiniteimal deformation verion 1. Elatic train redictor: etr ¼ e n þ D.. Elatic tre redictor: r tr ¼ ow e =o etr ; tr i ¼ i;n.. Check if yielding: F ðr tr ; tr i Þ P 0? No, et e = etr ; r = r tr ; i ¼ tr i and exit. 4. Ye, initialize Dk = 0 and iterate for Dk (te 5 7). 5. Platic corrector: e = etr Dkq, r = ow e /o e. 6. Udate latic internal variable i : (a) Cumulative train: = n + D. (b) Secific volume: v = v 0 (1 + tr). (c) Initialize i = i,n and iterate for i (te 6d f). (d) State arameter: w i ¼ v v c0 þ ~ k lnð i Þ. (e) Limit hardening latic variable: ( i ¼ exðaw i=mþ if N ¼ N ¼ 0; ð1 aw i N=MÞ ðn 1Þ=N if 0 6 N 6 N 6¼ 0. (f) Platic internal variable: i ¼ i;n f ð i i ÞDk. 7. Dicrete conitency condition: F(,q, i )=0.

10 514 R.I. Borja, J.E. Andrade / Comut. Method Al. Mech. Engrg. 195 (006) The hyerelatic equation take the following form indeendent of the dicrete latic multilier Dk (and hence, D 1 = D =0) ¼ 0 ex x 1 þ a 0 ~j e ; q ¼ ðl 0 a 0 0 ex xþ e. ð:51þ Thu, D 11 ¼ 0 ~j ex x 1 þ a 0 ~j e ; D ¼ l 0 a 0 0 ex x; D 1 ¼ D 1 ¼ 0a 0 e ex x. ~j We recall that D 1 = D 1 from the otulated exitence of an elatic tored energy function W e. The latic internal variable i i deely embedded in the evolution equation and i bet calculated iteratively, a hown in Box. Firt, from Ste no. 6(f), we contruct a calar reidual equation rð i Þ¼ i i;n þ f ð i i ÞDk; ð:5þ where i i calculated in ucceion from Ste no. 6(e,d) of Box uing the current etimate for i. Uing a ub-local Newton iteration, we determine the root that diiate thi reidual iteratively. The ub-local calar tangent oerator take the imle form " # i i r 0 ð i Þ¼1 þ f 0 ð i i ÞDk 1 ~ kað1 NÞ M aw i N ð:5þ ; N P 0. ð:54þ Having determined the converged value of i, we can then calculate the correonding value of w i and i and roceed with the following differentiation. From Box, Ste no. 6(f), we obtain the variation o i o e v ¼ f 0 ð i i ÞDk o i o i o e v o e v From Ste no. 6(e), we get o i ¼ o e v a i 1 N M aw i N owi From Ste no. 6(d), we obtain o e v. ð:55þ þ i D 11; N P 0. ð:56þ ow i ¼ ~ k o i. ð:57þ o e v i o e v Combining thee lat three equation give D 1 ¼ o i ¼ c 1 f 0 ð o e i i ÞDk i D 11 ; v " c ¼ 1 þ f 0 ð i i ÞDk 1 ~ # kað1 NÞ i. M aw i N i ð:58þ Note that c i the converged value of r 0 ( i ) when r = 0, cf. (.54). Following a imilar rocedure, we obtain D ¼ o i ¼ c 1 f 0 ð o e i i ÞDk i D 1. Again, uing the ame imlicit differentiation, we get D ¼ o i odk ¼ c 1 f ð i i Þ. For the hardening law adoted in [8], f 0 ð i i Þ reduce to the contant h. It i alo convenient to define the following vector oerator h i H ¼ ½H 1 H H Š ¼ o o q o i F. ð:59þ ð:60þ ð:61þ

11 R.I. Borja, J.E. Andrade / Comut. Method Al. Mech. Engrg. 195 (006) For the yield function at hand, the element of H are a follow. Firt, we obtain the firt derivative ( of o ¼ M lnð i=þ; if N ¼ 0; ðm=nþ½1 ð= i Þ N=ð1 NÞ Š; if N > 0; of oq ¼ 1; of ¼ M 1=ð1 NÞ ; N P 0. o i i Then, for N P 0, we have H 1 ¼ 1 N=ð1 NÞ M ; H ¼ 0; H ¼ 1 1=ð1 NÞ M. 1 N i 1 N i ð:6þ Finally, from the roduct formula induced by the chain rule, we define the vector oerator G, G ¼ HD ¼ ½G 1 G G Š. ð:64þ The algorithmic local tangent oerator for Newton iteration i then given by 1 þ DkbG 1 DkbG bðo F þ DkG Þ r 0 6 ðxþ ¼4 0 1 o q F 7 5. ð:65þ ðd 11 o þ D 1 o q þ D 1 o i ÞF ðd 1 o þ D o q þ D o i ÞF D o i F ð:6þ Remark. The numerical algorithm decribed above entail two level of neted Newton iteration to determine the local unknown. An alternative aroach would be to conider i a a fourth local unknown, along with e v, e and Dk, and olve them all iteratively in one ingle Newton loo. We have found that either aroach work well for the roblem at hand, and that either one demontrate about the ame comutational efficiency..7. Algorithmic tangent oerator The algorithmic tangent oerator c = or/o etr or/o i ued for the global Newton iteration of the finite element roblem. It ha been hown in [49,50] that it can alo be ued in lieu of the theoretically correct elatolatic contitutive oerator c e for detecting the onet of material intability, rovided the te ize i mall. To derive the algorithmic tangent oerator, conider the following exreion for the Cauchy tre tenor rffiffi r ¼ 1 þ q^n; ð:66þ where ^n ¼ =kk ¼e e =ke e k¼e etr =ke etr k from the co-axiality of the rincial direction. The chain rule then yield (ee [4]) c ¼ or o ¼ 1 D 11 o e v o þ D 1 o e o þ rffiffi ^n D 1 o e v o þ D o e o þ q etr I ^n ^n ; ð:67þ where I i the fourth-rank identity tenor with comonent I ijkl =(d ik d jl + d il d jk )/. Our goal i to obtain cloed-form exreion for the derivative o e v =o and oe =o. Uing the ame train invariant formulation of the reviou ection, we now write the ame local reidual vector a r ¼ rð etr v ; etr ; xþ, where x i the vector of local unknown. We recall that the trial elatic train were held fixed at the local level; however, at the global level they themelve are now the iterate. Conequently, at the converged tate where r = 0, we now write the train derivative of the reidual vector a or o ¼ or o þ x or ox etr v ; etr! ox o ¼ 0; which give a ox o ¼ or o ) ox or ¼ b x o o. ð:69þ x We recognize a a the ame tangent matrix r 0 (x) in(.65) evaluated at the locally converged tate, andb = a 1.In comonent form, we have ð:68þ

12 516 R.I. Borja, J.E. Andrade / Comut. Method Al. Mech. Engrg. 195 (006) o e v =o >< >= b 11 b 1 b 1 >< ð1 DkbhH Þ1 o e >: =o >; ¼ 6 7 ffiffiffiffiffiffiffi >= 4 b 1 b b 5 = ^n >: >; ; ð:70þ odk=o b 1 b b ho i F 1 in which o i /o = h1, and h ¼ c 1 Dkf 0 ð i i Þv 0 að1 NÞ i ð:71þ M aw i N i the linearization of the term aociated with the tate arameter w i. Thi facilitate the olution of the deired train derivative, o e v o ¼ ~ b 11 1 þ where rffiffi b 1^n; o e o ¼ ~ b 1 1 þ rffiffiffi b ^n; ~b 11 ¼ð1 DkbhH Þb 11 ðho i F Þb 1 ; ð:7þ ~b 1 ¼ð1 DkbhH Þb 1 ðho i F Þb. Defining the matrix roduct " # D 11 D 1 ¼ D " # 11 D 1 ~b11 b 1 ; ð:74þ D 1 D D 1 D ~b 1 b the conitent tangent oerator then become c ¼ D 11 q rffiffi q 1 1 þ D 9 etr 1 1 ^n þ D 1^n 1 þ ði ^n ^nþþ etr D ^n ^n. ð:75þ In the elatic regime the ubmatrix [b ij ] become an identity matrix, and hence D ij ¼ D ij for i,j = 1,. In thi cae, c reduce to the hyerelatic tangent oerator c e. Remark. A hown in Fig., the rooed yield and latic otential function create corner on the comaction ide of the hydrotatic axi. While the model i rimarily develoed to accurately cature dilative latic flow, and therefore i not exected to erform well in tre tate dominated by hydrotatic comaction, numerical roblem could till arie in general boundary-value roblem imulation when the tre ratio g a defined by (.7) goe to zero or even become negative. In order to avoid a negative g, we introduce a ca on the latic otential function uch that q þ g if g ¼ g P vm; Q ¼ ð:76þ if g ¼ g < vm; where v i a uer-ecified arameter controlling the oition of the latic otential function ca, e.g., v = For the cae where g < vm, the local reidual vector imlifie to 8 >< rðxþ ¼ >: e v etr v e etr F ð:7þ 9 Dk >= >;. ð:77þ The local tangent oerator i given by r ðxþ ¼ ð:78þ ðd 11 o þ D 1 o q ÞF ðd 1 o þ D o q ÞF 0 Finally, the train derivative of r holding x fixed reduce to 8 9 or >< 1 ffiffiffiffiffiffiffi >= o ¼ = ^n x >: >;. ð:79þ 0 Of coure, one can alo inert a mooth ca near the noe of the latic otential function a an alternative to the lanar ca.

13 . Finite deformation laticity; localization of deformation In the receding ection we have reformulated an infiniteimal rigid-latic contitutive model for and to accommodate non-aociated laticity and hyerelaticity. In thi ection we further generalize the model to accommodate finite deformation laticity. The final model i then ued to cature deformation and failure initiation in dene and, focuing on the effect of uneven void ditribution on the local and global reone..1. Entroy inequality Conider the multilicative decomoition of deformation gradient for a local material oint X [51 5] FðX ; tþ ¼F e ðx ; tþf ðx ; tþ. ð:1þ In the following we hall ue a a meaure of elatic deformation the contravariant tenor field b e reckoned with reect to the current lacement, called the left Cauchy Green deformation tenor, b e ¼ F e F et. ð:þ Aume then that the free energy i given by W ¼ WðX ; b e ; e Þ. ð:þ A in the infiniteimal model, we invetigate condition under which we could iolate an elatic tored energy function from the above free energy function. For the urely mechanical theory the local diiation function take the form D ¼ : d dwðx ; be ; e Þ P 0; ð:4þ dt where = Jr i the ymmetric Kirchhoff tre tenor, J = det(f), d = ym (l) i the rate of deformation tenor, and l i the atial velocity gradient. Uing the chain rule and invoking the tandard Coleman relation yield the contitutive equation [54] ¼ owðx ; be ; e Þ ob e b e ; ð:5þ along with the reduced diiation inequality D ¼ : d i _e P 0; ð:6þ where d i the latic comonent of the rate of deformation, d ¼ ymðl Þ; l :¼ F e L F e 1 ; L :¼ F _ F 1 ; ð:7þ and i ¼ owðx ; be ; e Þ oe ð:8þ i a tre-like latic internal variable equivalent to i of the infiniteimal theory. We aume that i evolve in the ame way a i, i.e., _ i ¼ /ð i i Þ_e ; ð:9þ where i deend not only on i but alo on b e. Integrating (.9) give i ¼ i0 Z e e 0 /ð i i Þde. Integrating once more, we get Z e Z WðX ; b e ; e Þ¼ e i0 de e 0 R.I. Borja, J.E. Andrade / Comut. Method Al. Mech. Engrg. 195 (006) e 0 Z e e 0 /ð i i Þde de þ We ðx ; b e ÞþW 0 ; where W e (X,b e ) i the elatic tored energy function. Finally, uing the contitutive Eq. (.5) once again, we get ¼ owe ðx ; b e Z Þ e Z e ob e þ / 0 ð i e e i Þ o i ob e de de be. ð:1þ fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} OðDe Þ ð:10þ ð:11þ

14 518 R.I. Borja, J.E. Andrade / Comut. Method Al. Mech. Engrg. 195 (006) The econd-order term in (.1) can be ignored at the initial tage of loading when De i mall, thu leaving the Kirchhoff tre varying with the elatic tored energy function alone. When De i large, /ð i i Þ vanihe at critical tate, and o the elatic and latic art of the free energy uncoule. In both cae the tree can be exreed in term of the elatic tored energy function alone, i.e., ¼ owe ðx ; b e Þ ob e b e. ð:1þ Once again, the erfectly latic behavior at critical tate i a key feature of the model that allow for the uncouling of the free energy... Finite deformation laticity model Conider the tre invariant ¼ 1 rffiffi tr; q ¼ knk; n ¼ 1. ð:14þ Then, a in the infiniteimal theory the yield function can be defined a where F ¼ q þ g 6 0; ( g ¼ M½1 þ lnð i=þš if N ¼ 0; ðm=nþ½1 ð1 NÞð= i Þ N=ð1 NÞ Š if N > 0. ð:15þ ð:16þ The material arameter M and N are imilar in meaning to thoe of the infiniteimal theory, although their value hould now be calibrated in the finite deformation regime. The flow rule may be written a before, d ¼ kq; _ q ¼ b rffiffiffi of o 1 þ of ^n; ^n ¼ n=knk; ð:17þ oq where b 6 1 i the non-aociativity arameter. We otulate a imilar hardening law in Kirchhoff tre ace given by (.9), with gn½/ð i i ÞŠ ¼ gn H ð:18þ to cature either a hardening or oftening reone deending on the oition of the tate oint relative to the limit hardening dilatancy. Box then ummarize the rate equation for the finite deformation laticity model. The model ummarized in Box ha ome noteworthy feature. Firt, the formulation aume that the latic in x i zero (ee [55] for ome dicuion on the ignificance of the latic in). Second, the fourth-order atial elatic tangent oerator a e can be determined from the exreion a e ¼ c e þ 1 þ 1; ð:19þ where ( 1) ijkl = jl d ik,( 1) ijkl = il d jk, and c e i a atial tangential elaticity tenor obtained from the uh-forward of all the indice of the econd tangential elaticity tenor defined in [5]. Box. Summary of rate equation for laticity model for and, finite deformation verion 1. Velocity gradient: l = l e + l.. Hyerelatic rate equation: _ ¼ a e : l e.. Flow rule: d ¼ ym ðl Þ¼ _ kq; x ¼ kwðl Þ¼0. 4. State arameter: _ w i ¼ _v þ ~ k _ i = i. 5. Hardening law: _ i ¼ /ð i i Þ_ k. 6. Conitency condition: f : _ H _ k ¼ 0, f = of/o. 7. Kuhn Tucker condition: _ k P 0, F 6 0, _ kf ¼ 0. Finally, the ecific volume varie according to the kinematical relation v ¼ Jv 0 ) _v ¼ _Jv 0 ¼ Jv 0 trðlþ ¼vtrðlÞ. ð:0þ

15 Thu, jut a in the infiniteimal theory where the rate equation may be viewed a driven by the train rate _, the rate equation hown in Box may be viewed a driven by the atial velocity gradient l... Numerical imlementation R.I. Borja, J.E. Andrade / Comut. Method Al. Mech. Engrg. 195 (006) For the roblem at hand we emloy a tandard elatic redictor-latic corrector algorithm baed on the roduct formula for b e, a ummarized in Box 4. Let b e n ¼ Fe n Fet n. ð:1þ Sureing latic flow, the trial elatic redictor for b e i b etr b etr nþ1 ¼ f nþ1 b e n f t nþ1 ; The latic corrector emanate from the exonential aroximation f nþ1 ¼ ox nþ1 ox n. ð:þ b e ¼ exð DkqÞb etr ; q q nþ1 ¼ oq o. From the co-axiality of latic flow, the rincial direction of q and coincide. Box 4. Return maing algorithm for laticity model for and, finite deformation verion 1. Elatic deformation redictor: b etr ¼ f nþ1 b e n f t nþ1.. Elatic tre redictor: tr ¼ ðow e =ob etr Þb etr ; tr i ¼ i;n.. Check if yielding: F ð tr ; q tr ; tr i Þ P 0? No, et b e ¼ b etr ; ¼ tr ; i ¼ tr i and exit. 4. Ye, initialize Dk = 0 and iterate for Dk (te 5 8). 5. Sectral decomoition: b etr ¼ P A¼1 ðketr A Þ m trðaþ. 6. Platic corrector in rincial logarithmic tretche: e e A ¼ lnðke A e e A ¼ eetr A Dkq A, A ¼ ow e =oe e A ; A ¼ 1; ;. 7. Udate latic internal variable i : (a) Total deformation gradient: F = f n+1 Æ F n. (b) Secific volume: v = det(f)v 0 = Jv 0. (c) Initialize i = i,n and iterate for i (Ste 7d f). (d) State arameter: w i ¼ v v c0 þ ~ k lnð i Þ. (e) Limit hardening latic variable: ( i ¼ exðaw i=mþ if N ¼ N ¼ 0; ð1 aw i N=MÞ ðn 1Þ=N if 0 6 N 6 N 6¼ 0. (f) Platic internal variable: i ¼ i;n /ð i i ÞDk. 8. Dicrete conitency condition: F(,q, i )=0. 9. Sectral reolution: b e ¼ P A¼1 ðke A Þ m trðaþ. Þ, eetr A ¼ lnðk etr A Þ, ð:þ Next we obtain a ectral decomoition of b e, b e ¼ X A¼1 ðk e A Þ m ðaþ ; m ðaþ ¼ n ðaþ n ðaþ ; ð:4þ where k e A are the elatic rincial tretche, n(a) are the unit rincial direction, and m (A) are the ectral direction. The correonding elatic logarithmic tretche are e e A ¼ lnðke AÞ; A ¼ 1; ;. ð:5þ From material frame indifference W e (X,b e ) only varie with e e A, and o we can write We ¼ W e ðx ; e e 1 ; ee ; ee Þ, which give ow e ob e ¼ 1 X A¼1 1 ow e m ðaþ. ðk e A Þ oe e A ð:6þ

16 510 R.I. Borja, J.E. Andrade / Comut. Method Al. Mech. Engrg. 195 (006) The elatic contitutive equation then write ¼ owe ob e b e ¼ X A¼1 A m ðaþ ; A ¼ owe ; ð:7þ oe e A imlying that the ectral direction of and b e alo coincide. Thu, b e and q are alo co-axial, and for (.) to hold, b e and b etr mut alo be co-axial, i.e., m ðaþ ¼ m trðaþ. ð:8þ Thi allow the latic corrector hae to take lace along the rincial axe, a hown in Box 4. Alternatively, we can utilize the algorithm develoed for the infiniteimal theory by working on the invariant ace of the logarithmic elatic tretch tenor. Let e e v ¼ ee 1 þ ee þ ee ; ee ¼ 1 q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½ðe e 1 ee Þ þðe e 1 ee Þ þðe e ee Þ Š denote the firt two invariant of the logarithmic elatic tretch tenor (imilar definition may be made for e etr v and ¼ 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð 1 þ þ Þ; q ¼ ½ð 1 Þ þð 1 Þ þð Þ Š= ð:9þ and e etr ), denote the firt two invariant of the Kirchhoff tre tenor. If we take the functional relationhi ¼ ðe e v ; ee Þ, q ¼ qðe e v ; ee Þ, and i ¼ i ðe e v ; ee ; DkÞ a before, uing the ame elatic tored energy function but now exreed in term of the logarithmic rincial elatic tretche, then the local reidual vector write 8 >< r ¼ rðxþ ¼ >: e e v e e eetr v eetr þ Dkbo F þ Dko q F F 9 >= >; ; x ¼ ð:0þ 8 9 >< e e v >= e e >: >;. ð:1þ Dk In thi cae, the local Jacobian r 0 (x) take a form identical to that develoed for the infiniteimal theory, ee (.65). Comaring Boxe and 4, we ee that the algorithm for finite deformation laticity differ from the infiniteimal verion only through a few additional te entailed for the ectral decomoition and reolution of the deformation and tre tenor. Contruction of b e from the ectral value require two te. The firt involve reolution of the rincial elatic logarithmic tretche from the firt two invariant calculated from return maing, e e A ¼ 1 rffiffiffi rffiffiffi ee v d A þ e e ^n e etr A A; ^n A ¼ ðeetr v =Þd A ; ð:þ e etr where d A = 1 for A = 1,,. The above tranformation entail caling the deviatoric comonent of the redictor tenor by the factor e e =eetr and adding the volumetric comonent. The econd te involve a ectral reolution from the rincial elatic logarithmic train (cf. (.4)) b e ¼ X A¼1 exðe e A ÞmtrðAÞ. The next ection demontrate that a cloed-form conitent tangent oerator i available for the above algorithm..4. Algorithmic tangent oerator For imlicity, we retrict to a quai-tatic roblem whoe weak form of the linear momentum balance over an initial volume B with urface ob read Z ðgradg : P q 0 g GÞdV B Z g t da ¼ 0; ob t ð:4þ where q 0 G i the reference body force vector, t = P Æ n i the nominal traction vector on ob t ob, n i the unit vector on ob t, g i the weighting function, P ¼ F t ð:5þ i the non-ymmetric firt Piola Kirchhoff tre tenor, and GRAD i the gradient oerator evaluated with reect to the reference configuration. We recall the internal virtual work ð:þ

17 Z W e INT ZB ¼ GRADg : P dv ¼ gradg : dv ð:6þ e B e for any B e B, where grad i the gradient oerator evaluated with reect to the current configuration. The firt variation give [56] dw e INT ¼ ZB e gradg : a : graddudv ; ð:7þ where u i the dilacement field, and a ¼ a 1; d ¼ a : graddu. ð:8þ Evaluation of a thu require determination of the algorithmic tangent oerator a. We alo recall the following ectral rereentation of the algorithmic tangent oerator a [56] a ¼ X A¼1 X B¼1 a AB m ðaþ m ðbþ þ X A¼1 X B6¼A k e tr B B A k e tr A k e tr B m ðabþ m ðabþ þ k etr A m ðabþ m ðbaþ ; ð:9þ where m (AB) = n (A) n (B), A 5 B. The coefficient a AB are element of the conitent tangent oerator obtained from a return maing in rincial axe, and i formally defined a a AB ¼ o A o A ; A; B ¼ 1; ;. ð:40þ oe etr B oe B The value of thee coefficient are ecific to the contitutive model in quetion, a well a deendent on the numerical integration algorithm utilized for the model. For the reent critical tate laticity theory a AB i evaluated a follow. The exreion r for ffiffi a rincial Kirchhoff tre i A ¼ d A þ q^n A ; A ¼ 1; ;. ð:41þ Differentiating with reect to a rincial logarithmic train give rffiffi a AB ¼ o A oe e v oe e ¼ d A D 11 þ D 1 þ oe B oe B oe B þ q d e etr AB 1 d Ad B ^n A^n B oe e v oe e ^n A D 1 þ D oe B oe B ; A; B ¼ 1; ; ; ð:4þ where d AB i the Kronecker delta. The coefficient D 11, D, and D 1 are identical in form to thoe hown in (.5) excet that the train invariant now take on logarithmic definition. A in the infiniteimal theory, we obtain the unknown train derivative above from the local reidual vector, whoe own derivative write or A ¼ or A oe B oe þ X B x C¼1 R.I. Borja, J.E. Andrade / Comut. Method Al. Mech. Engrg. 195 (006) or A ox C e etr v ;e etr fflfflfflfflffl{zfflfflfflfflffl} a AC ox C oe B ¼ 0; A; B ¼ 1; ; ; ð:4þ where the matrix ½a AB Š correond to the ame algorithmic local tangent oerator given in (.65). Letting [b AB ] denote the invere of ½a AB Š, we can then olve ox A ¼ X or C b AC oe B oe ; C¼1 B x A; B ¼ 1; ;. ð:44þ Thi latter equation rovide the deired train derivative, 8 oe e v =oe < A = b 11 b 1 b 1 < ð1 oe e =oe A : ; ¼ DkbhH ffiffiffiffiffiffiffi Þd A = 4 b 1 b b 5 = ^na ; : ; odk=oe A b 1 b b ho i F d A A ¼ 1; ; ; ð:45þ where h ¼ c 1 Dk/ 0 ð i að1 NÞ i Þv i M aw i N ; c ¼ 1 þ / 0 ð i i ÞDk 1 ~ kað1 NÞ i. M aw i N i ð:46þ

18 51 R.I. Borja, J.E. Andrade / Comut. Method Al. Mech. Engrg. 195 (006) Note that the finite deformation exreion for h utilize the current value of the ecific volume v wherea the infiniteimal verion ue the initial value v 0 (cf. (.71)). Inerting the exreion for oe e v =oe A and oe e =oe A back in (.4) yield the cloedform olution for a AB, which take an identical form to (.75): a AB ¼ D 11 q 9e etr d A d B þ rffiffi q D 1 d A^n B þ D 1^n A d B þ e etr See (.7) (.74) for ecific exreion for the coefficient D ij..5. Localization condition ðd AB ^n A^n B Þþ D ^n A^n B ; A; B ¼ 1; ;. ð:47þ Following [56,57], we ummarize the following alternative (and equivalent) exreion for the localization condition into lanar band. We denote the continuum elatolatic counterart of the algorithmic tenor a by a e ¼ X A¼1 X B¼1 a e AB mðaþ m ðbþ þ X A¼1 X B6¼A B A k e B ke A k e B mðabþ m ðabþ þ k e A mðabþ m ðbaþ ; ð:48þ where a e AB i the continuum elatolatic tangent tiffne matrix in rincial axe. (Note, thi formula aear in [46,49,50] with a factor 1/ before the in-term ummation, a tyograhical error.) Then, a e ¼ a e 1 ð:49þ define the continuum counterart of the fourth-order tenor a in (.8). Alternatively, we denote the contitutive elatolatic material tenor c e by the exreion [54] c e ¼ X A¼1 X B¼1 a e AB mðaþ m ðbþ þ X A¼1 A x ðaþ ; in which x ðaþ ¼ ½I b b e b e þ I b 1 A ð1 1 IÞþb Aðb e m ðaþ þ m ðaþ b e Þ I b 1 A ð1 mðaþ þ m ðaþ 1Þþwm ðaþ m ðaþ Š=D A ; ð:51þ where b A i the Ath rincial value of b e, I 1 and I are the firt and third invariant of b e, I b ¼ðb e b e þ b e b e Þ=; w ¼ I 1 b A þ I b 1 A 4b A ; ð:5þ and D A :¼ b A I 1b A þ I b 1 A. ð:5þ Note that c e ijkl ¼ F iaf jb F kc F ld C e ABCD i the atial uh-forward of the firt tangential elatolatic tenor Ce [5]. Then, a e ¼ c e þ 1 ð:54þ define an alternative exreion to (.49). Uing a e from either (.49) or (.54), we can evaluate the element of the Eulerian elatolatic acoutic tenor a a a ij ¼ n k a e ikjl n l; ð:55þ where n k and n l are element of the unit normal vector n to a otential deformation band reckoned with reect to the current configuration. Defining the localization function a F ¼ inf j n ðdet aþ; ð:56þ we can then infer the incetion of a deformation band from the initial vanihing of F. Though theoretically one need to ue the contitutive oerator a e or c e to obtain the acoutic tenor a, the algorithmic tangent tenor are equally accetable for bifurcation analye for mall te ize [50]. 4. Numerical imulation We reent two numerical examle demontrating the meo-cale modeling technique. To highlight the triggering of train localization via imoed material inhomogeneity, we only conidered regular ecimen (either rectangular or cubical) along with boundary condition favoring the develoment of homogeneou deformation (a in to arret rigid body mode and vertical roller at the to and bottom end of the ecimen). A common technique of erturbing the initial condition i to recribe a weak element; however, thi i unrealitic and arbitrary. In the following imulation we have ð:50þ

19 erturbed the initial condition by recribing a atially varying ecific volume (or void ratio) coniting of horizontal layer of relatively homogeneou denity but with ome variation in the vertical direction. Thi reulted in vein-like oil tructure in the denity field cloely reembling that hown in the hotograh of Fig. 1 and mimicked the lacement of and with a common laboratory technique called luviation. The ecific volume field were aumed to range from 1.60 to 1.70, with a mean value of 1.6. Thee value were choen uch that all oint remained on the dene ide of the CSL (w i < 0) Plane train imulation R.I. Borja, J.E. Andrade / Comut. Method Al. Mech. Engrg. 195 (006) A a firt examle we conidered a finite element meh 1 m wide and m tall and coniting of 4096 contant train triangular element hown in Fig. 5. The meh i comletely ymmetric to avoid any bia introduced by the triangle. The vertical ide were ubjected to reure load (natural boundary condition), wherea the to end wa comreed by moving roller uort (eential boundary condition). The load time function are hown in Fig. 6 with caling factor c = 100 kpa and b = 0.40 m for reure load G 1 (t) and vertical comreion G (t), reectively. The material arameter are ummarized in Table 1 and and roughly rereent thoe of dene Erkak and, ee [8]. The reconolidation tre wa et to c = 10 kpa and the reference ecific volume wa aumed to have a value v c0 = (uniform for all element). The ditribution of initial ecific volume i hown in Fig. 7. Fig. 8 how contour of the determinant function and the logarithmic deviatoric train at the onet of localization for the cae of finite deformation, which occurred at a nominal axial train of 1.6%. Localization for the infiniteimal model wa lightly delayed at 1.4%. The figure how a clear correlation between region in the ecimen where the determinant function vanihed for the firt time and where the deviatoric train were mot intene. Furthermore, the figure reveal an Fig. 5. Finite element meh for lane train comreion roblem. Fig. 6. Load time function.

20 514 R.I. Borja, J.E. Andrade / Comut. Method Al. Mech. Engrg. 195 (006) Table 1 Summary of hyerelatic material arameter (ee [44] for laboratory teting rocedure) Symbol Value Parameter ~j 0.0 Comreibility a 0 0 Couling coefficient l kpa Shear modulu kpa Reference reure e v0 0 Reference train Table Summary of latic material arameter (ee [8] for laboratory teting rocedure) Symbol Value Parameter ~k 0.04 Comreibility N 0.4 For yield function N 0. For latic otential h 80 Hardening coefficient Fig. 7. Initial ecific volume for lane train comreion roblem. Fig. 8. Contour of: (a) determinant function; and (b) deviatoric invariant of logarithmic tretche at onet of train localization. X-attern of hear band formation catured by both hear deformation and determinant function contour, reroducing thoe oberved in laboratory exeriment [,5]. Fig. 9 dilay contour of the latic modulu and the logarithmic volumetric train at the onet of localization. We recall that for thi laticity model the latic modulu i a function of the tate of tre, and in thi figure low value of

21 R.I. Borja, J.E. Andrade / Comut. Method Al. Mech. Engrg. 195 (006) Fig. 9. Contour of: (a) latic hardening modulu; and (b) volumetric invariant of logarithmic tretche at onet of train localization. hardening modulu correlate with area of highly localized hear train hown in Fig. 8. On the other hand, volumetric train at localization aear to reemble the initial ditribution of ecific volume hown in Fig. 7. In fact the blue ocket of high comreion extending horizontally at the to of the amle (Fig. 9b) i rereentative of that exerienced by the red horizontal layer of relatively low denity hown in Fig. 7. The calculated hear band were redominantly dilative. 4.. Three-dimenional imulation For the D imulation we conidered a cubical finite element meh hown in Fig. 10. The meh i 1 m wide by 1 m dee by m tall and conit of 000 eight-node brick element. All four vertical face of the meh were ubjected to reure load of 100 kpa (natural boundary condition). The to face at z = m wa comreed vertically by moving roller uort according to the ame load time function hown in Fig. 6 (eential boundary condition), effectively relicating a laboratory teting rotocol for triaxial comreion on a ecimen with a quare cro-ection. The initial ditribution of ecific volume i alo hown in Fig. 10 and roughly mimicked the rofile for lane train hown in Fig. 7. Fig. 11 and 1 comare the nominal axial tre and volume change behavior, reectively, of ecimen with and without imoed heterogeneitie. The homogeneou ecimen wa created to have a uniform ecific volume equal to the volume average of the ecific volume for the equivalent heterogeneou ecimen, or 1.6 in thi cae. We ued both the tandard numerical integration and B-bar method for the calculation [58 60], but there wa not much difference in the redicted reone. However, oftening within the range of deformation hown in thee figure wa detected by all olution excet by the homogeneou ecimen imulation. Furthermore, an earlier overall dilation from an initially contractive behavior wa detected by the heterogeneou ecimen imulation (Fig. 1). Thi reveral in volume change behavior from contractive to dilative i uually termed hae tranformation in the geotechnical literature, a feature that i not relicated Fig. 10. Finite element meh and initial ecific volume for D comreion roblem.

22 516 R.I. Borja, J.E. Andrade / Comut. Method Al. Mech. Engrg. 195 (006) Fig. 11. Nominal axial tre axial train reone for D comreion roblem. Fig. 1. Volume change-nominal axial train reone for D comreion roblem. Fig. 1. Stre ath for homogeneou ecimen imulation with finite deformation. by claical Cam-Clay model. Fig. 1 how that thi hae tranition wa catured by the contitutive model by firt yielding on the comreion ide of the yield urface, and later by yielding on the dilation ide.

23 R.I. Borja, J.E. Andrade / Comut. Method Al. Mech. Engrg. 195 (006) Fig. 14. Determinant function at onet of train localization. Fig. 14 how contour of the determinant function F at a nominal axial train of 8.78%, where the determinant vanihed for the firt time at a Gau oint located in the interior of the heterogeneou ecimen. Unlike the lane train olution, the otential hear band did not exhibit an X-attern in thi cae. Intead, the olution redicted a well-defined band extending acro the ecimen. Note that the horizontal lice hown in Fig. 14b have been rotated by 90 on the horizontal lane relative to the orientation of the olid volume hown in Fig. 14a for otimal D viualization (red region in Fig. 14a mut be matched with red region in Fig. 14b, etc.). A for the homogeneou ecimen, the comutation wa carried out u to 15% nominal axial train but the amle did not localize. Fig. 15 how contour of the deviatoric invariant of logarithmic tretche at the oint of initial localization. Again, the horizontal lice (Fig. 15b) are 90 rotated relative to the olid volume (Fig. 15a) for better viualization. Comaring Fig. 14 and 15, the olution clearly correlated region in the ecimen where the determinant function vanihed for the firt time with region where the deviatoric train were mot intene (a blue determinant region correlate with a red deviatoric train region, etc.). In contrat, Fig. 16 how contour of the volumetric invariant of logarithmic tretche reembling the initial ecific volume rofile of Fig. 10. In general, thee obervation are imilar to thoe oberved for the lane train examle. For comletene, Fig. 17 how the deformed FE meh for the heterogeneou amle at the intant of initial localization, uggeting that the ecimen moved laterally a well a twited torionally. Note once again that thi non-uniform deformation wa triggered by the imoed initial denity variation alone. Finally, Fig. 18 how the convergence rofile of global Newton iteration for the full D finite deformation imulation of a heterogeneou ecimen with B-bar integration. The iteration converged quadratically in all cae, uggeting otimal erformance. We emhaize that all of the reult reented above only ertain to the rediction of when and where a otential hear band will emerge. We have not urued the imulation beyond the oint of bifurcation due to meh Fig. 15. Deviatoric invariant of logarithmic tretche at onet of train localization.

24 518 R.I. Borja, J.E. Andrade / Comut. Method Al. Mech. Engrg. 195 (006) Fig. 16. Volumetric invariant of logarithmic tretche at onet of train localization. Fig. 17. Deformed finite element meh at onet of train localization (deformation magnified by a factor of ). Fig. 18. Convergence rofile of global Newton iteration: finite deformation imulation of heterogeneou ecimen with B-bar.