FINITE DEFORMATION ANALYSIS OF GEOMATERIALS

Transcription

1 REPORT NO. UCD/CGM-01/03 CENTER FOR GEOTECHNICAL MODELING FINITE DEFORMATION ANALYSIS OF GEOMATERIALS BY B. JEREMIC K. RUNESSON S. STURE DEPARTMENT OF CIVIL & ENVIRONMENTAL ENGINEERING COLLEGE OF ENGINEERING UNIVERSITY OF CALIFORNIA AT DAVIS SEPTEMBER 2001

2 Univrsity of Califorina at Davis, Cntr for Gotchnical Modling Rport UCD-CGM-01 Publishd in th Intrnational Journal for Numrical and Analytical Mthods in Gomchanics incorporating Mchanics of Cohsiv Frictional Matrials, Vol. 25, No. 8, pp , Finit Dformation Analysis of Gomatrials B. Jrmić 1 K. Runsson 2 S. Stur 3 1 Dpartmnt of Civil and Environmntal Enginring, Univrsity of California, Davis 95616, U.S.A. phon , fax , Jrmic@UCDavis.du. 2 Division of Solid Mchanics, Chalmrs Univrsity oftchnology, S Götborg, Swdn 3 Dpartmnt of Civil, Environmntal, and Architctural Enginring, Univrsity of Colorado, Bouldr, CO , U.S.A. 1

3 Abstract Th mathmatical structur and numrical analysis of classical small dformation lasto plasticity is gnrally wll stablishd. Howvr, dvlopmnt of larg dformation lastic plastic numrical formulation for dilatant, prssur snsitiv matrial modls is still a rsarch ara. In this papr w prsnt dvlopmnt of th finit lmnt formulation and implmntation for larg dformation, lastic plastic analysis of gomatrials. Our dvlopmnts ar basd on th multiplicativ dcomposition of th dformation gradint into lastic and plastic parts. A consistnt linarization of th right dformation tnsor togthr with th Nwton mthod at th constitutiv and global lvls lads toward an fficint and robust numrical algorithm. Th prsntd numrical formulation is capabl of accuratly modling dilatant, prssur snsitiv isotropic and anisotropic gomatrials subjctd to larg dformations. In particular, th formulation is capabl of simulating th bhavior of gomatrials in which igntriads of strss and strain do not coincid during th loading procss. Th algorithm is tstd in conjunction with th novl hyprlasto plastic modl trmd th B matrial modl, which is a singl surfac (singl yild surfac, affin singl ultimat surfac and affin singl potntial surfac) modl for dilatant, prssur snsitiv, hardning and softning gomatrials. It is spcifically dvlopd to modl larg dformation hyprlasto plastic problms in gomchanics. W prsnt an application of this formulation to numrical analysis of low confinmnt tsts on cohsionlss granular soil spcimns rcntly prformd in a SPACEHAB modul aboard th Spac Shuttl during th STS-89 mission. W compar numrical modling with tst rsults and show th significanc of addd confinmnt by th thin hyprlastic latx mmbran undrgoing larg strtching. Ky Words: Hyprlasto plasticity, Larg dformations, Gomatrials, Finit lmnt analysis 2

4 1 Background Thortical as wll as implmntation issus in matrial non linar finit lmnt analysis of solids and structurs ar incrasingly bcoming bttr undrstood for th cas of infinitsimal strain thoris. Likwis, larg dformation thoris and implmntations for matrials obying J 2 plasticity ruls ar fairly advancd. Larg strain analysis involving gomtric and matrial non linaritis or prssur snsitiv gomatrials ar still th subjct of activ rsarch. Th choic of appropriat strss and strain masurs, as wll as th issus prtaining to th intgration of lasto plastic constitutiv quations undr conditions of larg strain ar still disputd in th rsarch community. Th ky assumption in infinitsimal dformation lasto plasticity is th additiv dcomposition of strains into lastic and plastic parts. A numbr of gnralizd mid point numrical algorithms, ranging from purly xplicit to purly implicit schms was dvlopd and thir accuracy assssd (Crisfild [9], Kojić and Bath [20], Ortiz and Popov [35], Simo and Taylor [48], Ortiz and Simo [36], Runsson t. al. [41], Krig and Krig [22] to mntion a fw). Implicit, backward Eulr intgration schms hav in rcnt yars bn provn to b robust and fficint. Algorithmic tangnt stiffnss tnsors hav bn drivd (starting with th pionring work of Simo and Taylor [48] and Runsson and Samulsson [40]) for most of th intgration schms. It is important to not that strains ar non linar functions of displacmnts and thus additiv dcomposition of total strains into lastic and plastic componnts hold only for infinitsimal dformations (s mor in Lubarda and L [29] and Famiglitti and Prvost [13]) Morovr, a simpl xampl is prsntd, which illustrats diffrncs btwn larg and small dformation analysis. Th rspons of a solid in trms of small and larg dformations is compard. To this nd w us th dfinition of a dformation gradint F = x i;j and th Lagrangian strain tnsor E ) and compar it with th small dformation strain tnsor ffl. Clarly th diffrnc btwn E and ffl is in th nonlinar trm of displacmnt drivativs: E = 1 2 (u i;j + u j;i + u i;j u j;i ) ffl = 1 2 (u i;j + u j;i ) (1) Only vry small dformations can approximat E with ffl. Th rror xcds 10% aftr a nominal strain of 30%. Fig. 1 shows that by using th small dformation strain masur instad of th larg dformation strain, significant rror is introducd. Th arly xtnsions to larg dformation of rat basd numrical mthods for lasto 3

5 2500% shar strain 2000% 1500% 1000% 500% Φ E Error ε c) dformation (arctan Φ) Figur 1: Error introducd by using th small strain instad of Lagrangian strain tnsor. plastic analysis of solids was conductd in th Lagrangian form 1. Larg dformation principl of virtual work basd formulation for larg strain lastic plastic analysis of solids in th Lagrangian form was proposd by Hibbitt t al. [16]. Th Eulrian form of th solution to th problm was proposd by McMking and Ric [32]. Th disadvantag with this approach was in th ncssary us of incrmntally objctiv intgration algorithms that may b computationally xpnsiv. Hypolastic basd tchniqus, aimd at problms with small lastic strains wr also proposd by many othrs, (s for xampl Saran and Runsson [42]). Anumbr of problms ncountrd with diffrnt strss rats wr notd by Nagtgaal and d Jong [34], Kojić and Bath [21] and Szabó and Balla [52]. On th othr hand, hyprlastic basd tchniqus hav bn dvlopd rcntly for purly dviatoric plasticity, for xampl by Simo and Ortiz [47], [45], Bath t al. [2], Simo [43], [44], Etrovic and Bath [12], Prić t al. [39] and Cuitino and Ortiz [10]. Most of th multiplicativ split tchniqus ar basd on th arlir works of Hill [17], Bilby t al. [3] Krönr [23], L and Liu [26], Fox [14] and L [25]. Simo and Ortiz [47] whr th first to propos a computational approachntirly basd on th multiplicativ dcomposition of th dformation gradint. Thir strss updat algorithm, howvr, usd th cutting plan schm that has bn shown by d Borst and Fnstra [11] to yild rronous rsults for som yild critria. Bath t al. [2] hav usd th multiplicativ dcomposition with logarithmic stord nrgy function and an xponntial approximation of th flow rul for non linar analysis of mtals. Etrovic and Bath [12] includd kinmatic 1 Hypolasticity is prsntd in spatial format. Virtual work is normally statd in th matrial format. 4

6 hardning in thir dvlopmnt, but thy did not addrss th issu of tangnt stiffnss tnsors consistnt with th us of th Nwton schm for th solution of finit lmnt quations in th finit dformation rgim. Thy hav also xplord th us of a sris xpansion of th Hncky strain tnsor in thir numrical algorithm. Howvr, dvlopmnts wr mad for dviatoric plasticity only. Prić at al. [39] followd thir work and xprimntd with various rat forms and thir approximations. Thy also rstrictd th us of thir algorithm to th small lastic strain cas. Cuitino and Ortiz [10] proposd a mthod for xtnding small strain stat updat algorithms and thir corrsponding consistnt tangnt stiffnss moduli into th finit dformation rgim but, although thy claim that th mthod is applicabl to various matrial modls, thy stayd with th J 2 plasticity modl. Simo [43], [44], xplord a strain spac formulation. Th analysis was conductd for a linar hardning J 2 plasticity problm. In his latr work, Simo [45] consolidatd th thortical framwork and showd som xcllnt xampls of thr dimnsional larg dformation J 2 lasto plastic analysis. Limitd application of that work to gomatrials has bn shown by Simo and Mschk [46]. Thy applid th dvlopd framwork to th Cam Clay and gnral plasticity typ of modls, usd in gotchnics. Thy hav also xplord diffrnt implicit xplicit schms for intgration of th hardning law in ordr to bypass th hardning inducd non symmtric tangnt stiffnss moduli. Th shortcoming of that work was that an associatd flow rul was adoptd, thus rsulting in ovrstimation of dilatation. Morovr, loss of collinarity btwn strss and strain igntriads (occurring during non proportional loading of gomatrials) cannot b modld with this catgory of algorithm. Mor rcntly Lwis and Khoi [27] usd a rat basd total Langrangan formulation to th analysis of compactd powdrs. Prić and d Souza Nto [38] usd an oprator split algorithm in trms of principal strsss in conjunction with th Trsca modl. Armro [1] xtndd th multiplicativ algorithm (originally dvlopd by Simo) for a coupld poro plastic fully saturatd mdium. Borja and Alarcón [5] [8] usd multiplicativ dcomposition in principal coordinats (Simo's formulation) for th problm of larg dformation consolidation. Borja t al. [6], [4], [7] applid Simo's approach to th Cam Clay family of modls. Liu t al. [28] hav applid an arlir algorithm dvlopd by Simo [46] and addd a nw nonlinar lastic law for th analysis of tir sand composit matrial. Th abov dvlopmnts mak an implicit assumption on co linarity of principal dirctions of strss and strain tnsors, which rndrs thm unusabl for anisotropic hardning/softning matrial modls. 5

7 In th following, finit lmnt and constitutiv formulations for a gnral hyprlastic plastic gomatrial ar prsntd. Mor spcifically, sction 2 prsnts a larg dformation finit lmnt formulation with focus on th Lagrangian dscription. Sction 3 provids hyprlastic and hyprlastic plastic background dscriptions and dscribs th constitutiv intgration algorithm. Slctd rsults ar prsntd in sction 4. 2 Matrial and Gomtric Non Linar Finit Elmnt Formulation In th following w prsnt a dtaild formulation of a matrial and gomtric non linar static finit lmnt analysis schm. Th configuration of choic is matrial or Lagrangian. Th local form of quilibrium quations in Lagrangian format for th static cas can b writtn as: P ;j ρ 0 b i =0 (2) whr P = S kj (F ik ) t and S kj ar first and scond Piola Kirchhoff strss tnsors, rspctivly and b i ar body forcs. Th wak form of th quilibrium quations is obtaind by prmultiplying (2) with virtual displacmnts ffiu i and intgrating by parts with rfrnc to th initial configuration B 0 (initial volum V 0 ): V 0 ffiu i;j P dv = V 0 ρ 0 ffiu i b i dv S 0 ffiu i μt i ds (3) It provs bnficial to rwrit th lft hand sid of (3) by using th symmtric scond Piola Kirchhoff strss tnsor S : 1 ffiu i;j F jl S il dv = V 0 V 0 2 ((ffiu i;l + ffiu l;i )+(ffiu i;j u j;l + u l;j ffiu j;i )) S il dv (4) whr w hav usd th symmtry of S il and dfinition for dformation gradint F ki = ffi ki + u k;i. With a convnint dfinition of th diffrntial oprator ^Eil ( 1 u i ; 2 u i ) ^E il ( 1 u i ; 2 u i )= u i;l + 1 u l;i th virtual work quation (4) can b writtn as: 1 u l;j 2 u j;i + 2 u i;j 1 u j:l W int (ffiu i ;u (k) i )+W xt (ffiu i )=0 (6) 6 (5)

8 with: W int (ffiu i ; n+1 0 u(k) i ) = ^E (ffiu i ; n+1 0 u(k) i ) n+1 0 S(k) dv (7) W xt (ffiu i ) = ρ 0 ffiu i n+1 0 b i ffiu i n+1 0 t i ds (8) W choos a Nwton typ procdur for satisfying quilibrium. Givn th displacmnt fild u (k) i (X j ), in itration k, th itrativ chang u i = u (k+1) i u (k) i is obtaind from th linarizd virtual work xprssion W (ffiu i ;u (k+1) i ) ' W (ffiu i ;u (k) i )+ W(ffiu i ; u i ; u (k) i ) (9) Hr, W (ffiu i ;u (k) i ) is th virtual work xprssion W (ffiu i ;u (k) i )=W int (ffiu i ;u (k) i )+W xt (ffiu i ) (10) whr W (ffiu i ; u i ; u (k) i )isthlinarization of virtual work W (ffiu i ; u i ; u i ;u i +ffl u i ) i ) = lim = ^E (ffiu i ;u i )L kl ^Ekl ( u i ;u i )dv + Hr w hav usd ds =1=2L kl dc kl = L kl ^Ekl ( u i ;u i ). ^E (ffiu i ;u i )S dv In ordr to obtain xprssions for th stiffnss matrix w shall laborat furthr on (11). To this nd, (11) can b rwrittn by xpanding th dfinition for ^E as W (ffiu i ; u i ; u (k) i ) = (11) 1 ((ffiu j;i + ffiu i;j )+(u j;r ffiu r;i + ffiu i;r u r;j )) 4 L kl (( u k;l + u l;k )+(u k;s u s;l + u l;s u s;k )) dv ( u j;lffiu l;i + ffiu i;l u l;j ) S dv (12) Or, by convnintly splitting th abov quation w can writ 1W (ffiu i ; u i ; u (k) i ) = 1 ((ffiu j;i + ffiu i;j )+(u j;r ffiu r;i + ffiu i;r u r;j )) 4 L kl (( u k;l + u l;k )+(u k;s u s;l + u l;s u s;k )) dv (13) 7

9 2W (ffiu i ; u i ; u (k) 1 i )= 2 ( u j;lffiu l;i + ffiu i;l u l;j ) S dv (14) By furthr rorganizing (13) and collcting trms w can writ: 1W (ffiu i ; u i ; u (k) i )= (ffiu j;i + ffiu i;j ) 1 2 (ffiu j;i + ffiu i;j ) 1 1 L kl 2 ( u k;l + u l;k ) dv 1 L kl 2 (u k;s u s;l + u l;s u s;k ) 2 (u 1 j;rffiu r;i + ffiu i;r u r;j ) L kl 2 (u k;s u s;l + u l;s u s;k ) dv 1 2 (u j;rffiu r;i + ffiu i;r u r;j ) L kl 1 2 ( u k;l + u l;k ) dv dv (15) It should b notd that th Algorithmic Tangnt Stiffnss (ATS) tnsor L kl posssss both minor symmtris (L kl = L jikl = L lk ). Howvr, major symmtry cannot b guarantd. Non associatd flow ruls in lastoplasticity lad to th loss of major symmtry (L kl 6 =L kl ). Morovr, it can b shown (i.. [19]) that an algorithmically inducd symmtry loss is obsrvd vn for associatd flow ruls. Upon obsrving minor symmtry of L kl on can writ (15) as: 1W (ffiu i ; u i ; u (k) i )= ffiu i;j L kl u l;k dv ffiu i;j L kl u k;s u l;s dv ffiu i;r u r;j L kl u k;s u l;s dv ffiu i;r u r;j L kl u l;k dv (16) Similarly, by obsrving symmtry of th scond Piola Kirchhoff strss tnsor S writ w can 2W (ffiu i ; u i ; u (k) i )= ffiu i;l u l;j S dv (17) Th wak form of quilibrium xprssions for intrnal (W int ) and xtrnal (W xt ) virtual work, with th abov mntiond symmtry of S can b writtn as W int (ffiu i ; n+1 0 u(k) i )= ffiu i;j S dv + ffiu i;r u r;j S dv (18) W xt (ffiu i ) = ρ 0 ffiu i b i ffiu i t i ds (19) Standard finit lmnt discrtization of th displacmnt fild is: u i ß ^u i = H I μu Ii (20) 8

10 whr ^u i is th approximation displacmnt fild u i, H I ar FEM shap functions and μu Ii ar nodal displacmnts. With this approximation, w hav: 1W (ffiu i ; u i ; u (k) i )= (H I;j ffiμu Ii ) L kl (H I;j ffiμu Ii ) L kl (H Q;k μu Ql ) dv (H I;r ffiμu Ii )(H J;j μu Jr )L kl (H J;k μu Js )(H Q;s μu Ql ) dv (H J;k μu Js )(H Q;s μu Ql ) dv (H I;r ffiμu Ii )(H J;j μu Jr )L kl (H Q;k μu Ql ) dv (21) 2W (ffiu i ; u i ; u (k) i )= (H I;l ffiμu Ii )(H Q;j μu Ql ) S dv (22) W int (ffiu i ; n+1 0 u(k) i )= (H I;j ffiμu Ii ) S dv + (H I;r ffiμu Ii ) (H J;j μu Jr ) S dv (23) W xt (ffiu i ) = ρ 0 (H I ffiμu Ii ) b i (H I ffiμu Ii ) t i ds (24) Upon noting that virtual nodal displacmnts ffiu Ii ar any non zro, continuous displacmnts, and sinc thy occur in all xprssions for linarizd virtual work, thy can b factord out and aftr som rarngmnt can b writtn as (whil rmmbring that 1W + 1W + W xt + W int = 0): = H I;j L kl H Q;k dv + H I;r H J;j μu Jr L kl H J;k μu Js H Q;s dv H I;r H J;j μu Jr L kl H Q;k dv + (H I;j ) S dv + ρ 0 (H I ) b i dv + H I;j L kl H J;k μu Js H Q;s dv + H I;l H Q;j S dv (H I;r ) (H J;j μu Jr ) S dv μu (H I ) t i ds (25) Th global algorithmic tangnt stiffnss matrix (tnsor) is givn as K t = = + W (ffiu i ; u i ; u (k) i u i ) H I;j L kl H Q;k dv + H I;r H J;j μu Jr L kl H J;k μu Js H Q;s dv H I;r H J;j μu Jr L kl H Q;k dv + 9 H I;j L kl H J;k μu Js H Q;s dv H I;l H Q;j S dv (26)

11 Th global algorithmic tangnt stiffnss matrix contains both th linar strain incrmntal stiffnss matrix and th nonlinar gomtric and initial strss incrmntal stiffnss matrix. Th vctor of xtrnally applid load is thn R = ρ 0 (H I ) b i dv (H I ) t i ds (27) whil th load vctor from lmnt strsss is givn as F = (H I;j ) S dv + (H I;r ) (H J;j μu Jr ) S dv (28) It is important to not that th algorithmic tangnt stiffnss tnsor, vctor of xtrnally applid loads, and th vctor of lmnt strsss ar scond and fourth ordr tnsor. Convrsion from tnsors to matrics and vctors is prformd by th assmbly functions. It is also important to not that th tnsor of unknown displacmnts μu Ql is flattnd to a on dimnsional vctor ( u i ) through propr implmntation. Th itrativ chang in displacmnt vctor u i is obtaind by stting th linarizd virtual work to zro W (ffiu i ;u (k+1) i )=0 ) W(ffiu i ;u (k) i )= W(ffiu i ; u i ; u (k) i ) (29) In particular, th choic of th undformd configuration Ω 0 for a computational domain ( =Ω 0 ) yilds th Total Lagrangian (TL) formulation. Th itrativ displacmnt u i is obtaind from th quation whr and W (ffiu i ; n+1 u (k) i )= W(ffiu i ; u i ; n+1 u (k) i ) (30) W (ffiu i ; n+1 u (k) i ) = W (ffiu i ; u i ; n+1 u (k) i ) = ^E (ffiu i ; n+1 u (k) i ) n+1 S (k) dv ρ 0 ffiu i n+1 b i ffiu i n+1 t i ds (31) + ^E (ffiu i ; n+1 u (k) i ) n+1 L (k) kl ^E kl ( u i ; n+1 u (k) i ) dv d ^E (ffiu i ; u i ) n+1 S (k) dv (32) In th cas of hyprlastic plastic rspons, th scond Piola Kirchhoff strss n+1 S (k) obtaind by intgrating th constitutiv law, dscribd in sction 3. is It should b notd that by prforming th intgrations in th intrmdiat configuration, w obtain th Mandl 10

12 strss n+1 μ T and subsquntly th scond Piola Kirchhoff strss μ Skj = μ Cik 1 μ T. Th ATS tnsor μl kl is thn obtaind basd on μ Skj. In ordr to obtain th scond Piola Kirchhoff strss S kj and ATS tnsor in th initial configuration w nd to prform a pull-back from th intrmdiat configuration to th initial stat n+1 S = n+1 F p n+1 ip F p n+1μ jq Spq (33) n+1 L kl = n+1 F p n+1 im F p n+1 jn F p n+1 kr F p ls n+1μl mnrs (34) Th formulation prsntd abov is rathr gnral and rlvant to a larg st of nginring solids, both isotropic and anisotropic. This gnrality will b furthr nhancd in sction 3 with gnral, constitutiv lvl computations that can handl both isotropic and gnral anisotropic matrials. 3 Finit Dformation Hyprlasto Plasticity 3.1 Hyprlasticity A matrial is calld hyprlastic or Grn lastic, if thr xists an lastic potntial function W, also calld th strain nrgy function pr unit volum of th undformd configuration, which rprsnts a scalar function of strain of dformation tnsors, whos drivativs with rspct to a strain componnt dtrmins th corrsponding strss componnt. Th most gnral form of th lastic potntial function, is dscribd in quation (35), with rstriction to pur mchanical thory, by using th axiom of locality and th axiom of ntropy production 2 : W = W (X K ;F kk ) (35) By using th axiom of matrial fram indiffrnc 3, w conclud that W dpnds only on X K and C IJ, that is: W = W (X K ;C IJ ) or: W = W (X K ;c ) (36) 2 S Marsdn and Hughs [31] pp S Marsdn and Hughs [31] pp

13 In th cas of matrial isotropy, th strain nrgy function W (X K ;C IJ ) blongs to th class of isotropic, invariant scalar functions. It satisfis th rlation: W (X K ;C KL )=W X K ;Q KI C IJ (Q JL ) t (37) whr Q KI is th propr orthogonal transformation. If w choos Q KI = R KI, whr R KI is th orthogonal rotation transformation, dfind by th polar dcomposition thorm in quation (s Malvrn [30]), thn: W (X K ;C KL )=W(X K ;U KL )=W(X K ;v kl ) (38) Right and lft strtch tnsors, U KL, v kl hav th sam principal valus (principal strtchs) i ; i = 1; 3 so th strain nrgy function W can b rprsntd in trms of principal strtchs, or similarly in trms of principal invariants of th dformation tnsor: W = W (X K ; 1 ; 2 ; 3 ;)=W(X K ;I 1 ;I 2 ;I 3 ) (39) whr: I 1 I 2 I 3 df = C II df = 1 2 I 2 1 C IJ C JI df = dt (C IJ )= 1 6 IJK PQR C IP C JQ C KR = J 2 (40) Lft Cauchy Grn tnsors is dfind as C IJ = (F ki ) t F kj, and th spctral dcomposition thorm (s Simo and Taylor [49]) for symmtric positiv dfinit tnsors stats that C IJ = 2 A N (A) I N (A) J whr A = 1; 3 and N I ar th ignvctors (kn I k = 1) of C IJ. W can A thn calculat roots ( 2 A) of th charactristic polynomial P ( 2 A) df = 6 A + I 1 6 A I 2 4 A + I 3 =0 (41) It should b notd that no summation is implid ovr indics in parnthsis. For xampl, in th prsnt cas N (A) I actual quation C IJ = 2 A is th A th ignvctor with mmbrs N (A) 1, N (A) 2 and N (A) 3, so that th N (A) I N (A) J can also b writtn as C IJ = P A=3 A A=1 2 (A) N (A) I N (A) J. In ordr to follow th consistncy of indicial notation in this work, w shall mak an ffort to rprsnt all th tnsorial quations in indicial form. 12

14 Th mapping of th ignvctors is givn by (A) n (A) i = F ij N (A) J (42) whr kn (A) i k1. Th spctral dcomposition of F ij, R ij and b is thn givn by F ij = A n (A) i N (A) J A (43) R ij = 3X A=1 n (A) i N (A) J (44) b = 2 A n (A) i n (A) j A (45) Rcntly, Ting [53] and Morman [33] hav usd Srrin's rprsntation thorm in ordr to dvis a usful rprsntation for gnralizd strain tnsors E IJ through C m IJ. Aftr som tnsor algbra th Lagrangian igndyad N (A) I N (A) J N (A) I N (A) C J = 2 IJ I 1 2 (A) (A), can b writtn as ffi IJ + I 3 2 (A) (C 1 ) IJ 2 4 (A) I 1 2 (A) + I 3 2 (A) It should b notd that th dnominator in quation (46) can b writtn as: (46) 2 4 (A) I 1 2 (A) + I 3 2 (A) = 2 (A) 2 (B) 2 (A) 2 df (C) = D (A) (47) whr indics A; B; C ar cyclic prmutations of 1; 2; 3. It follows dirctly from th dfinition of D (A) in quation (47) that 1 6= 2 6= 3 ) D (A) 6= 0 for quation (46) to b valid. Similarly w can obtain: (C 1 ) IJ = 2 A N (A) I N (A) J A Th most gnral form of th isotropic strain nrgy function W in trms of of principal strtchs can b xprssd as: (48) W = W (X K ; 1 ; 2 ; 3 ;) (49) 13

15 In ordr to obtain th scond Piola Kirchhoff strss tnsor S IJ (and othr strss masurs) it is ncssary to calculat th =@C IJ. Morovr, th matrial tangnt stiffnss tnsor L IJKL rquir scond ordr drivativs of th strain nrgy 2 W=(@C KL ). In ordr to obtain ths quantitis w introduc a scond ordr tnsor M (A) IJ M (A) IJ df = 2 (A) N (A) I N (A) J (50) = (F ii ) 1 n (A) i = n (A) j (FjJ ) t 1 D (A) C IJ I 1 2 (A) ffi IJ + I 3 2 (A) (C 1 ) IJ from (46) whr D (A) was dfind by quation (47). With M (A) IJ C IJ = 4 A M (A) IJ A dfind by quation (50), w obtain (51) and it also follows (C 1 ) IJ = M (1) IJ + M (2) IJ + M (3) IJ (52) It can also b concludd that: ffi IJ = 2 (1) M (1) IJ + 2 (2) M (2) IJ + 2 (3) M (3) IJ = 2 A sinc, from th orthogonal proprtis of ignvctors ffi IJ = 3X A=1 N (A) I N (A) J = N (A) I A N (A) J A W also dfin th Simo Srrin fourth ordr tnsor M IJKL as: M (A) IJKL df IJ KL 1 I IJKL ffi KL ffi IJ + 2 (A) D (A) + I 3 2 (A) 2 (A) I 3 (C 1 ) IJ (C 1 ) KL M (A) IJ A ffi IJ M (A) KL + M (A) IJ ffi KL + (C 1 ) IK (C 1 ) JL +(C 1 ) IL (C 1 ) JK (C 1 ) IJ M (A) KL + M (A) IJ (C 1 ) KL D 0 (A) M(A) IJ A complt drivation of M IJKL is givn by Simo and Taylor [49]. W can thn dfin hyprlastic strss masurs as ffl 2. Piola Kirchhoff strss tnsor S IJ 14 M(A) KL (53) (54) (55)

16 ffl Mandl strss tnsor T IJ = C IK S KJ ffl 1. Piola Kirchhoff strss tnsor P ij = S IJ (F ii ) t ffl Kirchhoff strss tnsor fi ab = F ai (F bj ) t S IJ whr (A) IJ W( (A) IJ W ( (A) ) w A = 1 3 = 1 vol Th tangnt stiffnss oprator is dfind IJ J (C 1 ) IJ w A (M (A) IJ ) A iso W ( ~ (A) ~ B ~ B W ( ~ ~ (A) ~ (A) (57) L IJKL = vol L IJKL + iso L IJKL (58) with J 2@2vol (C 1 ) KL (C 1 ) IJ + vol L IJKL = (C 1 ) KL (C 1 ) IJ W(J) I (C 1 ) (59) iso L IJKL = Y AB (M (B) KL) B (M (A) IJ ) A +2w A (M (A) IJKL) A (60) 3.2 Multiplicativ Dcomposition Multiplicativ dcomposition of th dformation gradint is usd as a kinmatical basis for th dvlopmnts dscribd hr. Th motivation for th multiplicativ dcomposition can b tracd back to th arly works of Bilby t al. [3], and Krönr [23] on micromchanics of crystal dislocations and application to continuum modling. In th contxt of larg dformation 15

17 lastoplastic computations, th work by L and Liu [26], Fox [14] and L [25] gnratd an arly intrst in multiplicativ dcomposition. Th appropriatnss of multiplicativ dcomposition tchniqu for soils may b justifid from th particulat natur of th matrial. From th micromchanical point of viw, plastic dformation in soils ariss from slipping, crushing, yilding and plastic bnding 4 of granuls or platlts comprising th assmbly 5. It can crtainly b argud that dformations in soils ar prdominantly plastic, howvr, rvrsibl dformations could dvlop from th lasticity of individual soil grains, and could b rlativly larg, whn particls ar lockd in high dnsity spcimns. Ω Ω 0 σ F x Currnt Configuration x dx d X 1 X F F ē Rfrnc Configuration Intrmdiat F p Configuration X F p -1 Figur 2: Multiplicativ dcomposition of dformation gradint: schmatics. Th rasoning bhind multiplicativ dcomposition is a rathr simpl on. If an infinitsimal nighborhood of a body x i ;x i +dx i in an inlastically dformd body is cut out and unloadd to an unstrssd configuration, it would b mappd into ^x i ; ^x i + d^x i. Th transformation would b comprisd of a rigid body displacmnt 6 and purly lastic unloading. Th lastic unloading is fictitious, sinc in matrials with a strong Baushingr's ffct unloading will lad to loading in anothr strss dirction, and if thr ar rsidual strsss, th body must b cut out in small pics, and thn vry pic rlivd of strsss. Th unstrssd 4 For plat lik clay particls. 5 S also Borja and Alarc on [5] and Lamb and Whitman [24]. 6 Translation and rotation. 16 x u Ω d x

18 configuration is thus incompatibl and discontinuous. Th position ^x i is arbitrary, and w may assum a linar rlationship btwn dx i and d^x i,inthform 7 : d^x k =(F ik) 1 dx i (61) whr (F ik) 1 is not to b undrstood as a dformation gradint, sinc it may rprsnt th incompatibl, discontinuous dformation of a body. By considring th rfrnc configuration of a body dx i, thn th connction to th currnt configuration is: dx k = F ki dx i ) d^x k =(F ik) 1 F dx j (62) so that on can dfin: F p kj df = (F ik) 1 F ) F df = F ki F p kj (63) Th plastic part of th dformation gradint, F p kj rprsnts micro mchanically, th irrvrsibl procss of slipping, crushing dislocation and macroscopically th irrvrsibl plastic dformation of a body. Th lastic part, F ki rprsnts micro mchanically a pur lastic rvrsal of dformation for th particulat assmbly, macroscopically a linar lastic unloading toward a strss fr stat of th body, not ncssarily a compatibl, continuous dformation but rathr a fictitious lastic unloading of small cut outs of a dformd particulat assmbly or continuum body, 3.3 Constitutiv Rlations W propos th fr nrgy dnsity W, which is dfind in th intrmdiat configuration μω, as ρ 0 W ( μ C ;» ff)=ρ 0 W (μ C )+ρ 0 W p (» ff ) (64) whr W ( μ C ) rprsnts a suitabl hyprlastic modl in trms of th lastic right dformation tnsor μ C, whras W p (» ff ) rprsnts th hardning. inquality bcoms: D = μ T μ L p + X ff 7 rfrrd to sam Cartsian coordinat systm. 17 Th prtinnt dissipation μk ff _» ff 0 (65)

19 whr μ T is th Mandl strss and μ L p is th plastic vlocity gradint dfind on μω. W now dfin th lastic domain B as B = f μ T ; μ K ff j Φ( μ T ; μ K ff )» 0g (66) Whn yild function Φ is isotropic in μ T (which is th cas hr) in conjunction with lastic isotropy,w can conclud that μ T is symmtric and wmay rplac μ T by fi in yild function Φ. As to th choic of an lastic law, it is mphasizd that this is largly a mattr of convninc, sinc w shall b daling with small lastic dformations. Hr, th No Hookan lastic law isadoptd. Th constitutiv rlations can now b writtn as μl p := F _ p ik F p Λ jk = T μ = _μ M μ (67) μk ff = μ Kff (μ» fi ) (68) _μ» fi fi,» fi (0) = 0 (69) whr μ Kff ;ff = 1;2; is th hardning strss", Φ Λ (fi ; μ Kff ) is th plastic potntial, μ» fi is intrnal variabls, _μ is consistncy paramtr dtrmind from th loading conditions 8 and F p ik =(μ F li ) 1 F lk is th plastic part of th dformation gradint. 3.4 Implicit Intgration Algorithm Th incrmntal dformation and plastic flow ar govrnd by th systm of volution quations (67) and (69). Th flow rul (67) can b intgratd to giv By using th multiplicativ dcomposition and quation (70) w obtain n+1 F p = xp μ n+1 μ Mik n F p kj (70) F = μ F ik F p kj ) μ F ik = F F p kj 1 (71) n+1μ F = n+1 F im ( n Fmk) p 1 xp μ n+1 Mkj μ = n+1 F μ ;tr ik xp μ n+1 Mkj μ 8 Ths ar th Karush Kuhn Tuckr complmntary conditions in th spcial cas of fully associativ thory, dfining th Standard Dissipativ Matrial, cf. [15]. 18 (72)

20 whr w usd that Th lastic dformation is thn n+1μ F ;tr ik = n+1 F im ( n F p km) 1 (73) n+1μ C df = n+1μ T F n+1 μ im F mj n+1 μ F ;tr = xp μ n+1 M μ T ir rk = xp μ n+1 M μ T n+1 μ ir C ;tr T n+1 F μ ;tr kl xp μ n+1 Mlj μ rl xp μ n+1 Mlj μ (74) By rcognizing that th xponnt of a tnsor can b xpandd in Taylor sris 9 xp μ n+1 Mlj μ = ffilj μ n+1 Mlj μ + 1 n+1 μ Mls μ n+1 μ Msj μ + (75) 2 and by using th scond ordr xpansion in quation (74) and aftr som tnsor algbra w obtain n+1μ C = n+1 μ C ;tr rj μ n+1 μ Mir n+1μ C ;tr rj + ( μ)2 2 ( μ) ( μ)4 4 + n+1 μ C ;tr il n+1 μ Mis n+1 μ Msr n+1μ C ;tr rj n+1 Mis μ n+1 Msr μ n+1μ;tr C rl n+1 Mis μ n+1 Msr μ n+1μ C ;tr rl n+1 μ Mlj + +2 n+1 μ Mir n+1μ C ;tr rl n+1 Mlj μ 1 2 n+1 Mls μ n+1 Msj μ n+1 Mlj μ + n+1 Mls μ n+1 Msj μ n+1μ;tr C rl n+1 μ Mir n+1μ C ;tr rl n+1 μ Mls n+1 μ Msj + First ordr sris xpansion includs constant and linar (up to μ) mmbrs. Scond ordr xpansion includs th complt quation abov. Rmark 3.1 Th Taylor's sris xpansion in quation (75) is a propr approximation for th gnral nonsymmtric tnsor μ M lj. That is, th approximat solution givn by quation (76) is valid for a gnral anisotropic solid. This contrasts with th spctral dcomposition family of solutions 10 which ar rstrictd to isotropic solids. Rmark 3.2 In th limit, whn th dformations ar sufficintly small, th solution (76) collapss to lim ffi +2 n+1 ffl = + ffi +2 n+1 ffl ;tr F!ffi 9 S for xampl Parson [37]. 10 S Simo [45]. 19 (76)

21 ) n+1 ffl = μ n+1 μ M 2 μ n+1 μ Mir n+1 ffl tr rj μ n+1 μ M 2 μ n+1 ffl tr il n+1 μ Mlj + μ 2 n+1 μ Mil n+1 μ Mlj +2 μ 2 n+1 μ Mir n+1 ffl tr rln+1 μ Mlj = ffi +2 n+1 ffl ;tr 2 μ n+1 μ M n+1 ffl tr μ n+1 μ M (77) which rprsnts a small dformation lastic prdictor plastic corrctor quation in strain spac. In working out th small dformation countrpart (77) it was usd that lim F!ffi n+1μ C = ffi +2 n+1 ffl 2 μ n+1 ffl tr il n+1 μ Mlj fi n+1 μ M μ fi 1 (78) By nglcting th highr ordr trm with μ 2 in quation (76), th solution for th right lastic dformation tnsor n+1 C μ can b writtn as n+1μ C = n+1 C μ;tr μ n+1 Mir μ n+1μ C ;tr rj Th hardning rul (69) can b intgratd to giv + n+1 μ C ;tr il n+1 μ Mlj (79) n+1» ff = n» ff ff fi fifififin+1 (80) Th incrmntal problm is dfind by quations (79), (80), th constitutiv rlations and th Karush Kuhn Tuckr (KKT) conditions whr μ <0 ; n+1μ IJ fi fifififin+1 (81) n+1 K ff fi fifififin+1 (82) n+1 Φ» 0 ; μ n+1 Φ=0 (83) Φ=Φ(μ T ;K ff ) (84) 20

22 Rmark 3.3 Th Mandl strss tnsor μ T can b obtaind from th scond Piola Kirchhoff strss tnsor μ Skj and th right lastic dformation tnsor μ C ik as μ T = μ C ik This st of nonlinar quations will b solvd with a Nwton typ procdur, dscribd blow. For a givn n+1 F, or n+1 μ C ;tr, th upgradd quantitis n+1 μ SIJ and n+1 K ff can b found, thn th appropriat pull back toω 0 or push forward to Ω will giv n+1 S IJ and n+1 fi μ Skj n+1 S IJ = n+1 F p ii 1 n+1 μ SIJ n+1 F p jj T (85) Th lastic prdictor, plastic corrctor quation n+1 fi = n+1 F μ n+1μ n+1 1 ii SIJ FjJ (86) n+1μ C = n+1 μ C ;tr μ n+1 (87) is usd as a starting point for a Nwton itrativ algorithm. In th prvious quation, w hav introducd tnsor = μ n+1 μ Mir n+1μ C ;tr rj + n+1 μ C ;tr il n+1 μ Mlj (88) Th dfinition of abov assums us of first ordr xpansion in (76). Th trial right lastic dformation tnsor is dfind as n+1μ C ;tr = n+1μ F ;tr ri W introduc a tnsor of dformation rsiduals T n+1 F μ;tr rj = n+1 F rm ( n FiM) p 1 T n+1 F n 1 rs F p js R = C μ n+1μ C ;tr μ z} n+1 z } currnt BackwardEulr Th tnsor R rprsnts th diffrnc btwn th currnt right lastic dformation tnsor and th Backward Eulr right lastic dformation tnsor. Th trial right lastic dformation tnsor n+1 μ C ;tr (89) (90) is maintaind fixd during th itration procss. Th first ordr Taylor sris xpansion can b applid to th tnsor of rsiduals R in ordr to obtain th itrativchang, th nw rsidual R nw R nw = R old from th old R old + d μ C + d( μ) n+1 μ Tmn 21 d μ Tmn ff dk ff (91)

23 Furthrmor, sinc w can writ μt mn = μ C mk μ S kn ) μ C sk 1 μ Tsn = μ S kn (92) d μ Tmn = d μ C mk μ S kn + μ C mk d μ Skn = d μ C mk μ S kn μ C mk μ L knpq d μ C pq = d μ C mk μc sk 1 μ Tsn μ C mk μ L knpq d μ C pq (93) so that aftr stting R nw 0= R old + (ffi im ffi nj + μ Tik Upon introducing th notation = 0 and som tnsor algbra w obtain + d( μ) n+1 + dk ff ff μc sj 1 μ Tsk μ@n+1 μ Tpq μ C pk μ L kq ) d μ C (94) T mn = ffi im ffi nj + μ Tik w can solv for d μ C By using that it follows from (96) μc 1 μ sj Tsk + 1 mn 2 Tpq μ d μ C pq =(T mnpq ) 1 ψ R old mn d( μ) n+1 mn ff dk ff! dk ff fi d» fi = d( @K fi = d( μ) H fffi μ C pk μ L kq fi (97) ψ! d C μ pq =(T mnpq ) 1 R old mn d( μ) n+1 mn + d( μ) H fi A first ordr Taylor sris xpansion of a yild function togthr with (97) provids nw Φ( μ T ;K ff )= + old Φ( T μ ;K ff )+ T μ ;K ff ) T μ 1 pn sq Tsn μ + 1 μ T ;K ff μ T mn μ C mk μ L knpq! d μ C pq (98) d( μ T ;K ff ff H fi (99) 22

24 Upon introducing th following notation F pq μ T ;K ff Tpn μc sq 1 μ Tsn and with th solution for d μ C pq from (98), (99) μ T ;K ff Tmn μ C mk μ L knpq (100) nw Φ( T μ ;K ff )= old Φ( T μ ;K ff )+ ψ!! + F pq (T mnpq ) ψ R 1 old mn d( μ) n+1 mn + d( μ) Λ H fi d( μ T ;K ff ff H fi (101) Aftr stting nw Φ( μ T ;K ff ) = 0 w can solv for th incrmntal consistncy paramtr d( μ) d( μ) = old Φ F pq (T mnpq ) 1 Rmn old F pq (T mnpq ) 1 n+1 mn μ F pq (T mnpq ) n+1 Λ H fffi H fi (102) Rmark 3.4 In th limit, for small dformations, th incrmntal consistncy paramtr d( μ) bcoms d( μ) = n mn E mnpq old Φ (n mn E mnpq ) ψ ψ ffi mp ffi qn + E mn! 1 sinc in th limit, as dformations bcom small ffi pm ffi nq + E pq! 1 lim T mnpq = ffi pm ffi nq + mn F!ffi R old mn n+1 m ff H E fi (103) lim F pq = E mnpq F!ffi mn lim pq F!ffi = 2 m pq lim R pq F!ffi = 2 ffl pq (104) Upon noting that th rsidual R pq is dfind in strain spac, th incrmntal consistncy paramtr d( μ) compars xactly with it's small strain countrpart (Jrmić and Stur [19]). Th procdur dscribd blow summarizs th implmntation of th rturn algorithm. 23

25 3.4.1 Rturn Algorithm Givn th right lastic dformation tnsor n μ C pq and a st of hardning variabls n K ff at a spcific quadratur point in a finit lmnt, w comput th rlativ dformation gradint n+1 f for a givn displacmnt incrmnt n+1 u i n+1 f = ffi + u i;j (105) and th right dformation tnsor n+1μ C ;tr = n+1 f ir n F rk T n+1 f n kl F lj =( n Frk) T n+1 T f n+1 ir f n kl F lj (106) Thn w comput th trial lastic scond Piola Kirchhoff strss and th trial lastic Mandl strss tnsor n+1μ T ;tr n+1μ S ;tr =2 = n+1 μ C n+1μ C ;tr W thn valuat th yild function n+1 Φ tr ( T μ;tr ;K ff ), and st (107) n+1μ S ;tr lj (108) n+1μ C = n+1 C μ;tr n+1 K ff = n K ff n+1 T = n T ;tr and xit constitutiv intgration procdur. If n+1 Φ tr currnt incrmnt. If th yild critrion has bn violatd ( n+1 Φ tr > 0) procd stp 1. k th itration. Known variabls n+1μ C (k) ; n+1» (k) ff ; valuat th yild function and th rsidual n+1 K (k) ff ; Φ (k) = Φ( n+1 T μ(k) ; n+1 K (k) ff ) R (k) = n+1 C μ;(k) n+1μ C ;tr» 0 thr is no plastic flow in th n+1 T (k) ; n+1 μ (k) n+1 μ (k)n+1 (k) stp 2. Chck for convrgnc, Φ (k)» NTOL and kr (k) k»ntol. If convrgnc critrion is satisfid st n+1μ C = n+1 C μ(k) 24

26 n+1» ff = n+1» (k) ff n+1 K ff = n+1 K (k) ff n+1 T = n+1 T (k) n+1 μ = n+1 μ (k) Exit constitutiv intgration procdur. stp If convrgnc is not achivd comput th lastic stiffnss tnsor L kl μl (k) kl μ C 2 μ C (k) kl (109) stp 4. Comput th incrmntal consistncy paramtr d( μ (k+1) ) d( μ (k+1) )= μf (k) mn mn (k) μ (k) Φ (k) μf (k) mn R mn (k) μ F mn mn ff μ Hff ff μ Hff (k) (110) whr μh ff (k) = H fi ; μf mn (k) = F pq (k) T mnpq (k) 1 F pq T μ(k) ;K ff (k) Tpn mn T mn = ffi im ffi nj + T μ(k) ik μc ;(k) 1 μ sq T (k) sn μc ;(k) 1 μ(k) sj T sk stp 5. Updat th consistncy paramtr μ T μ(k) ;K ff (k) Tmn μ Tpq μ C ;(k) mk μ C ;(k) pk μl ;(k) knpq μl ;(k) kq μ (k+1) = μ (k) +d( μ (k+1) ) (111) stp 6. Calculat th incrmnts for th right dformation tnsor, th hardning variabl and th Mandl strss d μ C ;(k+1) pq = T (k) mnpq 1 ψ R (k) mn d( μ (k+1) ) n+1 (k) mn mn + ff d( μ (k+1) ) μ H (k) ff! (112) 11 From stp 3. to stp 9. all of th variabls ar in intrmdiat n +1 configuration. For th sak of brvity w ar omitting suprscript n

27 d μ T (k+1) mn d» (k+1) ff dk (k+1) ff = d μ C ;(k+1) mk = d( μ fi (113) = d( μ (k+1) ) H (k) Λ;(k) μc ;(k) 1 μ sk T (k) sn + 1 C 2 μ;(k) fi (114) μl ;(k) knpq d μ C ;(k+1) pq (115) stp 7. Updat th right dformation tnsor C μ;(k+1) pq, hardning variabl K ff (k+1) Mandl strss μ T (k+1) mn μc ;(k+1) pq = C μ;(k) pq + d( C μ;(k+1) pq )» (k+1) ff =» (k) ff + d(» (k+1) ff ) K (k+1) ff = K (k) ff + d(k (k+1) ff ) stp 8. valuat th yild function and th rsidual and μt (k+1) mn = μ T (k) mn + d( μ T (k+1) mn ) (116) R (k+1) stp 9. St k = k + 1 and and rturn to stp 2. = μ C ;(k+1) Φ (k+1) =Φ(μ T (k+1) μ C ;tr μ (k) = μ (k+1) μc ;(k) pq = C μ;(k+1) pq» (k) ff =» (k+1) ff K (k) ff = K (k+1) ff ;K (k+1) ff ) μ (k+1) (k+1) 3.5 Algorithmic Tangnt Stiffnss Tnsor Starting from th lastic prdictor plastic corrctor quation (117) μt (k) mn = μ T (k+1) mn (118) n+1μ C = n+1 μ C ;tr μ n+1 (119) 26

28 to which w apply a first ordr Taylor sris xpansion to obtain (aftr som tnsor algbra), ψ! d C μ =(T mn ) 1 d C μ;tr d( μ) + Λ H fffi fi whr T mn was dfind in (95) Nxt w us th first ordr Taylor sris xpansion of th yild function dφ( μ T ;K Tmn d μ ff dk ff = F pq d μ ff d( μ) H fi = 0 (121) with F pq dfind in (100). By using th solution for d μ C from (120) w can writ! F pq (T mnpq ) ψd 1 C μ;tr mn d( μ) mn + μd( Λ H @K ff d( μ) H fi = 0 (122) W ar now in th position to solv for th incrmntal consistncy paramtr d( μ) whr w hav usd to dnot Sinc d( μ) = F pq (T mnpq ) 1 d μ C ;tr mn (123) =F pq (T mnpq ) 1 n+1 mn μf pq (T mnpq ) n+1 Λ H Λ H fi (124) and by using (120) w can writ Thn d μ C pq = (T mnpq ) 1 ψ d μ Skn = 1 2 μ L knpq d μ C pq (125) ffi mv ffi nt F op (T rsop ) 1 ffi rv ffi st μ F op (T rsop ) 1 ffi rv ff H fi mn + d μ C ;tr vt (126) d μ C pq = μ P pqvt d μ C ;tr vt (127) 27

29 whr ψ μp pqvt =(T mnpq ) 1 ffi mv ffi nt F ψ!! ab(t vtab ) 1 mn Λ H fi Th algorithmic tangnt stiffnss tnsor μl AT S kl dfind as (128) (in intrmdiat configuration μω) is thn μl AT S knvt = μl knpq μ P pqvt (129) Pull back to th rfrnc configuration Ω 0 yilds th algorithmic tangnt stiffnss tnsor L kl in th rfrnc configuration Ω 0 n+1 L AT S kl = n+1 F p n+1 im F p n+1 jn F p n+1 kr F p ls n+1μl AT S mnrs (130) Rmark 3.5 In th limit, for small dformations and isotropic rspons, th Algorithmic Tangnt Stiffnss tnsor L AT S kl bcoms sinc lim F!ffi lim F!ffi μl AT S vtpq = E AT S vtpq = R knvt n cdr cdvt R knmr H mr μt mnpq = mnpq = ffi pm ffi nq + mn E rs lim F!ffi F ab = 1 2 n cde cdab lim H mn = m mn Λ H fffi fi lim =n ab R abmn H Λ H fffi fi It is notd that th Algorithmic Tangnt Stiffnss tnsor givn by (131) compars xactly with it's small strain countrpart (Jrmić and Stur [19]). 3.6 Matrial Modl A larg dformation matrial modl usd in computations is brifly dscribd hr. Th modl rlis on th dvlopmnt bhind th so calld MRS Lad modl (Stur t al. [51]) 28

30 and is subsquntly dnotd th B Modl. Th B Modl is a singl surfac modl, with uncoupld con portion and cap portion hardning. Vry low confinmnt rgion was carfully modld and th yild surfac was shapd in such a way to mimic rcnt findings obtaind during Micro Gravity Mchanics tsts aboard Spac Shuttl (Stur t al. [50]). Th larg dformation modl dfinition is basd on th us of th Mandl strss μ T for dscribing yild and potntial surfacs. A dtaild dscription of th modl is givn by Jrmić t al. [18]. 4 Numrical Simulations of Micro Gravity Mchanics In this sction w prsnt numrical modling of low confinmnt, microgravity larg dformation triaxial tst prformd during Spac Shuttl STS 79 mission in Sptmbr Figur 3 shows load displacmnt and volum displacmnt data for thr low confinmnt tsts. Th rspons curvs rprsnt load displacmnt data as thy wr masurd dur- vrtical forc [kn] MGM130 MGM052 MGM vrtical displacmnt [m] -6 3 volum chang (dilation) [10 m ] MGM052 MGM130 MGM displacmnt [m] Figur 3: Micro Gravity Mchanics, load displacmnt and volum displacmnt curvs for th thr tsts. ing th xprimnts. Th signal contains significant nois and th prsntd data ar in raw form. Th lastic rspons appars to b vry stiff (from unloading rloading loops). Dtaild dscription of th xprimntal stup is givn by Stur t al. [50]. Th thr dimnsional finit lmnt msh usd to modl th MGM tst is dpictd in Figurs 4. Instad of dvloping two dimnsional finit lmnt formulation, w hav optd for a full thr dimnsional implmntation. Although th stat of strss is triaxial, w modl th xprimnts with a 3D modl. Six quadratic 20 nod brick lmnts whr 29

31 0.075m m m Figur 4: Finit lmnt msh for th MGM spcimn. chosn to modl on ighth of th spcimn. Th analysis was prformd in two stags. First stag involvd isotropic comprssion to th dsign prssur. For th first stag only symmtry displacmnt boundary conditions wr in plac. Influnc of th mmbran was rmovd, sinc th mmbran dos not hav significant stiffnss in comprssion, and mmbran prstrssing had a minor ffct at this stag. During this stag th rspons was purly hyprlastic. Aftr th th first stag, th displacmnt boundary conditions wr changd by adding th movabl boundary at th top. Th top movabl boundary applid displacmnts to th top nods by mans of quivalnt forcs, obtaind through th partial invrsion of a stiffnss matrix. Th mmbran influnc was modld by adding quivalnt stiffnss (springs) to th boundary nods. Instad of using thin, highly distortd brick lmnts (mmbran is 0:3mm, distortion ratio would b (2 Λ37:5mm Λß=8)=(0:3mm) ß 100=1). w optd for th quivalnt spring mthod. Th output from th on lmnt xtnsion tsts on th hyprlastic latx rubbr spcimn whr usd to form a non linar spring of appropriat stiffnss. Consistnt intgration of th stiffnss trms for th quadratic brick lmnt thn supplid quivalnt spring stiffnss. Spcial attntion was givn to th spcimn nds, whr th latx mmbran 30

32 was wrappd around th nd platn and cratd a ring in th horizontal plan (paralll to nd platn) which was stiffr than th unstrtchd mmbran surrounding th spcimn. Th last row ofnodswas thus supportd by stiffr quivalnt mmbran lmnts. Th matrial paramtrs for th B Matrial Modl for all thr confining prssurs 12 wr kpt th sam xcpt for th Young's modulus. This consistncy in matrial paramtrs is important, sinc all thr spcimns containd th sam Ottawa F-75 sand at 85% rlativ dnsity initial confinmnt p = 0.05kPa vrtical forc [kn] MGM xprimnt -6 3 volum chang [10 m ] (a) vrtical displacmnt [m] (b) displacmnt [m] Figur 5: Mchanics of granular matrials rsponss, initial confinmnt (p 0 =0:05kPa) tst (a) load dformation and (b) volum dformation xprimnts and numrical rsults. (a) vrtical forc [kn] initial confinmnt p = 0.52kPa MGM xprimnt vrtical displacmnt [m] (b) -6 3 volum chang [10 m ] displacmnt [m] Figur 6: Mchanics of granular matrials rsponss, initial confinmnt (p 0 =0:52kPa) tst (a) load dformation and (b) volum dformation xprimnts and numrical rsults. 12 E = 300:0; 360:0; 700:0 kn=m 2 ; ν =0:2;p c = 1000:0 kn=m 2 ; p t =0:0kN=m 2 ; n =0:2;a=5:0;b =0:707 ; init =2:5;b 1 =1:0;d hard = 5000:0 ; hard =0:5; rs =0:15 ; pak =1:75 ; start =0:25 ; l =1:0;c con =0:030 ; r =1:00 ; c cap =0:30 ; p c;0 = 1000:0 kn=m 2 ; a s = 100:0 ;b s =0:707) 31

33 initial confinmnt p = 1.30kPa MGM xprimnt 90.0 vrtical forc [kn] volum chang [10 m ] (a) vrtical displacmnt [m] (b) displacmnt [m] Figur 7: Mchanics of granular matrials rsponss, initial confinmnt (p 0 =1:30kPa) tst (a) load dformation and (b) volum dformation xprimnts and numrical rsults. Figurs 5(a), 6(a) and 7(a) show comparison of numrical modling with th tst data for load displacmnt. Following obsrvations ar mad. Th initial (lastic) stiffnss is highr in th actual xprimnts. Th pak strngth is modld quit accuratly, whil th post pak bhavior is slightly stiffr in th numrical xprimnt. Th rsidual stiffnss is softr in th numrical modl than obsrvd in th MGM tsts. This can b xplaind by th stiffr spcimn nds in a physical tst. In othr words, th latx mmbran wrappd around th nd platns (th nd platns ar 30% widr than th spcimn) usually sticks to th nd platn aftr som radial displacmnts and thn acts as a full rstraint. Th friction btwn nd platns and th sand spcimn can also add to th whol spcimn stiffnss, howvr, th nd platns wr mad of highly polishd tungstn carbid, which has a vry low friction angl with quartz sand (3 ffi ), and w hav thus dcidd to nglct th influnc of nd platn friction on th ovrall rspons. It is of intrst to not that th maximum mobilizd friction angl is in th rang of 70 ffi and th dilatancy angls obsrvd in th arly parts of th xprimnts ar 30 ffi, which is unusually high. Figurs 5(b), 6(b) and 7(b) shows comparison of volumtric displacmnt data for xprimnts and numrical modling. In modling th lowst confinmnt (p 0 = 0:05kPa) lvl w corrctly prdict complt lack ofvolumtric comprssion. Numrical prdictions for two othr confinmnts (p 0 = 0:52kPa, p 0 = 1:20kPa) shows small amount of initial volum comprssion which was not obsrvd in xprimnts. Figur 8 shows atypical spcimn bfor and aftr th tst. Th latx ring formd by wrapping of mmbran around nd platns is visibl on both spcimn nds. 32

34 Figur 8: Th spcimn (p =1:30 kpa) bfor and aftr th tst. Th ffct of th latx mmbran on th load displacmnt bhavior of spcimn cannot b nglctd for th low confinmnt xprimnts. As a triaxial spcimn xpands, th mmbran xpands as wll. Th strtching of th hyprlastic mmbran producs additional strsss and incras th original confinmnt lvl. Figurs 9, 10 and 11 shows th influnc of latx mmbran on th spcimn bhavior. Th rspons without latx mmbran is softr, and it dos not lvl off in th post pak rgion. Th load displacmnt rspons has a flat portion, but starts hardning aftr approximatly 15% axial dformation for two highr confinmnt tsts (p = 0:52 kpa and p = 1:30 kpa), whil for th th vry low confinmnt tst (p = 0:05 kpa) it hardns monotonically. This can b xplaind by th larg displacmnt ffcts. For larg axial dformations, latral bulging is significant. As th axial dformation progrsss, th matrial (sand) movs from th spcimn cntr to th boundary rgion, thus crating a slight hardning ffct. Th incras in pak strngth du to th latx mmbran ffcts is not too pronouncd. Th post pak rgion, howvr, shows additional stiffning. For th lowst confinmnt tst, th influnc of th mmbran is substantial sinc th spcimn itslf (at only p =0:05 kpa) is quit soft. Figur 12 dpicts th dformd shap of a spcimn. Without th latx mmbran, th 33

35 vrtical forc [kn] with mmbran without mmbran -6 3 volum chang [10 m ] without mmbran with mmbran (a) vrtical displacmnt [m] (b) displacmnt [m] Figur 9: Mchanics of granular matrials rsponss, initial confinmnt (p 0 = 0:05kPa) (a) load dformation and (b) volum dformation numrical prdictions. Influnc of latx mmbran on th ovrall rspons with mmbran without mmbran vrtical forc [kn] without mmbran vrtical displacmnt [m] -6 3 volum chang [10 m ] with mmbran displacmnt [m] Figur 10: Mchanics of granular matrials rsponss, initial confinmnt (p 0 = 0:52kPa) (a) load dformation and (b) volum dformation numrical prdictions. Influnc of latx mmbran on th ovrall rspons. spcimn dforms uniformly. Th abov mntiond nd rstraint rsults in a diffus bulging dformd shap, shown in Figur Concluding Rmarks In this papr w hav prsntd a nw larg dformation constitutiv formulation for gomatrials. Constitutiv formulation was usd in conjunction with larg dformation Lagrangian finit lmnt mthod. Th formulation is capabl of simulating larg dformation hyprlastic plastic bhavior of gomatrials, vn whn collinarity btwn igntriads of 34

36 0.09 vrtical forc [kn] with mmbran without mmbran -6 3 volum chang [10 m ] without mmbran with mmbran vrtical displacmnt [m] displacmnt [m] Figur 11: Mchanics of granular matrials rsponss, initial confinmnt (p 0 = 1:30kPa) (a) load dformation and (b) volum dformation numrical prdictions. Influnc of latx mmbran on th ovrall rspons. Initial with mmbran without mmbran Figur 12: Uniform and bulging dformd shap of a spcimn. strss and strains is lost (for anisotropic and cyclic rspons). A dtaild constitutiv formulation has bn prsntd. Morovr, th rturn algorithm was outlind with implmntation dtails. Th dvlopd formulation and implmntation wr usd to simulat larg dformation tsts on sand prformd undr vry low confinmnt. To this nd, a consistnt st 35