Explicit integration scheme for generalized plasticity constitutive models with automatic error control

1222 EXPLICIT INTEGRATION SCHEME FOR GENERALIZED PLASTICITY CONSTITUTIVE MODELS WITH AUTOMATIC ERROR CONTROL MIGUEL M. STICKLE 1 *, PABLO DE LA FUENTE 2 AND CARLOS OTEO 3 1*: Applied Ma hema ics and Compu e Science Depa men ETSI Caminos, Canales y Pue os Uni e sidad Poli écnica de Mad id A d. P o eso A angu en s/n, 28040 Mad id, Spain e-mail: miguels ick[email p o ec ed]pm.es 2: Con inuum Mechanics and S uc u es Depa men ETSI Caminos, Canales y Pue os Uni e sidad Poli écnica de Mad id A d. P o eso A angu en s/n, 28040 Mad id, Spain e-mail: pdela @caminos.upm.es 3: P o esso on G ound Eng. C / To pede o Tucumán 26, 28016 Mad id, Spain e-mail: ca loso eo@ ele onica.ne Key wo ds: Gene alized Plas ici y, explici in eg a ion. Abs ac . An explici algo i hm o in eg a ing Gene alized Plas ici y cons i u i e models is p esen ed. This au oma ically di ides he applied s ain inc emen in o subinc emen s using an es ima e o he local e o con olling he global in eg a ion e o in he s ess. The algo i hm modi ies he well known S. W. Sloan subs epping scheme o accoun o Gene alized Plas ici y cons i u i e models, in which, unlike Classical Elas oplas ici y, he yield su ace is no explici ly de ined. The in eg a ion scheme is desc ibed and esul s a e p esen ed o a igid oo ing es ing on a laye o speci ic Gene alized Plas ici y model o sands, in which a hype elas ic o mula ion is in oduced o desc ibe he e e sible componen o he soil esponse ins ead o he hypoelas ic app oach o iginally p oposed. The explici algo i hm wi h au oma ic subs epping and e o con ol is shown o be eliable and e icien o hese complex cons i u i e laws. 1 INTRODUCTION Nowadays i is well ecognized ha he selec ion o an adequa e cons i u i e model, oge he wi h he use o accu a e, e icien and obus in eg a ion algo i hms o he elas oplas ic equa ions, is a key poin in ini e elemen analysis o geo echnical p oblems. As obse ed by Hughes [1], he in eg a ion o he cons i u i e equa ions a he local le el plays a c ucial pa in compu a ional plas ici y, since i s ongly a ec s he pe o mance o he cons i u i e equa ion in ac ual compu a ions. Implemen a ion o an ad anced elas oplas ic cons i u i e model in o a ini e elemen XI In e na ional Con e ence on Compu a ional Plas ici y. Fundamen als and Applica ions COMPLAS XI E. Oña e, D.R.J. Owen, D. Pe ic and B. Suá ez (Eds) 1223 Miguel M. S ickle, Pablo De la Fuen e, Ca los O eo. 2 p og am equi es he de elopmen o a obus and e icien nume ical p ocedu e in o de o pe o m he in eg a ion o he cons i u i e equa ions along a gi en loading pa h. In he con ex o classical plas ici y o mula ions, in which a yield unc ion is de ined in an explici manne and he en o cemen o he consis ency condi ion is a key ea u e o he in eg a ion algo i hm, a a ie y o implici and explici in eg a ion schemes migh be used [2, 3]. On he o he hand, based on he assump ion ha explici in eg a ion schemes o highly non-linea models may po en ially lead o inaccu acy and uns able beha io [4], implici in eg a ion algo i hms ha e been conside ed mos ly in he con ex o non-s anda d elas oplas ic models. The a o emen ioned assump ion canno be u he suppo ed, in he con ex o classical plas ici y o mula ions, i explici schemes a e endowed wi h e o con ol echniques [5, 6]. The same si ua ion is obse ed o Gene alized Plas ici y based models [7]. The ou line o he pape is as ollows. We i s p esen he undamen als o Gene alized Plas ici y, wi h pa icula a en ion paid o he SandPZ cons i u i e equa ions including he modi ica ions in he elas ic componen in oduced by Mi a and cowo ke s in 2009 pape . A no el explici algo i hm o in eg a ing Gene alized Plas ici y cons i u i e models is p esen ed in he ollowing sec ion. Finally, Resul s and conclusions a e p esen ed o a igid oo ing es ing on a sand laye modeled by a SandPZ cons i u i e ela ion. 2 GENERALIZED PLASTICITY FRAME WORK. MODIFIED SAND PZ MODEL. The Gene alized Plas ici y basic idea, in oduced by Zienkiewicz and M oz [8] la e ex ended by Pas o and cowo ke s [9, 10] and also by Mi a and cowo ke s [4], is ha no yield nei he plas ic po en ial su ace a e explici ly de ined, bu he g adien s o he unc ions hemsel es. The elas oplas ic beha io o he ma e ial wi hin he Gene alized Plas ici y heo y is desc ibed by he gene al inc emen al ela ionship, ( ) : ep ep ij ijkl kl d Dd d d σε ′′ =⋅ = σDε (1) In which he angen elas oplas ic s i ness ou o de enso ep D depends no only on he in e nal s a e a iables bu also on he cu en e ec i e s ess s a e ′ σ , on he s ain-s ess his o y and he di ec ion o he e ec i e s ess inc emen s d ′ σ . The dependence o ep D on he di ec ion o d ′ σ is exp essed by simply dis inguishing be ween wo di e en loading classes, namely Loading (L) and Unloading (U). The e o e a no malized di ec ion n is de ined in he e ec i e s ess space o any gi en ′ σ , de e mining loading/unloading/neu al loading condi ion. The e a e wo possibili ies o he angen elas oplas ic s i ness enso in (1) depending on whe he loading, ep L D o unloading ep U D is occu ing. To gua an ee con inui y be ween loading and unloading, ep L D and ep U D a e de ined as ( ) () [] ( ) () [] 11 11 1 1 ep e L L L ep e U U U H H −− −− = +⋅⊗ = +⋅⊗ D D mn D D mn (2) 1224 Miguel M. S ickle, Pablo De la Fuen e, Ca los O eo. 3 In exp ession (2), L m and U m a e di ec ions o uni no m ep esen ing he plas ic low di ec ion in loading (L) and unloading (U) condi ions espec i ely, L H and U H a e wo scala unc ions de ined as plas ic moduli while e D is he angen elas ic s i ness enso . By sui able manipula ion o (2), ep L D and ep U D can be ob ained gi ing: :: :: :: :: ee ep e L Le LL ee ep e U Ue UU H H ⊗ =− + ⊗ =− + D m nD DD nD m D m nD DD nD m (3) The s ain inc emen d ε can be decomposed in o elas ic and plas ic pa s as ep dd d =+ εε ε whe e ( ) 1 : 1 : o loading 1 : o unloading ee p L L p U U dd dd H dd H − ′ =  ′ = ⋅⊗    ′ = ⋅⊗   εDσ εmnσ εmnσ (4) The e o e, in a Gene alized Plas ici y app oach, he non-linea i e e sible beha io o soils can be ully desc ibed by simply speci ying h ee di ec ions, , L nm and U m , wo scala s, L H and U H and a ou h o de enso e D . Since he ha dening moduli L H and U H as well as he plas ic low di ec ions L m and U m a e ully de e mined wi hou e e ence o any yield su ace no plas ic po en ial, di e en exp essions can be selec ed o hem whe he he s ess inc emen implies loading o unloading. Mo eo e consis ency canno be en o ced and he consis ency pa ame e d λ is simply de ined as: :: :: e e LU LU d dH λ =+ nD ε nD m (5) Al hough no explici ly de ined, plas ic po en ial and yield su ace can be es ablished a pos e io i, by in eg a ing LU m and n , espec i ely. SandPZ model was de eloped by Pas o and cowo ke s [9] as a pa icula ype o Gene alized Plas ici y o mula ion wi h he aim o p edic ing g anula soil beha io unde bo h mono onic and cyclic loading. The model assumes an iso opic ma e ial esponse. As a esul , he plas ic low di ec ion m , as well as he loading di ec ion n , is exp essed in he in a ian space de ined by ,, pq θ ′ as 1225 Miguel M. S ickle, Pablo De la Fuen e, Ca los O eo. 4 mmm s ij ij ij pq θ θ σσσ ′ ∂∂∂ =++ ′′′ ∂∂∂ m (6) The alue o he coe icien s m ,m ,m s θ a e loading class (loading o unloading) dependen . In o de o ake in o accoun he main ea u es o sand esponse, i.e. he exis ence o a c i ical s a e condi ion, dila i e esponse a e peak, lique ac ion in loose sands, memo y o p e ious s ess pa h, Pas o and cowo ke s [9] p oposed o he plas ic modulus L H he ollowing ela ionship: ( ) 0 L s DM H H pH H H H ′ = ⋅⋅ ⋅ + ⋅ (7) Toge he wi h () 4 01 0 1 max 1 ; 1 ; exp 1 ; ; 1 1 s g p Dm s H HH MM H dd p M γ α α ηη ββ βξ α α ζη ξ ε ξζ ζα −  = − ⋅ =− = −   +     ′ = ===−⋅     +     ∫∫ In hese exp essions 001 ,,, H ββγ a e cons i u i e pa ame e s, ξ is he accumula ed de ia o ic plas ic s ain and max ζ s ands as he maximum alue o he mobilized s ess unc ion ζ accoun ing o he soil s ess his o y. In he case o unloading he plas ic modulus U H is gi en by: 0 0 o 1 o 1 u gg Uu uu g Uu u MM HH M HH γ ηη η  =>   =< (8) Fo whe e 0 u H is a cons i u i e pa ame e and u η , e e ed as unloading s ess a io, is he s ess a io qp ′ om which unloading akes place. Finally, he PZ model assumes a non-linea elas ic esponse o he soils. As in a la ge numbe o cons i u i e models, he non-linea e e sible beha io is desc ibed h ough a hypoelas ic app oach, in which he angen bulk modulus K and shea modulus G only depend on he hyd os a ic pa o he e ec i e s ess enso , acco ding o he ollowing ela ionships 00 00 , pp KK GG pp ′′ =⋅ =⋅ ′′ (9) Al hough widely used, one o he majo sho comings o such hypoelas ic o mula ion is ha i esul s in a non-conse a i e elas ic esponse and ene gy dissipa ion o e closed s ess 1226 Miguel M. S ickle, Pablo De la Fuen e, Ca los O eo. 5 pa hs [11]. An al e na i e is o desc ibe he elas ic esponse o soils wi hin a conse a i e amewo k adop ing he hype elas ic app oach based on he exis ence o an ene gy po en ial om which he e e sible esponse can be de i ed. This na u ally leads o a conse a i e elas ic esponse, gua an eed o obey he Fi s Law o The modynamics, and hus a oiding he p oblems on cycling desc ibed abo e [12, 13] Among he di e en o mula ions ecen ly p oposed in he geo echnical li e a u e, in his wo k he hype elas ic app oach desc ibed by Houslby and cowo ke s in 2005 ( om now on e e ed as HAR acco ding o au ho s’ ini ials) has been adop ed o desc ibe he e e sible componen o he soil esponse. This Gene alized Plas ici y PZ model o g anula soils has been p oposed i s ly by Mi a and cowo ke s [4] unde i s iaxial o mula ion, ex ending he model in he p esen wo k o deal wi h a gene al s ess o mula ion. The s o ed ene gy unc ion ϒ o he HAR model in gene al s ess o mula ion has wo di e en exp essions depending on he alue assigned o he dimensionless p essu e exponen HAR n , which go e ns he amoun o nonlinea i y in ol ed in he o mula ion. Fo 1 HAR n ≠ , ϒ akes he ollowing o m: () ()() ( )( ) 21 0 1 2 HAR HAR nn ea ij HAR HAR HAR HAR pkn kn ευ −− ϒ = ⋅ ⋅− ⋅  ⋅− (10) Whe e () () () 2 0 2 11 1 11 ee ij ij HAR ee ii jj HAR HAR HAR HAR HAR HAR g ee kn knkn υε ε             =+ ⋅+ + ⋅− ⋅− − , while he asymp o ic exp ession o 1 HAR n = is ( ) () a HAR e ee ii ij ij HAR HAR HAR k g k ee e ij pk e ε ε    ⋅+ ⋅ ⋅ ϒ= ⋅ . , HAR HAR kg a e dimensionless cons an s ep esen ing he shea and bulk s i ness ac o s, espec i ely, while a p is he a mosphe ic p essu e, adop ed as e e ence s ess. The e ec i e s ess enso ′ σ and he angen elas ic enso e D can be unambiguously de e mined by aking he i s and second o de de i a i es o (10), ob aining he ollowing exp ession () 0 2 0 1 12 3 HAR n ij kl e ijkl a HAR HAR HAR HAR ij kl HAR ik jl kl ij a p D p nk k n g pp σσ δδ δδ δδ ′′     =⋅ ⋅ ⋅ + − + −         (11) Whe e () ( ) 2 0 1 92 HAR HAR mn mn mm nn HAR k n ss pg σσ ⋅− ⋅ ′′ =+ Fo he p esen model he e a e 12 ma e ial pa ame e s equi ing de ini ion. Gene ally, all pa ame e s a e iden i ied om mono onic and cyclic iaxial es s, hough in ce ain cases some pa ame e s a e adop ed om p e ious expe iences i ull es eco ds a e una ailable. 3 EXPLICIT INTEGRATION OF GENERALIZED PLASTICITY MODELS. Du ing a ypical s ep o i e a ion o an elas oplas ic ini e elemen analysis, he o ces a e applied in inc emen s and he co esponding displacemen inc emen s a e ound om he 1227 Miguel M. S ickle, Pablo De la Fuen e, Ca los O eo. 6 global s i ness equa ions. Once he nodal displacemen inc emen s d u a e known, he s ain inc emen s a a disc e e numbe o in eg a ion poin s wi hin each elemen a e de e mined using he s ain-displacemen ela ion dd = ε Bu . I he s esses associa ed wi h an imposed s ain inc emen cause plas ic yielding, i is necessa y o sol e he sys em o i s o de o dina y di e en ial equa ions (12)-(13): :: ep ep ∆  ′==  ∆  ε σDDε ɺɺ (12) p LU d λ =⋅εm ɺ (13) Whe e :: :: ee LU ep e e LU LU H ⊗ =− + D m nD DD nD m (14) :: :: e e LU LU d dH λ =+ nD ε nD m (15) In hese exp essions, ′ σ deno es he e ec i e s ess enso , ε he small s ain enso and p ε he plas ic s ain enso . The supe io do ep esen s a de i a i e wi h espec o ime while ∆ is he ime in e al o e which he ex e nal o ces ha e been applied. Since he e ec i e s ess and he plas ic s ains a e kwon a he beginning o he ime in e al, and he known s ain a es may be assumed o be cons an h ough he ime in e al wi h alue ∆∆ ε , he equa ions (12) and (13) de ine an ini ial alue p oblem. In o de o in eg a e hese equa ions nume ically, i is con enien [14] o in oduce a pseudo ime, T , de ined by ( ) 0 T =− ∆ , whe e 0 is he ime a he s a o he load inc emen , while 0 +∆ is he ime a he end o he load inc emen , wi h 01 T ≤≤ . Since 1 dT d =∆ applica ion o he chain ule o ′ ɺ σ and p ɺ ε in (12) and (13) gi es :: : :: :: ee LU ep e ee LU e LU LU d dT H λ  ⊗ ′′ = ∆= − ∆=∆ −∆   +  D m nD σD εDεσ Dm nD m (16) p LU d dT λ =∆ ⋅ ε m (17) Whe e :: :: e e LU LU H λ ∆ ∆= + nD ε nD m (18) Equa ions (16) and (17) de ine a classical ini ial alue p oblem which needs o be 1228 Miguel M. S ickle, Pablo De la Fuen e, Ca los O eo. 7 in eg a ed o e he pseudo ime in e al om 0 T = o 1 T = , whe e he known alues known alues in hese ela ions a e he imposed s ain inc emen s, ∆ ε , oge he wi h he e ec i e s esses and plas ic s ain a he s a o he pseudo ime inc emen . The quan i ies / and LU mn a e e ec i e s ess unc ions, while pa ame e / LU H is a unc ion o bo h he e ec i e s ess and he plas ic s ain. In o de o sol e he sys em o i s o de o dina y di e en ial equa ions (16)-(17) Sloan de eloped a subs epping algo i hm [5] whe e he cons i u i e law is in eg a ed by au oma ically di iding he s ain inc emen in o a numbe o subs eps. An app op ia e size o each subs ep is ound h ough he use o modi ied Eule o Runge-Ku a-Do mand-P ince o mulae, which a e specially cons uc ed o p o ide an es ima e o he local e o . La e , Sloan and cowo ke s [6] gene alized he 1987 scheme, inco po a ing new algo i hms o handling elas oplas ic unloading, compu ing he yield in e sec ion poin , and es o ing he s ess o he yield su ace. Sloan and cowo ke s schemes we e de eloped exclusi ely o classical plas ici y based models, including classical and gene alized c i ical s a e models, whe e non-linea elas ic beha io inside he yield su ace is exhibi ed. In all hese models he admissible s a es in he s ess space a e cons ained o lie wi hin he in e io o he bounda y o he domain explici ly de ined by he yield su ace. As his is no he case o gene alized plas ici y based models, Sloan subs epping algo i hm should be adjus ed in o de o be able o in eg a e his kind o models. The p oposed in eg a ion scheme s a s wi h he known s ain inc emen , ∆ ε , he ini ial s ess 0 σ and ini ial plas ic s ain 0 p ε a he s a o he inc emen whe e 0 T = and 0 = . A he end o he in eg a ion p ocess he s esses and plas ic s ains a e ob ained a he end o he inc emen whe e 0 T = and 0 = . Conside a pseudo ime subinc emen in he ange 01 n T ≤∆ ≤ and le he subsc ip s 1 n − and n , deno e quan i ies e alua ed a he pseudo imes 1 n T − and 1 nn n TT T − = +∆ , espec i ely. Plas ic modulus and plas ic low di ec ion in exp essions (16)-(17) a e dependen on he di ec ion o he e ec i e s ess inc emen s he e o e di e en ia ion be ween he wo loading classes should be pe o med be o e he p ope in eg a ion p ocess s a s. By means o he s ain subinc emen nn T∆ =∆ ∆ εε he loading class is i s ly es ablished h ough he ollowing exp ession ( ) ( ) :: e n nn ′′ ∆ n σDσε (19) I exp ession (19) is posi i e an elas oplas ic loading p ocess is pe o med, i nega i e an elas oplasc ic unloading p ocess is implied, while an elas ic p ocess comes om a ze o alue. In he explici Eule me hod, he solu ion o , p ′ σε a he end o he pseudo ime s ep n T ∆ is ound om 11 11 nn pp p nn − − ′′ ′ = +∆ = +∆ σσ σ εε ε (20) Whe e 1229 Miguel M. S ickle, Pablo De la Fuen e, Ca los O eo. 8 ( ) () () 1 11 1 11 1 ,: ,, ep p nn n pp n n n LU n λ −− −− − ′′ ∆= ∆ ′ ∆ =∆ ∆ ⋅ σDσε ε ε σε εmσ (21) A mo e accu a e es ima e o he s ess and plas ic s ains a he end o he in e al n T ∆ can be ound using he modi ied Eule p ocedu e, which is gi en by () () 1 12 1 12 1 2 1 2 nn pp p p nn σσ εε − − ′′ ′ ′ = + ∆ +∆ = + ∆ +∆ σσ εε ⌢ ⌢ (22) Whe e 11 and p ′ ∆∆ σε a e ob ained om Eule scheme and ( ) () () 2 1 11 1 2 1111 11 ,: ,, ep p p nn n ppp n n n LU n λ −− −− − ′ ′′ ∆ = +∆ +∆ ∆ ′′ ′′ ∆ =∆ +∆ +∆ ∆ ⋅ +∆ σ D σ σε ε ε ε σ σε ε ε m σσ (23) Since he local unca ion e o [15] in he Eule and modi ied Eule solu ions is ( ) ( ) 23 and OT OT ∆∆ , espec i ely, he e o in n σ and p n ε can be es ima ed om () () 21 21 1 2 1 2 nn pp pp nn  ′′ ∆ −∆  ′′    −=         ∆ −∆   σσ σσ εε εε ⌢ ⌢ (24) Using any con enien no m, his quan i y can be used o compu e he ela i e e o measu e 21 21 1max , 2 pp np nn R  ′′ ∆ −∆ ∆ −∆  =    εε σσ σε (25) Following 1987 Sloan wo k, he cu en s ain subinc emen is accep ed i n R is no g ea e han some p esc ibed ole ance, STOL , and ejec ed o he wise. Rega dless o whe he he subinc emen is accep ed o ejec ed, he nex pseudo ime s ep is ound om he simple ela ion 1 nn T qT + ∆ = ⋅∆ (26) whe e q is chosen so ha 1 n R + sa is ies he cons ain 0.8 , 0.1 1.1 n q STOL R q ≤ ≤≤ (27) 1230 Miguel M. S ickle, Pablo De la Fuen e, Ca los O eo. 9 Two ypical con ols a e inally inco po a ed. A minimum absolu e s ep size, min T ∆ , and a s ep size is no allowed o g ow immedia ely a e a ailed subinc emen . 4 RESULTS AND CONCLUSIONS. The beha io o a smoo h igid s ip oo ing es ing on an elas oplas ic soil mass, go e ned by he modi ied SandPZ model p esen ed abo e, is conside in o de o analyze he pe o mance o he p oposed in eg a ion scheme. Due o he singula i y a he edge o he oo ing and he s ong o a ion o he p incipal s esses, his example is a good es o assessing he in eg a ion s a egy. As loading is p esc ibed in he o m o displacemen s, an equi alen uni o m p essu e is ound by summing he app op ia e nodal eac ions. To assess he accu acy o he scheme, an es ima e o he s ess in eg a ion e o is ound di ec ly om 2 2 e e o e σ ′′ − =′ σσ σ (28) Whe e ′ σ a e he e ec i e s esses ob ained by he p oposed in eg a ion scheme, e ′ σ a e he e e ence e ec i e s esses while 2 i is he Euclidean no m. The e e ence e ec i e s esses a e ob ained by he explici Do mand-P ince in eg a ion scheme wi h a s ess ole ance o 9 10 STOL − = . No e ha he e e ence s esses p o ide a e y accu a e se o s esses o he gi en mesh and loading sequence and all alues a e compu ed a he end o he las load inc emen . The esul s o he analyses wi h 10 load inc emen s o equal size a e p esen ed in Table 1. I can be obse ed how he uni o m p essu e o e he oo ing a e applying 4mm o e ical displacemen is simila o all o he speci ied s ess ole ances wi h alues a ying by less han 0.6% o he e e ence p essu e. Table 1: Smoo h igid s ip oo ing on SandPZ laye . 10 load s eps. S ess Tole ance Equi alen uni o m p essu e a e a e ical displacemen o 4mm [N/m 2 ] % o he equi alen p essu e ob ained unde Do mand-P ince in eg a ion scheme e o σ 2 10 STOL − = 115320 0.14% 3 1.7 10 − ⋅ 4 10 STOL − = 115450 0.02% 4 1.7 10 − ⋅ The e o in he compu ed s esses o he p oposed scheme, as de ined by equa ion (28), is less han he in eg a ion ole ance o 2 10 STOL − = and wi hin he o de o magni ude o 4 10 STOL − = . The e o e he ole ance STOL hus gi es a equi ed e o con ol. Figu e 1 shows he e o spa ial dis ibu ion induced by he p oposed local in eg a ion