Chapter 2 BASIC EQUATIONS OF NONLINEAR CONSTITUTIVE MODELS

Transcription

1 Chaptr BASIC EQUATIONS OF NONLINEAR CONSTITUTIVE MODELS TYPES OF NONLINEAR CONSTITUTIVE MODELS Gomatrials ar charactrizd by nonlinar strss-strain bhavior and, oftn, by tim-dpndnt dformations. Th nonlinar bhavior may b simulatd using svral approachs, including nonlinar lastic modls and lastoplastic modls. Tim- and rat-dpndnt bhavior can b dscribd by adding viscous ffcts to th modl quations, as is don in th cas of lastoviscoplasticity. This chaptr contains a dscription of th typs of constitutiv modls that ar availabl for gomatrials, and a dscription of th basic principls of lastoplastic constitutiv rlations. Nonlinar Elastic Modls Nonlinar lastic modls attmpt to simulat th nonlinar strss-strain bhavior of gomatrials by making th lastic constants (and consquntly th stiffnss) dpnd on strss stat and/or accumulatd strain. Svral nonlinar lastic modls hav bn usd to simulat th bhavior of gomatrials. Two of th mor common nonlinar lastic modls ar th K-G modl or Barron- Sandlr modl (Naylor t al., 1981) and th hyprbolic modl (Kondnr, 196) which was modifid for Mohr-Coulomb strngth paramtrs and applid to finit lmnt analysis by Duncan and Chang (1970). Both modls hav simpl hypolastic formulations in that th lastic moduli (bulk modulus K and shar modulus G for th K-G modl, and tangnt lastic modulus E t and tangnt Poisson s ratio ν t for th Duncan-Chang modl) ar functions of only th strss stat and modl paramtrs. Th cofficints usd to dtrmin th lastic moduli ar functions of only th failur critrion. Th Duncan-Chang modl gnrally can provid a bttr fit to laboratory data than th K-G modl bcaus th Duncan-Chang modl allows th Poisson s ratio to vary throughout th analysis. This fatur of th Duncan-Chang modl allows critical valus of dviator strss (i.., pak strngth and failur) and volum chang (i.., onst of dilatancy) to occur at diffrnt stags of th analysis, which is commonly obsrvd in ral gomatrials. Nonlinar lastic modls ar plagud by svral shortcomings as statd in th prvious chaptr: (1) strss history and path dpndncy cannot b takn into account during analysis. Thrfor, incrmntal strains ar a function of th incrmntal strsss, rathr than of th strss stat at which th incrmntal strss ar applid; () dilatant rspons during incrmntal 11

2 comprssion loading (.g., shar) violats thrmodynamic principls and cannot b simulatd within th framwork of lasticity; () unloading and othr changs in loading dirction cannot b proprly simulatd (xcpt by ad hoc modl modification) bcaus th stiffnss moduli ar dpndnt only on strss and/or strain stat. Elastoplastic Modls Classical lastoplastic modls ar formulatd using th concpts and principls dscribd in this chaptr: yild function, strain additivity, incrmntal lasticity, plastic flow rul, and plastic hardning rul. Elastoplastic modls hav thir basis in th following idas: (1) th st of allowabl strss stats in a matrial is limitd to som finit st, which is dfind by th yild surfac; () th bhavior of a matrial is lastic whn its strss stat is bnath th yild surfac, and is lastoplastic whn its strss stat is on th yild surfac; () th plastic or irrcovrabl rspons of a matrial dpnds not on its incrmntally applid load, but on its strss stat; and (4) th yild function and st of allowabl strss stats may chang as loading progrsss. This volution is a function of som plastic phnomnon, usually plastic strain or plastic work. Elastoplastic modls rang from simpl to complx. An xampl of a simpl modl is th nonhardning Mohr-Coulomb modl with constant plastic flow dirction and linar lasticity; this modl includs a singl yild surfac that is also a failur surfac. A mor complx lastoplastic modl is th Cam-Clay modl. This modl also has only a singl yild surfac, but is complicatd by incorporating yild surfac volution du to isotropic hardning and softning, strss stat-dpndnt plastic flow, and strss-dpndnt lasticity. Additional complications to lastoplastic modls can aris whn kinmatic hardning, multipl nstd yild surfacs, or bounding surfacs ar introducd into th formulation. On limitation of lastoplastic modls is that tim and rat of loading ffcts do not appar in th constitutiv quations; howvr, ths ffcts can b incorporatd into ths modls by rformulating th modl as an lastoviscoplastic modl. It is notabl that most constitutiv modls that hav bn proposd for chalk ar lastoplastic modls. Elastoviscoplastic Modls Elastoviscoplastic modls ar vry similar to lastoplastic modls xcpt that ffcts of loading rat ar includd in th formulation. Th concpts and principls on which lastoviscoplastic 1

3 modls ar basd ar th sam as thos for lastoplastic modls xcpt that for crtain formulations, irrcovrabl or inlastic phnomna may occur for rasons othr than mchanical loading. Also, for som lastoviscoplastic formulations, th ida that th st of allowabl strss stats in a matrial is limitd to som finit st, may b violatd tmporarily. Th original lastoviscoplastic approach (Przyna, 196) rprsnts only a slight modification to th classical lastoplastic modl in that idas,, and 4 of classical lastoplasticity apply. Howvr, strss stats ar allowd to li outsid th yild surfac tmporarily du to viscous proprtis of th matrial. As tim passs, plastic straining and plastic hardning occurs and th strss stat rturns to th yild surfac. In nwr lastoviscoplastic formulations, th concpt of purly lastic bhavior disappars. Irrvrsibl strains accumulat continuously du to aging of th matrial. Th quations incorporatd into lastoviscoplastic formulations ar prsntd latr in this chaptr. FORMULATION OF INCREMENTAL CONSTITUTIVE RELATIONS FOR ELASTOPLASTICITY Strss-strain rlations for gomatrials ar nonlinar. This nonlinar bhavior may b simulatd using svral approachs, including nonlinar lastic modls and lastoplastic modls. As dscribd abov, nonlinar lastic modls hav th advantag of bing asir to formulat and implmnt, but hav th disadvantag that lastic modls fail to simulat svral phnomna which occur during mchanical loading of gomatrials. Elastoplastic bhavior of gomatrials is path-dpndnt, bcaus th matrial bhavior dpnds on th strss history of th matrial. For this rason, constitutiv rlations cannot b xprssd in intgratd form but can only b xprssd in incrmntal form. Th incrmntal constitutiv rlations can b writtn using th proprtis of strain additivity, incrmntal lasticity, plastic flow rul, and plastic hardning rul. First, th invariants of strss and strain and th concpt of plastic yilding will b introducd. Th proprtis listd abov, which contribut to th constitutiv quations of classical rat-indpndnt lastoplasticity, will also b dscribd. Rat-indpndnc mans that bhavior dos not dpnd on tim or rat of loading. A brif discussion is also includd on mor complx formulations of plasticity, including lastoviscoplasticity and bounding surfac plasticity. 1

4 Strss and Strain Invariants Crtain strss- and strain-basd quantitis calld invariants hav bn dfind as dscribd blow. Ths quantitis ar usful bcaus thy do not dpnd on th choic of rfrnc axs and so ar invariant to th orintation of th rfrnc axs. Bcaus thy ar indpndnt of rfrnc axs, strss and strain invariants ar usful in constitutiv modling. Th various invariants which hav bn dfind for -dimnsional and -dimnsional spac ar listd in th following tabl. -D -D Strss invariants p, q p, q, θ Strain and strain incrmnt invariants dε v, dε s dε v, dε s, θ ε Th -dimnsional strss invariants which ar usd in this rport (man strss p, dviatoric strss q, and Lod angl θ) ar dfind as follows: 1 1 p = σii = ( σ11 + σ + σ ) (.1) [( σ σ ) + ( σ σ ) + ( σ σ ) + 6σ + 6σ + ] q 1 = J = = σ s s (.) θ = 1 sin 1 J 1.5 J (.) In ths dfinitions, J and J ar thmslvs strss invariants, and ar dfind as: [( σ σ ) + ( σ σ ) + ( σ σ ) + 6σ + 6σ + σ ] J 1 1 = = s s 6 (.4) 1 J = = s s jk ski s (.5) and s is th dviatoric strss tnsor: s = σ δ p (.6) Th quantity δ is th Kronckr dlta, which taks th valu δ = 1 if i = j, and = 0 if δ i j. Th summation convntion ovr rpatd indics is adoptd in th notations. 14

5 Ths strss invariants p, q, and θ will b usd in this proposal instad of anothr st of strss invariants bcaus ths quantitis can b visualizd graphically as shown in Figur.1, and bcaus lastoplastic constitutiv modls for gomatrials ar typically most fficintly dscribd in trms of ths invariants. Th man strss p is proportional to th distanc from th origin of strss spac along th hydrostatic axis to th π-plan in which th strss point lis (p has a constant valu on th π-plan); th dviatoric strss q is proportional to th distanc in th π- plan from th hydrostatic axis to th strss point; and th Lod angl θ shows th position of th strss point in th π-plan with rspct to th thr principal strsss, showing spcially th ffct of th intrmdiat principal strss σ. For triaxial comprssion conditions (σ 1 > σ = σ ), th Lod angl θ = -0º; for triaxial xtnsion conditions (σ 1 = σ > σ ), th angl θ = 0º; and for othr triaxial strss conditions, th Lod angl lis somwhr btwn ths two xtrms. Th -dimnsional strain incrmnt invariants (incrmntal volumtric strain dε v, incrmntal gnralizd shar strain dε s, and Lod angl θ ε ) ar dfind as: dε v = dεii = dε11 + dε + dε (.7) dε s = dd = 9 [( dε dε ) + ( dε dε ) + ( dε dε ) + 6dε + 6dε + dε ] (.8) θ 1 = sin J ' 1. J ' 1 ε 5 (.9) J and J ar thmslvs strain incrmnt invariants, and ar dfind as: J ' [( dε dε ) + ( dε dε ) + ( dε dε ) + 6dε + 6dε + ε ] 1 1 = d d = d (.10) 1 J ' = d d d = d (.11) jk ki and d is th dviatoric incrmntal strain tnsor, which is analogous to th dviatoric strss tnsor: d = dε 1 δdεv (.1) Th angl θ ε is analogous to th Lod angl θ in strss spac. 15

6 Th -dimnsional strss and strain invariants which ar usd in this rport ar similar to thos for -D: 1 1 p = σii = ( σ11 + σ ) (.1) [( σ σ ) + ] q 1 1 = J = = 11 4σ1 s s 4 (.14) dε = dε = dε + d v ii 11 ε (.15) dε = ( dε dε ) + d d d = ε s (.16) xcpt that th -dimnsional dviatoric incrmntal strain tnsor d is dfind diffrntly: d = dε 1 δdεv (.17) Th -dimnsional strss invariants rprsnt a spcial cas of th -dimnsional invariants, in which th Lod angl θ has a constant valu corrsponding to triaxial comprssion conditions. Yild Surfac and Yild Function In strss spac, it is common to divid strss stats into thos which ar stabl, unstabl (i.., at yild or failur), and impossibl (i.., in illgal strss spac). Th boundary btwn stabl strss stats and impossibl strss stats is calld th yild surfac. Strss stats which li insid th yild surfac ar stabl and ar rprsntd by lastic matrial bhavior. Strss stats which li outsid th yild surfac ar impossibl and cannot b attaind in th contxt of rat-indpndnt lastoplasticity (xcptions to this rul occur in th contxt of lastoviscoplasticity and will b discussd latr). Strss stats which li on th yild surfac ar at yild, and ar charactrizd by a combination of lastic and plastic bhavior. S Figur. for an xampl of a yild surfac in strss spac. Dpnding on th particular constitutiv modl, th yild surfac may b fixd in strss spac, or th yild surfac may chang siz, position, or shap in strss spac as plastic dformation accumulats. Th paramtrs which control th siz, position, and shap of th yild surfac ar th plastic hardning paramtrs q α. Th yild surfac is usually dscribd by an quation which dscribs th shap of th yild surfac in strss spac in trms of plastic hardning paramtrs and individual strss 16

7 componnts or strss invariants; strss invariants ar prfrrd so that th modl is not dpndnt on th rfrnc axs. It is possibl to thn writ th quation which dscribs a spcific yild surfac in strss spac in such a form that th quation rturns a valu f for any combination of strss componnts or invariants and plastic hardning paramtrs in which th quation is xprssd. Such an quation is calld a yild function: ( σ ) f = f, q (.18) A yild function is usually xprssd with th convntion that f = 0 if a strss point lis on th yild surfac, f < 0 if a strss point lis insid th yild surfac, and f > 0 if a strss point lis outsid th yild surfac, in impossibl strss spac. α Strain Additivity Th tim-indpndnt lastic and plastic strain incrmnts act indpndntly of ach othr. Elastic strains accompany dformation in which is th nrgy of dformation is stord, such that th lastic dformation and nrgy of lastic dformation ar fully rvrsibl and may b rcovrd during rmoval of th dformation-causing loads. Plastic strains accompany dformation in which th nrgy of dformation is dissipatd, so th plastic dformation and nrgy of plastic dformation ar not rcovrd whn th loads ar rmovd. A total strain incrmnt d may b additivly dcomposd into its rcovrabl or lastic ( ), and inlastic ε or irrcovrabl ( ir dε ), componnts. For classical lastoplasticity, th irrcovrabl strain p incrmnts ar qual to th plastic strain incrmnts ( d ): ε dε dε = dε + dε p (.19) Th rol of tim-dpndnt dformations and strains may b incorporatd into th additivity postulat using svral diffrnt formulations as discussd latr. Incrmntal Elasticity Th gnralizd Hook s Law rlats strss incrmnt dσ (of th Cauchy strss tnsor) to lastic strain incrmnt using th lastic constitutiv tnsor or th lastic complianc tnsor C : kl dε D kl 17

8 dσ = D kl dε kl (.0a) dε = C kl dσ kl (.0b) Ths tnsors ar invrss of ach othr: = ( C ) 1 D. kl kl Plastic Flow Rul Th dirction of plastic flow, which dtrmins th rlativ magnituds of th plastic strain componnts during lastoplastic loading, is rprsntd for ach yild mchanism by a plastic potntial surfac g. Plastic flow occurs in th dirction normal to th plastic potntial surfac. Th magnitud of plastic strain is proportional to th plastic multiplir ϕ. Thrfor, th plastic flow rul may b rprsntd as: p g dε = ϕ (.1) σ If th plastic potntial surfac is coincidnt with th yild surfac at all points (f = g), plastic flow is said to b associatd bcaus th dirction of plastic flow is associatd with th yild surfac. If th plastic potntial surfac is not coincidnt with th yild surfac, flow is said to b non-associatd. Figur. illustrats th diffrnc btwn associatd and non-associatd flow. Plastic Hardning Rul A yild surfac may b ithr hardning or non-hardning. A non-hardning yild surfac rmains in a fixd position in strss spac at all stags of loading, and thus for rat-indpndnt plasticity rprsnts a st of limiting strss stats which can nvr b xcdd. A hardning yild surfac may undrgo som typ of chang in position or shap in strss spac whn th matrial is subjctd to crtain loading conditions, such that th allowabl strss stats in th matrial chang as matrial loading progrsss. Hardning yild surfacs mov outward, in th loading dirction, as loading continus. Gnrally, sinc only th nrgy rlatd to th lastic componnt of lastoplastic dformation is rcovrabl, hardning of a yild surfac is rlatd to som aspct of th plastic componnt of lastoplastic dformation (.g., plastic strain or plastic work). 18

9 Th siz, position, and/or shap of th yild surfac in strss spac may chang as lastoplastic loading continus. Th paramtrs which dfin th siz, position, and/or shap of th yild surfac ar calld th plastic hardning paramtrs q α. Th plastic hardning function hα dscribs th volution of th plastic hardning paramtr dq strain or plastic work: Th plastic hardning function h and plastic hardning paramtr α α α α as a function of ithr plastic dq = ϕh (.) may tak various forms dpnding on th natur of th hardning function, so α may rplac a diffrnt numbr of indics. For th cas whr th hardning function affcts th siz of th yild surfac (i.., isotropic hardning), hα and dq α ar scalar quantitis; for th cas whr th hardning function affcts th position of th yild surfac in strss spac (i.., translational and rotational kinmatic hardning), ths ar scond-ordr tnsors; for th cas whr th hardning function affcts th shap of th yild surfac (i.., distortional kinmatic hardning), ths ar fourth-ordr tnsors. dqα Elastoviscoplasticity It is possibl to incorporat rat or tim ffcts into th framwork of lastoplasticity by modifying th rat-indpndnt constitutiv quations. Th simplst and most common framwork to incorporat rat-dpndnc is lastoviscoplasticity. In contrast to ratindpndnt lastoplasticity, lastoviscoplastic formulations ar commonly xprssd in trms of th constitutiv rat quations. In lastoviscoplasticity, additivity postulat is modifid such that th strain rat is additivly dcomposd. Th dcomposition of strain rats is modifid such that th irrvrsibl componnt bcoms th viscoplastic componnt ( ε & ): ε & = ε& + ε& (.) Th suprior dot indicats a tim drivativ. Elastoviscoplastic modls ar typically formulatd using ithr th ovrstrss approach or th rat-typ approach. Przyna (196) formulatd th viscoplastic strain rat using th ovrstrss approach with th following flow rul: g ε& = γ Φ( f ) (.4) σ 19

10 whr γ is th fluidity paramtr and Φ is a function of th yild function f; ths tak th plac of th plastic multiplir ϕ usd in rat-indpndnt lastoplasticity. Th Macauly brackts indicat that th bracktd quantity quals zro if th quantity insid th brackts is ngativ, and quals th quantity insid th brackts if that valu is nonngativ. According to th original viscoplastic formulation, th viscoplastic strain rat is zro if th strss point is locatd in th lastic zon blow th yild surfac, and is proportional to a function of th yild function if th strss point is locatd outsid th yild surfac; xcursions outsid th yild surfac into illgal strss spac ar tmporarily allowd in this formulation of lastoviscoplasticity. In th original ovrstrss approach of Przyna (196), viscoplastic bhavior only occurs for strss stats which li outsid th yild surfac. As tim passs, plastic straining and plastic hardning occurs and th strss stat rturns to th yild surfac. Ovrstrss modls hav bn modifid such that th ovrstrss is masurd rlativ to a rfrnc surfac which may not b th sam as th yild surfac. In this way, ovrstrss modls allow for tim-dpndnt dformations to occur at strss stats blow th yild surfac. Th ovrstrss approach is illustratd in Figur.4. Th rat-typ approach has its roots in th concpt of tim-lins and quivalnt tim (Bjrrum, 1967) and rprsnts a mor significant concptual chang to th classical lastoplastic modl. In this cas, th concpt of purly lastic bhavior disappars bcaus any strain incrmnt has both instant and dlayd componnts. Irrvrsibl strains accumulat continuously du to aging of th matrial, and th rat of irrvrsibl straining dpnds on its quivalnt ag t v rlativ to an quivalnt nwly dpositd matrial of minimum ag. Each timlin (Figur.5) rprsnts a matrial ag which corrsponds to a stady-stat viscoplastic strain rat. If a matrial is loadd at a constant rat, its strss-strain curv bcoms tangnt to a timlin. Hypoplasticity and Bounding Surfac Plasticity Hypoplasticity is a mor complx formulation than classical plasticity. In classical plasticity, th plastic strain rat dirction dpnds only on th strss stat. In hypoplasticity, th plastic strain rat dirction dpnds on th strss stat and on th strss rat dirction (Dafalias, 1986). On way to account for hypoplasticity in constitutiv modling is by using anisotropic hardning modls. 0

11 Anisotropic hardning constitutiv modls wr introducd to modling of gomatrials in an ffort to mor accuratly simulat th rspons of gomatrials, spcially to rvrsd and cyclic loading. A problm in anisotropic hardning modls occurs in dscribing th volution of th yild surfac and dtrmining th dirction of plastic flow for a givn loading stp. Mroz (1967) introducd multi-surfac plasticity to gomchanics, in which many nstd yild surfacs ar ncountrd and activatd in succssion during continud lastoplastic loading (Figur.6). Each yild surfac has its own yild function; if yild surfac i is activ, f i = 0, whil f i < 0 if it is inactiv. Ths nstd yild surfacs ar formulatd such that ach yild surfac can bcom tangnt to th nxt surfac, but may nvr cross it. Each activ yild surfac contributs to th plastic hardning modulus. In multi-surfac plasticity formulations, th hardning functions for all activ yild surfacs must b calculatd and kpt in mmory to dtrmin th instantanous hardning modulus. Bounding surfac plasticity was first applid to th study of mtal bhavior by Krig (1975) and latr applid to gomchanics by Mroz t al. (1978). Bounding surfac plasticity formulations hav bn commonly usd in gomchanics in rcnt yars (.g., Dafalias, 1986; Manzari and Nour, 1997; Borja t al., 001). Bounding surfac formulations ar mor fficint than gnral multi-surfac formulations in that only two surfacs in strss spac ar rquird to dscrib th bhavior of a matrial. Ths surfacs includ th loading surfac and bounding surfac, shown in Figur.7. Th loading surfac is analogous to th yild surfac in that a strss stat insid th loading surfac (if possibl) is charactrizd by lastic bhavior, whil a strss stat on th loading surfac is charactrizd by lastoplastic bhavior. Th loading surfac volvs during continud lastoplastic loading in th sam way that th yild surfac volvs for classical rat-indpndnt lastoplastic formulations. Th rol of th bounding surfac is to dtrmin th valu of th hardning modulus during lastoplastic loading. As for gnral multisurfac plasticity, th loading surfac may bcom tangnt to but nvr cross th bounding surfac; oftn, th loading surfac is similar to th bounding surfac. Th hardning modulus, and thrfor th volution of th loading surfac, is dtrmind by som function of th proximity of th conjugat strss points. Th conjugat strss points includ th strss point on th loading surfac and th imag strss point σ on th bounding surfac (Figur.7). Th form of a typical hardning function is: 1

12 h [( σ σ )( σ σ )] α = h α (.5) In addition to th volution of th loading surfac, th bounding surfac itslf may volv as a function of som xtrnal variabl such as a stat variabl. Incrmntal Rlations Th incrmntal rlations for an lastoplastic modl may b writtn using th proprtis of strain additivity, incrmntal lasticity, and plastic flow rul. Undr strss-controlld loading conditions, th strain incrmnt may b calculatd as a function of th lastic complianc tnsor, strss incrmnt, dirction of plastic flow, and plastic multiplir: g dε = Ckl dσ kl + ϕ (.6) σ Undr strain-controlld loading conditions, th strss incrmnt may b calculatd as a function of th lastic constitutiv tnsor, total strain incrmnt, dirction of plastic flow, and plastic multiplir: g d σ = ε ϕ Dkl d kl (.7) σ kl Intgration of th incrmntal constitutiv rlations ovr a finit loading stp is discussd in Chaptr 7. For lastoviscoplastic formulations, th constitutiv quations ar formulatd as rat quations, so th strain rat must b intgratd with rspct to tim to obtain th strain incrmnt.

13 REFERENCES Dafalias, Y.F. (1986). Bounding surfac plasticity I: Mathmatical foundation and hypoplasticity. ASCE Journal of Enginring Mchanics, 11(9), Duncan, J.M. and Chang, C.-Y. (1970). Nonlinar analysis of strss and strain in soils. ASCE Journal of th Soil Mchanics and Foundations Division, 96(SM5), Kondnr, R.L. (196). Hyprbolic strss-strain rspons: cohsiv soils. ASCE Journal of th Soil Mchanics and Foundations Division, 89(SM1), Krig, R.D. (1975). A practical two-surfac plasticity thory. ASME Journal of Applid Mchanics, 4, Manzari, M.T. and Nour, M.A. (1997). On implicit intgration of bounding surfac plasticity modls. Computrs and Structurs, 6(), Mroz, Z. (1967). On th dscription of anisotropic workhardning. Journal of Mchanics and Physics of Solids, 15, Mroz, Z., Norris, V.A., and Zinkiwicz, O.C. (1978). An anisotropic hardning modl for soils and its application to cyclic loading. Intrnational Journal for Numrical and Analytical Mthods in Gomchanics,, 0-1. Przyna, P. (196). Th constitutiv quations for work hardning and rat snsitiv plastic matrials. Procdings of Vibration Problms, 4(),

14 σ 1 σ q Hydrostatic axis: σ 1 = σ = σ p σ σ (a) θ 0º σ 1 σ q σ σ (b) Figur.1. Illustration of th -dimnsional strss invariants p, q, and θ: (a) in principal strss spac; (b) projctd on th π-plan. 4

15 Yild surfac: Illgal strss spac: f (σ,q α ) = 0 f (σ,q α ) > 0 Elastic rgion: f (σ,q α ) < 0 σ Figur.. Exampl yild surfac in strss spac, showing th yild surfac as th boundary btwn th lastic rgion and impossibl strss spac. Plastic flow dirction: g σ Plastic flow dirction: g σ Yild surfac: f (σ,q α ) = 0 Plastic potntial surfac: g (σ,q α ) = 0 Yild surfac: f (σ,q α ) = 0 Plastic potntial surfac: g (σ,q α ) = 0 (a) σ Figur.. Plastic potntial surfac and plastic flow dirction in strss spac, for (a) associatd flow; (b) non-associatd flow. (b) σ 5

16 Φ Viscoplastic potntial surfacs Φ 1 Φ 0 = 0 σ,1 g σ σ, Rfrnc surfac: Φ = 0 Φ < 0 Φ ε& σ,0 > Φ 1 = 0 ( σ ) > ε& ( σ ) > ε& ( σ ) = 0, > Φ 1 0,1 0,0 σ Figur.4. Main componnts of ovrstrss lastoviscoplastic formulation. Viscoplastic strain rat dpnds on distanc in strss spac from rfrnc surfac, and viscoplastic flow dirction is dtrmind by gradint to viscoplastic potntial surfac. σ tv, ε& tv1,ε& 1 Rfrnc tim-lin: tv0, ε& 0 ε kl t v ε& > t v1 < ε& 1 > t v0 < ε& Figur.5. Main componnts of rat-typ lastoviscoplastic formulation. Viscoplastic strain rat dpnds on ag of matrial compard to rfrnc ag. 0 6

17 dσ σ f 0 = f 1 = f = 0; f < 0 σ Yild surfac 0 Yild surfac 1 Yild surfac Yild surfac Figur.6. Multipl nstd yild surfacs in strss spac for a multi-surfac formulation. Surfacs 0, 1, and ar activ; surfac is not. dσ strss point σ imag strss point σ σ Loading surfac: f (σ,q α ) = 0 Bounding surfac: F (σ,q α ) = 0 Figur.7. Main componnts of bounding surfac formulation. Strss point and imag strss point ar locatd at similar points on loading surfac and bounding surfac. 7