On the Strain Saturation Conditions for Polycrystalline Ferroelastic Materials

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1 C. M. Landis Mm. ASME -mail: Dpatmnt of Mchanical Engining and Matials Scinc, MS 21, Ric Univsity, P.O. Box 1892, Houston, TX On th Stain Satuation Conditions fo Polycystallin Folastic Matials A phnomnological constitutiv law is dvlopd fo th dfomation of polycystallin folastic matials. Th modl is famd within a thmodynamic stting common to intnal vaiabl plasticity. Th two significant inputs to this modl a a switching (yild) sufac, and a hadning potntial. To maintain simplicity, th shap of th switching sufac is assumd to b sphical in a modifid dviatoic stss spac. In od to asctain th functional fom of th hadning potntial, micomchanical slf-consistnt simulations of multipl singl cystals, with ttagonal cystal stuctu, mbddd in an ffctiv polycystallin matix, a pfomd fo diffing loading paths in mannt (plastic) stain spac. As a sult of th asymmty in th tnsion vsus compssion bhavio of ths matials, it is shown that pu sha loading dos not sult in pu sha staining. This fatu of th matial bhavio is dmonstatd with th slfconsistnt simulations and pdictd by th phnomnological constitutiv law. Ultimatly, th phnomnological thoy is abl to captu th complx constitutiv bhavio of folastic matials pdictd by th micomchanical modl. DOI: / Intoduction Many smat matials, including folctics and shap mmoy alloys blow th matnsit finish tmpatu, M f ), hav a noncubic cystal stuctu. Th simplst of ths stuctus is ttagonal, but oth stuctus such as hombohdal and othohombic xist in tchnologically usful folctics with chmical compositions na a mophotopic phas bounday. A significant fatu of ths matials is that thy xhibit ivsibl dfomation though a switching mchanism,.g., a ttagonal vaiant ointd in th x diction can switch its ointation to th y diction. In shap mmoy alloys blow M f this switching is tmd twinning and dtwinning o twin ointation. In all cass this typ of switching, whn it is inducd by applid mchanical stss, is calld folasticity. An intsting fatu of folastic dfomation is that th xists an asymmty in th uniaxial tnsion and uniaxial compssion bhavio 1. In gnal, lag ivsibl stains can b attaind in tnsion than in compssion fo th common cas whn th c-axis of th unit cll is long and th a-axs. Consid a singl cystal of ttagonal matial. Assum that this cystal consists of qual quantitis of th ttagonal vaiants as illustatd in Fig. 1a, i.. on thid of th cystal has th c axis alignd in th x diction, on thid in th y, and on thid in th z. Within th cystal th vaiant typs a dividd into domains, with ach domain spaatd by a domain wall o twin bounday th tminology usd dpnds on th matial in considation. If w assum that th stain stat of a vaiant with its c axis alignd in th x, y, and z dictions a givn as xx 0, 0 /2, 0 /2, yy 0 /2, 0, 0 /2 and zz 0 /2, 0 /2, 0 spctivly, thn th initial volum avagd stain of th nti cystal vanishs. Now, apply a tnsil stss in th x diction. This stss will do positiv wok if th vaiants alignd in th y and z dictions switch thi ointations towads th x diction. Contibutd by th Applid Mchanics Division of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS fo publication in th ASME JOURNAL OF APPLIED ME- CHANICS. Manuscipt civd by th ASME Applid Mchanics Division, July 29, 2002; final vision, Dc. 10, Associat Edito: H. Gao. Discussion on th pap should b addssd to th Edito, Pof. Robt M. McMking, Dpatmnt of Mchanical and Envionmntal Engining Univsity of Califonia Santa Babaa, Santa Babaa, CA , and will b accptd until fou months aft final publication of th pap itslf in th ASME JOURNAL OF APPLIED MECHANICS. Gnally, th mchanism fo switching is th motion of domain walls/twin boundais. Switching will pocd until all of th vaiants a alignd in th x diction, laving th cystal with a stain of xx 0, yy 0 /2 and zz 0 /2. Now, if w apply a compssiv stss in th x diction, th stss will b abl to do positiv wok if th vaiants ointd in th x diction switch to ith th y o z dictions. Futhmo, th amount of wok don by ith of ths switchs is idntical. In gnal, th xact switching squnc will dpnd on th gomty of th domain walls; howv, if th cystal is lag nough it is asonabl to xpct that th final stat of th cystal will hav half of its vaiants alignd in th y diction and half in th z. Thfo, in compssion th avagd stain stat of th cystal is xx 0 /2, yy 0 /4, and zz 0 /4. Hnc, fo th singl cystal loadd along any of its 100 dictions th maximum ivsibl tnsil stain is 0 and th maximum compssiv stain is 0 /2, as illustatd in Fig. 1b. This obsvation has ld som sachs to popos a satuation condition fo folastic polycystals basd on a minimum pincipal mannt stain cition 4 6; namly, that th minimum pincipal mannt stain in th matial can nv b lss than 0 /2. This cition lads to a tnsion-compssion asymmty atio of 2:1 fo th polycystal loadd in any diction. Mo sophisticatd modls that tat th polycystal as an agggat of singl cystals hav found this asymmty atio to b about 1.7: Such a modl will b usd in this wok to invstigat th nti ang of stain satuation conditions. Fo xampl, what is th maximum valu of pu sha mannt stain that can b achivd? Aft th psntation of th micomchanical simulations a phnomnological modl fo folasticity will b dvlopd and compad to pdictions fom th micomchanical modl. 2 Micomchanical Computations Th micomchanical modl fo th polycystallin bhavio implmntd h is idntical to th modl psntd in Rf. 10 without th ffcts of lctic fild. Futhmo, this modl is analogous to th Hill-Hutchinson modl fo polycystal plasticity. Th fundamntal componnts to this modl a a tangnt constitutiv law fo th singl cystals and a slf-consistnt avaging mthod to pdict th poptis of th polycystal. 470 Õ Vol. 70, JULY 200 Copyight 200 by ASME Tansactions of th ASME

2 Fig. 1 a Th th possibl ointations of ttagonal vaiants within a singl cystal. Diffnt vaiants within a singl cystal will b spaatd by a domain wall o twin bounday. b Th uniaxial stss-stain spons of a modl singl cystal loadd along any of th Š100 dictions. Notic th asymmty in tnsion vsus compssion. Each singl cystal has a uniqu ointation and is tatd as a sphical inclusion mbddd in an infinit matix. Th tangnt poptis of th matix a takn to b a slf-consistnt avag of th incmntal bhavio of all of th singl cystals. Hnc, a stss o stain histoy can b applid to th polycystal and th modl can b usd to dtmin th cosponding stain o stss histoy. Th singl cystal constitutiv law is analogous to continuum slip plasticity modls with th addd ffct of stain satuation. In od to div th singl cystal constitutiv law it is assumd that th stss within th singl cystal is unifom and both th total stain and th mannt stain a th volum avags ov th nti cystal. Not that by assuming th stss within th singl cystal is unifom, w a nglcting lastic intactions btwn domains in th cystal. As illustatd in Fig. 1a, th ttagonal vaiants can xist in ach singl cystal. Hnc, th stain, mannt stain, and lastic complianc of th singl cystal a sc ij I1 c I (I) ij,,sc ij I1 c I (I) ij, and s sc ijkl I1 c I (I) s ijkl, (2.1) wh c I psnts th volum concntation of th Ith vaiant. Thn, th mannt stains of ach of th vaiants a givn by (1) ij 0 i1 j1 ij /2, (2) ij 0 i2 j2 ij /2, and () ij 0 i j ij /2, (2.2) wh th 1, 2, and coodinat dictions a paalll to th 100 cystal dictions, ij is th Konck dlta, and 0 has th sam maning as that fo th discussion in th Intoduction, and can b givn in tms of th lattic paamts as 0 2(c a)/(c2a). With ths dfinitions of th mannt stains fo th individual vaiants, th mannt stain of th nti cystal will b zo if th a qual volum concntations of ach of th vaiants. In od to dtmin how th volum concntations of th vaiants can chang with applid loading, w will fist consid th f ngy of th cystal. Und isothmal conditions, th Hlmholtz f ngy of th singl cystal, sc, is qual to th stod lastic ngy of th cystal and is givn as sc 1 2 c ijkl sc sc ij,sc ij sc kl,sc kl c sc ijkl s sc ijkl 1 and ij with sc sc sc c sc ijkl sc kl kl ij,sc. (2.) Not that th lastic complianc of th cystal, and hnc th lastic stiffnss, a allowd to chang as th volum concntations of th vaiants changs. Th dissipation at in th singl cystal is givn as th wok at du to applid stsss minus th f ngy at. By applying th pviously statd assumptions and an appopiat Lgnd tansfomation, th dissipation at can b shown to b ẇ D sc ij sc ij sc sc ij,sc ij 1 2 ṡ ijkl sc I1 I ċ sc ij ij (I) 1 2 s I ijkl sc ij kl sc sc ij kl sc. (2.4) Not that six diffnt switching tansfomations a possibl, i.., ach of th th vaiants can switch to th oth two. Futhmo, if a tansfomation is occuing, thn th volum concntations of th vaiants bing switchd to and fom incas and dcas, spctivly, at xactly th sam ats. By applying ths facts, it is thn possibl to wit th dissipation at and dfin th tansfomation diving focs, G,as 6 ẇ D ḟ sc ij () ij 1 2 s ijkl sc ij kl sc ḟ G, wh G sc ij () ij 1 2 s ijkl sc ij sc kl. (2.5) H, numbs th six possibl tansfomation systms and th psnts th diffnc in th following quantity btwn th vaiant bing tansfomd to and that bing tansfomd fom. Fo xampl, tak 1 to psnt th tansfomation fom vaiant 1 to vaiant 2. Thn, (1) ij 0 ( i2 j2 i1 j1 )/2 not that this is a pu sha stain, s 1 ijkl s (I2) ijkl s (I1) ijkl and th volum Jounal of Applid Mchanics JULY 200, Vol. 70 Õ 471

3 concntation ats a ċ (I1) ḟ (1) and ċ (I2) ḟ (1). Not that in gnal s (I1) ijkl and s (I2) ijkl will b diffnt du to diffncs in th ointations of th c axis of ach vaiant. It is assumd that if a quantity of a givn vaiant xists, i.., c I 0, thn it is possibl to incmntally switch that vaiant into on of th oth two vaiants. Futhmo, this switching btwn vaiants is assumd to occu only if such a tansfomation sults in a chaactistic at of nonngativ dissipation. Spcifically, a tansfomation systm is potntially activ if G 0 0 ḟ 0 (2.6) and th tansfomation systm is inactiv if G 0 0 ḟ 0. (2.7) If w nglct changs in th lastic poptis, thn th physical intptation of Eq. 2.6 is as follows; whn th solvd sha stss on a tansfomation systm achs th citical solvd sha stss, 0, thn tansfomation is allowd to occu on that systm. Not that if th volum faction of a givn vaiant vanishs, thn th tansfomation systms that duc th quantity of that vaiant cannot b activatd. This fatu nabls th modl to account fo stain satuation at th singl cystal lvl. In th absnc of hadning of th tansfomation systms, i.., 0 mains fixd duing switching, th singl cystal constitutiv law quis th following inputs: th lastic poptis of a singl ttagonal vaiant, th citical solvd sha stss to induc tansfomation 0, and th lattic paamts of a ttagonal vaiant that dtmin 0. Fom Eqs it is a mathmatical xcis to dtmin th tansfomation ats, ḟ, in tms of th applid stain ats, and thn foms fo th singl cystal tangnt moduli follow. This pocdu will not b givn h, but it is psntd claly in Rf. 10. Th sulting uniaxial stss-stain spons of th singl cystal loadd in any on of th 100 dictions is givn in Fig. 1b. Not howv, that th singl cystal spons is anisotopic, and loading along oth dictions will yild diffnt bhavios. Fo xampl, uniaxial stss applid in any of th 111 dictions dos not cat a diving foc on any of th tansfomation systms, and hnc 111 loading will sult in a pfctly lina lastic spons of th cystal. Using th singl cystal constitutiv law dscibd abov, a slf-consistnt modl is applid to comput th ovall stssstain bhavio fo a polycystal. Fo concptual simplicity th polycystal is viwd as an infinit collction of andomly ointd singl cystals subjctd to homognous stats of stss and stain. Th Catsian componnts of th macoscopic stss and stain incmnts of th polycystal a takn to b th volum avags of th Catsian componnts of th cosponding stss and stain incmnts in th singl cystals. Each individual singl cystal gion is modld as a sphical inclusion mbddd in an infinit ffctiv mdium matix. Th tangnt stiffnss of th ffctiv mdium is takn to b that of th polycystal. Sinc nighboing gains a not modld xplicitly, th constaint intactions btwn gains a not dtmind dictly in this modl. Instad, ach gain is constaind by th ffctiv mdium matix, and in this sns th modl accounts fo gain-to-gain constaints in an avagd sns. Ultimatly, th stss and stain stat in ach singl gain will dpnd on th applid loading histoy and th ointation of th cystal. Fo mo dtails of both th singl cystal constitutiv law and th slf-consistnt avaging mthod th ad is fd to Rfs Not that in Eqs th quantitis sc ij, sc,sc ij, and ij w usd to psnt stss, stain, and mannt stain in a singl cystal. Thoughout th maind of this pap, ij, ij, and ij will b usd to psnt stss, stain, and mannt stain in a polycystal. To isolat th bhavio of th polycystal satuation, th lastic poptis of a singl ttagonal vaiant w takn to b isotopic in this study, with sha modulus and Poisson s atio. Thn, a dimnsional analysis of th govning quations implis that th pdictions of th modl fo th nomalizd stsss ij / 0 vsus th nomalizd stains ij / 0 will only dpnd on th dimnsionlss paamts 0 / 0 and. Th goal of this invstigation is to map out th satuation conditions fo mannt stain stats btwn axisymmtic xtnsion and axisymmtic contaction. To do this th following mannt stain invaiants a intoducd: 2 ij ij 1/2 and J 4 1/ ij jk ki. (2.8) H ij is th mannt stain dviato, ij ij ij kk /. Of cous, sinc th dfomation pocsss of th singl cystals a volum consving, call that th tansfomation stains a pu shas and thfo kk 0; w hav ij ij and th intoduction of th mannt stain dviato appas to b unncssay. Howv, th thoy psntd in th nxt sction will qui divativs of ths invaiants, and th divativs will b affctd by this distinction. With th dfinition of ths two invaiants, a full ang of mannt stain satuation stats can b pobd by allowing th atio of J /J 2 to vay fom 1 axisymmtic contaction to 0 pu sha to 1 axisymmtic xtnsion. If w consid any volum consving mannt stain in th pincipal dictions, thn th mannt stain tnso and J /J 2 can b wittn as b 0 and 0 0 1b J )bb2 1/ 4 1/6 1bb 2 sgn, (2.9) wh b can b any abitay constant. In oth wods, vy multiaxial volum consving mannt stain stat can b dscibd by th atio of ths two invaiants, and this atio will always li in th ang 1J /J 2 1. Nxt, consid th poblm of finding th satuation conditions fo a pu sha mannt stain. At fist, on might attmpt to find this condition by applying a pu sha stss o pu sha total stain to th modl polycystallin matial. Howv, as will b shown, such a pocdu will not poduc a pu sha mannt stain stat. Du to th matial s ability to dfom mo in tnsion than in compssion, th atio of J /J 2 will appoach 1 as an applid pu sha stss is incasd. Thfo, it is ncssay to dvis a mo sophisticatd schm fo pobing th ang of J /J 2. Within th slf-consistnt modl, th tangnt moduli of th t polycystal, c ijkl, a computd at ach incmnt in th loading pocss. This knowldg of th instantanous tangnt moduli allows fo th adjustmnt of th loading path in applid stss spac, such that th atio J /J 2 mains constant duing th nti load xcusion. Mo spcifically, by manipulating th lationships t c ijkl ij c ijkl kl kl kl (2.10) th appopiat stss o stain incmnts that a quid to maintain a constant atio of J / duing loading can b dtmind. Figu 2 plots a st of sults fo th nomalizd ffctiv stss, / 0, vsus th nomalizd ffctiv mannt stain, J 2 / 0, fo th dimnsionlss matial atios of 0 / and Not that th ffctiv stss is dfind as 2 s ijs ij 1/2, wh s ij ij 1 kk ij. (2.11) Each plot on Fig. 2 psnt a diffnt loading path that was pscibd in such a way that th atio of th mannt stain invaiants, J /, maind constant thoughout th loading. Th 472 Õ Vol. 70, JULY 200 Tansactions of th ASME

4 Fig. 2 Slf-consistnt computations of th dfomation bhavio of th folastic matial fo diffnt popotional mannt staining paths. Th stsss a nomalizd by th citical solvd sha stss quid to caus switching and th ffctiv mannt stains a nomalizd by th tnsil satuation stain of a singl cystal. Not that J ÕJ 2 ÄÀ1,0,1 psnts axisymmtic contaction, pu sha mannt stain, and axisymmtic xtnsion, spctivly. Th inst is an xpandd viw of th gion cut off fom th lag plots. Not th lack of tnsion-compssion asymmty fo small stains, but th significant asymmty of th satuation stains. inst contains an xpandd viw of th gion xcludd fom th main gaph. Not that a matial with a switching yild sufac dscibd solly by th stss invaiant and a hadning bhavio dpndnt only on th mannt stain invaiant J 2 would hav cuvs on this gaph that a indpndnt of th atio J /J 2.At small mannt stains, ij , th satuation bhavio of th singl cystals has not fully dvlopd and has littl ffct on ths stss-stain cuvs. In fact, th diffncs in th cuvs appaing on th inst figu ais pimaily du to th fact that th switching sufac fo th modl matial follows a Tsca cition 15. As th mannt stain continus to incas, th satuation bhavio of th singl cystals bgins to hav an ffct on th polycystal, with th stss having to incas damatically in od to caus futh mannt staining. Ultimatly, fo ach load path J 2 appoachs som limiting valu. Th satuation valus ang fom fo axisymmtic contaction to fo axisymmtic xtnsion. Ths valus a in agmnt with oth modls with simila tansfomation systm switching citia, but lss sophisticatd polycystallin avaging tchniqus 7 9. In od to illustat th ffcts of th dimnsionlss atio 0 / 0 ; Fig. plots th nomalizd ffctiv stss vsus th nomalizd ffctiv mannt stain in a uniaxial compssion tst fo th valus of 0 / 0. Not that th cuvatu in th tansition fom -lik dfomation to satuation is lag as 0 / 0 dcass fo ths nomalizations. Also not that th ultimat satuation lvl of mannt stain is th sam fo ach lvl of 0 / 0. Th satuation stains a ultimatly a function of th undlying cystal stuctu gomty; hnc it is to b xpctd that th satuation lvls of mannt stain a indpndnt of th lastic poptis of th matial. In gnal, fo any givn untxtud polycystal, th satuation stain will dpnd only on th cystal stuctu of th vaiants, th switching cition fo th tansfomation systms, and th mannt stain invaiant atio J /J 2. Hnc, th sults fo th satuation stain lvl to b psntd in Fig. 4 a univsally valid fo ttagonal matials Fig. Slf-consistnt computations of th uniaxial compssion dfomation bhavio of th folastic matial fo diffnt lvls of th dimnsionlss atio 0 Õ 0. Not that th satuation stain fo ach tst is th sam but th shap of th stss-stain cuv diffs. with th c axis long than th a axis and tansfomation allowd at a citical solvd sha stss on th tansfomation systm. Lastly, Fig. 4 plots th satuation stain valus fo th modl polycystallin matial vsus th mannt stain atio J /. This sult is th ky componnt fom ths micomchanical simulations that will b applid to th phnomnological thoy fo folastic switching psntd in th nxt sction. Figu 4 illustats th nti ang of multiaxial mannt stain stats that a possibl in th modl matial. Spcifically, th gion blow th cuv psnts stain stats that a achivabl, and th gion abov th cuv consists of unattainabl mannt stains. Again, th obsvations that th low mannt stain gion is lativly indpndnt of J /, and th satuation lvls of Fig. 4 Th ffctiv satuation stain lvl as a function of th mannt stain invaiant atio J ÕJ 2. Not that J ÕJ 2 ÄÀ1,0,1 psnts axisymmtic contaction, pu sha mannt stain, and axisymmtic xtnsion spctivly. This figu illustats th anisotopic natu of th matial in spons to tnsion vsus compssion. Jounal of Applid Mchanics JULY 200, Vol. 70 Õ 47

5 mannt stain a dpndnt on J / will b usd to dvis a phnomnological constitutiv law fo ths matials in th nxt sction. Phnomnological Modl fo a Polycystallin Matial Th phnomnological modl follows th gnal fomulation dvlopd fo folctic matials dvlopd in Rf. 6. H w will focus only on puly folastic bhavio. Hnc this modl applis to unpold folctic matials loadd only by applid stss and not by lctic fild. Th modl also applis to twin ointation in shap mmoy alloys blow th matnsit finish tmpatu. Th modl will b cast within an isothmal, at indpndnt, small dfomation famwok. Th pimay assumption of th modl is that th intnal stat of th matial can b ntily chaactizd by th mannt, i.., ivsibl, stain stat of th matial ij. Hnc, w intoduc th Hlmholtz f ngy p unit volum of th polycystallin matial,, as s ij, ij ij. (.1) H, s psnts th stod lastic ngy p unit volum, ij a th componnts of th total stain, and psnts a contibution to th f ngy associatd only with th intnal stat of th matial. Th following dvlopmnt of th phnomnological thoy will dmonstat that givs is to back stsss. It is gnally accptd that th physical mchanism fo back stsss is th xistnc of sidual stsss in th matial du to inhomognous mannt/plastic stain fom gain to gain, i.., lockd-in ngy 18. Thfo, th physical intptation of is that it accounts fo th stod ngy du to th intnal sidual stsss in th matial. Not that th lastic poptis of th matial can dpnd on th mannt stain stat. Fo xampl, an initially unpold folctic will b lastically isotopic sinc th cystal vaiants with ttagonal lastic poptis a andomly ointd. Howv, if this matial is staind in tnsion in th x diction, most of th vaiants will align in th x diction and th lastic poptis will now b tansvsly isotopic about th x diction. Thn, assuming lina lastic bhavio about a fixd mannt stain stat, th stod f ngy can b wittn as s 1 2 c ijkl ij ij kl kl. (.2) H c ijkl a th componnts of th lastic stiffnss tnso that can dpnd on th mannt stain componnts ij. Th scond law of thmodynamics implis that th dissipation at must b non-ngativ, i.., ij ij 0. (.) A Lgnd tansfomation along with th following dfinitions: ij 1 2 s pqs ij B ij ij pq s, (.4), (.5) ˆ ij ij B ij ij, (.6) a usd to show that Eq.. can b wittn as ˆ ij ij 0. (.7) Not that th s ijkl a th componnts of th lastic complianc tnso, wh th lastic complianc is th invs of th lastic stiffnss, i.., s ijkl (c ijkl ) 1. Now, assum that a switching sufac (ˆ ij, ij )0 xists in ˆ ij spac such that switching can occu if th stat ˆ ij is on th sufac and th matial spons is lastic if th stat ˆ ij lis within th sufac. Stss stats outsid of th switching sufac a fobiddn. If th matial abids by th postulat of maximum plastic dissipation, such that (ˆ ij ˆ ij *) ij 0 fo any stss stat ˆ ij * on o within th switching sufac and th stss stat ˆ ij causs th mannt stain incmnt ij, thn th switching sufac must b convx and th flow law fo th mannt stain at is associativ. In oth wods, convxity implis that th Hssian 2 /ˆ ij ˆ kl is positiv dfinit, and th associativ flow ul implis that th mannt stain incmnt must b nomal to th switching sufac such that ij. (.8) ˆ ij H, is a positiv scala multipli that must b dtmind fom th consistncy condition. In addition to th convxity and nomality constaints, th scond law, Eq. 2.7, implis that th switching sufac must nclos th oigin in ˆ ij spac. Along with th following dfinitions U ijkl 1 2 s pqs 2 ij pq s, (.9) kl H ijkl 2 ij kl, (.10) ij s mnij ˆ ij ˆ mn, (.11) kl kl ij c ijkl kl, (.12) D H ˆ ijkl U ijkl ij ˆ kl ij ˆ ij, (.1) DD ij ij, (.14) and th consistncy condition 0, w can solv fo th multipli as 1 D ij ij 1 D ij ij (.15) and thn wit th incmntal constitutiv quations as ij s ijkl 1 D ij kl kl (.16) o ij c ijkl 1 D ij kl kl. (.17) Th dfinitions outlind in Eqs sv two puposs. Fist, ths dfinitions allow fo a compact notation, and scond, th nw vaiabls allow fo a simpl alization of th symmty of th tangnt moduli. 4 Fitting th Modl to th Matial Th matial spcific vaiabls that nd to b spcifid fo this modl includ th dpndnc of th lastic poptis on th mannt stain, th switching sufac, and th mannt potntial. Fo th sak of simplicity, w will assum that th lastic poptis a isotopic and indpndnt of th mannt stain fom h on. As such, ij and U ijkl vanish. This assumption is in agmnt with th slf-consistnt calculations sinc th individual cystallits w assumd to b lastically isotopic. Howv, al matials can xhibit significant anisotopy in thi lastic poptis and this assumption may nd modification. Again, th mphasis of this wok is to invstigat th satuation bhavio of ths matials so this simplification is of sconday impotanc. 474 Õ Vol. 70, JULY 200 Tansactions of th ASME

6 Du to th fact that th mannt stains ais du to th ointation of th cystal vaiants it is asonabl to assum that th mannt stain is volum consving, i.., kk 0. Thn th switching sufac must b a function of invaiants of th dviatoic stss, ŝ ij ˆ ij ij ˆ kk /. In gnal, th switching sufac can chang both siz and shap with continud mannt staining. Howv, again fo th sak of simplicity, w will assum th simpl fom fo th switching sufac, such that 2 ŝ ijŝ ij (4.1) As notd in Sc. 2, th initial switching sufac pdictd by th slf-consistnt modl follows th Tsca cition. Futhmo, th switching sufac will in gnal volv into vn mo complx shaps than th Tsca hxagon 10,15. Howv, it will b shown that this additional complxity nd not b addd to th phnomnological modl in od to captu th salint fatus of th dfomation bhavio of folastic matials, and that th switching sufac fom of Eq. 4.1 is adquat. Finally, w must spcify th fom of th hadning potntial. Of cous, this potntial must dpnd on th invaiants of th mannt stain. Futhmo, to fomally caus th back stss B ij to b puly dviatoic, w will wit in tms of invaiants of th mannt stain dviato, ij ij ij kk /. Byond this considation, dtmining bcoms a cuv fitting xcis. Howv, this cuv fitting must b don intlligntly and should b infomd by xpimntal obsvation and micomchanical modls. Rcall that th mannt stain satuation bhavio has a significant dpndnc on J. A fist stp to dtmining is to dtmin a asonabl fit to th satuation bhavio. Th satuation bhavio dpictd in Fig. 4 can b accuatly dtmind in th following way. Fist, dfin a function f of th mannt stain invaiant atio J /J 2 as 0.01 J fo J f J J f J J J J 0, (4.2) fo J 0. (4.) Not that this fit fo f has fist, scond, fouth and fifth divativs qual to zo at J / 0. Th impotanc of ths conditions on f at J / 0 will b discussd shotly. Also not that this is a function fo matials with ttagonal cystal stuctu only. Finally, on dividd by this function is abl to fit th nomalizd sults of Fig. 4 with nomalizd by c instad of 0 ) to within 0.04% accuacy fo J / 0 and 0.15% fo J / 0. Now, dfin a stainlik vaiabl as 21 J 2 f J /J 2. (4.4) Whn th mannt stain lvl chaactizd by achs th compssiv satuation lvl, c, th mannt stain will b satuatd, i.., c. Not that th micomchanical simulations dscibd in Sc. 2 found that c Thn Eqs. 4.2 and 4. imply that th ffctiv satuation lvl of mannt stain in tnsion is t c , and if th mannt stain stat is pu sha, thn th satuation lvl is s c Hnc, whn dvising a functional fom fo, th back stsss must bcom lag as appoachs c. Th scond significant obsvation fom th slf-consistnt computations is that, discounting th Tsca natu of th initial switching sufac, fo small mannt stain lvls th stss vsus mannt stain bhavio can b dscibd solly by th stain invaiant J 2, i.., it is indpndnt of J. This can b statd as: is only a function of J 2 as appoachs zo. Futhmo, it is asonabl to conclud that is a function of as appoachs c. Hnc, as pat of this cuv fitting pocdu, a nw stainlik vaiabl is intoducd, *1wJ 2 w, (4.5) wh w is a wighting function that must b zo whn 0 and on whn c. A simpl mthod fo gnating stplik wight functions is to tak w( )A p(1 ) q. Fo this study p, q 5 and th functional fom fo w is w c c 5 c 5 6 c c 1 9 c 9. (4.6) Finally, w tak to b a function of *, i.., (*). Th functional fom fo th divativ of with spct to * can b fittd to ith a uniaxial tnsion o compssion tst. Fo this study th functional fom was takn as d d* H 0 1 m 1*/ c 1. (4.7) Not that this function and hnc th back stss componnts appoach infinity as * c. Fom Eq. 4.7 th quid inputs to th phnomnological modl can b obtaind. Not that th pimd vaiabls dnot fist o scond divativs with spct to th agumnt of th function. and with B ij d d* * (4.8) ij H ijkl d2 * * d* 2 d d* ij kl 2 * (4.9) ij kl * 1w 2 ij ww wj ij, (4.10) 2 ij 2 * ij kl 1w 1 il jk ik jl 2 ij kl 4 9J 2 ij kl 2 w ij kl kl ij ww wj 2 2 2ww wj ij 2, (4.11) kl ij kl 2 f J f ij 4 f J 2 ik kj ij, (4.12) ij Jounal of Applid Mchanics JULY 200, Vol. 70 Õ 475

7 and 2 ij ij kl kl 4 J 2 9 J 4 f f J f 8 J J 5 f ik jl il jk 1 1 f J f J ij kl kl ij J f J 9 J 2 f ij kl 4 J 2 9 J 2 5 f J 2 f J f J ij km ml kl im mj 2 16 J J 5 f 8 9 J 4 f ik jl jl ik il jk jk il 2 1 J 2 f ij km ml im mj kl J 2 f J im mj kn nl J 2 J 4 f 9 J 5 f. (4.1) Not that Eq. 4.1 is th symmtic fom of a simila xpssion appaing in Rf. 6. Th appant complxity of ths xpssions is unfotunat, but is a sult of th simpl assumption that th intnal stat of th matial can b dscibd using only th mannt stain as an intnal vaiabl. Th utility of ths xpssions will b dmonstatd whn th thoy is compad to th mo dtaild micomchanical modl usd in Sc. 2. Not that th back stsss and hadning moduli dpnd dictly on th xpssions in Eqs and 4.1. In od fo th back stsss and hadning moduli to b finit, ctain stictions must b placd on f and its divativs at J / 0. Ths conditions can b obtaind by xpanding f into a pow sis about zo and quiing th xpssions in Eqs and 4.1 to b finit. Spcifically, f 0 I f 0 II f 0 IV f 0 V 0, wh f 0 n psnts th nth divativ of f with spct to J / at J / 0. Ths stictions must b tu fo any f dfining a stainlik vaiabl that is usd in th hadning potntial. Fo xampl, Landis 6 assumd that satuation occus whn th minimum pincipal mannt stain achs a citical valu. Th minimum pincipal mannt stain can b wittn as th pu sha stss tst is not qual to th J / 0 tst of Fig. 2. Fo a matial with a switching sufac and hadning potntial, th pu sha stss and J / 0 tsts would yild idntical sults. Not that th phnomnological thoy pdicts th satuation stain lvls accuatly. This is to b xpctd fo th tnsion, compssion, and in fact any of th popotional mannt stain path tsts sinc th functional fit to th satuation bhavio of Eqs. 4.2 and 4. has bn usd. Howv, th agmnt of th modl to th popotional stssing tsts is a tu pdiction of th phnomnological thoy. No paamts of th modl hav bn adjustd in od to fit ths micomchanical modl tsts. Futhmo, notic that vn th computation with th constant stss invaiant atio of 0.8 significant compssiv stss yilds mannt stains that a mo tnsil than th mannt stain contolld tst with J / 0.8. Not, of cous, that th tnsil mannt stain is not alignd with th compssiv stss diction, but ath th diction with th lagst stss dviato componnt. Th phnomnological modl is abl to captu this bhavio to asonabl accuacy. III 1 2 cos ) 2 sin, with accosj /. Not that th function in squa backts satisfis f 0 I f 0 II f 0 IV f 0 V 0, and futhmo f 0, f 0 III, f 0 VI 0. Th ultimat tst of this phnomnological thoy is to compa it to xpimntal masumnts on al matials. Howv, spcially fo folastic camics, it is difficult to invstigat th stain satuation bhavio du to th lag stsss quid. Th camic matial tnds to factu at lag stsss. Futhmo, as notd in Sc. 2, th stss paths quid to gnat tsts lik thos psntd in Fig. 2 a not simpl, and qui mo infomation than is possibl to obtain in a singl xpimnt. Hnc, th nxt bst tst of th thoy is to compa it to th pdictions of a mo dtaild micomchanical modl. Figu 5 is just such a compaison. On Fig. 5 th bold lins psnt th pdictions of th slfconsistnt modl and th thin lins a th pdictions of th phnomnological thoy fo diffnt popotional stss path loadings of th matial. Th paamts fo th phnomnological thoy w chosn to fit th uniaxial tnsion and compssion data. On this gaph a sults fo a uniaxial tnsion, uniaxial compssion, and two oth popotional stssing paths, including a pu sha stss tst. Th typ of popotional loading path is chaactizd by th stss invaiant atio / 9s ij s jk s ki /2, with 1, 0, and 1 cosponding to axisymmtic compssion, pu sha stssing, and axisymmtic tnsion spctivly. Oth computations fo stss invaiant atios of 0.5 and 0.5 w also pfomd, but not includd on Fig. 5 in od to avoid clutt aound th pu sha sults. Th compaison of th phnomnological thoy to ths sults was favoabl as wll. Notic that Fig. 5 A compaison of th phnomnological thoy to th slf-consistnt calculations fo diffnt popotional stssing paths. Th stsss and stains a now nomalizd by th paamts of th phnomnological modl. Not that th atio Õ 9s ij s jk s ki Õ2ÄÀ1,0,1 psnts axisymmtic compssion, pu sha stssing, and axisymmtic tnsion, spctivly. Notic that vn th stss path with significant compssion sults in an ultimat mannt stain stat with significant xtnsion. Of cous, th tnsil stain is not alignd with th compssiv stss in this situation. 476 Õ Vol. 70, JULY 200 Tansactions of th ASME

8 5 Discussion In this wok constitutiv modls fo polycystallin folastic matials with an undlying ttagonal cystal stuctu hav bn invstigatd and dvlopd. In Sc. 2 th slf-consistnt modl of Hub t al. 10 was implmntd to invstigat th mannt stain satuation bhavio of ths matials. It was found that th satuation stain fo axisymmtic xtnsion was gat than th satuation stain fo axisymmtic contaction by a facto of 1.7. Futhmo th nti ang of mannt stain satuation stats btwn axisymmtic xtnsion and contaction w chaactizd using th mannt stain invaiant atio J /. Equations 4.2 and 4. w poposd as an accuat fit fo th divation of th satuation cuv psntd in Fig. 4. In Sc. a phnomnological constitutiv famwok fo folastic matials was poposd. Th thoy assums that th intnal stat of th matial can b chaactizd compltly by th mannt stain. In oth wods, th mannt stain componnts a th intnal vaiabls that dtmin th lastic poptis, th switching sufac, and th nonlina hadning of th matial. Th thoy allows fo th lastic poptis of th matial to chang with mannt dfomation, and accounts fo ths changs in a thmodynamically consistnt fashion within th dfinition of th switching sufac spac. Convxity of th switching sufac in a modifid stss spac and an associatd flow ul fo th mannt stain incmnts follow fom assuming that th matial obys th postulat of maximum plastic dissipation. Equations.16 and.17 psnt th gnal fom fo th incmntal constitutiv lations fom th phnomnological thoy. Sction 4 of th pap is dvotd to dtmining functional foms fo th switching sufac and hadning potntial. Fo th micomchanical calculations of Sc. 2, th cystallits w assumd to b lastically isotopic. Hnc, th functional fom fo th lastic cofficints in th phnomnological modl w also takn to b isotopic and indpndnt of mannt stain. To maintain som simplicity, th switching sufac was takn to b a sph in th modifid dviatoic stss spac. It was notd that th initial switching sufac fo th slf-consistnt matial actually follows a Tsca cition. Futhmo, th pvious studis of Hutchinson 15 and Hub t al. 10 hav shown that th shap of th switching sufac volvs in a nontivial way with mannt dfomation. Howv, ths fatus w not invstigatd o accountd fo in this pap. Such modifications a lft fo possibl futu wok, but a xpctd to yild only incmntal impovmnts in th pdictions of th modl. Th mannt potntial was th last componnt quiing a fit fo th phnomnological modl. Th pimay ol of th mannt potntial in th phnomnological thoy is to account fo th complicatd dpndnc of th satuation stain on th mannt stain. Hnc th fist quimnt fo is to dfin a stainlik vaiabl that pdicts th stain satuation bhavio displayd in Fig. 4. Such a vaiabl is dfind in Eq. 4.4 with th function f givn by Eqs. 4.2 and 4.. This vaiabl,, must b lss than som citical valu c. Hnc, as appoachs c th mannt potntial must gow without bound. Thn, as a sult of th obsvation fom th micomchanical calculations that th initial dfomation is ssntially indpndnt of J /, a nw stainlik vaiabl, *, was intoducd in Eq Du to th wighting function w * is clos to fo small valus of, and appoachs as appoachs c. Finally, th ultimat spcification of quis a fit to a stss vsus mannt stain cuv. Th likly candidats fo this fit a ith a uniaxial tnsion o compssion cuv. To summaiz, th obsvations w usd to spcify. Fist, th function f was intoducd to fit th satuation stain as a function of J /, thn w was implmntd to fit th obsvation that small dfomations a indpndnt of J /, and finally d /d* was fit to a uniaxial stss vsus mannt stain cuv. Not that ach of th functions intoducd to fit spcific bhavios is in no way connctd to spcifid popotional stssing paths. Fo xampl, th loading path tavsd to dtmin th stain satuation condition whn J / 0isnot a pu sha loading path. In fact, th load path quid to kp th mannt stain atio qual to zo is not vn constant. Initially, th loading path is pu sha stssing. Rcall that pu sha is an xtnsion in on diction and an qual contaction in an othogonal diction. Sinc th matial is abl to xtnd mo asily than it can contact, in od to maintain pu sha mannt stain th compssiv load in th contaction diction must b gat than th than th tnsil load in th xtnsion diction as th stain appoachs satuation. In fact, as satuation is appoachd with th mannt stain atio maining constant at J / 0, th applid stss appoachs a uniaxial compssion. Thfo, compaing th phnomnological modl to th micomchanical modl fo diffnt popotional stssing paths is a valid tst of th phnomnological thoy. Figu 5 is a compaison of th phnomnological thoy to th slf-consistnt calculations, and th agmnt is quit good. Again, it is mphasizd that th slf-consistnt computations pfomd in Sc. 2 and th sults of that modl a valid fo matials with ttagonal cystal stuctu only. In od to invstigat oth cystal stuctus, lik othohombic o hombohdal, and th possibility of combinations of ths with ttagonal domains, th singl cystal constitutiv law usd in th slfconsistnt modl would nd to b modifid. Howv, th pimay sult that would b obtaind fom such calculations is a stain satuation cuv lik th on psntd in Fig. 4 fo ttagonal matials. Futhmo, simpl micomchanical modls that do not account fo gain to gain intactions can b usd to obtain this stain satuation cuv. Thn, th only significant chang to th phnomnological thoy would b to th functional fom of f dfind in Eqs. 4.2 and 4.. Th goal of this wok was to undstand th stain satuation bhavio of folastic matials and incopoat this infomation into a phnomnological constitutiv thoy. Th phnomnological modl is usful fo stuctual stss analyss on systms containing folastic matials. Such calculations will most likly b pfomd within a finit lmnt famwok. Hnc, it is impotant that th constitutiv spons can b intgatd apidly. This considation of computational spd uls out th mo dtaild slf-consistnt modl, sinc any on cuv asid fom th axisymmtic cass on Figs. 2,, o 5 quis a fw days to comput, whas th cosponding phnomnological thoy quis a fw sconds. Finally, whil th phnomnological thoy has bn shown to b in good agmnt with th mo dtaild micomchanical modl, th ultimat tst of this nw constitutiv law will b against caful multiaxial loading xpimnts. Acknowldgmnt This wok was suppotd by Contact No. DAAD fom th Amy Rsach Offic. Rfncs 1 Ftt, T., Müll, S., Munz, D., and Thun, G., 1998, Nonsymmty in th Dfomation Bhavio of PZT, J. Mat. Sci. Ltt., 17, pp Ftt, T., and Thun, G., 1998, Nonsymmtic Dfomation Bhavio of Lad Ziconat Titanat Dtmind in Bnding Tsts, J. Am. Cam. Soc., 81, pp Kaastamatis, T., Lupascu, D., Lucato, S., and Lynch, C. S., 2001, Th Effct of Gain Siz on th R-Cuv Bhavio of Lad Ziconat Titanat, Poc. SPIE, 4, pp Landis, C. M., and McMking, R. M., 1999, A Phnomnological Constitutiv Law fo Folastic Switching and a Rsulting Asymptotic Cack Tip Solution, J. Intll. Mat. Syst. Stuct., 10, pp Hub, J. E., and Flck, N. A., 2001, Multi-Axial Elctical Switching of a Folctic: Thoy vsus Expimnt, J. Mch. Phys. Solids, 49, pp Landis, C. M., 2002, Fully Coupld, Multi-Axial, Symmtic Constitutiv Laws fo Polycystallin Folctic Camics, J. Mch. Phys. Solids, 50, pp Jounal of Applid Mchanics JULY 200, Vol. 70 Õ 477

9 7 Lu, W., Fang, D. N., Li, C. Q., and Hwang, K. C., 1999, Nonlina Elctic- Mchanical Bhavio and Micomchanics Modlling of Folctic Domain Evolution, Acta Mat., 47, pp Fölich, A., 2001, Mikomchanischs Modll zu Emittlung Effktiv Matialignschaftn von Pizolktischn Polykistalln, Disstation thsis, Univsität Kalsuh, Gmany. 9 Kamlah, M., 2001, Folctic and Folastic Pizocamics Modling and Elctomchanical Hystsis Phnomna, Continuum Mch. Thmodyn., 1, pp Hub, J. E., Flck, N. A., Landis, C. M., and McMking, R. M., 1999, A Constitutiv Modl fo Folctic Polycystals, J. Mch. Phys. Solids, 47, pp Hill, R., 1965, Continuum Mico-Mchanics of Elastoplastic Polycystals, J. Mch. Phys. Solids, 1, pp Hill, R., 1965, A Slf-Consistnt Mchanics of Composit Matials, J. Mch. Phys. Solids, 1, pp Hill, R., 1966, Gnalizd Constitutiv Rlations fo Incmntal Dfomation of Mtal Cystals by Multislip, J. Mch. Phys. Solids, 14, pp Hill, R., 1967, Th Essntial Stuctu of Constitutiv Laws fo Mtal Composits and Polycystals, J. Mch. Phys. Solids, 15, pp Hutchinson, J. W., 1970, Elastic-Plastic Bhavio of Polycystallin Mtals and Composits, Poc. R. Soc. London, S. A, 19, pp Ric, J. R., 1970, On th Stuctu of Stss-Stain Rlations fo Tim- Dpndnt Plastic Dfomation in Mtals, J. Appl. Mch., 7, pp Ric, J. R., 1971, Inlastic Constitutiv Rlations fo Solids: An Intnal- Vaiabl Thoy and Its Applications to Mtal Plasticity, J. Mch. Phys. Solids, 19, pp Ric, J. R., 1975, Continuum Mchanics and Thmodynamics of Plasticity in Rlation to Micoscal Dfomation Mchanisms, in Constitutiv Rlations in Plasticity ditd by A. S. Agon, MIT Pss, Cambidg, Massachustts, pp Õ Vol. 70, JULY 200 Tansactions of th ASME

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