A Simple FEM Formulation Applied to Nonlinear Problems of Impact with Thermomechanical Coupling

Transcription

1 2439 A Smpl FEM Formulaton Appld to Nonlnar Problms of Impact wth Thrmomchancal Couplng Abstract Th thrmal ffcts of problms nvolvng dformabl structurs ar ssntal to dscrb th bhavor of matrals n fasbl trms. rfyng th transformaton of mchancal nrgy nto hat t s possbl to prdct th modfcatons of mchancal proprts of matrals du to ts tmpratur changs. Th currnt papr prsnts th numrcal dvlopmnt of a fnt lmnt mthod sutabl for nonlnar structurs coupld wth thrmomchancal bhavor; ncludng mpact problms. A smpl and ffctv altrnatv formulaton s prsntd, calld FEM postonal, to dal wth th dynamc nonlnar systms. Th dvlopd numrcal s basd on th mnmum potntal nrgy wrttn n trms of nodal postons nstad of dsplacmnts. Th ffcts of gomtrcal, matral and thrmal nonlnarts ar consdrd. Th thrmodynamcally consstnt formulaton s basd on th laws of thrmodynamcs and th Hlmholtz fr-nrgy, usd to dscrb th thrmolastc and th thrmoplastc bhavors. Th coupld thrmomchancal modl can rsult n scondary ffcts that caus rdstrbutons of ntrnal fforts, dpndng on th hstory of dformaton and matral proprts. Th numrcal rsults of th proposd formulaton ar compard wth xampls found n th ltratur. João Paulo d Barros Cavalcant a Danl Nlson Macl b, * Marclo Grco c José Nrs da Slva Flho a a Cvl Engnrng Dpartmnt, Fdral Unvrsty of Ro Grand do Nort, Lagoa Nova, , Natal, RN, Brazl. b School of Scncs and Tchnology, Fdral Unvrsty of Ro Grand do Nort, Lagoa Nova, , Natal, RN, Brazl. c Structural Engnrng Dpartmnt, Fdral Unvrsty of Mnas Gras, Avnda Prsdnt Carlos, , Blo Horzont, MG, Brazl. * Corrspondng author. E-mal addrss: dnmacl@ct.ufrn.br Kywords Postonal Formulaton; Nonlnar analyss; Thrmomchancal; Coupld problms; Impact. Rcvd In rvsd form Accptd Avalabl onln INTRODUCTION Th currnt papr prsnts th study of thrmomchancal rspons for frctonlss mpact problms, consdrng th nonlnar bhavor through a smpl formulaton basd on FEM, calld postonal.

2 244 J.P.B. Cavalcant t al. / A Smpl FEM Formulaton Appld to Nonlnar Problms of Impact wth Thrmomchancal Couplng Th formulaton s dscrbd n Total Lagrangan rfrnc usng as unknowns th nodal postons nstad of th dsplacmnt. Ths formulaton shows a smpl languag n rspct to nonlnar gomtrc approach whr th man advantag s th absnc of co-rotatonal axs (th formulaton can b carrd on drctly wth global axs only) (Coda and Grco, 24; Grco and Coda, 26; Carrazdo and Coda, 21; Coda and Paccola, 211; Coda t al., 213). Duhaml n 1837 and Numann n 1885 dntfd th modfcaton nd of th mchancal modls to consdr th tmpratur dffrncs to accuratly rprsnt th bhavor of th matrals (Shrf t al., 24). Th thrmomchancal thory nvstgats th ntracton btwn tmpratur and structural strans. In many cass, th nflunc of th strans n rlaton to th thrmal fld may b nglctd. Ths rsults n thrmomchancal uncoupld systm, for whch only th ffct of tmpratur modfs th strans fld. Th xprmntal studs and thortcal modls of thrmomchancal hav advancd smultanously. Th us of mor sophstcatd xprmnts rsults n mor accurat and ffcnt mathmatcal modls. Th thrmolastcty and thrmoplastcty ar dfnd as part of th thrmomchancal thory that dtrmns th bhavor of lastc and plastc bods, rspctvly, submttd to thrmal and mchancal fforts. Th frst studs wr dvotd to statc problms. Danlovskaya (195) was th frst rsarchr that trd to nclud th ffcts of nrta n thrmolastc transnt problms. Ths problms prsntd smpl natur, but th soluton xpland th procss of transmsson of th thrmal strsss. Th thory of uncoupld thrmolastcty may not b satsfactory for som transnt problms, bcaus xprmntal obsrvatons that shown th thrmal fld modfd by th strans fld. Svral xprmnts mphasz th nflunc of strans on thrmal fld (Rttl, 1998; Bnzrga t al., 25; Stanly, 28). Bot (1956) dvlopd th classcal thory of coupld thrmolastcty. Th quatons of lastcty and hat conducton ar ntrdpndnt. Th quatons usd to dscrb ths thory ar a combnaton btwn th lastcty thory and th laws of thrmodynamcs. Th quatons that rprsnt th hat conducton ar parabolc and hat propagaton vlocty n an lastc systm s nfnt. Th thory of gnralzd thrmolastcty cam to dal wth ths nconsstncs obsrvd n xprmntal rsults. Th gnral thory of thrmolastcty has bn modfd ovr th yars. Th frst thory proposd by Lord and Shulman (1967) consdrs that th hat conducton law s altrd to consdr th hat flux and ts rat. It was consdrd a tm paramtr rlaxaton to nsur fnt vlocty of propagaton of hat. Th scond thory dvlopd by Grn and Lndsay (1972) consdrs that th consttutv quatons ar modfd by nsrton of two tm paramtrs rlaxaton. Grn and Naghd (1993) prsntd a nw thory of gnralzd thrmolastcty basd on th balanc of nrgy and ntropy, whr nrgy dsspaton s not allowd. Subsquntly, Grn and Naghd (1995) prsntd th most gnral cas for th thory of gnralzd thrmolastcty, whr nrgy dsspaton was takn nto account. Th thors of thrmolastcty cannot accuratly dscrb th bhavor of lastoplastc bods. Thus, Dllon Jr. (1963) dvlopd th thory of thrmoplastcty wth th purpos of rprsnt th voluton of plastc strans. Snc thn, hat conducton problms wth lastoplastc charactrstcs hav bn studd xtnsvly and varous xact and numrcal solutons can b found n th ltratur, such as McKnght and Sobl (1977), Dargush and Banrj (1991), Hakansson t al. (25) and Carrazdo and Coda (21).

3 J.P.B. Cavalcant t al. / A Smpl FEM Formulaton Appld to Nonlnar Problms of Impact wth Thrmomchancal Couplng 2441 Although th soluton of a hat conducton problm dos not prsnt major complxty for most matrals, th modlng stratgy of thrmal and mchancal ntrdpndnc contnus to b a challng for nonlnar problms. Th study nonlnar thrmomchancal bhavor problms rmans subjct of many rsarchs n dffrnt flds, such as Rajagopal (1995), Rajagopal t al. (1996), Canadja and Brnc (29), Canadja and Brnc (21), Ozakn and Yavar (21) and Yavar and Gorly (213). 2 THE COUPLED THERMOMECHANICAL Th thrmolastcty has bn usd n svral aras, spcally for th applcaton of statc problms (Coptt, 1999; Shahan and Nabav, 27) and dynamc problms (Chn and Dargush, 1995; Norrs, 26; Shahan and Bashusqh, 214). Othr analyss mthods hav bn adoptd to dal wth coupld thrmolastc problms. As an xampl, Solr and Brull (1965) usd prturbaton tchnqus and mor rcntly Lychv t al. (21) dtrmnd a closd form soluton by an xpanson of functons gnratd by th hat conducton and quatons of moton. Svral studs dscrbd th thrmolastc bhavor of orthotropc matrals, such as Lu and Pstr (1975), ujosvc and Lubarda (22) Lubarda (24). Anothr fld of rsarch s th study of ansotropc matrals (Db t al., 1991; L, 1992; Clayton, 213; Mahmoud t al., 215). Numrcal approxmatons for thrmolastc quatons ar commonly found usng th FEM. Modls for transnt thrmolastc FEM ar dvlopd and compard wth analytcal solutons, as prsntd n Nckll and Sackman (1968) and Tng and Chn (1982). Rand and Gvol (1995) dvlopd a dynamc modl thrmolastc for FEM. Formulatons of fnt lmnts for thrmolastc dampng ar obtand from an rrvrsbl ntropy flux du to th hat fluxs causd by th varatons of volumtrc strsss, as prsntd n Srra and Bonald (29). Th dtrmnaton of th tmpratur fld n an lastc body s obtand by solvng th dffrntal quaton of hat conducton, subjct to ntal and boundary condtons. Th thrmomchancal bhavor of an lastc and hat conductng body s dscrbd by th hat conducton quaton (Eq. (1)) and local dynamc qulbrum (Eq. (2)), whch ar th man quatons of th thory of coupld thrmolastcty. 1 + n kq, = 2 aq r q G kk c -rr (1) - n é n 1 n ù æ ö 2 + d aqd G r = èç 1-2n 1-2n ø j kk j j F x êë, j ú (2) û whr G s th transvrs modulus of lastcty, n s th Posson coffcnt, r s th dnsty, ar th dformatons, d j s th Kronckr dlta and F ar th xtrnal forcs appld. Th Eqs. (1) and (2) also show th coffcnt of thrmal xpanson of matral (a ), spcfc hat ( c ), coffcnt of thrmal conductvty (k ), ntrnal hat sourc (R ), rfrnc tmpratur ( q ) and tmpratur varaton ( q ).

4 2442 J.P.B. Cavalcant t al. / A Smpl FEM Formulaton Appld to Nonlnar Problms of Impact wth Thrmomchancal Couplng Th hat varaton s dfnd by th hat flux and th hat gnratd ntrnally. Th quaton of thrmoplastcty frm n th frst and scond law of thrmodynamcs. In th form of th nqualty of Clausus-Duhm, on has: ru s q rr (3) - j j +, - = rq q S - rr + q, - q, ³ (4) q whr u s th ntrnal nrgy, s j ar th axal strss, q s th hat flux and S th ntropy. For all nrgy transformatons thr s an ncras n th ntropy. Th total nrgy nvolvd n th procss can not vary, and th ntropy ncras s a way to masur th nrgy that can b convrtd nto mchancal work. Th Hlmholtz fr-nrgy masurs th nrgy of a systm that can b transformd nto work. Th Eqs. (5) and (6) dfn, rspctvly, th Hlmholtz fr-nrgy (Φ ) and ts rat ( Φ ) as a functon of tmpratur (T ). (, ) Φ j T = U -TS (5) (, ) = - - = Φ + Φ T U TS TS T T Φ j j j (6) Combnng th concpt of Hlmholtz fr-nrgy wth Eqs. (3) and (4), t s possbl to obtan th quaton that dfns th nrgy consrvaton quaton n trms of th ntrnal dsspaton, dfnd by: Whr, rqs =L- q + rr (7), -rφ - rs q + s = L (8) j j Wth L rprsntng th ntrnal dsspaton of nrgy. 2.1 Intrnal arabls In accordanc wth xprmntal rsults t was obsrvd that part of th plastc work was convrtd nto hat (Farrn and Taylor, 1925; Taylor and Qunny, 1934). Dllon Jr. (1963) and Przyna and Sawzcuk (1973) prsntd th frst attmpts to dvlop consttutv modls consdrng th ntracton btwn th plastc work and thrmal ffcts. Subsquntly, formulatons hav bn dvlopd consdrng wth larg dformatons (Lmonds and Ndlman, 1986; Smo and Mh, 1992; Canadja and Brnc, 24). Smo and Mh (1992) prsnt an approach to thrmoplastcty analyss, consdrng a thrmodynamcally consstnt formulaton of th problm coupld and dtalng th prformanc and numrcal aspcts nvolvng th mplmntaton by FEM. Contnung th dvlopmnt of nrgy consrvaton quaton s ncssary to dvlop th ntrnal dsspaton trm. Two varabls ar

5 J.P.B. Cavalcant t al. / A Smpl FEM Formulaton Appld to Nonlnar Problms of Impact wth Thrmomchancal Couplng 2443 ntroducd: th plastc dformaton ( p j ) and th hardnng varabl ( x ). Th dcomposton of th stran tnsor s dfnd addtvly, whr th total stran s th rsult of th sum of lastc and plastc parts: p j j j (9) = - whr j s th lastc dformaton. From Eq. (9), th rat of Hlmholtz fr-nrgy can b xprssd by Eq. (1). For rasons of convnnc, from that momnt th tmpratur gradnt q, s dfnd by Θ. Φ Φ = - p Φ + x Φ + q Φ Φ + Θ j j x q Θ j j (1) Substtutng Eq. (1) nto (4) on has: æ ö æ ö s r r - Φ - + Φ q -r -r -r x - ³ ç è q Φ Φ p Φ q j j S Θ Θ ç è ø ø Θ j (11) x q j Accordng to Carrazdo and Coda (21), by dmandng that (11) holds for all admssbl thrmomchancal procss whch mans that t must satsfy all ndpndnt varatons of, q and Θ, Eq. (11) s satsfd f and only f: j Φ = = Φ Φ( j, x, q) (12) Θ Φ sj = r j (13) Φ S =- (14) q Consdrng Eqs. (12), (13) and (14) and substtutng Eq. (11) nto Eq. (7) gvs: Drvng th frst trm of Eq. (15), on has: Φ Φ - rq = s - r x - q + rr (15) q p j j, x æ ö Φ Φ Φ 2 p Φ j j j, ç è q q x q ø x j - rq + x + q = s - r x - q + r R (16) In th plastc rgm, a larg amount of plastc mchancal nrgy s dsspatd as hat. Howvr, th plastc work s not compltly transformd nto thrmal nrgy. Part of ths work s dsspatd du to ntracton btwn th ntrfacs of th mcrostructurs that consttut th matral. Th

6 2444 J.P.B. Cavalcant t al. / A Smpl FEM Formulaton Appld to Nonlnar Problms of Impact wth Thrmomchancal Couplng absorbd nrgy du to gnraton and rarrangmnt of mprfctons n th procss of plastcty s dfnd as stord nrgy of cold work ( E ), gvn by: 2 Φ q Φ E = - (17) x q x Basd on th law of Fourr, spcfc hat sttngs and stord nrgy of cold work, th Eq. (16) s rwrttn as follows: whr, s r q E c kq rr q s r x x (18) ( ) =, + + j + p j j j - q x H sj = q last j q (19) H = s p (2) plast j j E H = r ( x) x cw (21) x Th Eqs. (19), (2) and (21) rprsnt, rspctvly, th hat gnraton du to lastc dformaton, th dsspatd plastc workng and th stord nrgy of cold work. Th dsspaton mchansms ar of svral studs. Rfrrng th stord nrgy of cold work, s possbl to hghlght th thortcal and th xprmntal studs of Bvr t al. (1973), Olfruk t al. (1993), Rosaks t al. (2), Mroz and Olfruk (22), Rttl t al. (212) and Kolupava and Smnov (215). Ths studs mphasz th complxty of charactrzng E, bcaus t s dpndnt on th accumulatd plastc dformatons. In th absnc of nformaton on th mcrostructural bhavor of matrals about th varabls nflunc th procss and n what quantty, t s convnnt to us a constant factor to rprsnt th nrgy dsspaton. Thrfor, t s assumd that th rlatonshp btwn th plastc workng and stord nrgy of cold work s dfnd by a constant factor, hr dnotd by b (Smo and Mh, 1992; Zhou t al., 1996; Kapoor and Nmat-Nassr, 1998). Thus, w hav th followng smplfcaton: s p E - r ( x) x = bs x p j j j j (22) Combnng Eqs. (22) and (18), has th fnal xprsson that dfns th hat transfr to thrmoplastc problms, gvn by: s r q j = q, + r + q + bs q p c k R (23) j j j

7 J.P.B. Cavalcant t al. / A Smpl FEM Formulaton Appld to Nonlnar Problms of Impact wth Thrmomchancal Couplng Th Hat Conducton Dscrt Equaton Th nrgy balanc prsntd at Eq. (23) s solvd adoptng as a rfrnc th ntal confguraton of th structur, rwrttn as: In that kq rc q rr R (24), m = R m dfns th hat gnratd du to mchancal dformatons, xprssd as: 1 + n R =- 2 aq bs p m G kk j j (25) - n Th hat problm s solvd bfor th mchancal problm. Thrfor, mploy th hat sourc from th prvous tm stp t. Thus, for th currnt tm stp t +Dt, th xprsson (24) s rwrttn as follows: kq rc q rr R (26) t, m = as: Th numrcal procdur starts by substtutng th functon q by a fnt lmnt approxmaton q = q f (27) whr q and f dfnd tmpratur and shap functon n nod, rspctvly. To approxmat Eq. (26) th mthod of wghtd rsdus s adoptd hr. Spcfcally th Galrkn mthod, thus: whr, Wk d W c d W Rd WR d (28) ò ò ò ò t q, - r q + r + m = W = w j f j (29) whr s th volum, W th work, w j ar th wghtng functons and f j rprsnt arbtrary constants rlatd to nods j of th lmnts. Through Eqs. (27), (28) and (29), for any valu of w j and q bng constant, on has: ( qf ), f - r qff t ò j ò j + ò r fj + ò mfj = kk k d c d R d R d (3) Manpulatng Eq. (3), obtan a smlar xprsson, dfnd by: ò ò ( ) ò ò ò t qf, f, q f, f r qff r f f k j k k j, k j j m j k d - k d + c d - R d - R d = (31) Th applcaton of th dvrgnc thorm n th thrd trm of Eq. (31) gvs:

8 2446 J.P.B. Cavalcant t al. / A Smpl FEM Formulaton Appld to Nonlnar Problms of Impact wth Thrmomchancal Couplng ò ( ) ò k q f f d = k qf f da (32), k j, k A, n j Th Eq. (32) xprsss th thrmodynamc forcs appld on th boundary. Thrfor, combnng Eqs. (31) and (32) on has: whr, k d c d R d R d q da (33) ò ò ò ò ò t qf, kfj, k + r qff j - r fj - mfj + fj = A Dvlopng th volum ntgrals, Eq. (33) can b wrttn n a matrx form as: C jθ +Kjθ =F (34) C j = ò r c ff j d (35) K = ò d (36) j k f, kf j, k t ò ò ò 3 POSITIONAL FINITE ELEMENT METHOD F = rrfd + R fd - qfda (37) j m j A j In partcular, th study dvlopd uss an altrnatv formulaton, calld postonal FEM, consdrs nod coordnat postons as varabls nstad of dsplacmnts. Th postonal formulaton s classfd as Total Lagrang Formulaton (Wong and Tn-Lo, 199). Although rcnt, thr ar many studs that us postonal formulaton. Grco and Coda (26) prsnt a nonlnar dynamc analyss of on-dmnsonal structurs usng Nwmark's tmporal ntgraton algorthm. Coda and Paccola (28) study th gomtrc nonlnar analyss of shlls wth thcknss varaton and us of curvd lmnts. Carrazdo and Coda (21) appld th postonal formulaton n th study of th thrmomchancal couplng n nonlnar problms of mpact btwn trusss and rgd obstacl, through th postonal fnt lmnt mthod. Grco t al. (212) compars th numrcal rsults of th postonal and co-rotatonal formulaton for truss problms. It s also worth mntonng th followng studs that us th proposd formulaton n non-lnar problms: Grco t al. (26), Grco t al. (213), Rs and Coda (214), Sampao t al. (215) and Squra and Coda (216). Th postonal formulaton uss th Lagrangan dscrpton that dscrbs th knmatcs of th dformaton n trms of a coordnat systm, fxd n spac. Th prncpl of mnmum potntal nrgy, applyng a total Lagrangan dscrpton, s appld (Grco t al., 213). Th total potntal nrgy (П) s wrttn by: whr U s th stran nrgy, = U - P + KC + K A (38) K C s th kntc nrgy, K A s th loss of nrgy du to dampng and P s th potntal nrgy of concntratd forcs appld to th body. Th kntc nrgy s zro for statc problms.

9 J.P.B. Cavalcant t al. / A Smpl FEM Formulaton Appld to Nonlnar Problms of Impact wth Thrmomchancal Couplng 2447 Accordng to Eqs. (39) and (4), th total dformaton nrgy s dfnd by th ntgral of th spcfc stran nrgy ( u ) ovr th ntal volum and potntal nrgy of th appld forcs s xprssd as a functon of th xtrnal forcs appld ( F ) and poston ( X ). Th ndx rfrs to th dgr of frdom that ar assocatd forcs and postons. U u d (39) = ò whr, Th kntc nrgy s gvn by: Substtutng Eqs. (39), (4) and (41) nto (38) on has: P = FX (4) xx KC = ò r d (41) 2 xx = ò u d + ò r d + KA -FX (42) 2 KA KA = = r ò d ò c x d X X m (43) whr c m s th dampng coffcnt. Ths nrgy functon can b valuatd substtutng th xact poston fld for an approxmat non-dmnsonal fld ( x ). 1 2 = ò u ( x, ) + r ( x, ) + ( x, )- 2 ò X d x X d K A X F X (44) Thus, th poston of dynamc qulbrum s dfnd usng th mnmum potntal nrgy thorm, by dffrntatng Eq. (44) rgardng th gnrc nodal poston X, rsultng n: s ( ) ( ) ( ) u x, X x x, X K x, X = ò + ò ( x, ) + - = X X X X A d rx X d FS s s s s (45) Substtutng Eq. (43) nto (45) on has: u( x, X ) x ( x, ) ò ò ( ) X ò = d + rx x, X d + c rx ( x, X ) d - F = X X X m S s s s (46)

10 2448 J.P.B. Cavalcant t al. / A Smpl FEM Formulaton Appld to Nonlnar Problms of Impact wth Thrmomchancal Couplng Th qulbrum Eq. (28) s nonlnar rgardng X. Thus, to dsspat th rsdual forcs s nvtabl to us a numrcal stratgy to solv ths problm. In th currnt work, th Nwton-Raphson procdur was usd to rach th qulbrum. In ordr to solv t, a Taylor xpanson rgardng X s usd as follows: ( k ) 2 g X g( Xk +D Xk ) g( Xk ) + + O X k (47) Nglctng hghr-ordr rrors 2 O on has: ( ) æ ö g Xk D X = k ç è X ø k -2 ( ) g X k (48) whr, ( ) æ ( ) ( ) rç ò ( ) g ö ç X X X çè X X X X ø x ( x, X ) + ò cmr d - FS = X 2 2 u x, X x x, X x x, X x x, X = ò d + + x ( x, ) ç X d k k s k s k s k (49) In Eq. (49), th frst trm rprsnts th Hssan matrx. Thus, th dynamc nonlnar problm s achvd by combnng th tratv Nwton-Raphson procdur wth a tmporal ntgraton algorthm. Th total dformaton ( ) of th problm s gvn by th sum of th lastc ( ), plastc ( p ) and thrmal ( q ) parcls accordng to th quaton: p = + + q (5) Th total nrgy potntal stran, consdrng th nonlnar bhavor matral and thrmal ffcts s dfnd by: p q ( ) ò ò ò ò ò ò U = sd d = E d - E d - E d d æ1 ö 2 p q = ò E -E - E ç d è2 whr E s th lastc longtudnal modulus. In Eq. (51), q dscrbs th thrmal bhavor, whl th trm p dscrbs th plastc bhavor of th lmnt obtand from th consttutv matral modl. Hr, t wll b consdrd th mpact schm frctonlss dscussd n Smo t al. (1986), Grco t al. (24) and Carrazdo and Coda (21). Th so-calld schm of null-pntraton condton, havng as ts basc prncpl th poston lmtaton of ach nod of th structur that ar mpactd. It s usd th classc tm ntgraton algorthm of Nwmark. ø (51)

11 J.P.B. Cavalcant t al. / A Smpl FEM Formulaton Appld to Nonlnar Problms of Impact wth Thrmomchancal Couplng NUMERICAL EXAMPLES 4.1 Thrmal Loadng on Rod In ths xampl, prsntd by Coptt (22) s studd th thrmal bhavor of a on-dmnsonal bar wth thrmal loadng. Th bar was dvdd nto 1 lmnts of qual sz. Th author adopts th thrmal conductvty, th spcfc hat and th dnsty of th matral quvalnt to 1. Th tmporal dscrtzaton s prformd wth a tm ntrval qual to.1. Th Fgur 1 shows th thrmal load along th bar, dscrbd by Equaton: ( ) = 1 cos(2 p ) p x x (52) Th tmpratur and dsplacmnt ar constrand at th poston x = : ( ) q, t = 1 (53) ( ) u, t = (54) Th Fgurs 1 and 2 gv a summary of th tmpratur varatons ovr tm. Th rsults obtand ar smlar to th Coptt (22). Fgur 1: Tmpratur chang. Fgur 2: Tmpratur chang n dffrnt postons.

12 245 J.P.B. Cavalcant t al. / A Smpl FEM Formulaton Appld to Nonlnar Problms of Impact wth Thrmomchancal Couplng As shown n Fgur 1, bcaus th boundary condtons th tmpratur of all nods tnd to rman n qulbrum whn t = 4. It should b notd that ovr tm th thrmal load s dsspatd btwn th nods. Fgur 2 llustrats th path of th thrmal balanc for dffrnt postons of th bar, whr thy tnd to a common pont ( q = 1 ). 4.2 Tmpratur Evoluton n a Rod Ths xampl was orgnally prsntd by Kamlah and Haupt (1998), and subsquntly by Carrazdo and Coda (21), whch consst n a cylndrcal rod of 1 cm lngth undr lastoplastc loadng and a rfrnc tmpratur of 293 K, undr th followng ntal and boundary condtons: ( ) q x, = (55) ( ) q( ) Th followng matral proprts wr st: q, t = 1, t = (56) E = kgf m 2 k = 2 J ( K m s ) 3 r = 78 kg m c = 48 J ( K kg ) s = Y kgf m a =.16 m ( K m ) 8 2 It consdrd th knmatc hardnng, wth a valu of H = 31 kgf / m. It s assumd that all plastc work s convrtd nto hat, or nglcts to stord nrgy of cold work. An mportant consdraton usd by th authors s that th tmpratur dos not caus dformatons. Furthrmor, th loadng hstory s gvn by a non-monotonc stran hstory wth changng absolut valu of th stran rat: ( ) =.1-1 ( ) =-.5-1 ( ) =.2-1 t s for % 1.5 % s t 15 s t s for 1.5 % % 15 s t 21 s t s for % 1. % 21 s t 335 s Wth th ad of th mchancal modl rspons, ths procss can b dvdd furthr n prods of lastc and lastoplastc loadng: 1. s t 1 s Elastc tnson 2. 1 s t 15 s Elastoplastc tnson s t 154 s Elastc comprsson s t 21 s Elastoplastc comprsson s t 22 s Elastc tnson s t 335 s Elastoplastc tnson

13 J.P.B. Cavalcant t al. / A Smpl FEM Formulaton Appld to Nonlnar Problms of Impact wth Thrmomchancal Couplng 2451 Confrontng th rsults obtand wth th work mntond abov. Th dffrnt lastc and lastoplastc prods n th hstory of loadng can b asly dntfd n Fgur 3, whch llustrats th thrmomchancal hat sourc through tm. It s notd that th lastoplastc hardnng modl has a hat sourc largr than th prfct lastoplastc modl and thrfor mor rlvant tmpratur changs. Th Fgur 4 shows th tmpratur varaton n th cntr of rod. Fgur 3: Mchancal hat sourc. Fgur 4: Tmpratur varaton n tm. (a) Wth hardnng (b) Wthout hardnng Fgur 5: Tmpratur chang n th cntr of th bar.

14 2452 J.P.B. Cavalcant t al. / A Smpl FEM Formulaton Appld to Nonlnar Problms of Impact wth Thrmomchancal Couplng Th Fgur 5 shows th tmpratur varaton along th rod for dffrnt tm nstants. Changs n hatng and coolng phass ar dfnd by th stat of th body changs. As xpctd, t s notd that th lastoplastc modl wth stran hardnng knmatc prsnts hghr tmpratur varatons than th prfct lastoplastc modl, du to th lvls of strsss n th knmatc modl ar hghr. 4.3 Impact of a Bar on a Rgd Wall Ths xampl, approachd ntally by Armro and Ptocz (1998), conssts of unaxal mpact from a bar (wth a constant vlocty) and rgd wall, as shown n Fgur 6. Th problm s modfd to consdr th thrmomchancal ffcts. Th gomtrcal and matral charactrstcs of both lmnts ar gvn by: E = 1, L = 1, A = 1, r = 1, k = 1, c = 1, a =.17 and d =.5. Fgur 6: Impact of a rod on a rgd wall. Problm dfnton. Only th xtrmty nod of th bar s mpactd. It s nvstgatd th bhavor of contact forcs, vlocts, strsss and tmpratur changs of th bar. In ths xampl t s nvstgatd only gomtrc nonlnarty. Th tm dscrtzaton s don to 25 tm stps of.1. Th bar s dscrtzd wth 2 fnt lmnts of th sam dmnson. Th Fgurs 7 and 8 dscrb rspctvly th vlocty and contact forc, comparng th analytcal rspons to th mchancal and thrmomchancal numrc rsponss. In both fgurs, bfor th mpact, mchancal and thrmomchancal rspons ar qual and th structur dos not show dformatons. Fgur 7: locty on mpact pont.

15 J.P.B. Cavalcant t al. / A Smpl FEM Formulaton Appld to Nonlnar Problms of Impact wth Thrmomchancal Couplng 2453 Fgur 8: Contact forc on mpact pont. Th vlocty rsults of mchancal and thrmomchancal modls of th mpactor nod ar practcally dntcal up to th nstant of Aftr ths tm nstant, th dffrnc btwn th modls ncrass wth tm. As shown n Fgur 8, th hat gnraton rsultng n hghr contact forcs to mchancal rspons, and consquntly th contact tm s rducd. Th Fgur 9 shows th tmpratur fld to th undformd confguraton of th bar. It s consdrd th coupld and th uncoupld problm, rspctvly. It s obsrvd that th thrmomchancal couplng causs a scondary ffct and tmpratur varatons wr mor rlvant whn consdrng that th tmpratur changs gnrat dformatons. Th bar has maxmum tmpraturs btwn tms nstants of 1. and 1.5, approxmatvly. Th bar bgns to cool quckly aftr rflcton. (a) Wth hardnng (b) Wthout hardnng Fgur 9: Tmpratur fld ovr tm and spac.

16 2454 J.P.B. Cavalcant t al. / A Smpl FEM Formulaton Appld to Nonlnar Problms of Impact wth Thrmomchancal Couplng In th surfac of Fgur 1 t s possbl to compar th strsss on th bar. It s obsrvd a rdstrbuton of strsss causd by thrmomchancal couplng. In mchancal problm th strss vars btwn -.5 and.2. Whl th problm coupld, has hghr strss lvls n comprsson and tracton varyng from -.6 to.3. Consquntly, th thrmomchancal couplng ncrass th contact forc and rducs th contact tm. (a) Mchancal (b) Coupld Fgur 1: Strss fld ovr tm and spac. 4.4 Undrctonal Impact Btwn Two Bars In ths xampl t s studd th cas of mpact btwn two dntcal bars wth th sam ntal vlocty (Fgur 11), howvr, movng n oppost snss. Ths xampl s prsnt n th studs by Carpntr t al. (1991). Th gomtrcal and physcal charactrstcs of both lmnts ar gvn by: E = 3. ks, k = 8.9 BTU( ft h º F ) - 1, c =.12 BTU( lb º F ) - 1, r = lb s2 / n 4, b =. A = 1. n 2 and.8 Fgur 11: Problm dscrpton for on dmnsonal mpact xampl. Du to th symmtry of th problm th problm can b rducd to an mpact problm btwn a bar and rgd wall (Fgur 12).

17 J.P.B. Cavalcant t al. / A Smpl FEM Formulaton Appld to Nonlnar Problms of Impact wth Thrmomchancal Couplng 2455 Fgur 12: Impact of a rod on a rgd wall. Problm dfnton. Th bar was dscrtzd n 2 fnt lmnts and th ntal dstanc btwn th bars s d =.2 n. Plastcty s consdrd n th analyss. Adopts th modl of hardnng sotropc K = 15 ks wth s Y = 1 ks. Th rfrnc tmpratur s 68 ºF. Th numrc rsponss ar obtand wth tm ntrvals of D t =.5 s. In Fgur 13, vrfs th plastczng ffct of hat gnraton at x = 9.5 n. It s obsrvd that th voluton of tmpratur changs s vry dpndnt on th accumulaton of rrvrsbl dformatons. Th Fgurs 14, 15 and 16 compar th tmpratur changs obtand wth dffrnt coffcnt of thrmal xpanson. Fgur 13: Hardnng ffct on th tmpratur changs (coupld). (a) Uncoupld (b) Coupld Fgur 14: Tmpratur fld ovr tm and spac ( a =.96 ).

18 2456 J.P.B. Cavalcant t al. / A Smpl FEM Formulaton Appld to Nonlnar Problms of Impact wth Thrmomchancal Couplng (a) Uncoupld (b) Coupld Fgur 15: Tmpratur fld ovr tm and spac ( a =.46 ). (a) Uncoupld (b) Coupld Fgur 16: Tmpratur fld ovr tm and spac ( a =.96 ). It s notd that smallr coffcnts of thrmal xpanson rsult n smallr tmpratur changs and small dffrnc btwn th uncoupld and coupld rsponss. Th thrmomchancal coupld gnrats a scondary ffct whch causs largr varatons n tmpratur and changs n th structural bhavor. Th rsults prsntd n Fgur 16 show hgh lvls of tmpratur. In ths cas, du to rdstrbuton fforts, th fld of tmpraturs for coupld problm vars btwn -6.3 º F and 69.4 ºF, whl th uncoupld problm vars btwn -2.8 º F and 29.7 ºF. Th largr coffcnts of thrmal xpanson rsult n largr changs n tmpratur and contact forcs and, consquntly, th lowr th contact tm.

19 J.P.B. Cavalcant t al. / A Smpl FEM Formulaton Appld to Nonlnar Problms of Impact wth Thrmomchancal Couplng Impact Btwn a Plan Truss and a Rgd Wall Ths xampl s th mpact btwn a crcular truss and rgd wall (Fgur 17). Th structur has 264 bars and 97 nods. Th tm dscrtzaton s don through th Nwmark mthod wth D t =.1 s. Th gomtrcal and matral data of both lmnts ar gvn by: 11 2 E = N / m A =.36 m H = 5. 1 N / m 8 2 k = 27 J / ( K ms ) s Y = 1. 1 N / m c = 48 J / ( K kg ) 3 r = 785 kg / m a =.11 m / ( K m ) It s assumd that all plastc work s convrtd nto hat. It adopts a rfrnc tmpratur quvalnt to 3 K. Th structur movs wth vlocty of 35 m / s, wth d =.1 m. Fgur 17: Impact btwn a plan truss and a rgd wall. Problm dfnton. Fv nods ar mpactd (Fgur 17). Th Fgur 18 shows th voluton of tmpratur changs of th nods mpactng ovr tm. Th tmpratur of ths nods bgn to ncras rapdly from th momnt of mpact. It s obsrvd a small dcras n th varatons of tmpraturs from a crtan momnt, charactrzd by Gough-Joul ffct and th dsspaton of tmpraturs btwn nods. Th Fgur 19 shows for dffrnt tm, th tmpratur dstrbuton of th structur n th dformd confguraton. Thrfor, n Fgurs 18 and 19 mphasz th mportanc of consdrng th thrmomchancal bhavor n mpact problms bcaus th hgh stran rats can caus changs n th structural confguraton.

20 2458 J.P.B. Cavalcant t al. / A Smpl FEM Formulaton Appld to Nonlnar Problms of Impact wth Thrmomchancal Couplng Fgur 18: Tmpratur chang of mpact ponts. (a) t =.1 s (b) t =.2 s (c) t =.4 s (a) t =.5 s (a) t =.7 s (a) t =.1 s Fgur 19: Dstrbuton of tmpraturs.

21 J.P.B. Cavalcant t al. / A Smpl FEM Formulaton Appld to Nonlnar Problms of Impact wth Thrmomchancal Couplng CONCLUSIONS A smpl and ffctv altrnatv formulaton to dal wth th dynamc nonlnar systms, wrttn n rlaton to nodal postons has bn succssfully appld to coupld thrmomchancal problms. Th formulaton ncluds a complt tratmnt of th analyss of lastoplastc matrals for smpl confguratons. Through th numrcal xampls t s possbl to obsrv th dffrnc btwn th mchancal and th thrmomchancal rsponss. Thrfor, t s ncssary to consdr th ntracton btwn thrmal and mchancal bhavor, whch can caus rdstrbutons of strss. Th rfrnc tmpratur s a dtrmnng factor n thrmomchancal analyss, and hgh rfrnc tmpraturs mply hgh tmpratur varatons. Th study of structurs wth lastoplastc bhavor s mportant bcaus dpndng on th hstory of dformaton and accumulaton of rrvrsbl dformatons, has a sgnfcant part n th gnraton of hat. It mphaszs th consdraton of thrmomchancal couplng n mpact problms, bcaus th ntracton btwn th mchancal and thrmal rspons caus sgnfcant scondary ffcts du to hgh stran rats by modfyng th structural bhavor. Thrfor, dpndng on th matral, loadng and ntal condtons, th thrmomchancal couplng can rsult n prdomnant contrbutons n structural rspons. Acknowldgmnts Th authors would lk to thank th Brazlan Rsarch Fundng Agncs CAPES (Coordnaton of Improvmnt of Hghr Educaton Prsonnl), CNPq (Natonal Councl for Scntfc and Tchnologcal Dvlopmnt) and FAPEMIG (Foundaton for Rsarch Support of Mnas Gras Stat) for th fnancal supports. Rfrncs Armro, F., Ptocz, E., (1998). Formulaton and analyss of consrvng algorthms for frctonlss dynamc contact/mpact problms. Computr Mthods n Appld Mchancs and Engnrng 158: Bnzrga, A. A., Brcht, Y., Ndlman, A., an dr Gssn, E. (25). Th stord nrgy of cold work: Prdctons from dscrt dslocaton plastcty. Acta Matrala 53: Bvr, M. B., Holt, D. L., Ttchnr, A. L. (1973). Th stord nrgy of cold work. Mchancs of Matrals 17:1-19. Bot, M. A. (1956). Thrmolastcty and rrvrsbl thrmodynamcs. Journal of Appld Physcs 27: Canadja, M., Brnc, J. (24). Assocatv coupld thrmoplastcty at fnt strans wth tmpratur-dpndnt matral paramtrs. Intrnatonal Journal of Plastcty 2: Canadja, M., Brnc, J. (29). Nonlnar knmatc hardnng n coupld thrmoplastcty. Matrals Scnc and Engnrng A 499: Canadja, M., Brnc, J. (21). A dsspaton modl for cyclc non-assocatv thrmoplastcty at fnt strans. Mchancs Rsarch Communcatons 37: Carpntr, N. J., Taylor, R. L., Katona, M. G. (1991). Lagrang constrants for transnt fnt lmnt surfac contact. Intrnatonal Journal for Numrcal Mthods n Engnrng 32: Carrazdo, R., Coda, H. B. (21). Altrnatv postonal FEM appld thrmomchancal mpact of truss structurs. Fnt Elmnts n Analyss and Dsgn 46:

22 246 J.P.B. Cavalcant t al. / A Smpl FEM Formulaton Appld to Nonlnar Problms of Impact wth Thrmomchancal Couplng Chn, J., Dargush, G. F. (1995). Boundary lmnt mthod for dynamc porolastc and thrmolastc analyss. Intrnatonal Journal of Solds and Structurs 32: Clayton, J. D. (213). Nonlnar ulran thrmolastcty for ansotropc crystals. Journal of th Mchancs and Physcs of Solds 61: Coda, H. B., Grco, M. (24). A smpl FEM formulaton for larg dflcton 2D fram analyss basd on poston dscrpton. Computr Mthods n Appld Mchancs and Engnrng 193: Coda, H. B., Paccola, R. R. (28). A postonal FEM formulaton for gomtrcal non-lnar analyss of shlls. Latn Amrcan Journal of Solds and Structurs 5: Coda, H. B., Paccola, R. R. (211). A FEM procdur basd on postons and unconstrand vctors appld to nonlnar dynamc of 3D frams. Fnt Elmnts n Analyss and Dsgn 47: Coda, H. B., Paccola, R. R., Sampao, M. S. M. (213). Postonal dscrpton appld to th soluton of gomtrcally non-lnar plats and shlls. Fnt Elmnts n Analyss and Dsgn 67: Coptt, M. I. M. (1999). Fnt lmnt approxmaton to a contact problm n lnar thrmolastcty. Mathmatcs of computaton 68: Coptt, M. I. M. (22). A on-dmnsonal thrmolastc problm wth unlatral constrant. Mathmatcs and Computrs n Smulaton 59: Danlovskaya,. (195). Thrmal strsss n an lastc half-spac du to suddn hatng of ts boundary. Prkl. Mat. Mch. 14: Dargush, G. F., Banrj, P. K. (1991). Boundary lmnt mthods for thr-dmnsonal thrmoplastcty. Intrnatonal Journal of Solds and Structurs 28: Db, A., Hnry Jr., D. P., Wlson, R. B. (1991). Altrnatv BEM formulatons for 2- and 3-D ansotropc thrmolastcty. Intrnatonal Journal of Solds and Structurs 27: Dllon Jr., O. W. (1963). Coupld thrmoplastcty. Journal of th Mchancs and Physcs of Solds 11: Farrn, W. S., Taylor, G. I. (1925). Th hat dvlopd durng plastc xtnson of mtals. Procdngs of th Royal Socty of London 17: Grco, M., Coda, H. B., nturn, W. S. (24). An altrnatv contact/mpact dntfcaton algorthm for 2d structural problms. Computatonal Mchancs 34: Grco, M., Coda, H. B. (26). Postonal FEM formulaton for flxbl mult-body dynamc analyss. Journal of Sound and braton 29: Grco, M., Frrra, I. P., Barros, F. B. (213). A classcal tm ntgraton mthod appld for soluton of nonlnar quatons of a doubl-layr tnsgrty. Journal of th Brazlan Socty of Mchancal Scncs and Engnrng 35:41-5. Grco, M., Gsualdo, F. A. R., nturn, W. S., Coda, H. B. (26). Nonlnar postonal formulaton for spac truss analyss. Fnt Elmnts n Analyss and Dsgn 42: Grco, M., Mnn, R. C. G., Frrra, I. P., Barros, F. B. (212). Comparson btwn two gomtrcal nonlnar mthods for truss analyss. Structural Engnrng and Mchancs 41:735:75. Grn, A. E., Lndsay, K. A. (1972). Thrmolastcty. Journal of Elastcty 2:1-7. Grn, A. E., Naghd, P. M. (1993). Thrmolastcty wthout nrgy dsspaton. Journal of Elastcty 31: Grn, A. E., Naghd, P. M. (1995). A unfd procdur for constructon of thors of dformabl mda. I Classcal contnuum physcs, II Gnralzd contnua, III Mxturs of ntractng contnua. Procdngs of th Royal Socty: Mathmatcal, Physcal & Engnrng Scncs 448: Hakansson, P., Walln, M., Rstnmaa, M. (25). Comparson of sotropc hardnng and knmatc hardnng n thrmoplastcty; Intrnatonal Journal of Plastcty 21:

23 J.P.B. Cavalcant t al. / A Smpl FEM Formulaton Appld to Nonlnar Problms of Impact wth Thrmomchancal Couplng 2461 Kamlah, M., Haupt, P. (1998). On th macroscopc dscrpton of stord nrgy and slf hatng durng plastc dformaton. Intrnatonal Journal of Plastcty 13: Kapoor, R., Nmat-Nassr, S. (1998). Dtrmnaton of tmpratur rs durng hgh stran rat dformaton. Mchancs of Matrals 27:1-12. Kolupava, S., Smnov, M. (215). Th stord nrgy of plastc dformaton n crystals of fac-cntrd cubc mtals. Matrals Scnc and Engnrng 71:1-6. Lmonds, J., Ndlman, A. (1986). Fnt lmnt analyss of shar localzaton n rat and tmpratur dpndnt solds. Mchancs of Matrals 5: L, X. (1992). A gnralzd thory of thrmolastcty for an ansotropc mdum. Intrnatonal Journal of Engnrng Scnc 3: Lord, H. W., Shulman, Y. (1967). A gnralzd dynamcal thory of thrmolastcty. Journal of th Mchancs and Physcs of Solds 15: Lu, S. C. H., Pstr, K. S. (1975). Dcomposton of dformaton and rprsntaton of th fr nrgy functon for sotropc thrmolastc solds. Intrnatonal Journal of Solds and Structurs 11: Lubarda,. A. (24). Consttutv thors basd on th multplcatv dcomposton of dformaton gradnt Thrmolastcty, lastoplastcty, and bomchancs. ASME Appld Mchancs Rvw 57: Lychv, S. A., Manzhrov, A.., Joubrt, S.. (21). Closd solutons of boundary-valu problms of coupld thrmolastcty. Mchancs of Solds 45: Mahmoud, W., Ghalb, A. F., Rawy, E. K., Hassan, H. A. Z., Mosharafa, A. A. (215). Numrcal soluton to a nonlnar, on-dmnsonal problm of ansotropc thrmolastcty wth volum and hat suppl n a half-spac ntracton of dsplacmnts. Archv of Appld Mchancs 85: McKnght, R. L., Sobl, L. H. (1977). Fnt lmnt cyclc thrmoplastcty analyss by th mthod of subvolums. Computrs and Structurs 7: Mroz, Z., Olfruk, W. (22). Enrgy balanc and dntfcaton of hardnng modul n plastc dformaton procsss. Intrnatonal Journal of Plastcty 18: Nckll, R. E., Sackman, J. L. (1968). Approxmat solutons n lnar coupld thrmolastcty. Journal of Appld Mchancs 35: Norrs, A. N. (26). Dynamcs of thrmolastc thn plats a comparson of four thors. Journal of Thrmal Strsss 29: Olfruk, W., Swatnck, W. A., Grabsk, M. W. (1993). Rat of nrgy storag and mcrostructur voluton durng th tnsl dformaton of austntc stl. Matrals Scnc and Engnrng A 161: Ozakn, A., Yavar, A. (21). A gomtrc thory of thrmal strsss. Journal of Mathmatcal Physc 51:1-32. Przyna, P., Sawzcuk, A. (1973). Problms n thrmoplastcty. Nuclar Engnrng and Dsgn 24:1-55. Rajagopal, K. R. (1995). Boundary layrs n fnt thrmolastcty. Journal of Elastcty 36: Rajagopal, K. R., Manschy, C. E., Massoud, M. (1996). Inhomognous dformatons n fnt thrmos-lastcty. Intrnatonal Journal of Engnrng Scnc 34: Rand, O., Gvol, D. (1995). Rducton of th prodc thrmolastc dformaton n truss-structurs by dsgn rfnmnts and actv loads. Computrs and Structurs 54: Rs, M. C. J., Coda, H. B. (214). Physcal and Gomtrcal non-lnar analyss of plan frams consdrng lastoplastc sm-rgd connctons by th postonal FEM. Latn Amrcan Journal of Solds and Structurs 11: Rttl, D. (1998). Exprmntal nvstgaton of transnt thrmolastc ffcts n dynamc fractur. Intrnatonal Journal of Solds and Structurs 35:

24 2462 J.P.B. Cavalcant t al. / A Smpl FEM Formulaton Appld to Nonlnar Problms of Impact wth Thrmomchancal Couplng Rttl, D., Kdan, A. A., Alkhadr, M., nkrt, A., Landau, P., Ravchandran, G. (212). On th dynamcally stord nrgy of cold work n pur sngl crystal and polycrystalln coppr. Acta Matrala 6: Rosaks, P., Rosaks, A. J., Ravchandran, G., Hodowany, J. (2). A thrmodynamc ntrnal varabl modl for th partton of plastc work nto hat and stord nrgy n mtals. Journal of th Mchancs and Physcs of Solds 48: Sampao, M. S. M., Paccola, R. R., Coda, H. B. (215). A gomtrcally nonlnar FEM formulaton for th analyss of fbr rnforcd lamnat plats and shlls. Compost Structurs 119: Srra, E., Bonald, M. (29). A fnt lmnt formulaton for thrmolastc dampng analyss. Intrnatonal Journal for Numrcal Mthods n Engnrng 78: Shahan, A. R., Bashusqh, S. M. (214). Analytcal soluton of th thrmolastcty problm n a prssurzd thckwalld sphr subjctd to transnt thrmal loadng. Mathmatcs and Mchancs of Solds 19: Shahan, A. R., Nabav, S. M. (27): Analytcal soluton of th quas-statc thrmolastcty problm n a prssurzd thck-walld cylndr subjctd to transnt thrmal loadng. Appld Mathmatcal Modllng 9: Shrf, H. H., Hamza, F. A., Salh, H. A. (24). Th thory of gnralzd thrmolastc dffuson. Intrnatonal Journal of Engnrng Scnc 42: Smo, J. C., Mh, C. (1992). Assocatv coupld thrmoplastcty at fnt strans: Formulaton, numrcal analyss and mplmntaton. Computr Mthods n Appld Mchancs and Engnrng 98: Smo, J. C., Wrggrs, P., Schwzrhof, K. H., Taylor, R. L. (1986). Fnt dformaton post-bucklng analyss nvolvng nlastcty and contact constrants. Intrnatonal Journal for Numrcal Mthods n ngnrng 23: Squra, T. M., Coda, H. B. (216). Dvlopmnt of sldng connctons for structural analyss by a total lagrangan FEM formulaton. Latn Amrcan Journal of Solds and Structurs 13: Solr, A. L., Brull, M. A. (1965). On soluton to transnt coupld thrmolastc problms by prturbaton tchnqus. Journal of Appld Mchancs 32: Stanly, P. (28). Bgnnngs and arly dvlopmnt of thrmolastc strss analyss. Journal Complaton 44: Taylor, G. I., Qunny, H. (1934). Th latnt nrgy rmanng n a mtal aftr cold workng. Procdngs of th Royal Socty: Mathmatcal, Physcal and Engnrng Scncs 143: Tng, E. C., Chn, H. C. (1982). A unfd numrcal approach for thrmal strss wavs. Computrs and Structurs 15: ujosvc, L., Lubarda,. A. (22). Fnt-stran thrmolastcty basd on multplcatv dcomposton of dformaton gradnt. Thortcal and Appld Mchancs 28-29: Wong, M. B., Tn-Lo, F. (199). Gomtrcally nonlnar analyss of lastc framd structurs. Computrs and Structurs 34: Yavar, A., Gorly, A. (213). Nonlnar lastc nclusons n sotropc solds. Procdngs of th Royal Socty: Mathmatcal, Physcal and Engnrng Scncs 469:1-21. Zhou, M., Ravchandran, G., Rosaks, A. J. (1996). Dynamcally propagatng shar bands n mpact-loadd prnotchd plats II, Numrcal smulaton. Journal of th Mchancs and Physcs of Solds 44: