4. Hyprlastcty h Hyprlastc matral s xad n ths scton. 4..1 Consttutv Equatons h rat of chang of ntrnal nrgy W pr unt rfrnc volum s gvn by th strss powr, whch can b xprssd n a numbr of dffrnt ways (s 3.7.6): W & Jσ : d P : F& S : E& S : C& (4..1) h ntrnal nrgy s rgardd as a functon of a dformaton varabl. For xampl, h chang n nrgy s du to th dformaton whch taks plac, so tak W to b a functon of, say, th dformaton gradnt F (t), W (F). It s assumd that n th rfrnc confguraton th stran nrgy s zro, W ( I) 0, and that t grows wth dformaton, W 0 1. h chan rul gvs From 4..1, & ( F ) W W : F & (4..) F P F& ( ) : W : F& (4..3) Snc F and F & can tak on any valu ndpndnt of th othr, on must hav P (4..4) hs s a consttutv quaton rlatng th knmatc varabls to th forc varabls. From 4..1, altrnatv forms ar: ( E) S, S (4..5) h procdur usd n Eqns. 4..-4 cannot b usd for th strss powr rlaton J σ : d snc thr s no functon whos drvatv s th rat of dformaton. Instad, us th rlaton 3.5.6, σ J 1 PF, 4..4, P /, to gt 1 and that t tnds to nfnty as th matral s thr comprssd to a pont, J dt F 0 or xpandd to an nfnt rang, J dt F. 360
σ J F, σ J F m (4..6) An altrnatv rlaton can b obtand by frst drvng a rlatonshp btwn th partal drvatvs of th stran nrgy functon wth rspct to th dformaton gradnt, W /, and wth rspct to th rght Cauchy-Grn tnsor, W /. Suppos frst that th stran nrgy s a functon of th dformaton gradnt: F (4..7) h chan rul for W ( C(F) ) W gvs (4..8) Wth C F F, km F kn + F km kn δ δ F + δ δ F δ F + δ F k m kn k n km m n n (4..9) so that (and usng th fact that C s symmtrc), δ F F m n n F n nm m m + F + F + δ F n n n (4..10) or F, F k k (4..11) Now 4..6 can b r-wrttn as σ F J F (4..1) Slarly, Eqn. 4..4 cab b r-wrttn as P F (4..13) 361
h Stran Enrgy and th Rght Strtch nsor h rght strtch tnsor can b xprssd as C UU, whr U s th rght strtch, or, snc U s also symmtrc, C U U. On can s th slarty btwn ths rlaton and th rlaton C F F so, usng th sam arguts as gvn abov, on has for W W C(U), (s 1.11.36) ( ) U (4..14) U Snc U s symmtrc, U U U U U (4..15) showng that U and W / ar coaxal. 4.. Obctvty of th Consttutv Equatons * An obsrvr transformaton (s.8) rsults n F QF * * W ( C ) / /, and, so, from 4..1, and * C C, so * σ J QF F Q QσQ (4..16) whch s th obctvty rqurt for a spatal tnsor and so ths consttutv law satsfs th rqurt of matral fram-ndffrnc, whn W s a functon of C. hs vdntly holds tru also whn W s a functon of E. It dos not, howvr, hold tru n gnral whn W s a functon of F, as n th consttutv law P /. Howvr, wth W a functon of C, th consttutv quaton 4..13 can b sn to b obctv. h obctv consttutv quatons n ths scton ar ndcatd by a box around thm. 4..3 Elastcty nsors h total dffrntal of th PK strss can b wrttn as S( C) S( C) 1 1 ds : dc : dc C : dc (4..17) whr C s th fourth-ordr tnsor 36
S( C) S( E) C (4..18) and s a masur of th chang n strss whch occurs du to a chang n stran. It s calld th lastcty tnsor (n th matral dscrpton) or th tangnt modulus. Snc S and E ar symmtrc, C posssss th or symmtrs, C kl C kl Clk, and so has 36 ndpndnt componnts. Howvr, f hyprlastc condtons hold, so that S ( E) / /, thn W W ( E) C 4 (4..19) and so C posssss th maor symmtrs, componnts. C C, and only only 1 ndpndnt kl kl h rat form of th abov quatons s C C E S & d ( ) W ( ) C& W ( ) : E& dt : (4..0) or S & 1 C : C& : E& C (4..1) 4..4 A Not on th Stran Enrgy Functon Som confuson can ars n xprssons such as S, S (4..) As tond at th nd of 1.11.5, on can thr us th nrgy functon W * W *( E11, E1, E13, E1, E, E33), a functon of 6 ndpndnt varabls, n whch cas S 11 * 1 * 1 *, S1, S13, K (4..3) 11 1 13 us th nrgy functon W W ( E11, E1, E13, E1, E, E3, E31, E3, E33 ), a functon of 9 varabls, not all of thm ndpndnt, n whch cas S 11, S1, K S 1, K (4..4) 11 1 1 363
and th symmtry of W wth rspct to th strans, whch must b assumd hr, pls that th strss s also symmtrc. 4..5 Hyprlastcty wth Constrants Consdr a Hyprlastc matral whch s subct to th scalar constrant Wthout th constrant, on has ϕ ϕ 0 or & ϕ : F& 0 (4..5) & ( F ) W W : F& P : F& (4..6) h consttutv quaton P / can b drvd from 4..6 whn F & s arbtrary. Howvr, for th matral wth th constrant, th F & cannot b canclld out from ach sd; on has th condtons P : F& φ 0, : F& 0 (4..7) In gnral thn, P α (4..8) whr α s som arbtrary scalar. h strss s thrfor gvn by P and th rat of chang of ntrnal nrgy s + α (4..9) & : F + α : F& W& + αφ& (4..30) h stran nrgy s W + αφ( F) (4..31) h scond trm hr s vdntly zro and so dos not contrbut to th stran nrgy. Howvr, th strss s th drvatv of th stran nrgy wth rspct to a knmatc quantty, and th drvatv of ths last trm, th scond trm n 4..9, s not b zro. h scalar α s, or dtrs th magntud of, a worklss racton strss. It nsurs that th constrant s satsfd; t s not st a valu n th consttutv quaton, rathr, t s 364
dtrd by consdrng partcular problms, through qulbrum and boundary condtons. For N constrants φ 0, 1K N, 4..9 gnralss to P + α (4..3) N 1 Incomprssbl Matrals An ncomprssbl matral s on whos volum rmans constant throughout a moton, and so has th followng constrant: J dt F 1 or J & 0 (4..33) From 1.11.34, (dt A ) / A (dt A) A, on has J 1. hus th PK1 strsss can b wrttn as dj / df JF, whch quals F at P F pf (4..34) and th stran nrgy functon s of th form 1) W p( J (4..35) wth p α. Usng 3.5.9, S F P, on also has S pc (4..36) Usng.5.0 and.5.18b, and th strss powr s ( F EF & ) ( C E& tr ) C : E& 1 J J C C& J & Jtrd Jtr J : (4..37) 1 & & 1 p & S : C : C C : C & & P : F : F pf : F& W& pj& (4..38) consstnt wth 4..30. 365
h stran nrgy of 4..35 s xprssd n trms of F. hs can b r-xprssd n trms of C. Snc J dt F dt C IIIC, th ncomprssblty constrant can b xprssd as and th stran nrgy can b wrttn as III 0 ϕ (4..39) C 1 W p(iiic ) (4..40) h factor of 1/ has bn ncludd so that th p n 4..35 s th sam as th p n 4..40, snc, from 1.11.33, IIIC C (4..41) ladng to th sam xprsson for th PK strss as bfo, Eqn.4..36. Slarly, n trms of th Cauchy strss, on has{ Problm 1} σ F F pi (4..4) Bcaus of ts rol n ths quaton, th scalar p s calld th hydrostatc prssur. Inxtnsbl Constrant Consdr a matral whch s nxtnsbl n a drcton dfnd by a unt vctor  n th rfrnc confguraton. h constrant for ths matral s gvn by Eqn...60, hn, from 4..11, AC ˆ Aˆ 0 φ (4..43) F C FAˆ Aˆ FAˆ Aˆ (4..44) h strss s thn P F + αfaˆ Aˆ (4..45) 366
Post-contractng wth F (wth J 1 ) thn gvs σ F F + αfaˆ FAˆ (4..46) For nxtnsblty n two drctons, on has th strss Aˆ C Aˆ 0, φ Aˆ C Aˆ 0 φ (4..47) 1 1 1 σ ˆ ˆ α F F + α1fa1 FA1 + FA FA ˆ ˆ (4..48) 4..6 Problms 1. Us th quaton P pf to show that th consttutv quaton for an ncomprssbl hyprlastc matral can b wrttn n trms of th Cauchy strss as σ F F pi 367