Methods of Improving Constitutive Equations
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1 Mehods o mprovng Consuve Equaons Maxell Model e an mprove h ne me dervaves or ne sran measures. ³ ª º «e, d» ¼ e an also hange he bas equaon lnear modaons non-lnear modaons her Consuve Approahes Smple Maxell Model, shear Upper-Conveed Maxell Model, general Smple ereys Model, shear Upper-Conveed ereys Model, general ldroyd B Flud reardaon me
2 ereys Model - Mehanal Analog Maxell Model - Mehanal Analog Unorunaely, hs hange only modes G- ; he ereys Model s a GLVE model Smple ereys Model no rame-nvaran No, solvng or explly e oban, ³»» ¼ º ««ª d e G G her lnear modaons o he Maxell model movaed by sprngs and dashpos n seres and parallel mody G- bu do no oherse nrodue ne behavor. Mgh as ell use he Generalzed Maxell model
3 G Non-lnear modaons o he Maxell Model he-mezner Model ldroyd 8-Consan Model r UCM UCM erms UC The ldroyd 8-onsan onans many oher onsuve equaons as speal ases. The ldroyd 8-Consan model onans all erms lnear n sress ensor and a mos quadra n rae-o-deormaon ensor ha are also onssen h rame nvarane. r D Gesekus Model quadra n sress The only ay o hoose among hese nonlnear models s o ompare predons.
4 e an also mody negral models o add non-lneary and hus produe ne onsuve equaons. Faorzed vln-sayers Model Faorzed -BZ Model, are he nvarans o he Fnger or Cauhy sran ensors hese are relaed. ³ M, C, C U U ³ M C C d d Agan, he only ay o hoose among hese nonlnear models s o ompare predons see. G. Larson, Consuve Equaons or olymer Mels. Choosng Consuve Equaons e have xed all he obvous las n our onsuve equaons, and no e have oo many hoes! e ould make predons and ompare h expermenal daa, bu some o he models vln Sayer, -BZ have undened unons ha mus be speed. Ho o proeed? e need some gudane. All along e have aken a onnuum-mehans approah. e have run ha ourse all he ay hrough. No e mus go bak and seek some nsgh rom moleular deas o relaxaon and polymer dynams. 4
5 Moleular Approah o olymer Consuve Modelng moleular enson ore on arbrary surae ~ sress ensor e no aemp o alulae moleular ores by onsderng moleular models. olymer Dynams end-o-end veor, polymers may be modeled as random alks. olymer ol responds o deormaon A polymer han adops he mos random onguraon a equlbrum. end-o-end veor, hen deormed, he han res o reover ha mos random onguraon, gvng rse o a sprng-lke resorng ore. sprng o equlbrum lengh and orenaon e ll model he han dynams h a random alk. 5
6 Gaussan Sprngs Equlbrum onguraon dsrbuon unon - probably a alk has end-o-end dsane \ E S e E From an enropy alulaon on a random alk e an alulae he ore needed o deorm a Gaussan sprng kt e an relae hs ore o he arbrary ore on a surae, e an onne hese o moleular enson ore on arbrary surae ~ = τ sress ensor Moleular ore generaed by deormng han ~ Tenson ore on ³³³ Fore on surae due o hans o ETE d d d robably han o ETE rosses surae n ˆ see ex robably han has ETE \ d d d Fore exered by han / ETE kt 6
7 robably han o ETE rosses surae b a a nerseon h b d robably han o ETE rosses surae d volume Moleular ore generaed by deormng han ~ kt { ³³³ \ d d d BUT, rom beore... ~ moleular enson ore on arbrary surae n erms o Comparng hese o e onlude, kt Moleular ore generaed by deormng han 7
8 Ho an e onver hs equaon, kt Moleular ore generaed by deormng han hh relaes moleular ETE veor and sress, no a onsuve equaon, hh relaes sress and deormaon? e need a dea ha onnes ETE veor moon o marosop deormaon o a polymer neork or mel. Elas Crosslnked Sold Beeen every o rosslnks here s a polymer srand ha ollos a random alk o N seps o lengh a. Dsrbuon o ETE veors ETE = end-o-end veor 8
9 9 Ho an e relae hanges n end-o-end veor o marosop deormaon? ane-moon assumpon he marosop dmenson hanges are proporonal o he mrosop dmenson hanges ANSE beore aer Consder a general elongaonal deormaon F For ane moon e an relae he omponens o he nal and nal ETE veors as, ETE beore ETE aer
10 e are aempng o alulae he sress ensor h hs equaon kt { ³³³ \ d d d Bu, here do e ge hs? robably han has ETE beeen and +d Equlbrum onguraon dsrbuon unon \ \ d d d Conguraon dsrbuon unon e E S E Bu, he deormaon s ane, hen he number o ETE veors beeen and +d a me s equal o he number o veors h ETE beeen and +d a Conluson \ \ E E e S
11 No e are ready o alulae he sress ensor. kt { ³³³ \ ddd \ \ E E S e Fnal soluon kt eˆ eˆ Fnal soluon or sress kt eˆ eˆ Compare hs soluon h he Fnger Sran Tensor or hs lo. T C, F F Sne he Fnger ensor or any deormaon may be ren n dagonal orm symmer ensor our dervaon s vald or all deormaons. ktc hh s he same as he ne-sran Hooke s la dsussed earler, h G=kT.
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