Chapter 3

 

Constitutive Models MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqabKqzaiaeaa aaaaaaa8qacaWFtacaaa@3788@  Relations between Stress and Strain

 

 

 

3.10 Large Strain Viscoelasticity

 

This section describes constitutive equations that can be used to model large, irreversible deformations in polymers, and also to model biological tissue that is subjected to large shape changes.  Finite strain viscoelasticity is not as well developed as finite strain plasticity, and a number of different formulations exist. The model outlined here is based on Bergstrom and Boyce, J. Mech. Phys. Solids., Vol. 46, pp. 931-954, 1998.

 

The constitutive equation is intended to capture the following features of material behavior

1.      When the material is deformed very slowly (so that material behavior is fully reversible) it behaves like an ideal rubber, as described in Section 3.4;

2.      When deformed very quickly (so that there is no time for inelastic mechanisms to operate) it again behaves like an ideal rubber, but with different properties;

3.      At intermediate rates, the solid exhibits a rate dependent, hysteretic response.

 

In addition, we assume

 The material is isotropic.

 Material response to a pure volumetric strain ( ε 11 = ε 22 = ε 33 =δV/V MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew7aLnaaBaaaleaacaaIXaGaaGymaa qabaGccqGH9aqpcqaH1oqzdaWgaaWcbaGaaGOmaiaaikdaaeqaaOGa eyypa0JaeqyTdu2aaSbaaSqaaiaaiodacaaIZaaabeaakiabg2da9i abes7aKjaadAfacaGGVaGaamOvaaaa@4187@  with all other ε ij =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew7aLnaaBaaaleaacaWGPbGaamOAaa qabaGccqGH9aqpcaaIWaaaaa@35E2@  ) is perfectly elastic (with no time dependent behavior).

 The material is nearly incompressible;

 Hydrostatic stress has no effect on the deviatoric response of the solid.

.

The constitutive equations outlined in this section make use of many concepts from Sections 3.4, 3.6 and 3.7, so you may find it convenient to read these sections before the material to follow. 

 

 

3.10.1 Kinematics for finite strain viscoelasticity.

 

The description of shape changes in polymers follows closely the approach outlined in 3.7.1.  Let x i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadIhadaWgaaWcbaGaamyAaaqabaaaaa@327F@  be the position of a material particle in the undeformed solid. Suppose that the solid is subjected to a displacement field u i ( x k ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG1bWaaSbaaSqaaiaadMgaaeqaaO GaaiikaiaadIhadaWgaaWcbaGaam4AaaqabaGccaGGPaaaaa@3860@ , so that the point moves to y i = x i + u i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadMhadaWgaaWcbaGaamyAaaqabaGccq GH9aqpcaWG4bWaaSbaaSqaaiaadMgaaeqaaOGaey4kaSIaamyDamaa BaaaleaacaWGPbaabeaaaaa@38A7@ .  Define

 The deformation gradient and its jacobian

F ij = δ ij + u i x j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGgbWaaSbaaSqaaiaadMgacaWGQb aabeaakiabg2da9iabes7aKnaaBaaaleaacaWGPbGaamOAaaqabaGc cqGHRaWkdaWcaaqaaiabgkGi2kaadwhadaWgaaWcbaGaamyAaaqaba aakeaacqGHciITcaWG4bWaaSbaaSqaaiaadQgaaeqaaaaaaaa@4256@        J=det(F) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGkbGaeyypa0Jaciizaiaacwgaca GG0bGaaiikaiaahAeacaGGPaaaaa@398E@

 The velocity gradient

L ij = u ˙ i y j = F ˙ ik F kj 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadYeadaWgaaWcbaGaamyAaiaadQgaae qaaOGaeyypa0ZaaSaaaeaacqGHciITceWG1bGbaiaadaWgaaWcbaGa amyAaaqabaaakeaacqGHciITcaWG5bWaaSbaaSqaaiaadQgaaeqaaa aakiabg2da9iqadAeagaGaamaaBaaaleaacaWGPbGaam4AaaqabaGc caWGgbWaa0baaSqaaiaadUgacaWGQbaabaGaeyOeI0IaaGymaaaaaa a@43E5@

 The deformation gradient is decomposed into elastic and plastic parts as

F ij = F ik e F kj p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAeadaWgaaWcbaGaamyAaiaadQgaae qaaOGaeyypa0JaamOramaaDaaaleaacaWGPbGaam4Aaaqaaiaadwga aaGccaWGgbWaa0baaSqaaiaadUgacaWGQbaabaGaamiCaaaaaaa@3BE2@

 The velocity gradient is decomposed into elastic and plastic parts as

L ij = L ij e + L ij p , L ij e = F ˙ ik e F kj e1 , L ij p = F ik e F ˙ kl p F lm p1 F mj e1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadYeadaWgaaWcbaGaamyAaiaadQgaae qaaOGaeyypa0JaamitamaaDaaaleaacaWGPbGaamOAaaqaaiaadwga aaGccqGHRaWkcaWGmbWaa0baaSqaaiaadMgacaWGQbaabaGaamiCaa aakiaacYcacaaMc8UaaGPaVlaadYeadaqhaaWcbaGaamyAaiaadQga aeaacaWGLbaaaOGaeyypa0JabmOrayaacaWaa0baaSqaaiaadMgaca WGRbaabaGaamyzaaaakiaadAeadaqhaaWcbaGaam4AaiaadQgaaeaa caWGLbGaeyOeI0IaaGymaaaakiaacYcacaaMc8UaaGPaVlaadYeada qhaaWcbaGaamyAaiaadQgaaeaacaWGWbaaaOGaeyypa0JaamOramaa DaaaleaacaWGPbGaam4AaaqaaiaadwgaaaGcceWGgbGbaiaadaqhaa WcbaGaam4AaiaadYgaaeaacaWGWbaaaOGaamOramaaDaaaleaacaWG SbGaamyBaaqaaiaadchacqGHsislcaaIXaaaaOGaamOramaaDaaale aacaWGTbGaamOAaaqaaiaadwgacqGHsislcaaIXaaaaaaa@69FC@

 Define the elastic and plastic stretch rates and spin rates as

D ij e =( L ij e + L ji e )/2 W ij e =( L ij e L ji e )/2 D ij p =( L ij p + L ji p )/2 W ij p =( L ij p L ji p )/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaamiramaaDaaaleaacaWGPbGaam OAaaqaaiaadwgaaaGccqGH9aqpcaGGOaGaamitamaaDaaaleaacaWG PbGaamOAaaqaaiaadwgaaaGccqGHRaWkcaWGmbWaa0baaSqaaiaadQ gacaWGPbaabaGaamyzaaaakiaacMcacaGGVaGaaGOmaiaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaadEfadaqhaa WcbaGaamyAaiaadQgaaeaacaWGLbaaaOGaeyypa0JaaiikaiaadYea daqhaaWcbaGaamyAaiaadQgaaeaacaWGLbaaaOGaeyOeI0Iaamitam aaDaaaleaacaWGQbGaamyAaaqaaiaadwgaaaGccaGGPaGaai4laiaa ikdaaeaacaWGebWaa0baaSqaaiaadMgacaWGQbaabaGaamiCaaaaki abg2da9iaacIcacaWGmbWaa0baaSqaaiaadMgacaWGQbaabaGaamiC aaaakiabgUcaRiaadYeadaqhaaWcbaGaamOAaiaadMgaaeaacaWGWb aaaOGaaiykaiaac+cacaaIYaGaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8Uaam4vamaaDaaaleaacaWGPbGaamOAaa qaaiaadchaaaGccqGH9aqpcaGGOaGaamitamaaDaaaleaacaWGPbGa amOAaaqaaiaadchaaaGccqGHsislcaWGmbWaa0baaSqaaiaadQgaca WGPbaabaGaamiCaaaakiaacMcacaGGVaGaaGOmaaaaaa@BB50@

  Define the Left Cauchy-Green deformation tensor for the total and elastic deformation gradients

B ij = F ik F jk B ij e = F ik e F jk e MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGcbWaaSbaaSqaaiaadMgacaWGQb aabeaakiabg2da9iaadAeadaWgaaWcbaGaamyAaiaadUgaaeqaaOGa amOramaaBaaaleaacaWGQbGaam4AaaqabaGccaaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaamOq amaaDaaaleaacaWGPbGaamOAaaqaaiaadwgaaaGccqGH9aqpcaWGgb Waa0baaSqaaiaadMgacaWGRbaabaGaamyzaaaakiaadAeadaqhaaWc baGaamOAaiaadUgaaeaacaWGLbaaaaaa@5828@

  Define the invariants of B and B e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahkeadaahaaWcbeqaaiaadwgaaaaaaa@324A@  as

I ¯ 1 = B kk J 2/3 I ¯ 2 = 1 2 ( I ¯ 1 2 B ik B ki J 4/3 ) I ¯ 3 =detB= J 2 I ¯ 1 e = B kk e J e 2/3 I ¯ 2 e = 1 2 ( I ¯ 1 e2 B ik e B ki e J e 4/3 ) I ¯ 3 e =det B e = J e 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiqadMeagaqeamaaBaaaleaaca aIXaaabeaakiabg2da9maalaaabaGaamOqamaaBaaaleaacaWGRbGa am4AaaqabaaakeaacaWGkbWaaWbaaSqabeaacaaIYaGaai4laiaaio daaaaaaOGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7ceWGjbGbaebadaWgaaWcbaGaaG OmaaqabaGccqGH9aqpdaWcaaqaaiaaigdaaeaacaaIYaaaamaabmaa baGabmysayaaraWaa0baaSqaaiaaigdaaeaacaaIYaaaaOGaeyOeI0 YaaSaaaeaacaWGcbWaaSbaaSqaaiaadMgacaWGRbaabeaakiaadkea daWgaaWcbaGaam4AaiaadMgaaeqaaaGcbaGaamOsamaaCaaaleqaba GaaGinaiaac+cacaaIZaaaaaaaaOGaayjkaiaawMcaaiaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UabmysayaaraWaaSbaaSqaaiaaiodaaeqaaOGaeyypa0Ja ciizaiaacwgacaGG0bGaaCOqaiabg2da9iaadQeadaahaaWcbeqaai aaikdaaaaakeaaceWGjbGbaebadaqhaaWcbaGaaGymaaqaaiaadwga aaGccqGH9aqpdaWcaaqaaiaadkeadaqhaaWcbaGaam4AaiaadUgaae aacaWGLbaaaaGcbaGaamOsamaaDaaaleaacaWGLbaabaGaaGOmaiaa c+cacaaIZaaaaaaakiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlqadMeagaqeamaaDaaaleaacaaIYaaa baGaamyzaaaakiabg2da9maalaaabaGaaGymaaqaaiaaikdaaaWaae WaaeaaceWGjbGbaebadaqhaaWcbaGaaGymaaqaaiaadwgacaaIYaaa aOGaeyOeI0YaaSaaaeaacaWGcbWaa0baaSqaaiaadMgacaWGRbaaba GaamyzaaaakiaadkeadaqhaaWcbaGaam4AaiaadMgaaeaacaWGLbaa aaGcbaGaamOsamaaDaaaleaacaWGLbaabaGaaGinaiaac+cacaaIZa aaaaaaaOGaayjkaiaawMcaaiaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlqadMeagaqeamaaDaaaleaaca aIZaaabaGaamyzaaaakiabg2da9iGacsgacaGGLbGaaiiDaiaahkea daahaaWcbeqaaiaadwgaaaGccqGH9aqpcaWGkbWaa0baaSqaaiaadw gaaeaacaaIYaaaaaaaaa@C409@

*  Denote the principal stretches for B MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahkeaaaa@3133@  and B e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahkeadaahaaWcbeqaaiaadwgaaaaaaa@324A@  by λ i , λ i e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeU7aSnaaBaaaleaacaWGPbaabeaaki aacYcacqaH7oaBdaqhaaWcbaGaamyAaaqaaiaadwgaaaaaaa@37A9@  (these are the square roots of the eigenvalues of B MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahkeaaaa@3133@  and B e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahkeadaahaaWcbeqaaiaadwgaaaaaaa@324A@  ) and principal stretch directions by b i , b i e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahkgadaWgaaWcbaGaamyAaaqabaGcca GGSaGaaCOyamaaDaaaleaacaWGPbaabaGaamyzaaaaaaa@3617@ .

 

 

3.10.2 Stress measures for finite strain viscoelasticity.

 

Usually stress-strain laws are given as equations relating Cauchy stress (`true’ stress) σ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamyAaiaadQ gaaeqaaaaa@3692@  to left Cauchy-Green deformation tensor.  For some computations it may be more convenient to use other stress measures.  They are defined below, for convenience.

 Cauchy (“true”) stress represents the force per unit deformed area in the solid and is defined by

n i σ ij = Lim dA0 d P j (n) dA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGUbWaaSbaaSqaaiaadMgaaeqaaO Gaeq4Wdm3aaSbaaSqaaiaadMgacaWGQbaabeaakiabg2da9maaxaba baGaamitaiaadMgacaWGTbaaleaacaWGKbGaamyqaiabgkziUkaaic daaeqaaOWaaSaaaeaacaWGKbGaamiuamaaDaaaleaacaWGQbaabaGa aiikaiaah6gacaGGPaaaaaGcbaGaamizaiaadgeaaaaaaa@47F6@

 Kirchhoff stress  τ=Jσ τ ij =J σ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaqcaaSaaCiXdOGaeyypa0JaamOsaKaaal aaho8acaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlabes8a0PWaaSba aSqaaiaadMgacaWGQbaabeaakiabg2da9iaadQeacqaHdpWCdaWgaa WcbaGaamyAaiaadQgaaeqaaaaa@4819@

 

 

The constitutive model must specify relations between stress, the total deformation gradient F, the elastic part of the deformation gradient F e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahAeadaahaaWcbeqaaiaadwgaaaaaaa@324E@ , and the plastic part of the deformation gradient.

 

 

 

3.10.3 Relation between stress, deformation measures and strain energy density

 

Just as for hyperelastic materials, the instantaneous stress in a hyperviscoelastic solid is calculated from a strain energy density function U MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwfaaaa@3142@ .  For viscoelastic materials, the strain energy density is separated into two parts

U( I ¯ 1 , I ¯ 2 ,J, I ¯ 1 e , I ¯ 2 e )= U ( I ¯ 1 , I ¯ 2 ,J)+ U T ( I ¯ 1 e , I ¯ 2 e , J e ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwfacaGGOaGabmysayaaraWaaSbaaS qaaiaaigdaaeqaaOGaaiilaiqadMeagaqeamaaBaaaleaacaaIYaaa beaakiaacYcacaWGkbGaaiilaiqadMeagaqeamaaDaaaleaacaaIXa aabaGaamyzaaaakiaacYcaceWGjbGbaebadaqhaaWcbaGaaGOmaaqa aiaadwgaaaGccaGGPaGaeyypa0JaamyvamaaBaaaleaacqGHEisPae qaaOGaaiikaiqadMeagaqeamaaBaaaleaacaaIXaaabeaakiaacYca ceWGjbGbaebadaWgaaWcbaGaaGOmaaqabaGccaGGSaGaamOsaiaacM cacqGHRaWkcaWGvbWaaSbaaSqaaiaadsfaaeqaaOGaaiikaiqadMea gaqeamaaDaaaleaacaaIXaaabaGaamyzaaaakiaacYcaceWGjbGbae badaqhaaWcbaGaaGOmaaqaaiaadwgaaaGccaGGSaGaamOsamaaBaaa leaacaWGLbaabeaakiaacMcaaaa@5714@

Here

1.      U ( I ¯ 1 , I ¯ 2 ,J) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwfadaWgaaWcbaGaeyOhIukabeaaki aacIcaceWGjbGbaebadaqhaaWcbaGaaGymaaqaaaaakiaacYcaceWG jbGbaebadaqhaaWcbaGaaGOmaaqaaaaakiaacYcacaWGkbGaaiykaa aa@3A22@  specifies the strain energy density in the fully relaxed material.  It is represents the effect of a set of polymer chains in the solid which can only accommodate deformation by stretching to follow the total extension.  U =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwfadaWgaaWcbaGaeyOhIukabeaaki abg2da9iaaicdaaaa@34A9@  for a material that exhibits steady-state creep.

2.      U T ( I ¯ 1 e , I ¯ 2 e ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwfadaWgaaWcbaGaamivaaqabaGcca GGOaGabmysayaaraWaa0baaSqaaiaaigdaaeaacaWGLbaaaOGaaiil aiqadMeagaqeamaaDaaaleaacaaIYaaabaGaamyzaaaakiaacMcaaa a@39DF@  is an additional, transient contribution to the total strain energy.  This contribution gradually relaxes with time.   It represents a set of polymer chains which initially stretch with the solid, but with time are able to relax towards their preferred configuration.

 

The stress is related to the energy density by

σ ij = 2 J [ 1 J 2/3 ( U I ¯ 1 + I ¯ 1 U I ¯ 2 ) B ij I ¯ 1 3 U I ¯ 1 δ ij 1 J 4/3 U I ¯ 2 B ik B kj ]+ U J δ ij + 2 J e [ 1 J e 2/3 ( U T I ¯ 1 e + I ¯ 1 e U T I ¯ 2 e ) B ij e I ¯ 1 e 3 U T I ¯ 1 e δ ij 1 J e 4/3 U T I ¯ 2 e B ik e B kj e ]+ U T J e δ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGceaqabeaacqaHdpWCdaWgaaWcbaGaamyAai aadQgaaeqaaOGaeyypa0ZaaSaaaeaacaaIYaaabaGaamOsaaaadaWa daqaamaalaaabaGaaGymaaqaaiaadQeadaahaaWcbeqaaiaaikdaca GGVaGaaG4maaaaaaGcdaqadaqaamaalaaabaGaeyOaIyRaamyvamaa BaaaleaacqGHEisPaeqaaaGcbaGaeyOaIyRabmysayaaraWaaSbaaS qaaiaaigdaaeqaaaaakiabgUcaRiqadMeagaqeamaaBaaaleaacaaI XaaabeaakmaalaaabaGaeyOaIyRaamyvamaaBaaaleaacqGHEisPae qaaaGcbaGaeyOaIyRabmysayaaraWaaSbaaSqaaiaaikdaaeqaaaaa aOGaayjkaiaawMcaaiaadkeadaWgaaWcbaGaamyAaiaadQgaaeqaaO GaeyOeI0YaaSaaaeaaceWGjbGbaebadaWgaaWcbaGaaGymaaqabaaa keaacaaIZaaaamaalaaabaGaeyOaIyRaamyvamaaBaaaleaacqGHEi sPaeqaaaGcbaGaeyOaIyRabmysayaaraWaaSbaaSqaaiaaigdaaeqa aaaakiabes7aKnaaBaaaleaacaWGPbGaamOAaaqabaGccqGHsislda WcaaqaaiaaigdaaeaacaWGkbWaaWbaaSqabeaacaaI0aGaai4laiaa iodaaaaaaOWaaSaaaeaacqGHciITcaWGvbWaaSbaaSqaaiabg6HiLc qabaaakeaacqGHciITceWGjbGbaebadaWgaaWcbaGaaGOmaaqabaaa aOGaamOqamaaBaaaleaacaWGPbGaam4AaaqabaGccaWGcbWaaSbaaS qaaiaadUgacaWGQbaabeaaaOGaay5waiaaw2faaiabgUcaRmaalaaa baGaeyOaIyRaamyvamaaBaaaleaacqGHEisPaeqaaaGcbaGaeyOaIy RaamOsaaaacqaH0oazdaWgaaWcbaGaamyAaiaadQgaaeqaaaGcbaGa aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlabgUcaRm aalaaabaGaaGOmaaqaaiaadQeadaWgaaWcbaGaamyzaaqabaaaaOWa amWaaeaadaWcaaqaaiaaigdaaeaacaWGkbWaa0baaSqaaiaadwgaae aacaaIYaGaai4laiaaiodaaaaaaOWaaeWaaeaadaWcaaqaaiabgkGi 2kaadwfadaWgaaWcbaGaamivaaqabaaakeaacqGHciITceWGjbGbae badaqhaaWcbaGaaGymaaqaaiaadwgaaaaaaOGaey4kaSIabmysayaa raWaa0baaSqaaiaaigdaaeaacaWGLbaaaOWaaSaaaeaacqGHciITca WGvbWaaSbaaSqaaiaadsfaaeqaaaGcbaGaeyOaIyRabmysayaaraWa a0baaSqaaiaaikdaaeaacaWGLbaaaaaaaOGaayjkaiaawMcaaiaadk eadaqhaaWcbaGaamyAaiaadQgaaeaacaWGLbaaaOGaeyOeI0YaaSaa aeaaceWGjbGbaebadaqhaaWcbaGaaGymaaqaaiaadwgaaaaakeaaca aIZaaaamaalaaabaGaeyOaIyRaamyvamaaBaaaleaacaWGubaabeaa aOqaaiabgkGi2kqadMeagaqeamaaDaaaleaacaaIXaaabaGaamyzaa aaaaGccqaH0oazdaWgaaWcbaGaamyAaiaadQgaaeqaaOGaeyOeI0Ya aSaaaeaacaaIXaaabaGaamOsamaaDaaaleaacaWGLbaabaGaaGinai aac+cacaaIZaaaaaaakmaalaaabaGaeyOaIyRaamyvamaaBaaaleaa caWGubaabeaaaOqaaiabgkGi2kqadMeagaqeamaaDaaaleaacaaIYa aabaGaamyzaaaaaaGccaWGcbWaa0baaSqaaiaadMgacaWGRbaabaGa amyzaaaakiaadkeadaqhaaWcbaGaam4AaiaadQgaaeaacaWGLbaaaa GccaGLBbGaayzxaaGaey4kaSYaaSaaaeaacqGHciITcaWGvbWaaSba aSqaaiaadsfaaeqaaaGcbaGaeyOaIyRaamOsamaaBaaaleaacaWGLb aabeaaaaGccqaH0oazdaWgaaWcbaGaamyAaiaadQgaaeqaaaaaaa@DB0A@

 

You can use any of the hyperelastic strain energy density potentials listed in Section 3.4 to describe a particular material.  It is sensible to choose U T , U MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwfadaWgaaWcbaGaamivaaqabaGcca GGSaGaamyvamaaBaaaleaacqGHEisPaeqaaaaa@3578@  to have the same functional form (but with different material constants).   Note also that since the inelastic strains are assumed to be volume preserving (see below), J e =J MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadQeadaWgaaWcbaGaamyzaaqabaGccq GH9aqpcaWGkbaaaa@342C@ , and therefore once can take U T / J e =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgkGi2kaadwfadaWgaaWcbaGaamivaa qabaGccaGGVaGaeyOaIyRaamOsamaaBaaaleaacaWGLbaabeaakiab g2da9iaaicdaaaa@397F@  without loss of generality.

 

 

 

 

3.10.4 Strain relaxation

 

The strain rate dependence and irreversibility of a viscoelastic material can be modeled using the framework described in Section 3.8 for finite strain viscoplasticity.  The constitutive equations must specify the plastic stretch rate D ij p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadseadaqhaaWcbaGaamyAaiaadQgaae aacaWGWbaaaaaa@3430@  and plastic spin W ij p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadEfadaqhaaWcbaGaamyAaiaadQgaae aacaWGWbaaaaaa@3443@  as a function of stress.   The expressions given here follow Bergstrom and Boyce, J. Mech. Phys. Solids., Vol. 46, pp. 931-954, 1998.

 

Define the deviatoric Kirchhoff stress resulting from the elastic part of the deformation gradient F ij e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAeadaqhaaWcbaGaamyAaiaadQgaae aacaWGLbaaaaaa@3427@  as

τ ij =2[ 1 J e 2/3 ( U T I ¯ 1 e + I ¯ 1 e U T I ¯ 2 e ) B ij e I ¯ 1 e 3 U T I ¯ 1 e δ ij 1 J e 4/3 U T I ¯ 2 e B ik e B kj e ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGafqiXdqNbauaadaWgaaWcbaGaamyAai aadQgaaeqaaOGaeyypa0JaaGPaVlaaykW7caaIYaWaamWaaeaadaWc aaqaaiaaigdaaeaacaWGkbWaa0baaSqaaiaadwgaaeaacaaIYaGaai 4laiaaiodaaaaaaOWaaeWaaeaadaWcaaqaaiabgkGi2kaadwfadaWg aaWcbaGaamivaaqabaaakeaacqGHciITceWGjbGbaebadaqhaaWcba GaaGymaaqaaiaadwgaaaaaaOGaey4kaSIabmysayaaraWaa0baaSqa aiaaigdaaeaacaWGLbaaaOWaaSaaaeaacqGHciITcaWGvbWaaSbaaS qaaiaadsfaaeqaaaGcbaGaeyOaIyRabmysayaaraWaa0baaSqaaiaa ikdaaeaacaWGLbaaaaaaaOGaayjkaiaawMcaaiaadkeadaqhaaWcba GaamyAaiaadQgaaeaacaWGLbaaaOGaeyOeI0YaaSaaaeaaceWGjbGb aebadaqhaaWcbaGaaGymaaqaaiaadwgaaaaakeaacaaIZaaaamaala aabaGaeyOaIyRaamyvamaaBaaaleaacaWGubaabeaaaOqaaiabgkGi 2kqadMeagaqeamaaDaaaleaacaaIXaaabaGaamyzaaaaaaGccqaH0o azdaWgaaWcbaGaamyAaiaadQgaaeqaaOGaeyOeI0YaaSaaaeaacaaI XaaabaGaamOsamaaDaaaleaacaWGLbaabaGaaGinaiaac+cacaaIZa aaaaaakmaalaaabaGaeyOaIyRaamyvamaaBaaaleaacaWGubaabeaa aOqaaiabgkGi2kqadMeagaqeamaaDaaaleaacaaIYaaabaGaamyzaa aaaaGccaWGcbWaa0baaSqaaiaadMgacaWGRbaabaGaamyzaaaakiaa dkeadaqhaaWcbaGaam4AaiaadQgaaeaacaWGLbaaaaGccaGLBbGaay zxaaaaaa@7D0B@

1.      Define the effective stress

τ e = 3 τ ij τ ij /2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabes8a0naaBaaaleaacaWGLbaabeaaki abg2da9maakaaabaGaaG4maiqbes8a0zaafaWaaSbaaSqaaiaadMga caWGQbaabeaakiqbes8a0zaafaWaaSbaaSqaaiaadMgacaWGQbaabe aakiaac+cacaaIYaaaleqaaaaa@3E62@

2.      The plastic strain rate is then

D ij p = ε ˙ e ( τ e , I 1 e , I 2 e ,T) 3 τ ij 2 τ e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadseadaqhaaWcbaGaamyAaiaadQgaae aacaWGWbaaaOGaeyypa0JafqyTduMbaiaadaWgaaWcbaGaamyzaaqa baGccaGGOaGaeqiXdq3aaSbaaSqaaiaadwgaaeqaaOGaaiilaiaadM eadaqhaaWcbaGaaGymaaqaaiaadwgaaaGccaGGSaGaamysamaaDaaa leaacaaIYaaabaGaamyzaaaakiaacYcacaWGubGaaiykamaalaaaba GaaG4maiqbes8a0zaafaWaaSbaaSqaaiaadMgacaWGQbaabeaaaOqa aiaaikdacqaHepaDdaWgaaWcbaGaamyzaaqabaaaaaaa@4CD4@

3.      Here, ε ˙ e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqbew7aLzaacaWaaSbaaSqaaiaadwgaae qaaaaa@332E@  is the magnitude of the plastic strain rate, which is a function of temperature T, the effective stress τ e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabes8a0naaBaaaleaacaWGLbaabeaaaa a@3343@  and the elastic strain.  This function must be calibrated experimentally.  Bergstrom & Boyce suggest that the following function should describe approximately the relaxation dynamics of long-chain molecules

ε ˙ e = ε ˙ 0 ( I 1 e 3 ) n ( τ e τ 0 ) m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqbew7aLzaacaWaaSbaaSqaaiaadwgaae qaaOGaeyypa0JafqyTduMbaiaadaWgaaWcbaGaaGimaaqabaGcdaqa daqaamaakaaabaGaamysamaaDaaaleaacaaIXaaabaGaamyzaaaaae qaaOGaeyOeI0YaaOaaaeaacaaIZaaaleqaaaGccaGLOaGaayzkaaWa aWbaaSqabeaacaWGUbaaaOWaaeWaaeaadaWcaaqaaiabes8a0naaBa aaleaacaWGLbaabeaaaOqaaiabes8a0naaBaaaleaacaaIWaaabeaa aaaakiaawIcacaGLPaaadaahaaWcbeqaaiaad2gaaaaaaa@466C@

where ε ˙ 0 >0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqbew7aLzaacaWaaSbaaSqaaiaaicdaae qaaOGaeyOpa4JaaGimaaaa@34CA@ , 1<n<0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgkHiTiaaigdacqGH8aapcaWGUbGaey ipaWJaaGimaaaa@35C5@ , τ 0 >0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabes8a0naaBaaaleaacaaIWaaabeaaki abg6da+iaaicdaaaa@34DF@  and m>0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaad2gacqGH+aGpcaaIWaaaaa@331C@  are temperature dependent material properties.

 

Additional constitutive equations must specify W ij p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadEfadaqhaaWcbaGaamyAaiaadQgaae aacaWGWbaaaaaa@3443@ .   This has not been studied in detail: usually we just take W ij p =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadEfadaqhaaWcbaGaamyAaiaadQgaae aacaWGWbaaaOGaeyypa0JaaGimaaaa@360D@ .

 

 

 

3.10.5 Representative values for material parameters in a finite-strain viscoelastic model

 


Bergstrom and Boyce, J. Mech.Phys. Solids., Vol. 46, pp. 931-954, 1998 give experimental data for the rate dependent response of several rubbers, and fit material properties to their data.   They use the Arruda-Boyce potential for both U , U T MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwfadaWgaaWcbaGaeyOhIukabeaaki aacYcacaWGvbWaaSbaaSqaaiaadsfaaeqaaaaa@3578@

U = μ { 1 2 ( I ¯ 1 3)+ 1 20 β 2 ( I ¯ 1 2 9)+ 11 1050 β 4 ( I ¯ 1 3 27)+... }+ K 2 ( J1 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamyvamaaBaaaleaacqGHEisPaeqaaO Gaeyypa0JaeqiVd02aaSbaaSqaaiabg6HiLcqabaGcdaGadaqaamaa laaabaGaaGymaaqaaiaaikdaaaGaaiikaiqadMeagaqeamaaDaaale aacaaIXaaabaaaaOGaeyOeI0IaaG4maiaacMcacqGHRaWkdaWcaaqa aiaaigdaaeaacaaIYaGaaGimaiabek7aInaaDaaaleaacqGHEisPae aacaaIYaaaaaaakiaacIcaceWGjbGbaebadaqhaaWcbaGaaGymaaqa aiaaikdaaaGccqGHsislcaaI5aGaaiykaiabgUcaRmaalaaabaGaaG ymaiaaigdaaeaacaaIXaGaaGimaiaaiwdacaaIWaGaeqOSdi2aa0ba aSqaaiabg6HiLcqaaiaaisdaaaaaaOGaaiikaiqadMeagaqeamaaDa aaleaacaaIXaaabaGaaG4maaaakiabgkHiTiaaikdacaaI3aGaaiyk aiabgUcaRiaac6cacaGGUaGaaiOlaaGaay5Eaiaaw2haaiabgUcaRm aalaaabaGaam4saaqaaiaaikdaaaWaaeWaaeaacaWGkbGaeyOeI0Ia aGymaaGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaaaaa@671F@

U T = μ T { 1 2 ( I ¯ 1 3)+ 1 20 β T 2 ( I ¯ 1 2 9)+ 11 1050 β T 4 ( I ¯ 1 3 27)+... } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamyvamaaBaaaleaacaWGubaabeaaki abg2da9iabeY7aTnaaBaaaleaacaWGubaabeaakmaacmaabaWaaSaa aeaacaaIXaaabaGaaGOmaaaacaGGOaGabmysayaaraWaa0baaSqaai aaigdaaeaaaaGccqGHsislcaaIZaGaaiykaiabgUcaRmaalaaabaGa aGymaaqaaiaaikdacaaIWaGaeqOSdi2aa0baaSqaaiaadsfaaeaaca aIYaaaaaaakiaacIcaceWGjbGbaebadaqhaaWcbaGaaGymaaqaaiaa ikdaaaGccqGHsislcaaI5aGaaiykaiabgUcaRmaalaaabaGaaGymai aaigdaaeaacaaIXaGaaGimaiaaiwdacaaIWaGaeqOSdi2aa0baaSqa aiaadsfaaeaacaaI0aaaaaaakiaacIcaceWGjbGbaebadaqhaaWcba GaaGymaaqaaiaaiodaaaGccqGHsislcaaIYaGaaG4naiaacMcacqGH RaWkcaGGUaGaaiOlaiaac6caaiaawUhacaGL9baaaaa@5D58@

Material behavior is therefore characterized by values of the two shear moduli μ T , μ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeY7aTnaaBaaaleaacaWGubaabeaaki aacYcacqaH8oqBdaWgaaWcbaGaeyOhIukabeaaaaa@3730@ , the bulk modulus K, the coefficients β MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabek7aInaaBaaaleaacqGHEisPaeqaaa aa@33A6@ , β T MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabek7aInaaBaaaleaacaWGubaabeaaaa a@330E@ , and the parameters ε ˙ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqbew7aLzaacaWaaSbaaSqaaiaaicdaae qaaaaa@32FE@ , n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaad6gaaaa@315B@ , τ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabes8a0naaBaaaleaacaaIWaaabeaaaa a@3313@  and m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaad2gaaaa@315A@ , as outlined in the preceding section.

 

 

 

 

Material Parameters for a Nitrile Rubber (after Bergstrom & Boyce, 1998)

 

μ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeY7aTnaaBaaaleaacqGHEisPaeqaaa aa@33BB@

β MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabek7aInaaBaaaleaacqGHEisPaeqaaa aa@33A6@

μ T MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeY7aTnaaBaaaleaacaWGubaabeaaaa a@3323@

β T MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabek7aInaaBaaaleaacaWGubaabeaaaa a@330E@

K MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadUeaaaa@3138@

ε ˙ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqbew7aLzaacaWaaSbaaSqaaiaaicdaae qaaaaa@32FE@

τ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabes8a0naaBaaaleaacaaIWaaabeaaaa a@3313@

n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaad6gaaaa@315B@

m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaad2gaaaa@315A@

0.29MPa

6

0.73MPa

4

100MPa

7 sec-1

1MPa

-0.6

5.0