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

 

 

3.9 Large Strain, Rate Dependent Plasticity

 

This section describes the constitutive equations that are used to model large, permanent deformations in polycrystalline solids.  Representative applications include models of metal forming; crash simulations, and various military applications that are best left to the imagination.  The constitutive equations are used mostly in numerical simulations.  It is usually preferable to use a rate dependent,  viscoplasticity model for in computations, because they are less prone to instabilities than rate independent models.  The rate independent limit can always be approximated by using a high strain rate sensitivity.

 

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

 

 

 

3.9.1 Kinematics of finite strain plasticity

 

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 stretch rate and spin

D ij =( L ij + L ji )/2 W ij =( L ij L ji )/2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadseadaWgaaWcbaGaamyAaiaadQgaae qaaOGaeyypa0JaaiikaiaadYeadaWgaaWcbaGaamyAaiaadQgaaeqa aOGaey4kaSIaamitamaaBaaaleaacaWGQbGaamyAaaqabaGccaGGPa Gaai4laiaaikdacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caWGxbWaaSbaaSqaaiaadMgacaWGQbaabeaakiabg2 da9iaacIcacaWGmbWaaSbaaSqaaiaadMgacaWGQbaabeaakiabgkHi TiaadYeadaWgaaWcbaGaamOAaiaadMgaaeqaaOGaaiykaiaac+caca aIYaaaaa@7036@

 Recall that F ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAeadaWgaaWcbaGaamyAaiaadQgaae qaaaaa@333C@  relates infinitesimal material fibers d y i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWG5bWaaSbaaSqaaiaadMgaae qaaaaa@3369@  and d x i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWG4bWaaSbaaSqaaiaadMgaae qaaaaa@3368@  in the deformed and undeformed solid, respectively, as

d y i = F ij d x j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWG5bWaaSbaaSqaaiaadMgaae qaaOGaeyypa0JaamOramaaBaaaleaacaWGPbGaamOAaaqabaGccaWG KbGaamiEamaaBaaaleaacaWGQbaabeaaaaa@3A58@

 To decompose the deformation gradient into elastic and plastic parts, we borrow ideas from crystal plasticity.  The plastic strain is assumed to shear the lattice, without stretching or rotating it.  The elastic deformation rotates and stretches the lattice. We think of these two events occurring in sequence, with the plastic deformation first, and the stretch and rotation second, giving

d y i = F ij d x j = F ik e F kj p d x j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamizaiaadMhadaWgaaWcbaGaamyAaa qabaGccqGH9aqpcaWGgbWaaSbaaSqaaiaadMgacaWGQbaabeaakiaa dsgacaWG4bWaaSbaaSqaaiaadQgaaeqaaOGaeyypa0JaamOramaaDa aaleaacaWGPbGaam4AaaqaaiaadwgaaaGccaWGgbWaa0baaSqaaiaa dUgacaWGQbaabaGaamiCaaaakiaadsgacaWG4bWaaSbaaSqaaiaadQ gaaeqaaaaa@4670@

 To decompose the velocity gradient into elastic and plastic parts, note that

L ij = F ˙ ik F kj 1 =( F ˙ ik e F kl p + F ik e F ˙ kl p )( F lm p1 F mj e1 )= F ˙ ik e F kj e1 + 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 qaaOGaeyypa0JabmOrayaacaWaaSbaaSqaaiaadMgacaWGRbaabeaa kiaadAeadaqhaaWcbaGaam4AaiaadQgaaeaacqGHsislcaaIXaaaaO Gaeyypa0ZaaeWaaeaaceWGgbGbaiaadaqhaaWcbaGaamyAaiaadUga aeaacaWGLbaaaOGaamOramaaDaaaleaacaWGRbGaamiBaaqaaiaadc haaaGccqGHRaWkcaWGgbWaa0baaSqaaiaadMgacaWGRbaabaGaamyz aaaakiqadAeagaGaamaaDaaaleaacaWGRbGaamiBaaqaaiaadchaaa aakiaawIcacaGLPaaadaqadaqaaiaadAeadaqhaaWcbaGaamiBaiaa d2gaaeaacaWGWbGaeyOeI0IaaGymaaaakiaadAeadaqhaaWcbaGaam yBaiaadQgaaeaacaWGLbGaeyOeI0IaaGymaaaaaOGaayjkaiaawMca aiabg2da9iqadAeagaGaamaaDaaaleaacaWGPbGaam4Aaaqaaiaadw gaaaGccaWGgbWaa0baaSqaaiaadUgacaWGQbaabaGaamyzaiabgkHi TiaaigdaaaGccqGHRaWkcaWGgbWaa0baaSqaaiaadMgacaWGRbaaba GaamyzaaaakiqadAeagaGaamaaDaaaleaacaWGRbGaamiBaaqaaiaa dchaaaGccaWGgbWaa0baaSqaaiaadYgacaWGTbaabaGaamiCaiabgk HiTiaaigdaaaGccaWGgbWaa0baaSqaaiaad2gacaWGQbaabaGaamyz aiabgkHiTiaaigdaaaaaaa@78D0@

Thus the velocity gradient contains two terms, one of which involves only measures of elastic deformation, while the other contains measures of plastic deformation.  We use this to decompose L into elastic and plastic parts

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 aakiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaamitamaaDaaaleaacaWGPbGaamOAaa qaaiaadwgaaaGccqGH9aqpceWGgbGbaiaadaqhaaWcbaGaamyAaiaa dUgaaeaacaWGLbaaaOGaamOramaaDaaaleaacaWGRbGaamOAaaqaai aadwgacqGHsislcaaIXaaaaOGaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caWGmbWaa0baaSqaaiaadMgacaWGQbaabaGaamiCaaaa kiabg2da9iaadAeadaqhaaWcbaGaamyAaiaadUgaaeaacaWGLbaaaO GabmOrayaacaWaa0baaSqaaiaadUgacaWGSbaabaGaamiCaaaakiaa dAeadaqhaaWcbaGaamiBaiaad2gaaeaacaWGWbGaeyOeI0IaaGymaa aakiaadAeadaqhaaWcbaGaamyBaiaadQgaaeaacaWGLbGaeyOeI0Ia aGymaaaaaaa@8903@

 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@

 

Constitutive equations must specify relations between the stresses (as defined below) and the elastic and plastic parts of the deformation gradient.   The equations are usually written in rate form, in which case the elastic and plastic stretch rates and spin are related to the stress rate.

 

 

 

3.9.2 Stress measures for finite deformation plasticity

 

Stress measures that appear in descriptions of finite strain plasticity are summarized below:

 

 The 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 ij =J σ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaqcaaSaaGPaVlabes8a0PWaaSbaaSqaai aadMgacaWGQbaabeaakiabg2da9iaadQeacqaHdpWCdaWgaaWcbaGa amyAaiaadQgaaeqaaaaa@3CA6@

 Material stress for intermediate configuration   Σ ij =J F ik e1 σ kl F jl e1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeu4Odm1aaSbaaSqaaiaadMgacaWGQb aabeaakiabg2da9iaadQeacaWGgbWaa0baaSqaaiaadMgacaWGRbaa baGaamyzaiabgkHiTiaaigdaaaGccqaHdpWCdaWgaaWcbaGaam4Aai aadYgaaeqaaOGaamOramaaDaaaleaacaWGQbGaamiBaaqaaiaadwga cqGHsislcaaIXaaaaaaa@44F0@

 

Note that the material stress tensor is related to the Cauchy stress by a function of F e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaqcaaSaaGPaVRGaaCOramaaCaaaleqaba Gaamyzaaaaaaa@3513@ , not F as in the usual definition. This stress should be interpreted physically as a material stress associated with the intermediate configuration. This stress measure is introduced because the elastic constitutive equations require an internal force measure that is work-conjugate to an appropriate function of F e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaqcaaSaaGPaVRGaaCOramaaCaaaleqaba Gaamyzaaaaaaa@3513@ .

 

In addition, viscoplastic constitutive equations are often written in rate form (as in 3.7), relating stain rate to stress and (for the elastic part) stress rate.   Stress rates are difficult to work with in finite strain problems.  At first sight, it might appear that stress rate can be calculated by simply taking the time derivative of the stress components d σ ij /dt MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacqaHdpWCdaWgaaWcbaGaamyAai aadQgaaeqaaOGaai4laiaadsgacaWG0baaaa@37BC@ , but in fact this is not a useful measure of stress rate.  To see this, imagine applying a uniaxial tensile stress to a material, and then rotating the entire test apparatus (so the applied force and specimen rotate together).   The time derivatives of the stress components are nonzero, but the material actually experiences a time independent force per unit area.   As shown below, the correct stress rate is the Jaumann Rate with respect to the elastic spin, defined as

σ ij e = d σ ij dt W ik e σ kj + σ ik W kj e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaWaaCbiaeaacqaHdpWCdaWgaaWcbaGaam yAaiaadQgaaeqaaaqabeaacqGHhis0caWGLbaaaOGaeyypa0ZaaSaa aeaacaWGKbGaeq4Wdm3aaSbaaSqaaiaadMgacaWGQbaabeaaaOqaai aadsgacaWG0baaaiabgkHiTiaadEfadaqhaaWcbaGaamyAaiaadUga aeaacaWGLbaaaOGaeq4Wdm3aaSbaaSqaaiaadUgacaWGQbaabeaaki abgUcaRiabeo8aZnaaBaaaleaacaWGPbGaam4AaaqabaGccaWGxbWa a0baaSqaaiaadUgacaWGQbaabaGaamyzaaaaaaa@5035@

 

 

 

3.9.3 Elastic stress-strain relation for finite strain plasticity

 

Plastically deforming metals may experience large strains.  The stresses remain modest, however, and are usually substantially lower than the elastic modulus of the solid.   The elastic strains are small, but the  material may experience large rotations. Under these conditions, the small-strain elastic constitutive equations of 3.1 cannot be used, but the simple generalized Hooke’s law described in Section 3.3 can be used.  This law relates the elastic part of the deformation gradient to stress, as follows

1.      Define the Lagrangean elastic strain as E ij e =( F ki e F kj e δ ij )/2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadweadaqhaaWcbaGaamyAaiaadQgaae aacaWGLbaaaOGaeyypa0JaaiikaiaadAeadaqhaaWcbaGaam4Aaiaa dMgaaeaacaWGLbaaaOGaamOramaaDaaaleaacaWGRbGaamOAaaqaai aadwgaaaGccqGHsislcqaH0oazdaWgaaWcbaGaamyAaiaadQgaaeqa aOGaaiykaiaac+cacaaIYaaaaa@4438@

2.      Assume that the material stress is proportional to Lagrange strain, as Σ ij = C ijkl E kl e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqqHJoWudaWgaaWcbaGaamyAaiaadQ gaaeqaaOGaeyypa0Jaam4qamaaBaaaleaacaWGPbGaamOAaiaadUga caWGSbaabeaakiaadweadaqhaaWcbaGaam4AaiaadYgaaeaacaWGLb aaaaaa@3FE1@ , where C ijkl MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadoeadaWgaaWcbaGaamyAaiaadQgaca WGRbGaamiBaaqabaaaaa@351A@  are the components of the elastic stiffness tensor (as defined and tabulated in Section 3.1), for the material with orientation in the undeformed configuration. 

3.      For the special case of an elastically isotropic material, with Young’s modulus E and Poission ratio ν MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabe27aUbaa@3220@ , the stress-strain law is

Σ ij = E 1+ν { E ij e + ν 12ν E kk e δ ij } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfo6atnaaBaaaleaacaWGPbGaamOAaa qabaGccqGH9aqpdaWcaaqaaiaadweaaeaacaaIXaGaey4kaSIaeqyV d4gaamaacmaabaGaamyramaaDaaaleaacaWGPbGaamOAaaqaaiaadw gaaaGccqGHRaWkdaWcaaqaaiabe27aUbqaaiaaigdacqGHsislcaaI YaGaeqyVd4gaaiaadweadaqhaaWcbaGaam4AaiaadUgaaeaacaWGLb aaaOGaeqiTdq2aaSbaaSqaaiaadMgacaWGQbaabeaaaOGaay5Eaiaa w2haaaaa@4D75@

4.      The elastic stress-strain law is often expressed in rate form, as follows

τ ij e C ijkl e D kl e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaWaaCbiaeaacqaHepaDdaWgaaWcbaGaam yAaiaadQgaaeqaaaqabeaacqGHhis0caWGLbaaaOGaeyisISRaam4q amaaDaaaleaacaWGPbGaamOAaiaadUgacaWGSbaabaGaamyzaaaaki aadseadaqhaaWcbaGaam4AaiaadYgaaeaacaWGLbaaaaaa@426E@

where τ ij e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaWaaCbiaeaacqaHepaDdaWgaaWcbaGaam yAaiaadQgaaeqaaaqabeaacqGHhis0caWGLbaaaaaa@374B@  is the Jaumann rate of Kirchhoff stress; C ijkl e = F in e F jm e C nmpq F kp e F lq e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadoeadaqhaaWcbaGaamyAaiaadQgaca WGRbGaamiBaaqaaiaadwgaaaGccqGH9aqpcaWGgbWaa0baaSqaaiaa dMgacaWGUbaabaGaamyzaaaakiaadAeadaqhaaWcbaGaamOAaiaad2 gaaeaacaWGLbaaaOGaam4qamaaBaaaleaacaWGUbGaamyBaiaadcha caWGXbaabeaakiaadAeadaqhaaWcbaGaam4AaiaadchaaeaacaWGLb aaaOGaamOramaaDaaaleaacaWGSbGaamyCaaqaaiaadwgaaaaaaa@4B17@  (this can be thought of as the components of the elastic compliance tensor for material with orientation in the deformed configuration), and D ij e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadseadaqhaaWcbaGaamyAaiaadQgaae aacaWGLbaaaaaa@3425@  is the elastic stretch rate.  For the particular case of an isotropic material with Young’s modulus E and Poisson’s ratio ν MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH9oGBaaa@347E@ , the stress rate can be approximated further as

τ ij e E 1+ν { D ij e + ν 12ν D kk e δ ij } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaaxacabaGaeqiXdq3aaSbaaSqaaiaadM gacaWGQbaabeaaaeqabaGaey4bIeTaamyzaaaakiabgIKi7oaalaaa baGaamyraaqaaiaaigdacqGHRaWkcqaH9oGBaaWaaiWaaeaacaWGeb Waa0baaSqaaiaadMgacaWGQbaabaGaamyzaaaakiabgUcaRmaalaaa baGaeqyVd4gabaGaaGymaiabgkHiTiaaikdacqaH9oGBaaGaamiram aaDaaaleaacaWGRbGaam4AaaqaaiaadwgaaaGccqaH0oazdaWgaaWc baGaamyAaiaadQgaaeqaaaGccaGL7bGaayzFaaaaaa@510D@

 

Derivation of the rate form of the elastic stress-strain law: Our goal is to derive the expression in (4) above, starting from the stress-strain law in (2).  To this end:

1.      Take the time-derivative of the constitutive equation: d Σ ij dt = C ijkl d E kl e dt MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaWaaSaaaeaacaWGKbGaeu4Odm1aaSbaaS qaaiaadMgacaWGQbaabeaaaOqaaiaadsgacaWG0baaaiabg2da9iaa doeadaWgaaWcbaGaamyAaiaadQgacaWGRbGaamiBaaqabaGcdaWcaa qaaiaadsgacaWGfbWaa0baaSqaaiaadUgacaWGSbaabaGaamyzaaaa aOqaaiaadsgacaWG0baaaaaa@43AA@

2.      Take the time derivative of the formula relating material and Kirchhoff stress

τ ij = F ik e Σ kl F jl e d τ ij dt = d F ik e dt Σ kl F jl e + F ik e d Σ kl dt F jl e + F ik e Σ kl d F jl e dt MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGceaqabeaacqaHepaDdaWgaaWcbaGaamyAai aadQgaaeqaaOGaeyypa0JaamOramaaDaaaleaacaWGPbGaam4Aaaqa aiaadwgaaaGccqqHJoWudaWgaaWcbaGaam4AaiaadYgaaeqaaOGaam OramaaDaaaleaacaWGQbGaamiBaaqaaiaadwgaaaaakeaacqGHshI3 daWcaaqaaiaadsgacqaHepaDdaWgaaWcbaGaamyAaiaadQgaaeqaaa GcbaGaamizaiaadshaaaGaeyypa0ZaaSaaaeaacaWGKbGaamOramaa DaaaleaacaWGPbGaam4AaaqaaiaadwgaaaaakeaacaWGKbGaamiDaa aacqqHJoWudaWgaaWcbaGaam4AaiaadYgaaeqaaOGaamOramaaDaaa leaacaWGQbGaamiBaaqaaiaadwgaaaGccqGHRaWkcaWGgbWaa0baaS qaaiaadMgacaWGRbaabaGaamyzaaaakmaalaaabaGaamizaiabfo6a tnaaBaaaleaacaWGRbGaamiBaaqabaaakeaacaWGKbGaamiDaaaaca WGgbWaa0baaSqaaiaadQgacaWGSbaabaGaamyzaaaakiabgUcaRiaa dAeadaqhaaWcbaGaamyAaiaadUgaaeaacaWGLbaaaOGaeu4Odm1aaS baaSqaaiaadUgacaWGSbaabeaakmaalaaabaGaamizaiaadAeadaqh aaWcbaGaamOAaiaadYgaaeaacaWGLbaaaaGcbaGaamizaiaadshaaa aaaaa@76E3@

3.      Substitute for material stress in terms of Kirchoff stress

d τ ij dt = d F ik e dt F kl e1 τ lj + F ik e d Σ kl dt F jl e + τ il F lk e1 d F jk e dt MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaWaaSaaaeaacaWGKbGaeqiXdq3aaSbaaS qaaiaadMgacaWGQbaabeaaaOqaaiaadsgacaWG0baaaiabg2da9maa laaabaGaamizaiaadAeadaqhaaWcbaGaamyAaiaadUgaaeaacaWGLb aaaaGcbaGaamizaiaadshaaaGaamOramaaDaaaleaacaWGRbGaamiB aaqaaiaadwgacqGHsislcaaIXaaaaOGaeqiXdq3aaSbaaSqaaiaadY gacaWGQbaabeaakiabgUcaRiaadAeadaqhaaWcbaGaamyAaiaadUga aeaacaWGLbaaaOWaaSaaaeaacaWGKbGaeu4Odm1aaSbaaSqaaiaadU gacaWGSbaabeaaaOqaaiaadsgacaWG0baaaiaadAeadaqhaaWcbaGa amOAaiaadYgaaeaacaWGLbaaaOGaey4kaSIaeqiXdq3aaSbaaSqaai aadMgacaWGSbaabeaakiaadAeadaqhaaWcbaGaamiBaiaadUgaaeaa caWGLbGaeyOeI0IaaGymaaaakmaalaaabaGaamizaiaadAeadaqhaa WcbaGaamOAaiaadUgaaeaacaWGLbaaaaGcbaGaamizaiaadshaaaaa aa@6842@

4.      Recall that F ˙ ik e F kj e1 = L ij e = D ij e + W ij e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqadAeagaGaamaaDaaaleaacaWGPbGaam 4AaaqaaiaadwgaaaGccaWGgbWaa0baaSqaaiaadUgacaWGQbaabaGa amyzaiabgkHiTiaaigdaaaGccqGH9aqpcaWGmbWaa0baaSqaaiaadM gacaWGQbaabaGaamyzaaaakiabg2da9iaadseadaqhaaWcbaGaamyA aiaadQgaaeaacaWGLbaaaOGaey4kaSIaam4vamaaDaaaleaacaWGPb GaamOAaaqaaiaadwgaaaaaaa@4802@ , observe that W ij e = W ji e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadEfadaqhaaWcbaGaamyAaiaadQgaae aacaWGLbaaaOGaeyypa0JaeyOeI0Iaam4vamaaDaaaleaacaWGQbGa amyAaaqaaiaadwgaaaaaaa@3A05@ , D ij e = D ji e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadseadaqhaaWcbaGaamyAaiaadQgaae aacaWGLbaaaOGaeyypa0JaamiramaaDaaaleaacaWGQbGaamyAaaqa aiaadwgaaaaaaa@38F2@ , and substitute from (1)

d τ ij dt =( D ik e + W ik e ) τ kj + F ik e C klmn d E mn e dt F jl e + τ ik ( D kj e W kj e ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaWaaSaaaeaacaWGKbGaeqiXdq3aaSbaaS qaaiaadMgacaWGQbaabeaaaOqaaiaadsgacaWG0baaaiabg2da9maa bmaabaGaamiramaaDaaaleaacaWGPbGaam4AaaqaaiaadwgaaaGccq GHRaWkcaWGxbWaa0baaSqaaiaadMgacaWGRbaabaGaamyzaaaaaOGa ayjkaiaawMcaaiabes8a0naaBaaaleaacaWGRbGaamOAaaqabaGccq GHRaWkcaWGgbWaa0baaSqaaiaadMgacaWGRbaabaGaamyzaaaakiaa doeadaWgaaWcbaGaam4AaiaadYgacaWGTbGaamOBaaqabaGcdaWcaa qaaiaadsgacaWGfbWaa0baaSqaaiaad2gacaWGUbaabaGaamyzaaaa aOqaaiaadsgacaWG0baaaiaadAeadaqhaaWcbaGaamOAaiaadYgaae aacaWGLbaaaOGaey4kaSIaeqiXdq3aaSbaaSqaaiaadMgacaWGRbaa beaakmaabmaabaGaamiramaaDaaaleaacaWGRbGaamOAaaqaaiaadw gaaaGccqGHsislcaWGxbWaa0baaSqaaiaadUgacaWGQbaabaGaamyz aaaaaOGaayjkaiaawMcaaaaa@692D@

5.      Next, note that

d E mn e dt = F pm e D pq e F qn e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaWaaSaaaeaacaWGKbGaamyramaaDaaale aacaWGTbGaamOBaaqaaiaadwgaaaaakeaacaWGKbGaamiDaaaacqGH 9aqpcaWGgbWaa0baaSqaaiaadchacaWGTbaabaGaamyzaaaakiaads eadaqhaaWcbaGaamiCaiaadghaaeaacaWGLbaaaOGaamOramaaDaaa leaacaWGXbGaamOBaaqaaiaadwgaaaaaaa@43F3@

so

d τ ij dt =( D ik e + W ik e ) τ kj + F ik e F jl e C klmn F pm e F qn e D pq e + τ ik ( D kj e W kj e ) d τ ij dt + τ ik W kj e W ik e τ kj = C ijpq e D pq e + D ik e τ kj + τ ik D kj e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGceaqabeaadaWcaaqaaiaadsgacqaHepaDda WgaaWcbaGaamyAaiaadQgaaeqaaaGcbaGaamizaiaadshaaaGaeyyp a0ZaaeWaaeaacaWGebWaa0baaSqaaiaadMgacaWGRbaabaGaamyzaa aakiabgUcaRiaadEfadaqhaaWcbaGaamyAaiaadUgaaeaacaWGLbaa aaGccaGLOaGaayzkaaGaeqiXdq3aaSbaaSqaaiaadUgacaWGQbaabe aakiabgUcaRiaadAeadaqhaaWcbaGaamyAaiaadUgaaeaacaWGLbaa aOGaamOramaaDaaaleaacaWGQbGaamiBaaqaaiaadwgaaaGccaWGdb WaaSbaaSqaaiaadUgacaWGSbGaamyBaiaad6gaaeqaaOGaamOramaa DaaaleaacaWGWbGaamyBaaqaaiaadwgaaaGccaWGgbWaa0baaSqaai aadghacaWGUbaabaGaamyzaaaakiaadseadaqhaaWcbaGaamiCaiaa dghaaeaacaWGLbaaaOGaey4kaSIaeqiXdq3aaSbaaSqaaiaadMgaca WGRbaabeaakmaabmaabaGaamiramaaDaaaleaacaWGRbGaamOAaaqa aiaadwgaaaGccqGHsislcaWGxbWaa0baaSqaaiaadUgacaWGQbaaba GaamyzaaaaaOGaayjkaiaawMcaaaqaaiabgkDiEpaalaaabaGaamiz aiabes8a0naaBaaaleaacaWGPbGaamOAaaqabaaakeaacaWGKbGaam iDaaaacqGHRaWkcqaHepaDdaWgaaWcbaGaamyAaiaadUgaaeqaaOGa am4vamaaDaaaleaacaWGRbGaamOAaaqaaiaadwgaaaGccqGHsislca WGxbWaa0baaSqaaiaadMgacaWGRbaabaGaamyzaaaakiabes8a0naa BaaaleaacaWGRbGaamOAaaqabaGccqGH9aqpcaWGdbWaa0baaSqaai aadMgacaWGQbGaamiCaiaadghaaeaacaWGLbaaaOGaamiramaaDaaa leaacaWGWbGaamyCaaqaaiaadwgaaaGccqGHRaWkcaWGebWaa0baaS qaaiaadMgacaWGRbaabaGaamyzaaaakiabes8a0naaBaaaleaacaWG RbGaamOAaaqabaGccqGHRaWkcqaHepaDdaWgaaWcbaGaamyAaiaadU gaaeqaaOGaamiramaaDaaaleaacaWGRbGaamOAaaqaaiaadwgaaaaa aaa@A3D9@

6.      Finally, assume that τ ik D kj e << C ijpq e D pq e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabes8a0naaBaaaleaacaWGPbGaam4Aaa qabaGccaWGebWaa0baaSqaaiaadUgacaWGQbaabaGaamyzaaaakiab gYda8iabgYda8iaadoeadaqhaaWcbaGaamyAaiaadQgacaWGWbGaam yCaaqaaiaadwgaaaGccaWGebWaa0baaSqaaiaadchacaWGXbaabaGa amyzaaaaaaa@438E@  since the stresses are much less than the modulus. This shows that

τ ij e C ijkl e D kl e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaWaaCbiaeaacqaHepaDdaWgaaWcbaGaam yAaiaadQgaaeqaaaqabeaacqGHhis0caWGLbaaaOGaeyisISRaam4q amaaDaaaleaacaWGPbGaamOAaiaadUgacaWGSbaabaGaamyzaaaaki aadseadaqhaaWcbaGaam4AaiaadYgaaeaacaWGLbaaaaaa@426E@

 

 

 

3.9.4 Plastic constitutive law for finite strain viscoplasticity

 

Next, we turn to developing an appropriate plastic constitutive law for finite deformations.  The constitutive equations must specify a relationship between work conjugate measures of stress and strain MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  recall that τ ij L ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabes8a0naaBaaaleaacaWGPbGaamOAaa qabaGccaWGmbWaaSbaaSqaaiaadMgacaWGQbaabeaaaaa@371A@  is the rate of work done by stresses per unit reference volume.  Consequently, the constitutive equations must relate D p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8YjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaaCiramaaCaaaleqabaGaamiCaaaaaa a@32CC@ , W p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8YjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaaC4vamaaCaaaleqabaGaamiCaaaaaa a@32DF@  to τ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8YjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaaCiXdaaa@322D@  and its rate.

 

Usually, plastic constitutive laws for finite deformations are just simple extensions of small strain plasticity.  For example, for a finite strain, rate dependent, Mises solid with isotropic hardening power-law hardening we set

D ij p = ε ˙ 0 ( τ e σ 0 ) m 3 τ ij 2 τ e W ij p =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamiramaaDaaaleaacaWGPbGaamOAaa qaaiaadchaaaGccqGH9aqpcuaH1oqzgaGaamaaBaaaleaacaaIWaaa beaakmaabmaabaWaaSaaaeaacqaHepaDdaWgaaWcbaGaamyzaaqaba aakeaacqaHdpWCdaWgaaWcbaGaaGimaaqabaaaaaGccaGLOaGaayzk aaWaaWbaaSqabeaacaWGTbaaaOWaaSaaaeaacaaIZaGafqiXdqNbau aadaWgaaWcbaGaamyAaiaadQgaaeqaaaGcbaGaaGOmaiabes8a0naa BaaaleaacaWGLbaabeaaaaGccaaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaadE fadaqhaaWcbaGaamyAaiaadQgaaeaacaWGWbaaaOGaeyypa0JaaGim aaaa@6D94@

where τ ij = τ ij τ kk δ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqbes8a0zaafaWaaSbaaSqaaiaadMgaca WGQbaabeaakiabg2da9iabes8a0naaBaaaleaacaWGPbGaamOAaaqa baGccqGHsislcqaHepaDdaWgaaWcbaGaam4AaiaadUgaaeqaaOGaeq iTdq2aaSbaaSqaaiaadMgacaWGQbaabeaaaaa@41A0@  and 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@ .  The hardening rule is

σ 0 =Y ( 1+ ε e ε 0 ) 1/n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeq4Wdm3aaSbaaSqaaiaaicdaaeqaaO Gaeyypa0JaamywamaabmaabaGaaGymaiabgUcaRmaalaaabaGaeqyT du2aaSbaaSqaaiaadwgaaeqaaaGcbaGaeqyTdu2aaSbaaSqaaiaaic daaeqaaaaaaOGaayjkaiaawMcaamaaCaaaleqabaGaaGymaiaac+ca caWGUbaaaaaa@40A8@

where   ε 0 = ε ˙ 0 ( τ e / σ 0 ) m dt MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeqyTdu2aaSbaaSqaaiaaicdaaeqaaO Gaeyypa0Zaa8qaaeaacuaH1oqzgaGaamaaBaaaleaacaaIWaaabeaa kiaacIcacqaHepaDdaWgaaWcbaGaamyzaaqabaGccaGGVaGaeq4Wdm 3aaSbaaSqaaiaaicdaaeqaaOGaaiykamaaCaaaleqabaGaamyBaaaa kiaadsgacaWG0baaleqabeqdcqGHRiI8aaaa@43D6@

 

Finite strain plasticity models disagree on the correct way to prescribe W p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaaC4vamaaCaaaleqabaGaamiCaaaaaa a@32D1@ .  Many theories simply set W p =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaaC4vamaaCaaaleqabaGaamiCaaaaki abg2da9iaaicdaaaa@349B@ . Simple models of polycrystals give some support for this assumption, but it may not be appropriate in materials that develop a significant texture.  More complex models have also been developed.  For isotropically hardening solids, predictions are relatively insensitive to the choice of W p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaaC4vamaaCaaaleqabaGaamiCaaaaaa a@32D1@ , but any attempt to capture evolution of plastic anisotropy would need to specify W p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaaC4vamaaCaaaleqabaGaamiCaaaaaa a@32D1@  carefully.  Crystal plasticity based models provide a way out of this difficulty, because they have a clearer (but not completely unambiguous) definition of the plastic spin.