Problems for Chapter 3

 

Constitutive Models: Relations between Stress and Strain

 

 

3.9.  Viscoelasticity

 

3.9.1.      The figure shows a thin film of material that is deformed plastically during a pressure-shear plate impact experiment.  The goal of this problem is to derive the equations governing the velocity and stress fields in the specimen.  Assume that:

  • The film deforms in simple shear, and that the velocity v=v( y 2 ,t) e 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH2bGaeyypa0JaamODaiaacIcaca WG5bWaaSbaaSqaaiaaikdaaeqaaOGaaiilaiaadshacaGGPaGaaCyz amaaBaaaleaacaaIXaaabeaaaaa@3C8D@  and Kirchoff stress fields τ=q( y 2 ,t)( e 1 e 2 + e 2 e 1 )+( τ 11 e 1 e 1 + τ 22 e 2 e 2 + τ 33 e 3 e 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHepGaeyypa0JaamyCaiaacIcaca WG5bWaaSbaaSqaaiaaikdaaeqaaOGaaiilaiaadshacaGGPaGaaiik aiaahwgadaWgaaWcbaGaaGymaaqabaGccqGHxkcXcaWHLbWaaSbaaS qaaiaaikdaaeqaaOGaey4kaSIaaCyzamaaBaaaleaacaaIYaaabeaa kiabgEPielaahwgadaWgaaWcbaGaaGymaaqabaGccaGGPaGaey4kaS Iaaiikaiabes8a0naaBaaaleaacaaIXaGaaGymaaqabaGccaWHLbWa aSbaaSqaaiaaigdaaeqaaOGaey4LIqSaaCyzamaaBaaaleaacaaIXa aabeaakiabgUcaRiabes8a0naaBaaaleaacaaIYaGaaGOmaaqabaGc caWHLbWaaSbaaSqaaiaaikdaaeqaaOGaey4LIqSaaCyzamaaBaaale aacaaIYaaabeaakiabgUcaRiabes8a0naaBaaaleaacaaIZaGaaG4m aaqabaGccaWHLbWaaSbaaSqaaiaaiodaaeqaaOGaey4LIqSaaCyzam aaBaaaleaacaaIZaaabeaakiaacMcaaaa@6882@  are independent of x 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG4bWaaSbaaSqaaiaaigdaaeqaaa aa@34AA@
  • The material has mass density ρ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHbpGCaaa@3486@  and isotropic elastic response, with shear modulus μ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH8oqBaaa@347C@  and Poisson’s ratio ν MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH9oGBaaa@347E@
  • The film can be idealized as a finite strain viscoplastic solid with power-law Mises flow potential, as described in Section 3.9.  Assume that the plastic spin is zero.

3.9.1.1.            Calculate the velocity gradient tensor L, the stretch rate tensor D and spin tensor W for the deformation, expressing your answer as components in the e 1 , e 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHLbWaaSbaaSqaaiaaigdaaeqaaO GaaiilaiaahwgadaWgaaWcbaGaaGOmaaqabaaaaa@372B@  basis shown in the figure

3.9.1.2.            Find an expression for the plastic stretch rate, in terms of the stress and material properties

3.9.1.3.            Use the elastic stress rate-stretch rate relation τ ij = C ijkl D kl e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWfGaqaaiabes8a0naaBaaaleaaca WGPbGaamOAaaqabaaabeqaaiabgEGirdaakiabg2da9iaadoeadaWg aaWcbaGaamyAaiaadQgacaWGRbGaamiBaaqabaGccaWGebWaa0baaS qaaiaadUgacaWGSbaabaGaamyzaaaaaaa@41E5@   to obtain an expression for the time derivative of the shear stress q and the stress components τ 11 , τ 22 , τ 33 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabes8a0naaBaaaleaacaaIXaGaaGymaa qabaGccaGGSaGaeqiXdq3aaSbaaSqaaiaaikdacaaIYaaabeaakiaa cYcacqaHepaDdaWgaaWcbaGaaG4maiaaiodaaeqaaaaa@3C17@  in terms of v( x 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG2bGaaiikaiaadIhadaWgaaWcba GaaGOmaaqabaGccaGGPaaaaa@3709@ , τ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHepaaaa@3416@  and appropriate material properties

3.9.1.4.            Write down the linear momentum balance equation in terms of τ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHepaaaa@3416@  and v MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG2baaaa@33C1@ .

3.9.1.5.            How would the governing equations change if W p =W MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHxbWaaWbaaSqabeaacaWGWbaaaO Gaeyypa0JaaC4vaaaa@36B8@ ?