Problems for Chapter 3

 

Constitutive Models: Relations between Stress and Strain

 

 

3.10.    Large Strain Viscoelasticity

 

 

3.10.1.  A cylindrical specimen is made from a material that can be idealized using the finite-strain viscoelasticity model described in Section 3.10.   The specimen may be approximated as incompressible.

3.10.1.1.        Let L denote the length of the deformed specimen, and L 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadYeadaWgaaWcbaGaaGimaaqabaaaaa@321F@  denote the initial length of the specimen.  Write down the deformation gradient in the specimen in terms of λ=L/ L 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeU7aSjabg2da9iaadYeacaGGVaGaam itamaaBaaaleaacaaIWaaabeaaaaa@365D@

3.10.1.2.        Let λ= λ e λ p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeU7aSjabg2da9iabeU7aSnaaBaaale aacaWGLbaabeaakiabeU7aSnaaBaaaleaacaWGWbaabeaaaaa@38CB@  denote the decomposition of stretch in to elastic and plastic parts.  Write down the elastic and plastic parts of the deformation gradient in terms of λ e , λ p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeU7aSnaaBaaaleaacaWGLbaabeaaki aacYcacqaH7oaBdaWgaaWcbaGaamiCaaqabaaaaa@36C1@  and find expressions for the elastic and plastic parts of the stretch rate in terms of λ ˙ e , λ ˙ p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqbeU7aSzaacaWaaSbaaSqaaiaadwgaae qaaOGaaiilaiqbeU7aSzaacaWaaSbaaSqaaiaadchaaeqaaaaa@36D3@

3.10.1.3.        Assume that the material can be idealized using Arruda-Boyce potentials

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@

Obtain an expression for the stress in the specimen in terms of λ e , λ p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeU7aSnaaBaaaleaacaWGLbaabeaaki aacYcacqaH7oaBdaWgaaWcbaGaamiCaaqabaaaaa@36C1@ , using only the first two term in the expansion for simplicity.  Your answer should include an indeterminate hydrostatic part.

3.10.1.4.        Calculate the deviatoric stress measure

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@

in terms of λ e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeU7aSnaaBaaaleaacaWGLbaabeaaaa a@3332@ , and hence find an expression for λ ˙ p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqbeU7aSzaacaWaaSbaaSqaaiaadchaae qaaaaa@3346@  in terms of λ e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeU7aSnaaBaaaleaacaWGLbaabeaaaa a@3332@

3.10.1.5.        Suppose that the specimen is subjected to a harmonic cycle of nominal strain such that L=α L 0 sinωt MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadYeacqGH9aqpcqaHXoqycaWGmbWaaS baaSqaaiaaicdaaeqaaOGaci4CaiaacMgacaGGUbGaeqyYdCNaamiD aaaa@3B3D@ .  Use the results of 3.10.1.2 and 3.10.1.4 to obtain a nonlinear differential equation for λ e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeU7aSnaaBaaaleaacaWGLbaabeaaaa a@3332@

3.10.1.6.        Use the material data given in Section 3.10.5 to calculate (numerically) the variation of Cauchy stress in the solid with time induced by cyclic straining.  Plot the results as a curve of Cauchy stress as a function of true strain.  Obtain results for various values of α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg7aHbaa@3207@  and frequency ω MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeM8a3baa@3235@ .