Problems for Chapter 3

 

Constitutive Models: Relations between Stress and Strain

 

 

3.5.  Hyperelasticity

 

 

3.5.1.      Derive the stress-strain relations for an incompressible, Neo-Hookean material subjected to

3.5.1.1.            Uniaxial tension

3.5.1.2.            Equibiaxial tension

3.5.1.3.            Pure shear

Derive expressions for the Cauchy stress, the Nominal stress, and the Material stress tensors (the solutions for nominal stress are listed in the table in Section 3.5.6).  You should use the following procedure: (i) assume that the specimen experiences the length changes listed in 3.5.6; (ii) use the formulas in Section 3.5.5 to compute the Cauchy stress, leaving the hydrostatic part of the stress p as an unknown; (iii) Determine the hydrostatic stress from the boundary conditions (e.g. for uniaxial tensile parallel to e 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahwgadaWgaaWcbaGaaGymaaqabaaaaa@323D@  you know σ 22 = σ 33 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaaIYaGaaGOmaa qabaGccqGH9aqpcqaHdpWCdaWgaaWcbaGaaG4maiaaiodaaeqaaOGa eyypa0JaaGimaaaa@3A12@ ; for equibiaxial tension or pure shear in the e 1 , e 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahwgadaWgaaWcbaGaaGymaaqabaGcca GGSaGaaCyzamaaBaaaleaacaaIYaaabeaaaaa@34CD@  plane you know that σ 33 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaaIZaGaaG4maa qabaGccqGH9aqpcaaIWaaaaa@359B@  )

 

 

3.5.2.      Repeat problem 3.5.1 for an incompressible Mooney-Rivlin material. 

 

 

3.5.3.      Repeat problem 3.5.1 for an incompressible Arruda-Boyce material

 

 

3.5.4.      Repeat problem 3.5.1 for an incompressible Ogden material.

 

 

3.5.5.      Using the results listed in the table in Section 3.5.6, and the material properties listed in Section 3.5.7, plot graphs showing the nominal stress as a function of stretch ratio λ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeU7aSbaa@321C@  for each of (a) a Neo-Hookean material; (b) a Mooney-Rivlin material; (c) the Arruda-Boyce material and (c) the Ogden material when subjected to uniaxial tension, biaxial tension, and pure shear (for the latter case, plot the largest tensile stress S 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadofadaWgaaWcbaGaaGymaaqabaaaaa@3227@  ).  

 

 

3.5.6.      A foam specimen is idealized as an Ogden-Storakers foam with strain energy density

U ˜ = 2μ α 2 ( λ 1 α + λ 2 α + λ 3 α 3+ 1 β ( J αβ 1) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGabmyvayaaiaGaeyypa0ZaaSaaaeaaca aIYaGaeqiVd0gabaGaeqySde2aa0baaSqaaaqaaiaaikdaaaaaaOWa aeWaaeaacqaH7oaBdaqhaaWcbaGaaGymaaqaaiabeg7aHbaakiabgU caRiabeU7aSnaaDaaaleaacaaIYaaabaGaeqySdegaaOGaey4kaSIa eq4UdW2aa0baaSqaaiaaiodaaeaacqaHXoqyaaGccqGHsislcaaIZa Gaey4kaSYaaSaaaeaacaaIXaaabaGaeqOSdigaaiaacIcacaWGkbWa aWbaaSqabeaacqGHsislcqaHXoqycqaHYoGyaaGccqGHsislcaaIXa GaaiykaaGaayjkaiaawMcaaaaa@553C@

where μ,α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeY7aTjaacYcacqaHXoqyaaa@346D@  and β MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabek7aIbaa@3209@  are material properties.  Calculate:

3.5.6.1.            The Cauchy stress in a specimen subjected to a pure volume change with principal stretches λ 1 = λ 2 = λ 3 =λ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeU7aSnaaBaaaleaacaaIXaaabeaaki abg2da9iabeU7aSnaaBaaaleaacaaIYaaabeaakiabg2da9iabeU7a SnaaBaaaleaacaaIZaaabeaakiabg2da9iabeU7aSbaa@3D20@

3.5.6.2.            The Cauchy stress in a specimen subjected to volume preserving uniaxial extension

3.5.6.3.            The Cauchy stress in a specimen subjected to uniaxial tension, as a function of the tensile stretch ratio λ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeU7aSnaaBaaaleaacaaIXaaabeaaaa a@3303@ . (To solve this problem you will need to assume that the solid is subjected to principal stretches λ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeU7aSnaaBaaaleaacaaIXaaabeaaaa a@3303@  parallel to e 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahwgadaWgaaWcbaGaaGymaaqabaaaaa@323D@ , and stretches λ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeU7aSnaaBaaaleaacaaIYaaabeaaaa a@3304@  parallel to e 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahwgadaWgaaWcbaGaaGOmaaqabaaaaa@323E@  and e 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahwgadaWgaaWcbaGaaG4maaqabaaaaa@323F@ .  You will need to determine λ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeU7aSnaaBaaaleaacaaIYaaabeaaaa a@3304@  from the condition that σ 22 = σ 33 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaaIYaGaaGOmaa qabaGccqGH9aqpcqaHdpWCdaWgaaWcbaGaaG4maiaaiodaaeqaaOGa eyypa0JaaGimaaaa@3A12@  in a uniaxial tensile test.

 

 

3.5.7.      Suppose that a hyperelastic solid is characterized by a strain energy density U ¯ ( I ¯ 1 , I ¯ 2 ,J) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaaceWGvbGbaebacaGGOaGabmysayaara WaaSbaaSqaaiaaigdaaeqaaOGaaiilaiqadMeagaqeamaaBaaaleaa caaIYaaabeaakiaacYcacaWGkbGaaiykaaaa@3AEF@  where

I ¯ 1 = B kk J 2/3 I ¯ 2 = 1 2 ( I ¯ 1 2 B ik B ki J 4/3 )J= detB MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaaceWGjbGbaebadaWgaaWcbaGaaGymaa qabaGccqGH9aqpdaWcaaqaaiaadkeadaWgaaWcbaGaam4AaiaadUga aeqaaaGcbaGaamOsamaaCaaaleqabaGaaGOmaiaac+cacaaIZaaaaa aakiaaykW7caaMc8UaaGPaVlaaykW7ceWGjbGbaebadaWgaaWcbaGa aGOmaaqabaGccqGH9aqpdaWcaaqaaiaaigdaaeaacaaIYaaaamaabm aabaGabmysayaaraWaa0baaSqaaiaaigdaaeaacaaIYaaaaOGaeyOe I0YaaSaaaeaacaWGcbWaaSbaaSqaaiaadMgacaWGRbaabeaakiaadk eadaWgaaWcbaGaam4AaiaadMgaaeqaaaGcbaGaamOsamaaCaaaleqa baGaaGinaiaac+cacaaIZaaaaaaaaOGaayjkaiaawMcaaiaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaamOsaiab g2da9maakaaabaGaciizaiaacwgacaGG0bGaaCOqaaWcbeaaaaa@6634@

are invariants of the Left Cauchy-Green deformation tensor B ij = F ik F jk MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGcbWaaSbaaSqaaiaadMgacaWGQb aabeaakiabg2da9iaadAeadaWgaaWcbaGaamyAaiaadUgaaeqaaOGa amOramaaBaaaleaacaWGQbGaam4Aaaqabaaaaa@3C5A@ Suppose that the solid is subjected to an infinitesimal strain, so that B can be approximated as B ij δ ij + ε ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadkeadaWgaaWcbaGaamyAaiaadQgaae qaaOGaeyisISRaeqiTdq2aaSbaaSqaaiaadMgacaWGQbaabeaakiab gUcaRiabew7aLnaaBaaaleaacaWGPbGaamOAaaqabaaaaa@3D3D@ , where ε ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew7aLnaaBaaaleaacaWGPbGaamOAaa qabaaaaa@3418@  is a symmetric infinitesimal strain tensor.   Linearize the constitutive equations for ε ij <<1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew7aLnaaBaaaleaacaWGPbGaamOAaa qabaGccqGH8aapcqGH8aapcaaIXaaaaa@36E5@ , and show that the relationship between Cauchy stress σ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGPbGaamOAaa qabaaaaa@3434@  and infinitesimal strain ε ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew7aLnaaBaaaleaacaWGPbGaamOAaa qabaaaaa@3418@  is equivalent to the isotropic linear elastic constitutive equation.   Give formulas for the bulk modulus and shear modulus for the equivalent solid in terms of the derivatives of U ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaaceWGvbGbaebaaaa@33B8@ .

 

 

3.5.8.      The constitutive law for a hyperelastic solid is derived from a strain energy potential U( I 1 , I 2 , I 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGvbGaaiikaiaadMeadaWgaaWcba GaaGymaaqabaGccaGGSaGaamysamaaBaaaleaacaaIYaaabeaakiaa cYcacaWGjbWaaSbaaSqaaiaaiodaaeqaaOGaaiykaaaa@3B99@ , where

I 1 =trace(B)= B kk I 2 = 1 2 ( I 1 2 BB )= 1 2 ( I 1 2 B ik B ki ) I 3 =detB= J 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGjbWaaSbaaSqaaiaaigdaaeqaaO Gaeyypa0JaaeiDaiaabkhacaqGHbGaae4yaiaabwgacaGGOaGaaCOq aiaacMcacqGH9aqpcaWGcbWaaSbaaSqaaiaadUgacaWGRbaabeaaki aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaamysamaaBaaaleaacaaI Yaaabeaakiabg2da9maalaaabaGaaGymaaqaaiaaikdaaaWaaeWaae aacaWGjbWaa0baaSqaaiaaigdaaeaacaaIYaaaaOGaeyOeI0IaaCOq aiabgwSixlabgwSixlaahkeaaiaawIcacaGLPaaacqGH9aqpdaWcaa qaaiaaigdaaeaacaaIYaaaamaabmaabaGaamysamaaDaaaleaacaaI XaaabaGaaGOmaaaakiabgkHiTiaadkeadaWgaaWcbaGaamyAaiaadU gaaeqaaOGaamOqamaaBaaaleaacaWGRbGaamyAaaqabaaakiaawIca caGLPaaacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWGjb WaaSbaaSqaaiaaiodaaeqaaOGaeyypa0JaciizaiaacwgacaGG0bGa aCOqaiabg2da9iaadQeadaahaaWcbeqaaiaaikdaaaaaaa@76E2@

are the invariants of the Left Cauchy-Green deformation tensor B=F F T B ij = F ik F jk MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHcbGaeyypa0JaaCOraiabgwSixl aahAeadaahaaWcbeqaaiaadsfaaaGccaaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaamOqamaaBaaaleaacaWGPbGaamOAaa qabaGccqGH9aqpcaWGgbWaaSbaaSqaaiaadMgacaWGRbaabeaakiaa dAeadaWgaaWcbaGaamOAaiaadUgaaeqaaaaa@4DF0@ .

3.5.8.1.            Calculate the Cauchy stress induced in the solid when it is subjected to a rigid rotation, followed by an arbitrary homogeneous deformation.   Hence, demonstrate that the constitutive law is isotropic.

3.5.8.2.            Apply the simple check described in Section 3.1 to test whether the constitutive law is objective.

 

 


3.5.9.      The strain energy density of a hyperelastic solid is sometimes specified as a function of the right Cauchy-Green deformation tensor C ij = F ki F kj MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadoeadaWgaaWcbaGaamyAaiaadQgaae qaaOGaeyypa0JaamOramaaBaaaleaacaWGRbGaamyAaaqabaGccaWG gbWaaSbaaSqaaiaadUgacaWGQbaabeaaaaa@39FE@ , instead of  B ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadkeadaWgaaWcbaGaamyAaiaadQgaae qaaaaa@3338@  as described in Section 3.5.  (This procedure must be used if the material is anisotropic, for example)

3.5.9.1.            Suppose that the strain energy density has the general form W( C ij ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadEfacaGGOaGaam4qamaaBaaaleaaca WGPbGaamOAaaqabaGccaGGPaaaaa@3578@ .  Derive formulas for the Material stress, Nominal stress and Cauchy stress in the solid as functions of F ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAeadaWgaaWcbaGaamyAaiaadQgaae qaaaaa@333C@ , C ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadoeadaWgaaWcbaGaamyAaiaadQgaae qaaaaa@3339@  and W/ C ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgkGi2kaadEfacaGGVaGaeyOaIyRaam 4qamaaBaaaleaacaWGPbGaamOAaaqabaaaaa@3794@

3.5.9.2.            Apply the simple check described in Section 3.1 to demonstrate that the resulting stress-strain relation is objective.

3.5.9.3.            Calculate the Cauchy stress induced in the solid when it is subjected to a rigid rotation, followed by an arbitrary homogeneous deformation.   Hence, demonstrate that the constitutive law is not, in general, isotropic.

3.5.9.4.            Suppose that the constitutive law is simplified further by writing the strain energy density as a function of the invariants of C, i.e.  W( C ij )=U( I 1 , I 2 , I 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadEfacaGGOaGaam4qamaaBaaaleaaca WGPbGaamOAaaqabaGccaGGPaGaeyypa0JaamyvaiaacIcacaWGjbWa aSbaaSqaaiaaigdaaeqaaOGaaiilaiaadMeadaWgaaWcbaGaaGOmaa qabaGccaGGSaGaamysamaaBaaaleaacaaIZaaabeaakiaacMcaaaa@3F51@ , where

I 1 = C kk I 2 = 1 2 ( I 1 2 C ik C ki ) I 3 =detC= J 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGjbWaaSbaaSqaaiaaigdaaeqaaO Gaeyypa0Jaam4qamaaBaaaleaacaWGRbGaam4AaaqabaGccaaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaadMeadaWgaaWcbaGaaGOmaaqaba GccqGH9aqpdaWcaaqaaiaaigdaaeaacaaIYaaaamaabmaabaGaamys amaaDaaaleaacaaIXaaabaGaaGOmaaaakiabgkHiTiaadoeadaWgaa WcbaGaamyAaiaadUgaaeqaaOGaam4qamaaBaaaleaacaWGRbGaamyA aaqabaaakiaawIcacaGLPaaacaaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caWGjbWaaSbaaSqaaiaaiodaaeqaaOGaeyypa0Jaciiz aiaacwgacaGG0bGaaC4qaiabg2da9iaadQeadaahaaWcbeqaaiaaik daaaaaaa@6175@

Derive expressions relating the Cauchy stress components to U/ I j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgkGi2kaadwfacaGGVaGaeyOaIyRaam ysamaaBaaaleaacaWGQbaabeaaaaa@36AA@ .

3.5.9.5.            Demonstrate that the simplified constitutive law described in 3.5.9.4 characterizes an isotropic solid.