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

 

 

 

3.4 Generalized Hooke’s law MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqabKqzaiaeaa aaaaaaa8qacaWFtacaaa@3788@   elastic materials subjected to small stretches but large rotations

 

Recall that the stress-strain law for an anisotropic, linear elastic material (Sect 3.1) has the form

σ ij = C ijkl ε kl MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamyAaiaadQ gaaeqaaOGaeyypa0Jaam4qamaaBaaaleaacaWGPbGaamOAaiaadUga caWGSbaabeaakiabew7aLnaaBaaaleaacaWGRbGaamiBaaqabaaaaa@4012@

where σ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamyAaiaadQ gaaeqaaaaa@3692@  is stress (any stress measure you like), ε ij =( u i / x j + u j / x i )/2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaWgaaWcbaGaamyAaiaadQ gaaeqaaOGaeyypa0ZaaeWaaeaacqGHciITcaWG1bWaaSbaaSqaaiaa dMgaaeqaaOGaai4laiabgkGi2kaadIhadaWgaaWcbaGaamOAaaqaba GccqGHRaWkcqGHciITcaWG1bWaaSbaaSqaaiaadQgaaeqaaOGaai4l aiabgkGi2kaadIhadaWgaaWcbaGaamyAaaqabaaakiaawIcacaGLPa aacaGGVaGaaGOmaaaa@4ADE@  is the infinitesimal strain, and C ijkl MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGdbWaaSbaaSqaaiaadMgacaWGQb Gaam4AaiaadYgaaeqaaaaa@3778@  is the tensor of elastic moduli.

 

This stress-strain relation can only be used if the material is subjected to small deformations, and small rotations.  This is partly because the infinitesimal strain ε ij 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaWgaaWcbaGaamyAaiaadQ gaaeqaaOGaeyiyIKRaaGimaaaa@3901@  for a finite rotation MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  so the law predicts that a nonzero stress is required to rotate a solid.

 

There are some situations where a solid is subjected to small shape changes, but large rotations.  For example, the picture shows a long slender beam bent into a circle by moments applied to its ends.  The strains in the beam are of order h/R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGObGaai4laiaadkfaaaa@353D@ , where h is the thickness of the beam and R is its curvature.   The ends of the beam have rotated through a full 90 degrees, however.   The linear elastic constitutive equations would not predict the correct stress in the beam.

 

 It is easy to fix this problem: provided we choose a sensible (nonlinear) strain measure, together with the appropriate work-conjugate stress measure, we can still use a linear stress-strain relation.  To make this precise, suppose that a 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@ .  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 Lagrange strain

E ij = 1 2 ( F ki F kj δ ij )= 1 2 ( u i x j + u j x i + u k x i u k x j ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamyramaaBaaaleaacaWGPbGaamOAaa qabaGccqGH9aqpdaWcaaqaaiaaigdaaeaacaaIYaaaaiaacIcacaWG gbWaaSbaaSqaaiaadUgacaWGPbaabeaakiaadAeadaWgaaWcbaGaam 4AaiaadQgaaeqaaOGaeyOeI0IaeqiTdq2aaSbaaSqaaiaadMgacaWG QbaabeaakiaacMcacaaMc8UaaGPaVlabg2da9maalaaabaGaaGymaa qaaiaaikdaaaWaaeWaaeaadaWcaaqaaiabgkGi2kaadwhadaWgaaWc baGaamyAaaqabaaakeaacqGHciITcaWG4bWaaSbaaSqaaiaadQgaae qaaaaakiabgUcaRmaalaaabaGaeyOaIyRaamyDamaaBaaaleaacaWG QbaabeaaaOqaaiabgkGi2kaadIhadaWgaaWcbaGaamyAaaqabaaaaO Gaey4kaSYaaSaaaeaacqGHciITcaWG1bWaaSbaaSqaaiaadUgaaeqa aaGcbaGaeyOaIyRaamiEamaaBaaaleaacaWGPbaabeaaaaGcdaWcaa qaaiabgkGi2kaadwhadaWgaaWcbaGaam4AaaqabaaakeaacqGHciIT caWG4bWaaSbaaSqaaiaadQgaaeqaaaaaaOGaayjkaiaawMcaaaaa@6759@

* The Eulerian strain E ij * = 1 2 ( δ ij F ki 1 F kj 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaaMc8UaaGPaVlaaykW7caWGfbWaa0 baaSqaaiaadMgacaWGQbaabaGaaiOkaaaakiabg2da9maalaaabaGa aGymaaqaaiaaikdaaaGaaiikaiabes7aKnaaBaaaleaacaWGPbGaam OAaaqabaGccqGHsislcaWGgbWaa0baaSqaaiaadUgacaWGPbaabaGa eyOeI0IaaGymaaaakiaadAeadaqhaaWcbaGaam4AaiaadQgaaeaacq GHsislcaaIXaaaaOGaaiykaaaa@4C8E@

* The rotation tensor (see 2.1.13 for the best way to compute R in practice)

R ij = F ik U kj 1 U= ( F T F ) 1/2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGsbWaaSbaaSqaaiaadMgacaWGQb aabeaakiabg2da9iaadAeadaWgaaWcbaGaamyAaiaadUgaaeqaaOGa amyvamaaDaaaleaacaWGRbGaamOAaaqaaiabgkHiTiaaigdaaaGcca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaCyvaiabg2da9maabmaabaGaaCOramaaCaaaleqaba GaamivaaaakiaahAeaaiaawIcacaGLPaaadaahaaWcbeqaaiaaigda caGGVaGaaGOmaaaaaaa@560D@

* The Cauchy (“true”) stress σ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamyAaiaadQ gaaeqaaaaa@3692@ , defined so that  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@

* The Material stress (work conjugate to Lagrange strain)  Σ ij =J F ik 1 σ kl F jl 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeu4Odm1aaSbaaSqaaiaadMgacaWGQb aabeaakiabg2da9iaadQeacaWGgbWaa0baaSqaaiaadMgacaWGRbaa baGaeyOeI0IaaGymaaaakiabeo8aZnaaBaaaleaacaWGRbGaamiBaa qabaGccaWGgbWaa0baaSqaaiaadQgacaWGSbaabaGaeyOeI0IaaGym aaaaaaa@431C@

 

The Material stress-Lagrange strain relation can be expressed as

Σ ij = C ijkl E kl MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqqHJoWudaWgaaWcbaGaamyAaiaadQ gaaeqaaOGaeyypa0Jaam4qamaaBaaaleaacaWGPbGaamOAaiaadUga caWGSbaabeaakiaadweadaWgaaWcbaGaam4AaiaadYgaaeqaaaaa@3EF6@

where C ijkl MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGdbWaaSbaaSqaaiaadMgacaWGQb Gaam4AaiaadYgaaeqaaaaa@3778@  is the tensor of elastic moduli for the material with orientation in the undeformed configuration.  This is identical to the stress-strain relation for a linear elastic solid, except that the stress measure has been replaced by Material stress, and the strain measure has been replaced by Lagrange strain. You can therefore use all the matrix representations and tables of data given in Section 3.1 to apply the constitutive equation. The Cauchy (“true”) stress can be computed from the material stress as

σ ij = 1 J F ik Σ kl F jl R ik Σ kl R jl MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamyAaiaadQ gaaeqaaOGaeyypa0ZaaSaaaeaacaaIXaaabaGaamOsaaaacaWGgbWa aSbaaSqaaiaadMgacaWGRbaabeaakiabfo6atnaaBaaaleaacaWGRb GaamiBaaqabaGccaWGgbWaaSbaaSqaaiaadQgacaWGSbaabeaakiab gIKi7kaadkfadaWgaaWcbaGaamyAaiaadUgaaeqaaOGaeu4Odm1aaS baaSqaaiaadUgacaWGSbaabeaakiaadkfadaWgaaWcbaGaamOAaiaa dYgaaeqaaaaa@4DB1@

 

The Cauchy stress-Eulerian strain relation: Alternatively, the stress-strain relation can be expressed in terms of stress and deformation measures that characterize the deformed solid, as

σ ij = C ijkl * E kl * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamyAaiaadQ gaaeqaaOGaeyypa0Jaam4qamaaDaaaleaacaWGPbGaamOAaiaadUga caWGSbaabaGaaiOkaaaakiaadweadaqhaaWcbaGaam4AaiaadYgaae aacaGGQaaaaaaa@4093@

where C ijkl * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGdbWaa0baaSqaaiaadMgacaWGQb Gaam4AaiaadYgaaeaacaGGQaaaaaaa@3827@  is the tensor of elastic moduli for the material with orientation of the deformed configuration.  This tensor is related to C ijkl MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGdbWaaSbaaSqaaiaadMgacaWGQb Gaam4AaiaadYgaaeqaaaaa@3778@  by

C ijmn * = 1 J F ip F jq C pqkl F mk F nl R ip R jq C pqkl R mk R nl MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGdbWaa0baaSqaaiaadMgacaWGQb GaamyBaiaad6gaaeaacaGGQaaaaOGaeyypa0ZaaSaaaeaacaaIXaaa baGaamOsaaaacaWGgbWaaSbaaSqaaiaadMgacaWGWbaabeaakiaadA eadaWgaaWcbaGaamOAaiaadghaaeqaaOGaam4qamaaBaaaleaacaWG WbGaamyCaiaadUgacaWGSbaabeaakiaadAeadaWgaaWcbaGaamyBai aadUgaaeqaaOGaamOramaaBaaaleaacaWGUbGaamiBaaqabaGccqGH ijYUcaWGsbWaaSbaaSqaaiaadMgacaWGWbaabeaakiaadkfadaWgaa WcbaGaamOAaiaadghaaeqaaOGaam4qamaaBaaaleaacaWGWbGaamyC aiaadUgacaWGSbaabeaakiaadkfadaWgaaWcbaGaamyBaiaadUgaae qaaOGaamOuamaaBaaaleaacaWGUbGaamiBaaqabaaaaa@5D64@

 

For the special 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@

Σ ij = E 1+ν { E ij + ν 12ν E kk δ ij } σ ij = E 1+ν { E ij * + ν 12ν E kk * δ ij } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfo6atnaaBaaaleaacaWGPbGaamOAaa qabaGccqGH9aqpdaWcaaqaaiaadweaaeaacaaIXaGaey4kaSIaeqyV d4gaamaacmaabaGaamyramaaBaaaleaacaWGPbGaamOAaaqabaGccq GHRaWkdaWcaaqaaiabe27aUbqaaiaaigdacqGHsislcaaIYaGaeqyV d4gaaiaadweadaWgaaWcbaGaam4AaiaadUgaaeqaaOGaeqiTdq2aaS baaSqaaiaadMgacaWGQbaabeaaaOGaay5Eaiaaw2haaiaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqaHdpWCdaWg aaWcbaGaamyAaiaadQgaaeqaaOGaeyypa0ZaaSaaaeaacaWGfbaaba GaaGymaiabgUcaRiabe27aUbaadaGadaqaaiaadweadaqhaaWcbaGa amyAaiaadQgaaeaacaGGQaaaaOGaey4kaSYaaSaaaeaacqaH9oGBae aacaaIXaGaeyOeI0IaaGOmaiabe27aUbaacaWGfbWaa0baaSqaaiaa dUgacaWGRbaabaGaaiOkaaaakiabes7aKnaaBaaaleaacaWGPbGaam OAaaqabaaakiaawUhacaGL9baaaaa@8124@