Chapter 8

 

Theory and Implementation of the Finite Element Method

 

 

 

8.4 Finite element method for large deformations: hyperelastic materials
 

The finite element method can be used to solve problems involving large shape changes.  In this section, we show how to do this, using a solid made from a hyperelastic material as an example.


8.4.1 Summary of governing equations
 
To keep things as simple as possible we will devise a method to model a hyperelastic solid with a Neo-Hookean constitutive law as discussed in Section 3.4.

Given:

1.      The shape of the solid in its unloaded condition R 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamOuamaaBaaaleaacaaIWaaabeaaaa a@328C@  (this will be taken as the stress free reference configuration)

2.      A body force distribution b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8XjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaaCOyaaaa@31B8@  acting on the solid (Note that in this section we return to using b to denote force per unit mass)

3.      Boundary conditions, specifying displacements u * (x) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaaCyDamaaCaaaleqabaGaaiOkaaaaki aacIcacaWH4bGaaiykaaaa@350C@  on a portion 1 R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeyOaIy7aaSbaaSqaaiaaigdaaeqaaO GaamOuaaaa@33FD@  or tractions t * (x) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaaCiDamaaCaaaleqabaGaaiOkaaaaki aacIcacaWH4bGaaiykaaaa@350B@  on a portion 2 R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeyOaIy7aaSbaaSqaaiaaikdaaeqaaO GaamOuaaaa@33FE@  of the boundary of  the deformed solid (note that tractions are specified as force per unit deformed area MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  but we could also specify the tractions t 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaaCiDamaaCaaaleqabaGaaGimaaaaaa a@32B3@  per unit undeformed area acting on 2 R 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeyOaIy7aaSbaaSqaaiaaikdaaeqaaO GaamOuamaaBaaaleaacaaIWaaabeaaaaa@34E4@  if this is more convenient);

4.      The material constants μ 1 , K 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeqiVd02aaSbaaSqaaiaaigdaaeqaaO GaaiilaiaadUeadaWgaaWcbaGaaGymaaqabaaaaa@35DD@  for the Neo-Hookean constitutive law described in Section 3.4.5;

5.      The mass density of the solid in its reference configuration ρ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeqyWdi3aaSbaaSqaaiaaicdaaeqaaa aa@3375@

 

Calculate displacements u i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamyDamaaBaaaleaacaWGPbaabeaaaa a@32E3@ , deformation gradient tensor F ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAeadaWgaaWcbaGaamyAaiaadQgaae qaaaaa@333C@  and Cauchy stresses σ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeq4Wdm3aaSbaaSqaaiaadMgacaWGQb aabeaaaaa@349B@  satisfying the governing equations and boundary conditions

y i = x i + u i ( x k ) F ij = δ ij + u i x j J=det(F), B ij = F ik F jk σ ij y i +ρ b j =0 u i = u i * on 1 R σ ij n i = t j * on 2 R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGceaqabeaacaWG5bWaaSbaaSqaaiaadMgaae qaaOGaeyypa0JaamiEamaaBaaaleaacaWGPbaabeaakiabgUcaRiaa dwhadaWgaaWcbaGaamyAaaqabaGccaGGOaGaamiEamaaBaaaleaaca WGRbaabeaakiaacMcacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaamOramaaBaaaleaacaWGPb GaamOAaaqabaGccqGH9aqpcqaH0oazdaWgaaWcbaGaamyAaiaadQga aeqaaOGaey4kaSYaaSaaaeaacqGHciITcaWG1bWaaSbaaSqaaiaadM gaaeqaaaGcbaGaeyOaIyRaamiEamaaBaaaleaacaWGQbaabeaaaaGc caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caWGkbGaeyypa0JaciizaiaacwgacaGG0bGaaiikaiaahAea caGGPaGaaiilaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaamOqamaaBaaaleaacaWGPbGaamOAaaqabaGccqGH 9aqpcaWGgbWaaSbaaSqaaiaadMgacaWGRbaabeaakiaadAeadaWgaa WcbaGaamOAaiaadUgaaeqaaaGcbaWaaSaaaeaacqGHciITcqaHdpWC daWgaaWcbaGaamyAaiaadQgaaeqaaaGcbaGaeyOaIyRaamyEamaaBa aaleaacaWGPbaabeaaaaGccqGHRaWkcqaHbpGCcaWGIbWaaSbaaSqa aiaadQgaaeqaaOGaeyypa0JaaGimaiaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaamyD amaaBaaaleaacaWGPbaabeaakiabg2da9iaadwhadaqhaaWcbaGaam yAaaqaaiaacQcaaaGccaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa b+gacaqGUbGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqGHciITda WgaaWcbaGaaGymaaqabaGccaWGsbGaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlabeo8aZnaaBaaaleaacaWGPbGa amOAaaqabaGccaWGUbWaaSbaaSqaaiaadMgaaeqaaOGaeyypa0Jaam iDamaaDaaaleaacaWGQbaabaGaaiOkaaaakiaaykW7caaMc8UaaGPa VlaaykW7caaMc8Uaae4Baiaab6gacaaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlabgkGi2oaaBaaaleaacaaIYaaabeaakiaadkfaaaaa@F890@

with Cauchy stress related to left Cauchy-Green tensor through the neo-Hookean constitutive law

σ ij = μ 1 J 5/3 ( B ij 1 3 B kk δ ij )+ K 1 ( J1 ) δ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeq4Wdm3aaSbaaSqaaiaadMgacaWGQb aabeaakiabg2da9maalaaabaGaeqiVd02aaSbaaSqaaiaaigdaaeqa aaGcbaGaamOsamaaCaaaleqabaGaaGynaiaac+cacaaIZaaaaaaakm aabmaabaGaamOqamaaBaaaleaacaWGPbGaamOAaaqabaGccqGHsisl daWcaaqaaiaaigdaaeaacaaIZaaaaiaadkeadaWgaaWcbaGaam4Aai aadUgaaeqaaOGaeqiTdq2aaSbaaSqaaiaadMgacaWGQbaabeaaaOGa ayjkaiaawMcaaiabgUcaRiaadUeadaWgaaWcbaGaaGymaaqabaGcda qadaqaaiaadQeacqGHsislcaaIXaaacaGLOaGaayzkaaGaeqiTdq2a aSbaaSqaaiaadMgacaWGQbaabeaaaaa@5355@

 

 

8.4.2 Governing equations in terms of the principle of virtual work

 

As always the stress equilibrium equation is replaced by the equivalent principle of virtual work MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  which now has to be in a form appropriate for finite deformations.  The virtual work equation is given in terms of various stress and deformation measures in Section 2.4.5.  For our purposes, a slightly modified form of the version in terms of Kirchhoff stress is the most convenient.  This states that

R 0 τ ij δ L ij d V 0 R 0 ρ 0 b i δ v i d V 0 R t i * δ v i dA =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaWaa8quaeaacqaHepaDdaWgaaWcbaGaam yAaiaadQgaaeqaaOGaeqiTdqMaamitamaaBaaaleaacaWGPbGaamOA aaqabaGccaWGKbGaamOvamaaBaaaleaacaaIWaaabeaakiabgkHiTm aapefabaGaeqyWdi3aaSbaaSqaaiaaicdaaeqaaOGaamOyamaaBaaa leaacaWGPbaabeaakiabes7aKjaadAhadaWgaaWcbaGaamyAaaqaba GccaWGKbGaamOvamaaBaaaleaacaaIWaaabeaakiabgkHiTmaapefa baGaamiDamaaDaaaleaacaWGPbaabaGaaiOkaaaakiabes7aKjaadA hadaWgaaWcbaGaamyAaaqabaGccaWGKbGaamyqaaWcbaGaeyOaIyRa amOuaaqab0Gaey4kIipakiabg2da9iaaicdaaSqaaiaadkfadaWgaa adbaGaaGimaaqabaaaleqaniabgUIiYdaaleaacaWGsbWaaSbaaWqa aiaaicdaaeqaaaWcbeqdcqGHRiI8aaaa@5FA7@

for all virtual velocity fields δ v i ( x i ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeqiTdqMaamODamaaBaaaleaacaWGPb aabeaakiaacIcacaWG4bWaaSbaaSqaaiaadMgaaeqaaOGaaiykaaaa @380D@  and virtual velocity gradients δ L ij = v i / y j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeqiTdqMaamitamaaBaaaleaacaWGPb GaamOAaaqabaGccqGH9aqpcqGHciITcaWG2bWaaSbaaSqaaiaadMga aeqaaOGaai4laiabgkGi2kaadMhadaWgaaWcbaGaamOAaaqabaaaaa@3E15@  that satisfy δ v i =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeqiTdqMaamODamaaBaaaleaacaWGPb aabeaakiabg2da9iaaicdaaaa@3653@  on 1 R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeyOaIy7aaSbaaSqaaiaaigdaaeqaaO GaamOuaaaa@33FD@ .  Here τ ij =J σ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeqiXdq3aaSbaaSqaaiaadMgacaWGQb aabeaakiabg2da9iaadQeacqaHdpWCdaWgaaWcbaGaamyAaiaadQga aeqaaaaa@3A48@  is the Kirchhoff stress.  Some notes on this equation

1.      The volume integrals in the virtual work equation are taken over the reference configuration MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  this is convenient, because in a real problem we can take the given initial shape of the solid as reference, whereas the deformed configuration is unknown.

2.      The area integral is taken over the deformed configuration, but can be mapped back to the reference configuration by computing the inverse surface Jacobian η=dA/d A 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeq4TdGMaeyypa0Jaamizaiaadgeaca GGVaGaamizaiaadgeadaWgaaWcbaGaaGimaaqabaaaaa@3878@ .  One way (although not the best way in practice) to calculate η MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8XjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeq4TdGgaaa@3279@  would be through the relationship

n dA=Jm F 1 d A 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqabiaa=5gaca WFGaGaamizaiaadgeacqGH9aqpcaWGkbGaaCyBaiabgwSixlaa=zea daahaaWcbeqaaiabgkHiTiaaigdaaaGccaWGKbGaamyqamaaBaaale aacaaIWaaabeaaaaa@42A0@

where m is the normal to the surface in the reference configuration, and n is the normal to the surface in the deformed configuration.  Taking the magnitude of both sides gives

η=J m i F ik 1 F jk 1 m j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeq4TdGMaeyypa0JaamOsamaakaaaba GaamyBamaaBaaaleaacaWGPbaabeaakiaadAeadaWgaaWcbaGaamyA aiaadUgaaeqaaOWaaWbaaSqabeaacqGHsislcaaIXaaaaOGaamOram aaBaaaleaacaWGQbGaam4AaaqabaGcdaahaaWcbeqaaiabgkHiTiaa igdaaaGccaWGTbWaaSbaaSqaaiaadQgaaeqaaaqabaaaaa@4200@

Then the virtual work equation becomes

V 0 τ ij δ L ij d V 0 V 0 ρ 0 b i δ v i d V 0 2 V 0 t i * δ v i ηd A 0 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaWaa8quaeaacqaHepaDdaWgaaWcbaGaam yAaiaadQgaaeqaaOGaeqiTdqMaamitamaaBaaaleaacaWGPbGaamOA aaqabaGccaWGKbGaamOvamaaBaaaleaacaaIWaaabeaakiabgkHiTm aapefabaGaeqyWdi3aaSbaaSqaaiaaicdaaeqaaOGaamOyamaaBaaa leaacaWGPbaabeaakiabes7aKjaadAhadaWgaaWcbaGaamyAaaqaba GccaWGKbGaamOvamaaBaaaleaacaaIWaaabeaakiabgkHiTmaapefa baGaamiDamaaDaaaleaacaWGPbaabaGaaiOkaaaakiabes7aKjaadA hadaWgaaWcbaGaamyAaaqabaGccqaH3oaAcaWGKbGaamyqamaaBaaa leaacaaIWaaabeaaaeaacqGHciITdaWgaaadbaGaaGOmaaqabaWcca WGwbWaaSbaaWqaaiaaicdaaeqaaaWcbeqdcqGHRiI8aOGaeyypa0Ja aGimaaWcbaGaamOvamaaBaaameaacaaIWaaabeaaaSqab0Gaey4kIi paaSqaaiaadAfadaWgaaadbaGaaGimaaqabaaaleqaniabgUIiYdaa aa@6420@

 

 

8.4.3 Finite element equations

 

The finite element solution follows almost exactly the same procedure as before.  We first discretize the displacement field, by choosing to calculate the displacement field at a set of n nodes.  We will denote the coordinates of these special points in the reference configuration by x i a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamiEamaaDaaaleaacaWGPbaabaGaam yyaaaaaaa@33CD@ , where the superscript a ranges from 1 to n.  The unknown displacement vector at each nodal point will be denoted by u i a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamyDamaaDaaaleaacaWGPbaabaGaam yyaaaaaaa@33CA@ .

1.      The displacement field and virtual velocity field at an arbitrary point within the solid is again specified by interpolating between nodal values in some convenient way. 

u i (x)= a=1 n N a (x) u i a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamyDamaaBaaaleaacaWGPbaabeaaki aacIcacaWH4bGaaiykaiabg2da9maaqahabaGaamOtamaaCaaaleqa baGaamyyaaaakiaacIcacaWH4bGaaiykaiaadwhadaqhaaWcbaGaam yAaaqaaiaadggaaaaabaGaamyyaiabg2da9iaaigdaaeaacaWGUbaa niabggHiLdaaaa@4363@            δ v i (x)= a=1 n N a (x)δ v i a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeqiTdqMaamODamaaBaaaleaacaWGPb aabeaakiaacIcacaWH4bGaaiykaiabg2da9maaqahabaGaamOtamaa CaaaleqabaGaamyyaaaakiaacIcacaWH4bGaaiykaiabes7aKjaadA hadaqhaaWcbaGaamyAaaqaaiaadggaaaaabaGaamyyaiabg2da9iaa igdaaeaacaWGUbaaniabggHiLdaaaa@46AF@

Here, x denotes the coordinates of an arbitrary point in the reference configuration.  Note that the interpolation gives virtual velocity as a function of position x in the reference configuration, not y in the deformed configuration, so we have to be careful when computing the velocity gradient.

2.      Observe that we can compute the deformation corresponding to a given displacement field as

F ij = δ ij + u i x j = δ ij + a=1 n N a x j u i a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamOramaaBaaaleaacaWGPbGaamOAaa qabaGccqGH9aqpcqaH0oazdaWgaaWcbaGaamyAaiaadQgaaeqaaOGa ey4kaSYaaSaaaeaacqGHciITcaWG1bWaaSbaaSqaaiaadMgaaeqaaa GcbaGaeyOaIyRaamiEamaaBaaaleaacaWGQbaabeaaaaGccqGH9aqp cqaH0oazdaWgaaWcbaGaamyAaiaadQgaaeqaaOGaey4kaSYaaabCae aadaWcaaqaaiabgkGi2kaad6eadaahaaWcbeqaaiaadggaaaaakeaa cqGHciITcaWG4bWaaSbaaSqaaiaadQgaaeqaaaaakiaadwhadaqhaa WcbaGaamyAaaqaaiaadggaaaaabaGaamyyaiabg2da9iaaigdaaeaa caWGUbaaniabggHiLdaaaa@55C3@

3.      The derivatives of shape functions with respect to reference coordinates are computed exactly as for small strain problems.  Let N a ( ξ i ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamOtamaaCaaaleqabaGaamyyaaaaki aacIcacqaH+oaEdaWgaaWcbaGaamyAaaqabaGccaGGPaaaaa@36FF@  denote the shape functions in terms of local element coordinates ξ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8YjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeqOVdG3aaSbaaSqaaiaadMgaaeqaaa aa@33BA@ .  Then interpolate position within the element as

x i = a=1 N e N a ( ξ j ) x i a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamiEamaaBaaaleaacaWGPbaabeaaki abg2da9maaqahabaGaamOtamaaCaaaleqabaGaamyyaaaakiaacIca cqaH+oaEdaWgaaWcbaGaamOAaaqabaGccaGGPaGaamiEamaaDaaale aacaWGPbaabaGaamyyaaaaaeaacaWGHbGaeyypa0JaaGymaaqaaiaa d6eadaWgaaadbaGaamyzaaqabaaaniabggHiLdaaaa@43ED@

Define the Jacobian matrix

η ij = x i ξ j = a=1 N e N a ξ j x i a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeq4TdG2aaSbaaSqaaiaadMgacaWGQb aabeaakiabg2da9maalaaabaGaeyOaIyRaamiEamaaBaaaleaacaWG PbaabeaaaOqaaiabgkGi2kabe67a4naaBaaaleaacaWGQbaabeaaaa GccqGH9aqpdaaeWbqaamaalaaabaGaeyOaIyRaamOtamaaCaaaleqa baGaamyyaaaaaOqaaiabgkGi2kabe67a4naaBaaaleaacaWGQbaabe aaaaGccaWG4bWaa0baaSqaaiaadMgaaeaacaWGHbaaaaqaaiaadgga cqGH9aqpcaaIXaaabaGaamOtamaaBaaameaacaWGLbaabeaaa0Gaey yeIuoaaaa@4FF9@

then

N a x j = N a ξ k η kj 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaWaaSaaaeaacqGHciITcaWGobWaaWbaaS qabeaacaWGHbaaaaGcbaGaeyOaIyRaamiEamaaBaaaleaacaWGQbaa beaaaaGccqGH9aqpdaWcaaqaaiabgkGi2kaad6eadaahaaWcbeqaai aadggaaaaakeaacqGHciITcqaH+oaEdaWgaaWcbaGaam4Aaaqabaaa aOGaeq4TdG2aa0baaSqaaiaadUgacaWGQbaabaGaeyOeI0IaaGymaa aaaaa@45D8@

4.      Given the deformation gradients, we can compute any other deformation measure we need MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  we don’t need to spell out the details for now.   By substituting the appropriate deformation measure we could calculate the Kirchhoff stress.  Note that the Kirchhoff stress depends on displacements through the deformation gradient MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  we will express this functional relationship as τ ij [ F kl ( u i a ) ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeqiXdq3aaSbaaSqaaiaadMgacaWGQb aabeaakmaadmaabaGaamOramaaBaaaleaacaWGRbGaamiBaaqabaGc caGGOaGaamyDamaaDaaaleaacaWGPbaabaGaamyyaaaakiaacMcaai aawUfacaGLDbaaaaa@3DD9@

5.      Note also that the virtual velocity gradient can be calculated as

δ L ij = δ v i y j = δ v i x k x k y j = δ v i x k F kj 1 = a=1 n N a x k F kj 1 δ v i a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeqiTdqMaamitamaaBaaaleaacaWGPb GaamOAaaqabaGccqGH9aqpdaWcaaqaaiabgkGi2kabes7aKjaadAha daWgaaWcbaGaamyAaaqabaaakeaacqGHciITcaWG5bWaaSbaaSqaai aadQgaaeqaaaaakiabg2da9maalaaabaGaeyOaIyRaeqiTdqMaamOD amaaBaaaleaacaWGPbaabeaaaOqaaiabgkGi2kaadIhadaWgaaWcba Gaam4AaaqabaaaaOWaaSaaaeaacqGHciITcaWG4bWaaSbaaSqaaiaa dUgaaeqaaaGcbaGaeyOaIyRaamyEamaaBaaaleaacaWGQbaabeaaaa GccqGH9aqpdaWcaaqaaiabgkGi2kabes7aKjaadAhadaWgaaWcbaGa amyAaaqabaaakeaacqGHciITcaWG4bWaaSbaaSqaaiaadUgaaeqaaa aakiaadAeadaqhaaWcbaGaam4AaiaadQgaaeaacqGHsislcaaIXaaa aOGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8Uaeyypa0ZaaabCaeaadaWcaaqaaiabgkGi2kaad6eadaah aaWcbeqaaiaadggaaaaakeaacqGHciITcaWG4bWaaSbaaSqaaiaadU gaaeqaaaaakiaadAeadaqhaaWcbaGaam4AaiaadQgaaeaacqGHsisl caaIXaaaaOGaeqiTdqMaamODamaaDaaaleaacaWGPbaabaGaamyyaa aaaeaacaWGHbGaeyypa0JaaGymaaqaaiaad6gaa0GaeyyeIuoaaaa@8331@

6.      We can now substitute everything back into the virtual work equation

{ R 0 τ ij [ F pq ( u k b ) ] N a x m F mj 1 d V 0 V 0 ρ 0 b i N a d V 0 2 R 0 t i * N a ηd A 0 }δ v i a =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaWaaiWaaeaadaWdrbqaaiabes8a0naaBa aaleaacaWGPbGaamOAaaqabaGcdaWadaqaaiaadAeadaWgaaWcbaGa amiCaiaadghaaeqaaOGaaiikaiaadwhadaqhaaWcbaGaam4Aaaqaai aadkgaaaGccaGGPaaacaGLBbGaayzxaaWaaSaaaeaacqGHciITcaWG obWaaWbaaSqabeaacaWGHbaaaaGcbaGaeyOaIyRaamiEamaaBaaale aacaWGTbaabeaaaaGccaWGgbWaa0baaSqaaiaad2gacaWGQbaabaGa eyOeI0IaaGymaaaakiaadsgacaWGwbWaaSbaaSqaaiaaicdaaeqaaO GaeyOeI0Yaa8quaeaacqaHbpGCdaWgaaWcbaGaaGimaaqabaGccaWG IbWaaSbaaSqaaiaadMgaaeqaaOGaamOtamaaCaaaleqabaGaamyyaa aakiaadsgacaWGwbWaaSbaaSqaaiaaicdaaeqaaOGaeyOeI0Yaa8qu aeaacaWG0bWaa0baaSqaaiaadMgaaeaacaGGQaaaaOGaamOtamaaCa aaleqabaGaamyyaaaakiabeE7aOjaadsgacaWGbbWaaSbaaSqaaiaa icdaaeqaaaqaaiabgkGi2oaaBaaameaacaaIYaaabeaaliaadkfada WgaaadbaGaaGimaaqabaaaleqaniabgUIiYdaaleaacaWGwbWaaSba aWqaaiaaicdaaeqaaaWcbeqdcqGHRiI8aaWcbaGaamOuamaaBaaame aacaaIWaaabeaaaSqab0Gaey4kIipaaOGaay5Eaiaaw2haaiabes7a KjaadAhadaqhaaWcbaGaamyAaaqaaiaadggaaaGccqGH9aqpcaaIWa aaaa@779E@

7.      Since this must hold for all δ v i a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8YjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeqiTdqMaamODamaaDaaaleaacaWGPb aabaGaamyyaaaaaaa@357E@  we must ensure that

V 0 τ ij [ F pq ( u k b ) ] N a x m F mj 1 d V 0 V 0 ρ 0 b i N a d V 0 2 V 0 t i * N a ηd A 0 =0 {a,i}: x k a not on  1 R 0 u i a = u i * ( x i a ){a,i}: x k a on  1 R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGceaqabeaadaWdrbqaaiabes8a0naaBaaale aacaWGPbGaamOAaaqabaGcdaWadaqaaiaadAeadaWgaaWcbaGaamiC aiaadghaaeqaaOGaaiikaiaadwhadaqhaaWcbaGaam4Aaaqaaiaadk gaaaGccaGGPaaacaGLBbGaayzxaaWaaSaaaeaacqGHciITcaWGobWa aWbaaSqabeaacaWGHbaaaaGcbaGaeyOaIyRaamiEamaaBaaaleaaca WGTbaabeaaaaGccaWGgbWaa0baaSqaaiaad2gacaWGQbaabaGaeyOe I0IaaGymaaaakiaadsgacaWGwbWaaSbaaSqaaiaaicdaaeqaaOGaey OeI0Yaa8quaeaacqaHbpGCdaWgaaWcbaGaaGimaaqabaGccaWGIbWa aSbaaSqaaiaadMgaaeqaaOGaamOtamaaCaaaleqabaGaamyyaaaaki aadsgacaWGwbWaaSbaaSqaaiaaicdaaeqaaOGaeyOeI0Yaa8quaeaa caWG0bWaa0baaSqaaiaadMgaaeaacaGGQaaaaOGaamOtamaaCaaale qabaGaamyyaaaakiabeE7aOjaadsgacaWGbbWaaSbaaSqaaiaaicda aeqaaaqaaiabgkGi2oaaBaaameaacaaIYaaabeaaliaadAfadaWgaa adbaGaaGimaaqabaaaleqaniabgUIiYdGccqGH9aqpcaaIWaaaleaa caWGwbWaaSbaaWqaaiaaicdaaeqaaaWcbeqdcqGHRiI8aaWcbaGaam OvamaaBaaameaacaaIWaaabeaaaSqab0Gaey4kIipakiaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7cqGHaiIicaGG7bGaamyyaiaacYcacaWGPbGaaiyFaiaaykW7caaM c8UaaGPaVlaacQdacaaMc8UaaGPaVlaaykW7caWG4bWaa0baaSqaai aadUgaaeaacaWGHbaaaOGaaGPaVlaab6gacaqGVbGaaeiDaiaabcca caqGVbGaaeOBaiaabccacqGHciITdaWgaaWcbaGaaeymaaqabaGcca WGsbWaaSbaaSqaaiaaicdaaeqaaaGcbaGaamyDamaaDaaaleaacaWG PbaabaGaamyyaaaakiabg2da9iaadwhadaqhaaWcbaGaamyAaaqaai aacQcaaaGccaGGOaGaamiEamaaDaaaleaacaWGPbaabaGaamyyaaaa kiaacMcacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqGHaiIicaGG7b GaamyyaiaacYcacaWGPbGaaiyFaiaaykW7caaMc8UaaGPaVlaacQda caaMc8UaaGPaVlaadIhadaqhaaWcbaGaam4AaaqaaiaadggaaaGcca aMc8UaaGPaVlaab+gacaqGUbGaaeiiaiabgkGi2oaaBaaaleaacaqG Xaaabeaakiaadkfaaaaa@E362@

 

This is a set of n nonlinear equations in n unknowns, very similar to those we obtained for hypoelastic problems MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  except that now we have to deal with all the additional geometric terms associated with finite deformations.  The procedure for solving these equations is outlined in the following sections.

 

 

 

8.4.4 Solution using Consistent Newton Raphson Iteration

 

As before, we can solve the nonlinear virtual work equation using Newton-Raphson iteration, as follows

1.      Start with some initial guess for u i a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamyDamaaDaaaleaacaWGPbaabaGaam yyaaaaaaa@33CA@  - say w i a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaam4DamaaDaaaleaacaWGPbaabaGaam yyaaaaaaa@33CC@  (we can start with zero displacements, or for incremental solutions we can use the solution at the end of the preceding increment).  This solution will not satisfy the governing equation (unless you are very lucky)

2.      Next, attempt to correct this guess to bring it closer to the proper solution by setting w i a w i a +d w i a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaam4DamaaDaaaleaacaWGPbaabaGaam yyaaaakiabgkziUkaadEhadaqhaaWcbaGaamyAaaqaaiaadggaaaGc cqGHRaWkcaWGKbGaam4DamaaDaaaleaacaWGPbaabaGaamyyaaaaaa a@3D92@ .  Ideally, we would want the correction to satisfy

V 0 τ ij [ F pq ( w k b +d w k b ) ] N a x m (F+dF) mj 1 d V 0 V 0 ρ 0 b i N a d V 0 2 V 0 t i * N a (η+dη)d A 0 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaWaa8quaeaacqaHepaDdaWgaaWcbaGaam yAaiaadQgaaeqaaOWaamWaaeaacaWGgbWaaSbaaSqaaiaadchacaWG XbaabeaakiaacIcacaWG3bWaa0baaSqaaiaadUgaaeaacaWGIbaaaO Gaey4kaSIaamizaiaadEhadaqhaaWcbaGaam4AaaqaaiaadkgaaaGc caGGPaaacaGLBbGaayzxaaWaaSaaaeaacqGHciITcaWGobWaaWbaaS qabeaacaWGHbaaaaGcbaGaeyOaIyRaamiEamaaBaaaleaacaWGTbaa beaaaaGccaGGOaGaamOraiabgUcaRiaadsgacaWGgbGaaiykamaaDa aaleaacaWGTbGaamOAaaqaaiabgkHiTiaaigdaaaGccaWGKbGaamOv amaaBaaaleaacaaIWaaabeaakiabgkHiTmaapefabaGaeqyWdi3aaS baaSqaaiaaicdaaeqaaOGaamOyamaaBaaaleaacaWGPbaabeaakiaa d6eadaahaaWcbeqaaiaadggaaaGccaWGKbGaamOvamaaBaaaleaaca aIWaaabeaaaeaacaWGwbWaaSbaaWqaaiaaicdaaeqaaaWcbeqdcqGH RiI8aaWcbaGaamOvamaaBaaameaacaaIWaaabeaaaSqab0Gaey4kIi pakiabgkHiTmaapefabaGaamiDamaaDaaaleaacaWGPbaabaGaaiOk aaaakiaad6eadaahaaWcbeqaaiaadggaaaGccaGGOaGaeq4TdGMaey 4kaSIaamizaiabeE7aOjaacMcacaWGKbGaamyqamaaBaaaleaacaaI WaaabeaaaeaacqGHciITdaWgaaadbaGaaGOmaaqabaWccaWGwbWaaS baaWqaaiaaicdaaeqaaaWcbeqdcqGHRiI8aOGaeyypa0JaaGimaaaa @7E35@

where F+dF MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaaCOraiabgUcaRiaadsgacaWHgbaaaa@3438@  denotes the deformation gradient for the updated solution. This equation cannot be solved for d w k b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamizaiaadEhadaqhaaWcbaGaam4Aaa qaaiaadkgaaaaaaa@34B8@  in its present form.

3.      To make progress, linearize in d w k b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamizaiaadEhadaqhaaWcbaGaam4Aaa qaaiaadkgaaaaaaa@34B8@ , just as for the hypoelastic problem discussed in the preceding section.  The linearization (derived in detail below) yields a system of linear equations

K aibk d w k b + R i a F i a =0{a,i}: x k a not on  1 V 0 u i a = u i * ( x i a ){a,i}: x k a on  1 V MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGceaqabeaacaWGlbWaaSbaaSqaaiaadggaca WGPbGaamOyaiaadUgaaeqaaOGaamizaiaadEhadaqhaaWcbaGaam4A aaqaaiaadkgaaaGccqGHRaWkcaWGsbWaa0baaSqaaiaadMgaaeaaca WGHbaaaOGaeyOeI0IaamOramaaDaaaleaacaWGPbaabaGaamyyaaaa kiabg2da9iaaicdacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlabgcGiIiaacUhacaWGHbGaaiilaiaadMgacaGG 9bGaaGPaVlaaykW7caaMc8UaaiOoaiaaykW7caaMc8UaaGPaVlaadI hadaqhaaWcbaGaam4AaaqaaiaadggaaaGccaaMc8UaaeOBaiaab+ga caqG0bGaaeiiaiaab+gacaqGUbGaaeiiaiabgkGi2oaaBaaaleaaca qGXaaabeaakiaadAfadaWgaaWcbaGaaGimaaqabaaakeaacaWG1bWa a0baaSqaaiaadMgaaeaacaWGHbaaaOGaeyypa0JaamyDamaaDaaale aacaWGPbaabaGaaiOkaaaakiaacIcacaWG4bWaa0baaSqaaiaadMga aeaacaWGHbaaaOGaaiykaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl abgcGiIiaacUhacaWGHbGaaiilaiaadMgacaGG9bGaaGPaVlaaykW7 caaMc8UaaiOoaiaaykW7caaMc8UaamiEamaaDaaaleaacaWGRbaaba GaamyyaaaakiaaykW7caaMc8Uaae4Baiaab6gacaqGGaGaeyOaIy7a aSbaaSqaaiaabgdaaeqaaOGaamOvaaaaaa@B25A@

K aibk = V 0 τ ij F kl N b x l N a x m F mj 1 d V 0 V 0 τ ij N a x m F mk 1 N b x p F pj 1 d V 0 2 V 0 t i * N a η w k b d A 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaam4samaaBaaaleaacaWGHbGaamyAai aadkgacaWGRbaabeaakiabg2da9maapefabaWaaSaaaeaacqGHciIT cqaHepaDdaWgaaWcbaGaamyAaiaadQgaaeqaaaGcbaGaeyOaIyRaam OramaaBaaaleaacaWGRbGaamiBaaqabaaaaOWaaSaaaeaacqGHciIT caWGobWaaWbaaSqabeaacaWGIbaaaaGcbaGaeyOaIyRaamiEamaaBa aaleaacaWGSbaabeaaaaGcdaWcaaqaaiabgkGi2kaad6eadaahaaWc beqaaiaadggaaaaakeaacqGHciITcaWG4bWaaSbaaSqaaiaad2gaae qaaaaakiaadAeadaqhaaWcbaGaamyBaiaadQgaaeaacqGHsislcaaI XaaaaaqaaiaadAfadaWgaaadbaGaaGimaaqabaaaleqaniabgUIiYd GccaWGKbGaamOvamaaBaaaleaacaaIWaaabeaakiabgkHiTmaapefa baGaeqiXdq3aaSbaaSqaaiaadMgacaWGQbaabeaakmaalaaabaGaey OaIyRaamOtamaaCaaaleqabaGaamyyaaaaaOqaaiabgkGi2kaadIha daWgaaWcbaGaamyBaaqabaaaaOGaamOramaaDaaaleaacaWGTbGaam 4AaaqaaiabgkHiTiaaigdaaaaabaGaamOvamaaBaaameaacaaIWaaa beaaaSqab0Gaey4kIipakmaalaaabaGaeyOaIyRaamOtamaaCaaale qabaGaamOyaaaaaOqaaiabgkGi2kaadIhadaWgaaWcbaGaamiCaaqa baaaaOGaamOramaaDaaaleaacaWGWbGaamOAaaqaaiabgkHiTiaaig daaaGccaWGKbGaamOvamaaBaaaleaacaaIWaaabeaakiabgkHiTmaa pefabaGaamiDamaaDaaaleaacaWGPbaabaGaaiOkaaaakiaad6eada ahaaWcbeqaaiaadggaaaGcdaWcaaqaaiabgkGi2kabeE7aObqaaiab gkGi2kaadEhadaqhaaWcbaGaam4AaaqaaiaadkgaaaaaaOGaamizai aadgeadaWgaaWcbaGaaGimaaqabaaabaGaeyOaIy7aaSbaaWqaaiaa ikdaaeqaaSGaamOvamaaBaaameaacaaIWaaabeaaaSqab0Gaey4kIi paaaa@9204@

R i a = V 0 τ ij N a x m F mj 1 d V 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamOuamaaDaaaleaacaWGPbaabaGaam yyaaaakiabg2da9maapefabaGaeqiXdq3aaSbaaSqaaiaadMgacaWG QbaabeaakmaalaaabaGaeyOaIyRaamOtamaaCaaaleqabaGaamyyaa aaaOqaaiabgkGi2kaadIhadaWgaaWcbaGaamyBaaqabaaaaOGaamOr amaaDaaaleaacaWGTbGaamOAaaqaaiabgkHiTiaaigdaaaGccaWGKb GaamOvamaaBaaaleaacaaIWaaabeaaaeaacaWGwbWaaSbaaWqaaiaa icdaaeqaaaWcbeqdcqGHRiI8aaaa@4AC1@   F i a = V 0 ρ 0 b i N a d V 0 + 2 V 0 t i * N a ηd A 0 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamOramaaDaaaleaacaWGPbaabaGaam yyaaaakiabg2da9maapefabaGaeqyWdi3aaSbaaSqaaiaaicdaaeqa aOGaamOyamaaBaaaleaacaWGPbaabeaakiaad6eadaahaaWcbeqaai aadggaaaGccaWGKbGaamOvamaaBaaaleaacaaIWaaabeaaaeaacaWG wbWaaSbaaWqaaiaaicdaaeqaaaWcbeqdcqGHRiI8aOGaey4kaSYaa8 quaeaacaWG0bWaa0baaSqaaiaadMgaaeaacaGGQaaaaOGaamOtamaa CaaaleqabaGaamyyaaaakiabeE7aOjaadsgacaWGbbWaaSbaaSqaai aaicdaaeqaaaqaaiabgkGi2oaaBaaameaacaaIYaaabeaaliaadAfa daWgaaadbaGaaGimaaqabaaaleqaniabgUIiYdGccqGH9aqpcaaIWa aaaa@5425@

which can be solved for d w k b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamizaiaadEhadaqhaaWcbaGaam4Aaa qaaiaadkgaaaaaaa@34B8@ .

4.      If you prefer, you can use a slightly simpler set of formulas for the stiffness matrix and force vector

K aibk = V 0 C e ijkl N a y j N b y l d V 0 V 0 τ ij N a y k N b y j d V 0 2 V 0 t i * N a η w k b d A 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaam4samaaBaaaleaacaWGHbGaamyAai aadkgacaWGRbaabeaakiabg2da9maapefabaGaam4qamaaCaaaleqa baGaamyzaaaakmaaBaaaleaacaWGPbGaamOAaiaadUgacaWGSbaabe aakmaalaaabaGaeyOaIyRaamOtamaaCaaaleqabaGaamyyaaaaaOqa aiabgkGi2kaadMhadaWgaaWcbaGaamOAaaqabaaaaOWaaSaaaeaacq GHciITcaWGobWaaWbaaSqabeaacaWGIbaaaaGcbaGaeyOaIyRaamyE amaaBaaaleaacaWGSbaabeaaaaaabaGaamOvamaaBaaameaacaaIWa aabeaaaSqab0Gaey4kIipakiaadsgacaWGwbWaaSbaaSqaaiaaicda aeqaaOGaeyOeI0Yaa8quaeaacqaHepaDdaWgaaWcbaGaamyAaiaadQ gaaeqaaOWaaSaaaeaacqGHciITcaWGobWaaWbaaSqabeaacaWGHbaa aaGcbaGaeyOaIyRaamyEamaaBaaaleaacaWGRbaabeaaaaaabaGaam OvamaaBaaameaacaaIWaaabeaaaSqab0Gaey4kIipakmaalaaabaGa eyOaIyRaamOtamaaCaaaleqabaGaamOyaaaaaOqaaiabgkGi2kaadM hadaWgaaWcbaGaamOAaaqabaaaaOGaamizaiaadAfadaWgaaWcbaGa aGimaaqabaGccqGHsisldaWdrbqaaiaadshadaqhaaWcbaGaamyAaa qaaiaacQcaaaGccaWGobWaaWbaaSqabeaacaWGHbaaaOWaaSaaaeaa cqGHciITcqaH3oaAaeaacqGHciITcaWG3bWaa0baaSqaaiaadUgaae aacaWGIbaaaaaakiaadsgacaWGbbWaaSbaaSqaaiaaicdaaeqaaaqa aiabgkGi2oaaBaaameaacaaIYaaabeaaliaadAfadaWgaaadbaGaaG imaaqabaaaleqaniabgUIiYdaaaa@809F@    R i a = V 0 τ ij N a y j d V 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamOuamaaDaaaleaacaWGPbaabaGaam yyaaaakiabg2da9maapefabaGaeqiXdq3aaSbaaSqaaiaadMgacaWG QbaabeaakmaalaaabaGaeyOaIyRaamOtamaaCaaaleqabaGaamyyaa aaaOqaaiabgkGi2kaadMhadaWgaaWcbaGaamOAaaqabaaaaOGaamiz aiaadAfadaWgaaWcbaGaaGimaaqabaaabaGaamOvamaaBaaameaaca aIWaaabeaaaSqab0Gaey4kIipaaaa@4634@

where we have defined

C e ijkl = τ ij F km F lm =J σ ij δ kl +J σ ij F km F lm MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaam4qamaaCaaaleqabaGaamyzaaaakm aaBaaaleaacaWGPbGaamOAaiaadUgacaWGSbaabeaakiabg2da9maa laaabaGaeyOaIyRaeqiXdq3aaSbaaSqaaiaadMgacaWGQbaabeaaaO qaaiabgkGi2kaadAeadaWgaaWcbaGaam4Aaiaad2gaaeqaaaaakiaa dAeadaWgaaWcbaGaamiBaiaad2gaaeqaaOGaeyypa0JaamOsaiabeo 8aZnaaBaaaleaacaWGPbGaamOAaaqabaGccqaH0oazdaWgaaWcbaGa am4AaiaadYgaaeqaaOGaey4kaSIaamOsamaalaaabaGaeyOaIyRaeq 4Wdm3aaSbaaSqaaiaadMgacaWGQbaabeaaaOqaaiabgkGi2kaadAea daWgaaWcbaGaam4Aaiaad2gaaeqaaaaakiaadAeadaWgaaWcbaGaam iBaiaad2gaaeqaaaaa@5BB4@              N a y i = N a x j F ji 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaWaaSaaaeaacqGHciITcaWGobWaaWbaaS qabeaacaWGHbaaaaGcbaGaeyOaIyRaamyEamaaBaaaleaacaWGPbaa beaaaaGccqGH9aqpdaWcaaqaaiabgkGi2kaad6eadaahaaWcbeqaai aadggaaaaakeaacqGHciITcaWG4bWaaSbaaSqaaiaadQgaaeqaaaaa kiaadAeadaqhaaWcbaGaamOAaiaadMgaaeaacqGHsislcaaIXaaaaa aa@442E@

Note that the formula for stiffness is very similar to the result for small strain problems, except for two additional terms.  These additional terms are called the `geometric stiffness’ because they arise as a result of accounting properly for finite geometry changes.  In addition, note that while the first integral in the stiffness is symmetric, the second and third are not.  There is therefore some additional computational cost associated with finite strain problems, since it is necessary to store and solve an unsymmetric system of equations. 

5.      After solving the system of equations in (3) for d w k b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamizaiaadEhadaqhaaWcbaGaam4Aaa qaaiaadkgaaaaaaa@34B8@ , check for convergence (you can use the magnitude of d w k b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamizaiaadEhadaqhaaWcbaGaam4Aaa qaaiaadkgaaaaaaa@34B8@  or the magnitude of the force vector R i a F i a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadkfadaqhaaWcbaGaamyAaaqaaiaadg gaaaGccqGHsislcaWGgbWaa0baaSqaaiaadMgaaeaacaWGHbaaaaaa @3703@  as a measure of error).  If the solution has not yet converged, go back to (3) and correct the solution again.

 

Linearizing the virtual work equation:  This is a tedious, but straightforward calculation.  Start with

V 0 τ ij [ F pq ( w k b +d w k b ) ] N a x m (F+dF) mj 1 d V 0 V 0 ρ 0 b i N a d V 0 2 V 0 t i * N a (η+dη)d A 0 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaWaa8quaeaacqaHepaDdaWgaaWcbaGaam yAaiaadQgaaeqaaOWaamWaaeaacaWGgbWaaSbaaSqaaiaadchacaWG XbaabeaakiaacIcacaWG3bWaa0baaSqaaiaadUgaaeaacaWGIbaaaO Gaey4kaSIaamizaiaadEhadaqhaaWcbaGaam4AaaqaaiaadkgaaaGc caGGPaaacaGLBbGaayzxaaWaaSaaaeaacqGHciITcaWGobWaaWbaaS qabeaacaWGHbaaaaGcbaGaeyOaIyRaamiEamaaBaaaleaacaWGTbaa beaaaaGccaGGOaGaamOraiabgUcaRiaadsgacaWGgbGaaiykamaaDa aaleaacaWGTbGaamOAaaqaaiabgkHiTiaaigdaaaGccaWGKbGaamOv amaaBaaaleaacaaIWaaabeaakiabgkHiTmaapefabaGaeqyWdi3aaS baaSqaaiaaicdaaeqaaOGaamOyamaaBaaaleaacaWGPbaabeaakiaa d6eadaahaaWcbeqaaiaadggaaaGccaWGKbGaamOvamaaBaaaleaaca aIWaaabeaaaeaacaWGwbWaaSbaaWqaaiaaicdaaeqaaaWcbeqdcqGH RiI8aaWcbaGaamOvamaaBaaameaacaaIWaaabeaaaSqab0Gaey4kIi pakiabgkHiTmaapefabaGaamiDamaaDaaaleaacaWGPbaabaGaaiOk aaaakiaad6eadaahaaWcbeqaaiaadggaaaGccaGGOaGaeq4TdGMaey 4kaSIaamizaiabeE7aOjaacMcacaWGKbGaamyqamaaBaaaleaacaaI WaaabeaaaeaacqGHciITdaWgaaadbaGaaGOmaaqabaWccaWGwbWaaS baaWqaaiaaicdaaeqaaaWcbeqdcqGHRiI8aOGaeyypa0JaaGimaaaa @7E35@

Note that

F ij w k a = N a x j δ ik MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaWaaSaaaeaacqGHciITcaWGgbWaaSbaaS qaaiaadMgacaWGQbaabeaaaOqaaiabgkGi2kaadEhadaqhaaWcbaGa am4AaaqaaiaadggaaaaaaOGaeyypa0ZaaSaaaeaacqGHciITcaWGob WaaWbaaSqabeaacaWGHbaaaaGcbaGaeyOaIyRaamiEamaaBaaaleaa caWGQbaabeaaaaGccqaH0oazdaWgaaWcbaGaamyAaiaadUgaaeqaaa aa@4535@

We also have that

F ij F jk 1 = δ ik F ij w n a F jk 1 + F ij F jk 1 w n a =0 F pk 1 w n a = F pi 1 F ij w n a F jk 1 ( F+dF ) mn 1 F mn 1 F mi 1 F ij w k b F jn 1 d w k b = F mn 1 F mk 1 N b x j F jn 1 d w k b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGceaqabeaacaWGgbWaaSbaaSqaaiaadMgaca WGQbaabeaakiaadAeadaqhaaWcbaGaamOAaiaadUgaaeaacqGHsisl caaIXaaaaOGaeyypa0JaeqiTdq2aaSbaaSqaaiaadMgacaWGRbaabe aakiabgkDiEpaalaaabaGaeyOaIyRaamOramaaBaaaleaacaWGPbGa amOAaaqabaaakeaacqGHciITcaWG3bWaa0baaSqaaiaad6gaaeaaca WGHbaaaaaakiaadAeadaqhaaWcbaGaamOAaiaadUgaaeaacqGHsisl caaIXaaaaOGaey4kaSIaamOramaaBaaaleaacaWGPbGaamOAaaqaba GcdaWcaaqaaiabgkGi2kaadAeadaqhaaWcbaGaamOAaiaadUgaaeaa cqGHsislcaaIXaaaaaGcbaGaeyOaIyRaam4DamaaDaaaleaacaWGUb aabaGaamyyaaaaaaGccqGH9aqpcaaIWaGaeyO0H49aaSaaaeaacqGH ciITcaWGgbWaa0baaSqaaiaadchacaWGRbaabaGaeyOeI0IaaGymaa aaaOqaaiabgkGi2kaadEhadaqhaaWcbaGaamOBaaqaaiaadggaaaaa aOGaeyypa0JaeyOeI0IaamOramaaDaaaleaacaWGWbGaamyAaaqaai abgkHiTiaaigdaaaGcdaWcaaqaaiabgkGi2kaadAeadaWgaaWcbaGa amyAaiaadQgaaeqaaaGcbaGaeyOaIyRaam4DamaaDaaaleaacaWGUb aabaGaamyyaaaaaaGccaWGgbWaa0baaSqaaiaadQgacaWGRbaabaGa eyOeI0IaaGymaaaaaOqaaiabgkDiEpaabmaabaGaamOraiabgUcaRi aadsgacaWGgbaacaGLOaGaayzkaaWaa0baaSqaaiaad2gacaWGUbaa baGaeyOeI0IaaGymaaaakiabgIKi7kaadAeadaqhaaWcbaGaamyBai aad6gaaeaacqGHsislcaaIXaaaaOGaeyOeI0IaamOramaaDaaaleaa caWGTbGaamyAaaqaaiabgkHiTiaaigdaaaGcdaWcaaqaaiabgkGi2k aadAeadaWgaaWcbaGaamyAaiaadQgaaeqaaaGcbaGaeyOaIyRaam4D amaaDaaaleaacaWGRbaabaGaamOyaaaaaaGccaWGgbWaa0baaSqaai aadQgacaWGUbaabaGaeyOeI0IaaGymaaaakiaadsgacaWG3bWaa0ba aSqaaiaadUgaaeaacaWGIbaaaOGaeyypa0JaamOramaaDaaaleaaca WGTbGaamOBaaqaaiabgkHiTiaaigdaaaGccqGHsislcaWGgbWaa0ba aSqaaiaad2gacaWGRbaabaGaeyOeI0IaaGymaaaakmaalaaabaGaey OaIyRaamOtamaaCaaaleqabaGaamOyaaaaaOqaaiabgkGi2kaadIha daWgaaWcbaGaamOAaaqabaaaaOGaamOramaaDaaaleaacaWGQbGaam OBaaqaaiabgkHiTiaaigdaaaGccaWGKbGaam4DamaaDaaaleaacaWG RbaabaGaamOyaaaaaaaa@BFB3@

In addition,

τ ij [ F pq ( w k b +d w k b )] τ ij + τ ij F kl F kl w n b d w n b = τ ij + τ ij F kl N b x l d w k b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeqiXdq3aaSbaaSqaaiaadMgacaWGQb aabeaakiaacUfacaWGgbWaaSbaaSqaaiaadchacaWGXbaabeaakiaa cIcacaWG3bWaa0baaSqaaiaadUgaaeaacaWGIbaaaOGaey4kaSIaam izaiaadEhadaqhaaWcbaGaam4AaaqaaiaadkgaaaGccaGGPaGaaiyx aiabgIKi7kabes8a0naaBaaaleaacaWGPbGaamOAaaqabaGccqGHRa WkdaWcaaqaaiabgkGi2kabes8a0naaBaaaleaacaWGPbGaamOAaaqa baaakeaacqGHciITcaWGgbWaaSbaaSqaaiaadUgacaWGSbaabeaaaa GcdaWcaaqaaiabgkGi2kaadAeadaWgaaWcbaGaam4AaiaadYgaaeqa aaGcbaGaeyOaIyRaam4DamaaDaaaleaacaWGUbaabaGaamOyaaaaaa GccaWGKbGaam4DamaaDaaaleaacaWGUbaabaGaamOyaaaakiabg2da 9iabes8a0naaBaaaleaacaWGPbGaamOAaaqabaGccqGHRaWkdaWcaa qaaiabgkGi2kabes8a0naaBaaaleaacaWGPbGaamOAaaqabaaakeaa cqGHciITcaWGgbWaaSbaaSqaaiaadUgacaWGSbaabeaaaaGcdaWcaa qaaiabgkGi2kaad6eadaahaaWcbeqaaiaadkgaaaaakeaacqGHciIT caWG4bWaaSbaaSqaaiaadYgaaeqaaaaakiaadsgacaWG3bWaa0baaS qaaiaadUgaaeaacaWGIbaaaaaa@797D@

Substituting these expansions in the virtual work equation, and retaining linear terms in dw leads to the results given in step (3) above.

 

 

8.4.5 Tangent stiffness for the neo-Hookean material

 

The tangent stiffness is defined as

C e ijkl = τ ij F km F lm =J σ ij δ kl +J σ ij F km F lm MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaam4qamaaCaaaleqabaGaamyzaaaakm aaBaaaleaacaWGPbGaamOAaiaadUgacaWGSbaabeaakiabg2da9maa laaabaGaeyOaIyRaeqiXdq3aaSbaaSqaaiaadMgacaWGQbaabeaaaO qaaiabgkGi2kaadAeadaWgaaWcbaGaam4Aaiaad2gaaeqaaaaakiaa dAeadaWgaaWcbaGaamiBaiaad2gaaeqaaOGaeyypa0JaamOsaiabeo 8aZnaaBaaaleaacaWGPbGaamOAaaqabaGccqaH0oazdaWgaaWcbaGa am4AaiaadYgaaeqaaOGaey4kaSIaamOsamaalaaabaGaeyOaIyRaeq 4Wdm3aaSbaaSqaaiaadMgacaWGQbaabeaaaOqaaiabgkGi2kaadAea daWgaaWcbaGaam4Aaiaad2gaaeqaaaaakiaadAeadaWgaaWcbaGaam iBaiaad2gaaeqaaaaa@5BB4@

The Neo-Hookean solid has a stress-strain relation given by

σ ij = μ 1 J 5/3 ( B ij 1 3 B kk δ ij )+ K 1 ( J1 ) δ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeq4Wdm3aaSbaaSqaaiaadMgacaWGQb aabeaakiabg2da9maalaaabaGaeqiVd02aaSbaaSqaaiaaigdaaeqa aaGcbaGaamOsamaaCaaaleqabaGaaGynaiaac+cacaaIZaaaaaaakm aabmaabaGaamOqamaaBaaaleaacaWGPbGaamOAaaqabaGccqGHsisl daWcaaqaaiaaigdaaeaacaaIZaaaaiaadkeadaWgaaWcbaGaam4Aai aadUgaaeqaaOGaeqiTdq2aaSbaaSqaaiaadMgacaWGQbaabeaaaOGa ayjkaiaawMcaaiabgUcaRiaadUeadaWgaaWcbaGaaGymaaqabaGcda qadaqaaiaadQeacqGHsislcaaIXaaacaGLOaGaayzkaaGaeqiTdq2a aSbaaSqaaiaadMgacaWGQbaabeaaaaa@5355@

Evaluating the derivatives is a tedious but straightforward exercise in index notation. The following identity is helpful

J F km =J F mk 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaWaaSaaaeaacqGHciITcaWGkbaabaGaey OaIyRaamOramaaBaaaleaacaWGRbGaamyBaaqabaaaaOGaeyypa0Ja amOsaiaadAeadaqhaaWcbaGaamyBaiaadUgaaeaacqGHsislcaaIXa aaaaaa@3DB4@

giving

C e ijkl = μ 1 J 2/3 ( δ ik B jl + B il δ jk 2 3 { B ij δ kl + B kl δ ij }+ 2 3 B qq 3 δ ij δ kl )+ K 1 (2J1)J δ ij δ kl MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaam4qamaaCaaaleqabaGaamyzaaaakm aaBaaaleaacaWGPbGaamOAaiaadUgacaWGSbaabeaakiabg2da9maa laaabaGaeqiVd02aaSbaaSqaaiaaigdaaeqaaaGcbaGaamOsamaaCa aaleqabaGaaGOmaiaac+cacaaIZaaaaaaakmaabmaabaGaeqiTdq2a aSbaaSqaaiaadMgacaWGRbaabeaakiaadkeadaWgaaWcbaGaamOAai aadYgaaeqaaOGaey4kaSIaamOqamaaBaaaleaacaWGPbGaamiBaaqa baGccqaH0oazdaWgaaWcbaGaamOAaiaadUgaaeqaaOGaeyOeI0YaaS aaaeaacaaIYaaabaGaaG4maaaadaGadaqaaiaadkeadaWgaaWcbaGa amyAaiaadQgaaeqaaOGaeqiTdq2aaSbaaSqaaiaadUgacaWGSbaabe aakiabgUcaRiaadkeadaWgaaWcbaGaam4AaiaadYgaaeqaaOGaeqiT dq2aaSbaaSqaaiaadMgacaWGQbaabeaaaOGaay5Eaiaaw2haaiabgU caRmaalaaabaGaaGOmaaqaaiaaiodaaaWaaSaaaeaacaWGcbWaaSba aSqaaiaadghacaWGXbaabeaaaOqaaiaaiodaaaGaeqiTdq2aaSbaaS qaaiaadMgacaWGQbaabeaakiabes7aKnaaBaaaleaacaWGRbGaamiB aaqabaaakiaawIcacaGLPaaacqGHRaWkcaWGlbWaaSbaaSqaaiaaig daaeqaaOGaaiikaiaaikdacaWGkbGaeyOeI0IaaGymaiaacMcacaWG kbGaeqiTdq2aaSbaaSqaaiaadMgacaWGQbaabeaakiabes7aKnaaBa aaleaacaWGRbGaamiBaaqabaaaaa@7CE4@

 

 

8.4.6 Evaluating the boundary traction integrals

 

Finally, we need to address how to calculate the factor η MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8XjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeq4TdGgaaa@3279@  and its derivative in the surface integrals.

 

There are two common cases we need to deal with.  In some problems, we find it convenient to specify the nominal traction (force per unit undeformed area) acting on part of a solid.  For example, if you were to model the behavior of a bar under uniaxial tension, you might know the force you are going to apply to the bar.  Since you know the cross sectional area of the undeformed bar, you could easily calculate nominal traction.  However, in this case you would have no idea what the true traction acting on the bar is MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  to calculate that, you have to know the cross sectional area of the deformed bar.

 

In other problems you need to be able to impose a certain force per unit deformed area MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  i.e. to specify the true traction distribution.  This would be the case if you wanted to model fluid or aerodynamic forces acting on part of the solid.

 

We will deal with both cases.  The first case is easy MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  note that the nominal and true traction are related by t 0 =η t * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaaCiDamaaCaaaleqabaGaaGimaaaaki abg2da9iabeE7aOjaahshadaahaaWcbeqaaiaacQcaaaaaaa@3747@ .  The expression for the external forcing can therefore be written

F i a = V 0 ρ 0 b i N a d V 0 + 2 V 0 t i 0 N a d A 0 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamOramaaDaaaleaacaWGPbaabaGaam yyaaaakiabg2da9maapefabaGaeqyWdi3aaSbaaSqaaiaaicdaaeqa aOGaamOyamaaBaaaleaacaWGPbaabeaakiaad6eadaahaaWcbeqaai aadggaaaGccaWGKbGaamOvamaaBaaaleaacaaIWaaabeaaaeaacaWG wbWaaSbaaWqaaiaaicdaaeqaaaWcbeqdcqGHRiI8aOGaey4kaSYaa8 quaeaacaWG0bWaa0baaSqaaiaadMgaaeaacaaIWaaaaOGaamOtamaa CaaaleqabaGaamyyaaaakiaadsgacaWGbbWaaSbaaSqaaiaaicdaae qaaaqaaiabgkGi2oaaBaaameaacaaIYaaabeaaliaadAfadaWgaaad baGaaGimaaqabaaaleqaniabgUIiYdGccqGH9aqpcaaIWaaaaa@52A5@

and since this expression does not involve η MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8XjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeq4TdGgaaa@3279@  the last term in the expression for stiffness vanishes.

 

The second case is a pain.  It is simplest to treat the surface integrals directly.  Consider a general 3D element face, with nodal coordinates (in the deformed configuration) x i a = X i a + u i a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamiEamaaDaaaleaacaWGPbaabaGaam yyaaaakiabg2da9iaadIfadaqhaaWcbaGaamyAaaqaaiaadggaaaGc cqGHRaWkcaWG1bWaa0baaSqaaiaadMgaaeaacaWGHbaaaaaa@3BA2@   for a=1..n. Introduce a convenient interpolation scheme to define the shape of the element face in terms of its nodal coordinates

x i ( ξ α )= a=1 n M a ( ξ α ) x i a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamiEamaaBaaaleaacaWGPbaabeaaki aacIcacqaH+oaEdaWgaaWcbaGaeqySdegabeaakiaacMcacqGH9aqp daaeWbqaaiaad2eadaahaaWcbeqaaiaadggaaaGccaGGOaGaeqOVdG 3aaSbaaSqaaiabeg7aHbqabaGccaGGPaGaamiEamaaDaaaleaacaWG PbaabaGaamyyaaaaaeaacaWGHbGaeyypa0JaaGymaaqaaiaad6gaa0 GaeyyeIuoaaaa@4896@

where 1 ξ α 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeyOeI0IaaGymaiabgsMiJkabe67a4n aaBaaaleaacqaHXoqyaeqaaOGaeyizImQaaGymaaaa@3A34@  with α=1,2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8XjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeqySdeMaeyypa0JaaGymaiaacYcaca aIYaaaaa@3599@  denote a suitable set of local coordinates that will specify position within an element face, and M a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamytamaaCaaaleqabaGaamyyaaaaaa a@32B4@  are a set of interpolation functions.

 

We now evaluate the surface integral as

Ω e t i * N a ηd A 0 = 1 +1 1 +1 t i * N a ( ξ α ) η ˜ ( ξ α )d ξ 1 d ξ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaWaa8quaeaacaWG0bWaa0baaSqaaiaadM gaaeaacaGGQaaaaOGaamOtamaaCaaaleqabaGaamyyaaaakiabeE7a OjaadsgacaWGbbWaaSbaaSqaaiaaicdaaeqaaaqaaiabgkGi2kabfM 6axnaaBaaameaacaWGLbaabeaaaSqab0Gaey4kIipakiabg2da9maa pehabaWaa8qCaeaacaWG0bWaa0baaSqaaiaadMgaaeaacaGGQaaaaO GaamOtamaaCaaaleqabaGaamyyaaaakiaacIcacqaH+oaEdaWgaaWc baGaeqySdegabeaakiaacMcacuaH3oaAgaacaiaacIcacqaH+oaEda WgaaWcbaGaeqySdegabeaakiaacMcacaWGKbGaeqOVdG3aaSbaaSqa aiaaigdaaeqaaOGaamizaiabe67a4naaBaaaleaacaaIYaaabeaaae aacqGHsislcaaIXaaabaGaey4kaSIaaGymaaqdcqGHRiI8aaWcbaGa eyOeI0IaaGymaaqaaiabgUcaRiaaigdaa0Gaey4kIipaaaa@640C@

where η MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8XjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeq4TdGgaaa@3279@  must be computed by finding the two natural basis vectors

p i α = x i ξ α = a M a ξ α x i a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamiCamaaDaaaleaacaWGPbaabaGaeq ySdegaaOGaeyypa0ZaaSaaaeaacqGHciITcaWG4bWaaSbaaSqaaiaa dMgaaeqaaaGcbaGaeyOaIyRaeqOVdG3aaSbaaSqaaiabeg7aHbqaba aaaOGaeyypa0ZaaabuaeaadaWcaaqaaiabgkGi2kaad2eadaahaaWc beqaaiaadggaaaaakeaacqGHciITcqaH+oaEdaWgaaWcbaGaeqySde gabeaaaaGccaWG4bWaa0baaSqaaiaadMgaaeaacaWGHbaaaaqaaiaa dggaaeqaniabggHiLdaaaa@4D88@

and then using dA=| p α × p β |d ξ 1 d ξ 2 = η ˜ d ξ 1 d ξ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamizaiaadgeacqGH9aqpdaabdaqaai aahchadaahaaWcbeqaaiabeg7aHbaakiabgEna0kaahchadaahaaWc beqaaiabek7aIbaaaOGaay5bSlaawIa7aiaadsgacqaH+oaEdaWgaa WcbaGaaGymaaqabaGccaWGKbGaeqOVdG3aaSbaaSqaaiaaikdaaeqa aOGaeyypa0Jafq4TdGMbaGaacaWGKbGaeqOVdG3aaSbaaSqaaiaaig daaeqaaOGaamizaiabe67a4naaBaaaleaacaaIYaaabeaaaaa@4F8A@  .  A straightforward exercise shows that

η ˜ = ( p k 1 p k 1 )( p m 2 p m 2 ) ( p k 1 p k 2 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGafq4TdGMbaGaacqGH9aqpdaGcaaqaam aabmaabaGaamiCamaaDaaaleaacaWGRbaabaGaaGymaaaakiaadcha daqhaaWcbaGaam4AaaqaaiaaigdaaaaakiaawIcacaGLPaaadaqada qaaiaadchadaqhaaWcbaGaamyBaaqaaiaaikdaaaGccaWGWbWaa0ba aSqaaiaad2gaaeaacaaIYaaaaaGccaGLOaGaayzkaaGaeyOeI0Yaae WaaeaacaWGWbWaa0baaSqaaiaadUgaaeaacaaIXaaaaOGaamiCamaa DaaaleaacaWGRbaabaGaaGOmaaaaaOGaayjkaiaawMcaamaaCaaale qabaGaaGOmaaaaaeqaaaaa@4B22@

With this in hand, we can calculate

η ˜ u j b = 1 η ˜ { ( p m 2 p m 2 ) p k 1 p k 1 u j b +( p m 1 p m 1 ) p k 2 p k 2 u j b ( p m 1 p m 2 )( p k 2 p k 1 u j b + p k 1 p k 2 u j b ) } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaWaaSaaaeaacqGHciITcuaH3oaAgaacaa qaaiabgkGi2kaadwhadaqhaaWcbaGaamOAaaqaaiaadkgaaaaaaOGa eyypa0ZaaSaaaeaacaaIXaaabaGafq4TdGMbaGaaaaWaaiWaaeaada qadaqaaiaadchadaqhaaWcbaGaamyBaaqaaiaaikdaaaGccaWGWbWa a0baaSqaaiaad2gaaeaacaaIYaaaaaGccaGLOaGaayzkaaGaamiCam aaDaaaleaacaWGRbaabaGaaGymaaaakmaalaaabaGaeyOaIyRaamiC amaaDaaaleaacaWGRbaabaGaaGymaaaaaOqaaiabgkGi2kaadwhada qhaaWcbaGaamOAaaqaaiaadkgaaaaaaOGaey4kaSYaaeWaaeaacaWG WbWaa0baaSqaaiaad2gaaeaacaaIXaaaaOGaamiCamaaDaaaleaaca WGTbaabaGaaGymaaaaaOGaayjkaiaawMcaaiaadchadaqhaaWcbaGa am4AaaqaaiaaikdaaaGcdaWcaaqaaiabgkGi2kaadchadaqhaaWcba Gaam4AaaqaaiaaikdaaaaakeaacqGHciITcaWG1bWaa0baaSqaaiaa dQgaaeaacaWGIbaaaaaakiabgkHiTmaabmaabaGaamiCamaaDaaale aacaWGTbaabaGaaGymaaaakiaadchadaqhaaWcbaGaamyBaaqaaiaa ikdaaaaakiaawIcacaGLPaaadaqadaqaaiaadchadaqhaaWcbaGaam 4AaaqaaiaaikdaaaGcdaWcaaqaaiabgkGi2kaadchadaqhaaWcbaGa am4AaaqaaiaaigdaaaaakeaacqGHciITcaWG1bWaa0baaSqaaiaadQ gaaeaacaWGIbaaaaaakiabgUcaRiaadchadaqhaaWcbaGaam4Aaaqa aiaaigdaaaGcdaWcaaqaaiabgkGi2kaadchadaqhaaWcbaGaam4Aaa qaaiaaikdaaaaakeaacqGHciITcaWG1bWaa0baaSqaaiaadQgaaeaa caWGIbaaaaaaaOGaayjkaiaawMcaaaGaay5Eaiaaw2haaaaa@8660@

where

p i α u j b = M b ξ α δ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaWaaSaaaeaacqGHciITcaWGWbWaa0baaS qaaiaadMgaaeaacqaHXoqyaaaakeaacqGHciITcaWG1bWaa0baaSqa aiaadQgaaeaacaWGIbaaaaaakiabg2da9maalaaabaGaeyOaIyRaam ytamaaCaaaleqabaGaamOyaaaaaOqaaiabgkGi2kabe67a4naaBaaa leaacqaHXoqyaeqaaaaakiabes7aKnaaBaaaleaacaWGPbGaamOAaa qabaaaaa@4783@

so that the last term in the stiffness can be evaluated as

Ω e t i * N a η w k b d A 0 = 1 +1 1 +1 t i * N a ( ξ α ) η ˜ u k b d ξ 1 d ξ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaWaa8quaeaacaWG0bWaa0baaSqaaiaadM gaaeaacaGGQaaaaOGaamOtamaaCaaaleqabaGaamyyaaaakmaalaaa baGaeyOaIyRaeq4TdGgabaGaeyOaIyRaam4DamaaDaaaleaacaWGRb aabaGaamOyaaaaaaGccaWGKbGaamyqamaaBaaaleaacaaIWaaabeaa aeaacqGHciITcqqHPoWvdaWgaaadbaGaamyzaaqabaaaleqaniabgU IiYdGccqGH9aqpdaWdXbqaamaapehabaGaamiDamaaDaaaleaacaWG PbaabaGaaiOkaaaakiaad6eadaahaaWcbeqaaiaadggaaaGccaGGOa GaeqOVdG3aaSbaaSqaaiabeg7aHbqabaGccaGGPaWaaSaaaeaacqGH ciITcuaH3oaAgaacaaqaaiabgkGi2kaadwhadaqhaaWcbaGaam4Aaa qaaiaadkgaaaaaaOGaamizaiabe67a4naaBaaaleaacaaIXaaabeaa kiaadsgacqaH+oaEdaWgaaWcbaGaaGOmaaqabaaabaGaeyOeI0IaaG ymaaqaaiabgUcaRiaaigdaa0Gaey4kIipaaSqaaiabgkHiTiaaigda aeaacqGHRaWkcaaIXaaaniabgUIiYdaaaa@6AE5@

 

 

8.4.7 Example hyperelastic finite element code

 

It is evidently quite straightforward to extend a nonlinear small-strain finite element code to account for finite strains.  The only changes necessary are:

(1)   The general finite deformation measures must be calculated;

(2)   The material tangent stiffness is now a function of strain;

(3)   Two additional geometric terms must be added to the stiffness matrix MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  one of these is a volume integral over all the elements, the second is an integral over the boundary;

(4)   We have to deal with an unsymmetric stiffness matrix.

 

An example code is provided in the file FEM_2Dor3D_hyperelastic_static.mws.  For simplicity, the example is coded to apply a fixed nominal traction to the boundary (the geometric terms in the surface integral outlined above are not included).

 

An input file Hyperelastic_quad4.txt is also provided: the file sets up a simple 1 element problem.