Chapter 8

 

Theory and Implementation of the Finite Element Method

 

 

 

8.3 Finite element method for nonlinear (hypoelastic) materials:
 

The finite element method can also solve boundary value problems for inelastic solids.  In this section, we show how to extend the finite element method to nonlinear problems. For the time being, we restrict attention to small deformations.  Furthermore, before attempting to solve problems involving the rather complex (load history dependent) plastic stress-strain relations, we begin by setting up the finite element method for static, hypoelastic problems.   We will idealize the stress-strain behavior of the material using the simple hypoelastic constitutive law described in Section 3.2.

 

 

8.3.1 Summary of governing equations

 

We shall solve the following boundary value problem.  Given:

1.      The shape of the solid in its unloaded condition R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8XjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamOuaaaa@31A4@

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 will use b to denote force per unit volume rather than force per unit mass, to avoid having to write out the mass density all the time)

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 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 R;

4.      The material constants n, σ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeq4Wdm3aaSbaaSqaaiaaicdaaeqaaa aa@3378@  and ε 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeqyTdu2aaSbaaSqaaiaaicdaaeqaaa aa@335C@  for the hypoelastic constitutive law described in Section 3.5;

Calculate displacements, strains and stresses u i , ε ij , σ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhadaWgaaWcbaGaamyAaaqabaGcca GGSaGaeqyTdu2aaSbaaSqaaiaadMgacaWGQbaabeaakiaacYcacqaH dpWCdaWgaaWcbaGaamyAaiaadQgaaeqaaaaa@3B6C@  satisfying the following equations

1.      displacement equation ε ij = 1 2 ( u i / x j + u j / x i ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeqyTdu2aaSbaaSqaaiaadMgacaWGQb aabeaakiabg2da9maalaaabaGaaGymaaqaaiaaikdaaaGaaiikaiab gkGi2kaadwhadaWgaaWcbaGaamyAaaqabaGccaGGVaGaeyOaIyRaam iEamaaBaaaleaacaWGQbaabeaakiabgUcaRiabgkGi2kaadwhadaWg aaWcbaGaamOAaaqabaGccaGGVaGaeyOaIyRaamiEamaaBaaaleaaca WGPbaabeaakiaacMcaaaa@48CF@

2.      The equation of static equilibrium for stresses σ ij / x i + b j =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeyOaIyRaeq4Wdm3aaSbaaSqaaiaadM gacaWGQbaabeaakiaac+cacqGHciITcaWG4bWaaSbaaSqaaiaadMga aeqaaOGaey4kaSIaamOyamaaBaaaleaacaWGQbaabeaakiabg2da9i aaicdaaaa@3EF3@

3.      The boundary conditions on displacement and stress

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 biaabeqaaiqabaWaaaGcbaGaamyDamaaBaaaleaacaWGPbaabeaaki abg2da9iaadwhadaqhaaWcbaGaamyAaaqaaiaacQcaaaGccaaMc8Ua aGPaVlaab+gacaqGUbGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cq GHciITdaWgaaWcbaGaaGymaaqabaGccaWGsbGaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8Uaeq4Wdm3aaSbaaSqaaiaadMgacaWGQbaabeaakiaad6ga daWgaaWcbaGaamyAaaqabaGccqGH9aqpcaWG0bWaa0baaSqaaiaadQ gaaeaacaGGQaaaaOGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaab+gacaqGUbGaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7cqGHciITdaWgaaWcbaGaaGOmaaqabaGccaWGsbaaaa@7A78@

4.      The hypoelastic constitutive law, which relates stress to strain as follows

σ ij = S ij + σ kk δ ij /3 S ij = 2 3 σ e e ij ε e σ kk = E 12ν 1 3 ε kk MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeq4Wdm3aaSbaaSqaaiaadMgacaWGQb aabeaakiabg2da9iaadofadaWgaaWcbaGaamyAaiaadQgaaeqaaOGa ey4kaSIaeq4Wdm3aaSbaaSqaaiaadUgacaWGRbaabeaakiabes7aKn aaBaaaleaacaWGPbGaamOAaaqabaGccaGGVaGaaG4maiaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaadofadaWgaaWcbaGa amyAaiaadQgaaeqaaOGaeyypa0ZaaSaaaeaacaaIYaaabaGaaG4maa aacqaHdpWCdaWgaaWcbaGaamyzaaqabaGcdaWcaaqaaiaadwgadaWg aaWcbaGaamyAaiaadQgaaeqaaaGcbaGaeqyTdu2aaSbaaSqaaiaadw gaaeqaaaaakiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Uaeq 4Wdm3aaSbaaSqaaiaadUgacaWGRbaabeaakiabg2da9maalaaabaGa amyraaqaaiaaigdacqGHsislcaaIYaGaeqyVd4gaamaalaaabaGaaG ymaaqaaiaaiodaaaGaeqyTdu2aaSbaaSqaaiaadUgacaWGRbaabeaa aaa@8C79@

where

e ij = ε ij 1 3 ε kk δ ij ε e = 2 3 e ij e ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamyzamaaBaaaleaacaWGPbGaamOAaa qabaGccqGH9aqpcqaH1oqzdaWgaaWcbaGaamyAaiaadQgaaeqaaOGa eyOeI0YaaSaaaeaacaaIXaaabaGaaG4maaaacqaH1oqzdaWgaaWcba Gaam4AaiaadUgaaeqaaOGaeqiTdq2aaSbaaSqaaiaadMgacaWGQbaa beaakiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlab ew7aLnaaBaaaleaacaWGLbaabeaakiabg2da9maakaaabaWaaSaaae aacaaIYaaabaGaaG4maaaacaWGLbWaaSbaaSqaaiaadMgacaWGQbaa beaakiaadwgadaWgaaWcbaGaamyAaiaadQgaaeqaaaqabaaaaa@64F1@

σ e σ 0 ={ 1+ n 2 (n1) 2 ( n n1 ε e ε 0 ) 2 1 n1 ε e ε 0 ( ε e ε 0 ) 1/n ε e ε 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaWaaSaaaeaacqaHdpWCdaWgaaWcbaGaam yzaaqabaaakeaacqaHdpWCdaWgaaWcbaGaaGimaaqabaaaaOGaeyyp a0ZaaiqaaeaafaqabeGabaaabaWaaOaaaeaadaWcaaqaaiaaigdacq GHRaWkcaWGUbWaaWbaaSqabeaacaaIYaaaaaGcbaGaaiikaiaad6ga cqGHsislcaaIXaGaaiykamaaCaaaleqabaGaaGOmaaaaaaGccqGHsi sldaqadaqaamaalaaabaGaamOBaaqaaiaad6gacqGHsislcaaIXaaa aiabgkHiTmaalaaabaGaeqyTdu2aaSbaaSqaaiaadwgaaeqaaaGcba GaeqyTdu2aaSbaaSqaaiaaicdaaeqaaaaaaOGaayjkaiaawMcaamaa CaaaleqabaGaaGOmaaaaaeqaaOGaaGPaVlaaykW7caaMc8UaeyOeI0 IaaGPaVlaaykW7caaMc8UaaGPaVpaalaaabaGaaGymaaqaaiaad6ga cqGHsislcaaIXaaaaiaaykW7caaMc8UaaGPaVlaaykW7cqaH1oqzda WgaaWcbaGaamyzaaqabaGccqGHKjYOcqaH1oqzdaWgaaWcbaGaaGim aaqabaaakeaadaqadaqaamaalaaabaGaeqyTdu2aaSbaaSqaaiaadw gaaeqaaaGcbaGaeqyTdu2aaSbaaSqaaiaaicdaaeqaaaaaaOGaayjk aiaawMcaamaaCaaaleqabaGaaGymaiaac+cacaWGUbaaaOGaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqaH1o qzdaWgaaWcbaGaamyzaaqabaGccqGHLjYScqaH1oqzdaWgaaWcbaGa aGimaaqabaaaaaGccaGL7baaaaa@DFEB@

and E=n σ 0 / ε 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamyraiabg2da9iaad6gacqaHdpWCda WgaaWcbaGaaGimaaqabaGccaGGVaGaeqyTdu2aaSbaaSqaaiaaicda aeqaaaaa@3985@  is the slope of the uniaxial stress-strain curve at ε e =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeqyTdu2aaSbaaSqaaiaadwgaaeqaaO Gaeyypa0JaaGimaaaa@3556@

 

 

The uniaxial stress-strain curve for this material is illustrated in the figure.  The material is elastic, in that it is perfectly reversible, but the stresses are related to strains by a nonlinear function (a power-law in this case).  This material model does not describe any actual material, but is sometimes used to approximate the more complicated stress-strain laws for plastic materials.

 

 

 

 

8.3.2 Governing equations in terms of the Virtual Work Principle

 

As in all FEM analysis, the stress equilibrium equation is replaced by the equivalent statement of the principle of  virtual work.  Thus, u i , ε ij , σ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhadaWgaaWcbaGaamyAaaqabaGcca GGSaGaeqyTdu2aaSbaaSqaaiaadMgacaWGQbaabeaakiaacYcacqaH dpWCdaWgaaWcbaGaamyAaiaadQgaaeqaaaaa@3B6C@  are determined as follows. 

1.      First, calculate a displacement field that satisfies

R σ ij [ u k ] δ v i x j dV R b i δ v i dV 2 R t i * δ v i dA=0 u i = u i * on 1 R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGceaqabeaadaWdrbqaaiabeo8aZnaaBaaale aacaWGPbGaamOAaaqabaGccaGGBbGaamyDamaaBaaaleaacaWGRbaa beaakiaac2fadaWcaaqaaiabgkGi2kabes7aKjaadAhadaWgaaWcba GaamyAaaqabaaakeaacqGHciITcaWG4bWaaSbaaSqaaiaadQgaaeqa aaaakiaadsgacaWGwbGaeyOeI0Yaa8quaeaacaWGIbWaaSbaaSqaai aadMgaaeqaaOGaeqiTdqMaamODamaaBaaaleaacaWGPbaabeaakiaa dsgacaWGwbGaeyOeI0Yaa8quaeaacaWG0bWaa0baaSqaaiaadMgaae aacaGGQaaaaOGaeqiTdqMaamODamaaBaaaleaacaWGPbaabeaakiaa dsgacaWGbbGaeyypa0JaaGimaaWcbaGaeyOaIy7aaSbaaWqaaiaaik daaeqaaSGaamOuaaqab0Gaey4kIipaaSqaaiaadkfaaeqaniabgUIi YdaaleaacaWGsbaabeqdcqGHRiI8aaGcbaGaamyDamaaBaaaleaaca WGPbaabeaakiabg2da9iaadwhadaqhaaWcbaGaamyAaaqaaiaacQca aaGccaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8Uaae4Baiaab6gacaaMc8UaaGPaVlabgkGi2oaa BaaaleaacaaIXaaabeaakiaadkfaaaaa@7FCB@

for all virtual velocity fields δ v i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeqiTdqMaamODamaaBaaaleaacaWGPb aabeaaaaa@3489@  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, the notation σ ij [ u k ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGPbGaamOAaa qabaGccaGGBbGaamyDamaaBaaaleaacaWGRbaabeaakiaac2faaaa@381E@  is used to show that the stress in the solid depends on the displacement field (through the strain-displacement relation and the constitutive equations).

2.      Compute the strains from the definition ε ij = 1 2 ( u i / x j + u j / x i ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeqyTdu2aaSbaaSqaaiaadMgacaWGQb aabeaakiabg2da9maalaaabaGaaGymaaqaaiaaikdaaaGaaiikaiab gkGi2kaadwhadaWgaaWcbaGaamyAaaqabaGccaGGVaGaeyOaIyRaam iEamaaBaaaleaacaWGQbaabeaakiabgUcaRiabgkGi2kaadwhadaWg aaWcbaGaamOAaaqabaGccaGGVaGaeyOaIyRaamiEamaaBaaaleaaca WGPbaabeaakiaacMcaaaa@48CF@

3.      Compute the stresses from the stress-strain law σ ij = C ijkl ε kl MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeq4Wdm3aaSbaaSqaaiaadMgacaWGQb aabeaakiabg2da9iaadoeadaWgaaWcbaGaamyAaiaadQgacaWGRbGa amiBaaqabaGccqaH1oqzdaWgaaWcbaGaam4AaiaadYgaaeqaaaaa@3E1B@

The stress will automatically satisfy the equation of equilibrium, so all the field equations and boundary conditions will be satisfied.

 

The procedure to solve the equations is conceptually identical to the linear elastic solution found in Section 8.1.1 and 8.1.2.  The only complication is that the stress is now a nonlinear function of the strains, so the virtual work equation is a nonlinear function of the displacement field.  It must therefore be solved by iteration.

 

 

8.3.3 Finite element equations

 

The finite 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 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@ .  The finite element equations are then set up as follows

1.      The displacement 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@

2.      Observe that we can compute the stress corresponding to a given displacement field by first computing the strain

ε ij = 1 2 ( u i x j + u j x i )= 1 2 a=1 n ( N a x j u i a + N a x i u j a ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeqyTdu2aaSbaaSqaaiaadMgacaWGQb aabeaakiabg2da9maalaaabaGaaGymaaqaaiaaikdaaaWaaeWaaeaa daWcaaqaaiabgkGi2kaadwhadaWgaaWcbaGaamyAaaqabaaakeaacq GHciITcaWG4bWaaSbaaSqaaiaadQgaaeqaaaaakiabgUcaRmaalaaa baGaeyOaIyRaamyDamaaBaaaleaacaWGQbaabeaaaOqaaiabgkGi2k aadIhadaWgaaWcbaGaamyAaaqabaaaaaGccaGLOaGaayzkaaGaeyyp a0ZaaSaaaeaacaaIXaaabaGaaGOmaaaadaaeWbqaamaabmaabaWaaS aaaeaacqGHciITcaWGobWaaWbaaSqabeaacaWGHbaaaaGcbaGaeyOa IyRaamiEamaaBaaaleaacaWGQbaabeaaaaGccaWG1bWaa0baaSqaai aadMgaaeaacaWGHbaaaOGaey4kaSYaaSaaaeaacqGHciITcaWGobWa aWbaaSqabeaacaWGHbaaaaGcbaGaeyOaIyRaamiEamaaBaaaleaaca WGPbaabeaaaaGccaWG1bWaa0baaSqaaiaadQgaaeaacaWGHbaaaaGc caGLOaGaayzkaaaaleaacaWGHbGaeyypa0JaaGymaaqaaiaad6gaa0 GaeyyeIuoaaaa@6673@

and then using the constitutive law to compute the stresses. We write this functional relationship as

σ ij = σ ij [ ε kl ( u i a ) ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeq4Wdm3aaSbaaSqaaiaadMgacaWGQb aabeaakiabg2da9iabeo8aZnaaBaaaleaacaWGPbGaamOAaaqabaGc daWadaqaaiabew7aLnaaBaaaleaacaWGRbGaamiBaaqabaGccaGGOa GaamyDamaaDaaaleaacaWGPbaabaGaamyyaaaakiaacMcaaiaawUfa caGLDbaaaaa@438F@

3.      Substituting into the principle of virtual work, we see that

{ R σ ij [ ε kl ( u i a ) ] N a x j dV R b i N a dV R t i * N a dA }δ v i a =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaWaaiWaaeaadaWdrbqaaiabeo8aZnaaBa aaleaacaWGPbGaamOAaaqabaGcdaWadaqaaiabew7aLnaaBaaaleaa caWGRbGaamiBaaqabaGccaGGOaGaamyDamaaDaaaleaacaWGPbaaba GaamyyaaaakiaacMcaaiaawUfacaGLDbaadaWcaaqaaiabgkGi2kaa d6eadaahaaWcbeqaaiaadggaaaaakeaacqGHciITcaWG4bWaaSbaaS qaaiaadQgaaeqaaaaakiaadsgacaWGwbGaeyOeI0Yaa8quaeaacaWG IbWaaSbaaSqaaiaadMgaaeqaaOGaamOtamaaCaaaleqabaGaamyyaa aakiaadsgacaWGwbGaeyOeI0Yaa8quaeaacaWG0bWaa0baaSqaaiaa dMgaaeaacaGGQaaaaOGaamOtamaaCaaaleqabaGaamyyaaaakiaads gacaWGbbaaleaacqGHciITcaWGsbaabeqdcqGHRiI8aaWcbaGaamOu aaqab0Gaey4kIipaaSqaaiaadkfaaeqaniabgUIiYdaakiaawUhaca GL9baacqaH0oazcaWG2bWaa0baaSqaaiaadMgaaeaacaWGHbaaaOGa eyypa0JaaGimaaaa@68D8@

and since this must hold for all δ v i a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeqiTdqMaamODamaaDaaaleaacaWGPb aabaGaamyyaaaaaaa@3570@  we must ensure that

R σ ij [ ε kl ( u i a ) ] N a x j dV R b i N a dV R t i * N a dA =0{a,i}: x k a not on  1 R 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 biaabeqaaiqabaWaaaGceaqabeaadaWdrbqaaiabeo8aZnaaBaaale aacaWGPbGaamOAaaqabaGcdaWadaqaaiabew7aLnaaBaaaleaacaWG RbGaamiBaaqabaGccaGGOaGaamyDamaaDaaaleaacaWGPbaabaGaam yyaaaakiaacMcaaiaawUfacaGLDbaadaWcaaqaaiabgkGi2kaad6ea daahaaWcbeqaaiaadggaaaaakeaacqGHciITcaWG4bWaaSbaaSqaai aadQgaaeqaaaaakiaadsgacaWGwbGaeyOeI0Yaa8quaeaacaWGIbWa aSbaaSqaaiaadMgaaeqaaOGaamOtamaaCaaaleqabaGaamyyaaaaki aadsgacaWGwbGaeyOeI0Yaa8quaeaacaWG0bWaa0baaSqaaiaadMga aeaacaGGQaaaaOGaamOtamaaCaaaleqabaGaamyyaaaakiaadsgaca WGbbaaleaacqGHciITcaWGsbaabeqdcqGHRiI8aaWcbaGaamOuaaqa b0Gaey4kIipaaSqaaiaadkfaaeqaniabgUIiYdGccqGH9aqpcaaIWa GaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlabgcGiIiaacUhacaWGHbGaaiilaiaadMga caGG9bGaaGPaVlaaykW7caaMc8UaaiOoaiaaykW7caaMc8UaaGPaVl aadIhadaqhaaWcbaGaam4AaaqaaiaadggaaaGccaaMc8UaaeOBaiaa b+gacaqG0bGaaeiiaiaab+gacaqGUbGaaeiiaiabgkGi2oaaBaaale aacaqGXaaabeaakiaadkfaaeaacaWG1bWaa0baaSqaaiaadMgaaeaa caWGHbaaaOGaeyypa0JaamyDamaaDaaaleaacaWGPbaabaGaaiOkaa aakiaacIcacaWG4bWaa0baaSqaaiaadMgaaeaacaWGHbaaaOGaaiyk aiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlabgcGiIiaacUhacaWGHb GaaiilaiaadMgacaGG9bGaaGPaVlaaykW7caaMc8UaaiOoaiaaykW7 caaMc8UaamiEamaaDaaaleaacaWGRbaabaGaamyyaaaakiaaykW7ca aMc8Uaae4Baiaab6gacaqGGaGaeyOaIy7aaSbaaSqaaiaabgdaaeqa aOGaamOuaaaaaa@E19D@

 

This is a set of n equations in n unknowns, very similar to those we obtained for linear elastostatic problems, but now the equations are nonlinear, because the stress is a nonlinear function of the unknown nodal displacements u i a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8YjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamyDamaaDaaaleaacaWGPbaabaGaam yyaaaaaaa@33D8@ .

 

 

8.3.4 Solving the finite element equations using Newton Raphson Iteration

 

We can solve the nonlinear equations using Newton-Raphson iteration, as follows.

 

1.      We 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).

2.      We then 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, of course, we would want the correction to satisfy

σ ij [ ε kl ( w i a +d w i a ) ] N a x j dV b i N a dV t i * N a dA =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaWaa8quaeaacqaHdpWCdaWgaaWcbaGaam yAaiaadQgaaeqaaOWaamWaaeaacqaH1oqzdaWgaaWcbaGaam4Aaiaa dYgaaeqaaOGaaiikaiaadEhadaqhaaWcbaGaamyAaaqaaiaadggaaa GccqGHRaWkcaWGKbGaam4DamaaDaaaleaacaWGPbaabaGaamyyaaaa kiaacMcaaiaawUfacaGLDbaadaWcaaqaaiabgkGi2kaad6eadaahaa WcbeqaaiaadggaaaaakeaacqGHciITcaWG4bWaaSbaaSqaaiaadQga aeqaaaaakiaadsgacaWGwbGaeyOeI0Yaa8quaeaacaWGIbWaaSbaaS qaaiaadMgaaeqaaOGaamOtamaaCaaaleqabaGaamyyaaaakiaadsga caWGwbGaeyOeI0Yaa8quaeaacaWG0bWaa0baaSqaaiaadMgaaeaaca GGQaaaaOGaamOtamaaCaaaleqabaGaamyyaaaakiaadsgacaWGbbaa leaacqGHciITcqWIDesOaeqaniabgUIiYdaaleaacqWIDesOaeqani abgUIiYdaaleaacqWIDesOaeqaniabgUIiYdGccqGH9aqpcaaIWaaa aa@689B@

but since we can’t solve this, we linearize in d w i a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamizaiaadEhadaqhaaWcbaGaamyAaa qaaiaadggaaaaaaa@34B5@  and set

R { σ ij [ ε kl ( w i b ) ]+ σ ij ε lm ε lm u k b d w k b } N a x j dV R b i N a dV R t i * N a dA =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaWaa8quaeaadaGadaqaaiabeo8aZnaaBa aaleaacaWGPbGaamOAaaqabaGcdaWadaqaaiabew7aLnaaBaaaleaa caWGRbGaamiBaaqabaGccaGGOaGaam4DamaaDaaaleaacaWGPbaaba GaamOyaaaakiaacMcaaiaawUfacaGLDbaacqGHRaWkdaWcaaqaaiab gkGi2kabeo8aZnaaBaaaleaacaWGPbGaamOAaaqabaaakeaacqGHci ITcqaH1oqzdaWgaaWcbaGaamiBaiaad2gaaeqaaaaakmaalaaabaGa eyOaIyRaeqyTdu2aaSbaaSqaaiaadYgacaWGTbaabeaaaOqaaiabgk Gi2kaadwhadaqhaaWcbaGaam4AaaqaaiaadkgaaaaaaOGaamizaiaa dEhadaqhaaWcbaGaam4AaaqaaiaadkgaaaaakiaawUhacaGL9baada WcaaqaaiabgkGi2kaad6eadaahaaWcbeqaaiaadggaaaaakeaacqGH ciITcaWG4bWaaSbaaSqaaiaadQgaaeqaaaaakiaadsgacaWGwbGaey OeI0Yaa8quaeaacaWGIbWaaSbaaSqaaiaadMgaaeqaaOGaamOtamaa CaaaleqabaGaamyyaaaakiaadsgacaWGwbGaeyOeI0Yaa8quaeaaca WG0bWaa0baaSqaaiaadMgaaeaacaGGQaaaaOGaamOtamaaCaaaleqa baGaamyyaaaakiaadsgacaWGbbaaleaacqGHciITcaWGsbaabeqdcq GHRiI8aaWcbaGaamOuaaqab0Gaey4kIipaaSqaaiaadkfaaeqaniab gUIiYdGccqGH9aqpcaaIWaaaaa@7D1B@

3.      Recall that

ε lm = 1 2 a=1 n N a x m u l a + N a x l u m a ε lm u k b = 1 2 a=1 n N a x m δ ab δ lk + N a x l δ ab δ mk MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGceaqabeaacqaH1oqzdaWgaaWcbaGaamiBai aad2gaaeqaaOGaeyypa0ZaaSaaaeaacaaIXaaabaGaaGOmaaaadaae WbqaamaalaaabaGaeyOaIyRaamOtamaaCaaaleqabaGaamyyaaaaaO qaaiabgkGi2kaadIhadaWgaaWcbaGaamyBaaqabaaaaOGaamyDamaa DaaaleaacaWGSbaabaGaamyyaaaakiabgUcaRmaalaaabaGaeyOaIy RaamOtamaaCaaaleqabaGaamyyaaaaaOqaaiabgkGi2kaadIhadaWg aaWcbaGaamiBaaqabaaaaOGaamyDamaaDaaaleaacaWGTbaabaGaam yyaaaaaeaacaWGHbGaeyypa0JaaGymaaqaaiaad6gaa0GaeyyeIuoa aOqaaiabgkDiEpaalaaabaGaeyOaIyRaeqyTdu2aaSbaaSqaaiaadY gacaWGTbaabeaaaOqaaiabgkGi2kaadwhadaqhaaWcbaGaam4Aaaqa aiaadkgaaaaaaOGaeyypa0ZaaSaaaeaacaaIXaaabaGaaGOmaaaada aeWbqaamaalaaabaGaeyOaIyRaamOtamaaCaaaleqabaGaamyyaaaa aOqaaiabgkGi2kaadIhadaWgaaWcbaGaamyBaaqabaaaaOGaeqiTdq 2aaSbaaSqaaiaadggacaWGIbaabeaakiabes7aKnaaBaaaleaacaWG SbGaam4AaaqabaGccqGHRaWkdaWcaaqaaiabgkGi2kaad6eadaahaa WcbeqaaiaadggaaaaakeaacqGHciITcaWG4bWaaSbaaSqaaiaadYga aeqaaaaakiabes7aKnaaBaaaleaacaWGHbGaamOyaaqabaGccqaH0o azdaWgaaWcbaGaamyBaiaadUgaaeqaaaqaaiaadggacqGH9aqpcaaI XaaabaGaamOBaaqdcqGHris5aaaaaa@83A9@

Note also that

σ ij ε ml = σ ij ε lm MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaWaaSaaaeaacqGHciITcqaHdpWCdaWgaa WcbaGaamyAaiaadQgaaeqaaaGcbaGaeyOaIyRaeqyTdu2aaSbaaSqa aiaad2gacaWGSbaabeaaaaGccqGH9aqpdaWcaaqaaiabgkGi2kabeo 8aZnaaBaaaleaacaWGPbGaamOAaaqabaaakeaacqGHciITcqaH1oqz daWgaaWcbaGaamiBaiaad2gaaeqaaaaaaaa@46AF@

so

σ ij ε lm ε lm u k b = σ ij ε km N b x m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaWaaSaaaeaacqGHciITcqaHdpWCdaWgaa WcbaGaamyAaiaadQgaaeqaaaGcbaGaeyOaIyRaeqyTdu2aaSbaaSqa aiaadYgacaWGTbaabeaaaaGcdaWcaaqaaiabgkGi2kabew7aLnaaBa aaleaacaWGSbGaamyBaaqabaaakeaacqGHciITcaWG1bWaa0baaSqa aiaadUgaaeaacaWGIbaaaaaakiabg2da9maalaaabaGaeyOaIyRaeq 4Wdm3aaSbaaSqaaiaadMgacaWGQbaabeaaaOqaaiabgkGi2kabew7a LnaaBaaaleaacaWGRbGaamyBaaqabaaaaOWaaSaaaeaacqGHciITca WGobWaaWbaaSqabeaacaWGIbaaaaGcbaGaeyOaIyRaamiEamaaBaaa leaacaWGTbaabeaaaaaaaa@5744@

and finally

R σ ij ε kl N a x j N b x l dVd w k b + R σ ij [ ε kl ( w i b ) ] N a x j dV R b i N a dV R t i * N a dA =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaWaa8quaeaadaWcaaqaaiabgkGi2kabeo 8aZnaaBaaaleaacaWGPbGaamOAaaqabaaakeaacqGHciITcqaH1oqz daWgaaWcbaGaam4AaiaadYgaaeqaaaaakmaalaaabaGaeyOaIyRaam OtamaaCaaaleqabaGaamyyaaaaaOqaaiabgkGi2kaadIhadaWgaaWc baGaamOAaaqabaaaaOWaaSaaaeaacqGHciITcaWGobWaaWbaaSqabe aacaWGIbaaaaGcbaGaeyOaIyRaamiEamaaBaaaleaacaWGSbaabeaa aaGccaWGKbGaamOvaiaadsgacaWG3bWaa0baaSqaaiaadUgaaeaaca WGIbaaaaqaaiaadkfaaeqaniabgUIiYdGccqGHRaWkdaWdrbqaaiab eo8aZnaaBaaaleaacaWGPbGaamOAaaqabaGcdaWadaqaaiabew7aLn aaBaaaleaacaWGRbGaamiBaaqabaGccaGGOaGaam4DamaaDaaaleaa caWGPbaabaGaamOyaaaakiaacMcaaiaawUfacaGLDbaadaWcaaqaai abgkGi2kaad6eadaahaaWcbeqaaiaadggaaaaakeaacqGHciITcaWG 4bWaaSbaaSqaaiaadQgaaeqaaaaakiaadsgacaWGwbGaeyOeI0Yaa8 quaeaacaWGIbWaaSbaaSqaaiaadMgaaeqaaOGaamOtamaaCaaaleqa baGaamyyaaaakiaadsgacaWGwbGaeyOeI0Yaa8quaeaacaWG0bWaa0 baaSqaaiaadMgaaeaacaGGQaaaaOGaamOtamaaCaaaleqabaGaamyy aaaakiaadsgacaWGbbaaleaacqGHciITcaWGsbaabeqdcqGHRiI8aa WcbaGaamOuaaqab0Gaey4kIipaaSqaaiaadkfaaeqaniabgUIiYdGc cqGH9aqpcaaIWaaaaa@83FD@

4.      This is evidently a system of linear equations for the correction d w i a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8YjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamizaiaadEhadaqhaaWcbaGaamyAaa qaaiaadggaaaaaaa@34C3@  of the form

K aibk d w k b + R i a F i a =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaam4samaaBaaaleaacaWGHbGaamyAai aadkgacaWGRbaabeaakiaadsgacaWG3bWaa0baaSqaaiaadUgaaeaa caWGIbaaaOGaey4kaSIaamOuamaaDaaaleaacaWGPbaabaGaamyyaa aakiabgkHiTiaadAeadaqhaaWcbaGaamyAaaqaaiaadggaaaGccqGH 9aqpcaaIWaaaaa@42BA@

with

K aibk = R σ ij ε kl N a x j N b x l dV R i a = R σ ij [ ε kl ( w i b ) ] N a x j dV F i a = R b i N a dV+ R t i * N a dA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaam4samaaBaaaleaacaWGHbGaamyAai aadkgacaWGRbaabeaakiabg2da9maapefabaWaaSaaaeaacqGHciIT cqaHdpWCdaWgaaWcbaGaamyAaiaadQgaaeqaaaGcbaGaeyOaIyRaeq yTdu2aaSbaaSqaaiaadUgacaWGSbaabeaaaaGcdaWcaaqaaiabgkGi 2kaad6eadaahaaWcbeqaaiaadggaaaaakeaacqGHciITcaWG4bWaaS baaSqaaiaadQgaaeqaaaaakmaalaaabaGaeyOaIyRaamOtamaaCaaa leqabaGaamOyaaaaaOqaaiabgkGi2kaadIhadaWgaaWcbaGaamiBaa qabaaaaOGaamizaiaadAfaaSqaaiaadkfaaeqaniabgUIiYdGccaaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWGsbWaa0baaSqaai aadMgaaeaacaWGHbaaaOGaeyypa0Zaa8quaeaacqaHdpWCdaWgaaWc baGaamyAaiaadQgaaeqaaOWaamWaaeaacqaH1oqzdaWgaaWcbaGaam 4AaiaadYgaaeqaaOGaaiikaiaadEhadaqhaaWcbaGaamyAaaqaaiaa dkgaaaGccaGGPaaacaGLBbGaayzxaaWaaSaaaeaacqGHciITcaWGob WaaWbaaSqabeaacaWGHbaaaaGcbaGaeyOaIyRaamiEamaaBaaaleaa caWGQbaabeaaaaGccaWGKbGaamOvaaWcbaGaamOuaaqab0Gaey4kIi pakiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWG gbWaa0baaSqaaiaadMgaaeaacaWGHbaaaOGaeyypa0Zaa8quaeaaca WGIbWaaSbaaSqaaiaadMgaaeqaaOGaamOtamaaCaaaleqabaGaamyy aaaakiaadsgacaWGwbGaey4kaSYaa8quaeaacaWG0bWaa0baaSqaai aadMgaaeaacaGGQaaaaOGaamOtamaaCaaaleqabaGaamyyaaaakiaa dsgacaWGbbaaleaacqGHciITcaWGsbaabeqdcqGHRiI8aaWcbaGaam Ouaaqab0Gaey4kIipaaaa@9E0F@

 

These expressions are almost identical to the equations we needed to solve for linear elastostatic problems.  There are only two differences: the stiffness contains the (strain dependent) material tangent moduli σ ij / ε kl MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeyOaIyRaeq4Wdm3aaSbaaSqaaiaadM gacaWGQbaabeaakiaac+cacqGHciITcqaH1oqzdaWgaaWcbaGaam4A aiaadYgaaeqaaaaa@3BD8@  instead of the elastic constants C ijkl MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaam4qamaaBaaaleaacaWGPbGaamOAai aadUgacaWGSbaabeaaaaa@3581@ , and we need to compute an extra term R i a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamOuamaaDaaaleaacaWGPbaabaGaam yyaaaaaaa@33A7@  in the residual force vector.  This is a straightforward exercise MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  the integral is divided up into contributions from each element, and evaluated numerically using Gaussian quadrature.

 

 

8.3.5 Tangent moduli for the hypoelastic solid

 

One painful aspect of nonlinear FEM is that the material tangent moduli σ ij / ε kl MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeyOaIyRaeq4Wdm3aaSbaaSqaaiaadM gacaWGQbaabeaakiaac+cacqGHciITcqaH1oqzdaWgaaWcbaGaam4A aiaadYgaaeqaaaaa@3BD8@  must be calculated.  This is usually a tedious algebraic exercise.  For the hypoelastic constitutive law used in our example, one can show (if I got the calculation right, that is)

d σ ij d ε kl = 4 9 ( E t E s ) e ij e kl ε e 2 + 2 3 E s ( δ ik δ jl + δ il δ jk 2 δ ij δ kl 3 )+ E 9(12ν) δ kl δ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaWaaSaaaeaacaWGKbGaeq4Wdm3aaSbaaS qaaiaadMgacaWGQbaabeaaaOqaaiaadsgacqaH1oqzdaWgaaWcbaGa am4AaiaadYgaaeqaaaaakiabg2da9maalaaabaGaaGinaaqaaiaaiM daaaWaaeWaaeaacaWGfbWaaSbaaSqaaiaadshaaeqaaOGaeyOeI0Ia amyramaaBaaaleaacaWGZbaabeaaaOGaayjkaiaawMcaamaalaaaba GaamyzamaaBaaaleaacaWGPbGaamOAaaqabaGccaWGLbWaaSbaaSqa aiaadUgacaWGSbaabeaaaOqaaiabew7aLnaaDaaaleaacaWGLbaaba GaaGOmaaaaaaGccqGHRaWkdaWcaaqaaiaaikdaaeaacaaIZaaaaiaa dweadaWgaaWcbaGaam4CaaqabaGccaGGOaWaaSaaaeaacqaH0oazda WgaaWcbaGaamyAaiaadUgaaeqaaOGaeqiTdq2aaSbaaSqaaiaadQga caWGSbaabeaakiabgUcaRiabes7aKnaaBaaaleaacaWGPbGaamiBaa qabaGccqaH0oazdaWgaaWcbaGaamOAaiaadUgaaeqaaaGcbaGaaGOm aaaacqGHsisldaWcaaqaaiabes7aKnaaBaaaleaacaWGPbGaamOAaa qabaGccqaH0oazdaWgaaWcbaGaam4AaiaadYgaaeqaaaGcbaGaaG4m aaaacaGGPaGaey4kaSYaaSaaaeaacaWGfbaabaGaaGyoaiaacIcaca aIXaGaeyOeI0IaaGOmaiabe27aUjaacMcaaaGaeqiTdq2aaSbaaSqa aiaadUgacaWGSbaabeaakiabes7aKnaaBaaaleaacaWGPbGaamOAaa qabaaaaa@7BB3@

where E s = σ e / ε e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamyramaaBaaaleaacaWGZbaabeaaki abg2da9iabeo8aZnaaBaaaleaacaWGLbaabeaakiaac+cacqaH1oqz daWgaaWcbaGaamyzaaqabaaaaa@3A20@  is the secant modulus, and E t =d σ e /d ε e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamyramaaBaaaleaacaWG0baabeaaki abg2da9iaadsgacqaHdpWCdaWgaaWcbaGaamyzaaqabaGccaGGVaGa amizaiabew7aLnaaBaaaleaacaWGLbaabeaaaaa@3BF3@  is the tangent modulus of the uniaxial stress-strain curve, as shown in the picture.  You can calculate formulas for the tangent and secant moduli using the relationships between σ e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGLbaabeaaaa a@3341@  and ε e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew7aLnaaBaaaleaacaWGLbaabeaaaa a@3325@  given in 8.3.1.

 

If you try to re-derive these expressions yourself, you may end up with a different answer for d S ij /d ε kl MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWGtbWaaSbaaSqaaiaadMgaca WGQbaabeaakiaac+cacaWGKbGaeqyTdu2aaSbaaSqaaiaadUgacaWG Sbaabeaaaaa@398C@ .  This is because derivatives of a symmetric tensor with respect to another symmetric tensor are not unique (there is an indeterminate skew part).  It is convenient to make the tangent modulus symmetric, so that the element (and global) stiffness matrices are symmetric.

 

 

 

8.3.6 Summary of the Newton-Raphson procedure for hypoelastic solids

 

It is evidently quite straightforward to extend a linear elasticity code to nonlinear problems.  The Newton-Raphson loop looks like this:

1.      Start with an initial guess for the solution w i a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaam4DamaaDaaaleaacaWGPbaabaGaam yyaaaaaaa@33CC@

2.      For the current guess, compute K aibk MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaam4samaaBaaaleaacaWGHbGaamyAai aadkgacaWGRbaabeaaaaa@3576@ , R i a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamOuamaaDaaaleaacaWGPbaabaGaam yyaaaaaaa@33A7@  and F i a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamOramaaDaaaleaacaWGPbaabaGaam yyaaaaaaa@339B@ , where

K aibk = R σ ij ε kl N a x j N b x l dV R i a = R σ ij [ ε kl ( w i b ) ] N a x j dV F i a = R b i N a dV+ R t i * N a dA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaam4samaaBaaaleaacaWGHbGaamyAai aadkgacaWGRbaabeaakiabg2da9maapefabaWaaSaaaeaacqGHciIT cqaHdpWCdaWgaaWcbaGaamyAaiaadQgaaeqaaaGcbaGaeyOaIyRaeq yTdu2aaSbaaSqaaiaadUgacaWGSbaabeaaaaGcdaWcaaqaaiabgkGi 2kaad6eadaahaaWcbeqaaiaadggaaaaakeaacqGHciITcaWG4bWaaS baaSqaaiaadQgaaeqaaaaakmaalaaabaGaeyOaIyRaamOtamaaCaaa leqabaGaamOyaaaaaOqaaiabgkGi2kaadIhadaWgaaWcbaGaamiBaa qabaaaaOGaamizaiaadAfaaSqaaiaadkfaaeqaniabgUIiYdGccaaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaadkfadaqhaaWcbaGaamyAaa qaaiaadggaaaGccqGH9aqpdaWdrbqaaiabeo8aZnaaBaaaleaacaWG PbGaamOAaaqabaGcdaWadaqaaiabew7aLnaaBaaaleaacaWGRbGaam iBaaqabaGccaGGOaGaam4DamaaDaaaleaacaWGPbaabaGaamOyaaaa kiaacMcaaiaawUfacaGLDbaadaWcaaqaaiabgkGi2kaad6eadaahaa WcbeqaaiaadggaaaaakeaacqGHciITcaWG4bWaaSbaaSqaaiaadQga aeqaaaaakiaadsgacaWGwbGaaGPaVlaaykW7caaMc8oaleaacaWGsb aabeqdcqGHRiI8aOGaaGPaVlaadAeadaqhaaWcbaGaamyAaaqaaiaa dggaaaGccqGH9aqpdaWdrbqaaiaadkgadaWgaaWcbaGaamyAaaqaba GccaWGobWaaWbaaSqabeaacaWGHbaaaOGaamizaiaadAfacqGHRaWk daWdrbqaaiaadshadaqhaaWcbaGaamyAaaqaaiaacQcaaaGccaWGob WaaWbaaSqabeaacaWGHbaaaOGaamizaiaadgeaaSqaaiabgkGi2kaa dkfaaeqaniabgUIiYdaaleaacaWGsbaabeqdcqGHRiI8aaaa@97E3@

and formulas for the tangent σ ij / ε kl MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeyOaIyRaeq4Wdm3aaSbaaSqaaiaadM gacaWGQbaabeaakiaac+cacqGHciITcqaH1oqzdaWgaaWcbaGaam4A aiaadYgaaeqaaaaa@3BD8@  are given in the preceding section.

3.      Modify the system of equations to enforce any displacement boundary constraints

4.      Solve K aibk d w k b = R i a + F i a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaam4samaaBaaaleaacaWGHbGaamyAai aadkgacaWGRbaabeaakiaadsgacaWG3bWaa0baaSqaaiaadUgaaeaa caWGIbaaaOGaeyypa0JaeyOeI0IaamOuamaaDaaaleaacaWGPbaaba GaamyyaaaakiabgUcaRiaadAeadaqhaaWcbaGaamyAaaqaaiaadgga aaaaaa@41F6@

5.      Let 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 yyaaaakiabg2da9iaadEhadaqhaaWcbaGaamyAaaqaaiaadggaaaGc cqGHRaWkcaWGKbGaam4DamaaDaaaleaacaWGPbaabaGaamyyaaaaaa a@3CAB@

6.      Check for convergence (more on this below)  - go to 2 if the solution has not yet converged.

 

Two methods may be used to check for convergence.

1.      You can check and see if the residual forces R i a + F i a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeyOeI0IaamOuamaaDaaaleaacaWGPb aabaGaamyyaaaakiabgUcaRiaadAeadaqhaaWcbaGaamyAaaqaaiaa dggaaaaaaa@384C@  are sufficiently small (they should vanish for an equilibrium stress field).  You could find the maximum value at any node, and ensure that it falls below a user defined tolerance, or you could use the rms error

e= 1 n a=1 n ( R i a F i a )( R i a F i a ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamyzaiabg2da9maakaaabaWaaSaaae aacaaIXaaabaGaamOBaaaadaaeWbqaamaabmaabaGaamOuamaaDaaa leaacaWGPbaabaGaamyyaaaakiabgkHiTiaadAeadaqhaaWcbaGaam yAaaqaaiaadggaaaaakiaawIcacaGLPaaadaqadaqaaiaadkfadaqh aaWcbaGaamyAaaqaaiaadggaaaGccqGHsislcaWGgbWaa0baaSqaai aadMgaaeaacaWGHbaaaaGccaGLOaGaayzkaaaaleaacaWGHbGaeyyp a0JaaGymaaqaaiaad6gaa0GaeyyeIuoaaSqabaaaaa@4AD0@

(where n is the number of nodes in the mesh) as an error measure.

2.      Alternatively, you can check the magnitude of the correction d w i a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamizaiaadEhadaqhaaWcbaGaamyAaa qaaiaadggaaaaaaa@34B5@  and stop iterating when either the maximum correction at any node or the rms correction falls below some specified tolerance.

In practice both criteria are often used.

 

 

 

8.3.7 What to do if the Newton-Raphson iterations don’t converge

 

There is no guarantee that the Newton-Raphson method will converge.  It will converge quadratically to the exact solution if the initial guess is sufficiently close to the correct answer, but if you are unlucky it may diverge or spiral hopelessly around forever without finding the correct solution.

 

The most common way to fix this is to apply the load in a series of increments instead of all at once.  The solution at the end of the preceding increment is used as the initial guess for the solution at the end of the next.  Convergence can sometimes be accelerated by extrapolating the solution from preceding time steps to find a better initial guess for the solution.

 

The other approach (used in desperation) is to update the approximation to the solution as w i a = w i a +αd w i a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaam4DamaaDaaaleaacaWGPbaabaGaam yyaaaakiabg2da9iaadEhadaqhaaWcbaGaamyAaaqaaiaadggaaaGc cqGHRaWkcqaHXoqycaWGKbGaam4DamaaDaaaleaacaWGPbaabaGaam yyaaaaaaa@3E4A@ , where α<1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8XjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeqySdeMaeyipaWJaaGymaaaa@342B@  is a numerical relaxation factor.  This slows convergence of the Newton-Raphson iterations, but increases the radius of convergence.

 

 

 

8.3.8 Variations on Newton-Raphson iteration

 

There are several variants on the fully consistent Newton Raphson scheme outlined in the preceding section MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  some are obvious and some are subtle.  These techniques are known collectively as `Quasi-Newton’ methods.

 

All these variations attempt to address the two major limitations of the fully consistent Newton method, which are (i) that the tangent matrix K aibk MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaam4samaaBaaaleaacaWGHbGaamyAai aadkgacaWGRbaabeaaaaa@3576@  must be recomputed during each iteration, and (ii) it is necessary to solve the system of equations K aibk d w k b = R i a + F i a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaam4samaaBaaaleaacaWGHbGaamyAai aadkgacaWGRbaabeaakiaadsgacaWG3bWaa0baaSqaaiaadUgaaeaa caWGIbaaaOGaeyypa0JaeyOeI0IaamOuamaaDaaaleaacaWGPbaaba GaamyyaaaakiabgUcaRiaadAeadaqhaaWcbaGaamyAaaqaaiaadgga aaaaaa@41F6@  repeatedly to obtain a convergent solution. 

 

A simple fix is not to bother recomputing K aibk MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaam4samaaBaaaleaacaWGHbGaamyAai aadkgacaWGRbaabeaaaaa@3576@ , or to recompute it occasionally.  This gives a quasi-Newton method as follows

1.      Start with an initial guess for the solution w i a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaam4DamaaDaaaleaacaWGPbaabaGaam yyaaaaaaa@33CC@

2.      Compute K aibk MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaam4samaaBaaaleaacaWGHbGaamyAai aadkgacaWGRbaabeaaaaa@3576@  

3.      Compute R i a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamOuamaaDaaaleaacaWGPbaabaGaam yyaaaaaaa@33A7@  for the current solution

4.      Modify the system of equations to enforce any displacement boundary constraints

5.      Solve K aibk d w k b = R i a + F i a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaam4samaaBaaaleaacaWGHbGaamyAai aadkgacaWGRbaabeaakiaadsgacaWG3bWaa0baaSqaaiaadUgaaeaa caWGIbaaaOGaeyypa0JaeyOeI0IaamOuamaaDaaaleaacaWGPbaaba GaamyyaaaakiabgUcaRiaadAeadaqhaaWcbaGaamyAaaqaaiaadgga aaaaaa@41F6@

6.      Let 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 yyaaaakiabg2da9iaadEhadaqhaaWcbaGaamyAaaqaaiaadggaaaGc cqGHRaWkcaWGKbGaam4DamaaDaaaleaacaWGPbaabaGaamyyaaaaaa a@3CAB@

7.      Check for convergence - go to 3 if the solution has not yet converged; go to 2 if you feel like doing some extra work to speed up convergence.

 

In this method equation solution can be speeded up further, by computing and storing the LU decomposition of K aibk MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaam4samaaBaaaleaacaWGHbGaamyAai aadkgacaWGRbaabeaaaaa@3576@  instead of K aibk MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaam4samaaBaaaleaacaWGHbGaamyAai aadkgacaWGRbaabeaaaaa@3576@  itself.  Equation solution then just involves back-substitution, which can be accomplished quite quickly.

 

A more subtle approach is to obtain a succession of improved approximations to K aibk 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaam4samaaDaaaleaacaWGHbGaamyAai aadkgacaWGRbaabaGaeyOeI0IaaGymaaaaaaa@371F@  directly, from the changes in residual F i a R i a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamOramaaDaaaleaacaWGPbaabaGaam yyaaaakiabgkHiTiaadkfadaqhaaWcbaGaamyAaaqaaiaadggaaaaa aa@376A@   and solution increments d w i a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamizaiaadEhadaqhaaWcbaGaamyAaa qaaiaadggaaaaaaa@34B5@  during successive iterations.   A very efficient implementation of the so-called BFGS algorithm is described by Matthies and Strang, Int J. Num. Meth in Eng. 14, 1613, (1979).

 

 

 

 

8.3.9 Example hypoelastic FEM code

 

As always, we provide a simple example FEM code to illustrate the actual implementation. 

 

1.      The code is in the file FEM_2Dor3D_hypoelastic_static.mws

2.      An example input file is in Hypoelastic_quad4.txt

 

Some notes on the example:

1.      The code and sample input file are set up to solve the problem illustrated on the right.

2.       The element deforms in plane strain and has the hypoelastic constitutive response described earlier with σ 0 =10, ε 0 =0.001n=10ν=0.3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8YjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeq4Wdm3aaSbaaSqaaiaaicdaaeqaaO Gaeyypa0JaaGymaiaaicdacaGGSaGaaGPaVlaaykW7caaMc8UaeqyT du2aaSbaaSqaaiaaicdaaeqaaOGaeyypa0JaaGimaiaac6cacaaIWa GaaGimaiaaigdacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaad6ga cqGH9aqpcaaIXaGaaGimaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaeqyVd4Maeyypa0JaaGimaiaac6cacaaIZaaaaa@5A57@ .

3.      The program applies load in a series of 5 increments, and uses consistent Newton-Raphson iteration to solve the nonlinear equations at each step.  When  you get to the appropriate part of the code,  you will see it printing out values for the error measures at each iteration. 

4.      Element strains and stresses are printed to a file on completion, and the stress-v-displacement curve for the element is plotted, as shown in the figure.

 

HEALTH WARNING: This demonstration code uses fully integrated elements and will in general perform very poorly because of volumetric locking.  For details of locking and how to avoid it see section 8.6.