Neo-Hookean Material

Constitutive Law

This MPM Material is an isotropic, elastic material in large strains using a hyperelastic formulation. It is a neo-Hookean material. Although a Mooney-Rivilin material is a neo-Hookean material of G₂=0, this material gives an alternatre neo-Hookean formation with slightly different stresses. In a neo-Hookean material, the stored stain energy is given by the expression:

      [math]\displaystyle{ W =U(J) + {G\over 2 } (I_{1}-3-2\ln J) }[/math]

where J (= det F) is relative volume change, G is low strain shear modulus, and [math]\displaystyle{ I_{1} }[/math] is the strain invariant

      [math]\displaystyle{ I_{1} = B_{xx}+B_{yy}+B_{zz} }[/math]

where B is the left Cauchy-Green strain tensor. This material allows three options for the U(J) term:

      [math]\displaystyle{ U(J) ={\lambda\over 2 }({1\over 2 }(J^2-1)-\ln J) \qquad({\rm UJOption}=0) }[/math]

      [math]\displaystyle{ U(J) ={\lambda\over 2 }(J-1)^2 \qquad\qquad\qquad({\rm UJOption}=1) }[/math]

      [math]\displaystyle{ U(J) ={\lambda\over 2 }(\ln J)^2 \qquad\qquad\qquad({\rm UJOption}=2) }[/math]

where [math]\displaystyle{ \lambda }[/math] is Lame modulus. At low strains, this material is equivalent to a linear elastic, isotropic material with shear modulus G and bulk modulus [math]\displaystyle{ \kappa = \lambda+2G/3 }[/math].

The Cauchy (or true stress) stress tensor is determined by differentiating the strain energy function:

      [math]\displaystyle{ \mathbf{\sigma} ={\lambda\over 2}\left(J-{1\over J}\right)\mathbf{I} + {G \over J } (\mathbf{B}-\mathbf{I}) }[/math]

The above stress is for UJOption=0; for the other two options, the first term changes to:

      [math]\displaystyle{ \lambda(J-1) \quad({\rm UJOption}=1) \qquad{\rm and}\qquad \lambda {\ln J\over J} \quad({\rm UJOption}=2) }[/math]

Note that UJOption=0 is the default option because it is the only pressure that correctly becomes infinite for both J approaching 0 and J approaching infinity.

The stresses can be alternative be divided into pressure and deviatoric stress, [math]\displaystyle{ \mathbf{\sigma} = -p \mathbf{I} + \bar \mathbf{\sigma} }[/math] resulting in:

      [math]\displaystyle{ p = {\lambda\over 2}\left(J-{1\over J}\right) + {G\over J}\left({I_1\over 3}-1\right) }[/math]

      [math]\displaystyle{ \bar\mathbf{\sigma} = {G \over J }{\rm dev}(\mathbf{B}) }[/math]

The first term in the pressure changes for the other two UJOption settings.

Residual Stresses

The stresses in the presence of residual stresses are evaluated by the process described for the Mooney-Rivlin material The results are:

      [math]\displaystyle{ p = {\lambda\over 2}\left(J_{eff}-{1\over (J_{eff}}\right) + {G\over (J_{eff}}\left({I_1\over 3J_{res}^{2/3}}-1\right) }[/math]

      [math]\displaystyle{ \bar\mathbf{\sigma} = {G J_{res}^{1/3}\over J }{\rm dev}(\mathbf{B}) }[/math]

where

      [math]\displaystyle{ J_{res} = \lambda_{res}^3 }[/math]