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{ \mathbf{\sigma} = {G \over J }{\rm dev}(\mathbf{B}) }[/math]
The first term in the pressure changes for the other two UJOption settings.