Mooney Material

Constitutive Law

The Mooney-Rivlin material Mooney-Rivlin is an isotropic, elastic, hyperelastic material. The constitutive law derived from a strain energy function that is a function of the deformation gradient tensor F.

Regarding to the objectivity conditions and using the representation theorem,strain energy is represented by the invariants of a strain tensor, such as the left Cauchy-Green strain tensor.

      [math]\displaystyle{ W ={G_{1} \over 2 } (\bar I_{1}-3)+{G_{2} \over 2 }(\bar I_{2}-3)+{K\over 2 }(J-1)^2 }[/math]

where [math]\displaystyle{ G_{1} }[/math], [math]\displaystyle{ G_{2} }[/math] and K are material properties, [math]\displaystyle{ \bar I_{1} }[/math], [math]\displaystyle{ \bar I_{2} }[/math] and J are the invariant of the chosen strain tensor, with J=det F and

[math]\displaystyle{ \bar I_{1} ={B_{xx}+B_{yy}+B_{zz} \over J^{2/3}} }[/math]

[math]\displaystyle{ \bar I_{2} ={1 \over 2} (\bar I_{1}^2-{{B_{xx}^2+B_{yy}^2+B_{zz}^2+2B_{xy}^2+2B_{xz}^2+2B_{yz}^2} \over J^{2/3}}) }[/math]

For low strains, this material is equivalent for a linear elastic, isotropic material with shear modulus G1 + G2 and bulk modulus K. If G2 = 0, the material is a neo-Hookean material. See below for an alternate compressibility terms. Some hyperelastic rubber models assume incompressible materials, which corresponds to K → ∞; such models do not work in dynamic code (because wave speed is infinite).  The Cauchy (or true stress) is found by differentiating the strain energy to get

[math]\displaystyle{ \mathbf{\sigma} ={G_{1} \over J^{5/3} } (\mathbf{B}-{I_{1} \over3}\mathbf{I})+{G_{2} \over J^{7/3}} (I_{1} \mathbf{B}-\mathbf{B^2}-{2I_{2} \over3}\mathbf{I})+K(J-1)\mathbf{I} }[/math]

where I = J 2/3 I ̄ and I = J 4/3 I ̄ . The stress components can be divided into pressure,

deviatoric stress, s = σ + P, whichexplicitly evaluate to: