Mooney–Rivlin solid

In continuum mechanics, a Mooney–Rivlin solid is a hyperelastic material model where the strain energy density function ${\displaystyle W\,}$ is a linear combination of two invariants of the left Cauchy–Green deformation tensor ${\displaystyle {\boldsymbol {B}}}$. The model was proposed by Melvin Mooney in 1940 and expressed in terms of invariants by Ronald Rivlin in 1948.

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The strain energy density function for an incompressible Mooney–Rivlin material is

${\displaystyle W=C_{1}({\bar {I}}_{1}-3)+C_{2}({\bar {I}}_{2}-3),\,}$

where ${\displaystyle C_{1}}$ and ${\displaystyle C_{2}}$ are empirically determined material constants, and ${\displaystyle {\bar {I}}_{1}}$ and ${\displaystyle {\bar {I}}_{2}}$ are the first and the second invariant of ${\displaystyle {\bar {\boldsymbol {B}}}=(\det {\boldsymbol {B}})^{-1/3}{\boldsymbol {B}}}$ (the unimodular component of ${\displaystyle {\boldsymbol {B}}}$):

${\displaystyle {\begin{aligned}{\bar {I}}_{1}&=J^{-2/3}~I_{1},\quad I_{1}=\lambda _{1}^{2}+\lambda _{2}^{2}+\lambda _{3}^{2},\\{\bar {I}}_{2}&=J^{-4/3}~I_{2},\quad I_{2}=\lambda _{1}^{2}\lambda _{2}^{2}+\lambda _{2}^{2}\lambda _{3}^{2}+\lambda _{3}^{2}\lambda _{1}^{2}\end{aligned}}}$

where ${\displaystyle {\boldsymbol {F}}}$ is the deformation gradient and ${\displaystyle J=\det({\boldsymbol {F}})=\lambda _{1}\lambda _{2}\lambda _{3}}$. For an incompressible material, ${\displaystyle J=1}$.