Mooney–Rivlin solid

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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.

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, ${\displaystyle J=1}$.