Strain energy density function

A strain energy density function or stored energy density function is a scalar valued function that relates the strain energy density of a material to the deformation gradient.

Equivalently,

where ${\displaystyle {\boldsymbol {F}}}$ is the (two-point) deformation gradient tensor, ${\displaystyle {\boldsymbol {C}}}$ is the right Cauchy-Green deformation tensor, ${\displaystyle {\boldsymbol {B}}}$ is the left Cauchy-Green deformation tensor, and ${\displaystyle {\boldsymbol {R}}}$ is the rotation tensor from the polar decomposition of ${\displaystyle {\boldsymbol {F}}}$.

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