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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 F {\displaystyle {\boldsymbol {F}}} is the deformation gradient tensor, C {\displaystyle {\boldsymbol {C}}} is the right Cauchy–Green deformation tensor, B {\displaystyle {\boldsymbol {B}}} is the left Cauchy–Green deformation tensor,and R {\displaystyle {\boldsymbol {R}}} is the rotation tensor from the polar decomposition of F {\displaystyle {\boldsymbol {F}}}.
For an anisotropic material, the strain energy density function W ^ {\displaystyle {\hat {W}}} depends implicitly on reference vectors or tensors that characterize internal material texture. The spatial representation, W ~ {\displaystyle {\tilde {W}}} must further depend explicitly on the polar rotation tensor R {\displaystyle {\boldsymbol {R}}} to provide sufficient information to convect the reference texture vectors or tensors into the spatial configuration.