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. W = W ^ ( C ) = W ^ ( F T ⋅ F ) = W ¯ ( F ) = W ¯ ( B 1 / 2 ⋅ R ) = W ~ ( B , R ) {\displaystyle W={\hat {W}}({\boldsymbol {C}})={\hat {W}}({\boldsymbol {F}}^{T}\cdot {\boldsymbol {F}})={\bar {W}}({\boldsymbol {F}})={\bar {W}}({\boldsymbol {B}}^{1/2}\cdot {\boldsymbol {R}})={\tilde {W}}({\boldsymbol {B}},{\boldsymbol {R}})} Equivalently, W = W ^ ( C ) = W ^ ( R T ⋅ B ⋅ R ) = W ~ ( B , R ) {\displaystyle W={\hat {W}}({\boldsymbol {C}})={\hat {W}}({\boldsymbol {R}}^{T}\cdot {\boldsymbol {B}}\cdot {\boldsymbol {R}})={\tilde {W}}({\boldsymbol {B}},{\boldsymbol {R}})} where F {\displaystyle {\boldsymbol {F}}} is the (two-point) 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}}} .

Source: Wikipedia — Strain energy density function (CC BY-SA 4.0)

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. W = W ^ ( C ) = W ^ ( F T ⋅ F ) = W ¯ ( F ) = W ¯ ( B 1 / 2 ⋅ R ) = W ~ ( B , R ) {\displaystyle W={\hat {W}}({\boldsymbol {C}})={\hat {W}}({\boldsymbol {F}}^{T}\cdot {\boldsymbol {F}})={\bar {W}}({\boldsymbol {F}})={\bar {W}}({\boldsymbol {B}}^{1/2}\cdot {\boldsymbol {R}})={\tilde {W}}({\boldsymbol {B}},{\boldsymbol {R}})} Equivalently, W = W ^ ( C ) = W ^ ( R T ⋅ B ⋅ R ) = W ~ ( B , R ) {\displaystyle W={\hat {W}}({\boldsymbol {C}})={\hat {W}}({\boldsymbol {R}}^{T}\cdot {\boldsymbol {B}}\cdot {\boldsymbol {R}})={\tilde {W}}({\boldsymbol {B}},{\boldsymbol {R}})} where F {\displaystyle {\boldsymbol {F}}} is the (two-point) 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}}} .

Source: Wikipedia "Strain energy density function" · CC BY-SA 4.0

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