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.

$W = \bar{W}(\boldsymbol{F}) = \hat{W}(\boldsymbol{C}) = \tilde{W}(\boldsymbol{B})$

where $\boldsymbol{F}$ is the (two-point) deformation gradient tensor, $\boldsymbol{C}$ is the right Cauchy-Green deformation tensor, and $\boldsymbol{B}$ is the left Cauchy-Green deformation tensor .

For an isotropic material, the deformation gradient can be expressed uniquely in terms of the principal stretches or in terms of the invariants of the left Cauchy-Green deformation tensor or right Cauchy-Green deformation tensor and we have

$W = \hat{W}(\lambda_1,\lambda_2,\lambda_3) = \tilde{W}(I_1,I_2,I_3) = \bar{W}(\bar{I}_1,\bar{I}_2,J) = U(I_1^c, I_2^c, I_3^c)$

with

$\begin{align} \bar{I}_1 & = J^{-2/3}~I_1 ~;~~ I_1 = \lambda_1^2 + \lambda_2 ^2+ \lambda_3 ^2 ~;~~ J = \det(\boldsymbol{F}) \\ \bar{I}_2 & = J^{-4/3}~I_2 ~;~~ I_2 = \lambda_1^2 \lambda_2^2 + \lambda_2^2 \lambda_3^2 + \lambda_3^2 \lambda_1^2 \end{align}$

A strain energy density function is used to define a hyperelastic material by postulating that the stress in the material can be obtained by taking the derivative of $W$ with respect to the strain. For an isotropic, hyperelastic material the function relates the energy stored in an elastic material, and thus the stress-strain relationship, only to the three strain (elongation) components, thus disregarding the deformation history, heat dissipation, stress relaxation etc.

The strain energy density function relates to the Helmholtz free energy function $\psi$ ,

$W = \rho_0 \psi \;.$

Examples of strain energy density functions

Some examples of hyperelastic constitutive equations are

• Saint Venant–Kirchhoff
• Neo-Hookean
• Generalized Rivlin
• Mooney–Rivlin
• Ogden
• Yeoh
• Arruda–Boyce model
• Gent

See also

• Helmholtz and Gibbs free energy in thermoelasticity
• Hyperelastic material
• Finite strain theory