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 formula_3 is the (two-point) deformation gradient tensor, formula_4 is the right Cauchy-Green deformation tensor, formula_5 is the left Cauchy-Green deformation tensor,
and formula_6 is the rotation tensor from the polar decomposition of formula_3.
For an anisotropic material, the strain energy density function formula_8 depends implicitly on reference vectors or tensors (such as the initial orientation of fibers in a composite) that characterize internal material texture. The spatial representation, formula_9 must further depend explicitly on the polar rotation tensor formula_6 to provide sufficient information to convect the reference texture vectors or tensors into the spatial configuration. 
For an isotropic material, consideration of the principle of material frame indifference leads to the conclusion that the strain energy density function depends only on the invariants of formula_4 (or, equivalently, the invariants of formula_5 since both have the same eigenvalues). In other words, the strain energy density function 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:
For isotropic materials,
with
For linear isotropic materials undergoing small strains, the strain energy density function specializes to
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 formula_16 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. 
For isothermal elastic processes, the strain energy density function relates to the specific Helmholtz free energy function formula_17,
For isentropic elastic processes, the strain energy density function relates to the internal energy function formula_19,
Examples.
Some examples of hyperelastic constitutive equations are: