Hyperelastic material

Remarks

For a frame-indifferent material, $W(QF) = W(F)$ for all rotations $Q$. If the material is also isotropic, $W$ is a function of the three principal invariants $I_1, I_2, I_3$ of the right Cauchy–Green tensor $C = F^T F$. For an incompressible material the constraint $\det F = 1$ adds a pressure term: $P = \partial W / \partial F - p F^{-T}$, with $p$ a Lagrange multiplier. Example: the incompressible neo-Hookean solid, with $W = \frac{\mu}{2}(I_1 - 3)$. Non-example: take the response $P = \mu F + AF$ for a fixed nonzero antisymmetric matrix $A$. Its stress depends only on $F$, but the derivative of $F \mapsto P$ is not symmetric. So $P$ is not the derivative of any function $W$. The total work of the stress around any closed path of deformations is zero for a hyperelastic material, and this property characterises hyperelasticity among Cauchy elastic materials.