Saint Venant–Kirchhoff model

Example

The Saint Venant–Kirchhoff model is the hyperelastic material with the strain energy density function below. $$W = \frac{\lambda}{2}(\tr E)^2 + \mu \tr(E^2), \qquad E = \tfrac12(F^T F - I).$$ Here $F$ is the deformation gradient, $E$ is the Green–Lagrange strain tensor and $\lambda, \mu$ are the Lamé parameters. Its second Piola–Kirchhoff stress is $S = \partial W / \partial E = \lambda (\tr E) I + 2\mu E$. Check: $W$ depends on $F$ only through $F^T F$, so $W(QF) = W(F)$ for every rotation $Q$, and the model is frame-indifferent. For small displacement gradients $E$ is close to the small strain, and the model reduces to isotropic linear elasticity. It behaves badly under strong compression: as $F \to 0$, $E \to -\tfrac12 I$ and $W$ stays finite. So the energy does not prevent a volume element from being crushed to zero volume.