Saint Venant-Kirchhoff model

{{#include ../../../../conspire.rs/src/constitutive/solid/hyperelastic/saint_venant_kirchhoff/model.md}}

Helmholtz free energy density

$$a(\mathbf{F})=\mu \mathrm{tr}({\mathbf{E}}^2)+\frac{1}{2}\left(\kappa -\frac{2}{3}\mu \right)\mathrm{tr}(\mathbf{E}{)}^2$$

Second Piola-Kirchhoff stress

$$\mathbf{S}(\mathbf{F})=2\mu {\mathbf{E}}^{\prime }+\kappa \mathrm{tr}(\mathbf{E})1$$

Second Piola-Kirchhoff tangent stiffness

$${\mathcal{G}}_{IJkL}(\mathbf{F})=\mu {\delta }_{JL}{F}_{kI}+\mu {\delta }_{IL}{F}_{kJ}+\left(\kappa -\frac{2}{3}\mu \right){\delta }_{IJ}{F}_{kL}$$