Saint Venant-Kirchhoff model

The Saint Venant-Kirchhoff hyperelastic constitutive model.

Parameters

• The bulk modulus $\kappa$.
• The shear modulus $\mu$.

External variables

• The deformation gradient $\mathbf{F}$.

Internal variables

• None.

Notes

• The Green-Saint Venant strain measure is given by $\mathbf{E}=\frac{1}{2}(\mathbf{C}-1)$.

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

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