Under high deformations the linear elasticity is not applicable, the Hook tensor depends on the deformation. Generally the deformation energy ($W$) depends on the strain tensor.

\[ \sigma_{ij} = \frac{\partial W(\epsilon)}{\partial \epsilon_{ij}} \]

The Newton law for the volume element without damping is: \[ \rho \frac{\partial^2 \vec{u}}{\partial t^2} = \nabla \cdot \sigma \]

If the $W(\epsilon)$ relation in known, then the equation can be solved.

In the FEM solver the Neo Hookean model is implemented:

\[ W = C_1 \cdot (J^{-1} I_1 -2) + K \cdot (J-1)^2 \]

Where $C_1$, $K$ are material parameters, $J$ is the Jacobi matrix, and $I_1$ is the first invariant of the right Cauchy-Green deformation tensor.

\[ J = (1+\epsilon_{11})(1+\epsilon_{22})(1+\epsilon_{33}) \] \[ I_1 = (1+\epsilon_{11})^2+(1+\epsilon_{22})^2+(1+\epsilon_{33})^2 \]


2D Simulations

Stationary study