This material is based on the following uncoupled hyperelastic strain energy function [11]: $$\Psi\left(\mathbf{C},\lambda_{1},\lambda_{2},\lambda_{3}\right)=\tilde{\Psi}_{iso}\left(\mathbf{\tilde{C}}\right)+\sum\limits _{i=1}^{3}\tilde{\Psi}_{i}^{TC}\left(\tilde{\lambda}_{i}\right)+U\left(J\right)\,.$$ The isotropic strain energy $\tilde{\Psi}_{iso}$ and the dilatational energy $U$ are the same as for the Mooney-Rivlin material. The tension-compression term is defined as follows: $$\tilde{\Psi}_{i}^{TC}\left(\tilde{\lambda}_{i}\right)=\begin{cases} \xi_{i}\left(\tilde{\lambda}_{i}-1\right)^{\beta_{i}} & \tilde{\lambda}_{i}>1\\ 0 & \tilde{\lambda}_{i}\leqslant1 \end{cases}\quad\xi_{i}\geqslant0\quad\left(\text{no sum over }i\right)\,.$$ The $\tilde{\lambda}_{i}$ parameters are the deviatoric fiber stretches of the local material fibers: $$\tilde{\lambda}_{i}=\left(\mathbf{a}_{i}^{0}\cdot\mathbf{\tilde{C}}\cdot\mathbf{a}_{i}^{0}\right)^{1/2}\,.$$ The local material fibers are defined (in the reference frame) as an orthonormal set of vectors $\mathbf{a}_{i}^{0}$ . See Section 4.1.1↑ for more information. As with all uncoupled materials, this material uses the three-field element formulation.

A complete example for this material follows.

<material id="7" name="cartilage" type="TC nonlinear orthotropic">
  <c1>1.0</c1>
  <c2>0.0</c2>
  <k>100</k>
  <beta>4.3,4.3,4.3</beta>
  <ksi>4525, 4525, 4525</ksi>
  <mat_axis type="local">0,0,0</mat_axis>
</material>