The stress $\tilde{\boldsymbol{\sigma}}$ for this material is given by [38, 7, 4]: $$\tilde{\boldsymbol{\sigma}}=\int_{0}^{2\pi}\int_{0}^{\pi}H\left(\tilde{I}_{n}-1\right)\tilde{\sigma}_{n}\left(\mathbf{n}\right)\sin\varphi\,d\varphi\,d\theta.$$ $\tilde{I}_{n}=\tilde{\lambda}_{n}^{2}=\mathbf{N}\cdot\mathbf{\tilde{C}}\cdot\mathbf{N}$ is the square of the fiber stretch $\tilde{\lambda}_{n}$ , $\mathbf{N}$ is the unit vector along the fiber direction (in the reference configuration), which in spherical angles is directed along $\left(\theta,\varphi\right)$ , $\mathbf{n}=\mathbf{\tilde{F}}\cdot\mathbf{N}/\tilde{\lambda}_{n}$ , and $H\left(.\right)$ is the unit step function that enforces the tension-only contribution. The fiber stress is determined from a fiber strain energy function in the usual manner, $$\tilde{\sigma}_{n}=\frac{2\tilde{I}_{n}}{J}\frac{\partial\tilde{\Psi}}{\partial\tilde{I}_{n}}\mathbf{n}\otimes\mathbf{n}\,,$$ where in this material, $$\tilde{\Psi}\left(\mathbf{n},\tilde{I}_{n}\right)=\xi\left(\mathbf{n}\right)\left(\tilde{I}_{n}-1\right)^{\beta\left(\mathbf{n}\right)}\,.$$ The materials parameters $\beta$ and $\xi$ are determined from: $$\begin{aligned}\xi\left(\mathbf{n}\right) & =\left(\frac{\cos^{2}\theta\sin^{2}\varphi}{\xi_{1}^{2}}+\frac{\sin^{2}\theta\sin^{2}\varphi}{\xi_{2}^{2}}+\frac{\cos^{2}\varphi}{\xi_{3}^{2}}\right)^{-1/2}\\ \beta\left(\mathbf{n}\right) & =\left(\frac{\cos^{2}\theta\sin^{2}\varphi}{\beta_{1}^{2}}+\frac{\sin^{2}\theta\sin^{2}\varphi}{\beta_{2}^{2}}+\frac{\cos^{2}\varphi}{\beta_{3}^{2}}\right)^{-1/2} \end{aligned} \,.$$ The orientation of the material axis can be defined as explained in detail in Section 4.1.1↑.

Example:

<material id="1" type="uncoupled solid mixture">
  <mat_axis type="local">0,0,0</mat_axis>
  <k>1000</k>
  <solid type="Mooney-Rivlin">
    <c1>1</c1>
    <c2>0</c2>
  </solid>
  <solid type="EFD uncoupled">
    <ksi>10, 12, 15</ksi>
    <beta>2.5, 3, 3</beta>
  </solid>
</material>