$\newcommand{\Dev}{\operatorname{Dev}}$ $\newcommand{\dev}{\operatorname{dev}}$ $\newcommand{\grad}{\operatorname{grad}}$ $\newcommand{\Grad}{\operatorname{Grad}}$ $\newcommand{\divg}{\operatorname{div}}$ $\newcommand{\Ei}{\operatorname{Ei}}$ $\newcommand{\tr}{\operatorname{tr}}$ $\newcommand{\Divg}{\operatorname{Div}}$ $\newcommand{\cay}{\operatorname{cay}}$ $\newcommand{\rot}{\operatorname{rot}}$

This constitutive model describes a material that is composed of an ellipsoidal continuous fiber distribution in an uncoupled formulation. The deviatoric part of the stress is given by [5, 8, 47], $$\begin{equation} \tilde{\boldsymbol{\sigma}}=\int_{0}^{2\pi}\int_{0}^{\pi}H\left(\tilde{I}_{n}-1\right)\tilde{\boldsymbol{\sigma}}_{n}\left(\mathbf{n}\right)\sin\varphi\,d\varphi\,d\theta\,,\label{eq465} \end{equation}$$ and the corresponding elasticity tensor is $$\begin{equation} \tilde{\boldsymbol{\mathcal{C}}}=\int_{0}^{2\pi}\int_{0}^{\pi}H\left(\tilde{I}_{n}-1\right)\tilde{\boldsymbol{\mathcal{C}}}_{n}\left(\mathbf{n}\right)\sin\phi\,d\phi\,d\theta\,.\label{eq466} \end{equation}$$ $\tilde{I}_{n}=\tilde{\lambda}_{n}^{2}=\mathbf{N}\cdot\mathbf{\tilde{C}}\cdot\mathbf{N}$ is the square of the fiber stretch $\mathbf{F}$ , $\mathbf{N}$ 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 $H\left(\cdot\right)$ 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: $$\begin{equation} \tilde{\boldsymbol{\sigma}}_{n}\left(\mathbf{n}\right)=2J^{-1}\tilde{I}_{n}\frac{\partial\tilde{\Psi}}{\partial\tilde{I}_{n}}\mathbf{n}\otimes\mathbf{n}\,,\label{eq467} \end{equation}$$ whereas the fiber elasticity tensor is $$\begin{equation} \tilde{\boldsymbol{\mathcal{C}}}_{n}\left(\mathbf{n}\right)=4J^{-1}\tilde{I}_{n}^{2}\frac{\partial^{2}\tilde{\Psi}}{\partial\tilde{I}_{n}^{2}}\mathbf{n}\otimes\mathbf{n}\otimes\mathbf{n}\otimes\mathbf{n}\,,\label{eq468} \end{equation}$$ where in this material $$\begin{equation} \tilde{\Psi}\left(\mathbf{n},\tilde{I}_{n}\right)=\xi\left(\mathbf{n}\right)\left(\tilde{I}_{n}-1\right)^{\beta\left(\mathbf{n}\right)}\,.\label{eq469} \end{equation}$$ The materials parameters $\beta$ and $\xi$ are determined from: $$\begin{equation} \begin{aligned}\xi\left(\mathbf{n}\right) & =\left(\frac{\cos^{2}\theta\sin^{2}\varphi}{\xi_{1}^{2}}+\frac{\sin^{2}\theta\sin^{2}\varphi}{\xi_{2}^{2}}+\frac{\cos^{2}\varphi}{\xi_{3}^{2}}\right)^{-1/2}\,,\\ \beta\left(\mathbf{n}\right) & =\left(\frac{\cos^{2}\theta\sin^{2}\varphi}{\beta_{1}^{2}}+\frac{\sin^{2}\theta\sin^{2}\varphi}{\beta_{2}^{2}}+\frac{\cos^{2}\varphi}{\beta_{3}^{2}}\right)^{-1/2}\,. \end{aligned} \label{eq470} \end{equation}$$ Since fibers can only sustain tension, this material is not stable on its own. It must be combined with a material that acts as the ground matrix. The total stress is then given by the sum of the fiber stress and the ground matrix stress: $$\begin{equation} \tilde{\boldsymbol{\sigma}}=\tilde{\boldsymbol{\sigma}}_{m}+\tilde{\boldsymbol{\sigma}}_{f}\,.\label{eq471} \end{equation}$$