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

This material model is based on the following uncoupled hyperelastic strain energy function [12]: $$\begin{equation} \Psi\left(\mathbf{C},\lambda_{1},\lambda_{2},\lambda_{3}\right)=\tilde{\Psi}_{\text{iso}}\left(\mathbf{\tilde{C}}\right)+\sum\limits _{i=1}^{3}\tilde{\Psi}_{i}^{TC}\left(\tilde{\lambda}_{i}\right)+U\left(J\right)\,.\label{eq480} \end{equation}$$ The isotropic strain energy $\tilde{\Psi}_{\text{iso}}$ and the dilatational energy $U$ are the same as for the Mooney-Rivlin material (Section 5.3.1↑). The tension-compression term is defined as follows: $$\begin{equation} \tilde{\Psi}_{i}^{TC}\left(\tilde{\lambda}_{i}\right)=\begin{cases} \xi_{i}\left(\tilde{\lambda}_{i}-1\right)^{\beta_{i}} & \tilde{\lambda}_{i}>1\\ 0 & \tilde{\lambda}_{i}\leqslant1 \end{cases}\,,\xi_{i}\geqslant0\quad\mbox{(no sum over }i\text{).}\label{eq481} \end{equation}$$ The $\tilde{\lambda}_{i}$ parameters are the deviatoric fiber stretches of the local material fibers, $$\begin{equation} \tilde{\lambda}_{i}=\left(\mathbf{A}_{i}\cdot\mathbf{\tilde{C}}\cdot\mathbf{A}_{i}\right)^{1/2}\,.\label{eq482} \end{equation}$$ The local material fibers are defined (in the reference frame) as an orthonormal set of vectors $\mathbf{A}_{i}$ . The corresponding deviatoric part of the Cauchy stress is $$\begin{equation} \tilde{\boldsymbol{\sigma}}=J^{-1}\sum\limits _{i=1}^{3}\frac{1}{\tilde{\lambda}_{i}}\frac{\partial\tilde{\Psi}}{\partial\tilde{\lambda}_{i}}\mathbf{a}_{i}\otimes\mathbf{a}_{i}\,,\label{eq483} \end{equation}$$ and the spatial elasticity tensor is $$\begin{equation} \tilde{\boldsymbol{\mathcal{C}}}=J^{-1}\sum\limits _{i=1}^{3}\frac{1}{\tilde{\lambda}_{i}}\frac{\partial}{\partial\tilde{\lambda}_{i}}\left(\frac{1}{\tilde{\lambda}_{i}}\frac{\partial\tilde{\Psi}}{\partial\tilde{\lambda}_{i}}\right)\mathbf{a}_{i}\otimes\mathbf{a}_{i}\otimes\mathbf{a}_{i}\otimes\mathbf{a}_{i}\,,\label{eq484} \end{equation}$$ where $\mathbf{a}_{i}=\mathbf{\tilde{F}}\cdot\mathbf{A}_{i}$ .