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

The hyperelastic constitutive equations discussed so far are unrestricted in their application. Isotropic material symmetry is defined by requiring the constitutive behavior to be independent of the material axis chosen and, consequently, $\Psi$ must only be a function of the invariants of $\mathbf{C}$ , $$\begin{equation} \Psi\left(\mathbf{C}\left(\mathbf{X}\right),\mathbf{X}\right)=\Psi\left(I_{1},I_{2},I_{3},\mathbf{X}\right)\,,\label{eq62} \end{equation}$$ where the invariants of C are defined here as, $$\begin{equation} I_{1}=\tr\mathbf{C}=\mathbf{C}:\mathbf{I},\,I_{2}=\frac{1}{2}\left[\left(\tr\mathbf{C}\right)^{2}-\tr\mathbf{C}^{2}\right],\,I_{3}=\det\mathbf{C}=J^{2}\,.\label{eq63} \end{equation}$$ As a result of the isotropic restriction, the second Piola-Kirchhoff stress tensor can be written as, $$\begin{equation} \mathbf{S}=2\frac{\partial\Psi}{\partial\mathbf{C}}=2\frac{\partial\Psi}{\partial I_{1}}\frac{\partial I_{1}}{\partial\mathbf{C}}+2\frac{\partial\Psi}{\partial I_{2}}\frac{\partial I_{2}}{\partial\mathbf{C}}+2\frac{\partial\Psi}{\partial I_{3}}\frac{\partial I_{3}}{\partial\mathbf{C}}\,.\label{eq64} \end{equation}$$ The second order tensors formed by the derivatives of the invariants with respect to C can be evaluated as follows: $$\begin{equation} \frac{\partial I_{1}}{\partial\mathbf{C}}=\mathbf{I}\,,\,\frac{\partial I_{2}}{\partial\mathbf{C}}=I_{1}\mathbf{I}-\mathbf{C}\,,\,\frac{\partial I_{3}}{\partial\mathbf{C}}=I_{3}\mathbf{C}^{-1}\,.\label{eq65} \end{equation}$$ Introducing expressions (2.4.1-4) into equation (2.4.1-3) enables the second Piola-Kirchhoff stress to be evaluated as, $$\begin{equation} \mathbf{S}=2\left\{ \left(\Psi_{1}+I_{1}\Psi_{2}+I_{2}\Psi_{3})\right)\mathbf{I}-\left(\Psi_{2}+I_{1}\Psi_{3}\right)\mathbf{C}\right\} +\Psi_{3}\mathbf{C}^{2}\,,\label{eq66} \end{equation}$$ where $\Psi_{I}=\partial\Psi/\partial I_{1}$ , $\Psi_{2}=\partial\Psi/\partial I_{2}$ , and $\Psi_{3}=\partial\Psi/\partial I_{3}$ .