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

When the constitutive behavior is only a function of the current state of deformation, the material is elastic. In the special case when the work done by the stresses during a deformation is only dependent on the initial state and the final state, the material is termed hyperelastic and its behavior is path-independent. As a consequence of the path-independence a strain energy function per unit undeformed volume can be defined as the work done by the stresses from the initial to the final configuration: $$\begin{equation} \Psi\left(\mathbf{F}\left(\mathbf{X}\right),\mathbf{X}\right)=\int\limits _{t_{0}}^{t}\mathbf{P}\left(\mathbf{F}\left(\mathbf{X}\right),\mathbf{X}\right):\mathbf{\dot{F}}\,dt\,.\label{eq57} \end{equation}$$ The rate of change of the potential is then given by $$\begin{equation} \dot{\Psi}\left(\mathbf{F}\left(\mathbf{X}\right),\mathbf{X}\right)=\mathbf{P}:\mathbf{\dot{F}}\,.\label{eq58} \end{equation}$$ Or alternatively, $$\begin{equation} P_{iJ}=\sum\limits _{i,J=1}^{3}\frac{\partial\Psi}{\partial F_{iJ}}\dot{F}_{iJ}\,.\label{eq59} \end{equation}$$ Comparing (2.4-2) with (2.4-3) reveals that $$\begin{equation} \mathbf{P}\left(\mathbf{F}\left(\mathbf{X}\right),\mathbf{X}\right)=\frac{\partial\Psi\left(\mathbf{F}\left(\mathbf{X}\right),\mathbf{X}\right)}{\partial\mathbf{F}}.\label{eq60} \end{equation}$$ This general constitutive equation can be further simplified by observing that, as a consequence of the objectivity requirement, $\Psi$ may only depend on $\mathbf{F}$ through the stretch tensor $\mathbf{U}$ and must be independent of the rotation component $\mathbf{R}$ . For convenience, however, $\Psi$ is often expressed as a function of $\mathbf{C}=\mathbf{U}^{2}=\mathbf{F}^{T}\cdot\mathbf{F}$ . Noting that $\frac{1}{2}\mathbf{\dot{C}}=\mathbf{\dot{E}}$ is work conjugate to the second Piola-Kirchhoff stress $\mathbf{S}$ , establishes the following general relationships for hyperelastic materials: $$\begin{equation} \dot{\Psi}=\frac{\partial\Psi}{\partial\mathbf{C}}:\mathbf{\dot{C}}=\frac{1}{2}\mathbf{S}:\mathbf{\dot{C}},\quad\boxed{\mathbf{S}\left(\mathbf{C}\left(\mathbf{X}\right),\mathbf{X}\right)=2\frac{\partial\Psi}{\partial\mathbf{C}}=\frac{\partial\Psi}{\partial\mathbf{E}}}\,.\label{eq61} \end{equation}$$