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For isotropic materials, the principal directions of the strain and stress tensors are the same. Let the eigenvalues of $\mathbf{C}$ be denoted by $\lambda_{i}^{2}$ ($i=1,2,3)$ , then the strain energy density may be given as a function of these eigenvalues, $\Psi\left(\lambda_{1}^{2},\lambda_{2}^{2},\lambda_{3}^{2}\right)$ . To derive the expression for the stress, recognize that $$\begin{equation} \frac{\partial\lambda_{i}^{2}}{\partial\mathbf{C}}=\mathbf{N}_{i}\otimes\mathbf{N}_{i}\equiv\mathbf{A}_{i}\,,\label{eq68} \end{equation}$$ where the $\mathbf{N}_{i}$ are the eigenvectors of $\mathbf{C}$ . It follows that the second Piola-Kirchhoff stress may be represented as $$\begin{equation} \mathbf{S}=\sum\limits _{i=1}^{3}S_{i}\mathbf{A}_{i}\,,\label{eq69} \end{equation}$$ where $$\begin{equation} S_{i}=2\frac{\partial\Psi}{\partial\lambda_{i}^{2}}\,.\label{eq70} \end{equation}$$ To evaluate the material elasticity tensor, recognize that $$\begin{equation} \frac{\partial\mathbf{A}_{i}}{\partial\mathbf{C}}=\frac{1}{\lambda_{i}^{2}-\lambda_{j}^{2}}\left(\mathbf{A}_{i}\,\underline{\overline{\otimes}}\,\mathbf{A}_{j}+\mathbf{A}_{j}\,\underline{\overline{\otimes}}\,\mathbf{A}_{i}\right)+\frac{1}{\lambda_{i}^{2}-\lambda_{k}^{2}}\left(\mathbf{A}_{i}\,\underline{\overline{\otimes}}\,\mathbf{A}_{k}+\mathbf{A}_{k}\,\underline{\overline{\otimes}}\,\mathbf{A}_{i}\right)\,,\label{eq71} \end{equation}$$ where $i,j,k$ form a permutation over $1,2,3$ . Then it can be shown that the material elasticity tensor is given by $$\begin{equation} \begin{aligned}\boldsymbol{\mathbb{C}} & =\sum\limits _{i=1}^{3}4\frac{\partial^{2}\Psi}{\partial\lambda_{i}^{2}\partial\lambda_{i}^{2}}\mathbf{A}_{i}\otimes\mathbf{A}_{i}\\ & +\sum\limits _{i=1}^{3}\sum\limits _{j=i+1}^{3}4\frac{\partial^{2}\Psi}{\partial\lambda_{i}^{2}\partial\lambda_{j}^{2}}\left(\mathbf{A}_{i}\otimes\mathbf{A}_{j}+\mathbf{A}_{j}\otimes\mathbf{A}_{i}\right)\\ & +\sum\limits _{i=1}^{3}\sum\limits _{j=i+1}^{3}2\frac{S_{i}-S_{j}}{\lambda_{i}^{2}-\lambda_{j}^{2}}\left(\mathbf{A}_{i}\,\overline{\underline{\otimes}}\,\mathbf{A}_{j}+\mathbf{A}_{j}\,\overline{\underline{\otimes}}\,\mathbf{A}_{i}\right) \end{aligned} \,.\label{eq72} \end{equation}$$ When eigenvalues coincide, L'Hospital's rule may be used to evalue the coefficient in the last term, $$\begin{equation} \lim\limits _{\lambda_{j}^{2}\to\lambda_{i}^{2}}2\frac{S_{i}-S_{j}}{\lambda_{i}^{2}-\lambda_{j}^{2}}=4\left(\frac{\partial^{2}\Psi}{\partial\lambda_{j}^{2}\partial\lambda_{j}^{2}}-\frac{\partial^{2}\Psi}{\partial\lambda_{i}^{2}\partial\lambda_{j}^{2}}\right)\,.\label{eq73} \end{equation}$$ The double summations in (2.4.2-5) are arranged such that the summands represent fourth-order tensors with major and minor symmetries.