The Cauchy stress tensor for this material may be obtained using the general formula for isotropic elasticity in principal directions given in (2.4.2-7), with $$\begin{equation} \sigma_{i}=c_{p}\left(J-1\right)+\sum\limits _{k=1}^{N}\frac{1}{J}\frac{c_{k}}{m_{k}}\left(\lambda_{i}^{m_{k}}-1\right)\,.\label{eq426} \end{equation}$$ Similarly, the spatial elasticity tensor is given by $$\begin{equation} \begin{aligned}\boldsymbol{\mathcal{C}} & =\sum\limits _{i=1}^{3}\left(c_{p}+\sum\limits _{k=1}^{N}\frac{1}{J}\frac{c_{k}}{m_{k}}\left[\left(m_{k}-2\right)\lambda_{i}^{m_{k}}+2\right]\right)\mathbf{a}_{i}\otimes\mathbf{a}_{i}\\ & +\sum\limits _{i=1}^{3}\sum\limits _{j=i+1}^{3}c_{p}\left(2J-1\right)\left(\mathbf{a}_{i}\otimes\mathbf{a}_{j}+\mathbf{a}_{j}\otimes\mathbf{a}_{i}\right)\\ & +\sum\limits _{i=1}^{3}\sum\limits _{j=i+1}^{3}2\frac{\lambda_{j}^{2}\sigma_{i}-\lambda_{i}^{2}\sigma_{j}}{\lambda_{i}^{2}-\lambda_{j}^{2}}\left(\mathbf{a}_{i}\,\overline{\underline{\otimes}}\,\mathbf{a}_{j}+\mathbf{a}_{j}\,\overline{\underline{\otimes}}\,\mathbf{a}_{i}\right)\,, \end{aligned} \label{eq427} \end{equation}$$ where $\mathbf{a}_{i}=\mathbf{n}_{i}\otimes\mathbf{n}_{i}$ and $\mathbf{n}_{i}$ are the eigenvectors of $\mathbf{b}$ . In the limit when eigenvalues coincide, $$\begin{equation} \lim\limits _{\lambda_{j}\to\lambda_{i}}2\frac{\sigma_{i}\lambda_{j}^{2}-\sigma_{j}\lambda_{i}^{2}}{\lambda_{i}^{2}-\lambda_{j}^{2}}=2c_{p}\left(1-J\right)+\sum\limits _{k=1}^{N}\frac{1}{J}\frac{c_{k}}{m_{k}}\left[2+\left(m_{k}-2\right)\lambda_{i}^{m_{k}}\right]\,.\label{eq428} \end{equation}$$ In the reference configuration the elasticity tensor reduces to $$\begin{equation} \left.\boldsymbol{\mathcal{C}}\right|_{\mathbf{b}=\mathbf{I}}=c_{p}\mathbf{I}\otimes\mathbf{I}+\left(\sum\limits _{k=1}^{N}c_{k}\right)\mathbf{I}\,\overline{\underline{\otimes}}\,\mathbf{I}\,,\label{eq429} \end{equation}$$ which has the form of Hooke's law for infinitesimal isotropic elasticity (see Section 5.1↑), with equivalent Lamé coefficients $c_{p}\equiv\lambda$ and $2\mu\equiv\sum\nolimits _{k=1}^{N}c_{k}$ .