In the current configuration, the porosity is given by $$\varphi^{w}=\frac{J-1+\varphi_{r}^{w}}{J}\,.$$ We may define a new variable, $$\bar{J}\equiv\frac{J-1+\varphi_{r}^{w}}{\varphi_{r}^{w}}=\frac{J-\varphi_{r}^{s}}{1-\varphi_{r}^{s}}\,,$$ which represents the pore volume ratio. It is equal to $1$ when $J=1$ and is equal to $J$ when $\varphi_{r}^{w}=1$ (or $\varphi_{r}^{s}=1-\varphi_{r}^{w}=0$ ). Now, $$\varphi^{w}=\varphi_{r}^{w}\frac{\bar{J}}{J}\,,$$ and $$\frac{\partial\bar{J}}{\partial J}=\frac{1}{\varphi_{r}^{w}}\,.$$ Pore closure occurs when $\varphi^{w}=0$ , which corresponds to $J=\varphi_{r}^{s}$ and $\bar{J}=0$ .

Let us also define a modified deformation gradient, $$\bar{\mathbf{F}}=\left(\frac{\bar{J}}{J}\right)^{1/3}\mathbf{F}\,,$$ such that $\det\bar{\mathbf{F}}=\bar{J}$ . Let the corresponding modified right Cauchy-Green tensor be given by $$\bar{\mathbf{C}}=\bar{\mathbf{F}}^{T}\cdot\bar{\mathbf{F}}=\left(\frac{\bar{J}}{J}\right)^{2/3}\mathbf{C}\,,$$ so that $$\frac{\partial\bar{\mathbf{C}}}{\partial\mathbf{C}}=\left(\frac{\bar{J}}{J}\right)^{2/3}\left(\frac{\varphi_{r}^{s}}{3\left(J-\varphi_{r}^{s}\right)}\mathbf{C}\otimes\mathbf{C}^{-1}+\mathbf{I}\,\underline{\overline{\otimes}}\,\mathbf{I}\right)\,.$$

The constitutive relation for the strain energy density of the compressible porous neo-Hookean material may be given by $$\Psi_{r}=\frac{\mu}{2}\left(\bar{I}_{1}-3\right)-\mu\ln\bar{J}\,,$$ where $\bar{I}_{1}=\text{tr}\bar{\mathbf{C}}$ . This relation shows that the material develops an infinite strain energy density as $\bar{J}$ approaches zero. From this expression, the 2nd Piola-Kirchhoff stress is given by $$\mathbf{S}=2\frac{\partial\Psi_{r}}{\partial\mathbf{C}}=\mu\left[\left(\frac{\bar{J}}{J}\right)^{2/3}\mathbf{I}+\frac{1}{J-\varphi_{r}^{s}}\left(\varphi_{r}^{s}\left(\frac{\bar{J}}{J}\right)^{2/3}\frac{I_{1}}{3}-J\right)\mathbf{C}^{-1}\right]\,.$$ When $\mathbf{C}=\mathbf{I}$ we can verify that $\mathbf{S}=\mathbf{0}$ . The corresponding Cauchy stress is $$\boldsymbol{\sigma}=\frac{\mu}{J}\left[\left(\frac{\bar{J}}{J}\right)^{2/3}\mathbf{b}+\frac{1}{J-\varphi_{r}^{s}}\left(\varphi_{r}^{s}\left(\frac{\bar{J}}{J}\right)^{2/3}\frac{I_{1}}{3}-J\right)\mathbf{I}\right]\,,$$ where $\mathbf{b}$ is the left Cauchy-Green tensor.

The material elasticity tensor is given by $$\begin{aligned}\mathbb{C} & =2\frac{\partial\mathbf{S}}{\partial\mathbf{C}}\\ & =\frac{2}{3}g\left(J\right)\left(\mathbf{I}\otimes\mathbf{C}^{-1}+\mathbf{C}^{-1}\otimes\mathbf{I}\right)+\left(J\frac{dg}{dJ}\frac{I_{1}}{3}+J\frac{dh}{dJ}\right)\mathbf{C}^{-1}\otimes\mathbf{C}^{-1}\\ & -2\left[g\left(J\right)\frac{I_{1}}{3}+h\left(J\right)\right]\mathbf{C}^{-1}\,\underline{\overline{\otimes}}\,\mathbf{C}^{-1} \end{aligned} \,,$$ where $$\begin{aligned}f\left(J\right) & =\mu\left(\frac{\bar{J}}{J}\right)^{2/3}\\ g\left(J\right) & =\frac{\varphi_{r}^{s}}{J-\varphi_{r}^{s}}f\left(J\right)\\ h\left(J\right) & =-\mu\frac{J}{J-\varphi_{r}^{s}} \end{aligned} \,,$$ and $$\begin{aligned}J\frac{dg}{dJ} & =\mu\frac{\left(2\varphi_{r}^{s}-3J\right)\varphi_{r}^{s}}{3\left(J-\varphi_{r}^{s}\right)^{2}}\left(\frac{\bar{J}}{J}\right)^{2/3}\\ J\frac{dh}{dJ} & =\mu\frac{J\varphi_{r}^{s}}{\left(J-\varphi_{r}^{s}\right)^{2}} \end{aligned} \,.$$ Then, the spatial elasticity tensor may be evaluated as $$\boldsymbol{\mathcal{C}}=J^{-1}\left[\frac{2}{3}g\left(J\right)\left(\mathbf{b}\otimes\mathbf{I}+\mathbf{I}\otimes\mathbf{b}\right)+\left(J\frac{dg}{dJ}\frac{I_{1}}{3}+J\frac{dh}{dJ}\right)\mathbf{I}\otimes\mathbf{I}-2\left[g\left(J\right)\frac{I_{1}}{3}+h\left(J\right)\right]\mathbf{I}\,\underline{\overline{\otimes}}\,\mathbf{I}\right]\,.$$

In the limit of infinitesimal strains and rotations, when $\mathbf{b}=\mathbf{I}$ and $J=1$ , we find that $$\begin{aligned}f\left(1\right) & =\mu & J\frac{dg}{dJ} & =\mu\frac{\left(2\varphi_{r}^{s}-3\right)\varphi_{r}^{s}}{3\left(1-\varphi_{r}^{s}\right)^{2}}\\ g\left(J\right) & =\mu\frac{\varphi_{r}^{s}}{1-\varphi_{r}^{s}} & J\frac{dh}{dJ} & =\mu\frac{\varphi_{r}^{s}}{\left(1-\varphi_{r}^{s}\right)^{2}}\\ h\left(J\right) & =-\mu\frac{1}{1-\varphi_{r}^{s}}\mu \end{aligned}$$ and $$\boldsymbol{\mathcal{C}}=\frac{2\mu}{3}\left(\frac{1}{\left(\varphi_{r}^{w}\right)^{2}}-1\right)\mathbf{I}\otimes\mathbf{I}+2\mu\mathbf{I}\,\underline{\overline{\otimes}}\,\mathbf{I}\,.$$ Thus, by comparison to a standard neo-Hookean material, this porous neo-Hookean material has an effective Young's modulus equal to $$E=\frac{3\mu}{1+\frac{1}{2}\left(\varphi_{r}^{w}\right)^{2}}\,,$$ and an effective Poisson's ratio equal to $$\nu=\frac{1-\left(\varphi_{r}^{w}\right)^{2}}{2+\left(\varphi_{r}^{w}\right)^{2}}\,.$$ The two material properties that need to be provided are $E$ and the referential porosity $\varphi_{r}^{w}$ (or referential solid volume fraction $\varphi_{r}^{s}=1-\varphi_{r}^{w}$ ). Poisson's ratio in the limit of infinitesimal strains is dictated by the porosity according to the above formula. In particular, a highly porous material ($\varphi_{r}^{w}\to1$ ) has an effective (infinitesimal strain) Poisson ratio that approaches zero ($\nu\to0$ ) and $E\to2\mu$ . A low porosity material ($\varphi_{r}^{w}\to0$ ) has $\nu\to\frac{1}{2}$ and $E\to3\mu$ , which is the expected behavior of an incompressible neo-Hookean solid. Note that setting $\varphi_{r}^{w}=0$ would not produce good numerical behavior, since the Cauchy stress in an incompressible material would need to be supplemented by a pressure term (a Lagrange multiplier that enforces the incompressibility constraint). Nevertheless, this compressible porous neo-Hookean material behaves well even for values of $\varphi^{w}$ as low as $\sim0.015$ .