This model describes an unconstrained neo-Hookean material [19] whose Young's modulus is a power-law function of the referential apparent density $\rho_{r}^{\sigma}$ of a solid-bound molecule. It is derived from the following hyperelastic strain-energy function: $$\Psi_{r}=\frac{E_{Y}}{2\left(1+\nu\right)}\left[\frac{1}{2}\left(\tr\mathbf{C}-3\right)-\ln J+\frac{\nu}{1-2\nu}\left(\ln J\right)^{2}\right].$$ Here, $\mathbf{C}$ is the right Cauchy-Green deformation tensor and $J$ is the determinant of the deformation gradient tensor.

Young's modulus depends on $\rho_{r}^{\sigma}$ according to a power law [20, 21], $$E_{Y}=E_{0}\left(\frac{\rho_{r}^{\sigma}}{\rho_{0}}\right)^{\gamma}\,.$$ This type of material references a solid-bound molecule that belongs to a multiphasic mixture. Therefore this material may only be used as the solid (or a component of the solid) in a multiphasic mixture (Section 4.10↓). The solid-bound molecule must be defined in the <Globals> section (Section 3.4.3↑) and must be included in the multiphasic mixture using a <solid_bound> tag. The parameter sbm must refer to the global index of that solid-bound molecule. The value of $\rho_{r}^{\sigma}$ is specified within the <solid_bound> tag. If a chemical reaction is defined within that multiphasic mixture that alters the value of $\rho_{r}^{\sigma}$ , lower and upper bounds may be specified for this referential density within the <solid_bound> tag to prevent $E_{Y}$ from reducing to zero or achieving excessively elevated values.

Example:

<material id="1" name="solid matrix" type="multiphasic">
  <phi0>0</phi0>
  <solid_bound sbm="1">
    <rho0>1</rho0>
    <rhomin>0.1</rhomin>
    <rhomax>5</rhomax>
  </solid_bound>
  <fixed_charge_density>0</fixed_charge_density>
  <solid type="Carter-Hayes">
    <sbm>1</sbm>
    <E0>10000</E0>
    <rho0>1</rho0>
    <gamma>2.0</gamma>
    <v>0</v>
  </solid>
  <permeability type="perm-const-iso">
    <perm>1</perm>
  </permeability>
  <osmotic_coefficient type="osm-coef-const">
    <osmcoef>1</osmcoef>
  </osmotic_coefficient>
  <reaction name="solid remodeling" type="mass-action-forward">
    <Vbar>0</Vbar>
    <vP sbm="1">1</vP>
    <forward_rate type="Huiskes reaction rate">
      <B>1</B>
      <psi0>0.01</psi0>
    </forward_rate>
  </reaction>
</material>