Carter-Hayes

Module: solid

Category: material

Type string: "Carter-Hayes"

Parameters

Name	Description	Default	Units
density	density	1	[M/L^3]
E0	E0	0	[P]
rho0	rho0	1	[M/L^3]
gamma	gamma	0	[]
v	v	-6.27744e+66	[]
sbm	sbm	-1	[]

Description

This model describes an unconstrained neo-Hookean material 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 ^{1 2}, \(E_{Y}=E_{0}\left(\frac{\rho_{r}^{\sigma}}{\rho_{0}}\right)^{\gamma}\,\).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. The solid-bound molecule must be defined in the Globals section 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>

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1. Carter, D R and Hayes, W C, "Bone compressive strength: the influence of density and strain rate", Science 194, 4270 (1976), pp. 1174-6. ↩

2. Carter, D R and Hayes, W C, "The compressive behavior of bone as a two-phase porous structure", J Bone Joint Surg Am 59, 7 (1977), pp. 954-62. ↩