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Holmes-Mow

Module: solid

Category: material

Type string: "Holmes-Mow"

Parameters

Name Description Default Units
density density 1 [M/L^3]
E Young's modulus -6.27744e+66 [P]
v Poisson's ratio -6.27744e+66 []
beta power exponent -6.27744e+66 []

Description

The material type for the Holmes-Mow material 1 is Holmes-Mow. This isotropic hyperelastic material has been used to represent the solid matrix of articular cartilage 12 and intervertebral disc 3.

The coupled hyperelastic strain-energy function for this material is given by 1:

\[ W\left(I_{1},I_{2},J\right)=\frac{1}{2}c\left(e^{Q}-1\right), \]

where \(I_{1}\) and \(I_{2}\) are the first and second invariants of the right Cauchy-Green tensor and Jis the jacobian of the deformation gradient.

Furthermore,

\[ \begin{aligned}Q & =\frac{\beta}{\lambda+2\mu}\left[\left(2\mu-\lambda\right)\left(I_{1}-3\right)+\lambda\left(I_{2}-3\right)-\left(\lambda+2\mu\right)\ln J^{2}\right]\\ c & =\frac{\lambda+2\mu}{2\beta} \end{aligned} \,, \]

and \(\lambda\) and \(\mu\) are the Lamé parameters which are related to the more familiar Young's modulus and Poisson's ratio in the usual manner:

\[ \begin{aligned}\lambda & =\frac{E\nu}{\left(1+\nu\right)\left(1-2\nu\right)}\\ \mu & =\frac{E}{2\left(1+\nu\right)} \end{aligned} \,. \]

Example: 

<material id="3" type="Holmes-Mow">
  <E>1</E>
  <v>0.35</v>
  <beta>0.25</beta>
</material>

  1. Holmes, M. H. and Mow, V. C., "The nonlinear characteristics of soft gels and hydrated connective tissues in ultrafiltration", J Biomech 23, 11 (1990), pp. 1145-56. ↩↩↩

  2. Ateshian, G. A., Warden, W. H., Kim, J. J., Grelsamer, R. P., and Mow, V. C., "Finite deformation biphasic material properties of bovine articular cartilage from confined compression experiments", J Biomech 30, 11-12 (1997), pp. 1157-64. ↩

  3. Iatridis, J. C., Setton, L. A., Foster, R. J., Rawlins, B. A., Weidenbaum, M., and Mow, V. C., "Degeneration affects the anisotropic and nonlinear behaviors of human anulus fibrosus in compression", J Biomech 31, 6 (1998), pp. 535-44. ↩