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Fung orthotropic

Module: solid

Category: material

Type string: "Fung orthotropic"

Parameters

Name Description Default Units
density density 1 [M/L^3]
k bulk modulus 0 [P]
pressure_model pressure_model 0 []
E1 E1 -6.27744e+66 [P]
E2 E2 -6.27744e+66 [P]
E3 E3 -6.27744e+66 [P]
G12 G12 -6.27744e+66 [P]
G23 G23 -6.27744e+66 [P]
G31 G31 -6.27744e+66 [P]
v12 v12 -6.27744e+66 []
v23 v23 -6.27744e+66 []
v31 v31 -6.27744e+66 []
c c -6.27744e+66 [P]
mat_axis []

Description

The material type for orthotropic Fung elasticity 1 2 is Fung orthotropic.

The hyperelastic strain energy function is given by 3,

\[ \Psi=\frac{1}{2}c\left(e^{\tilde{Q}}-1\right)+U\left(J\right)\,, \]

where,

\[ \tilde{Q}=c^{-1}\sum\limits_{a=1}^{3}\left[2\mu_{a}\mathbf{M}_{a}:\mathbf{\tilde{E}}^{2}+\sum\limits_{b=1}^{3}\lambda_{ab}\left(\mathbf{M}_{a}:\mathbf{\tilde{E}}\right)\left(\mathbf{M}_{b}:\mathbf{\tilde{E}}\right)\right]. \]

Here, \(\mathbf{\tilde{E}}=\left(\mathbf{\tilde{C}}-\mathbf{I}\right)/2\) and \(\mathbf{M}_{a}=\mathbf{V}_{a}\otimes\mathbf{V}_{a}\) where \(\mathbf{V}_{a}\) defines the initial direction of material axis \(a\). The Lamé constants \(\mu_{a} (a=1,2,3)\) and \(\lambda_{ab} (a,b=1,2,3, \lambda_{ba}=\lambda_{ab})\) are related to Young's moduli \(E_{a}\), shear moduli \(G_{ab}\) and Poisson's ratios \(\nu_{ab}\) via

\[ \begin{aligned} & \left[\begin{array}{cccccc} \lambda_{11}+2\mu_{1} & \lambda_{12} & \lambda_{13} & 0 & 0 & 0\\ \lambda_{12} & \lambda_{22}+2\mu_{2} & \lambda_{23} & 0 & 0 & 0\\ \lambda_{13} & \lambda_{23} & \lambda_{33}+2\mu_{3} & 0 & 0 & 0\\ 0 & 0 & 0 & \frac{1}{2}\left(\mu_{1}+\mu_{2}\right) & 0 & 0\\ 0 & 0 & 0 & 0 & \frac{1}{2}\left(\mu_{2}+\mu_{3}\right) & 0\\ 0 & 0 & 0 & 0 & 0 & \frac{1}{2}\left(\mu_{1}+\mu_{3}\right) \end{array}\right]^{-1}\\ & =\left[\begin{array}{cccccc} \frac{1}{E_{1}} & -\frac{\nu_{12}}{E_{1}} & -\frac{\nu_{13}}{E_{1}} & 0 & 0 & 0\\ -\frac{\nu_{21}}{E_{2}} & \frac{1}{E_{2}} & -\frac{\nu_{23}}{E_{2}} & 0 & 0 & 0\\ -\frac{\nu_{31}}{E_{3}} & -\frac{\nu_{32}}{E_{3}} & \frac{1}{E_{3}} & 0 & 0 & 0\\ 0 & 0 & 0 & \frac{1}{G_{12}} & 0 & 0\\ 0 & 0 & 0 & 0 & \frac{1}{G_{23}} & 0\\ 0 & 0 & 0 & 0 & 0 & \frac{1}{G_{31}} \end{array}\right] \end{aligned} .\]

The orthotropic Lamé parameters should produce a positive definite stiffness matrix.

Example: 

<material id="3" type="Fung orthotropic">
  <E1>124</E1>
  <E2>124</E2>
  <E3>36</E3>
  <G12>67</G12>
  <G23>40</G23>
  <G31>40</G31>
  <v12>-0.075</v12>
  <v23>0.87</v23>
  <v31>0.26</v31>
  <c>1</c>
  <k>120000</k>
</material>

  1. Fung, Y. C., Fronek, K., and Patitucci, P., "Pseudoelasticity of arteries and the choice of its mathematical expression", Am J Physiol 237, 5 (1979), pp. H620-31. ↩

  2. Fung, Y. C., Biomechanics : mechanical properties of living tissues 2nd (New York: Springer-Verlag, 1993). ↩

  3. Ateshian, G. A. and Costa, K. D., "A frame-invariant formulation of Fung elasticity", J Biomech 42, 6 (2009), pp. 781-5. ↩