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coupled trans-iso Mooney-Rivlin

Module: solid

Category: material

Type string: "coupled trans-iso Mooney-Rivlin"

Parameters

Name Description Default Range Units
density density 1 \(\ge 0\) M/L^3
c1 c1 0 \(\gt 0\) P
c2 c2 0 \(\in \mathbb{R}\) P
c3 c3 0 \(\in \mathbb{R}\) P
c4 c4 0 \(\in \mathbb{R}\)
c5 c5 0 \(\in \mathbb{R}\)
lam_max lam_max 1 \(\ge 1\)
k k 0 \(\gt 0\)
fiber fiber N/A

Description

This material describes a transversely isotropic Mooney-Rivlin material using a fully-coupled formulation. It is defined through the coupled trans-iso Mooney-Rivlin material type.

The strain-energy function for this constitutive model is defined by

\[ W=c_{1}\left(I_{1}-3\right)+c_{2}\left(I_{2}-3\right)-2\left(c_{1}+2c_{2}\right)\ln J+F\left(\lambda\right)+U\left(J\right) \]

The first three terms define the coupled Mooney-Rivlin matrix response. The third term is the fiber response which is a function of the fiber stretch \(\lambda\) and is defined as in 1. For U, the following form is chosen in FEBio.

\[ U\left(J\right)=\frac{1}{2}k\left(\ln J\right)^{2} \]

where \(J=\det\mathbf{F}\) is the Jacobian of the deformation.

Example: 

<material id="1" type="coupled trans-iso Mooney-Rivlin">
  <c1>1</c1>
  <c2>0.1</c2>
  <c3>1</c3>
  <c4>1</c4>
  <c5>1.34</c5>
  <lam_max>1.3</lam_max>
  <k>100</k>
</material>

  1. Weiss, J.A., Maker, B.N., and Govindjee, S., "Finite element implementation of incompressible, transversely isotropic hyperelasticity", Computer Methods in Applications of Mechanics and Engineering 135 (1996), pp. 107-128. ↩