Mooney-Rivlin¶

Module: solid

Category: material

Type string: "Mooney-Rivlin"

Parameters¶

Name	Description	Default	Units
`density`	density	1	[M/L^3]
`k`	bulk modulus	0	[P]
`pressure_model`	pressure_model	0	[]
`c1`	c1	0	[P]
`c2`	c2	0	[P]

Description¶

This material model is a hyperelastic Mooney-Rivlin type with uncoupled deviatoric and volumetric behavior. The strain-energy function is given by:

\[ \Psi=C_{1}\left(\tilde{I}_{1}-3\right)+C_{2}\left(\tilde{I}_{2}-3\right)+\frac{1}{2}K\left(\ln J\right)^{2}\,. \]

\(C_{1}\) and \(C_{2}\) are the Mooney-Rivlin material coefficients. The variables \(\tilde{I}_{1}\) and \(\tilde{I}_{2}\) are the first and second invariants of the deviatoric right Cauchy-Green deformation tensor \(\mathbf{\tilde{C}}\). The coefficient \(K\) is a bulk modulus-like penalty parameter and \(J\) is the determinant of the deformation gradient tensor. When \(C_{2}=0\), this model reduces to an uncoupled version of the neo-Hookean constitutive model.

This material model uses a three-field element formulation, interpolating displacements as linear field variables and pressure and volume ratio as piecewise constant on each element ¹.

This material model is useful for modeling certain types of isotropic materials that exhibit some limited compressibility, i.e. \(100 < (K/C_{\mathrm{1}}) < 10000\).

Example:

<material id="2" type="Mooney-Rivlin">
  <c1>10.0</c1>
  <c2>20.0</c2>
  <k>1000</k>
</material>

Simo, J.C. and Taylor, R.L., "Quasi-incompressible finite elasticity in principal stretches: Continuum basis and numerical algorithms", Computer Methods in Applied Mechanics and Engineering 85 (1991), pp. 273-310. ↩