Ogden

Module: solid

Category: material

Type string: "Ogden"

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]
c3	c3	0	[P]
c4	c4	0	[P]
c5	c5	0	[P]
c6	c6	0	[P]
m1	m1	1	[]
m2	m2	1	[]
m3	m3	1	[]
m4	m4	1	[]
m5	m5	1	[]
m6	m6	1	[]

Description

This material describes an incompressible hyperelastic Ogden material ¹.

The uncoupled hyperelastic strain energy function for this material is given in terms of the eigenvalues of the deformation tensor:

\[ \Psi=\sum\limits_{i=1}^{N}\frac{c_{i}}{m_{i}^{2}}\left(\tilde{\lambda}_{1}^{m_{i}}+\tilde{\lambda}_{2}^{m_{i}}+\tilde{\lambda}_{3}^{m_{i}}-3\right)+U\left(J\right). \]

Here, \(\tilde{\lambda}_{i}^{2}\) are the eigenvalues of \(\mathbf{\tilde{C}}\), \(c_{i}\) and \(m_{i}\) are material coefficients and \(N\) ranges from 1 to 6. Note that you only have to include the material parameters for the terms you intend to use.

Example:

<material id="1" type="Ogden">
  <m1>2.4</m1>
  <c1>1</c1>
  <k>100</k>
</material>

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1. 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. ↩