Ogden unconstrained¶

Module: solid

Category: material

Type string: "Ogden unconstrained"

Parameters¶

Name	Description	Default	Units
`density`	density	1	[M/L^3]
`cp`	cp	0	[P]
`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 unconstrained hyperelastic Ogden material ¹.

The hyperelastic strain energy function for this material is given in terms of the eigenvalues of the right or left stretch tensor:

\[ W\left(\lambda_{1},\lambda_{2},\lambda_{3}\right)=\frac{1}{2}c_{p}\left(J-1\right)^{2}+\sum\limits_{i=1}^{N}\frac{c_{i}}{m_{i}^{2}}\left(\lambda_{1}^{m_{i}}+\lambda_{2}^{m_{i}}+\lambda_{3}^{m_{i}}-3-m_{i}\ln J\right). \]

Here, \(\lambda_{i}^{2}\) are the eigenvalues of the right or left Cauchy deformation tensor, \(c_{p}\), \(c_{i}\) and \(m_{i}\) are material coefficients and \(N\) ranges from 1 to 6. Any material parameters that are not specified by the user are assumed to be zero.

Example:

<material id="1" type="Ogden unconstrained">
  <m1>2.4</m1>
  <c1>1</c1>
  <cp>2</cp>
</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. ↩