Ogden¶
Module: solid
Category: material
Type string: "Ogden"
Parameters¶
| Name | Description | Default | Range | Units |
|---|---|---|---|---|
density |
density | 1 | \(\ge 0\) | M/L^3 |
k |
bulk modulus | 0 | \(\ge 0\) | P |
pressure_model |
pressure_model | 0 | \([0, 3]\) | |
c1 |
c1 | 0 | \(\in \mathbb{R}\) | P |
c2 |
c2 | 0 | \(\in \mathbb{R}\) | P |
c3 |
c3 | 0 | \(\in \mathbb{R}\) | P |
c4 |
c4 | 0 | \(\in \mathbb{R}\) | P |
c5 |
c5 | 0 | \(\in \mathbb{R}\) | P |
c6 |
c6 | 0 | \(\in \mathbb{R}\) | P |
m1 |
m1 | 1 | \(\neq 0\) | |
m2 |
m2 | 1 | \(\neq 0\) | |
m3 |
m3 | 1 | \(\neq 0\) | |
m4 |
m4 | 1 | \(\neq 0\) | |
m5 |
m5 | 1 | \(\neq 0\) | |
m6 |
m6 | 1 | \(\neq 0\) |
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>
-
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. ↩