Ogden-Roxburgh model

The Ogden-Roxburgh model ^[1] is an approach which extends hyperelastic material models to allow for the Mullins effect. It is used in several commercial finite element codes.

The basis of pseudo-elastic material models is a hyperelastic second Piola–Kirchhoff stress $\boldsymbol{S}_0$, which is derived from a suitable strain energy density function $W(\boldsymbol{C})$:

$\boldsymbol{S} = 2 \frac{\partial W}{\partial \boldsymbol{C}} \quad .$

The key idea of pseudo-elastic material models is that the stress during the first loading process is equal to the basic stress $\boldsymbol{S}_0$. Upon unloading and reloading $\boldsymbol{S}_0$ is multiplied by a positive softening function $\eta$. The function $\eta$ thereby depends on the strain energy $W(\boldsymbol{C})$ of the current load and its maximum $W_{max}(t) := \max\{W(\tau), \tau \le t\}$ in the history of the material:

$\boldsymbol{S} = \eta(W, W_{max}) \boldsymbol{S}_0, \quad \text{where } \eta \begin{cases} = 1, \quad & W = W_{max},\\ < 1, & W < W_{max} \end{cases} \quad .$

It was shown^[2] that this idea can also be used to extend arbitrary inelastic material models for softening effects.

References

1. Ogden, R. W; Roxburgh, D. G. (1999). "A pseudo–elastic model for the Mullins effect in filled rubber.". Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 455 (1988): 2861–2877. doi:10.1098/rspa.1999.0431.
2. Naumann, C.; Ihlemann, J. (2015). "On the thermodynamics of pseudo-elastic material models which reproduce the Mullins effect". International Journal of Solids and Structures. 69-70: 360–369. doi:10.1016/j.ijsolstr.2015.05.014.

L. Mullins, Rubber Chemistry and Technology, 42, 339 (1969).