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 ${\displaystyle {\boldsymbol {S}}_{0}}$, which is derived from a suitable strain energy density function ${\displaystyle W({\boldsymbol {C}})}$:

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

The key idea of pseudo-elastic material models is that the stress during the first loading process is equal to the basic stress ${\displaystyle {\boldsymbol {S}}_{0}}$. Upon unloading and reloading ${\displaystyle {\boldsymbol {S}}_{0}}$ is multiplied by a positive softening function $\eta$. The function $\eta$ thereby depends on the strain energy ${\displaystyle W({\boldsymbol {C}})}$ of the current load and its maximum ${\displaystyle W_{max}(t):=\max\{W(\tau ),\tau \leq t\}}$ in the history of the material:

${\displaystyle {\boldsymbol {S}}=\eta (W,W_{max}){\boldsymbol {S}}_{0},\quad {\text{where }}\eta {\begin{cases}=1,\quad &W=W_{max},\\<1,&W<W_{max}\end{cases}}\quad .}$

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

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).