Ogden–Roxburgh model

The Ogden–Roxburgh model^[1] is an approach published in 1999 which extends hyperelastic material models to allow for the Mullins effect.^[2] It is used in several commercial finite element codes, and is named after R.W. Ogden and D. G. Roxburgh. The fundamental idea of the approach can already be found in a paper by De Souza Neto et al. from 1994.^[3]

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 ${\displaystyle \eta }$. The function ${\displaystyle \eta }$ 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.^[4]