View source for Ogden–Roxburgh model

{{Use dmy dates|date=October 2017}}
The '''Ogden–Roxburgh model'''<ref>{{cite journal|last=Ogden|first=R. W|author2=Roxburgh, D. G.|title=A pseudo–elastic model for the Mullins effect in filled rubber.|journal=Proceedings of the Royal Society of London A|year=1999|issue=1988|volume=455|pages=2861–2877|doi=10.1098/rspa.1999.0431|bibcode=1999RSPSA.455.2861W}}</ref> is an approach published in 1999 which extends [[Hyperelastic material|hyperelastic material models]] to allow for the [[Mullins effect]].<ref>{{cite journal |last1=Mullins |first1=L. |title=Softening of Rubber by Deformation |journal=Rubber Chemistry and Technology |date=1969 |volume=42 |issue=1 |pages=339–362 |doi=10.5254/1.3539210 |url=https://doi.org/10.5254/1.3539210 |access-date=16 September 2023}}</ref> It is used in several commercial [[finite element]] codes, and is named after [[Raymond Ogden|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.<ref>{{cite journal|last=De Souza Neto|first=E. A.|author2=Perić, D.|author3=Owen, D. R. J.|title=A phenomenological three-dimensional rate-independent continuum damage model for highly filled polymers : Formulation and computational aspects|journal=Journal of the Mechanics and Physics of Solids|year=1994|issue=10|volume=42|pages=1533–1550|doi=10.1016/0022-5096(94)90086-8}}</ref>

The basis of pseudo-elastic material models is a hyperelastic [[Piola–Kirchhoff stress tensor|second Piola–Kirchhoff stress]] <math>\boldsymbol{S}_0</math>, which is derived from a suitable [[strain energy density function]] <math>W(\boldsymbol{C})</math>:
:<math>
   \boldsymbol{S} = 2 \frac{\partial W}{\partial \boldsymbol{C}} \quad .
 </math>
The key idea of pseudo-elastic material models is that the stress during the first loading process is equal to the basic stress <math>\boldsymbol{S}_0</math>. Upon unloading and reloading <math>\boldsymbol{S}_0</math> is multiplied by a positive softening function <math>\eta</math>. The function <math>\eta</math> thereby depends on the strain energy <math>W(\boldsymbol{C})</math> of the current load and its maximum <math>W_{max}(t) := \max\{W(\tau), \tau \le t\}</math> in the history of the material:
:<math>
 \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 .
</math>

It was shown that this idea can also be used to extend arbitrary inelastic material models for softening effects.<ref>{{cite journal|last=Naumann|first=C.|author2=Ihlemann, J.|title=On the thermodynamics of pseudo-elastic material models which reproduce the Mullins effect|journal=International Journal of Solids and Structures|year=2015|volume=69–70|pages=360–369|doi=10.1016/j.ijsolstr.2015.05.014|doi-access=free}}</ref>

== References ==
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{{DEFAULTSORT:Ogden-Roxburgh model}}
[[Category:Continuum mechanics]]
[[Category:Elasticity (physics)]]
[[Category:Rubber properties]]
[[Category:Solid mechanics]]