Misplaced Pages

Ogden–Roxburgh model

Article snapshot taken from[REDACTED] with creative commons attribution-sharealike license. Give it a read and then ask your questions in the chat. We can research this topic together.
(Redirected from Ogden-Roxburgh model)

The Ogden–Roxburgh model is an approach published in 1999 which extends hyperelastic material models to allow for the Mullins effect. 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.

The basis of pseudo-elastic material models is a hyperelastic second Piola–Kirchhoff stress S 0 {\displaystyle {\boldsymbol {S}}_{0}} , which is derived from a suitable strain energy density function W ( C ) {\displaystyle W({\boldsymbol {C}})} :

S = 2 ∂ W ∂ 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 S 0 {\displaystyle {\boldsymbol {S}}_{0}} . Upon unloading and reloading S 0 {\displaystyle {\boldsymbol {S}}_{0}} is multiplied by a positive softening function η {\displaystyle \eta } . The function η {\displaystyle \eta } thereby depends on the strain energy W ( C ) {\displaystyle W({\boldsymbol {C}})} of the current load and its maximum W m a x ( t ) := max { W ( τ ) , τ ≤ t } {\displaystyle W_{max}(t):=\max\{W(\tau ),\tau \leq t\}} in the history of the material:

S = η ( W , W m a x ) S 0 , where  η { = 1 , W = W m a x , < 1 , W < W m a x . {\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.

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 A. 455 (1988): 2861–2877. Bibcode:1999RSPSA.455.2861W. doi:10.1098/rspa.1999.0431.
  2. Mullins, L. (1969). "Softening of Rubber by Deformation". Rubber Chemistry and Technology. 42 (1): 339–362. doi:10.5254/1.3539210. Retrieved 16 September 2023.
  3. De Souza Neto, E. A.; Perić, D.; Owen, D. R. J. (1994). "A phenomenological three-dimensional rate-independent continuum damage model for highly filled polymers : Formulation and computational aspects". Journal of the Mechanics and Physics of Solids. 42 (10): 1533–1550. doi:10.1016/0022-5096(94)90086-8.
  4. 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.
Categories: