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Rheology pp 397-403 | Cite as

Theoretical and Numerical Studies of Anelastic Materials

  • Jehuda Rosenberg
  • Yu Chen

Abstract

The paper deals with the formulation of a constitutive equation for anelastic materials for which the stress depends on the current deformation gradient \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{F}\) and its rate \({\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\dot{F}}}\). Coleman and Mizel1 developed a thermodynamic theory for such a class of materials. The stress tensor is decomposed into an elastic omponent \({\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\sigma } ^e}\) and a dissipative component \({\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\sigma } ^d}\). Thus, let \({\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\sigma }} = {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\hat{\sigma }}}(F,\dot{F})\) and \({\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\sigma } ^e} = {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\sigma } ^e}(F) = \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\sigma } (F,0)\) and \({{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\sigma }}}^{d}} = {{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\sigma }}}^{d}}(F,\dot{F}) = \hat{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\sigma }}}(F,\dot{F}) - {{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\sigma }}}^{\varepsilon }}(F)\). The Clausius-Duhem inequality leads to \({\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{T} ^e} = {\sigma _O}\partial \psi /\partial \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{F}\) where \({\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{T} ^e}\) is the first Piola-Kirch-hoff elastic stress tensor, U is the free energy function. The, dissipative component should satisfy the inequality.

Keywords

Deformation Gradient Space Discretization Free Energy Function Prony Series Incremental Variable 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Coleman, B. D. and Mizel V. S., “Existence of Caloric Equation of State in Thermodynamics,” J. Chem. Phys. 40 (1964)Google Scholar
  2. 2.
    Parton, Y. “The Dilatational and Distortional Partition of the the Energy Function in Finite Strain,” Israel J. of Tech. 4 (1966)Google Scholar
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    Derman, D., Zaphir, Z. and Bodner, S. R., “Nonlinear Anelastic Behavior of Synthetic Rubber at Finite Strains,” J. of Rheology, 22 (1978)Google Scholar
  4. 4.
    Rosenberg, J., “Theoretical and Numerical Analyses of Anelastic Materials at Finite Strain,” Ph.D. Thesis. Department of Mechanics and Materials Science, Rutgers U. (1980)Google Scholar
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    Dill, E. H., “The Finite Element Method of Nonlinear Field Theories of Mechanics,” Proceedings of the Third IMACS International Symp. on Computer Methods for PDE (1979)Google Scholar
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    Zienkiewicz, O. C. “The Finite Element Method,” 3rd. Ed. McGraw-Hill, UK, (1977)Google Scholar
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    Flügge, W., “Viscoelasticity,” Springer-Verlag, N.Y. (1975)Google Scholar

Copyright information

© Plenum Press, New York 1980

Authors and Affiliations

  • Jehuda Rosenberg
    • 1
  • Yu Chen
    • 1
  1. 1.Department of Mechanics and Materials ScienceRutgers UniversityPiscatawayUSA

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