An equivalence between the rank 1 convexity and
polyconvexity of energy functions on SL+(3)

Jérémie Gaston Sambou; Edouard Diouf

doi:https://doi.org/10.14445/22315373/IJMTT-V66I9P504

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 66 | Issue 9 | Year 2020 | Article Id. IJMTT-V66I9P504 | DOI : https://doi.org/10.14445/22315373/IJMTT-V66I9P504

An equivalence between the rank 1 convexity and polyconvexity of energy functions on SL+(3)

Jérémie Gaston Sambou, Edouard Diouf

Abstract

In this research paper, we have proposed as a work to demonstrate the equivalence between the rank 1 convexity and the polyconvexity of deformation energy functions. The study is done in the three-dimensional deformation with a case of a cylindrical hyperelastic incompressible isotropic tube. To achieve our objective, the kinematics with an isotropic and incompressible strain energy function was used. We have state and demonstrate three propositions which allowed us to show that there is an equivalence between the rank 1 convexity and the polyconvexity of the energy potential which is a function of the gradient tensor. We also obtain from this study, a new theorem on the convexity and polyconvexity of energy functions in three dimensions.

Keywords

Incompressible deformation, isotropic elementary invariants, eigenvalues, Frobenius norm, energy function, Rank 1 convexity, polyconvexity.

References

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Citation :

Jérémie Gaston Sambou, Edouard Diouf, "An equivalence between the rank 1 convexity and polyconvexity of energy functions on SL+(3)," International Journal of Mathematics Trends and Technology (IJMTT), vol. 66, no. 9, pp. 17-25, 2020. Crossref, https://doi.org/10.14445/22315373/IJMTT-V66I9P504