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Necessary and sufficient conditions for isotropic rank-one convex functions in dimension 2

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Abstract

We provide some new necessary and sufficient conditions for regular isotropic rank-one convex functions on M 2 +={2×2 matrices such that det M≥0}. It is well known that isotropic functions W (M) can be written as W (M)=G(λ1, λ2) where λi are the singular values of M. One of these conditions allows us to understand better the gap between the rank-one convexity and the quasiconvexity.

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References

  1. E. Acerbi and N. Fusco, Semicontinuity problems in the calculus of variations. Arch. Ration. Mech. Anal. 86 (1984).

  2. J.J. Alibert and B. Dacorogna, An example of a quasiconvex function not polyconvex in dimension two. Arch. Ration. Mech. Anal. 117 (1992).

  3. G. Aubert, On a counterexample of a rank-one convex function which is not polyconvex in the case n=2. Proc. Roy. Soc. Edim. A. 106 (1987).

  4. G. Aubert, Quelques remarques sur les notions de 1-rang convexité et de polyconvexité en dimensions 2 et 3. Math. Model and Num. Anal. M 2 AN. Vol. 22, no1 (1988).

    Google Scholar 

  5. G. Aubert and R. Tahraoui, Sur la faible fermeture de certains ensembles de contraintes en elasticité non linéaire plane; Arch. Ration. Mech. Anal. 97 (1987).

  6. J.M. Ball, Existence theorems in non linear elasticity; Arch. Ration. Mech. Anal. 63 (1976).

  7. B. Dacorogna, Direct methods in the calculus of variations; Springer Verlag, Berlin (1990).

    Google Scholar 

  8. J.K. Knowles and E. Sternberg, On the failure of ellipticity of the equations of finite elasticity. Arch. Ration. Mech. Anal. 8 (1978).

  9. J.D. Davies, A simple derivation of necessary and sufficient conditions for strong ellipticity of isotropic hyperelastic materials in plane strain. J. Elasticity 26 (1991).

  10. C.B. Morrey, Multiple integrals in the calculus of variations; Springer Verlag, Berlin (1966).

    Google Scholar 

  11. P. Rosakis and H.C. Simpson, On the relation between polyconvexity and rank-one convexity in nonlinear elasticity. J. Elasticity 37 (1995) 113–137.

    Google Scholar 

  12. H.C. Simpson and S.J. Spector, On compositive matrices and strong ellipticity for isotropic material. Arch. Ration. Mech. Anal. 84 (1983).

  13. V. Sverak, Rank-one convexity does not imply quasiconvexity. Proc. Roy. Soc. Edinburgh. 120 (1992).

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Authors and Affiliations

  1. Laboratoire de Mathématiques J. A. Dieudonné, URA 168, Université de Nice-Sophia Antipolis, Parc Valrose, B.P. 71, F-06108, Nice cedex 02, France

    Gilles Aubert

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  1. Gilles Aubert
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Aubert, G. Necessary and sufficient conditions for isotropic rank-one convex functions in dimension 2. J Elasticity 39, 31–46 (1995). https://doi.org/10.1007/BF00042440

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  • DOI: https://doi.org/10.1007/BF00042440

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