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When Rank-One Convexity Meets Polyconvexity: An Algebraic Approach to Elastic Binodal

  • Yury Grabovsky
  • Lev Truskinovsky
Article

Abstract

In the variational problems involving non-convex integral functionals, finding the binodal, the boundary of validity of the quasiconvexity of the energy density, is of central importance. We develop a systematic methodology for identifying a part of the binodal corresponding to simple laminates by showing that in this case the supporting null-Lagrangians, establishing polyconvexity, can be constructed explicitly. We present a nontrivial example from nonlinear elasticity where this approach allows one to obtain the entire quasiconvex envelope.

Keywords

Quasiconvexity Polyconvexity Rank-one convexity Jump set Elastic stability Binodal 

Mathematics Subject Classification

74A50 74G65 49K40 49S05 

Notes

Acknowledgements

The authors wish to thank Bob Kohn for discussions and suggestions, while handling the review process. This work was conducted during the stay of YG at ESPCI Paris supported by Chair Joliot. This material is based upon work supported by the National Science Foundation under Grant No. DMS-1714287. LT was also supported by the French Government under the Grant No. ANR-10-IDEX-0001-02 PSL.

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Copyright information

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Authors and Affiliations

  1. 1.Department of MathematicsTemple UniversityPhiladelphiaUSA
  2. 2.PMMHESPCIParisFrance

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