An optimization algorithm applied to the Morrey conjecture in nonlinear elasticity
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Date
2007
Journal Title
Journal ISSN
Volume Title
Publisher
PERGAMON-ELSEVIER SCIENCE LTD
Abstract
For a long time it has been studied whether rank-one convexity and quasiconvexity give rise to different families of constitutive relations in planar nonlinear elasticity. Stated in 1952 the Morrey conjecture says that these families are different, but no example has come forward to prove it. Now we attack this problem by deriving a specialized optimization algorithm based on two ingredients: first, a recently found necessary condition for the quasiconvexity of fourth-degree polynomials that distinguishes between both classes in the three dimensional case, and secondly, upon a characterization of rank-one convex fourth-degree polynomials in terms of infinitely many constraints.
After extensive computational experiments with the algorithm, we believe that in the planar case, the necessary condition mentioned above is also necessary for the rank-one convexity of fourth-degree polynomials. Hence the question remains open. (c) 2006 Elsevier Ltd. All rights reserved.
After extensive computational experiments with the algorithm, we believe that in the planar case, the necessary condition mentioned above is also necessary for the rank-one convexity of fourth-degree polynomials. Hence the question remains open. (c) 2006 Elsevier Ltd. All rights reserved.
Description
Keywords
rank-one convexity, quasiconvexity, Morrey conjecture, cutting-plane method, constitutive relations in nonlinear elasticity, QUASICONVEXITY