From microstructure-independent formulas for composite materials to rank-one convex, non-quasiconvex functions
نویسنده
چکیده
Examples of non-quasiconvex functions that are rank-one convex are rare. In this paper we construct a family of such functions by means of the algebraic methods of the theory of exact relations for polycrystalline composite materials, developed to identify G-closed sets of positive codimensions. The algebraic methods are used to construct a set of materials of positive codimension that is closed under lamination but is not G-closed. The well-known link between G-closed sets and quasiconvex functions and sets closed under lamination and rank-one convex functions is then used to construct a family of rotationally invariant, nonnegative, and 2-homogeneous rank-one convex functions, that are not quasiconvex.
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