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1. Introduction. Let Sym denote the linear space of all symmetric second-order tensors on an n-dimensional real vector space Vect with scalar product. (If Vect is identified with R n , then Sym may be identified with the set of all symmetric n-by-n matrices.) A function f : Sym → R is said to be isotropic if f (A) = f (QAQ T) for all A ∈ Sym and all Q(More)
For interfacial interactions of “separable type” the existence is proved of stable multiphase equilibrium states minimizing the total energy which includes a sharp interface contribution along interfaces separating the phases. The second gradients of deformation do not occur; the theory is based on interfacial null lagrangians as determined in [11–12]. The(More)
A version of Cauchy’s stress theorem is given in which the stress describing the system of forces in a continuous body is represented by a tensor valued measure with weak divergence a vector valued measure. The system of forces is formalized in the notion of an unbounded Cauchy flux generalizing the bounded Cauchy flux by Gurtin & Martins [12]. The main(More)
Let f be a rotationally invariant (with respect to the proper orthogonal group) function defined on the set M2×2 of all 2 by 2 matrices. Based on conditions for the rank 1 convexity of f in terms of signed invariants of A (to be defined below), an iterative procedure is given for calculating the rank 1 convex hull of a rotationally invariant function. A(More)
The sharp interface limit of a diffuse interface theory of phase transitions is considered in static situations. The diffuse interface model is of the Allen–Cahn type with deformation, with a parameter ε measuring the width of the interface. Equilibrium states of a given elongation and a given interface width are considered and the asymptotics for ε r 0 of(More)
The differentiability of the metric projection P onto a closed convex set K in Rn is examined. The boundary ãK can have singular points of orders k ̈ −1Ù 0Ù 1ÜÙ n − 1. Here k ̈ −1 corresponds to the interior points of K , k ̈ 0 to regular points of the boundary (i.e., faces), k ̈ 1ÙÜ Ù n − 2 to edges and k ̈ n − 1 to vertices. It is assumed that for(More)
This note proposes to modify the definition of quasiconvexity of a function f Úìm˜n r Ï ñ Ú ̈ ñ T −ðÙð( on the spaceìm˜n of m ˜ n matrices in such a way that (i) the polyconvexity implies quasiconvexity without any additional measurability or continuity assumption on f and (ii) the pointwise supremum of any family of quasiconvex functions is a quasiconvex(More)