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The lower quasiconvex envelope of the stored energy function for an elastic crystal. (English) Zbl 0718.73075
The author studies minimization problems for integrals of the form \(E_{\Omega}(u):=\int_{\Omega}W(\nabla u(x))dx\), where \(\Omega \subset {\mathbb{R}}^ m\) is open and u: \(\Omega \to {\mathbb{R}}^ n\). Applying some general observations to the case when \(m=n\) and W is the stored-energy function of a homogeneous elastic material, she shows that in this case the \(W^{1,\infty}\)-quasiconvex envelope QW of W is rank 1-convex. For certain models of elastic crystals (having a “large” symmetry group) she deduces that \(QW(F)=h(\det F)\) for some function h. Related conclusions have been obtained by M. Chipot and D. Kinderlehrer [Arch. Ration. Mech. Anal. 103, No.3, 237-277 (1988; Zbl 0673.73012)].

74A60 Micromechanical theories
74M25 Micromechanics of solids
74B20 Nonlinear elasticity
49J52 Nonsmooth analysis
49J27 Existence theories for problems in abstract spaces
74E15 Crystalline structure
74S30 Other numerical methods in solid mechanics (MSC2010)
74P10 Optimization of other properties in solid mechanics