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On the relaxation of a three-well energy. (English) Zbl 0960.74027

The authors study the relaxation of multi-well non-convex energies in the context of infinitesimal (geometrically nonlinear) elasticity or analogous problem for gradient fields. A dual variational principle of Hashin-Shtrikman type is developed for the relaxed energy under the assumption that the original energy function is a difference of a quadratic function and of another convex function. In the particular case of linear elastic wells sharing the same elastic modulus, this reduces to the Kohn approach of [see, e.g., K. Bhattacharya, R. V. Kohn and S. Kozlov, Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 455, No. 1982, 567-583 (1999; Zbl 0926.74099)] and leads to the minimization of an energy functional with respect to matrix measures on the unit sphere (\(H\)-measures), which are subject to certain constraints. The main result: for the ‘three-well’ linearized energy the minimizing measure subject to these constraints can be chosen as a sum of no more than three Dirac masses. The authors also describe a subclass of such three-point measures realizable by microstructures. It is demonstrated by means of examples that the minimizing measures do fall within this subclass in some cases, thereby providing an exact value for the three-well non-convex relaxed energy. More generally, the minimization algorithm leads to an improved lower bound for the relaxed energy. If the three phases are pairwise compatible, the relaxed energy is a convexification of the original energy.

MSC:

74G65 Energy minimization in equilibrium problems in solid mechanics
74A40 Random materials and composite materials
74Q20 Bounds on effective properties in solid mechanics

Citations:

Zbl 0926.74099
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