Symmetry in the Problem of Shear of Composites

Mathematical models of the micromechanical problem of shear of symmetric composites on their representative cells are considered. Within the context of the Curie principle, two types of symmetry elements of mechanical fields are distinguished — induced and produced ones. The sufficient conditions of produced symmetry were determined earlier by the present author. Employment of the corresponding theorem and its consequences allowed the author to formulate the boundary value problem of micromechanics on the minimum representative cell — the symmetry cell. By using symmetry and the Lagrange and Castigliano variational principles in a generalized form, it is proved that solutions of the problem on an “infinite” cell give the lower and upper bounds for the exact shear modulus and that these bounds lie within the Voigt and Reuss bounds.