| We calculate the effective properties of magnetostrictive composites by minimizing the total free-energy of the composite system. The Gamma-convergence method and two-scale convergence method are used to account for the multiple scales of the composite systems. There are two cases. In the first case of dilute limit, the composite consists of well separated identical ellipsoidal particles of magnetostrictive material surrounded by an elastic matrix. By using the constrained theory (DeSimone and James, 2002) and the Eshelby's solution (1957), we show that the energy minimization problem for the composite can be cast as a quadratic programming problem. In the second case of finite volume fraction, we assume fine identical magnetostrictive particles embedded periodically inside the elastic matrix. We successfully derive a homogenized theory for this case. We further show that the energy minimization problem can be again cast as a quadratic programming problem if the shape of the magnetostrictive particles in the periodic cell enjoys the same special property as an ellipsoid does in the infinite space.; Some special shapes/inclusions have played a central role in the modeling of magnetostrictive composites. The existence of these special inclusions can be addressed by a related free-boundary obstacle problem. In this way, we construct the so-called "E-inclusions". In physical terms, E-inclusions have the remarkable property that constant magnetization of the inclusion, subjected to periodic boundary conditions, induces constant magnetic field on the inclusions. These inclusions apparently enjoy many interesting properties with respect to homogenization and energy minimization. For instance, we use them to give new results on (a) explicit optimal bounds of the effective moduli of two-phase composites; (b) energy-minimizing microstructures; and (c) the characterization of the Gtheta-closure of two well-ordered composites. |
