Two-Phase Isotropic Composites of Extremal Moduli. The Inverse Homogenization Problem

This work deals with the inverse homogenization problem: for given two well-ordered elastic and isotropic materials characterized by the bulk and shear moduli ($\kappa_1$, $\mu_1$), ($\kappa_{12}$, $\mu_2$) and the volume fraction $\rho$ of the second material reconstructing the layout of the most second- rank orthogonal laminates within a hexagonal 2D periodicity cell $ \mathit{Y}$ corresponding to the predefined values of moduli ($\kappa^*$, $\mu^*$) of the effective isotropic composite. The used algorithm follows from imposing the finite element (FE) approximation on the solution to the basic cell problems of the homogenization theory [7] along with periodicity assumptions. The material properties of each element are described by three independent parameters. Thus, the formulated inverse problem is solved numerically by the gradient method. The adopted cell structure, i.e., the hexagonal cell with the rotational symmetry of $120^\circ$ angle guarantees the isotropic effective properties of the composite, and thus the optimization problem is greatly simplified. Isotropic constraints do not appear in the formulated optimization problem.