Effective elastic moduli of a soft medium with hard polygonal inclusions and extremal behavior of effective Poisson's ratio

We consider two-dimensional, two-phase, elastic composites consisting of a soft isotropic medium into which hard elastic inclusions have been placed, requiring that the inclusions be interconnected only at corner points. Denoting by δ the ratio of Young's modulus for the soft and hard phases, we show that the leading term in the asymptotic expansion as δ→0 for the effective moduli can be calculated from a finite-dimensional algebraic minimization problem. For several composites with either hexagonal symmetry or orthotropic symmetry, we explicitly solve this algebraic problem. In particular, from the above constituents we construct an isotropic material with maximal positive Poisson's ratio, as well as an orthotropic material with Poisson's ratio less than -1. We also recover in a simple way, Milton's isotropic composite with Poisson's ratio close to -1.