Composite materials with poisson's ratios close to — 1

Abstract

A f amily of two-dimensional, two-phase, composite materials with hexagonal symmetry is found with Poisson's ratios arbitrarily close to — 1. Letting k∗, k 1, k 2 and μ∗,μ 1,μ 2 denote the bulk and shear moduli of one such composite, stiff inclusion phase and compliant matrix phase, respectively, it is rigorously established that when k 1 = K 2 r and μ 1 = μ 2 r there exists a constant c depending only on k 2, μ 2 and the geometry such that k∗/μ∗ < c√ r for all sufficiently small stiffness ratios r (specifically for r <frcase|1/9). This implies that the Poisson's ratio approaches — 1 as r → 0 and in this limit it is conjectured that the material deforms conlbrmally on a macroscopic scale. By introducing additional microstructure on a smaller length scale a second family of composites is obtained with substantially lower Poisson's ratios, each satisfying k/μ∗ < crThese two families provided conclusive proof that isotropic materials with negative Poisson's ratio exist within the framework of continuum elasticity. It is also shown that elastically isotropic two- and three-dimensional composites with Poisson's ratio approaching — 1 as r → 0 can be generated simply by layering the component materials together in different directions on widely separated length scales.