COMPOSITE-MATERIALS WITH POISSON RATIOS CLOSE TO -1

A FAMILY of two-dimensional, two-phase, composite materials with hexagonal symmetry is found with Poisson's ratios arbitrarily close to -1. Letting kappa*, kappa-1, kappa-2 and mu*, mu-1, mu-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 kappa-1 = kappa-2/r and mu-1 = mu-2/r there exists a constant c depending only on kappa-2, mu-2 and the geometry such that kappa*/mu* < c square-root s for all sufficiently small stiffness ratios r (specifically for r < 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 conformally 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 kappa*/mu* < c'r. These two families provide 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.