Effective elastic properties of composites of ellipsoids (II). Nearly disk- and needle-like inclusions

A general analytic solution has been obtained to effective Poisson ratio and Youngʹs modulus of an isotropic two-phase disordered composite composed of an incompressible matrix and elliptic or ellipsoidal inclusions, each having a Poisson ratio of −1, in the mean-field approximation, which yields a further result that as long as the inclusion area or volume fraction exceeds 0.33, 0.83, 0.42, 0.61 and 0.85 for nearly disc-, blade-, sphere-, disk- and neddle-like inclusions, respectively, it is always possible to make the resulting composite auxetic. Analytic expansions of the effective elastic moduli in the parameters characterizing a small deviation of inclusion shapes from blades or disks or needles have been developed, giving good approximations when truncated at second order. Similar analytic expansions to fifth order of the depolarizing or demagnetizing factors have also been presented. For a matrix having a non-negative Poisson ratio, it is found that auxeticity windows exist only for auxetic inclusions, and a maximum effective Youngʹs modulus occurs at a certain value of volume fraction of auxetic inclusions that are not far from blade- or disk- or needle-like. This maximum-Youngʹs-modulus effect may be advantageously used to produce technologically important high-strength auxetic composites as in the case of nearly disc-like or spherical inclusions studied before.