Effective elastic properties of composites of ellipsoids (I). Nearly spherical inclusions

Exact formulation for calculating effective elastic moduli of an isotropic two-phase disordered composite with ellipsoidal or elliptic inclusions are given in the mean-field approximation, which yields simple analytic expansions of effective Poisson ratio and Youngʹs modulus to second order in the small asphericity parameters for nearly disc-like and spherical inclusions. Analytic expansions to fifth order in these parameters of the depolarizing or demagnetizing factors for nearly spherical ellipsoids have also been obtained, as have those to second order of the critical parameters of the auxeticity windows in the case of rigid auxetic inclusions randomly embedded in an incompressible matrix. For a matrix having a non-negative Poisson ratio, it is found that auxeticity windows for both inclusion volume or area fraction and the ratio of Youngʹs modulus of inclusion to that of a matrix 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 disc-like or spherical. This maximum Youngʹs modulus effect may be exploited to produce technologically important high-strength auxetic composites.