Cubic inclusion arrangement: Effects on stress and effective properties

A 3D stress analysis method based on the theory of eigenstrains and Eshelby’s equivalency principle is established. Multiple inclusion interaction is taken into account, thus the eigenstrains in each inclusion are no longer assumed to be uniform. The multiple inclusion problem is solved from the governing elasticity equations to give a set of coupled singular integral equations in the unknown eigenstrains of each inclusion. The set of coupled singular integral equations are rewritten using numerical integration, to give a set of algebraic equations in the unknown eigenstrains. Once the equivalent eigenstrains are obtained both the local stress/strain and the effective elastic properties can be calculated. For illustrative purposes the inclusions are dispersed in a cubic arrangement. Four different inclusion separation distances are considered and in each of the four situations three different inclusion stiffnesses are considered. The obtained stresses from the analysis are seen to be highly influenced by inclusion separation. The effective elastic properties display the symmetry required by the dispersion and obeys the Avellaneda [J. Appl. Math. 47 (1987) 1216] bounds for dispersions with cubic symmetry.