Plane-strain elastic problem for a square array of disks. I. Elastic field in a composite with soft inclusions

The stress-strain elastic field in a square array of $N$ non-overlapping circular inclusions is described by approximate analytical formulas. In particular, soft inclusions are studied by an asymptotic analysis. The case with $N=1$ yields a regular square array of disks of radius $r$ embedded in an elastic matrix. The computations of Natanzon and Filshtinsky are based on an infinite system of linear algebraic equations solved by the truncation method. The infinite system determines the Taylor series coefficients of the Kolosov–Muskhelishvili complex potentials. A method of functional equations is used to write the series coefficients in symbolic form up to terms of the order of $O(r^{2s})$ at a fixed value of $s$. Approximate analytical formulas for local elastic fields are derived.