Two-periodic problem of loading isotropic plane with a grid of square inclusions

This paper presents the construction of a solution to a plane doubly periodic loading problem for an infinite elastic isotropic plane with a lattice of square inclusions. The plane is subject to one of two loads: either tension at an angle to the X-axis or pure shear at infinity. A regular square periodic cell contains a single square inclusion with sides perpendicular to the cell boundaries. The size of the inclusion is much larger than the plate thickness. The stresses are located near the stress concentrator—at the boundary between the inclusion and the matrix. The solution to the problem is reduced to finding complex functions from the boundary conditions obtained from the equality of the normal forces and displacements of the matrix and inclusion, using conformal mappings and integration by Muskhelishvili's method. The influence of non-central inclusions is expressed using the small parameter method. As a result, a system of linear algebraic equations was obtained for solving the doubly periodic problem under consideration, and solutions were found for several special cases. The result was verified against a numerical solution using the finite element method in the Abaqus software package. The solution to this problem involves modeling the loading of a fiber composite, making it highly relevant. Relatively few works have been published on the topic of fiber composites in mechanics, most of which are devoted to the analysis of experimental studies or numerical solutions, so this analytical solution has significant scientific value.

УДК 539.375

Keywords: two-periodic problem, complex functions, conformal mapping, small parameter method, integration by the method of Muskhelishvili, square inclusions.