Two-periodic problem of loading an isotropic plane with a hexagonal packing of elliptical inclusions

This paper presents a solution to the plane doubly periodic problem of loading an infinite, isotropic, elastic plane with a hexagonal packing of elliptical inclusions. The plane is subjected to one of three types of remote loading: uniaxial tension along one of the inclusion’s axes or pure shear. The regular hexagonal unit cell contains a single elliptical inclusion with its axes perpendicular to the cell boundaries. The size of the inclusion is significantly larger than the plate thickness. The solution is constructed by reducing the problem to finding complex potentials from the boundary conditions, which are derived from the equality of normal forces and displacements in the matrix and the inclusion. This is achieved using conformal mapping techniques and the method of Muskhelishvili. The influence of non-central inclusions is accounted for via a small parameter method. As a result, a system of linear algebraic equations is derived for solving the considered doubly periodic problem, and solutions for several specific cases are obtained. The results were verified against a numerical solution obtained by the finite element method in the Abaqus software package. The solution to this problem serves as a model for the loading of a fibrous composite, which makes it highly relevant. A relatively small number of works in mechanics are devoted to fibrous composites, with most publications focusing on the analysis of experimental studies or numerical solutions. Therefore, this analytical solution possesses significant scientific value.