Abstract
We construct a solution of a plane two-period problem on loading an infinite elasticisotropic plane with a grid of square inclusions. The plane is under one of two loads. It is eitherstretched at some angle to the \(X\) -axis or has a pure shear at the infinity. Each regular square periodicity cellcontains a single square inclusion whose sides are perpendicular to the sides of the cell. The size ofeach inclusion is considerably greater than the thickness of the plate. The stresses are located neara stress concentrator at the frontier between the inclusion and matrix. Solution of the problem isreduced to finding complex-valued functions on the basis of boundary conditions obtained fromthe equalities of the normal forces and displacements of the matrix and inclusions. We useconformal mappings and integrate by the Muskhelishvili method. The effect of noncentralinclusions is expressed by using the small parameter method. As a result, we obtain a system oflinear algebraic equations for solving the two-period problem under consideration and find itssolutions for several partial cases. We compare our result with the numerical solution obtainedwith the use of the Abaqus software (which is based on the finite element method). Solution ofsuch a problem is actual because it models loading of a fiber composite. There are comparativelyfew articles on fiber composites in mechanics. The majority of them is devoted to analysis ofeither experiments or numerical solutions. Therefore, the presented analytic solution is ofsignificant scientific value.
