An elastic medium weakened by a doubly periodic system of circular holes filled with washers of foreign elastic material, the surface of which is covered with a cylindrical film, is considered. The medium (binder) is weakened by doubly periodic systems of straight through cracks. An external load xy∞τ in such an environment around the holes is formed by zones of increased stress, the location of which has a doubly periodic character. The presented stresses and their displacements are expressed in terms of an analytical function. General representations of solutions are constructed that describe the class of problems with a doubly periodic stress distribution outside circular holes and straight-line cracks. For the solution, the well-known position is used that the displacement in the case of a transverse shear is a harmonic function. A known representation of the solution in each area is applied through the corresponding complex analytical function. The three analytical functions are represented by Laurent series. Satisfying the boundary condition on the contours of holes and crack faces, the problem is reduced to two infinite algebraic systems with respect to the sought coefficients and to one singular integral equation. Then the singular integral equation is reduced to a finite algebraic system of equations by the Multopp – Kalandia method. A procedure for calculating the stress intensity factors is presented. For the numerical implementation of the described method, the cases of the location of the holes at the vertices of the triangular and square meshes were taken. The results of calculations of the critical load depending on the crack length and elastic geometric parameters of the perforated medium are presented. The relevance of such studies is due to the widespread use in engineering of structures and products made of composite materials. Research on the development of mathematical models of the theoretically described stress-strain state of the reinforced composite near the inclusion at shear and cracks is practically small. The transverse shear of a linearly reinforced medium with three mutually perpendicular planes of symmetry in a shear state in a plane perpendicular to the orientation of the fibers is considered. Due to the symmetry of the medium, its deformations along the orientation of the filler are absent, and the stress – strain state is a function of only the variables 2x and 3x; it is obvious that the shears of the medium in the plane under consideration will be independent of the shear deformations in the reinforcement plane. All of the above conditions reduce to the fulfillment of the equalities 12 13 110, 0, 0= = =γγε and 10u =. As always, the 1Ox axis is directed along the fiber orientation.

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