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On a longitudinal shift modelling of fibrous composites, reinforce with one-directional orthotropic fibers,weakened by rectangular cracks

Mamedov A.T., Mehtiev R.K.

#### Abstract:

We consider a doubly periodic lattice with circular holes in a plane filled with washers without interference from an orthotropic elastic material whose surface is uniformly coated with a homogeneous film and each fiber is weakened by rectilinear crack. In each washer there is a centrally located crack, the length of which is smaller than the diameter of the washer. The external load in such an environment around the openings creates zones of increased stresses, the location of which has a periodic pattern. The stresses and their displacements are expressed through an analytic function. For the solution, the well-known proposition is used that the displacement in the case of antiplane shift is a harmonic function. A well-known representation of the solution in each region is used through the corresponding complex analytic function. Three analytic functions are represented by Laurent series. Satisfying the boundary condition on the contours of the holes and the shores of the cracks, the problem reduces to two infinite algebraic systems with respect to the unknown coefficients and to one singular integral equation. Then the singular integral equation by the Multpop-Kalandia method is reduced to a finite algebraic system of equations. The procedure for calculating the stress intensity factors is given. For the numerical realization of the above method, the cases of the arrangement of holes in the vertices of triangular and square grids were taken. The results of calculations of the critical load as a function of crack length and elastic geometric parameters of a perforated medium are presented. The relevance of such studies is caused by the extensive use in the technology of structures and products made of composite materials. Studies on the development of mathematical models of the theoretically described stress-strain state of a reinforced composite near the inclusion under shear and crack are practically absent.

#### Keywords:

circular orifice,

critical loading,

lengthwise displacement,

ortotropic elastic comprehension,

perforated sheet,

rectilinear crack,

критическая нагрузка,

круговое отверстие,

ортотропное упругое включение,

перфорированная плоскость,

продольный сдвиг,

прямолинейная трещина
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