LONGITUDINAL SHIFT OF THE BINDER AND INCLUSIONS OF A COMPOSITE WEAKENED BY TWO-PERIODIC RIGHT LINEAR CRACKS

In this paper, we consider antiplane deformation for an isotropic elastic material consisting of an infinite system of parallel identical circular cylindrical fibers covered with a uniform cylindrical film uniformly covering the surface of each fiber and a bonding medium weakened by a doubly periodic system of rectilinear cracks. Each washer has a centrally located crack that is less than the diameter of the washer. The presented stresses and their displacements are expressed in terms of an analytical function. For the solution, the well-known position is used that the displacement in the case of an antiplane shear is a harmonic function. A known representation of the solution in each area is applied through the corresponding complex analytical function. Three analytic 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 two singular integral equations with a Cauchy-type kernel. Then the singular integral equation is reduced to a finite algebraic system of equations by the Multopp – Kalandia method. The procedure for calculating the stress intensity coefficients is given. The numerical implementation of the described method is given at IBM. The results of calculations of the critical load depending on the crack length and elastic geometric parameters of the perforated medium are presented. Keywords: isotropically elastic material, doubly periodic lattice, rectilinear cracks, stress intensity factor, mean stresses, critical load, circular hole, longitudinal shear.

Anti-Plane Deformation Uniformly Piecewise Homogeneous Space with a Periodic System of Semi-Infinite Internal Cracks

In this paper, we have constructed a solution for the problem of antiplane deformation of a uniformly piecewise homogeneous space of two alternately repeating heterogeneous layers of equal thickness from different materials, which are relaxed on their median planes by two semi-infinite, periodic parallel tunneling cracks. A system of defining equations of the problem is derived in the form of a system of two singular equations of the first kind, with respect to contact stresses acting in the contact zones on the median planes of heterogeneous layers, the solution of which, in the general case, is constructed by the method of mechanical quadrature. In the particular case when the cracks in the heterogeneous layers are the same, the solution of the problem is reduced to the solution of two independent equations and their closed solutions are constructed. The defining singular integral equation of the problem is also obtained in the case when there are no cracks in one of the heterogeneous layers. In the general case, a numerical calculation was carried out and patterns of changes in contact stresses and intensity factors of destructive stresses at the end points of cracks were determined depending on the physical and mechanical and geometric parameters of the problem, which are the ratios of the shear moduli of the layers and the ratio of the layer thickness and crack lengths.

On the Application of the Method of Hypersingular Integral Equations to Solving Problems for an Elastic Plane with a Collinear System of Cracks

In the present paper, using the method of hypersingular integral equations, based on the formulas of the inversion of the corresponding singular integral equations, the exact quadrature solution of the classical problems of the mechanics of an elastic plane with a collinear system of cracks is constructed. The elastic plane is in a state of antiplane deformation or plane deformation; in case of antiplane deformation, crack edges are symmetrically loaded by tangential forces, while in case of plane deformation, they are again loaded symmetrically but by normal forces. Mixed boundary-value problems for an elastic half-plane equivalent to these problems are formulated. Under plane deformation, the mixed boundary-value problem for an elastic half-plane is discussed as well when the plane boundary is reinforced by two similar and symmetrically located semi-infinite stringers between which a system of an arbitrarily final number of stringers is situated. It is considered that the stringers are absolutely rigid for expansion and compression and absolutely flexible for bending. A particular case of two similar symmetrically located cracks is considered more in detail. In this case, the exact solution to the problem is also constructed by the method of Chebyshev orthogonal polynomials.

Effect of surface tension on the antiplane deformation of bimaterial with a thin interface microinclusion

Within the framework of the concept of micromechanics, a method for taking into account the effect of surface energy for a thin interface micro-inclusion in the bimaterial under conditions of longitudinal shear has been proposed. The possibility of non-ideal contact between inclusion and matrix is provided, in particular, tension contact. This significantly extends the scope of applicability of the results. A generalized model of a thin inclusion with arbitrary elastic mechanical properties was built. Based on the application of the theory of functions of a complex variable and the jump function method, the stress field in the vicinity of the inclusion during its interaction with the screw dislocation was calculated. Several effects have been identified that can be used to optimize the energy parameters of the problem.

On the Interaction of Cracks and Stringers with a Heterogeneous Wedge-Shaped Body, Manufactured from Different Materials at an Antiplane Deformation

The problem of stress state of an elastic piecewise-homogeneous wedge-shaped body at an antiplane deformation, consisting of heterogeneous wedges with different shear modules and opening of apex angles is considered, when a system of arbitrary finite number of collinear cracks is located on the interface line of the heterogeneous materials and the boundary faces of the compound wedge are reinforced with stringers of finite lengths. The solution of the problem is reduced to solving a system of three singular integral equations (SIE) using the Mellin integral transform, which based on quadrature formulas Gauss for calculating SIE with Cauchy kernel and ordinary integrals reduces to a system of systems of linear algebraic equations (SLAE). As a result, the characteristics of the problem are expressed by explicit simple structures algebraic formulas.