Equilibrium internal fractures in elastic bodies supported by thin flexible coatings

Objective. In this paper, the authors study problems of a plane strain of elastic bodies containing internal rectilinear fractures. In each case, the margins of the considered areas are supported by thin flexible coatings. The first part of the paper is devoted to the problem of an infinite elastic wedge, the faces of which are free from the outside and reinforced with a thin flexible material, and the bisector contains a rectilinear fracture with regular forces applied to the margins, and to the study of the stress concentration at the fracture vertices. In the second part of the paper, the authors consider the problem of an equilibrium radial internal fracture in the cross-section of a round pipe. The inner surface of the pipe experiences hydrostatic pressure; the outer surface is reinforced with a thin flexible coating. The purpose of the study in each of the presented tasks is to determine the values of the influence factor. Methods. Both problems are united by a single approach, in which the presence of a coating is modeled mathematically, using special marginal conditions obtained based on an asymptotic analysis of the exact solution for a strip or ring flexible coating of small relative thickness. In the first issue, the singular integral equation is derived using the Mellin transform, which allows proceeding to the solution of a system of ordinary differential equations and obtaining a singular integral equation relative to the derivative of the discontinuity function of the first kind with a Cauchy kernel. In the second issue, discontinuous solutions are constructed using the Fourier series, resulting in a singular integral equation of a similar structure. Previously, similar ideas were successfully implemented by the authors in the study of the problem of the equilibrium state of a strip with a coating weakened by an internal transverse fracture under arbitrary conditions on the lower edge of the strip. Conclusion. Singular integral equations for the considered problems are obtained. The collocation method is used to construct solutions of singular integral equations for various combinations of geometric and physical characteristics of issues. In all the considered cases, the values of the influence factor were calculated. The analysis of changes in the influence factor depending on various combinations of geometric parameters and mechanical characteristics of problems is carried out. It is noted that with increasing rigidity of the coating and increasing its thickness, the values of the influence factor decrease; the increase in the value of the influence factor is provided by approaching the fracture to the body margin and increasing its relative length.

FEATURES OF STRESSES AT THE APEX OF AN ELASTIC WEDGE, SUPPORTED BY A THIN FLEXIBLE COATING ON THE SIDES

Objectives To study the problem of determining the degree of stress at the apex of a wedge-shaped area in cases where the sides (or one of them) are covered with a thin flexible coating.Method It is assumed that the coating is not stretchable. On the other side of the wedge-shaped area, the same coating is assumed to be present; it is either fixed, stress-free or in smooth contact with a rigid base. Mathematically, the problem is reduced to the task of determining the roots of characteristic transcendental equations arising from the existence of a nontrivial solution to the system of linear homogeneous equations.Results Values for the specific characteristics of the radial component of a stress tensor are determined for different combinations of boundary conditions and solution angles. In particular, the angles at which the singular behaviour of stresses occurs are determined. The case is considered when a special boundary condition is given on the edge surface, simulating the overlay. Characteristic equations are obtained to determine the index of the degree dependency of the asymptotic solution in its vicinity for four variants of boundary conditions. In two cases, transcendental equations are obtained, which are solved numerically.Conclusion Calculations of the first positive roots of the equations depending on the angle of the edge solution and Poisson's ratio are presented. The values of the angles, at which the singular behaviour of stresses occurs, are determined. In the case of a combination of boundary conditions (III – IV), the singular stress behaviour is observed for the angle ???? = ????/8, while in the case of (III – III) this value is equal to ????/4.