Two classical problems of the theory of elasticity are considered in the paper. The first is the Kelvin problem for an infinite space loaded with a concentrated force. The classical solution is singular and specifies an infinitely high displacement of the point of the force application which has no physical meaning. To obtain a physically consistent solution, the nonlocal theory of elasticity is used, which, in contrast to the classical theory, is based on the equations derived for an element of continuum that has small but finite dimensions, and allows one to obtain regular solutions for traditional singular problems. The equations of the nonlocal theory include an additional experimental constant, which has the dimension of length and cannot be determined for a space problem. Consequently, the second problem for an infinite plane loaded with two concentrated forces lying on the same straight line and acting in the opposite directions is considered. The classical solution of this problem is also singular and specifies an infinitely high elongation of the distance between the forces, irrespective of their magnitude. The solution of this problem is also obtained within the framework of the nonlocal theory of elasticity, which specifies a regular dependence of this distance on the forces magnitude. This solution also includes an additional constant which is determined experimentally for a plane problem.

The article deals with the problem of stability of an infinitely long cylindrical shell, on which absolutely flexible threads are wound under tension, compressing the shell so that it can lose stability. Using the equations of the nonlinear theory of shells, an exact solution is obtained that determines the critical tension of the threads. It has been established that the critical tension is determined by a universal formula that includes the radius and flexural rigidity of the shell. Using the concept of continuing loading, the obtained solution is generalized to the case of an elastic-plastic shell material. The problem of pressure on an elastic shell by layers of elastic threads wound with tension is considered. The solution is compared with the results of an experiment carried out on shells made of steel, aluminum and thermoplastic.

The article deals with the problem of stability of an infinitely long cylindrical shell, located in an absolutely rigid medium, compressing the shell so that it can lose stability only by deforming into an inner cavity. Using the equations of the nonlinear theory of shells, which takes into account from nonlinear effects only changes in the radii of curvature of the middle surface of the shell during deformation, an exact solution is obtained that determines the critical pressure or the limiting value of the subcritical deformation of the shell. It was found that the critical pressure and deformation largely depend on the connection between the shell and the external environment in the annular direction. Two limiting cases are investigated: a shell rigidly bound to the medium and a shell free from tangential annular surface load. The solution obtained is compared with the results of an experiment carried out on composite shells with a metal and polymer inner layer.

A classical plane problem of the theory of elasticity about a crack in a stretched orthotropic elastic unbounded plane is considered, which leads to a singular solution for stresses in the vicinity of the crack edge. The relations of the generalized theory of elasticity, including a small scale parameter, are given. The equations of the generalized theory are of a higher order than the equations of the classical theory and allow eliminating the singularity of the classical solution. The scale parameter is determined experimentally. The results obtained determine the effect of the crack length on the bearing capacity of the plate and are compared with the experimental results for plates made of fiberglass and carbon fiber reinforced plastic.

The paper is concerned with the classical Flamant problem of the theory of elasticity. The classical solution of this problem obtained by A. Flamant in the nineteenth century specifies the stresses and the displacements induced in a half-plane by a concentrated force applied to the half-plane boundary in the normal direction. Being presented in numerous textbooks in the theory of elasticity, this solution provides results that can be qualified as paradoxical and which traditionally are left without proper comments in the literature. Particularly, the half-plane displacements are singular at the point of the force application and are infinitely increasing with a distance from this point. The displacement of the boundary in the boundary direction is not continuous and experiences a jump at the point of the force application. A boundary element under any force, irrespective of how low it is, rotates at the point of the force application by 90 $$^{\circ }$$ which does not correspond to basic assumptions of the linear theory of elasticity; hence, the classical solution cannot be qualified as consistent. The consistent solution of the problem is constructed in the paper on the basis of the generalized theory of elasticity the equations of which are obtained for the solid element that has small but not infinitesimal dimensions. As a result, these equations provide regular solutions of the problems that are singular in the classical theory of elasticity. The obtained generalized solution of the Flamant problem demonstrates regular behavior of the displacements which are not singular at the point of the force application. The solution is supported by experimental results obtained for the plate of silicon rubber simulating the half-plane.

The problem of stretching a plate with a central or lateral crack is considered. The previously obtained solution, which determines the stresses in the plate in the vicinity of the central crack, is used to predict the load at which plastic deformation appears in the plate with a lateral crack. The Mises condition is used as a criterion for plasticity. The calculation results are compared with the experiment carried out on plates made of aluminum alloy, brass and steel.

The problem of the strength of a plate made of a brittle material with through mode I cracks is discussed. In contrast to the approach based on the singular solution of the classical theory of elasticity for a plane with a crack and on linear fracture mechanics, we propose to use nonsingular solutions obtained within the generalized elasticity theory and, as a result, to implement a method, conventional for the strength estimation of solids with stress concentration, based on the maximum stress criterion. The maximum stress is determined from a nonsingular solution of the generalized elasticity equations for a plane with a crack. The reported experimental results for plates with cracks under tension and bending confirm the solution obtained by the proposed method and allow it to be compared with a solution based on linear fracture mechanics. In fact, a new concept of fracture mechanics is put forward, which is free of singular solutions and allows the problems of fracture mechanics to be treated as problems of stress concentration. Comparison of the obtained analytical solutions with the experimental data has shown that the scale factor of generalized elasticity determines the critical state in fracture mechanics with no less accuracy than the critical stress intensity factor and therefore can be used as a fracture criterion. The resulting explicit nonsingular solutions allow the prediction of the stress concentration caused by a crack.

A new technological process for the synthesis of composite materials based on carbon fibers and the combination of a thermoset and a thermoplastic matrix is described. The mechanical characteristics of a two-matrix composite are presented. The scope of its application is discussed.