№ 5 (2024)

For the first time, an exact solution is given to the contact problem of the non-stationary action of a wedge-shaped, right-angled stamp occupying the first quadrant, which act on a deformable multilayer base. The base, which is affected by a rigid stamp in the shape of a quarter plane, can be a multilayer anisotropic composite material. It is assumed that it is possible to construct a Green’s function for it, which makes it possible to construct an integral equation of the contact problem. The geometric Cartesian coordinates of the first quadrant and the time parameter, which varies along the entire axis, are taken as parameters describing the integral equation. It is assumed that time in the boundary value problem under consideration follows from negative infinity, crosses the origin and grows to infinity, covering the entire time interval. Thus, there is no requirement in the formulation of the Cochet problem when it is necessary to set initial conditions. In this formulation, the problem is reduced to solving the three-dimensional Wiener-Hopf integral equation. The authors are not aware of any attempts to solve this problem analytically or numerically. The investigation and solution of the contact problem was carried out using block elements in a variant applicable to integral equations. It is proved that the constructed solution exactly satisfies the integral equation. The properties of the constructed solution are studied. In particular, it is shown that the solution of the non-stationary contact problem has a higher concentration of contact stresses at the edges of the stamps and at the angular point of the stamp, compared with a static case. This corresponds to the observed in practice more effective non-stationary effect of rigid bodies on deformable media, for their destruction, compared with static. The results may be useful in engineering practice, seismology, in assessing the impact of incoming waves on foundations, in the areas of using Wiener-Hopf integral equations in probability theory and statistics, and other areas.

This article presents a method for solving the boundary-value problem for an elliptic boundary layer occurring in thin-walled shells of revolution under impact loads of normal type applied to the face surfaces. The elliptic boundary layer is formed in the vicinity of the conditional front of Rayleigh surface waves and is described by elliptic equations with boundary conditions determined by hyperbolic equations. In the general case of shells of revolution, methods for solving elliptic boundary layer equations developed for shells of revolution with zero Gaussian curvature cannot be applied. The previously considered approach using Laplace and Fourier integral transforms fails because the governing equations become equations with variable coefficients. The method proposed in this article for solving the equations of the elliptic boundary layer is based on the use of asymptotic representations of the Laplace-transformed solutions (in time) in exponential form. Numerical calculations of normal stresses based on the obtained analytical solutions are provided for the case of a spherical shell.

The article discusses problems of dynamic bending for beams of semi-infinite length. To solve such problems, the article uses a method based on the implementation of the conservation laws, namely, the law of energy conservation, the law of change in momentum and the law of change in angular momentum. The results obtained are compared with the analytical solution for the problem of a semi-infinite beam motion loaded at the free end with a transverse force. The peculiarity of this solution is that the change in the stress-strain state of the rod is characterized by a wave front. It is considered that all changes in the state of the beam occur at an infinite speed. All designed solutions are characterized by the presence of a wave front in the beam. It is shown that, in contrast to the transfer of longitudinal disturbances along the length of the beam, which occur at a constant speed, bending disturbances propagate at a variable speed, and, with increasing time, this speed decreases and tends to zero at an infinitely distant point of the beam. It was discovered that the propagation velocity of the wave front during the transfer of concentrated force and concentrated moment differs from each other. In this case, the speed of transverse force transfer is almost twice as high as the speed of the wave front from the bending moment.

The stress-strain state is considered and the time to fracture of a composite tensile rod during creep in an active medium is determined. The influence of the active medium is determined by a non-classical diffusion process, with the active substance penetrating into the material in two states: free and bound. The process of such diffusion is described by a modified diffusion equation that takes into account the two-phase state of the active substance in the material. A system of equations has been obtained that models the creep of a composite rod, in which its parts are rigidly connected to each other without slipping, and also includes kinetic equations for the accumulation of damage in parts of the rod. The influence of the active medium is taken into account by introducing into the indicated kinetic equations the function of the influence of the active medium - a function of the integral average concentration. Stress distributions and damage accumulation processes over time in various parts of the composite rod are analyzed. Calculations were carried out in two cases, namely, classical and non-classical diffusion processes are considered. The setting of these differences is determined by the choice of appropriate parameters in the diffusion model under consideration. Dependences of damage accumulation and stress distribution in parts of the rod over time were obtained. As a result, it was determined that the destruction of a composite rod in the classical case occurs earlier than in the case of the considered non-classical diffusion process.

For the tensor-matrix FEM system of equations describing the final state of large deformations of an incompressible elastic body, a development to solve axisymmetric problems is obtained. Also, analytical expressions for the components of the partial derivatives matrix of the system are obtained. Examples of calculating the everted state of a circular cylinder, as well as analysis of sealing rings are described.

Based on the enthalpy formulation of a three-dimensional transient nonlinear thermal conductivity problem for a multiphase system, mathematical modeling of the selective laser melting process of titanium and aluminum alloy powders was conducted to produce metallic components. The geometric parameters of a single track, as well as single-layer and multi-layer systems of overlapping tracks, were determined as functions of laser beam power and speed. This enabled the evaluation of the structure and types of defects arising during the layer-by-layer printing of samples. To investigate the influence of single and multiple defects on the fatigue strength of printed samples under high-frequency loading, a previously proposed multi-mode cyclic damage model was used. It was demonstrated that the internal heterogeneity of the microstructure of materials printed using selective laser melting can lead to earlier subsurface initiation of fatigue cracks, significantly reducing fatigue strength and durability. This effect is more pronounced in systems with multiple defects. The proposed models and computational algorithms enable the calculation of the fatigue strength and durability of samples for various defect systems in the microstructure, corresponding to the specified characteristics of the moving laser beam. They also make it possible to identify process parameters for selective laser melting that achieve the best fatigue strength performance under high-frequency loading.