The paper presents a model of steady-state oscillations of an inhomogeneous body with a prestressed exfoliating coating based on a general linearized statement of the problem of the motion of a prestressed-strained elastic body. On its basis, the statement of the problem of oscillations of an inhomogeneous strip consisting of a substrate and a prestressed coating is formulated, between which there is a delamination in a certain region. Steady oscillations are caused by a load applied to the upper boundary of the coating. To calculate the oscillations of the two-dimensional structure under consideration, the Fourier transform in the longitudinal coordinate was used and the original problem was reduced to solving a number of auxiliary boundary value problems with respect to transformants of the desired functions. From the conditions that the stress functions vanish (the cover is modeled as a mathematical section) of the substrate and the coating, the operator relations are constructed in the area of delamination to calculate the opening functions. The kernels of these operator relations are singular and are integrals over an infinite interval. A study was made of the behavior of their integrands at infinity, on the basis of which special approaches were used to calculate the kernels. As a result of solving the obtained hypersingular equations with difference kernels, for which the collocation method is used, the originals of the disclosure functions are constructed. Using a similar approach for inverting the Fourier transformations, we constructed relations to calculate the originals of the displacement functions at the upper boundary of the coverage. Based on the computational experiments, an analysis is made of the influence of the initial geometric and mechanical parameters of the substrate and coating on the values of the disclosure functions in the delamination region and the displacement functions at the upper boundary of the layer. The influence of the prestress level on the amplitude-frequency characteristics (AFC) was also investigated. It was found that the most significant effect on the frequency response is in the vicinity of the frequencies of the thick resonances. Based on the information on the displacement fields, it is possible to construct schemes for identifying delamination characteristics.

A grinding process using a free impact breakage mechanism is used in industries. In order to make calculations, predict grinding results, and evaluate mills functioning, it is necessary to assess the parameters of the grinding process and interrelations between the process parameters, mills parameters and materials properties, i.e. it is necessary to use an adequate mechanical-mathematical model of the process. However it is difficult to model due to some phenomena occurring in this process. Nowadays, various researchers have established the basis for the structure of the grinding process, but the application of the existing hypotheses and methods to evaluate the grinding process is quite difficult. This paper solves the problem of a spherical shape particle impacting an absolutely rigid half-space. It proposes a refined mechanical and mathematical model describing the process of destruction of the particle using the free direct impact breakage mechanism on an absolutely rigid, stationary, and flat surface. By using the Hertz-Staerman's classical analytical dependencies on the force contact interaction of the spherical bodies and the technical theory of the longitudinal waves’ propagation in the elastic continuous medium, we obtained a new refined solution of the applied dynamic problem related to a direct impact of a ball simulating a particle of a feeding material (an absolutely rigid surface simulating the working body of the mill) taking into account local physically linear deformations, the time parameter and radial particle size. The improved theoretical model of the spherical particle destruction was brought to applicable analytical calculations, tested and illustrated by a numerical example. It made it possible to describe the fracture of the material particles, predict the result and calculate the grinding process depending on its parameters providing the required quality of grinding by regulating and selecting characteristics, designing and selecting the grinding equipment, and modeling the grinding process using the free impact breakage mechanism.

The work considers rods with a constant cross-section. The deformation law of each layer of the rod is adopted as an approximation by a polynomial of the second order. The method of determining the coefficients of the indicated polynomial and the limit deformations under compression and tension of the material of each layer is described with the presence of three traditional characteristics: modulus of elasticity, limit stresses at compression and tension. On the basis of deformation diagrams of the concrete grades B10, B30, B50 under tension and compression, these coefficients are determined by the method of least squares. The deformation diagrams of these concrete grades are compared on the basis of the approximations obtained by the limit values and the method of least squares, and it is found that these diagrams approximate quite well the real deformation diagrams at deformations close to the limit. The main problem in this work is to determine if the rod is able withstand the applied loads, before intensive cracking processes in concrete. So as a criterion of the conditional limit state this work adopts the maximum permissible deformation value under tension or compression corresponding to the points of transition to a falling branch on the deformation diagram level in one or more layers of the rod. The Kirchhoff-Lyav classical kinematic hypotheses are assumed to be valid for the rod deformation. The cases of statically determinable and statically indeterminable problems of bend of the rod are considered. It is shown that in the case of statically determinable loadings, the general solution of the problem comes to solving a system of three nonlinear algebraic equations which roots can be obtained with the necessary accuracy using the well-developed methods of computational mathematics. The general solution of the problem for statically indeterminable problems is reduced to obtaining a solution to a system of three nonlinear differential equations for three functions - deformation and curvatures. The Bubnov-Galerkin method is used to approximate the solution of this equation on the segment along the length of the rod, and specific examples of its application to the Maple system of symbolic calculations are considered.

The complex influence of the surface plastic hardening technology, residual stresses, and boundary conditions on the bending of a hardened beam of EP742 alloy was performed. A phenomenological method of restoring the fields of residual stress and plastic deformations performed by its experimental verification in the particular case of ultrasonic hardening is given. The correspondence of the calculated and experimental data for the residual stresses is observed. For assess the influence of the formed residual stresses on convex cylinders, the calculation methods are used for initial strains based on using analogies between the initial (residual) plastic strains and temperature strains in an inhomogeneous temperature field. This allowed us to reduce the consideration of the problem to the problem of thermoelasticity, which was further solved by numerical methods. The effect of four types of boundary conditions for fixing the ends of the beams (rigid fastening and articulation of the ends and ribs in various combinations, cantilever) on the shape and size of the bending of the beam 10×10×100 mm after ultrasonic hardening is studied in detail. It was found that the minimum deflection is observed with a hard seal of both ends of the beam. The effect of the thickness of the beam, which varied from 2 to 10 mm, on their buckling under the same distribution of residual stresses in the hardened layer was studied, and the nonlinear nature of the increase in the deflection boom with decreasing thickness for all types of boundary conditions was established. It is shown that under all boundary conditions, the curvature along the length of the beam practically does not change, therefore it can be considered constant. The consequence of this is the preservation of the hypothesis of flat sections after the hardening procedure, which is confirmed by the calculated profile of the beam section in plane symmetry, close to a straight line. The influence of the anisotropy of surface plastic hardening on the buckling of the beam was found to be significant, which can serve as the basis for choosing the optimal hardening procedure. The performed parametric analysis of the task is presented in the form of graphical and tabular information on the results of the calculations.

Two methods for calculating the phase-structural deformations of shape memory alloy (SMA) structures under complex stress conditions are considered. They both are based on the one-dimensional phenomenological model, which is built upon the relationship between the direct transformation and martensitic inelasticity diagrams, which makes it possible to uniformly describe strains in the phase and structural transformations, since both of the strain components are associated with the formation of oriented martensite. The ability of the model to describe a number of basic macromechanical effects caused by martensitic transformations in SMA was shown in our previous work. After the generalization to the case of a complex stress state it can successfully be used for solving certain engineering problems. The generalization of the model can be accomplished in two ways. The first method involves the construction of three-dimensional constitutive relations, proceeding from the previously developed one-dimensional relations and some simplifying hypotheses, and the numerical implementation of these relations by the finite element method. The second is the structural method, applicable to structures, in which the stress-strain state is described by one kinematic and one force parameter. This method suggests the use of structural diagrams of direct transformation and martensitic inelasticity, which are similar to the corresponding material diagrams, but establish the dependence of the phase-structural component of the kinematic parameter on the force parameter (not the dependence of phase-structural strains on the stress). Although the structural method is associated with the necessity to experimentally determine the structural diagrams, it has the advantage of significantly reducing the computational costs. Additionally, the article presents a comparison of two methods for describing the tension-compression asymmetry, and also develops a method taking finite deformations into account.

The paper formulates the initial-boundary-value problem of the viscoelastic-plastic bending behavior of cylindrical circular shells cross-reinforced along equidistant surfaces. The instant elastoplastic deformation of the shell composition components is described by the governing equations of the theory of plastic flow with isotropic hardening. The viscoelastic deformation of these materials is described by the defining relations of the Maxwell - Boltzmann model of body. The geometric nonlinearity of the problem is taken into account in the Karman approximation. The used system of two-dimensional resolving equations and the corresponding initial and boundary conditions make it possible to determine displacements and stress-strain state (including residual one) in materials of the composition of flexible cylindrical shells with varying degrees of accuracy. In this case, the weak resistance of the considered composite structures to transverse shears is taken into account. In the first approximation, the equations are used, the initial and boundary conditions correspond to the relations of the widely used non-classical Reddy theory. A numerical solution of the initial-boundary-value problem posed is constructed using an explicit step-by-step "cross" scheme. The elastoplastic and viscoelastic-plastic dynamic deformation of a relatively thin long circular cylindrical shell is investigated. The structure is rationally reinforced in the circumferential direction and is loaded with an internal pressure of an explosive type. It has been demonstrated that under intense short-term loading even of a relatively thin cylindrical reinforced shell by internal pressure, the traditional Reddy theory does not guarantee that the maximum residual deflection and the intensity of residual deformations of the components of the composition are accurate to within 10% compared to calculations performed by the refined theory. The difference in the results of the corresponding calculations increases with an increase in the relative thickness of the composite shell. It was found that after plastic deformation of a long reinforced cylindrical shell in its residual state, not only appear zones of edge effects, but also a local zone of an intense deformation located in the vicinity of the central section of the shell. The length of the local central zone is comparable with the length of the zones of edge effects. It is shown that the amplitude of the transverse vibrations of the reinforced shell in the vicinity of the initial moment of time significantly (by an order of magnitude) exceeds the value of the maximum modulus of the residual deflection. Therefore, the calculations performed in the framework of the theory of elastoplastic deformation of composition materials do not allow a very approximate determination of the magnitude of the residual displacements and the magnitude of the residual deformed state of the components of the composition of the cylindrical shell.