Two-dimensional thermo-viscoelastic waves in layered media Dynamic problems of deformation of solids have been the subject of numerous studies in the CIS and abroad. The rejection of a number of simplifying assumptions made in the cited and other published works leads to the need for further refinement and improvement of mechanical and mathematical models describing the kinematics and stress state of both the drummer and the barrier. Further, the axisymmetric collision of a cylindrical indenter with an obstacle in the form of a package of isotropic plates containing free cavities and rigid inclusions is numerically investigated within the framework of the coupled theory of thermoviscoelasticity. Various formulations of the problems of the theory of elasticity and thermo-viscoelasticity are possible. However, the used formulation in velocities and stresses is one of the most universal, since it allows solving the main boundary value problems (including mixed ones) by a uniform way. The paper gives a grid-characteristic scheme and its convergence. In accordance with the theory of A.A. Samarskii, the stability in the energy norm of the grid problem is proved.

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In article considered the problem of curved rod fluctuation with Jung†TM s module. Shown the civility of the problem formulation. For solution used integro-interpolation method. Constructed implicit differential scheme, which realized by five-point sweep method. Conducted numerical calculations showed coincidence of theoretical calculation values of solution. Calculations conducted on the system of computer algebra Wolfram Mathematica. Results of calculations are given for two cases of fixing ends of the rod: both ends are fixed and one end is fixed other is free.

In this paper, was performed by numerical work according to the difference scheme. Analysis of the numerical results showed: one of the important issues of contact interaction is to determine the duration of the impact of the colliding bodies. Obviously, under the condition of a hard clutch, sticking of the striker from the barrier will not occur. To study the process of complete breakage of mechanical contact (appearance of separation zones), we will use boundary conditions that simulate a perfectly smooth impact. Analysis of the dynamics of contact resistance has shown that its magnitude and features of evolution over time substantially depend on the geometric and physicomechanical parameters of the deformable system, as well as on the type of boundary conditions. An increase in the acoustic rigidity of the impactor leads to an increase in the amplitude and duration of the impact. The impact of a less rigid punch or the presence in the barrier of a shielding layer of a polymeric material reduces the contact resistance of the plate, but the force interaction between the impacted bodies is longer. As the analysis of the results shows, the evolution of contact stresses is characterized by a number of specific features. For example, there is a direct correlation between the height of the cylinder and the time of its complete detachment from the obstacle, which corresponds to the vanishing of the function   tk  . An increase in the acoustic rigidity of the impactor leads to a sharp increase in the amplitude of the total resistance and an increase in the duration of the contact interaction. Thus, the contours of the isolines provide a visual representation of the configuration of the areas at which points the stresses develop, immediately preceding the appearance of elastoplastic deformations for spall fractures (for brittle materials).

Approaching of the solution of a static compressible medium to the solution of an incompressible medium A well-known analogy of the flow of viscous incompressible fluid and incompressible elastic medium. According to this analogy, the solution of the equations of the elasticity theory with the Poisson's ratio ν = 0, 5 and for any fixed shear modulus µ can be interpreted as a motion of a viscous incompressible fluid with viscosity µ. Thus, we can consider the usual static linear elasticity task with Hooke's law at λ → ∞, as a mathematical model of approaching to incompressible medium. In this paper, we obtained the asymptotic λ → ∞. Estimation of the proximity of the solution of an elastic static problem with Hooke's law to the solution of incompressible medium (Stokes problem). The final estimate allows to use well-known difference schemes and algorithms for an elastic compressible medium to solve incompressible medium. In this paper, an estimate of the proximity of the solutions of these problems is proved at λ → ∞, i.e. u→u H λ→∞ λ div u→−p λ→∞ σ→σ H λ→∞ . To substantiate this fact in [1-3], various methods for the first boundary value problem were investigated. For the static problem of the theory of elasticity, there is currently a whole series of papers devoted to numerical implementation using difference schemes. In paper [4], the estimate O(λ −α ) where k = 1 2 was obtained, in the proposed paper the estimate O(λ −1 ), and in further work we will show that this estimate is best possible in order.