Mechanics of Solids

Influence of planets of the solar system on the precession of the orbit of Mercury is studied in the framework of classical mechanics. It is shown that the average perihelion shift of the orbit of Mercury, calculated in the framework of the planar limited problem, is 556.5 arc seconds per century and coincides with the observed one with a relative accuracy of 2.5%. The incomplete coincidence between the calculated average displacement and the observational data is explained by the presence in the observed displacement of vibrational components with a total amplitude of up to 80 arc seconds and periods from several years to tens of years.

An exact analytical solution to the problem of the wave motion of a tubular semi-infinite rod interacting with its surrounding elastic medium according to the Coulomb’s law of dry friction under the action of an exponentially falling dynamic load on its end (modeling of explosive charge detonation) and in the case of a finite mass impact by a rigid body along this end is obtained. The pattern of wave motion depending on the loading parameters is investigated: the leading and trailing edges of the elastic wave are determined, and the distribution of residual strains as it passes along the rod is found. The obtained results can be used in calculating the deformation of oil and gas pipelines in the case of shock loads.

The article deals with the problem on setting boundary conditions for asymmetric problems in the mechanics of growing solids. Firstly, we study the conditions on the growing surface that are most important from the point of view of the theory completeness. When deriving relations on the growing surface, we use the results known from the algebra of rational invariants. Geometrically and mechanically consistent differential constraints that are valid for a very wide range of materials and metamaterials are obtained on the growing surface. Several variants of constitutive equations on the growing surface of different levels of complexity are investigated. The formulated differential constraints imply the experimental identification of several defining functions. Thus, the results obtained can serve as a general basis in applied research on the mechanics of growing solids with an asymmetric stress tensor.

Within the framework of simplified nonlinear models, the initial nonlinear-elastic stage of obtaining hollow gear parts from the circular shells by the method of hydrostatic compression from circular circles is studied by analytical and numerical methods. It is shown that a suitable choice of a rigid rod inside the shell allows one to control the corrugation of the cross section of the pipe billet, to prevent the formation of unwanted folds and ribs on the surfaces of gear products.

Spatial nonlinear vibrations of a pipeline section supported at the ends are studied. A pipe bent under both its own weight and constant pressure of the fluid contained therein is subjected to hydraulic shock. The model of bending and rotational motions of the pipeline is used. The gravity forces and the Coriolis (inertia) forces as well as the mutual effect of internal pressure and changes in the curvature of the axial line of the pipe are taken into account. Oscillatory movements of the pipeline are described by a system of two nonlinear differential equations. The first form of oscillation is considered. For an approximate analysis of the dynamics of the pipeline deformation, the inertial and inertial-elastic stages are introduced. At the first stage, only pressure in the fluid and inertial forces are taken into account. The second stage of the bending and rotational motions of the pipeline is a continuation of its inertial stage. At the end of the first stage, the action of the shock load ceases. The Cauchy problem with zero initial conditions is also solved by using the numerical Runge—Kutta method. A comparison of the results of approximate analytical and numerical solutions is given. Changes in the bending of the midpoint of span and the angle of rotation of the steel pipeline are presented as a function of time for different amplitudes of dynamic pressure.

A method is proposed for solving inverse flaw detection problems for rods performing longitudinal oscillations. Based on the modeling of the cross-sectional defect as a known function, the main parameters characterizing it, such as the location and volume of the two lowest oscillation frequencies of the free and cantilevered rods, will be approximately determined. Using numerical simulations, it is shown that to satisfactorily determine the properties of a defect, it is sufficient to use several lower frequencies. A method for identifying a defect by one lowest frequency of a free rod is also proposed, provided the known location of the defect is applied, for which it is necessary to determine the characteristics of the oscillation modes. The results of an experimental study are presented.

Using the Kotelnikov–Study transference principle, a generalization of the Hamilton–Ishlinskii solid angle theorem for spatial motion of a solid that is a composition of translational and rotational motions and the dual conjugate theorem for the body motion are presented. An example of the studied spatial motion of a solid is considered. Possible applications of the dual solid angle theorem in the theory of spatial mechanisms and the mechanics of robotic manipulators are pointed out. Its application for the inertial navigation problem for determining the orientation and apparent velocity of a moving object is given. In considering the example and application, biquaternionic kinematic equations and their analytical solutions are used for the solid’s spatial motions under consideration. In the present study, the results obtained earlier by the author of the article are developed and generalized.

This article presents results on studying effect of multi-walled carbon nanotubes (CNTs) on the tribological properties of frost-resistant rubber based on epichlorohydrin rubber. The sliding friction coefficient has been determined using a laboratory tribometer according to the “rubber ring-smooth steel disk” contact scheme. Rubber samples have been tested under conditions of dry friction for ranges of normal pressures of 0.1 … 0.4 MPa, sliding velocities of 1 … 100 mm/s for two values of the bulk temperature: 23 and −25° C. The wear rate of the samples has been determined at a temperature of 23° C, pressure of 0.4 MPa, and sliding speed of 10 mm/s. As a counterbody in experiments on the wear resistance assessment, sandpaper made of silicon carbide with a grain size of 120 and 250 microns has been used. It has been found that using CNTs as a filler for studied rubbers leads to an increase in their wear resistance. There is a multiple decrease in the wear rate when filling rubber with a small amount of CNTs (1–2 pts. wt. per 100 pts. wt. of rubber). Moreover, an improvement in physical and mechanical characteristics while maintaining high frost resistance is established. The addition of CNTs to the studied rubbers does not affect the static coefficient of friction.