Abstract— The problem of the dynamics of a rigid body in the form of a parallelepiped on a plane in the presence of two-dimensional dry friction (generalized model of dry friction Coulomb-Contensou [1]) is considered. Since in this model of friction there are no discontinuities of the first kind in velocity, the problem is mathematically simpler than problems with one-dimensional (according to Coulomb)

Abstract— In connection with the 120th anniversary of the publication of the last volume of A. Poincaré’s book “New Methods of Celestial Mechanics”, the following methods are considered that have arisen since then. 1. The normal form method, which allows one to study regular perturbations near a stationary solution, near a periodic solution, etc. 2. The method of truncated systems obtained with the

Abstract— The method of analytical construction of the control of the spatial motion of a rigid body (control of screw motion, equivalent to the composition of angular (rotational) and translational movements) is being developed in a nonlinear dynamic setting using Clifford biquaternions and dual matrices. The controls provide asymptotic stability in large for any selected programmed spatial motion

Abstract— Based on the analysis of the results of experimental studies of 12Х18Н10Т stainless steel specimens under a rigid (controlled deformation) process of deformation, which includes a sequence of monotonic and cyclic loading modes, some features and differences in isotropic and anisotropic hardening processes under monotonic and cyclic loading are revealed. To describe these features within the

Abstract— The article proposes a numerical method for solving spatial problems of fracture mechanics (method of discontinuous displacements). The advantage of the method is the representation of the solution in the form of a finite series of expansion in terms of the found analytically presented functions. The expansion coefficients are determined from the conditions for fulfilling the boundary conditions

Abstract— To study the influence and selection of rational elastic-dissipative parameters for under wing pylon-mounted engine suspension, a mathematical model of aeroelasticity of a large-sized transport aircraft has been developed by taking into account the kinetic moment of the engine rotors and specially designed engine attachment points to the pylons. It is shown that the elastic-dissipative parameters

Abstract— The need to improve existing and develop new effective technologies for processing metal materials by pressure (including those with severe plastic deformations that make it possible to obtain products with an ultrafine-grained structure), constantly increasing requirements for the accuracy of strength calculations, emerging new materials with complex properties (including gradient materials)

Abstract— The article presents the results of a computational and experimental study on the impact interaction of structured punches corresponding to the design of Russian and foreign small arms ammunition with homogeneous steel and composite armor. Particular attention has been paid to the careful modeling of the behavior of the materials of the punch and the plate under dynamic loading. To confirm

Abstract— We study the stability of a one-dimensional laminated beam which consists of two identical layers of uniform thickness. An adhesive of small thickness is bonding the two layers and creating a restoring force thereby producing a damping called adhesive or structural damping at the interface. It is well-known in the literature that this single frictional damping is not strong enough to stabilize

Abstract— A pseudodifferential operator is constructed that describes the stress field in an anisotropic medium caused by a dislocation with a variable Burgers vector. Under the assumption \({\mathbf{b}} \in {{H}_{{1/2}}}(\Pi ,{{R}^{3}})\), analytical expressions for the energy of circular dislocations with a variable Burgers vector in an elastic medium with general anisotropy are obtained for the

Abstract— This article is the second part of the work, published in the form of two articles. The case, which remained unexplored in the first part, is considered, when for an unperturbed value of the constant of the cyclic integral, the mechanical reduced potential energy is constant with respect to the internal cardan angle. It is proved that in this case, for the majority of device designs, instability

Abstract— The fundamentals of the theory of objective tensor mechanical characteristics are presented: the general concept of objectivity, types of objectivity, simple tensor analogs of mechanical characteristics and simple commutative diagrams connecting them. For objective tensors of the second rank, a general classification of simple diagrams is given, examples of diagrams are given. The mappings

Abstract— This article proposes a model that generalizes the well-known equations of the nonlinear theory of oscillations (Van der Pol and Rayleigh equations), the limit cycles of which are the curves of the phase plane, determined by the total energy of oscillations in the absence of dissipation/energy inflow into the system. By changing the parameters of the system and the type of force action, it

Abstract The synthetization of high-precision analytical approximations of the geodesic longitude dependence on the reduced latitude on the spheroid geodesic line is considered. The obtained approximations are relevant for both geodetic and navigation problems providing the capability of significant reduction in computational costs in high-precision geodetic and navigation calculations.

Abstract Small transverse oscillations near the equilibrium vertical position are studied for an extra-long flexible inextensible cable of a space elevator. The linear approximation shows that the oscillations in the two planes, equatorial and meridional, are separated and have identical sets of eigenfunctions, while their eigenfrequencies are connected by a simple relation. A Sturm–Liouville problem

Abstract— The simplest model of a tippe-top is a nonhomogeneous dynamically symmetric ball whose center of mass lies on the axis of dynamic symmetry, but does not coincide with the geometric center. Local analysis of this model is presented in [1, 2] and global qualitative analysis is presented in [3–5]. Numerical studies for multicomponent dry friction were carried out in [6]. A comparative analysis

Abstract Analytical solutions are obtained for an uncoupled quasi-static cyclic thermoelasticity problem for regular solids: a half-space, a plate, a space with a cylindrical channel, a cylinder, a space with a spherical cavity, and a sphere. The solutions, which are trigonometric Fourier series of temporary variable, are transformed by changing the spatial variable into a form convenient for analysis

Abstract In this paper, we describe a method for solving the problem for calculating the acoustic radiation of a deformable solid oscillating in a compressible fluid by using the mode analysis procedures found in mechanoacoustic systems. The method is based on an approximate representation of the acoustic radiation loss function and modifications of the Lanczos method. This allows constructing an efficient

Abstract In spite of great progress in the rocket technology and practical experience in the development and production of a number of rocket space systems (i.e., launch vehicles – LV), the task of optimizing the LV structure is still essential. Even in the fundamental work of well-known American scientists [1] in 1965, where the task of optimization of propellant distribution between the stages of

Abstract The problem to determine the mechanical field in a homogeneous half-plane supported by a finite homogeneous stringer, material of which obeys the nonlinear Hooke’s law, is considered. The contact between the plate and stringer is realized by a thin glue layer. The posed problem is reduced to a nonlinear singular integrodifferential equation. Using the Schauder fixed-point principle, we prove

Abstract In this paper, a model is developed for the geomechanical state of coal-rock mass containing the following: the seam and the hydraulic fracture for effectively applying a method of directional hydraulic fracturing of solid rocks. The model is based on the provisions of solid mechanics and Griffiths–Irwin linear fracture mechanics and is implemented using the method of boundary integral equations

Abstract— The optimal control problem regarding the reorientation of an asymmetric rigid body has been investigated. The integral-quadratic functional consistent with the inertial symmetry of the body, which characterizes the total energy consumption, has been chosen as the criterion. Control is considered to be the main moment of applied external forces. An explicit description of a family of extremals

Abstract— Problems of plane cracks of normal fracture (mathematical cuts) in the middle plane of a transversely isotropic elastic layer, the outer faces of which are under conditions of a sliding support, are considered. Isotropic planes are parallel or perpendicular to layer faces. Using the Fourier integral transform, the problems are reduced to integro-differential equations for crack opening, from

Abstract In this review, we analyze the basic equations and assumptions for the design of critical state models used in the mechanics of non-cohesive media. The relationship of the modified cam clay model with related models of the plasticity theory with isotropic hardening described by closed plasticity surfaces is noted. The state equations of modified cam clay models in the elastic zone are analyzed;

Abstract— The article considers the motion problem of three connected bodies in a homogeneous gravitational field (the generalized motion problem of a gyroscope in a gimbal). All stationary motions of the system, conditions of their stability and branching are found. Based on the analysis performed, it is shown that the complete atlas of Smale bifurcation diagrams for this problem consists of 24 maps

Abstract The effect of asymmetric orthotropic friction exerted on a solid body bearing up against a narrow rectangular area and sliding across a plane is studied. As an example, the motion of a uniform rod across a plane with asymmetric orthotropic friction is considered. The obtained results could find wide practical applications in the development of algorithms for the processing of material contacting

Abstract The evolution of the planetary orbits of the Solar System is studied as part of a planetary problem. The displacement of the orbital elements is found by the numerical integration of the equations of perturbed motion written in non-degenerate variables. The results are presented as tables for the average displacements of the orbital elements and their oscillatory components for the epoch J2000

Abstract The structural model of hybrid composites angle-ply that were reinforced parallel to some plane is constructed for the analytical determination of the composition’s yield loci while considering the plane stress state in all components. The materials of the components are homogeneous and isotropic, as well as have different yield strengths in tension and compression. Their mechanical behavior

Abstract A junction of N semi-infinite cylindrical acoustic waveguides is considered. The asymptotics of the scattering matrix is constructed for small frequencies. The effects of almost complete reflection and complete transmission of waves are found only in the cases N = 1 and N = 2 under condition of equal cross section areas, while for other cases, there are no anomalies of the wave diffraction

Abstract— We have considered the case of a restricted three-body problem (material points), when the masses of the two main attracting bodies are equal. Their orbits have been assumed to be ellipses. The problem regarding the third body’s motion of negligible mass under the influence of the main body’s gravitational attraction (the Sitnikov problem) allows particular solutions for the following: the

Abstract The problem of detecting and localizing a cavity, a crack, or an inclusion in an elastic body by means of partially overdetermined data given on its outer boundary is considered. The input data can be obtained in a single static experiment. A new approach for solving this problem is proposed. The capabilities of the proposed approach are illustrated by the example of the problem for detecting

Abstract A contact problem is considered for a heterogeneous fluid-saturated half-space considering friction forces in the contact domain originated from the movement of a die with a flat or and parabolic base. For to consider the internal microstructure of the base, Biot’s model is used. The boundary problem is reduced with the help of the Fourier transformation to a first-kind integral equation with

An Erratum to this paper has been published: https://doi.org/10.3103/S0025654420660012

Abstract— The general relations of the torsion theory of inhomogeneous rods made of an ideal rigid-plastic material are considered. In the case of the linearized yield criterion, integrals are obtained that determine the stressed and deformed states of an ideal rigid-plastic inhomogeneous rod during torsion. The field of characteristics of the basic relations is constructed, the lines of stress rupture

Abstract— The article deals with the determination of the stress state for an unbounded space in the vicinity of a cylindrical cavity (thick-walled slab with a hole), which has a cross-section of a shape close to a regular polygonal (Galin – Ivlev problem). As a model of the space material, we have used a model of the medium that takes into account the aging elastic-viscoplastic properties. The solution

Abstract— In the framework of the linear theory of micropolar shells, existence and uniqueness theorems for weak solutions of boundary value problems describing small deformations of elastic micropolar shells connected to a system of absolutely rigid bodies are proved. The definition of a weak solution is based on the principle of virial movements. A feature of this problem is non-standard boundary

Abstract— Tensor fields of arbitrary rank in multidimensional space are naturally extended to the concepts of divergence, gradient, curl, deformer operators, as well as their second-order superpositions. Two options for generalizing the rotor as an external product are presented. Differential operators of the second order that do not change the rank of the tensor to which they are applied are considered

Abstract— The paper considers a sequence of solutions to the one-dimensional problem of irreversible deformation of a functionally graded material under conditions of uneven thermal expansion. Numerical solutions are obtained for the problems of heating an elastoplastic sphere, the material constants of which are linear functions of the radius, and exact solutions, in which the material constants are

Abstract— The article deals with problems on determining the effective moduli of porous piezoceramic materials with substances that have extreme properties and are deposited on the pore boundaries. The argumentation for the method of effective moduli is given. The reciprocity relation, from which the properties of effective moduli that are similar to known ones for moduli of piezoelectric materials

Abstract— The stress-strain state of a thin adhesion layer in a layered composite is investigated under shear loading taking into account its possible elastoplastic deformation. The layer thickness is treated as a linear parameter. Analyzed is the analytical solution obtained on the basis of a simplified formulation of the problem in differential form. For small layer thicknesses, a generalized criterion

Abstract— The influence of the orientation of the reinforcing fibers on the deformation and damage of the shape memory composite during bending and torsion, as well as the restoration of the initial shape of the sample upon activation of the shape memory effect (SME), has been studied. The composite is a three-layer carbon fiber impregnated with a shape memory polymer, the activating factor of which

Abstract— Some features of the behavior of ideal tips of transverse shear and tear cracks (“shear-tear” crack tips) propagating together with the shear wave fronts are investigated. It is shown that in the quasi-static approximation, the ideally plastic behavior of the material near the tear crack has a spatial character and has characteristic planes oriented at an angle of 45 degrees to the directions

Abstract— The article is devoted rapidly rotating metal disks layered by thickness. The deformation law of each layer of the disk is adopted as an approximation by a second-order polynomial. For rotating thin bimetallic disks, the basic relations are given for the study of critical deformed states arising from small technological deviations in the fabrication of the canvas or contour forms.

Abstract— By the example of a local inhomogeneity in the form of a horizontal crack, a numerical and experimental confirmation of the existence of resonance effects in an inhomogeneous elastic strip, investigated by I. I. Vorovich, is given. Examples of the use of this phenomenon for detecting delamination in composite plates and determining their size and depth based on the data of ultrasonic monitoring

Abstract— An approach to modeling a ferroelectric heterostructure with a prestressed coating made of a two-component functional-gradient material is proposed. Various types of PZT-based ceramics with close values of elastic moduli and significant differences in piezoelectric and dielectric parameters are considered as coating materials. The initial stress state (ISS) of the coating is caused by the

Abstract— A method for solving the boundary value problem of a residual stress relaxation in a surface-hardened solid cylinder under creep conditions with initially defined and rigidly fixed axial deformation and torsion angle has been developed. It includes a phenomenological methodology for reconstructing the stress-strain state after hardening and its kinetics during relaxation of axial load and

Abstract— In this article, we analyze axisymmetric contact problems on the interaction of a rigid stamp with a poroelastic layer using the equations of the Cowin–Nunziato theory of poroelastic bodies. It is assumed that the stamp base has a flat or parabolic shape and there is no friction in the contact zone. The posed problems are reduced to integral equations, the main kernel part of which is the

Abstract— The article discusses and analyzes the issues of applicability and the limits of applicability of some of the main hypotheses of the general mathematical theory of plasticity. In the theory of elastoplastic deformation processes, this is the postulate of the isotropy of initially isotropic bodies, in which the invariance of orthogonal transformations of the process images is established when

Abstract— The article deals with the dynamic equations of the linear micropolar theory of elasticity. They are related with respect to vector fields of displacements and microrotations. Coupled equations are also obtained for vortex vector potentials of displacements and microrotations. Using (or not) calibration conditions, it is possible to formulate uncoupled equations that include the same main

Abstract— On the basis of a numerical-analytical approach, the disturbed motion of the Earth’s pole is investigated. The existence of various modes of the oscillatory process is shown that affect the accuracy of predicting its position. Perturbations are found that lead to a change in the average speed of the Earth’s pole and to a change in its oscillatory mode. A method is proposed for constructing

Abstract— In the paper, in the form of series in the Papkovich–Fadle eigenfunctions, exact solutions of two inhomogeneous boundary value problems of elasticity theory for a half-strip with free long sides are constructed: (1) the half-strip end is free, (2) the half-strip end is rigidly clamped. To solve the inhomogeneous problem in a free strip, the Papkovich orthogonality relation is used. The exact

Abstract— The determining problem of the variable characteristics of an elastic rod made of a functionally graded material based on acoustic sounding data is considered. The amplitude-frequency characteristics of the rod during longitudinal and bending vibrations are used as additional information. A system of nonlinear integral equations is derived, the solution of which is based on an iterative scheme

Abstract— The general representation of a symmetric isotropic tensor-function of the second rank in three-dimensional space, depending on two tensor arguments of the second rank, is investigated. This representation includes eight scalar material functions of ten invariants of dependent tensors, including four joint invariants. The tensor function is interpreted as the constitutive relation of an isotropic

Abstract— The article deals with flows of the perfectly plastic compressible media for stress states corresponding to the facets of a piecewise linear yield criterion. Similar flows are observed, in particular, in loosely bonded Coulomb–Mohr media for plane strain states. It is assumed that the intermediate principle stress has no effect on the yielding or transition to the limit state. Under these

Abstract— The article deals with some problems of the theory of elasticity for domains bounded by non-concentric circles by taking into account surface effects such as surface elasticity and surface stresses. The solutions are obtained by expanding variables written in a bipolar coordinate system into Fourier series. The values of surface stresses and stress concentrations that are of our interest

Abstract— It is proved in the article that the recently revealed cracks of a new type, which have a different mechanism of destruction of the medium than the Griffith cracks, cannot be obtained as a result of the convergence of the sides of the wedge-shaped cavity in the plate. Thus, cracks of a new type are obtained as a result of the virtual convergence of the lateral sides of a rectangular cavity

Abstract— A model of planar oscillations of a plane region in the presence of prestress fields is formulated. Various formulations of two-dimensional inverse problems on the restoration of the preliminary stress state are presented. Methods and algorithms for solving the stated inverse problems are proposed and discussed.

Abstract— In this paper, asymmetric equations for an elliptical boundary layer in the vicinity of the conditional front of Rayleigh surface waves, which occurs in shells of revolution under shock end impacts of normal type are constructed. The technique of asymptotic derivation of these equations, based on the use of the symbolic Lurie method and the introduction of special coordinates that distinguish

Abstract— The flexural rigidity of a thin elastic multilayer plate with transversely isotropic layers is considered. If the rigidity of the layers is very different, the classical model based on the hypothesis of a straight normal is not applicable and the effect of lateral shear must be taken into account. Two models for taking into account the effect of transverse shear for a multilayer plate are

Abstract The paper discusses the dynamic propagation of cracks in brittle materials under various loads. The crack propagation is studied under both quasi-static and the shock-pulse loading conditions. Particular attention is paid to the dependences characterizing the crack propagation and having a non-stationary character. Thus, for the case of crack propagation under quasi-static loading, the crack