Recent Papers

ON THE BASIC EXISTENCE THEOREMS OF SPACE ELASTICITY THEORY

Investigation of nonlinear vibrations of thin plates with consideration of damping

ON ONE METHOD OF DESCRIBING THE VISCOELASTIC BEHAVIOR OF POLYMERIC BODIES

This paper develops an operator method for describing the linear viscoelastic behavior of polymeric bodies using Volterra kernels of the form of a power law multiplied by an exponential factor. Extending Rabotnov’s fractional-exponential operator theory, it proves composition rules for these kernels…

ON THE EQUATIONS OF SHORT WAVES IN A VISCOUS HEAT-CONDUCTING GAS

This paper derives asymptotic equations for short wave flows in a viscous heat conducting gas, focusing on narrow regions near shock fronts where flow parameters vary rapidly. Starting from the continuity, Navier Stokes, energy, and state equations, the author introduces scaled moving coordinates an…

THEORY OF THE MOTION OF DROPLETS OF SOLUTIONS IN A DIFFUSING BINARY GAS MIXTURE

This paper develops a continuum theory for the diffusiophoretic motion of large droplets of dilute solutions in a diffusing binary gas mixture, for Knudsen numbers much less than unity. The authors model a partition of widely spaced spherical droplets, solve the hydrodynamic and stationary diffusion…

UNSTEADY OUTFLOWS OF GAS IN A CONSTANT GRAVITATIONAL FIELD

This paper treats the one-dimensional unsteady outflow of an ideal polytropic gas initially at rest in a constant gravitational field, with a nonuniform initial pressure and sound-speed distribution induced by gravity. Using characteristic relations and general solutions for the gas-dynamic equation…

A DOUBLY PERIODIC PROBLEM IN THE THEORY OF A CIRCULAR CLOSED CYLINDRICAL SHELL

The paper formulates and solves a doubly periodic elasticity problem for a circular cylindrical shell weakened by a lattice of identical circular holes and subjected to symmetric edge loads, internal pressure, and uniform tension at infinity. Using complex variable representations with Weierstrass e…

The group of symmetric transformations of a generalized Gyldén problem

A generalized Gylden problem is considered, i.e., the problem of the motion of a point of variable mass in a nonstationary central force field whose reduced force law (the ratio of force to mass) is expressed by an arbitrary function $f_{\text{pr}}(t,r)$ of time $t$ and the distance $r$ of the point…

ON THE FATIGUE OF METALS

This paper proposes a nonholonomic continuum model for metal fatigue in which stress, strain, temperature, and an additional history dependent parameter describe the transition from elastic behavior to plastic deformation under cyclic loading. The analysis links the fatigue threshold to specimen hea…

ON THE THEORY OF CONSTRUCTING MODELS OF CONTINUOUS MEDIA

The paper develops a general variational framework for constructing relativistic models of continuous media in a pseudo-Riemannian space, allowing the material Lagrangian to depend on tensor or spinor parameters, their derivatives up to arbitrary finite order, motion variables, and the metric Lamé t…

A CONTACT ELASTIC-PLASTIC PROBLEM FOR PLATES

This note derives an exact solution for a frictionless contact problem involving ideal elastic-plastic bodies in plane stress that satisfy the Tresca-Saint-Venant yield condition. The argument shows that plastic regions occur as zero-thickness lines on the contact boundary in the more plastic body, …

ON BOUNDARY-VALUE PROBLEMS OF THE CONTINUUM THEORY OF DISLOCATIONS

This paper formulates boundary-value problems for the continuum theory of dislocations in an anisotropic elastic medium. Using generalized functions concentrated on volumes, surfaces, and contours, it derives a Green formula for internal stresses and applies it to dislocation sources, including volu…

ON AN EXTREMAL PROBLEM OF THERMOELASTICITY FOR AN INFINITE CYLINDRICAL SHELL

This paper formulates an extremal thermoelasticity problem for an infinite isotropic cylindrical shell subjected to an axisymmetric temperature field that is uniform through the thickness and vanishes at infinity. The elastic strain energy is minimized under prescribed values of displacement-related…

An investigation of a system of equations describing the motion of a spherical pendulum in the case of the presence of a resistance

The system of differential equations \begin{gather}\theta=x,\notag\\\dot{x}=-\alpha x-\frac{g}{l}\sin\theta+y^2\sin\theta\cos\theta+L,\tag{1}\\\dot{y}=-\alpha y+2xy\operatorname{ctg}\notag\theta\end{gather} is investigated using Lyapunov functions. It is shown that for $0<\alpha<2\sqrt{\frac{q}{l}}$…

ON THE DIRECT PROBLEM OF PLANE SYMMETRIC FLOW PAST A SMOOTH CONVEX PROFILE WITH A DETACHED SHOCK WAVE

The paper studies plane adiabatic supersonic flow of an ideal gas past a smooth convex symmetric profile with a detached shock wave, focusing on the neighborhood of a sonic point on the contour. Using characteristic relations, entropy monotonicity near the critical streamline, and hodograph-plane ma…

On the Relationship Between Volumetric and Shear Plastic Deformations and Shock Waves in Soft Soils

This paper examines the coupling between volumetric and shear plastic deformations in soft soils, challenging models that treat hydrostatic compaction and shear flow as independent. Drawing on triaxial test evidence for sands, clayey soils, and lead shot, it formulates an idealized rigid dilatant mo…

FLOW PAST A SPHERE BY A RAREFIED GAS STREAM

This paper reports an experimental study of supersonic rarefied gas flow past a 5 mm radius sphere in the transition regime between continuum and free molecular behavior. Using multiple beam interferometry and photometric interpretation with axial symmetry, the authors measured density fields, shock…

Corrections to the Article “On a Problem of Stochastic Linear Programming and Its Stability” (DAN, Vol. 162, No. 1, 1965)

CALCULATION OF FLOW IN A LAVAL NOZZLE

The paper addresses the inverse problem of Laval nozzle theory, in which streamlines and gas dynamic parameters are determined from a prescribed velocity distribution on the axis of symmetry. It formulates the problem for irrotational isentropic ideal gas flow in stream function coordinates and prop…

ON THE CONSTRUCTION OF SOLUTIONS OF THE SYSTEM OF EQUATIONS OF BOUNDARY-LAYER THEORY BY THE METHOD OF LINES

The paper develops a method of lines construction for solutions of the Prandtl boundary-layer equations, principally for axisymmetric three-dimensional nonstationary incompressible flows and for related two-dimensional cases. After transforming variables by using the normalized velocity and introduc…

TOWARD AN INVESTIGATION OF THE STABILITY OF DISCONTINUOUS SOLUTIONS OF THE GAS-DYNAMIC EQUATIONS

The problem of transverse vibrations of a viscoelastic rod

This paper considers the problem of free transverse vibrations of a finite visco-elastic rod where one end is fixed and the other is free. Mathematically, the problem is formulated as seeking a solution to the equation: \begin{equation} \frac{\partial^2u}{\partial t^2}+a\frac{\partial^4u}{\partial x…

Initial Stage of Two-Dimensional Unsteady Gas Flow

This paper analyzes the initial stage of plane unsteady outflow of a polytropic compressed gas from a tube after removal of a wall, with attention to the formation of lateral rarefaction waves in addition to the central one-dimensional Riemann wave. Using characteristic equations and self-similar va…

THE VELOCITY DISTRIBUTION FUNCTION OF GAS MOLECULES NEAR A SOLID WALL IN A NONUNIFORMLY HEATED GAS

This paper examines the velocity distribution function of gas molecules near a solid wall when a temperature gradient is applied tangentially to the wall, a problem relevant to the calculation of thermal slip. The authors solve the stationary Boltzmann kinetic equation more directly than in their ea…

ON THE QUESTION OF CREEP OF METALS

A constitutive equation is proposed for describing metal creep under conditions that also account for the limiting dynamic stress strain diagram and the transition between elastic and plastic deformation. The formulation generalizes a previously used dynamic loading equation by introducing experimen…

SOLUTION OF THE GENERALIZED PROBLEM OF I. Ya. SHTAERMAN

The paper treats a generalized version of Shtaerman’s contact problem for two elastic cylinders of nearly equal radii, incorporating friction and possible regions of slip and rigid adhesion. Using Kolosov-Muskhelishvili potentials, the author derives contact equations from equality of boundary curva…

DISSIPATION OF ENERGY IN A TURBULENT GAS CONTAINING SUSPENDED PARTICLES

This paper analyzes energy dissipation in a turbulent gas containing dilute suspended solid particles at large Reynolds number, assuming local homogeneity, isotropy, stationarity, and particle sizes small enough for Stokes drag to apply. It derives estimates for the mean square relative velocity of …

DAMPING OF DISTURBANCES INTRODUCED BY A BODY OF REVOLUTION INTO A SUPERSONIC FLOW OF A VISCOUS HEAT-CONDUCTING GAS

This paper analyzes the far-field damping of disturbances produced by a body of revolution in a supersonic flow of a viscous, heat-conducting gas. Starting from the axisymmetric continuity, Navier-Stokes, and heat-transfer equations, the authors introduce coordinates aligned with the characteristics…

ON BOUNDARY CONDITIONS FOR THE NAVIER—STOKES EQUATIONS IN A RAREFIED GAS

The paper examines boundary conditions for the Navier-Stokes equations in a monatomic rarefied gas when the gas-surface interaction law is treated as an arbitrary boundary transform rather than a prescribed specular-diffuse model. Using a Grad-type representation of the distribution function and req…

ON A LOWER ESTIMATE OF THE FREQUENCIES OF VIBRATIONS OF ELASTIC SYSTEMS

The paper develops a method for obtaining lower bounds on vibration frequencies for a class of connected elastic systems. The approach replaces a given system by a sequence of less rigid auxiliary systems constructed from the original one, expresses their influence kernels through determinant formul…

VARIATIONAL PRINCIPLES OF THE DYNAMICS OF SHELL THEORY

The paper addresses the variational formulation of dynamic problems in elastic shell theory when the motion is specified by initial conditions rather than by prescribed terminal positions as in Hamilton’s principle. For a linear Timoshenko-type shell model, it introduces kinematic variables, stress …

VARIATIONAL PRINCIPLES

This paper formulates variational principles for equations of gas kinetic theory by treating the gas as a continuous medium in phase space and varying phase trajectories under conservation of phase density. Starting from Hamilton’s principle for an N-particle system, the author derives a variational…

ON THE PROBABILITY DISTRIBUTION OF THE SQUARE OF THE TEMPERATURE DIFFERENCE AT TWO POINTS OF A TURBULENT FLOW

This paper experimentally investigates the probability distribution of the square of the temperature difference between two closely spaced points in a turbulent atmospheric flow, motivated by theories of intermittency and lognormal dissipation in high Reynolds number turbulence. Temperature differen…

ON SOME TWO-DIMENSIONAL SELF-SIMILAR FLOWS OF A POLYTROPIC GAS WITH VARIABLE ENTROPY

This paper studies plane self-similar nonisentropic flows of a polytropic gas with variable entropy, using a hodograph formulation in which density and entropy are treated as functions of the velocity components. It derives a closed system of partial differential equations for these functions and sh…

THERMODYNAMIC RELATIONS ON THE SURFACE OF A STRONG DISCONTINUITY IN AN ELASTIC MEDIUM UNDER FINITE DEFORMATIONS

This paper derives thermodynamic compatibility relations for a strong discontinuity, or velocity-discontinuity wave, propagating in an elastic medium under finite deformation, using Eulerian variables and the Almansi strain tensor. Conservation of mass, momentum, and energy are combined with the sec…

On a Condition for the Breakdown of a Region of Continuous Supersonic Flow in the Flow Past a Convex Profile with a Detached Shock Wave

The paper examines conditions under which a continuous supersonic region behind a detached shock wave breaks down in symmetric flow of an ideal gas past a convex profile. Using characteristic compatibility relations and assumptions on uniqueness, entropy monotonicity, and continuity in the minimal d…

ON ONE MODEL OF A CONTINUOUS MEDIUM

This paper proposes a particle-based model for nondissipative continuous media in which mass, energy, and momentum components are treated symmetrically as conserved quantities satisfying continuity equations. The author formulates trajectories for finite particles representing each conserved quantit…

Unsteady vibrations of a free three-layer strip

The paper studies unsteady flexural oscillations of an unbounded free three-layer strip under assumptions that reduce the plate model to a single equation for a displacement function. For the special case in which the relative flexural stiffness parameter equals one, the governing equation is transf…

On a class of motions of a pendulum in a medium with large resistance

DISPLACEMENT OF A RIGID SPHERE UNDER THE ACTION OF AN ACOUSTIC PRESSURE WAVE

This paper examines the displacement of a rigid sphere immersed in an ideal compressible fluid under the action of an acoustic pressure shock wave. Building on earlier expressions for reflected and radiated wave potentials near curved bodies, it formulates the sphere’s equation of motion and compare…

ON RECTILINEAR MOTIONS OF AN IDEALLY PLASTIC MEDIUM

The paper studies rectilinear stationary and nonstationary motions of an ideally plastic medium using variational formulations. For stationary pipe flow and for cylinders moving in a plastic medium, it analyzes the minimization of a nonreflexive, nonstrictly convex functional and shows that below cr…

KINETIC EQUATIONS FOR THE VORTEX FIELD

This paper formulates a statistical description of turbulence using the vorticity field as the basic variable for an incompressible viscous fluid. Starting from the hydrodynamic equations, it derives kinetic chains for Lagrangian multiparticle distribution densities and for Eulerian multipoint vorti…

ON NONLINEAR OSCILLATIONS OF THIN PLATES WITH ACCOUNT OF ROTATIONAL INERTIA

The paper studies nonlinear oscillations of thin plates while incorporating rotational inertia, addressing the non-wave character and related uniqueness difficulties of earlier plate equations. It formulates a boundary-initial value problem for the modified von Kármán-type system, defines generalize…

Estimates of the domain of solvability for the system of equations of a moment-free reticulated cylindrical shell in the case of the first boundary value problem

The paper investigates a system of three equations of strongly elliptic type, supplemented by a boundary condition for one variable and a periodicity condition for the second. This system corresponds to the problem of determining the displacements of a momentless lattice cylindrical shell from an in…

DEFORMATION OF AN ORTHOTROPIC ELASTIC–VISCOUS BODY UNDER CONDITIONS OF A PLANE PROBLEM

This paper studies plane deformation of orthotropic elastic viscoelastic bodies, with particular attention to fiberglass plastics whose creep behavior can be represented by power law kernels. By applying operational transforms, the viscoelastic plane problem is reduced formally to the corresponding …

ON A PARTICULAR SOLUTION OF THE PROBLEM OF THE ROTATION OF A HEAVY RIGID BODY ABOUT A FIXED POINT

This note studies a special case of the rotation of a heavy rigid body about a fixed point, under the condition that the center of gravity lies on the perpendicular to a circular section of the gyration ellipsoid. Starting from the standard equations of motion and their known first integrals, the au…

On the Bifurcation of Rotational Fluid Flows

The paper studies bifurcation and stability of stationary axisymmetric rotational flows of a viscous incompressible fluid between coaxial rotating cylinders, including Couette flow as a special case. The Navier-Stokes problem is reformulated as a compact operator equation in a Hilbert space, and the…

EXACT THEORY OF STEADY WAVES ON THE FREE SURFACE AND ON THE INTERFACE OF TWO LIQUIDS

This paper studies the existence and construction of steady finite amplitude waves on the free surface of a finite layer of light liquid over a heavy liquid of infinite depth, together with waves on their interface. Using complex potentials and conformal variables, the hydrodynamic boundary conditio…

ON SLIP SURFACES

This paper examines kinematic conditions on slip surfaces in three dimensional rigid plastic bodies, especially for incompressible materials under the Tresca plasticity condition. Using discontinuity relations for velocities and strain rates, together with Cauchy strain rate formulas in surface coor…

FURTHER ON THE BREAKING OF WAVES IN SHALLOW WATER

This paper revisits the breaking of long waves in shallow water and seeks analytical support for an earlier hypothesis that instantaneous phase velocity can be approximated by substituting the instantaneous depth, rather than the still-water depth, into classical wave formulas. For gently sloping ti…