Hertz problem on compression of anisotropic bodies : PMM vol. 38, n 6, 1974, pp. 1079–1083

On the basis of results in [1], a derivation is given of the fundamental Hertz relationships for the compression of anisotropic (orthotropic) bodies which differs from [2]. It is shown that if the elastic constants satisfy some additional conditions, then the domain of contact is a circle in the compression of axisymmetric bodies along their common axes of geometric symmetry.     

Inverse problems of the plane theory of elasticity : PMM vol. 38, n 6, 1974, pp. 963–979

In connection with the fact that failure of a structure ordinarily starts at sites of the most acute stress concentrations near cavitiies, it is of interest to determine the shape of the equally strong outlines of holes on which the technologically inevitable stress concentration would be least as compared with all other outlines.An effective exact solution of some inverse plane problems of the theory of elasticity concerning the determination of equally strong outlines of holes is proposed. A formulation of the problem is given first and the fundamental relationships are presented. Then the general problem for any number of holes in an infinite plane is reduced to a standard Dirichlet problem for the exterior of the same number of parallel slits on a parametric plane. An effective exact solution is found by this method for the case of one and two holes as well as for the case of periodic and doubly-periodic series of holes. The question of application of the solutions obtained to the theory of a minimum weight structure is considered.     

On the reissner-naghdi elasticity relationships : PMM vol. 38, n 6, 1974, pp. 1063–1071

Shell theory equations are constructed by the method in [1] to the accuracy of quantities of the order of h_*^2+k, where and k = 2−4t for (h_* is the relative semithickness of the shell and t is the index of the state of stress variation). Without being within the framework of the Lovetype theory, the equations obtained are compared with the Reissner-Naghdi equations. [2, 3] in which the transverse shear is taken into account, and it is shown that from the asymptotic viewpoint these latter are inconsistent. It is also shown that if the shell resists shear weakly, then from the asymptotic viewpoint the Reissner-Naghdi theory is completely well founded.The three-dimensional equations of elasticity theory are reduced to two-dimensional equations in [1] by using an asymptotic method, i.e. all members of the same order relative to the small parameter h_* are taken into account at each stage of the calculations. It has been shown that without going outside the framework of the ordinary concepts of the Love-type theory of shells (in particular, without taking account of transverse shear), the shell theory equations can be constructed to the accuracy of quantities of the order of h^2−2t_*, but it is impossible to exceed this limit without a qualitative complication in the theory.     

On a case of resonance for nonlinear systems : PMM vol. 38, n 6, 1974, pp. 986–990

We examine the case of resonance for systems close to nonlinear systems, admitting of a parametric periodic solution. Among the eigenvalues of the matrix of the system's linear part there are zero and pure imaginary ones. We have proved (under certain conditions) the absence of a periodic solution for the original system for which the generating solution is trivial.     

Stability of flow of a liquid crystal layer on an inclined plane : PMM vol. 38, n 6, 1974, pp. 1031–1040

The framework of the linear mechanics of liquid crystal media [1] is used to study propagation of waves in a layer of a nematic liquid crystal (NLC) on an inclined plane, in a magnetic field, for three different cases of orientation of the anisotropy axis, namely orthogonal to the inclined plane, parallel to the inclined plane and orthogonal to the plane of flow. Such orientations of the anisotropy axis are realized in practice in the course of special machining of solid surfaces [2]. Exact solutions of the equations of motion are obtained describing the steady flow of the layer, and the behavior of small plane perturbations is studied. It is shown that two types of plane waves can propagate in a layer of the nematic mesophase, namely, the surface and the orientational waves. In the case of long surface waves the formulas for the critical Reynolds number are obtained. For the orientational waves a sufficient criterion of stability of the flow in the layer is obtained for two cases. The influence of the magnetic field and of the rheological parameters of NLC on the character of propagation of the first and second type waves is investigated.From amongst the papers dealing with wave propagation in NLC, we draw the readers' attention to [3] which deals with the longitudinal, shear and torsional waves in a liquid crystal domain and obtains the corresponding dispersion relationships.     

Fresh water lens produced by uniform infiltration : PMM vol. 38, n 6, 1974, pp. 1048–1055

The plane model proposed by N. N. Verigin for a stabilized fresh water lens produced by uniform infiltration is investigated in hydrodynamic formulation in the case of equidistant horizontal slit drains. Formulas are obtained for the separation boundary, the depression curve, and characteristic dimesions of the lens.     

Three-dimensional contact problem for a prestressed elastic body : PMM vol. 42, no. 6, 1978, pp 1080–1084

A half-space of an incompressible neo — Hookean [1,2] material subjected to a homogeneous bi-axial tension or compression along its boundary, is considered. A small deformation caused by the action of a smooth rigid stamp on the boundary of the half-space is superimposed on the initial finite deformation. An integral equation is obtained for the contact pressure. A solution of this equation is obtained for an inclined elliptic stamp with a flat base, and for an elliptic stamp with a curved base, for the cases when the extension coefficients in two directions are either identical, or differ little from each other. The influence of the inital loading on the distribution of the contact pressure, the displacement of the stamp and the form of the contact zone, is analysed.     

Dual integral equations related to the kontorovich-lebedev transform : PMM vol. 38, n 6, 1974, pp. 1090–1097

We study the dual integral equations related to the Kontorovich-Lebedev integral transforms arising in the course of solution of the problems of mathematical physics, in particular of the mixed boundary value problems for the wedge-shaped regions. We show that the solutions of these equations can be expressed in quadratures, using the auxilliary functions satisfying the integral Fredholm equation of second kind with a symmetric kernel.At present, the dual equations investigated in most detail are those connected with the Fourier and Hankel integral transforms. The results obtained and their applications are given in [1–3]. A large number of papers also deal with the theory and applications of the dual integral equations connected with the Mehler-Fock integral transform and its generalizations [4–11]., The dual integral transforms considered in the present paper belong to a more complex class than those listed above, and so far, no effective solution has been obtained for them. The only relevant results known to the authors are those in [12, 13]. In [12] a method of solving the equations (1.2) is given for a single particular value of the parameter γ = π/2, while in [13] the dual equations of the type under consideration are reduced to a solution of an infinite system of linear algebraic equations.     

On a self-similar solution of a two-dimensional filtration problem in regions with moving boundaries : PMM vol. 38, n 6, 1974, pp. 1136–1139

We determine a family of self-similar solutions of a two-dimensional problem involving the filtration of an incompressible liquid in regions with moving boundaries. Our work is based on a method developed by Galin for solving the problem of settling of water cones in a gravitational field [1 – 3]. Following this method, we reduce the problem to one of finding an analytic function of a complex variable and the time, which effects a conformal mapping of the filtration region onto a strip and satisfies a special nonlinear condition on the boundary. For the solution of a problem of this kind Galin proposed the method of successive approximations.     

Asymptotic analysis of three-dimensional dynamic equations of a thin plate : PMM vol. 38, n 6, 1974, pp. 1072–1078

Two-dimensional dynamic equations of thin plate vibrations are obtained from the three-dimensional dynamic equations of elasticity theory on the basis of an asymptotic method [1 – 3], Such an approach permits establishing the limits of applicability of the two-dimensional dynamic equations and the corresponding boundary and initial conditions, and indicating the means of obtaining refined results.The question of the construction of an inner state of stress of a thin plate under dynamic conditions is examined herein. The possibility of considering states of stress with distinct variability in time and in the coordinates and with a distinct relationship between the displacement intensities, is taken into account.     

On the determination of the sound field in a layer : PMM vol. 38, n 6, 1974, pp. 1129–1132

We consider a method for determining the sound field in a two-dimensional layer. The method we present combines the usual method of reflected plane waves with a summation from graphs. It makes it comparatively easy to take into account the complex interference pattern due to the transformation of the various waves at the boundaries of the layer and to obtain integral relations for the sound potentials. When the layer thickness tends to infinity, the problem reduces to one concerning the reflection of sound waves at the interface of two media. We study the potentials of normal waves in the case of a harmonic source in a solid.     

Separation of the elasticity theory equations with radial inhomogeneity : PMM vol. 38, n 6, 1974, pp. 1139–1144

The separation of a system of three elasticity theory equations in the static case to a system of two equations and one independent equation for a space with a radial inhomogeneity is presented in a spherical coordinate system. These equations are solved by separation of variables for specific kinds of radial inhomogeneity. In particular, solutions are found for the Lamé coefficients μ = const, λ (ifr) is an arbitrary function, μ = μ_or^β, λ = λ_or^β.While methods of solving problems associated with the equilibrium of an elastic homogeneous sphere have been studied sufficiently [1], problems with spherical symmetry of the boundary conditions have mainly been solved for an inhomogeneous sphere [2, 3],For a particular kind of inhomogeneity dependent on one Cartesian coordinate, the equations have been separated completely in [4], A system of three equations with a radial inhomogeneity in a spherical coordinate system is separated below by a method analogous to [4].     

Explosive instability in nonlinear waves in media with negative viscosity : PMM vol. 38, n 6, 1974, pp. 991–995

The propagation of a wave of a finite amplitude in a medium with a nonlinearity of the second degree and negative viscosity, is examined. It is shown that in a finite time singularities appear in the solution. The exact solution of the Cauchy problem is given for a specific case. Recently the effects of negative viscosity which cause an increase in the energy of the wave motion have been studied intensively in electrodynamics, plasma physics, the Earth's atmosphere, in the theory of the circulation of the oceans and of flow in open channels [1–4], Wave amplification caused by an energy transfer from turbulent to regular motions, is possible in any medium having space-time fluctuations, provided the correlation time is sufficiently small [5, 6]. As the wave amplitude increases, nonlinear effects become important; they have been taken into account in cases where the interaction of a finite number of harmonics [2, 4] and the structure of steady motions have been examined [3].It is shown in this paper that in a medium with negative viscosity and a second degree dynamic nonlinearity, a solution of the Cauchy problem for an arbitrary “good” form of the initial perturbation, exists over a finite time interval. An example of such a solution is given.     

On the “equivalence” rule for flows of perfect gas : PMM vol. 38, n 6, 1974, pp. 1004–1014

An extension of the equivalence of “area” rule [1, 2] is presented. The rule was initially derived for stationary flows of perfect (inviscid and non-heat-conducting) gas past slender fine pointed bodies (or blunted bodies in the hypersonic flow case) whose transverse dimensions are small in comparison with their length. According to that rule the wave drag of a three-dimensional body is equal to the wave drag of an axisymmetric body with the same distribution of cross-sectional areas along the axis. The rule is extended here to stationary and nonstationary flows past nonslender bodies and to internal flows, using the procedure of averaging with respect to the angular variable of a cylindrical system of coordinates. That procedure is, strictly speaking, valid for nearly axisymmetric bodies. However the numerical solutions obtained by the authors for a fairly wide range of external and internal problems show that the generalized equivalence rule is applicable to substantially nonaxisymmetric configurations (^*) (see next page).     

On the existence of a field of stress rates in a hardening elastic-plastic medium : PMM vol. 38, n 6, 1974, pp. 1114–1121

The boundary value problem for the stress rates and rates of change fields in the quasi-static motion of a volume V of an elastic-plastic medium [1] consists of finding the pairs σ_ij^., _ij^. related by the governing equations of an appropriate model; here the σ_ij^. should be statically admissible i.e. should satisfy the equations and boundary conditions σ_ij=−X/._i; /.σ_ijn_j|S_p=p_i and _ij should be kinematically admissible i.e. 2/._ij = v_ij + v_ji, where v_i|S_u = u_io Here S_p and S_u are nonintersecting parts of the boundary of the volume V, X_i, p_i, u_i/.o are specified functions. The question of the existence of a solution of this problem reduces to the question of the functional reaching the lower bound in a set of kinematically admissible /._ij^o and statically admissible σ_ij/.^/*. However, its lower bound may not be reached if in the minimization we limit ourselves only to smooth fields. It is proposed to augment the set of admissible fields σ_ij/.^/*,_ij/.^o by closing them in the norm L₂ (for v_i^o this corresponds to closure in the norm II¹). Some properties of the functional I (σ_ij^*,_ij/.) are considered in the augmented set of admissible fields. It is shown that the equivalence of the two problems is conserved, where I (σ_ij^*,_ij⁰ can be minimized in σ_ij^/*,_ij^o or in σ_ij^/*,_ij/.^o, The lower bound is reached in each of three cases, at a single point. From the fact that u_i^o belongs to the Sobolev space W₂⁽¹⁾, there results the absence of surfaces of velocity discontinuity. Variational principles have been used in plasticity theory to construct models [2] and to investigate the existence and properties of solutions [1, 3].     

Optimum control of the forced motions of systems with continuously distributed parameters : PMM vol. 38, n 6, 1974, pp. 1056–1062

We propose one of the possible versions of the optimum control of the forced motions of elastic systems of the type of rods, plates, and shells. We apply the procedure developed to elementary problems on the transition of a freely-supported rod or plate from an initial state φ, ψ to the rest state in the least possible time T in the presence of a constraint on the forcing load. We use the elementary results of theory of the l-problem of moments of Krein [1–3].     

Contact problem for a layer with two stamps : PMM vol. 42, no. 6, 1978, pp. 1068–1073

A problem of impressing coaxial stamps of circular cross section into the upper and lower surface of a homogeneous elastic layer is studied. The bases of the stamps have axial symmetry. The parts of the layer surfaces lying oustide the contact zone are stress-free, there is no friction or coupling between the layer and the stamps. A system of two integral equations with two unknown functions is obtained, and provides a solution of the problem. The method of separating the singularities provides the way of reducing this system to the Fredholm equations of second kind. An approximate solution of the equations is obtained for the case of flat stamps under the assumptions that the two parameters entering the system are sufficiently small.

Problems of a layer with various boundary conditions were formulated and solved in many papers and books, e.g. [1, 2]. However, to the best of the author's knowledge, in all these problems the conditions at the boundary were assumed different only on one side of the layer; in the present problem the boundary conditions are mixed at both sides of the layer, and this results in a system of two integral equations.     

On the stability of weakly inhomogeneous states with a small addition of white noise : PMM vol. 38, n 6, 1974, pp. 996–1003

Using the concepts developed in [1] we investigate, in the presence of certain restrictions, the stability of a weakly inhomogeneous state parametrically perturbed by a small random addition of white noise. We show that when the characteristic wavelength is arbitrarily small as compared with the distance over which it varies substantially, then the mechanism of formation of the eigenfunctions responsible for the stability of the state is analogous to the mechanism given in [1]. In the present case it is not the boundaries that act as reflectors, as in [1], but the points at which the condition of existence of the global eigenfunction for the homogeneous problem holds. We obtain the criterion of stability of the state in question and discuss the problem of application of the results obtained to the case in which the ratio of the characteristic wavelength to the distance over which it varies substantially, cannot be taken as arbitrarily small.     

Nonstationary mass transfer of a reacting spherical particle in laminar stream of a viscous fluid : PMM vol. 38, n 6, 1974, pp. 1041–1047

The process of the formation of a stationary mass transfer mode for a moving reacting particle is examined. An analytic expression valid for a nonstationary distribution of the concentration of matter in a steady stream of viscous fluid, flowing past a spherical particle, was obtained for the case when at a certain instant a chemical reaction of the first order begins at the surface of the sphere. The problem is solved for small finite Reynolds and Péclet numbers. The solution of the corresponding stationary problem has been obtained in [1]. Paper [2] examined a nonstationary heat transfer of a fluid spherical drop in an inviscid flow with spasmodic change of initial temperature at high Péclet numbers. Paper [3] contains an analysis of the problem of a nonstationary heat transfer of a rigid spherical particle for small Reynolds and Péclet numbers at spasmodic change of temperature of the particle surface. The results obtained in [3] can be used to describe the mass transfer for a moving reacting particle only in the case of a diffusion mode of the chemical reaction.     

The game problem on the dolichobrachistochrone : PMM vol.40, n 6, 1976, pp. 1003–1013

The capture and evasion sets, the players' optimal strategies and the game value determined for the game problem on the dolichobrachistochrone, analysed within the framework of a position formalism similar to [1]. Singularities inherent in the game of the minimax-maximin time to contact [1, 2] become apparent; they are determined in the given problem by the specific behavior of the optimal paths close to the target set. Isaacs [4] examined the game problem on the dolichobrachistochrone, being the game analog of the classical variational problem on the brachistochrone [3]. However, as was shown in [5], the solution proposed by Isaacs contains erroneous statements.