State of stress of an anisotropic plate with a closed rod pressed into a curvilinear hole in it

A method is proposed for solving the problem of elastic equilibrium in the case of an anisotropic plate with a closed rod pressed into a curvilinear hole in it (or stretched over an anisotropic disk). This method is based on representing the boundary conditions in the form of contour integrals of an arbitrary function holomorphic within the region of that plate. The normal magnitude of the jump of the displacement vector at the contact line is given as the function of the arc. Friction at the contact line is assumed negligible. The stress-strain state of the rod (ring) is described by the equations in the theory of thin curvilinear beams.I. V. Franko L'vov State University. Translated from Mekhanika Polimerov, No. 2, pp. 304–309, March–April, 1976.     

Concentration of electroelastic fields in a composite piezoceramic plate with a hole intersecting the interface of materials

A plane problem of electroelasticity is considered for an infinite compound plate with a hole located in both constituents of the plate. The corresponding boundary value problems is reduced to a system of singular integral equations of second kind, which is solved in numerical quadratures. Calculation results are presented that describe the concentration of electroelastic fields near the hole upon action at infinity of the vectors of mechanical stresses and electric field strength.Translated from Mekhanika Kompozitnykh Materialov, Vol. 35, No. 3, pp. 359–366, May–June, 1999.     

Relative stiffness of irregularly perforated plates

Approximation of the average field is used in order to calculate the relative elastic properties of plates irregularly perforated with a large number of different hole. Relative properties are calculated for a plate perforated with a different kind of elliptical hole. It is shown that the Young's modulus and shear modulus of a plate perforated with identical round holes calculated in this approximation have values similar to those calculated accurately for the case of a regularly perforated plate.Translated from Dinamicheskie Sistemy, No. 5, pp. 3–11, 1986.     

Free bending vibrations of an isotropic plate with free holes

A solution of the problem of the free bending vibrations of a thin finite isotropic plate with stiffly clamped outer edge is obtained by using functions of a complex variable. Results of numerical investigations are presented for an elliptic plate with acircular hole.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 20, pp. 67–70, 1989.     

The effect of a stiffening ring plate on an acoustic field scattered by a thin spherical shell

We study the axisymmetric problem of diffraction of an acoustic pulse on a spherical shell containing a thin elastic ring plate. The solution of the problem is obtained using the Fourier transform with respect to time and the method of superposition of normal modes. We analyze the solution in the stationary and nonstationary cases. We classify the resonances of the spectral density and the sequence of pulses of peripheral waves in the echo-signal. We study the effect of a stiffening ring on the spatial distribution of the acoustic field.Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 3, 1997, pp. 71–77.     

On the contact of a rigid sphere and a thin plate

We consider the problem of the contact between a rigid sphere and a thin initially flat plate. After reviewing some plate theory, we establish that a deformation where a finite piece of the plate takes the shape of the sphere is physically unrealisable, and that the contact region must be a ring. However, for both small deflections using classical linear elastic theory and large deflections using von Kármán theory, looking at some typical parameter values we find that the radius of the ring is so small that for practical purposes it should be considered as a point load.     

Analysis of the stress-strain state of a ring-reinforced cylindrical section of an underground pipe

A method is presented for solving the problem of determining the stress-strain state of closed circular cylindrical shells in an elastic medium. The problem relates to the design of underground pipelines. The work of cylindrical shells is examined from the viewpoint of the theory of thin-walled three-dimensional systems, with allowance being made for the unilateral character of the interaction with the elastic medium. The stress-strain state of a cylindrical section of an underground pipe reinforced in the middle by a ring is investigated. It is shown that different factors influence the stress-strain state of the shell of the pipe.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 18, pp. 66–72, 1987.     

Problem of the infusion of a fluid in an elastic deformable medium

The problem of the infusion of a fluid in an elastic coal seam via a hole drilled from the daylight surface is solved. Analytical expressions that make it possible to investigate the effect of the stress—strain state, physicomechanical and collecting properties of the coal, and the rate of injection on the process of fluid infusion under prescribed conditions, are obtained.Donetsk. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 21, pp. 112–116, 1190.     

Transition functions of the Il'yushin approximation method

Transition functions are constructed whose direct substitution for combinations of elastic constants in solutions of problems of elasticity theory permits writing the solution of the corresponding viscoelastic problems. Examples of using the transition functions for obtaining solutions of viscoelastic problems are given: circularly symmetric solid and anular plates, plate with a small circular hole, and viscoelastic half-plane loaded by a concentrated force.M. M. Gubkin Moscow Institute of the Petrochemical and Gas Industry. Moscow Institute of Electronic Engineering. Translated from Mekhanika Polimerov, Vol. 9, No. 3, pp. 405–416, May–June, 1973.     

An axially symmetric contact problem for a half-space with an elastically reinforced cylindrical cavity

We consider the pressure of a plate on a half-space with a round cylindrical cavity. The surface of the cavity is reinforced by elastic elements that are modeled by very general operators. The problem is reduced to a Fredholm integral equation of second kind. A detailed study is made of the case of reinforcement described by the Winkler law. An approximate solution is obtained in the form of the asymptotics with respect to the radii of the plate and the cavity.Translated fromDinamicheskie Sistemy, No. 6, 1987, pp. 80–85.     

Elastoplastic stability of glass-reinforced plastic plates with a central metal layer

The problem of the stability of a three-layer plate with a central plastic layer of metal sandwiched between elastic glass-reinforced plastic outer layers is considered. The presence of a metal layer restrains the development of creep strains in the glass-reinforced plastic and makes it possible to neglect the viscous strain components. The general equations of the problem are obtained, and the approximate Il'yushin formulation [1] is considered. An example is presented for a rectangular plate in pure shear. It is shown that the elastic anisotropic layers play the part of a load-relieving system for the central plastic layer [3], which results in an increase in the over-all critical load for the layered plate.Kalinin Polytechnic Institute. Translated from Mekhanika Polimerov, No. 5, pp. 909–915, September–October, 1969.     

A class of contact problems for thin anisotropic reinforced-plastic plates

Certain problems of the interaction of perfectly rigid smooth bodies and elastic reinforced-plastic plates are considered. The state of stress of the plate is described by the equations of orthotropic plates resisting interlaminar shear. Solutions are obtained in the case of a stamp in the shape of a rigid paraboloid of revolution indenting a circular plate with ring reinforcement and in the case of a parabolic cylinder indenting a unidirectionally reinforced plate in cylindrical bending. The expressions obtained for the contact pressure and the relation between the load P, the size of the contact region, and the settlement of the stamp are investigated with respect to their dependence on the orthotropy, wall thickness, and shear compliance parameters.Physicomechanical Institute, Academy of Sciences of the Ukrainian SSR, L'vov. Ivano-Frankovskii Institute of Petroleum and Gas. Translated from Mekhanika Polimerov, No. 2, pp. 346–350, March–April, 1972.     

Influence of geometric parameters on the stress concentration in an elastoplastic cylindrical shell

The elastoplastic stress-strain state of cylindrical shells weakened by a curvilinear (circular) hole is investigated. Numerical results are given for shells with a reinforced hole under the action of an internal pressure of specified intensity. The influence of the geometric parameters of the shell on the stress distribution in the concentration zone is investigated in the elastic and inelastic stages of deformation. The variation in flexure along the hole contour is given for linear and nonlinear problems, and the distribution of the maximum stress-concentration coefficients is also given for various geometric parameters.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 19, pp. 100–103, 1988.     

Coupled Frequencies of a Fluid in Circular Cylindrical Tank with an Elastic Inclusion in its Roof

A closed upright rigid circular cylindrical tank is filled with a compressible and inviscid fluid as a part of the tank roof is an elastic plate. It is supposed that the tank roof and the elastic plate are eccentrically and the plate is clamped. The problem about the determination of the free coupling vibrations of the received hydroelastic system is considered. Using the Bubnov‐Galerkin method, the frequency equation is obtained. Some numerical examples are made and the results show the influence of the sizes of the radius of the elastic plate as well as of the other parameters of the hydroelastic system on its coupled frequencies.     

Computation of anisotropic plates with curvilinear holes reinforced by prestressed rods

We consider the stressed state of an anisotropic plate with an elliptic hole whose boundary is reinforced by a prestressed curvilinear rod of arbitrary cross section symmetric with respect to the middle plane of the plate. The elastic equilibrium of the rod is described by the equations of the theory of curvilinear rods. The solution of the problem is reduced to a system of linear algebraic equations. We show the influence of prestressing on the stressed states of the bodies.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 36, 1992, pp. 76–80.     

Limiting equilibrium of a plane with a circular hole

The elastoplastic state of a compressible isotropic plane with a circular hole is studied by the method of a small parameter. An unknown boundary separating the domain of limiting equilibrium and the elastic domain is determined. We construct the complex Kolosov-Muskhelishvili functions that describe the elastic state of a plane and compare these with the solutions of Galin's problem.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 7, pp. 980–981. July, 1993.     

Equations for the vibrations of a plate with general anisotropy

The exact form of the displacement field for an elastic layer with general anisotropy and nonuniform thickness is constructed. The low-frequency part of the solution, which satisfies the equations for the vibrations of a plate, is found.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 165, pp. 122–135, 1987.     

Optical theorem for the scattering of waves in an elastic plate

The scattering of waves in an elastic plate is investigated. A generalization of the optical theorem is found for the case of a noncompact scatterer. The formulas obtained can be used for indirect control of the correctness of computing diffraction fields.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 104, pp. 20–23, 1981.The author is grateful to D. P. Kouzov for his constant attention to the work.     

The action of an unsteady external load on a circular elastic plate floating on shallow water

A solution of the transient problem of the behaviour of a circular elastic plate floating on the free surface of a liquid under the action of an external load is constructed. It is assumed that the liquid is ideal and incompressible and that its depth is small compared with the radius of the plate. The compatible motion of the plate and the liquid is treated within the framework of linear theory. The flow of the liquid is assumed to be irrotational. The behaviour of the plate under different loads is investigated and it is shown that the bounded dimensions of the elastic plate have a substantial effect on its unsteady behaviour.     

State of stress of flexible plastic shells with an opening

The state of stress of flat flexible shells with an opening is investigated with allowance for the viscoelastic properties of the material. The equilibrium equations and boundary conditions are written in finite-difference form. A nonlinear system of algebraic equations is solved by successive approximations. A method of accelerating the convergence of slowly converging iteration processes is proposed. The effect of the viscoelastic properties of the shell material on its state of stress is investigated with reference to the example of a polymethyl methacrylate shell. The variations of the ring moment and ring forces at the free edge of the shell are plotted for various moments of time, load values, and flatness parameters. It is shown that as soon as the viscosity factor begins to take effect, the state of stress and strain of the shell changes sharply; the concentration of forces and moments increases in the flexible viscoelastic (as compared with the elastic) shell.Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Mekhanika Polimerov, No. 6, pp. 1071–1075, November–December, 1973.