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.     

Rectangular anisotropic (orthotropic) plates on a tensionless elastic foundation

The behavior of anisotropic (orthotropic) elastic plates of rectangular shape on a tensionless Winkler foundation is analyzed. The tensionless character of the foundation is taken into account by using an auxiliary function. The displacement function of the plate is approximated by using the eigenfunctions of the completely free beam. The difference between the free-end boundary conditions of the plate and the beam is compensated for by considering a differential operator in addition to the governing equation of the plate. Using Galerkin's method, the problem is reduced to the solution of a system of algebraic equations. The governing equations of the plate are derived under action of external uniformly distributed load, concentrated load, and moments. However, the influence of the mechanical properties on the configurations of the contact region and on the distribution of the displacements is investigated for concentrated load and moments for various values of the mechanical properties characterizing the anisotropy of the plate material. Considered problems are solved within the framework of Kirchhoff-Love hypothesis.Presented at the Ninth International Conference on the Mechanics of Composite Materials (Riga, October 1995).Published in Mekhanika Kompozitnykh Materialov, Vol. 31, No. 3, pp. 378–386, May–June, 1995.     

Displacement of a rigid rod in a viscous medium

The spatial displacement of an absolutely rigid vertical rod in a viscous medium is considered; the rod is hinged at the upper end to a platform moving on the surface of the viscous medium over a specified curvilinear trajectory. A load is attached to the lower end of the rod. Using a variational Lagrangian equation, a nonlinear system of ordinary differential equations in terms of the angles of rod rotation determining the position of points of this rod at any time is obtained. As an example, the problem is solved for conditions of acceleration, uniform motion, and deceleration of the reference point moving over a trajectory consisting of rectilinear sections and portions of circles. The differential equations obtained may be used in determining the position of rod-type elements suspended in a viscous medium.Donetsk. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 21, pp. 108–112, 1990.     

Vibration of a Reissner–Mindlin–Timoshenko plate–beam system

In this paper, we consider a plate–beam system in which the Reissner–Mindlin plate model is combined with the Timoshenko beam model. Natural frequencies and vibration modes for the system are calculated using the finite element method. The interface conditions at the contact between the plate and beams are discussed in some detail. The impact of regularity on the enforcement of certain interface conditions is an important feature of the paper.     

A method for determining the dynamic Young's modulus of curvilinear specimens

On the basis of a numerical solution of the problem of the natural frequencies of a rod shaped as the are of a circle, the authors suggest a method for determining the dynamic Young's modulus of a curvilinear specimen. They derive an approximate expression suitable for practical use. The results agree well with experiment.Moscow. Translated from Mekhanika Polimerov, No. 2, pp. 335–339, March–April, 1974.     

Stress distribution near a reinforced elliptical hole in a spherical shell at the elastoplastic stage of deformation

On the basis of Lagrange's variational equation the authors obtained nonlinear resolvent equations and coefficients taking into account the effect of a reinforcing element (a rod) on the state of stress and strain of a spherical shell weakened by a curvilinear (elliptical) hole. The article explains the method of numerical investigation of the inelastic state of the shell based on the application of the variational difference method in combination with the method of elastic solutions. The inelastic state of a shell with a reinforced hole was numerically investigated.Kiev. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 21, pp. 80–83, 1990.     

Propagation of Creeping Waves Along the Curvilinear Surface of an Anisotropic Elastic Body

An analytic expression for the wave field of a creeping wave propagating along the curvilinear surface of an anisotropic elastic body is constructed by the boundary layer method. Bibliography: 14 titles.Dedicated to L. A. Molotkov on the occasion of his seventieth birthday__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 297, 2003, pp. 9–29.     

Application of singular integral equations in two-dimensional problems of the theory of elasticity for piecewise-uniform bodies with cracks

Using the method of singular integral equations we solve a two-dimensional problem of the theory of elasticity for an infinite plate containing an elastic inclusion of arbitrary configuration and a system of curvilinear incisions. The numerical solution is found by the method of mechanical quadratures for the case of an elliptic inclusion and a single polygonal crack.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 27, 1988, pp. 93–98.     

Action of a concentrated load on an elastic ring pressed into a circular hole in an anisotropic plate

A solution is obtained for the elastic equilibrium problem for an anisotropic plate with a circular hole into which is pressed a ring of arbitrary cross section symmetrical about the middle surface of the plate. Before deformation the outside radius of the ring differs from the radius of the hole in the plate by the amount of the permissible elastic displacements. A normal concentrated load is applied to the ring. The ring is analyzed in accordance with the theory of thin curved bars. A numerical example is given.I. Franko L'vov State University. Translated from Mekhanika Polimerov, No. 6, pp. 1054–1059, November–December, 1973.     

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.     

Extension of a piezoceramic bimorph with a crack crossing the interface

We consider the boundary-value problem of electroelasticity for a composite plate weakened by a crack crossing the line joining the media. The initial boundary-value problem, is reduced to a mixed system of singular integral and algebraic equations. We present the calculation results characterizing the variation in the stress intensity factors as a function of the opening angle of a segmented crack for different types of loading.Translated from Mekhanika Kompozitnykh Materialov, Vol. 33, No. 4, pp. 482–488, July–August 1997.     

Stress analysis for a rectangular thick plate of a composite material with a spatially locally curved structure under forced vibration

Natural and forced vibrations of a thick rectangular plate fabricated from a composite material with a spatially locally curved structure are investigated with the use of exact three-dimensional equations of motion of the theory of elastic anisotropic bodies. The investigations are carried out within the framework of the continuum approach developed by Akbarov and Guz. It is supposed that the plate is clamped at all its edges and is loaded on the upper face with uniformly distributed normal forces periodically changing with time. The influence of the parameters of local curving on the fundamental frequency of the plate and on the distribution of the normal stress acting in the thickness direction under forced vibration is studied. The corresponding boundary-value problems are solved numerically by employing the three-dimensional FEM modelling.Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 40, No. 6, pp. 779–790, November–December, 2004.     

The thermoelastic problem for a halfspace with a semiinfinite elastic thermal insulator

Numerical conditions are given in an infinite and semiinfinite plate (heat insulator), which is connected by a vertical two-sided connection only with an elastic halfspace, in the interior of which is a concentrated source of heat, which generates a stationary heat field. The problem is reduced to the solution of an integral-differential equation of the Wiener-Hopf type with respect to the Fourier transform of the contact stress. Its exact solution is constructed using the factorization method, and the final solution is represented by a series with respect to Chebyshev-Laguerre polynomials. Calculations of bending moments and transverse forces are given in an infinite plate, semiinfinite, and infinite beam-rolling plates.Translated from Dinamicheskie Sistemy, No. 7, pp. 114–123, 1988.     

Contact Problem for a Pneumatic Tire Interacting with a Rigid Foundation

Based on mixed finite-element approximations, a numerical algorithm is developed for solving a geometrically nonlinear contact problem for a prestressed multilayered Timoshenko-type shell undergoing arbitrarily large displacements and rotations. As unknowns, six displacements of faces of the shell are taken, which allows one to use principally new relationships for components of the Green–Lagrange strain tensor in curvilinear orthogonal coordinates, exactly representing arbitrarily large displacements of the shell as a rigid body. As an example, a tire interacting with a rigid foundation is considered.     

Effect of shear stiffness and concrete wedge cone failure on fracture of a fiber-reinforced plastic rod at an intersection with a concrete crack

Experimental results of model speciments in which FRP rods fractured due to local deformation at a crack intersection in a concrete member were analyzed by a 3D nonlinear finite element method in which orthogonal anisotropy of the FRP rod was considered. The analytical results indicated that accurate prediction of shear modulus of the FRP rod and size of concrete wedge cone failure around the FRP rod was significant to predict deformation and fracture of the FRP rod. FRP rods as reinforcement in concrete members, the small shear modulus, because of the orthogonal anisotropy and the wedge cone failure, may prevent the FRP rod from fracturing at a very low tensile stress due to the local deformation at the crack intersection.Presented at the Ninth International Conference on the Mechanics of Composite Materials, Riga, October, 1995.Published in Mekhanika Kompozitnykh Materialov, Vol. 21, No. 2, pp. 158–166, March–April, 1996.     

Torsion of multiconnected bodies with cracks

A general expression is obtained for a multiconnected anisotropic (isotropic) body with rectilinear cracks, for the complex torsion potential that exactly satisfies conditions on the cracks and contains unknown functions determined from the boundary conditions on closed contours. A solution is given for the torsion problem of an elliptical rod with a crack. Results are presented of investigation on the clarification of the influence of geometric and elastic characteristics of the rod on the magnitude of the stress intensity coefficient near an outer edge.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 18, pp. 34–38, 1987.     

Ultrasonic method of determining the elastic characteristics of anisotropic materials

An ultrasonic method of determining the elastic characteristics, developed for isotropic materials and based on the measurement of the angles of total internal reflection of ultrasound at a liquid-solid interface, is applied to an anisotropic Compreg of the DSPA type. The results obtained for the shear modulus using the ultrasonic method at 2.5 MHz and an independent method based on measuring the frequency of the damped torsional vibrations of a prismatic bar at 10–30 Hz are compared. It is shown that the shear modulus varies only slightly (10–12%) on the frequency interval investigated.Leningrad Kirov Forest Engineering Academy. Translated from Mekhanika Polimerov, No. 4, pp. 728–731, July–August, 1969.     

Fundamental solution of a plane problem of elasticity theory for a composite anisotropic medium

The plane problem on the action of an arbitrarily oriented concentrated force, applied at some point of an elastic plane, composed of two different anisotropic half-planes, is considered. By a special choice of a particular solution the problem reduces to a well-known differential equation of the anisotropic theory of elasticity with discontinuous coefficients. The latter reduces, by the method of the integral Fourier transform, to the Riemann boundary value problem. Expressions for the stresses and displacement derivatives at an arbitrary point of the plane are obtained. The application of the obtained results is illustrated on the example of a problem on an elastic linear inclusion (strap).Translated from Dinamicheskie Sistemy, No. 4, pp. 40–45, 1985.     

Control of damped vibrations of a rod by means of a concentrated force or moment loading

The optimal local loading regimes of a vibrating rod (one endpoint of the rod is fixed, while the second is stress-free), which ensure the damping of certain modes of elastic vibration over a minimal time interval, for a minimal norm of the control function and also under the simultaneous minimization of the time and the norm of the load function are given.Translated from Matematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 25, pp. 71–75, 1987.     

The uniform heating and bending of a two-layer plate

The problem of the uniform heating of a two-layer plate is solved. The transversely isotropic layer considered (a soft plate) is in ideal contact with a rigid isotropic thin elastically deformed layer. The ends of the plate are load-free. A boundary layer of the soft plate (a thin contact layer) is introduced, which enables the boundary conditions on the ends of the plate to be formulated in such a way that the problem has a bounded smooth solution [1]. The two-layer plate, generally speaking, is bounded along the axis perpendicular to the axes directed along the length and thickness of the plate. The resultant force and the resultant moment, applied to the end transverse sections, are equal to zero. The exact solution of the temperature problem is sought using the equations of the theory of elasticity. The plane problem of the bending of a two-layer plate acted upon by a uniformly distributed pressure applied to the side surface of an anisotropic layer is solved by a similar method. The ends of the rigid isotropic layer are clamped.