Flexure of transversely isotropic plates with a low resistance to transverse deformation

This study is concerned with the stress field around holes in plates of a transversely isotropic material, such as a reinforced plastic, with a low resistance to transverse, normal, and tangential deformation. The problem is solved on the basis of the general theory according to I. N. Vekua. It is shown that in the case of a strongly anisotropic material disregarding the transverse deformation will lead to a large error in the determination of the stress field.Institute of Metallurgy, Dnepropetrovsk. Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Mekhanika Polimerov, No. 1, pp. 89–92, January–February, 1974.     

Stresses in elastic conical tubes of transversely isotropic materials with spherical anisotropies under temperature and force loadings

Analytic solutions are proposed for a number of new problems on determining the state of stress of a transversely-isotropic hollow cone with spherical anisotropy. An exact solution of the problem of the axisymmetric deformation of a long conical tube (or continuous cone) from an elastic transversely-isotropic material with spherical anisotropy subjected to an axial force is obtained in a spherical coordinate system R, , θ, the material axis of symmetry is directed along the spherical radius R. A rigorous solution is given of the problem of the uniform heating of a conical tube of transversely-isotropic material with spherical anisotropy for particular values of Poisson's ratios; the material axis of symmetry is directed along the θ-axis. For arbitrary Poisson's ratios an asymptotic solution is found for the temperature problem for a tube with small conicity.     

Effect of material properties on the stability of transversely isotropic shells with an elastic filler subjected to the action of mechanical loads and temperature

The effect of the low shear strength of the material (glass-reinforced plastic) on the stability of cylindrical shells with an elastic filler is investigated in relation to axial compression, external pressure, and heating. The equations of the thermoelastic problem of the theory of monotropic shells, constructed with allowance for the effect of tangential shearing stresses, are used in the calculations.Physicomechanical Institute, Academy of Sciences of the Ukrainian SSR, L'vov. L'vov Polytechnic Institute. Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 5, pp. 903–907, September–October, 1970.     

Fluctuations of forced vibrations in a medium with resistance

For a differential oscillation equation with coefficients perturbed by Gaussian delta-correlated random processes with a random external force, we obtain closed moment equations. In a special case, the mathematical expectation and the covariance function are found.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 1, pp. 132–135, January, 1993.     

Simulation of the transverse vibrations of a cantilever beam with an eccentric tip mass in the axial direction using integral transforms

The main purpose of the current work is to employ an integral transform approach based on eigenfunction expansion and on an implicit filter scheme in order to solve the governing equations for the transverse vibrations of a cantilever beam clamped at one end and with an eccentric tip mass in the axial direction at the other end. Numerical results are obtained for both the undamped and damped natural frequencies of the system, as well as for its transverse displacement due to arbitrarily time-varying load and imposed displacement at the clamped end. The numerical results reported in the current work are highly accurate and new in the literature. New exact results are also provided for the transient displacement and its higher-order spatial derivatives to allow computation of bending stresses and strains. The relative merits of the proposed approach are finally pointed out.     

Investigation of the tangency of front sets of two transversal waves in transversely isotropic elastic media

A special case of a transversely isotropic elastic medium is considered. This medium is excited by a point force perpendicular to the anisotropy axis. The wave field in this medium is constructed and investigated. The front sets of SV and SH waves are in contact with one another at a point. In order to investigate a neighborhood of this point, we derive simpler expressions for the wave field and establish a special function describing the tangency of two front sets. Bibliography: 8 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 342, 2007, pp. 206–216.     

Effect of anisometry of a platelike nanofiller on the elastic constants of a transversely isotropic composite

Analysis results for the elastic properties of a composite with a small amount of coplanarly arranged platelike filler particles are presented. The geometrical form of the particles is described by an oblate ellipsoid of revolution. The calculations are performed by formulas obtained by using the Eshelby approach for media with a low concentration of inclusions. The effect of anisometry of the ellipsoidal particles and of the ratio between the elastic moduli of the filler and matrix on the effective elastic constants of the composite is discussed. Calculation results are compared with experimental data for the elastic moduli of a nanocomposite containing completely exfoliated particles of an unmodified montmorillonite. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 44, No. 4, pp. 493–504, July–August, 2008.     

On the natural frequency of transversely isotropic magneto-electro-elastic plates in contact with fluid

In this paper, the vibration characteristics of transversely isotropic magneto-electro-elastic (MEE) rectangular plates in contact with fluid are investigated. The mathematical formulation on the determination of added virtual mass for water-contacting MEE rectangular plates with uniform thickness is performed. Based on the recently proposed differential equation governing the dynamical responses of the MEE rectangular plates, a fluid-structure interaction model is established and analyzed. First of all, the added virtual mass incremental (AVMI) factor of the system is calculated by using the proposed approach and the added virtual mass can then be obtained, furthermore, the natural frequencies of the MEE rectangular plates in contact with fluid with different boundary conditions are also investigated. It is noted that the natural frequencies based on the proposed method are very useful for those engineers or researchers who are engaged in the vibration analysis and design of the MEE plate in contact with fluid.     

Stability of a layered cylindrical shell with a transversely isotropic core in axial compression

The problem of the stability of a cylindrical shell with a transversely isotropic core in axial compression is considered. A solution is obtained for the axisymmetric case.     

Effect of shear compliance in bending and torsion on the stress concentration in a transversely isotropic plate with a hole whose edge is supported

The stress concentration in shear-compliant, transversely isotropic plates with a hole whose edge is supported has been investigated in bending and torsion. Two variants of the boundary conditions are examined. It is shown that the use of the classical Kirchhoff model may lead to a number of qualitative discrepancies.Mathematical Physics Branch, Institute of Mathematics, Academy of Sciences of the Ukrainian SSR. Translated from Mekhanika Polimerov, No. 3, pp. 458–463, May–June, 1975.     

Determination of contact stresses in problems on bending of a package of transversely isotropic bars

A mechanomathematical model for bending of packages of transversely isotropic bars of rectangular cross section is proposed. Adhesion, slippage, and separation zones between the bars are considered. The resolving equations for deflections and tangential displacements are supplemented with a system of linear differential equations for determining the normal and tangential contact stresses, and boundary conditions are formulated. A scheme for analytical solution of two contact problems—a package under the action of a distributed load and a round stamp—is considered. For these packages, a transition is performed from the initial system of differential equations for determining the contact stresses, where the unknown functions are interrelated by recurrent relationships, to one linear differential equation of fourth order and then to a system of linear algebraic equations. This transition allows us to integrate the initial system and get expressions for the contact stresses.Translated from Mekhanika Kompozitnykh Materialov, Vol. 40, No. 6, pp. 761–778, November–December, 2004.     

Reflection of magneto-thermoelastic waves with two relaxation times and temperature dependent elastic moduli

The model of the equations of generalized magneto-thermoelasticity with two relaxation times in an isotropic elastic medium under the effect of reference temperature on the modulus of elasticity is established. The modulus of elasticity is taken as a linear function of reference temperature. Reflection of magneto-thermoelastic waves under generalized thermoelasticity theory is employed to study the reflection of plane harmonic waves from a semi-infinite elastic solid in a vacuum. The expressions for the reflection coefficients, which are the relations of the amplitudes of the reflected waves to the amplitude of the incident waves, are obtained. Similarly, the reflection coefficients ratios variations with the angle of incident under different conditions are shown graphically. A comparison is made with the results predicted by the coupled theory and with the case where the modulus of elasticity is independent of temperature.     

Optimal control of the transverse vibrations of a beam with a bound on the potential energy

The problem of the optimal control of the transverse vibrations of a uniform beam with a bound on the potential energy is considered. Approximation theory is used. It is shown that the optimal control exists, and a simple method for its computation is suggested.     

Influence of the temperature of dissipative heating on the damping of forced resonance vibrations of a simply supported viscoelastic cylindrical panel with the help of piezoelectric actuators

We consider the problem of active damping of the forced resonance vibrations of a viscoelastic cylindrical panel with the help of piezoelectric actuators. The end faces of the panel are assumed to be simply supported and heat-insulated. An analytical solution of the problem of electromechanics is obtained. On this basis, the dissipative function is calculated, and the energy equation is solved. We also analyze the influence of the temperature of dissipative heating on the efficiency of active damping of the forced vibrations of a cylindrical panel.     

Mixed problems for the equation of longitudinal vibrations of a heterogeneous rod and for the equation of transverse vibrations of a heterogeneous string consisting of two segments with different densities and elasticities

Doklady Mathematics -     

The property of orthogonality and energy transfer by three-dimensional eigenwaves in transversely isotropic laminated plates with and without contact with a liquid

The properties of eigenwaves in laminated plates with anisotropy in a transforse directon properties are studied. The most general form of the solutions, dispersion relations, power flows and generalized orthogonality relations are analysed. The similarity and difference in the properties of the waves as compared with isotropic media and ideal fluids, as well as the extension to the case of layered spaces and half-spaces, is investigated. A method of determining the coefficients for the eigenwaves radiated in the plate is proposed in the case of a problem with dynamical sources of finite size. A method of summing the series over eigenwaves is suggested.     

Effect of errors in the spatial geometry for temperature dependent Stokes flow

A priori bounds are established for the solution to the problem of Stokes flow in a bounded domain, for a viscous, heat conducting, incompressible fluid, when changes in the spatial geometry are admitted. These bounds demonstrate how the velocity field and the temperature field depend on changes in the spatial geometry and also yield a convergence theorem in terms of boundary perturbations. The results have a direct bearing on an error analysis for a numerical approximation to non-isothermal Stokes flow when the boundary of a complicated domain is approximated by a simpler one, e.g., in the procedure of triangulation combined with finite elements.     

Dynamic vibrations of a damageable viscoelastic beam in contact with two stops

A model for the material damage, due to dynamic vibrations of a Kelvin‐Voigt viscoelastic beam whose tip is constrained to move between two stops, is presented and numerically analyzed. The contact of the free tip with the stops is described by the normal compliance condition. The evolution of damage of the beam's material, which measures the reduction of its load carrying capacity, is modeled with a parabolic inclusion. The existence of the unique local solution is stated. A numerical algorithm is presented, in which spatially it is approximated by finite elements, and the time derivatives are discretized with the Euler scheme. Error estimates are derived for sufficiently regular solutions, and four numerical simulations are shown. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Modelling of thermoelastic Rayleigh waves in a solid underlying a fluid layer with varying temperature

The present paper is aimed at to study the propagation of surface waves in a homogeneous isotropic, thermally conducting and elastic solid underlying a layer of viscous liquid with finite thickness in the context of generalized theories of thermoelasticity. The secular equations for non-leaky Rayleigh waves, in compact form are derived after developing the mathematical model. The amplitude ratios of displacements and temperature change in both media at the surface (interface) are also obtained. The liquid layer has successfully been modeled as thermal load in addition to normal (hydrostatic pressure) one, which is the distinctive feature of the present study and missing in earlier researches. Finally, the numerical solution is carried out for aluminum-epoxy composite material solid (half-space) underlying a viscous liquid layer of finite thickness. The computer simulated results for dispersion curves, attenuation coefficient profiles, amplitude ratios of surface displacements and temperature change have been presented graphically, in order to illustrate and compare the theoretical results. The present analysis can be utilized in electronics and navigation applications in addition to surface acoustic wave (SAW) devices.