Thermoelastic deformation of a transversely isotropic compressed spheroid

An exact solution is presented for a static thermoelastic problem for a transversely isotropic compressed spheroid when an arbitrary temperature distribution is assigned on its surface. It is assumed that the surface of the spheroid is free of external forces. The general solution is expressed through four potential functions, each of which is harmonic in a certain coordinate system. The external and internal problems for the spheroid are investigated. The solution is constructed in the form of double series in products of trigonometric functions and associated Legendre functions.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 19, pp. 39–46, 1988.     

Existence and uniqueness of a rayleigh surface wave propagating along the free boundary of a transversely isotropic elastic half space

In this article the existence and the uniqueness of a Rayleigh surface wave propagating along the free boundary of a transversely isotropic elastic half space are proved by a spectral method.     

Strong evaporation into a half space

Evaporation of a liquid into a vacuum occupying a half space is investigated on the basis of the one-dimensional BGK model linearized about a drifting Maxwellian distribution. A unique solution is shown to exist if the downstream speed remains subsonic. Exact analysis is used, and numerical results are given.

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Zusammenfassung Auf der Basis eines eindimensionalen BGK-Modells, das über eine driftende Maxwell-Verteilung linearisiert ist, wird die Verdampfung einer Flüssigkeit in ein einen Halbraum ausfüllendes Vakuum untersucht. Es wird gezeigt, daß nur eine einzige Lösung existiert, wenn die Geschwindigkeit im Unterschallbereich bleibt. Exakte Analysis wird verwendet und numerische Resultate angegeben.

     

Semi-analytic modelling of transversely isotropic magneto-electro-elastic materials under frictional sliding contact

Functionally graded magneto-electro-elastic (FGMEE) materials has been increasingly used in engineering applications, particularly in smart material or intelligent structure systems. This paper proposes a semi-analytical approach for sliding frictional contact problem between a rigid insulating sphere and a transversely isotropic FGMEE film and half-space based on frequency response functions (FRFs). Multilayered approximation is used to model the functionally graded material (FGM), and the FRFs for each MEE layer are derived explicitly. The unknown coefficients in FRFs are formulated by two matrix equations, and their efficient solution process is proposed. Based on the obtained FRFs, a highly efficient semi-analytical model (SAM) is developed which is able to solve the three-dimensional frictional contact of FGMEE materials with arbitrary layer designs. The model is validated with finite element method and the literature. Furthermore, the pressure/stress distribution and electric/magnetic potential are studied in different FGM designs to investigate the influence of material layout.     

Thermoelastic strain of a transversally isotropic cone

The exact solution to the problem of statical thermoelasticity for a transversally isotropic cone in the case where the characteristic equation has multiple roots, the surface of the cone is free of the action of external forces, and the temperature on the boundary can be represented in one variable by the Mellin integral and in a second variable, by a trigonometric series, is presented. Bibliography: 6 titles. Translated fromObchyslyuval’na ta Prykladna Matematyka, No. 76, 1992, pp. 23–32.     

Elastostatic bounds for the stiffness of an elliptical disc inclusion embedded at a transversely isotropic bi-material interface

The present paper examines the problem related to the axial translation of a rigid elliptical disc inclusion which is embedded in bonded contact at the interface of a transversely isotropic bi-material elastic medium. The result for the stiffness of the elliptical disc inclusion is presented in the form of a set of bounds. These bounds are developed, in exact closed form, by imposing displacement and traction constraints at the bi-material interface.

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Resumé On examine le problème d'une inclusion rigide de form elliptique soumise à une translation axiale. Cette inclusion est située à l'interface d'un milieu élastique bi-matériel à isotropie transversale et elle est en contact lié avec se dernier. La rigidité de l'inclusion elliptique est exprimée par un ensemble de limites et la forme exacte de ces limites à été obtenue en imposant des contraintes de déplacement et traction à l'interface du bi-matériau.

     

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.     

Analyticity of the resolvent for elastic waves in a perturbed isotropic half space

In this article the analytic and asymptotic properties of the resolvent for elastic waves in a three dimensional domain perturbed from the isotropic half space R ³₊ are studied. In this case, the asymptotic expansion of the resolvent at the origin has logarithmic terms. This property is totally different from the case of the whole 3‐dimensional space. The existence of the surface waves like the Rayleigh waves makes this difference. As an application of the asymptotic properties of the resolvent, the rate of the local energy decay estimates for the dynamical equations is obtained. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Elementary solution of contact problems for a transversely isotropic elastic layer bonded to a rigid foundation

The contact problem for an arbitrary punch acting on a transversely isotropic elastic layer bonded to a rigid foundation is solved by the generalized images method developed by the author earlier. The problem is reduced to that of an electrostatic problem of infinite row of coaxial charged disks in the shape of the domain of contact. The solution can be obtained by the method of iteration, collocations or any other standard procedure for solving integral equations. Exact inversion can be obtained in the case of a circular domain of contact. The mean value theorem can be used for estimation of the resultant force and tilting moment acting on a punch of arbitrary shape and circular domain of contact. A limiting case of general solution gives the solution for an isotropic layer. (Received: August 11, 2003)     

Elementary solution of contact problems for a transversely isotropic elastic layer bonded to a rigid foundation

The contact problem for an arbitrary punch acting on a transversely isotropic elastic layer bonded to a rigid foundation is solved by the generalized images method developed by the author earlier. The problem is reduced to that of an electrostatic problem of infinite row of coaxial charged disks in the shape of the domain of contact. The solution can be obtained by the method of iteration, collocations or any other standard procedure for solving integral equations. Exact inversion can be obtained in the case of a circular domain of contact. The mean value theorem can be used for estimation of the resultant force and tilting moment acting on a punch of arbitrary shape and circular domain of contact. A limiting case of general solution gives the solution for an isotropic layer.     

Interference transverse sh waves in a transversally,isotropic, inhomogeneous half space

In the paper Love waves in a transversally, Isotropic, inhomogeneous half space are investigated in the case where the plane of isotropy does not coincide with the boundary plane. The dependence on the angle between them is established to the solution obtained.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 104, pp. 228–234, 1981.     

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.     

Dynamic analysis of anisotropic half space containing an elliptical inclusion under SH waves

Based on the methods of complex function, conformal mapping, and multipolar coordinate system, dynamic response of an elliptical inclusion embedded in an anisotropic half space is investigated. In order to find the solution of SH waves, the governing equation is transferred into its normalized form. Then, the scattering wave induced by the inclusion and the standing wave in the inclusion is deduced. Different incident wave angles and the corresponding anisotropy of the half space are considered to obtain the reflected waves. The elliptical inclusion is transferred into a unit circle by conformal mapping method, and then the undetermined coefficients in scattering wave and standing wave are solved by using the continuous condition at the boundary of the inclusion. Subsequently, the dynamic stress concentration factor (DSCF) around the inclusion is calculated and analyzed. Numerical results demonstrate that the distribution of the DSCF is mainly influenced by the incident wave angle and the incident wave number. It is affected by anisotropic parameters as well.     

An approximate solution concerning strong evaporation into a half space

Evaporation of a liquid into a vacuum occupying a half space is investigated on the basis of the one-dimensional BGK model linearized about a drifting Maxwellian distribution. TheF _N method is used to deduce accurate numerical results for the perturbations in the density and temperature.

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Zusammenfassung Auf der Basis eines eindimensionalen BGK-Modells, das über eine driftende Maxwell-Verteilung linearisiert ist, wird die Verdampfung einer Flüssigkeit in ein einen Halbraum ausfüllendes Vakuum untersucht. DieF _N-Methode wird verwendet zur Ableitung von genauen numerischen Ergebnissen für die Störungen in Dichte und Temperatur.

     

The interaction of punches on a transversely isotropic half-space

Three-dimensional contact problems on the interaction of two similar punches on an elastic transversely isotropic half-space (five elastic constants) are investigated, when the isotropy planes are perpendicular to the boundary of the half-space. In this connection the stiffness of the half-space boundary depends on the direction. The kernel of the integral equation of the contact problems is represented in a quadrature-free form using the theory of generalized functions. This form of the kernel enables it to be regularized at singular points and enables Galanov's method to be used to solve the contact problem with an unknown contact area.     

Stability of a transversely isotropic cylindrical shell in axial compression

The problem of the stability of a transversely isotropic cylindrical shell subjected to axial compression is considered with the aid of three-dimensional linearized equations for small subcritical strains. Transcendental equations are obtained for the critical loads corresponding to axisymmetric and nonaxisymmetric deformation. It is shown that in the case of the axisymmetric buckling mode the Kirchhoff-Love hypothesis is asymptotically exact irrespective of the properties of the material. The dependence of the critical load on the properties of the shell material is investigated numerically. Graphs of the variation of the critical load with relative shear modulus are presented.Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Mekhanika Polimerov, No. 6, pp. 1064–1068, November–December, 1969.     

The indentation of a transversely isotropic half-space by a rigid punch

Zusammenfassung Der Verfasser gibt eine allgemeine Lösung für die Verteilung des Druckes zwischen einem axialsymmetrischen Stempel und einem transversal-isotropen Halbraum. Es wird gezeigt, dass die Verteilung des Druckes für den flachen Stempel mit allgemeiner Belastung unabhängig ist von den elastischen Eigenschaften des Halbraums und auch genau dieselbe, als ob der Halbraum isotrop wäre.