Thermoelastic problem for an isotropic sphere with temperature dependent properties

Zusammenfassung Es werden Wärmespannungen infolge polarsymmetrischer stationärer Temperaturfelder in einer isotropen Kugel mit temperaturabhängigen Stoffwerten untersucht. Das Material wird als inkompressibel vorausgesetzt, was einem oberen Grenzwert der viskoelastischen Spannungen in einem Maxwellschen Körper entspricht. Diese Annahme erlaubt, die Lösung in einer geschlossenen Form und für jede Temperaturverteilung und Temperaturabhängigkeit der Materialkonstanten und des Wärmeausdehnungskoeffizienten aufzustellen. Es werden allgemeine Formeln für die Spannungs- und Verschiebungsfelder in einer vollen Kugel und in einer kugelförmigen Schale gegeben und die Einwirkung der variablen Wärmeleitzahl auf das Temperaturfeld untersucht. Ein numerisches Beispiel für lineare Temperaturabhängigkeit des Elastizitätsmoduls und des Wärmeausdehnungskoeffizienten wird berechnet.

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This research was sponsored by the United States Army under Contract No. DA-11-022-ORD-2059     

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.     

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.     

Asymptotic and numerical analysis of thermoelastic waves in an isotropic plate

^** Email: n.scott{at}uea.ac.uk This paper considers the propagation of a plane thermoelasticwave in an infinite homogeneous isotropic plate subjected toeither isothermal or thermally insulated traction-free boundaryconditions. The primary concerns are the derivation and numericalexamination of the dispersion relations along with the developmentof asymptotic models for the fundamental branches of the dispersionrelations. Thermal dissipation is also discussed.     

Longitudinal vibrations of a composite isotropic plate

Approximate integration of the longitudinal vibrations equations for composite plates is considered. A numerical investigation is performed of the potential energy distribution in a two-layered ring when a pulsating pressure acts on the inner contour.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 20, pp. 63–67, 1989.     

Thermal stresses in a plate with transversely isotropic material

In this paper the thermal stresses in a plate with transversely isotropic material have been obtained by the method of Hankel transforms. Three cases of surface temperature over a circular region of exposure with flux of heat, paraboloidal distribution and constant temperature with surface radiation have been considered. Numerical results are presented for the case of surface radiation.     

Forced vibrations of a multiply connected isotropic plate

On the basis of A. I. Aleksandrovich's investigations the present authors worked out an approximate method of solving problems of forced flexural vibrations of a multiply connected isotropic plate. For an elliptic plate with circular holes numerical results are presented.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 18, pp. 105–111, 1987.     

Water waves diffraction by a circular plate

The interaction of water waves with circular plate within the framework of a linear theory is considered. The plate lies on the free surface in water of finite depth. The integral transform technique is used to solve this problem. The problem is reduced to a system of dual integral equations for a spectral function. The way to solve these equations consists in converting them into Fredholm integral equation of the second kind. The asymptotic solutions of this equation are obtained. Representations for diffraction field and for the forces on the plate are given.     

Flow of a viscous incompressible fluid between two parallel plates,one in uniform motion and the other at rest with uniform suction at the stationary plate

In the present paper an exact solution of the Navier-Stoke’s equations reduced to third and second order non-linear differential equations with appropriate boundary conditions is obtained. The longitudinal and transverse velocity profiles for λ=0·1, 1 and R=100 are drawn. It is noted that for large values of\(\bar x\), an adverse pressure gradient is developed which causes a back flow. There is increase in pressure even for very small fluid suction along the stationary plate. The skin friction and the flow coefficient decrease with reference to the suction velocity and the distance along the stationary plate. For λ=0, the results transform to the known results for plane couette flow without suction.     

Thermoelastic contact during pressing of a heated elliptical cross-section die into a transversely isotropic piezoceramic half space

The three-dimensional thermoelastic contact as a nonplanar elliptical cross-section die is pressed into a transversely isotropic half space is examined under conditions such that the die has a previously specified temperature while the boundary of the half space outside the die is kept at zero temperature. Contact between the die and the half space is assumed to be ideal. The stressed state in the elastic medium is characterized by potential functions which are constructed explicity. Expressions are given for determining the translational displacement of points on the base of the stamp as well as the major half axis and eccentricity of the contact area. Some possible forms of boundary conditions for the electrical variables at the contact area are examined. Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 30, pp. 37–53, 1999.     

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.     

Magnetohydrodynamic free convection flow about a semi-infinite plate at whose surface the heat flux is uniform

Summary The magnetohydrodynamic free convection flow of an electrically conducting fluid past a vertical, semi-infinite plate in a strong cross field has been examined when a uniform heat flux is present at the plate. Formulation in terms of a characteristic length has enabled a full numerical solution to be obtained providing details of skin friction and heat transfer at all stations along the plate for a range of Prandtl numbers appropriate to liquid metal coolants. Series solutions are also presented which give good estimates of the skin friction and heat transfer coefficients.

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Zusammenfassung Die magneto-hydrodynamische Strömung mit freier Konvektion in einer elektrisch leitenden Flüssigkeit entlang einer vertikalen halbunendlichen Platte mit einem gleichförmigen Wärmefluß ist in einem starken Querfeld untersucht worden. Die Einführung einer charakteristischen Länge hat eine vollständige numerische Lösung ermöglicht, die die Reibungskraft und den Wärmeübergang entlang der Platte ergibt, für einen Bereich von Prandtlzahlen geeignet für Flüssigmetallkühler, Reihenlösungen, die gute Schätzungen von Reibungskraft und Wärmeübergangs-Koeffizienten ermöglichen.

     

Dispersion of normal waves in a transtropic plate subject to different conditions at its flat sides

The dispersion relations for potential, homogeneous, three-dimensional oscillations of transversely isotropic plates are studied. Possible characteristic effects of the boundary conditions on the phase and group velocities are demonstrated. Donetsk State University. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 30, pp. 20–25, 1999.     

The growth and decay of sonic waves in a radiating gas at high temperature

The propagation of a sonic discontinuity in an optically thick gray gas at temperature 10⁵°K or higher has been studied. The effects of radiation pressure and radiation energy density have been taken into account, while the profiles structured by radiant heat transfer are imbedded in the discontinuities under high temperature conditions of an optically thick medium. When the sonic discontinuity is propagating into a gas at rest, its velocity of propagation is found to be a constant which is the effective speed of sound in a radiating gas. The fundamental differential equations governing the growth of the sonic discontinuity are obtained and solved. It is concluded that if the sonic discontinuity is a compressive wave of order 1, then it terminates into a shock wave after a critical timet _c which has been determined. But on the other hand, when the sonic discontinuity is an expansion wave of order 1, then it will decay and will vanish ultimately. Particular cases of interest have been studied in details.     

The elastic equilibrium of a multiply connected isotropic plate with cracks

Donetsk. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 21, pp. 34–45, 1990.