Using spline-approximation to solve problems of axisymmetric free vibration of thick-walled orthotropic cylinders

The three-dimensional theory of elasticity is used to study the free vibrations of an anisotropic hollow cylinder with different boundary conditions at the ends. The relevant problem is solved by a numerical-and-analytic method. Spline approximation and collocation is used to reduce the partial differential equations of elasticity to a boundary-value problem for a system of ordinary differential equations of high order for the radial coordinate, which is solved using the stable discrete-orthogonalization and incremental-search methods. The calculated results for an orthotropic inhomogeneous cylinder with boundary conditions of several types are presented Translated from Prikladnaya Mekhanika, Vol. 44, No. 10, pp. 74–85, October 2008.     

Investigation of supersonic three-dimensional flow past pointed axisymmetric bodies

The flow pattern near bodies of revolution with very long cylindrical and pointed nose sections is studied in the framework of an ideal gas model by means of a numerical method based on MacCormack's difference scheme. The existence of internal shock waves, oriented in both the longitudinal and the transverse directions, in the shock layer is established. The variation of the aerodynamic coefficients of the configuration with its length, angle of attack, and free stream Mach number is investigated. The calculated and experimental data are compared, and the connection between the flow parameters on the body surface and the position of the separation line of the boundary layer on its lateral face is established. A method of calculating the influence of the boundary layer on the values of the aerodynamic coefficients of bodies of revolution of large aspect ratio at small angles of attack is proposed. Axisymmetric flow near blunt bodies has been studied in detail in [1].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 127–133, September–October, 1986.The author expresses his gratitude to A. N. Pokrovskii for his help in calculating the boundary layer parameters on the surfaces of the considered configurations.     

Solution of three-dimensional problems of the dynamic theory of elasticity for bodies with cracks using hypersingular integrals

Expressions are derived for the singular kernels of a system of integral equations that describe a three-dimensional dynamic problem for a cracked elastic body under harmonic loading. The order of singularity of the kernels of the integral equations is analyzed. Relations that allow reducing hypersingular surface integrals to ordinary contour ones are obtained. Test calculations are conducted. Translated from Prikladnaya Mekhanika, Vol. 36, No. 1, pp. 88–94, January, 2000.     

Interactions of three-dimensional vicous axisymmetric vortex rings with a free surface

The unsteady nonlinear interaction of three-dimensional vortices with a free surface is a great challenge in fluid mechanics, which has deep theoretical significance and important practical background. Applying the three-dimensional VOF method, the interactions of three-dimensional axisymmetric vortex rings with a free surface in an incompressible viscous fluid are numerically simulated. The influence of the Froude number and the surface tension are studied and the evolution of the vorticity, the trajectories of the vortex rings and the baroclinic vorticity on the surface are obtained. The results agreed well with the experiments reported in the literature. The project supported by the National Natural Science Foundation of China     

Solution of three-dimensional static problems of thermoelasticity

Zaporozh Institute of Mechanical Engineering. Translated from Prikladnaya Mekhanika, Vol. 30, No. 3, pp. 25–34, March, 1994.     

Solution of axisymmetric dynamic problems for cylindrical shells on an elastic foundation

A dynamic problem for a cylindrical shell on an elastic foundation is formulated. A numerical algorithm for solving this problem is outlined. The results obtained are analyzed __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 12, pp. 103–109, December 2007.     

Solution of plane axisymmetric problems for an ideal plastic body

Chelyabinsk Science Center, Urals Branch, Russian Academy of Sciences. Translated from Prikladnaya Mekhanika, Vol. 28, No. 9, pp. 29–35, September, 1992.     

On three-dimensional elastic problems of one-dimensional hexagonal quasicrystal bodies

Based on rigorous operator theory, a general solution of the three-dimensional elasticity equations of equilibrium for 1D hexagonal quasicrystals is obtained. The solution is expressed in terms of four quasiharmonic functions, which is very simple and useful. The point phonon (phason) force solution of an infinite 1D hexagonal quasicrystal body is derived, all in terms of elementary functions. They can play an important role in numerical simulations such as boundary element method.     

Simultaneous heat and mass transfer in unsteady free convection from two-dimensional and axisymmetric bodies

The unsteady laminar free convection boundary layer flows around two-dimensional and axisymmetric bodies placed in an ambient fluid of infinite extent have been studied when the flow is driven by thermal buoyancy forces and buoyancy forces from species diffusion. The unsteadiness in the flow field is caused by both temperature and concentration at the wall which vary arbitrarily with time. The coupled nonlinear partial differential equations with three independent variables governing the flow have been solved numerically using an implicit finite-difference scheme in combination with the quasilinearization technique. Computations have been performed for a circular cylinder and a sphere. The skin friction, heat transfer and mass transfer are strongly dependent on the variation of the wall temperature and concentration with time. Also the skin friction and heat transfer increase or decrease as the buoyancy forces from species diffusion assist and oppose, respectively, the thermal buoyancy force, whereas the mass transfer rate is higher for small values of the ratio of the buoyancy parameters than for large values. The local heat and mass transfer rates are maximum at the stagnation point and they decrease progressively with increase of the angular position from the stagnation point.     

Numerical solution of three-dimensional thermoplasticity problems for bodies of complex shape

Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Prikladnaya Mekhanika, Vol. 24, No. 8, pp. 24–30, August, 1988.     

Rotation of axisymmetric bodies in hydromagnetics

Summary This paper deals with the distrubance due to steady rotation of axisymmetric bodies in a viscous incompressible fluid of infinite conductivity in which the uniform ambient flow field is collinear with the uniform magnetic field. The known results of Sowerby¹) for the problem Couple on a rotating spheroid in a slow stream in an incompressible viscous fluid are generalized to apply to all ellipsoids of revolution and a special case of the circular disc has also been investigated in detail. The conditions of flow are assumed to be such as to permit the use of Oseen's approximation in the equations of motion. The coefficient of couple has been found to the first order approximation in terms ofR(1–S) orR(S–1) whereR is the Reynolds number andS=H ₀ ² /4W ² the pressure number which plays an important part. The couple exerted on the body is computed for various values ofS andR.     

On the solution of three-dimensional boundary-value problems for layered bodies with nonplanar interfaces

The interfaces play an important role in various buildup bodies, and also in the composite materials and structural elements. Special monographs [7, 8] have been devoted to this question, presenting the results of scientific studies of the physical and chemical phenomena on the interfaces, the mechanical behavior, and the role of the interfaces in the damage processes, and also their influence on the basic mechanical properties of the composites. In many cases the interfaces deviate from the ideal geometric shapes: planar (in the layered composites), circular cylindrical (in the fibrous composites), and spherical (in the granular composites). Numerous theoretical and experimental studies confirm this. Thus, in the explosive welding of metals (and nonmetals) there form wavy surfaces, the sections of which may be close to sinusoids, for example in the welding of niobium and copper [9]. If the densities of the materials differ significantly, then the sinusoidal nature of the interface distorts as illustrated in [12] for the example of the welding of lead and steel. In addition, in view of the nature of the technological processes [10] the interfaces may become curved in the layered composite materials and deviate locally or periodically from the ideal coordinate planes. Theoretical and experimental studies have shown that the shape of the interface has a significant influence on the physical and mechanical processes and phenomena (bond strength, stress concentration, wave diffraction, thermal conduction, and so on). Numerous publications that are cited in the survey works [1, 3, 11] confirm this. A second variant of the boundary shape perturbation method was developed in [4, 5] for the solution of the three-dimensional boundary-value problems for nonorthogonal surfaces that are close to the coordinate planes. It was assumed that the equations of the interfaces are linear relative to the small parameter characterizing the degree of deviation from the coordinate planes. This narrowed significantly the class of the examined boundary-value problems and their practical importance. In the present work we examine the three-dimensional boundary-value problems of the mechanics of layered bodies with interfaces that are described by nonlinear equations relative to a small parameter. We construct in general form the recurrence relations and the differential operators of the boundary conditions, making it possible to solve the three-dimensional boundary-value problems with the accuracy that is required for applications. We examine particular cases and present one of the possible criteria for evaluating the accuracy of the approximate solutions that are obtained with the aid of the described variant of the boundary shape perturbation method.S. P. Timoshenko Institute of Mechanics, Ukrainian Academy of Sciences, Kiev. Translated from Prikladnaya Mekhanika, Vol. 30, No. 2, pp. 23–32, February, 1994.     

Distribution of radiant heat fluxes over the surface of three-dimensional and axisymmetric bodies in a supersonic ideal-gas flow

Numerous studies have been devoted to the calculation of supersonic radiating flow past bodies (see the bibliography in [1, 2]). In almost all these studies the gas flows investigated are plane or axisymmetric. Three-dimensional flows, however, have received little attention. The flow of air past three-dimensional bodies was considered in [3], where the chief object was to investigate the accuracy with which a real radiating volume can be simulated by the widely used plane-layer approximation. In [4] the flow of a hydrogen-helium mixture past three-dimensional bodies was investigated in the hypersonic approximation in the neighborhood of the stagnation point and in two planes of symmetry with allowance for the screening of the radiation of vaporized material.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 92–97, January–February, 1986.The authors wish to thank V. V. Lunev for his useful observations and advice.     

Solution of three-dimensional quasistatic problems of nonlinear viscoelasticity in stresses

Samarkand Institute of Architectural Engineering. Translated from Prikladnaya Mekhanika, Vol. 26, No. 6, pp. 19–25, June, 1990.