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2013 year, number 1
N. M. Zubarev, O. V. Zubareva
Keywords: free surface, conducting fluid, magnetic field, linear currentcarrying conductor, exact solutions
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This paper considers the problem of the possible equilibrium configurations of the free surface of a perfectly conducting fluid deformed by a nonuniform magnetic field. A family of exact solutions of the problem is obtained using conformal mappings; equilibrium is achieved due to the balance of capillary and magnetic pressures. According to these solution, the surface strain amplitude increases with increasing current and the hole is transformed into a twodimensional bubble covering the linear conductor.

G. Hussaina^{}, A. Hameedb^{}, J. G. Hetheringtonb^{}, A. Q. Malika^{}, K. Sanaullaha^{}
Keywords: velocity, yield stress, pressure, compression, divergence, stability
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Hydrocode simulations are carried out using Ansys Autodyn (version 11.0) to study the effects of the liner material (mild steel, copper, armco iron, tantalum, and aluminum) on the shape, velocity, traveled distance, pressure, internal energy, temperature, divergence orstability, density, compression, and lengthtodiameter ratio of explosively formed projectiles. These parameters are determined at the instants of the maximum as well as stable velocity during the flight towards the target. The results of these parameters present the potential capability of each liner material used to fabricate explosively formed projectiles. An experimental analysis is performed to study the velocity status and the lengthtodiameter ratio of explosively formed projectiles.

S. V. Meleshko^{}, O. Sumrum^{}, E. Schulz^{}
Keywords: stochastic equations, equations of gas dynamics, Navier–Stokes equations, group analysis, invariant solutions
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Group analysis is used to study stochastic equations of fluid dynamics. Determining equations for admitted Lie groups of transformation involving independent and dependent variables and Wiener processes are obtained. It is shown that, as in the case of deterministic differential equations, admitted groups make it possible to reduce invariant solutions of stochastic differential equations to solutions with a smaller number of independent variables.

A. M. Elnaggar, K. M. Khalil
Keywords: nonlinear oscillator, saddlenode bifurcation, feedback controller, superharmonic resonance, multiple scales
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A weakly nonlinear oscillator is modeled by a differential equation. A superharmonic resonance system can have a saddlenode bifurcation, with a jumping transition from one state to another. To control the jumping phenomena and the unstable region of the nonlinear oscillator, a combination of feedback controllers is designed. Bifurcation control equations are derived by using the method of multiple scales. Furthermore, by performing numerical simulations and by comparing the responses of the uncontrolled system and the controlled system, we clarify that a good controller can be obtained by changing the feedback control gain. Also, it is found that the linear feedback gain can delay the occurrence of saddlenode bifurcations, while the nonlinear feedback gain can eliminate saddlenode bifurcations. Feasible ways of further research of saddlenode bifurcations are provided. Finally, we show that an appropriate nonlinear feedback control gain can suppress the amplitude of the steadystate response.

A. G. Petrov
Keywords: Navier–Stokes equations, exact solutions, moving parallel plates
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In this paper, we study exact solutions of the Navier–Stokes equations for a layer between parallel plates, the distance between which varies according to an arbitrary power law and whose boundary has a noslip condition, are under study. A solution in the form of a power series of the Reynolds number is obtained. Comparison with the exact solutions is performed, and high accuracy of expansions for Reynolds numbers Re = 1÷10 is shown. An accurate estimate of the error of the Reynolds thin layer approximation is obtained.

A. G. Petrova^{}, I. I. Potapovb^{}
Keywords: bottom perturbations, turbulent fluid flow, sediment transport
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The problem of linear perturbations of the sandy bottom in a rectangular channel with a heavy incompressible fluid is formulated. The turbulent viscosity of the flow is defined as a drag coefficient function, and the hydrodynamic equations are written in the longwave Boussinesq approximation. In the expression for the hydrostatic pressure, a correction is applied to the Boussinesq approximation that changes the sediment discharge. The problem of the development of bottom perturbations is solved taking into account the modified formula of sediment discharge, resulting in analytical expressions for the velocity of bottom perturbations and the wavelength of the fastestgrowing bottom perturbations at small Froude numbers.

S. Sadria, M. Babaelahia
Keywords: laminar boundary layer, porous flat plate, optimal homotopy asymptotic method (OHAM), suction, injection, heat transfer
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An analysis is performed to study a laminar boundary layer flow over a porous flat plate with injection or suction imposed at the wall. The basic equations of this problem are reduced to a system of nonlinear ordinary differential equations by means of appropriate transformations. These equations are solved analytically by the optimal homotopy asymptotic method (OHAM), and the solutions are compared with the numerical solution (NS). The effect of uniform suction/injection on the heat transfer and velocity profile is discussed. A constant surface temperature in thermal boundary conditions is used for the horizontal flat plate.

V. V. Bulatov, Yu. V. Vladimirov
Keywords: internal gravity waves, method of geometrical optics, stratified medium
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A problem of wave dynamics of internal gravity waves in a variabledepth stratified medium is considered. By using a modified method of geometrical optics (vertical modes–horizontal rays), wave modes of higher approximations of asymptotic solutions are constructed. It is demonstrated that the main contributions to the solution in real stratified media are made by the first terms of the corresponding asymptotic presentations.

I. G. Khusainov
Keywords: acoustic sounding, perforated wellbore, perforating tunnels
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The reflection and transmission of harmonic waves and waves of finite duration through the boundary of the perforated zone of a cased wellbore filled with a fluid are studied. A model of the plane fluid flow in the well (in a quasionedimensional approximation) and filtration absorption of the fluid by the porous medium surrounding the well is proposed. The effect of the quality of the perforation (length of perforation tunnels) on the evolution of the waves reflected from the boundary of the perforated zone of the well is studied.

A. A. Bochkarev, V. I. Polyakova
Keywords: Langmuir sorption model, surface diffusion, nucleation, Kelvin–Thomson effect, condensation
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Conditions of unstable equilibrium of singlelayer islands of the adsorbate and empty adsorption vacancies are studied. An analog of the Kelvin–Thomson effect for these islands is found. Appropriate corrections are made in the classical theory of nucleation.

M. A. A. Mahmoud, A. M. Megahed
Keywords: nonNewtonian fluid, mixed convection, porous medium, thermal radiation, thermal diffusion and diffusionthermo effects
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Thermal radiation, thermal diffusion, and diffusionthermo effects on heat and mass transfer by mixed convection of nonNewtonian powerlaw fluids over a vertical permeable surface embedded in a saturated porous medium are investigated. The governing equations describing the problem are nondimensionalized and transformed into a nonsimilar form. The transformed equations are solved by using the local nonsimilarity method combined with the shooting technique. The effects of the physical parameters of the problem on the fluid temperature and concentration are illustrated graphically and analyzed. Also, the effects of the pertinent parameters on the local Nusselt number and the local Sherwood number are presented.

V. G. Bazhenov, V. K. Lomunov, S. L. Osetrov, E. V. Pavlenkova
Keywords: cylindrical shell, tension, experimental and computational approach, true strain diagrams, edge effects, stability, plastic strain localization, neck
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An experimental and computational method of constructing true strain diagrams of steel tubular specimens under tension to rupture at large deformations is developed. Experimental and theoretical studies were performed to investigate the effect of the geometric parameters of cylindrical shells, initial imperfections of the geometry, and edge effects on strain localization, the point of necking, and the critical loads.

L. B. Zuev
Keywords: plasticity, autowaves, localization of deformation, dislocations, selforganization
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This paper considers the basic laws of localized plastic flow in solids obtained from an experimentally established relation invariant for plastic and elastic deformation that determine the propagation velocities of localized plasticity autowaves, the dispersion of these waves, and the dependence of the autowave length on the grain size. The relationship of the equations of localized plasticity and the equations of dislocation dynamics is established.

E. A. Moiseichik
Keywords: crack, slip bands, physicochemical processes, dislocation tunneling, deformation heat generation, break
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The localization of heat release during deformation of a notched steel plate is studied. It is shown that the source of heat generation in the metal specimen is not the whole region of plastic deformation near the tip of the defect (crack), but only the slip bands occupying the relatively small part of the zone (Chernov–Luders bands), in which the deformation initiates physical and chemical processes. It was found that in the slip bands, the metal temperature increases by several tens of degrees (or more). The deformation of notched specimens is characterized by uneven development of plastic deformation in the volume of the material and a high velocity of heatwave propagation in the direction of the maximum slip stresses.

A. Kacia^{}, K. Bakhtia^{}, H. Hebalia^{}, A. Tounsia^{}
Keywords: sigmoid functionally graded materials, nonlinear behavior, plate
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Problems of nonlinear cylindrical bending of sigmoid functionally graded plates in which material properties vary through the thickness are considered. The variation of the material properties follows two powerlaw distributions in terms of the volume fractions of constituents. The nonlinear straindisplacement relations in the von Kármán sense are used to study the effect of geometric nonlinearity. The governing equations are reduced to a linear differential equation with nonlinear boundary conditions, yielding a simple solution procedure. Numerical results are presented to show the effect of the material distribution on the deflections and stresses.

A. M. Akhtyamov^{}, M. A. Il'gamov^{}
Keywords: inverse problem, beam, eigenfrequencies, notch
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Matching conditions simulating a transverse notch in a beam are proposed. These conditions are used to determine the location and size of the notch. The notch is identified from beam deflections at several points in the static case and from the first eigenfrequencies of the beam in the dynamic case. Dependences of the first eigenfrequencies on the characteristic parameters of the problem are obtained. The effect of the relative error of frequency measurements on the relative error of calculation of the notch parameters is studied. It is shown that the use of the first eigenfrequencies of flexural vibrations of the beam with respect to different axis provides a more accurate identification than the use of eigenfrequencies of flexural vibrations relative to one axis. Since local defects such as hollows, local corrosion, and an open crack can be simulated by a notch, the results can also be used to identify these defects.

V. A. Morozova^{}, G. G. Savenkovb^{}
Keywords: fractal crack, speed of sound, time scale, dimension
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A model of a fractal crack is considered. It is found that the limiting velocity of crack propagation is determined by the fractal dimension of the crack contour. It is shown that for commercial steels, the limiting crack velocity is in the range V_{lim} = (0.155 ÷ 0.537)c_{1} (c_{1} is the speed of sound).

L. V. Kovtanyuk
Keywords: elasticity, viscoplasticity, residual stresses
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Analytical solutions of a number of onedimensional quasistatic problems that describe the processes of elastic deformation of the material of a hollow sphere and of generation and development of the plastic flow in this material with increasing pressure on the external boundary are presented. The process of unloading during slow removal of the loading pressure is studied. Stress fields, fields of elastic and plastic strains in the material of the spherical layer, the law of motion of the elastoplastic boundary, and residual stresses are determined. It is demonstrated that (in contrast to the ideal plasticity case) the allowance for the viscous properties of the material during its plastic flow eliminates the possibility of plastic flow emergence during unloading.

V. D. Kurguzov, V. M. Kornev
Keywords: quasibrittle and quasiductile fractures, necessary and sufficient criteria, prefracture zone
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Materials with a regular structure characterized by quasibrittle and quasiductile fractures are considered in the case where the characteristic linear dimension of the structural element is known. Necessary and sufficient fracture criteria are constructed using the Neuber–Novozhilov approach. A modified Leonov–Panasyuk–Dugdale model for an opening mode crack is proposed where the width of the prefracture zone coincides with the width of the plasticity zone. For the critical parameters of quasibrittle fracture (tensile stress, length of prefracture zones, stress intensity factors), relations are obtained that allow material fracture to be considered in the case where the crack length is negligible compared to the characteristic linear dimension of the structural element. A fracture diagram obtained using the critical stresses calculated from the necessary and sufficient criteria is considered in a wide range of crack lengths. The elastoplastic problem of extension of a plate with a central crack is solved using the finiteelement method. The dimensions and shape of the plastic zone near the crack tip are determined for different levels of loads corresponding to quasibrittle and quasiductile fracture. The obtained results are analyzed to estimate the width of the prefracture zone and the critical crack opening.

