Planar Flexural Vibrations and Dissipative Heating of Laminated Rectangular Plates

The coupled thermomechanical dynamic behavior of a heterogeneous solid is investigated by an example of laminated inelastic rectangular plates subjected to forced vibration and dissipative heating. The problem is solved in two formulations: complete and approximate. The complete formulation uses generalized flow theory to describe the inelastic properties of the material. The approximate formulation employs a model based on the idea of complex moduli. Special distributions of the field variables and amplitude–frequency responses of the rectangular plate are analyzed. The effect of the loading level on the mode shape and stress distribution is studied as well. A good agreement between the results obtained in the complete and approximate formulations is pointed out.     

Tomographic diagnostics of technical materials and biological tissues using electric current

A mathematical model for impedance computer tomography methods is considered. The continuum formulation of the main problem is studied. Resolving integral equations are derived. A solution algorithm based on the Bubnov—Galerkin method with linearization of nonlinear resolving equations is developed. A numerical example is given, and numerical results are analyzed. Some drawbacks of the model are considered together with methods for avoiding them.     

Study of the Resonant Vibrations and Dissipative Heating of Thin-Walled Elements Made of a Physically Nonlinear Material

An approximate formulation is given to a dynamic coupled thermomechanical problem for physically nonlinear inelastic thin-walled structural elements within the framework of a geometrically linear theory and the Kirchhoff–Love hypotheses. A simplified model is used to describe the vibrations and dissipative heating of inhomogeneous physically nonlinear bodies under harmonic loading. Nonstationary vibroheating problem is solved. The dissipative function obtained from the solution for steady-state vibrations is used to simulate internal heat sources. For the partial case of forced vibrations of a beam, the amplitude–frequency characteristics of the field quantities are studied within a wide frequency range. The temperature characteristics for the first and second resonance modes are compared.     

Pseudotrajectories generated by a discretization of a parabolic equation

A semiimplicit discretization of a parabolic equation is considered. The resulting diffepmorphism is shown to be generically Morse-Smale. Uniform bounds for the dimension of its attractor are given and numerical trajectories—including round-off errors—are shown to approximate the attractor.     

On Approximate Specification of the Functionals of the Thermoviscoplastic Equations for Complex Loading Along Plane Paths

A technique is proposed for approximate specification of the functionals of the thermoviscoplastic equations for complex loading of isotropic materials along arbitrary plane paths. The lag in the vector and scalar properties of the material is taken into account. Instantaneous deformation and creep diagrams plotted in baseline experiments are used. The technique is used to analyze a specific deformation process. The calculated results agree well with experimental data     

Stability of flow of a viscous liquid down an inclined plane

Using the Navier-Stokes equation the stability of a layer of viscous liquid flowing down a solid surface under gravity is studied in the linear formulation. The effect of surface tension and the inclination of the solid surface on the limits of stability are examined also. Curves are calculated for the neutral stability with respect to two types of perturbations — surface waves and shear waves.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskol Fiziki, No. 2, pp. 172–176, March–April, 1975.     

Nonlinear Antiplane Deformation of an Elastic Body

The antiplane elastic deformation of a homogeneous isotropic prestretched cylindrical body is studied in a nonlinear formulation in actual–state variables under incompressibility conditions, the absence of volume forces, and under constant lateral loading along the generatrix. The boundary–value problem of axial displacement is obtained in Cartesian and complex variables and sufficient ellipticity conditions for this problem are indicated in terms of the elastic potential. The similarity to a plane vortex–free gas flow is established. The problem is solved for Mooney and Rivlin—Sonders materials simulating strong elastic deformations of rubber–like materials. Axisymmetric solutions are considered.     

A large strain thermoviscoplastic formulation for the solidification of S.G. cast iron in a green sand mould

This paper presents a large strain thermoviscoplastic formulation for the analysis of the solidification process of spheroidal graphite (S.G.) cast iron in a green sand mould. This formulation includes two different non-associate constitutive models in order to describe the thermomechanical behaviour of each of such materials during the whole process. The performance of these models is evaluated in the analysis of a solidification test.     

Modelling the Stationary Vibrations and Dissipative Heating of Thin-Walled Inelastic Elements with Piezoactive Layers

A coupled dynamic problem of thermoelectromechanics for thin-walled multilayer elements is formulated based on a geometrically nonlinear theory and the Kirchhoff–Love hypotheses. In the case of harmonic loading, an approximate formulation is given using the concept of complex moduli to characterize the cyclic properties of the material. The model problem on forced vibrations of sandwich beam, whose core layer is made of a passive physically nonlinear material, and face layers, of a viscoelastic piezoactive material, is considered as an example to demonstrate the possibility of damping the vibrations by applying harmonic voltage to the oppositely polarized layers of the beam. Substantiation is given for a linear control law with a complex coefficient for the electric potential, which provides damping of vibrations in the first symmetric mode at the linear and nonlinear stages of deformation. The stress–strain state and dissipative-heating temperature are studied     

Hydrodynamic interaction of cascades of plates during their relative motion

The problem of the hydrodynamic interaction taking place between the profiled plates of a (turbine) cascade is considered. This problem has hitherto been little studied — existing papers [1, 2] consider the model of effectively steady-state flow, the flow of the liquid at any instant of time being regarded as free from eddies, while time plays the part of a parameter; however, under practical conditions the field of velocities around moving cascades changes rapidly with time, so casting doubt on the validity of the steady-flow mode. The model proposed in this paper allows for high-frequency pulsations in the flow. The approximate solution here derived determines the interference between the profiles on the basis of a limiting solution with infinitely large Strouhal numbers. A typical calculation is presented.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 49–54, January–February, 1974.     

Mechanical model of lung parenchyma as a two-phase porous medium

The anatomy and geometry of the lung at the micro- and macroscopic level have been described briefly. A notion of lung parenchyma — a macroscopically continuous medium whose mechanical properties result from those of microstructural components — has been adapted. Simplifying assumptions propounded in the constitutive model have been discussed. Two phases have been distinguished in the medium: the solid phase — a highly deformable, nonlinearly elastic skeleton in the form of a thin-walled tissue structure on the micro-scale — and the fluid phase — perfect gas (air) filterating through the structure. General constitutive relations for both phases and their mechanical interactions have ben formulated. Further, the fundamental set of differential equations of the quasi-static coupled problem has been developed. Large deformations, material nonlinearities, and dependence of permeability on skeleton deformation have been included. Matrix formulation of the problem has been presented from the point of view of the finite element method. An implicit iterative time integration scheme has been proposed. The algorithm has been illustrated with results of simple numerical tests.     

Spatial Thermoviscoplastic Problems

Methods and results of studies of the three-dimensional viscoplastic stress–strain state of engineering structures under thermomechanical loading are presented. The following classes of thermoviscoplastic problems are considered: axisymmetric problems, nonaxisymmetric problems for bodies of revolution, three-dimensional problems for bodies of arbitrary shapes, and three-dimensional problems for anisotropic bodies of revolution     

Mid–Span Deflection and End–Shortening of a Rod after Buckling

The load—displacement relations governing the postbuckling behavior are expressed in terms of elementary functions. An approximate solution of the elastica problem with modified expressions for the curvature is given. Equations of the elastic curve are obtained with the use of an approximate determination of elliptic integrals.     

The determination of the crater for an explosion in ground with angular and curvilinear free boundaries

One of the simple mathematical models in the theory of the deformation of continuous media in an explosion is the solid—liquid model [1, 2]. This does not describe the dynamics of the ground and so enables us to determine only approximate characteristics of the crater. This model has now been used to study a wide range of problems in determining a crater in a continuous medium with various tensile properties and various positions of the explosive [3–5]. We consider below within the framework of the solid—liquid model boundary-value problems in determining a crater in the explosions of point explosives and uniformly distributed explosives on the surface and deep within an isotropic ground with angular and curvilinear free boundaries. The desirability of studying problems such as these was pointed out by Il'inski.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 3–9, January–February, 1985.     

Concentration of stresses and strains in a notched cyclinder of a viscoplastic material under harmonic loading

Certain aspects of the correct definitions of stress and strain concentration factors for elastic-viscoplastic solids under cyclic loading are discussed. Problems concerning the harmonic kinematic excitation of cylindrical specimens with a lateral V-notch are examined. The behavior of the material of a cylinder is modeled using generalized flow theory. An approximate model based on the concept of complex moduli is used for comparison. Invariant characteristics such as stress and strain intensities and maximum principal stress and strain are chosen as constitutive quantities for concentration-factor definitions. The behavior of time-varying factors is investigated. Concentration factors calculated in terms of the amplitudes of the constitutive quantities are used as representative characteristics over the cycle of vibration. The dependences of the concentration factors on the loads are also studied. The accuracy of Nueber's and Birger's formulas is evaluated. The solution of the problem in the approximate formulation agrees with its solution in the exact formulation. The possibilities of the approximate model for estimating low-cycle fatigue are evaluated. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 35, No. 2, pp. 15–22, February, 1999.     

Comparison of fast algorithms for estimating large-scale permeabilities of heterogeneous media

The problem of upscaling permeability data from the core to the reservoir grid block scale for input into flow simulators is addressed. Two fast, approximate algorithms which have been suggested for this purpose—one based on random walks and the other on real-space renormalisation group methods—are compared using the results of numerical tests performed on 30 different heterogeneous permeability realisations in 3-D. The results show that random walks outperform renormalisation for this problem, being computationally more efficient and demonstrating significantly better accuracy for particular cases.     

On flows of granular materials

This article reviews the behavior of materials made up of a large assemblage of solid particles under rapid and quasi static deformations. The focus is on flows at relatively high concentrations and for conditions when the interstitial fluid plays an insignificant role. The momentum and energy exchange processes are then primarily governed by interparticle collisions and Coulomb-type frictional contact. We first discuss some physical behavior —dilatancy, internal friction, fluidization and particle segregation — that are typical to the understanding of granular flows. Bagnold's seminal Couette flow experiments and his simple stress analysis are then used to motivate the first constitutive theories that use a microstructural variable — the fluctuation energy or granular temperature — governing the subscale fluctuating motion. The kinetic theories formalize the derivation of the field equations of bulk mass, momentum and energy, and permit derivation of constitutive relations for stress, flux of fluctuation energy and its dissipation rate for simple particle assemblages and when frictional rubbing contact can be ignored. These statistical considerations also show that formulation of boundary conditions needs special attention. The frictional-collisional constitutive behavior in which both Coulomb-type rubbing contact and collisional encounters are significant are discussed. There is as yet no rigorous formulation. We finally present a phenomenological approach that describes rapid flows of granular materials under simultaneous transport of heat and close with a summary of stability analyses of the basic flow down an inclined plane.Dedicated to Professor Dr.-Ing. Franz Gustav Kollmann on the occasion of his sixtieth brithday     

The Thermoviscoelastoplastic State of Shells of Revolution Under Axisymmetric Deformation Along Various Flat Paths

The basic results obtained at the S. P. Timoshenko Institute of Mechanics of the National Academy of Sciences of Ukraine in developing a thermoviscoelastoplastic theory of thin-walled shells of revolution subject to arbitrary axisymmetric loading are analyzed. The theory includes the constitutive thermoviscoplastic equations describing the deformation of an isotropic body along arbitrary flat paths with functionals made specific in base experiments and the solutions of boundary-value problems, which indicate the influence of the geometry of strain paths on the stress–strain state of shells     

Similarity autooscillations in a plane jet

Experimental investigations into the stability of a plane jet [1, 2] show that after the stationary flow has lost its stability a stable autooscillatory regime arises. In the present paper, an autooscillatory flow in a jet is studied theoretically on the basis of a plane-parallel flow in a fairly wide channel in the presence of a field of external forces. The external forces are such that at zero amplitude of the autooscillations they produce a Bickley—Schlichting velocity profile. The excitation of the secondary regimes is studied by the methods of bifurcation theory [3].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 26–32, May–June, 1979.We thank M. A. Gol'dshtik and V. N. Shtern for discussing the formulation of the problem and the results.     

Spontaneous swirling in axisymmetric MHD flows of an ideally conducting fluid with closed streamlines

The region of instability of the Hill-Shafranov viscous MHD vortex with respect to azimuthal axisymmetric perturbations of the velocity field is determined numerically as a function of the Reynolds number and magnetization in a linear formulation. An approximate formulation of the linear stability problem for MHD flows with circular streamlines is considered. The further evolution of the perturbations in the supercritical region is studied using a nonlinear analog model (a simplified initial system of equations that takes into account some important properties of the basic equations). For this model, the secondary flows resulting from the instability are determined. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 3, pp. 40–50, May–June, 2007.