Edge crack opening in an elastic plane with a wedge-like cut in contact with a punch

The equilibrium of an elastic plane with a wedge-like cut and an internal or edge crack on the symmetry axis was studied in [1] in the case of punch indentation in the lateral faces of the cut at a distance from the cut tip. In [1], the systemof singular integral equations of the problemwas solved numerically by the mechanical quadrature method. In this paper, the generalized Wiener-Hopf method [2] is used to obtain the analytic solution of a similar problem in the case of an edge crack under punch pressure on parts of the cut lateral faces adjacent to the cut tip. Some special cases of this problem were considered earlier without a crack [3, 4] or a punch [5, 6].     

Heat exchange between a selectively emitting liquid and a laminar gas flow in the presence of an external source of radiation

An investigation is conducted in the solution of a number of practical problems of the radiative and combined heat exchange in nonuniform systems having widely different physical properties. The processes of thermal interaction between the ocean and the atmosphere have been treated in the paper [1], the effect of thermal radiation on the melting and solidification of semitransparent crystals has been investigated in [2], the flow of a selectively emitting gas around the lateral surface of an object evaporating under the action of radiative heating has been discussed in [3], and heat transfer from a jet to the molten mass of glass in a glassmaking furnace tank has been investigated in [4]. The radiative and combined heat exchange between a selectively emitting liquid and a transparent heat-conducting laminar gas flow in the case of a specified external thermal radiation field is discussed in this paper. The energy conservation equations are set up taking into account the heat transfer by radiation, convection, and molecular thermal conduction. A differential approximation is used in calculating the values of the radiation fluxes. A system of fundamental computational equations is solved by the method of finite differences and iterations and by the Runge-Kutta method. The results of the calculations are presented in the form of graphs.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 116–122, May–June, 1976.     

On the stability of nearly cylindrical long shells of revolution

In contrast to [1–4], where the stability problem was studied for shells of medium length, in the present paper we study the stability problem for nearly cylindrical long shells under the action of meridian forces uniformly distributed over their ends and under the action of the normal pressure distributed over the entire lateral surface of the shell. We consider the shells whose generating midsurface shape is determined by a parabolic function. The study is performed for nonaxisymmetric buckling modes by using an equation refined as compared with the equation given in [1]. We consider shells of both positive and negative Gaussian curvature. We assume that the shell ends are freely supported and obtain formulas for the critical load under both separate and joint action of the meridian forces and the pressure. In the specific case of a cylindrical shell under the action of longitudinal compression, the formulas thus obtained imply both the Euler formula and the Southwell-Timoshenko formula [5]. When solving the problem, we use the Bubnov-Galerkin method combined with the optimal approximation method [6].     

The random choice method and the free-surface water profile after the collapse of a dam wall

The random choice (RC) method due to Glimm [1], subsequently modified by Chorin [2] and Sod [3], is applied to the shallow water equations of Stoker [4], in order to determine the free-surface profile of water, as a function of position and time, when a dam wall suddenly collapses. The numerical results obtained by this method are compared with the analytical results due to Stoker for this classical problem [5], and with numerical results obtained by a further modification of the Glimm-Chorin scheme involving a smoothing operation. The improvement brought about by the smoothing, and the generally close agreement with the exact result that is then obtained, is a feature of this modified approach.     

Analytical approach for the prediction of steady state tyre forces and moments under different normal pressure distributions

In general, all primary as well as disturbing forces acting on a vehicle (with the exception of aerodynamic forces) are generated at the tyre–road interface. The small size pneumatic tyres presented in this work are mostly used in lightweight vehicles like scooters, motor cycles and 3-wheelers that are extensively used for intercity transportation applications in India and other developing countries. Analytical approach is more advantage than experimental work due to many factors like it involves less cost, minimum time and little effort. Hence, Analytical tyre models are developed based on various normal pressure distributions with tyre contact patch as rectangle at the middle with part of circles at the ends. The tyre longitudinal and lateral force characteristic equations for proposed tyre models are developed. In addition, the equations for self – aligning moment and overturning moment characteristics for all tyre models are also derived. Results of proposed tyre models in the present study are compared with well-established earlier tyre models [4], [5], [6]. From the force and moment characteristics of all tyre models, unsymmetrical trapezoidal pressure distribution tyre model gives better performance than other models due to which better lateral stability of the vehicle can be obtained.     

Relaxation of anharmonic molecules

The inclusion of anharmonic effects is important in vibrational population inversions in CO-lasers [1, 2], in relaxation processes in jets [3], in thermal dissociation [1], in the kinetics of chemical reactions with high thresholds [4], etc. Usually these effects are studied by including anharmonic corrections to the kinetic constants in the discrete model of single-quantum transitions or in the diffusion approximation [1, 2]. In [5] a method was given of solving the relaxation equations fro arbitrary forms of the rate constants and the spectrum of the molecule. The method is valid when the ratio of the population densities of neighboring levels varies smoothly with quantum number. It was shown in [5] that this approximation can be used to construct analytical solutions for a wide class of problems. In the present paper the method of [5] is extended to the case of equations with variable coefficients. The properties of the solutions for VT-relaxation of anharmonic molecules are analyzed, and the inclusion of sources is considered. A simple method of taking into account multiple-quantum transitions is given, as well as an extension of the method to an arbitrary mixture of gases. The population densities are calculated and the possibility of using our solutions in relaxation gas dynamics is discussed.Translated from Zhurnal Prikladnoi Mekhaniki Tekhnicheskoi Fiziki, No. 3, pp. 22–31, May–June, 1986.     

Asymptotic analysis of laminated plates and shallow shells

It was noted long ago [1] that the material strength theory develops both by improving computational methods and by widening the physical foundations. In the present paper, we develop a computational technique based on asymptotic methods, first of all, on the homogenization method [2, 3]. A modification of the homogenization method for plates periodic in the horizontal projection was proposed in [4], where the bending of a homogeneous plate with periodically repeating inhomogeneities on its surface was studied. A more detailed asymptotic analysis of elastic plates periodic in the horizontal projection can be found, e.g., in [5, 6]. In [6], three asymptotic approximations were considered, local problems on the periodicity cell were obtained for them, and the solvability of these problems was proved. In [7], it was shown that the techniques developed for plates periodic in the horizontal projection can also be used for laminated plates. In [7], this was illustrated by an example of asymptotic analysis of an isotropic plate symmetric with respect to the midplane.     

Turbulent boundary layer on a catalytic surface in a nonequilibrium dissociating gas

The majority of the studies which consider the flow of a dissociating gas in a turbulent boundary layer are devoted to the investigation of either frozen or equilibrium flows on a flat plate.The frozen turbulent boundary layer has been studied by Dorrance [1], Kutateladze and Leont'ev [2], and Lapin and Sergeev [3]. A study of the effect of catalytic recombination processes at the plate surface on the heat transfer in a frozen turbulent boundary layer was made by Lapin [4].Kosterin and Koshmarov [5], Ginzburg [6], Dorrance [7], and Lapin [8] have studied the turbulent boundary layer on a plate in equilibrium dissociating gas.The calculation of the heat transfer in a turbulent boundary layer on a catalytic plate surface with nonequilibrium dissociation was made by Kulgein [9]. In this study the nonequilibrium nature of the dissociation process was taken into account only in the laminar sublayer, while the flow in the turbulent core was considered frozen. The solution was found numerically using a computer by means of a laborious iteration process.The present paper reports a method for calculating the turbulent boundary layer on a flat catalytic plate with arbitrary dissociation rate. The method, constructed using the assumptions customary for turbulent boundary layer theory, is a successive approximation method. Good convergence of the method is assured by the fact that the effect of the nonequilibrium nature of the dissociation process on the parameter distribution in the boundary layer and, consequently, on the friction and heat transfer may be allowed for merely by finding corrections, usually relatively small, to the distribution of these parameters in the equilibrium or frozen flows. The basis of the study is the two-layer scheme of the turbulent boundary layer. The Prandtl and Schmidt numbers and also their turbulent analogs are taken equal to unity. As the model of the dissociating gas we use the Lighthill model of the ideal dissociating gas [10], extended by Freeman [11] to nonequilibrium flows.     

Steady flow of a viscous fluid between coaxial rotating cylinders after loss of stability

Viscous fluid flow between rotating cylinders is the best known case in which a secondary steady (equilibrium) flow develops and reaches equilibrium after loss of stability. This flow, consisting of vortices which are periodic along the axis of rotation, the so-called Taylor vortices, is the result of essentially nonlinear interactions in the flow. It arises for sufficiently high rotational velocity of the inner cylinder. The first attempt at theoretical calculation of the flow was undertaken by Stuart [1], in which the form of solution was assumed from linear stability theory and the amplitude was found from the equation expressing the energy balance in integral form. The Stuart solution was improved by Davey [2], who took into account the appearance in the solution of the next harmonic and the distortion of the fundamental mode. Concrete calculations were carried out under the assumption that the vortex dimension equals the distance between the cylinders. The results agree in general with the experimental data. Individual calculations using the method of nets were made in [3], more detailed calculations weie made in [4], and the perturbation method was applied to this problem in [5].In the following, the method of [6, 7] is applied to the study of secondary flow of a viscous fluid between cylinders. The solution is found from a single system of nonlinear differential equations, which are derived, with a definite approximation, from the equations of motion (without account for the special relation for the amplitude).     

Convection and heat transfer in a vertical layer with allowance for radiation from nonisothermic walls

Many studies (for example, [1–5]) consider motion and heat transfer in closed vertical cavities with given different temperatures of the lateral boundaries. The majority of studies cover the case of convection, but of late studies have appeared (for example, [4]) in which joint radiative—convective heat transfer is taken into account. In the present study we consider motion and heat transfer in a rectangular cavity separating two media with given different temperatures. In contrast to [4], the temperature of the lateral boundaries is determined from the condition for interaction with the surrounding medium, and the air in the cavity is assumed to be transparent for the heat radiation of the walls. The problem considered is a mathematical model of the heat transfer through windows, and is necessary for the analysis of methods of improving the heat proofing of buildings.Translated-from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 25–30, 1987.     

A class of variational difference schemes for a singular perturbation problem

In this paper, a singularly perturbed boundary value problem for second order self-adjoint ordinary differential equation is discussed. A class of variational difference schemes is constructed by the finite element method. Uniform convergence about small parameter is proved under a weaker smooth condition with respect to the coefficients of the equation. The schemes studied in refs. [1], [3], [4] and [5] belong to the class.     

Volumic method for the variational sum of a 2D discrete model

The geometric complexity of some heterogeneous materials (for example, fibers distributed randomly or deterministically with high conductivity [5], [2]) can make it difficult to model their macroscopic behavior. In some cases, it is convenient to simplify the geometry by cutting it into “simple” elements, so that the first study is performed only on these items. The difficulties arise from the reconstruction of the material. In such study, we describe a method for reconstructing a material cut into thin plates having undergone a size reduction (see [6] and [5], for example). The method used is of variational summation limit.     

Application of the angular superposition method to the contact problem on the compression of an elastic cylinder

The angular superposition method is used to construct an approximate solution of the contact problem on the compression of an elastic cylinder by two rigid plates. The solution thus obtained has a closed-form analytic expression and can be used in the entire domain of the cylinder cross-section. We analyze the absolute error, which takes the largest value near the points of contact between the plates and the cylinder, where the boundary conditions are discontinuous. According to the von Mises criterion, when moving into the depth of the cylinder from the contact site along the symmetry axis, the second invariant J ₂ of the stress deviator tensor first decreases and then, after attaining a minimum, increases and attains the largest value at a small depth, which agrees with Johnson’s photoelastic experiments and Dinnik’s computations. We present the graphs of the displacement and normal stress distributions over the contact site, the dependence of the compressing force on the displacements of rigid plates, and the dependence of the invariant J ₂ on the coordinate along the symmetry axis. If 640 computation points are chosen on the cylinder boundary and the Hertz law for the normal pressure on the contact site is used, then the error in the approximate solution near the endpoint of the contact site is approximately 55%, and if the proposed two-parameter normal law is used, then the error is of the order of 4%. On the free lateral surface of the cylinder boundary, we find the critical pointM*, which separates the cylinder contraction and extension parts.The contact problems are the most difficult problems, and their solution is complicated by the discontinuous boundary conditions [1–5]. In [6], the contact problem is solved by the Fourier method, which can be used only for bodies of classical shapes. In such cases, the problem can be reduced to solving coupled integral equations [7]. The interaction between the bandage and a cylindrical body is considered in [2, 6, 7]. In [8], the possibility of using the finite element method is investigated in the case of contact problems for a differential wheel with roughness of the contacting surfaces taken into account. In [9, 10], the method of homogeneous solutions is used to consider contact problems for a finite-dimensional elastic cylinder loaded on its end surfaces. Note that only error estimates are given in the literature cited above; the absolute error over the entire domain of the elastic body is not studied, although this is one of the important characteristics of the obtained approximate solution. A sufficiently complete survey of the literature in the field of contact interactions of elastic bodies is given in [3–5].In what follows, we propose to solve contact problems by the angular superposition method [11]. This method can be used for bodies of nonclassical shapes, which can be multiply connected, and the friction on the contact site can be taken into account. In the present paper, as a first example of applied character, we show how this method can be used in the simplest case. The multiple connectedness and the curvilinearity of the shape of the body, as well as taking into account the friction on the boundary, do not create new essential difficulties in this method.     

Numerical solution of nonlinear ordinary differential equation for a turning point problem

By using the method in[3],several useful estimations of the derivatives of the solutionof the boundary value problem for a nonlinear ordinary differential equation with a turningpoint are obtained.With the help of the technique in[4],the uniform convergence on thesmall parameterεfor a difference scheme is proved.At the end of this paper,a numericalexample is given.The numerical result coincides with theoretical analysis.     

Propagation of shock waves in a medium with exponentially varying density

The results of Raizer [1], Hays [2], and Chernous'ko [3] are generalized to-the case of self-similar propagation of shock waves in a gas with exponentially varying density and constant pressure. A solution is found by the method of successive approximations. The zero-order approximation coincides with the Whitham method [4]. The first-order approximation is in good agreement with numerical calculations in [2]. The non-selfsimilar motion of a weak shock wave is investigated in the framework of linear theory.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 48–54, November–December, 1970.     

Draft of a nozzle with the circulating outflow of a vortical flow of gas

The presence of circulation in an outflowing gas leads to a change in the working parameters of a nozzle. The question of the mass flow rate and the draft of a nozzle without a diffusor (a point) for twisted flows has been studied theoretically and experimentally [1–6]. The use of nozzles with a supersonic part introduces a considerable degree of complication into the method for the analytical calculation of the draft characteristics and the program for their experimental investigation. In [2, 7], a theory of a nozzle is formulated for a model of a potential circulating flow of gas; in [5, 8], an electronic computer was used to solve the complete system of the equations of gasdynamics for the motion of a rotating flow along a nozzle; in [7, 9], an investigation was made of a variational problem of the shaping of a diffusor for a circulation flow. The calculation of the draft, carried out in the above-mentioned communications (with the exception of [2], in which a study was made of a partial model of an eddyless rotational motion), is bound up with labor-consuming computer calculations. In the present article, in a development of [3, 6], a quasi-one-dimensional theory of a supersonic nozzle for a vortical flow of gas is formulated and verified experimentally.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 145–149, September–October, 1975.     

Bounds for effective elastic moduli of inhomogeneous solid bodies

In connection with the extensive use of various kinds of inhomogeneous materials (glass, carbon and boron reinforced plastics, cermets, concrete, reinforced materials, etc.) in technology, there arises a need to calculate the elastic properties of such systems. Here in each case it is necessary to work out specific methods for finding both elastic fields and effective moduli. Since, as a rule, such methods do not take into account the character of distribution of inhomogeneities in space, which is reflected on the form of the central moment functions [1], they can be referred to a single class and, consequently, can be obtained by a common method [2], In the given paper, by means of the method of solution of stochastic problems for microinhomogeneous solid bodies proposed in the work of the author [2], we find elastic fields and effective moduli in an arbitrary approximation. Depending on the choice of parameters, the latter form bounds within which there lie the exact values of the effective moduli. It is shown that the conditions used earlier for finding these parameters [3] are not the best ones. The effective elastic moduli of an inhomogeneous medium are calculated, and bounds, narrower than the bounds formed in [3], are found for them.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhniki, No. 5, pp. 144–150, September–October, 1973.     

Calculation of the stability of the flow in a circular pipe with injection

An experimental investigation of the transition of a laminar flow regime into a turbulent one has been carried out in [1] for a flow in a circular pipe which is organized due to injection through the porous lateral surface with a jammed leading end of the pipe. It was established as a result that injection leads to an increase in stability of the laminar flow regime and increases the Reynolds number of the transition to 10,000 instead of the value 2300 which is characteristic of flow in a circular pipe with impenetrable walls. A similar effect was discovered in [2], in which it was also obtained that the Reynolds number of stability loss under the action of injection can take values significantly larger than in pipes with impenetrable walls. The phenomenon of relaminarization of a turbulent flow in the initial section of a circular pipe under the action of injection has been experimentally detected at the entrance for relatively low Reynolds numbers in [3, 4]. Theoretical investigations of stability of flow with injection have been performed only for a plane channel [5, 6]. A calculation is made in this paper of the stability of a hydrodynamically developed flow in a circular pipe with injection through a porous lateral surface.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 82–86, May–June, 1984.     

Fracture analysis of concrete using singular fractal functions with lattice beam network and confirmation with acoustic emission study

In the present study singular fractal functions (SFF) were used to generate stress-strain plots for quasi-brittle material like concrete and cement mortar and subsequently stress-strain plot of cement mortar obtained using SFF was used for modeling fracture process in concrete. The fracture surface of concrete is rough and irregular. The fracture surface of concrete is affected by the concrete’s microstructure that is influenced by water cement ratio, grade of cement and type of aggregate [1], [2], [3] and [4]. Also the macrostructural properties such as the size and shape of the specimen, the initial notch length and the rate of loading contribute to the shape of the fracture surface of concrete. It is known that concrete is a heterogeneous and quasi-brittle material containing micro-defects and its mechanical properties strongly relate to the presence of micro-pores and micro-cracks in concrete [1], [2], [3] and [4]. The damage in concrete is believed to be mainly due to initiation and development of micro-defects with irregularity and fractal characteristics. However, repeated observations at various magnifications also reveal a variety of additional structures that fall between the ‘micro’ and the ‘macro’ and have not yet been described satisfactorily in a systematic manner [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [15], [16] and [17]. The concept of singular fractal functions by Mosolov was used to generate stress-strain plot of cement concrete, cement mortar and subsequently the stress-strain plot of cement mortar was used in two-dimensional lattice model [28]. A two-dimensional lattice model was used to study concrete fracture by considering softening of matrix (cement mortar). The results obtained from simulations with lattice model show softening behavior of concrete and fairly agrees with the experimental results. The number of fractured elements are compared with the acoustic emission (AE) hits. The trend in the cumulative fractured beam elements in the lattice fracture simulation reasonably reflected the trend in the recorded AE measurements. In other words, the pattern in which AE hits were distributed around the notch has the same trend as that of the fractured elements around the notch which is in support of lattice model.