Interface crack with a contact zone in an isotropic bimaterial under thermomechanical loading

A plane problem for a thermally insulated interface crack with a contact zone in an isotropic bimaterial under tension–shear mechanical loading and a temperature flux is considered. The expressions for the stresses and the electrical flux as well as for the derivatives of the displacement and the temperature jumps at the material interfaces via sectionally holomorphic mechanical and thermal potential functions are given. After the solution of the thermal problem the inhomogeneous combined Dirichlet–Riemann boundary value problem is formulated and solved exactly. The stresses at the interface and the stress intensity factors at the singular points are presented in a clear analytical form. Special attention is devoted to the case of a small contact zone when the stress intensity factors can be presented in form similar to the associated presentation for an “open” crack model. A transcendental equation and an asymptotic analytic formula for the determination of the real contact zone length are derived. It is shown that for a certain bimaterial this length as well as the correspondent stress intensity factor are defined by a single parameter which depends on the normal-shear loading and the heat flux.     

Numerical simulation of the displacement of oil by water

If the mobility of a displacing fluid is greater than the mobility of the displaced fluid, the displacement is unstable (see, for example, [1–3]), and the originally plane displacement front is broken up into irregular tongues. It follows from the linear analysis of stability that initially the amplitude of the perturbation increases exponentially, and according to [1] the extended tongues that are formed move with constant velocity relative to the displaced fluid. The intermediate stages in the development of the instability, like questions relating to a more precise formulation of the problem (which involves giving up the piston displacement approximation) remain unstudied. A natural approach to their study is through numerical simulation, which was realized for the first time in [4, 5]. Some of the results of such an investigation are presented in the present paper. In contrast to [4], the main attention is devoted to the development of regular perturbations. It is shown that for the investigated mobility ratios the development of the perturbations follows the linear theory unexpectedly long, and then arrives at a stationary asymptotic regime. We also investigate the influence of the loss of displacement stability on waterless oil extraction in the case of displacement in homogeneous and inhomogeneous strata.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 58–63, September–October, 1979.We thank L. A. Chudov for advice and discussions.     

Contact Interaction of the Faces of a Rectangular Crack under Normally Incident Tension–Compression Waves

A three-dimensional problem on the contact interaction between the faces of a rectangular crack under a normally incident harmonic tension–compression wave is considered. The problem is solved by using the method of boundary integral equations and an iterative algorithm. The contact forces and the discontinuity in the displacement of the crack faces are studied. The results obtained are compared with those for a finite plane crack.     

Motion of the interface of two fluids with formation of a region of two-phase flow

A study is made of the problem of the displacement of one fluid by another with the formation of a region of combined flow in the case of an elastic flow regime in the region of the displaced fluid. A self-similar solution is constructed for the flow equations averaged with respect to the vertical coordinate. A numerical algorithm is developed for determining the saturation in the region of the mixture, the pressure, and also the position and shape of the interface.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 8, pp. 78–83, October–December, 1981.We thank A. A. Barmin for a helpful discussion of the work.     

Experience in making rheological measurements at high pressures

Summary We review the difficulties encountered in the design and operation of apparatus for rheological studies on liquids at pressures up to 900 MN m^–2 and temperatures from –30 to 120°C. Such rheological information is required in connection with elastohydrodynamic lubrication in which high pressures and shear rates are encountered.With 3 figures     

Influence of porosity of the base on a plane standing wave of a heavy liquid

A study is made of the simple problem of the contact of two plane-parallel potential flows of incompressible fluid when one takes place in a layer of finite thickness and the other in a semiinfinite space of a porous medium. At the interface, which is taken to be a plane, the same conditions are used as earlier in problems of the contact of two wave flows of fluids with different densities and the contact of a wave motion in a layer of compressible fluid and wave motions in an elastic semi-infinite space. These conditions reduce to equalities of the pressures and projections of the velocity vectors onto the normal to the interface.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 160–163, July–August, 1984.     

The symmetrical adhesive contact problem for circular elastic cylinders

The rolling contact problem involving circular cylinders is at the heart of numerous industrial processes, and critical to any elastohydrodynamic lubrication analysis is an accurate knowledge of the associated contact pressure for the static dry problem. In a recent article [1] the authors have obtained new horizontal pressure distributions, both exact and approximate for various problems involving the symmetrical contact of circular elastic cylinders. In [1] it is assumed that only the circumferential horizontal displacement is prescribed in the contact region while the vertical circumferential displacement is left arbitrary and is assumed to take on whatever value is predicted by the deformation. The advantage of this assumption is that the problem reduces to a single singular integral equation which by transformations can be simplified to an integral equation involving the standard finite Hilbert transform. Here we consider the more general displacement boundary value problem within the contact region, and to be specific we examine the problem with zero vertical circumferential displacement and prescribed horizontal circumferential displacement. The solution of this problem involves two coupled singular integral equations for the horizontal and vertical pressure distributions. Basic equations and some approximate analytical solutions are obtained for symmetrical contact of circular elastic cylinders by both parallel plates and circular cylinders which are either rigid or elastic. Numerical results for the approximate analytical solutions are given for contact by rigid parallel plates and rigid circular cylinders.     

Study of Iterative Algorithms for Solution of Dynamic Contact Problems for Elastic Cracked Bodies

A plane problem is solved for the contact interaction between the faces of a rectilinear crack under the action of a normally incident harmonic tension–compression wave. Iterative algorithms are presented to solve the problem for both given initial distribution of contact forces and given initial discontinuity in the displacements of the crack faces. The convergence rates of the algorithms, the maximum contact forces, and displacement discontinuities are compared.     

润滑力学中非牛顿流动的普遍Reynolds方程

本文导出了润滑力学中关于非牛顿流动的普遍 Reynolds 方程。这一方程适用于多种非牛顿流动模型,可以用于解算热流体动力润滑或热弹性流体动力润滑膜的压力分布。本文给出了一种同时求出剪应力、剪切率、速度和等效粘度的解法,并以两种润滑力学中常用的流变模型为例,应用这一方程得到了线接触热弹性流体动力润滑问题的数值解。     

Inclined entry of a blunt profile into an ideal fluid

A study is made of the two-dimensional unsteady motion of an ideal incompressible fluid due to the entry into it of a blunt profile at a given angle of attack. In the initial stage of the process, when the penetration depth is relatively small, the problem can be investigated by the methods of asymptotic analysis. The dimensionless time t plays the part of the small parameter. It is shown that to 0(t²) as t 0 the displacement field of the fluid particles does not depend on the angle of attack and is determined by the solution to the problem of vertical entry. The asymptotic behaviors of the principal vector and principal moment of the forces exerted on the profile by the fluid at short times are found. The asymptotic behavior of the principal moment of the forces is proportional to the distance traversed by the body along the surface of the fluid.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 145–150, May–June, 1988.     

Micromechanics of residual saturation formation during immiscible displacement in a porous medium

The process of displacement of a nonwetting fluid has been studied experimentally on a transparent model of a porous medium for various percolation velocities in the stable front regime, when the viscosity of the displacing fluid is greater than that of the fluid displaced. The flow structures in the final displacement regime, when the nonwetting phase is distributed in the pore space in the form of individual drops or ganglia, have been visually investigated. Imbibition is numerically modeled on a two-dimensional network model with allowance for various microdisplacement mechanisms. The effect of the initial displacing phase saturation on the magnitude and structure of the residual displaced fluid saturation is demonstrated. The fractal dimensionality of the displacement boundary is measured.Novosibirsk. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.3, pp. 116–121, May–June, 1994.     

Motion of two pulsating spheres in an ideal incompressible fluid

The problem of the interaction of two pulsating spheres in an ideal incompressible fluid was first investigated in detail by Bjerknes [1]. However, his and subsequent studies on this subject [2–5] were restricted to the interaction forces between the spheres, whereas the law of their motion was not considered because of the much greater complexity of the corresponding problem. The aim of the present paper is to find an approximate analytic solution to the problem of the motion of two pulsating spheres in an ideal incompressible fluid filling the entire space exterior to the spheres under the assumption that the flow of the fluid is irrotational.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 159–162, May–June, 1983.     

Quasi-three-dimensional calculation of flow in blade passages of turbines

The flow in turbomachines is currently calculated either on the basis of a single successive solution of an axisymmetric problem (see, for example, [1-A]) and the problem of flow past cascades of blades in a layer of variable thickness [1, 5], or by solution of a quasi-three-dimensional problem [6–8], or on the basis of three-dimensional models of the motion [9–11]. In this paper, we derive equations of a three-dimensional model of the flow of an ideal incompressible fluid for an arbitrary curvilinear system of coordinates based on averaging the equations of motion in the Gromek–Lamb form in the azimuthal direction; the pulsation terms are taken into account in the equations of the quasi-three-dimensional motion. An algorithm for numerical solution of the problem is described. The results of calculations are given and compared with experimental data for flows in the blade passages of an axial pump and a rotating-blade turbine. The obtained results are analyzed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 69–76, March–April, 1991.I thank A. I. Kuzin and A. V. Gol'din for supplying the results of the experimental investigations.     

Experimental investigation of an elastic cylinder entering water at high velocity

With reference to the end-on penetration of a cylinder, an attempt is made to deal with the water entry problem as a whole and present a picture of the various processes involved and their interrelationships. The penetration process is considered from the moment of initial contact of the cylinder with the surface of the fluid to transition to the quasisteady regime.Based on a paper read at the Seventh Congress on Theoretical and Applied Mechanics, Moscow, August 1991. Presented by Yu. L. Yakimov.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.5, pp. 20–31, September–October, 1992.     

A case of jet flow of a gravity fluid

The problem of plane steady ideal heavy fluid flow bounded by an impermeable polygonal section, a curvilinear arc section, and a finite section of free surface is investigated in an exact nonlinear formulation. Hydrodynamic singularities may exist in the stream. A large class of captation problems of jet theory reduces to studying this kind of flow. The unique solvability of the problem under investigation is proved for sufficiently large Froude numbers and small arc curvature. A method of solution is given and an example is computed. Such problems have been solved earlier by numerical methods [1–3]. Some problems about jet flows of a gravity fluid with polygonal solid boundaries have been investigated by an analogous method in [4, 5].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 140–143, May–June, 1975.     

Nonisothermal displacement of oil by a solution of an active additive

In earlier work [1, 2] mathematical models have been constructed for processes of displacement of oil from a porous medium by a solution of an active additive, i.e., an additive capable of changing the hydro-dynamic characteristics of the fluid and the medium. An additive of this kind that was considered was a polymer that in the dissolved state influences the properties of the displacing fluid and in the adsorbed state the permeability of the porous medium. Self-similar solutions were obtained corresponding to the problem of frontal displacement from a homogeneous porous medium, and a number of numerical calculations were made. It is natural to generalize this treatment by introducing into the problem a second active factor, which is here taken to be the temperature of the injected fluid. The analysis of the nonisothermal displacement of oil by a solution of an active additive can be transferred without significant modifications to the general problem of displacement of oil by a solution carrying two active agents. The names additive and temperature are retained here only for convenience of exposition.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 90–107, November–December, 1980.We thank A. A. Barmin, A. G. Kulikovskii, and L. A. Chudov for helpful discussions.     

Estimate of the pressure distribution for the nonlinear verigin problem in the two-dimensional radial case

A study is made of the problem of two-dimensional radial displacement by gas of liquid from an unbounded uniform stratum. A comparison theorem is used to obtain lower and upper bounds for the pressure distribution. Calculations made using the obtained expressions showed that the difference between the upper and lower bounds is a fraction of a percent. Thus, in a calculation of the pressure a high degree of accuracy is achieved by using the lower bound for it.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 153–155, May–June, 1983.     

Impact of a figure of revolution in a stream of incompressible fluid with separation of a jet

The impact interaction of bodies with a fluid in a flow with jet separation has been considered in [1–3], for example. This investigation was in the two-dimensional formulation. The present paper considers the three-dimensional problem of impact of a figure of revolution in a stream of an ideal incompressible fluid with separation of a jet in accordance with Kirchhoff's scheme. A boundary-value problem is formulated for the impact flow potential and solved by the Green's function method. A method for constructing the Green's function is described. Expressions are given for the coefficients of the apparent masses. The results are given of computer calculations of these coefficients in the case of a cone using the flow geometry of the corresponding two-dimensional problem.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 176–180, November–December, 1980.     

An implicit method for frictional contact, impact and rolling

In this paper an implicit method for frictional contact, impact and rolling is suggested. A nonclassical formulation of a two-dimensional hyperelastic body unilaterally constrained to rigid supports is proposed by following the ideas of Moreau and Jean. A total Lagrangian formulation of the system is given. The elastic properties are defined by coupling the second Piola–Kirchhoff stress to the Green–Lagrange strain via the Kirchhoff–St. Venant law. The equation of motion is written in the spirit of Moreau by using the mean value impulses introduced by Jean. The mean value impulses appear explicitly in the equation of motion. In such manner the treatment of nonconstant kinematic transformation matrices becomes straightforward. The rigid supports are described by smooth functions. By utilizing these functions and the mean value impulses, new contact/impact laws of Signorini and Coulomb type are formulated. The governing equations are solved by a nonsmooth Newton method. This is performed by following the augmented Lagrangian approach and deriving the consistent stiffness matrix as well as the contact stiffness matrices. Three two-dimensional examples are solved by the method: a contact problem, an impact problem and a rolling contact problem.