Hydrodynamics of the interaction between bodies in a viscous fluid in the vicinity of their contact at low and high Reynolds numbers

An interaction between bodies in the vicinity of their contact is investigated. Stresses determining the main part of the forces acting on the bodies in motion are produced in a narrow gap between the bodies in the vicinity of their contact. In many cases the velocity and pressure fields in the vicinity of the contact can be determined and the main asymptotics for the hydrodynamic interaction force can be constructed in the small distance between the surfaces. An overview of the problems solved using this approach is presented. For certain problems new formulations are given. The plausibility of the results is confirmed by comparing with the available exact particular solutions and the experimental data. Owing to the restrictions on the size of the paper, only two-dimensional problems are considered, although the approach developed can be applied to the solution of three-dimensional problems as well.     

Central deflection of a transversely struck beam

Experimental investigations were performed in order to verify the maximum impact force and the time function of the central deflection which were derived by a simplified theory. This theory is only applicable in the short time interval after contact when the stresses in the vicinity of the contact point are only a function of the interaction between the colliding bodies. It will be assumed that the reflected elastic waves have no influence on these stresses. The maximum impact force can be expected to occur in this time interval.     

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.     

Stick-slip analysis of a circular point contact between a rigid sphere and a flat unidirectional composite with cylindrical fibers

The stick-slip contact problem is investigated here when at least one of the contacting bodies behaves as an ideal composite material with long fibers perpendicular to the direction of movement. Cylindrical inhomogeneous inclusions within a homogeneous media and with axes parallel to the contact surface are considered. The Eshelby’s equivalent inclusion method is used to solve the problem numerically. Interactions between close inclusions are taken into account in the numerical procedure, as well as the coupling between the normal and tangential contact problems. It is found that the presence of heterogeneities in the vicinity of the surface contact affects significantly the contact pressure distribution and subsequently the distribution of shear and slip at the interface.     

Modeling of motion mechanism in the intermediate layer between contacting bodies in compression shear

The conditions of force interaction and friction on the contact between bodies are related to the intermediate layer structure in the contact region, i.e., to the appearance of fracture products or intensive deformation in this region. The subsequent interaction between the bodies occurs through elements of the intermediate layer structure. In the present paper, we determine conditions and the basic mechanism controlling the formation of the interface structure when the interaction between the bodies is implemented through structure elements of the intermediate layer (balls) which are capable of rolling.     

介观双粗糙弹塑性流体动力润滑界面法向接触刚度模型

弹性流体动力润滑状态通常出现在机械高副零部件的点/线接触部位,如齿轮、轴承和蜗轮蜗杆等. 宏观上点/线接触在介观层面表现为两粗糙表面的接触,在微观层面上则又表现为微凸体间的接触. 由于在中/重载荷作用下,粗糙表面上的微凸体发生接触后会产生弹塑性/塑性变形,从而使得两粗糙表面的弹流润滑接触转变为弹塑性流体动力润滑接触. 此外,界面的接触刚度决定了机械装备的整机刚度. 为了精确获得弹性流体动力润滑状态下界面法向接触刚度及其主要影响因素,基于界面的法向接触刚度由固体接触刚度和润滑油膜刚度两部分构成的思想,根据固体弹塑性理论和流体动力学理论,分别对界面间微凸体侧接触及部分膜流体动力润滑进行分析,从微观入手揭示双粗糙表面弹塑性流体动力润滑接触机理,进而建立考虑微凸体侧接触弹塑性变形的流体动力润滑界面法向接触刚度模型. 通过仿真分析,揭示了法向载荷、卷吸速度、表面粗糙度及润滑介质特性等因素对润滑界面法向接触刚度的影响规律. 研究表明:在相同速度、粗糙度及润滑油黏度的工况下,固体接触刚度和油膜接触刚度均随着法向接触载荷的增加呈非线性增大;在相同载荷、速度及润滑油黏度的工况下,接触表面粗糙度越大,表面形貌对于润滑状态的影响较强,固体接触刚度占界面总刚度的主要部分,界面主要由固体承载;在相同载荷、粗糙度及润滑油黏度工况下,随着卷吸速度的增大,固体接触刚度逐渐减小,油膜刚度占界面总刚度的主要部分;在相同载荷、粗糙度及速度工况下,随着润滑油黏度的增大,油膜刚度基本保持不变,固体接触刚度基本不受润滑油黏度的影响. 通过理论建模准确获得单位面积弹塑性流体动力润滑结合面法向接触刚度,对改善机械装备动态性能、提高机械装备的可靠性具有重要的理论和实际意义.      

Diffraction of weak shock waves by deformed bodies submerged in a liquid

Conclusion On the basis of an analysis of theoretical and experimental data obtained up to now by various investigators, we can note the following major advances in the field of the interaction of shock waves with barriers submerged in a liquid:Exact solutions have been obtained for problems in the diffraction of acoustic shock waves by rigid and stationary bodies of specified shape (plates, wedges, cones, parabolic, elliptical, and circular cylinders, spheres, paraboloids of revolution); approximate schemes have been worked out for estimating hydrodynamic loads, making it possible to investigate various stages of the interaction of shock waves with elastic shells of revolution and solid bodies; studies have been conducted in the exact formulation of the interaction of plane (spherical) nonstationary waves with elastic barriers (unbounded plates, plates in a screen, infinitely long thin-walled and thick-walled cylindrical shells, closed thin-walled and thick-walled spherical shells); an exact solution has been found for the internal problems in the case of cavities (circular and elliptical cylinders, spheres, spheroids) and elastic shells of revolution (infinitely long cylindrical and closed spherical shells); methods have been worked out for the approximate determination of the parameters of objects (elastic thin-walled infinitely long cylindrical and closed spherical shells) from reflected echo signals; estimates have been given for the influence of the structural characteristics of an object (support, concentrated masses), the nonlinear properties of interacting media, cavitation in liquid, and plastic deformations in the barrier material on the process of hydrodynamic interaction.We should also mention the main lines of further investigation and the problems which require solution: designing new experimental apparatus and measuring complexes for studying the nonstationary behavior of deformed bodies and structures in a liquid; solution of problems in diffraction by oonical and cylindrical shells of finite length, and by compound structures of complicated form in which account is taken of the structural characteristics and the internal elements; calculation of three-layer and multilayer shells acted upon by shock waves, taking account of the transverse compression of the filler; construction of more exact schemes (models) for the nonlinear and cavitation-type interaction of waves with barriers; development of numerical and combined methods for the solution of the problems in hydroelasticity.Mechanics Institute, Moscow State University. Translated from Prikladnaya Mekhanika, Vol. 16, No. 5, pp. 3–11, May, 1980.     

接触-碰撞算法研究进展

接触-碰撞广泛存在于实际工程问题中,是影响数值计算效率与计算精度的重要因素。本文针对变形体间接触-碰撞问题的显式有限元计算,介绍接触-碰撞算法近30年来取得的主要进展。首先,简要介绍接触-碰撞问题的界面离散模型;然后,从全局接触搜索、局部搜索、接触约束施加以及接触计算的并行化四方面详细阐述目前主要算法的基本思想与特点,并分析其优势与不足;最后,对接触-碰撞算法相关研究方向给出建议。     

Contact problems for a circular plate with sliding fixation at the end face

Two mixed elasticity problems of punch indentation into a circular plate placed without clearance in a rigid cylindrical holder with smooth walls are considered. In the first problem, the plate lies without friction on a rigid base, and in the second problem, the plate is rigidly fixed to the base. The problems are solved by a method that was developed for bodies of finite dimensions and is based on the properties of closed systems of orthogonal functions. Each of the problems is reduced to two integral equations, namely, a Volterra integral equation of the first kind for the contact pressure function and a Fredholm integral equation of the first kind for the derivatives of the displacement of the plate upper surface outside the punch. The displacement function is sought as the sum of a trigonometric series and a power function with a root singularity. After truncation, the obtained illposed system of linear algebraic equation has a stable solution. A method for solving Volterra integral equations is given. The contact pressure distribution function and the dimensionless indentation force are determined. Examples of calculation of the plate interaction with the plane punch are given. Contact problems were earlier studied for a rectangle and a circular plate with a stress-free end both without taking account of their fixation [1, 2] and with regard for their fixation [3, 4]. The solution method described here was used to study the interaction of elastic hollow cylinder of finite length with a rigid bandage and a rigid insert [5, 6]. Other papers dealing with contact problems for bodies of finite dimensions, in particular, for a circular plate, should also be mentioned. In these papers, the problems under study were solved by the method of homogeneous solutions [7, 8] and by the method of coupled series-equations [9].     

Modeling of friction at different scale levels

We construct a model for studying the common influence of the imperfect elasticity of actual bodies, the microgeometry of their surfaces, and their adhesive interaction on the contact characteristics (the contact pressure distribution, the region of actual contact) and on the sliding friction force. The model is based on the solution of a plane contact problem of sliding of a rigid body with a regular relief on the boundary of a viscoelastic foundation with surface molecular attraction in the gap between the surfaces taken into account. We analyze the influence of the surface microgeometry parameters at different scale levels on the character of the surface interaction (the saturated or discrete contact) and the friction force for different sliding velocities of the contacting bodies.     

Motion of a circular cylinder in a rectangular channel

When bodies move in fluids, the parameters of their motion depend strongly on the interaction of the bodies with the surrounding fluid [1, 2]. The present paper is devoted to determination of the hydrodynamic forces that act on a cylinder moving in an infinite rectangular channel in an ideal incompressible fluid that is at rest.     

Contact Simulation for Many Particles Considering Adhesion

Abstract

This article deals with the calculation and administration of contact between many moving bodies in the planar case. The main issue is the contact determination between bodies of either round or polygonal shape. The modeling is done by means of molecular dynamics to investigate the motion of many bodies very efficiently. Besides elastic normal forces, which prevent penetration of the bodies, damping forces and adhesion is described and considered for the computations. Our interest is directed toward collision detection. By means of sorting algorithms, neighboring bodies are found easily and, therefore, systems consisting of a large number of bodies can be determined efficiently.     

Semianalytic Finite-Element Method in Problems of Nonlinear Continuum Mechanics

A technique for solving problems of nonlinear continuum mechanics associated with contact interaction, plastic distortion, and continuous and discrete fracture of spatial bodies is developed based on the semianalytic finite-element method generalized to noncanonical bodies. Solutions are obtained to new applied problems in various branches of technology. Compared with the traditional FEM, the technique is highly efficient—the amount of computation needed to solve spatial problems reduces by several orders of magnitude     

Separation of Contacting Surfaces in Thermoelastic Interaction of Two Cylinders with Time-Dependent Heat Release due to Friction

The problem of a thermoelastic interaction of two cylinders with separation of their contact surfaces due to local loading of lateral surfaces is formulated and solved. The effect of the multiply connected contact region is shown to exist under a certain relation between the linear thermal expansion coefficients of the bodies.     

Singularities of the boundary layer equations and the structure of the flow in the vicinity of the convergence plane on conical bodies

The singularities of the boundary layer equations and the laminar viscous gas flow structure in the vicinity of the convergence plane on sharp conical bodies at incidence are analyzed. In the outer part of the boundary layer the singularities are obtained in explicit form. It is shown that in the vicinity of a singularity a boundary domain, in which the flow is governed by the shortened Navier-Stokes equations, is formed; their regular solutions are obtained. The viscous-inviscid interaction effect predominates in a region whose extent is of the order of the square root of the boundary layer thickness, in which the flow is described by a two-layer model, namely, the Euler equations in the slender-body approximation for the outer region and the three-dimensional boundary layer equations; the pressure is determined from the interaction conditions. On the basis of an analysis of the solutions for the outer part of the boundary layer it is shown that interaction leads to attenuation of the singularities and the dependence of the nature of the flow on the longitudinal coordinate, but does not make it possible to eliminate the singularities completely.     

Contact interaction of cylindrical bodies with allowance for the surface roughness factors

The problem of the elastic contact interaction between a rough disk and a rough plane with a round cut-out is solved with allowance for the microgeometry of their surfaces. This makes it possible to clarify the effect of the main parameters of the problem, on the stress state in the examined joint. A comparison of results from an analysis of the stress state in the contact region for various combinations of the elastic characteristics of interacting smooth bodies and results of well-known studies confirms the high effectiveness of the approach proposed. Belarussian State Polytechnical Academy, Minsk 220027. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 6, pp. 139–143, November–December, 1999.     

OVERLAPPING GRIDS FOR THE SIMULATION OF PARTICLE INTERACTION AT THE MICRO SCALE

This paper describes the use of overlapping grids for the calculation of flow around single and multipleparticle configurations at the micro scale. The basic equations for calculation are those for conservation of mass and momentum which are solved using a common Finite-Volume formulation. The hydrodynamic particle-particle and particle-wall interaction can be calculated by using an overlapping or Chimera grid scheme. With the grid structuring procedure it is possible to use simple and structured grids around the particles and the overall main grid geometry. The particle grids are lapped over the main grid such that they can move independently after each time step without remeshing the whole geometry. The paper gives results for the validation of the code developed for general test cases, for a rotating ellipsoid in simple shear flow, the flow around particles attached to a wall, the motion of a particle in the vicinity of a wall and some results for the flow through a packed bed configuration.     

Contact analysis in presence of spherical inhomogeneities within a half-space

This paper presents a fast method of solving contact problems when one of the mating bodies contains multiple heterogeneous inclusions, and numerical results are presented for soft or stiff inhomogeneities. The emphasis is put on the effects of spherical inclusions on the contact pressure distribution and subsurface stress field in an elastic half-space. The computing time and allocated memory are kept small, compared to the finite element method, by the use of analytical solution to account for the presence of inhomogeneities. Eshelby’s equivalent inclusion method is considered in the contact solver. An iterative process is implemented to determine the displacements and stress fields caused by the eigenstrains of all spherical inclusions. The proposed method can be seen as an enrichment technique for which the effect of heterogeneous inclusions is superimposed on the homogeneous solution in the contact algorithm. 3D and 2D Fast Fourier Transforms are utilized to improve the computational efficiency. Configurations such as stringer and cluster of spherical inclusions are analyzed. The effects of Young’s modulus, Poisson’s ratio, size and location of the inhomogeneities are also investigated. Numerical results show that the presence of inclusions in the vicinity of the contact surface could significantly changes the contact pressure distribution. From a numerical point of view the role of Poisson’s ratio is found very important. One of the findings is that a relatively ‘soft’ and nearly incompressible inclusion – for example a cavity filled with a liquid – can be more detrimental for the stress state within the matrix than a very hard inclusion with a classical Poisson’s ratio of 0.3.     

Study of the dynamic contact interaction of deformable bodies

A new algorithm for solving dynamic contact problems involving deformable bodies is proposed. The algorithm is based on formulation of the boundary conditions for the contact interaction with allowance for Coulomb friction in the form of quasivariational inequalities. The algorithm is numerically stable and satisfies geometric constraints in the a priori unknown contact region and conditions specifying that the normal pressure be nonnegative and that the vectors describing tangential velocity and shear stress during slip be oppositely directed. Results are presented from calculations performed for a contact problem for an elastoplastic body in a two-dimensional formulation. Computer Center, Siberian Division, Russian Academy of Sciences, Krasnoyarsk 660036. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 4, pp. 167–173, July–August, 1998.