Three-dimensional problems of the hydrodynamic interaction between bodies in a viscous fluid in the vicinity of their contact

The study is devoted to the hydrodynamic interaction between bodies near their contact. The stresses produced in a narrow gap between the bodies in the vicinity of their contact determine the main part of forces acting on the bodies in motion. 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 in the small spacing between the surfaces can be derived. An overview of the three-dimensional problems solved using this method is presented and for certain problems new formulations are given. The reliability of the results is confirmed by the comparison with available exact particular solutions and experimental data.     

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.     

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.     

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.     

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.     

Exact Analytical Solutions of Static Electroelastic and Thermoelectroelastic Problems for a Transversely Isotropic Body in Curvilinear Coordinate Systems

An approach to the solution of three-dimensional static problems for a transversely isotropic (rectilinear anisotropy) body is expounded and the solutions for piezoceramic canonical bodies are systematized. The result of the study is explicit analytical solutions of three-dimensional problems. Bodies are examined whose boundary surface is the coordinate surfaces in coordinate systems that permit the separation of the variables in the three-dimensional Laplace equation. The stress concentration in bodies near necks, cavities, inclusions, and cracks is investigated. The stress intensity factors of the force field and electric induction near elliptic and parabolic cracks are determined. The contact interaction of a piezoceramic half-space with elliptic and parabolic dies is studied. The bodies are under various mechanical, thermal, and electric loads     

桩侧扰动分区饱和土中桩竖向振动特性研究

考虑沉桩过程中桩周土受到扰动产生软化或硬化的现象,建立了饱和土中径向分区桩竖向振动简化模型,通过分离变量方法,得到了桩竖向振动频域解析解和时域半解析解。分析了分区土对桩竖向振动特性的影响,数值计算结果表明,土层模量比、分区半径和桩底支承系数对桩竖向振动有较明显的影响。研究结果还表明,在较强的振动作用下,由于分区软化或强...     

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].     

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.     

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     

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.     

Soft-impact dynamics of deformable bodies

Systems constituted by impacting beams and rods of non-negligible mass are often encountered in many applications of engineering practice. The impact between two rigid bodies is an intrinsically indeterminate problem due to the arbitrariness of the velocities after the instantaneous impact and implicates an infinite value of the contact force. The arbitrariness of after-impact velocities is solved by releasing the impenetrability condition as an internal constraint of the bodies and by allowing for elastic deformations at contact during an impact of finite duration. In this paper, the latter goal is achieved by interposing a concentrate spring between a beam and a rod at their contact point, simulating the deformability of impacting bodies at the interaction zones. A reliable and convenient method for determining impact forces is also presented. An example of engineering interest is carried out: a flexible beam that impacts on an axially deformable strut. The solution of motion under a harmonic excitation of the beam built-in base is found in terms of transverse and axial displacements of the beam and rod, respectively, by superimposition of a finite number of modal contributions. Numerical investigations are performed in order to examine the influence of the rigidity of the contact spring and of the ratio between the first natural frequencies of the beam and the rod, respectively, on the system response, namely impact velocity, maximum displacement, spring stretching and contact force. Impact velocity diagrams, nonlinear resonance curves and phase portraits are presented to determine regions of periodic motion with impacts and the appearance of chaotic solutions, and parameter ranges where the functionality of the non-structural element is at risk.     

Frictional contact of two rotating elastic disks

We study the problem of constrained uniform rotation of two precompressed elastic disks made of different materials with friction forces in the contact region taken into account. The exact solution of the problem is obtained by the Wiener-Hopf method.An important stage in the study of rolling of elastic bodies is the Hertz theory [1] of contact interaction of elastic bodies with smoothly varying curvature in the contact region under normal compression. Friction in the contact region is assumed to be negligible. If there are tangential forces and the friction in the contact region is taken into account, then the picture of contact interaction of elastic bodies changes significantly. Although the normal contact stress distribution strictly follows the Hertz theory for bodies with identical elastic properties and apparently slightly differs from the Hertz diagram for bodies made of different materials, the presence of tangential stresses results in the splitting of the contact region into the adhesion region and the slip region. This phenomenon was first established by Reynolds [2], who experimentally discovered slip regions near points of material entry in and exit from the contact region under constrained rolling of an aluminum cylinder on a rubber base. The theoretical justification of the partial slip phenomenon in the contact region, discovered by Reynolds [2], can be found in Carter [3] and Fromm [4]. Moreover, Fromm presents a complete solution of the problem of constrained uniform rotation of two identical disks. Apparently, Fromm was the first to consider the so-called “clamped” strain and postulated that slip is absent at the point at which the disk materials enter the contact region.Ishlinskii [5, 6] gave an engineering solution of the problem on slip in the contact region under rolling friction. Considering the problem on a rigid disk rolling on an elastic half-plane, we model this problem by an infinite set of elastic vertical rods using Winkler-Zimmermann type hypotheses. Numerous papers of other authors are surveyed in Johnson’s monograph [7].The exact solution of the problem on the constrained uniform rotation of precompressed rigid and elastic disks under the assumptions of Fromm’s theory is contained in the papers [8, 9]. In the present paper, we generalize the solution obtained in [8, 9] to the case of two elastic disks made of different materials.     

Fracture initiation in the contact region under shear

In the contact region between sliding elastic bodies, there are subregions where the interacting shores are bonded and subregions where they can slide along each other. It is convenient to interpret the latter as transverse shear cracks with slip resistance forces acting on their closed shores. In the end regions of such a crack, stress concentration may lead to fracture initiation in the contacting bodies. Experimental results and an analytic model of the phenomenon are given for a situation where the fracture intersects the contact plane tilted with respect to the direction of the loads.     

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.     

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.     

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.