Propagation of discontinuity waves in complex rheological media with viscous properties

The propagation of discontinuity waves of various order in rheological media is examined. It is assumed that the region of discontinuity of values can be represented by an intermediate layer of infinitesimal thickness. By means of this representation, results can be obtained for a rather wide class of continuous media with viscous properties, which generalize Duhem's results. The first integrals of the laws of momentum and energy conservation are obtained, which hold inside the intermediate layer at a shock wave.It is shown that when viscosity elements are introduced in a special way into the rheological model of a continuous medium, discontinuity waves of any order are propagated in the medium, and that at the surface of a strong discontinuity in a heat-conducting medium, the temperature is continuous. Additional conditions for strain discontinuities at the viscosity elements are obtained. For certain inclusions of the viscosity elements into the rheological model discontinuity waves do not propagate; instead there is merely a weak discontinuity surface which acts as an interface between the flow region of the continuous medium and the region in the state of rest. Contact discontinuities can occur in any continuous medium.The possible existence of a geometrical discontinuity surface in a viscous gas was examined first by Duhem [1]. He established that singluar strong-discontinuity surfaces cannot take place in a viscous gas. However, if one assumes that the velocity and temperature are continuous in the passage through a singular surface, only contact discontinuities are possible [2].     

Thermoelastic contact instability of a functionally graded layer and a homogeneous half-plane

The conductive heat transfer between two elastic bodies in the static contact can cause the system to be unstable due to the interaction between the thermoelastic distortion and pressure-dependent thermal contact resistance. This paper investigates the thermoelastic contact instability of a functionally graded material (FGM) layer and a homogeneous half-plane using the perturbation method. The FGM layer and half-plane are exposed to a uniform heat flux and are pressed together by a uniform pressure. The material properties of the FGM layer vary exponentially along the thickness direction. The characteristic equation governing the thermoelastic stability behavior is obtained to determine the stability boundary. The effects of the gradient index, layer thickness and material combination on the critical heat flux are discussed in detail through a parametric study. Results indicate that the thermoelastic stability behavior can be modified by adjusting the gradient index of the FGM layer.     

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.     

二维激波与剪切层相互作用的直接数值模拟研究

采用五阶weighed esseritially non-oscillatory (WENO) 格式和三阶total variation diminishing (TVD) Runge-Kutta 格式, 通过求解二维非定常Navier-Stokes 方程, 直接数值模拟了激波与剪切层相互作用, 目的在于揭示激波与剪切层相互作用过程中噪声产生的机理. 研究发现:(1) 当入射激波穿过剪切层时, 剪切层中心位置向下层区域偏移;(2) 入射激波穿过剪切层产生小激波, 在小激波与剪切层接触点处产生声波并向外辐射;(3) 反射激波穿过剪切层后形成了分段弧状激波;(4) 当反射激波穿过剪切层时, 激波在鞍点处泄漏并向外辐射声波, 这是一种激波泄漏机制.     

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.     

ANALYSIS OF SHIP MANEUVERING PERFORMANCES AND ICE LOADS ON SHIP HULL WITH DISCRETE ELEMENT MODEL IN BROKEN-ICE FIELDS

采用离散元模型对碎冰区浮冰与船舶结构的相互作用进行了数值研究。碎冰由三维圆盘单元构成,并考虑其在海流作用下的浮力、拖曳力和附加质量。船体结构由一系列三角形单元组合构造。通过海冰与船体单元间的接触判断和接触力计算,确定海冰与船体结构之间的相互作用。采用以上离散单元模型对不同冰况（冰速、冰厚、冰块尺寸和密集度）以及航速条件下,海冰对船体的动力作用过程进行了数值分析,对比分析了以上因素对船体冰载荷的影响,可为冰区船舶的安全运行和结构设计提供一定的借鉴作用。     

Particle image velocimetry measurements of a shock wave/turbulent boundary layer interaction

Particle image velocimetry is used to investigate the interaction between an incident shock wave and a turbulent boundary layer at Mach 2.1. A particle response assessment establishes the fidelity of the tracer particles. The undisturbed boundary layer is characterized in detail. The mean velocity field of the interaction shows the incident and reflected shock wave pattern, as well as the boundary layer distortion. Significant reversed flow is measured instantaneously, although, on average no reversed flow is observed. The interaction instantaneously exhibits a multi-layered structure, namely, a high-velocity outer region and a low-velocity inner region. Flow turbulence shows the highest intensity in the region beneath the impingement of the incident shock wave. The turbulent fluctuations are found to be highly anisotropic, with the streamwise component dominating. A distinct streamwise-oriented region of relatively large kinematic Reynolds shear stress magnitude appears within the lower half of the redeveloping boundary layer. Boundary layer recovery towards initial equilibrium conditions appears to be a gradual process.     

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.     

Contact of Transversely Isotropic Bodies in the Hertz Theory

Within the framework of the anisotropic theory of elasticity, a three-dimensional contact problem of interaction of two massive transversely isotropic bodies, whose dimensions substantially exceed the size of the contact region, is investigated. In this case, the isotropy planes of contacting elastic bodies are mutually perpendicular. Exact and numerical solutions of the problem are determined. Calculations for various transversely isotropic materials are carried out.     

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.     

Boundary element method for moving and rolling contact of 2D elastic bodies with defects

A scheme of boundary element method for moving contact of two-dimensional elastic bodies using conforming discretization is presented. Both the displacement and the traction boundary conditions are satisfied on the contacting region in the sense of discretization. An algorithm to deal with the moving of the contact boundary on a larger possible contact region is presented. The algorithm is generalized to rolling contact problem as well. Some numerical examples of moving and rolling contact of 2D elastic bodies with or without friction, including the bodies with a hole-type defect, are given to show the effectiveness and the accuracy of the presented schemes. The project supported by the National Natural Science Foundation of China (19772025)     

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.     

Finite-element simulation of contact interaction with the use of concepts of contact pseudomedium mechanics

In the present paper, we analyze an algorithm for solving problems of elastic and elastoplastic contact between rigid deformable bodies taking into account the presence of additional media with various properties in the contact region. These may be nonlinear properties of surface rough layers and films. To take into account the influence of such layers, the finite-element scheme uses contact finite elements with nonlinear properties. We describe an algorithm developed for solving problems with various types of nonlinearities. Using the example of experimental studies described in detail in [1, 2], we identify the nonlinear properties of contact finite elements. Several examples are used to demonstrate the algorithm operating capability.     

A frictional receding contact plane problem between a functionally graded layer and a homogeneous substrate

This paper investigates the plane problem of a frictional receding contact formed between an elastic functionally graded layer and a homogeneous half space, when they are pressed against each other. The graded layer is assumed to be an isotropic nonhomogeneous medium with an exponentially varying shear modulus and a constant Poisson’s ratio. A segment of the top surface of the graded layer is subject to both normal and tangential traction while rest of the surface is devoid of traction. The entire contact zone thus formed between the layer and the homogeneous medium can transmit both normal and tangential traction. It is assumed that the contact region is under sliding contact conditions with the Coulomb’s law used to relate the tangential traction to the normal component. Employing Fourier integral transforms and applying the necessary boundary conditions, the plane elasticity equations are reduced to a singular integral equation in which the unknowns are the contact pressure and the receding contact lengths. Ensuring mechanical equilibrium is an indispensable requirement warranted by the physics of the problem and therefore the global force and moment equilibrium conditions for the layer are supplemented to solve the problem. The Gauss–Chebyshev quadrature-collocation method is adopted to convert the singular integral equation to a set of overdetermined algebraic equations. This system is solved using a least squares method coupled with a novel iterative procedure to ensure that the force and moment equilibrium conditions are satisfied simultaneously. The main objective of this paper is to study the effect of friction coefficient and nonhomogeneity factor on the contact pressure distribution and the size of the contact region.     

An FE-MPEC approach for limit load evaluation in the presence of contact and displacement constraints

This paper describes a mathematical programing based approach for the direct limit load evaluation of a structural system under simultaneous contact and limited displacement conditions. The contact model we adopt can simulate either a classical unilateral (nonassociative) Coulomb friction situation or a cohesive fracture idealization at the potential discontinuity interface between contacting bodies. The discrete FE model is constructed using locking free mixed finite elements. The main feature and novelty of our proposed approach is to compute in a single step the maximum load capacity of the structure, such that both the imposed displacement limitations and the contact conditions are satisfied. In essence, the formulation is a nontrivial extension of classical limit analysis. The analysis is cast in its most natural form, namely in mixed static-kinematic variables, and leads to, what is known in the mathematical programing literature, as a mathematical program with equilibrium constraints or MPEC. A nonlinear programing (NLP) based algorithm is proposed to solve the MPEC. Two examples are provided to illustrate application of the proposed scheme, and some comments regarding the various advantages of the adopted mathematical programing framework are made.     

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.     

Finite element modeling of contact conditions in multibody system dynamics

In this paper a new method is developed for the dynamic analysis of contact conditions in flexible multibody systems undergoing a rolling type of motion. The relative motion between the two contacting bodies is treated as a constraint condition describing their kinematic and geometric relations. Equations of motion of the system are presented in a matrix form making use of Kane's equations and finite element method. The method developed has been implemented in a general purpose program called DARS and applied to the simulation and analysis of a rotating wheel on a track. Both the bodies are assumed flexible and discretized using a three dimensional 8-noded isoparametric elements. The time variant constraint conditions are imposed on the nodal points located at the peripheral surfaces of the bodies under consideration. The simulation is carried out under two different boundary conditions describing the support of the track. The subsequent constraint forces associated with the generalized coordinates of the system are computed and plotted. The effects of friction are also discussed.     

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.