Contact Problems For Bodies Research Articles

A contact model for the functionally graded coated elastic structures: Extension of the Hertz theory to the contact of beam structures

The Hertzian displacement assumption is widely used in analyzing the contact problems of non-uniform elastic bodies. It is essential to account for the support conditions of the elastic body as they significantly influence the contact stiffness and distribution of contact pressure. To address this, the deformation of beam structures is integrated into the Hertzian displacement assumption, which leads to the development of an extended contact mechanics model suitable for the elastic bodies of a beam structure which can be non-uniform and with functionally graded coatings. The problem is solved by using a numerical method based on the Gauss–Chebyshev quadrature for the singular integral equation of Cauchy type. In the contact problems of the doubly simply supported (SS-SS) and cantilever beams, the contact pressure and contact stiffness in conjunction with the interactions between the indentation and contact bodies are discussed. An in-depth study on the coupling effects between the structural deformation and functionally graded coatings is presented.

Explicit frictional stick–slip dynamics of elastic contact problem incorporating the LuGre model

Dynamical contact problem with friction is of continuous concern and have been investigated over the years. The effective and accurate description of the development of the friction force is the key to this problem. However, the stick–slip phenomenon, as typical dynamical behaviour, has rarely been considered when developing contact algorithms for frictional contact problems of elastic bodies. The conventionally used Coulomb’s model, with differentiating the static and kinetic coefficient of friction, can lead to inaccurate results due to errors generated when switch between the stick or slip states. In this work, a three-dimensional mortar-based frictional contact algorithm is proposed in an explicit scheme. The proposed algorithm incorporates the LuGre model to account for dynamical frictional behaviours such as stick–slip. The effectiveness and accuracy of the proposed algorithm are validated through several simplified two-dimensional cases. The sticking time ratio and the evolution of the global coefficient of friction with time of a three-dimensional contact problem with a randomly rough contact interface are investigated as an application.

Approximate contact solutions for non-axisymmetric homogeneous and power-law graded elastic bodies: A practical tool for design engineers and tribologists

In two recent papers, approximate solutions for compact non-axisymmetric contact problems of homogeneous and power-law graded elastic bodies have been suggested, which provide explicit analytical relations for the force–approach relation, the size and the shape of the contact area, as well as for the pressure distribution therein. These solutions were derived for profiles, which only slightly deviate from the axisymmetric shape. In the present paper, they undergo an extensive testing and validation by comparison of solutions with a great variety of profile shapes with numerical solutions obtained by the fast Fourier transform (FFT)-assisted boundary element method (BEM). Examples are given with quite significant deviations from axial symmetry and show surprisingly good agreement with numerical solutions.

Variable boundary contact problem between pulley and flexible rope

The constraint method (analyzing the constraints between the contacts) has a better computational efficiency than the force method (analyzing the forces between the contacts) in contact dynamics. According to the constraint method, co-position and tangential conditions are proposed and applied to both rigid and flexible body contact problems. A pure rolling problem on the rigid elliptical disk is studied. And a typical slow-release mechanism, including winch, pulley and rope, is also analyzed by decomposing the contact-induced system into several continuous components. The boundary points among the parts can be quickly obtained by the constraint method, and the Kinematic continuity conditions on contact boundary point are deduced.

Optimal location of a finite set of rigid inclusions in contact problems for inhomogeneous two-dimensional bodies

The 2D-model of an elastic body with a finite set of rigid inclusions is considered. We assume that the body can come in frictionless contact on a part of its boundary with a rigid obstacle. On the remaining part of the body’s boundary a homogeneous Dirichlet boundary condition is imposed. For a family of corresponding variational problems, we analyze the dependence of their solutions on locations of the rigid inclusions. Continuous dependency of the solutions on location parameters is established. The existence of a solution of the optimal control problem is proven. For this problem, a cost functional is defined by an arbitrary continuous functional on the solution space, while the control is given by location parameters of the rigid inclusions.

Contact between rough rock surfaces using a dual mortar method

The mechanical behavior of fractures in rocks has strong implications for reservoir engineering applications. Deformations, and the corresponding change in contact area and aperture field, impact rock fracture stiffness and permeability, thus altering the reservoir properties significantly. Simulating contact between fractures is numerically difficult as the non-penetration constraints lead to a nonlinear problem and the surface meshes of the solid bodies on the opposing fracture sides may be non-matching. Furthermore, due to the complex geometry, the non-penetration constraints must be updated throughout the solution procedure. Here we present a novel implementation of a dual mortar method for contact. It uses a non-smooth sequential quadratic programming method as solver, and is suitable for parallel computing. We apply it to a two body contact problem consisting of realistic rock fracture geometries from the Grimsel underground laboratory in Switzerland. The contributions of this article are: (1) a novel, parallel implementation of a dual mortar method with a non-smooth sequential quadratic programming method, (2) realistic rock geometries with rough surfaces, and (3) numerical examples, which prove that the dual mortar method is capable of replicating the nonlinear closure behavior of fractures observed in laboratory experiments.

Contact problems for bodies with complex coatings

The paper deals with various statements and mathematical models of contact and contact‐wear problems for bodies with coatings. It is shown that the mathematical models for a number of such problems can be represented as a mixed integral equation or a system of mixed integral equations with additional conditions. It is also shown that these equations contain rapidly varying or even discontinuous functions in the case of interaction between bodies of complex shape and with some surface properties. Therefore, it is necessary to use a special approach for constructing efficient analytic solutions. Its implementation is demonstrated by an example.

Solution of the 3-D elastodynamic contact problem for body with cracks using the BIEM and constrained optimization algorithm

Mathematical formulation and numerical solution of the 3-D elastodynamic contact problem for body with cracks have been considered here. Diffraction of the normally incident harmonic tension-compression P-waves and shear deformation SH- waves on the penny sharped crack in the 3-D space is studied with considering contact interaction of the crack edges. Boundary element method (BEM) along with constrained optimization algorithm which is based on variational principles and nonsmooth analysis has been used for the considered problem solution. Influence of the crack edges contact interaction on the stress intensity factor (SIF) have been studied.

Multiple contact problem for bodies with nonuniform coatings of variable thickness and regular systems of punches with complex base shape

Plane problem of multiple contact interaction for a viscoelastic foundation with nonuniform coating of variable thickness and regular system of rigid punches with complex base shape is considered. We obtain basic system of integral equations and additional conditions for this problem. The analytic solutions for one version of the problem is presented.

Decomposition method for dynamic contact problems of several deformable bodies

The iterative method of Schwarz domain decomposition (DDS) was used as a dynamic contact algorithm for two and more contacting 3D bodies allowing the consideration of the solution of the stress-strain state problem in the standard statement for each of the contacting body and the high-level parallelization of the solution process. According to the offered iteration algorithm, two steps are performed in each iteration, where the contact conditions for the displacement and stresses are fulfilled in turn for the nodes on the contact surface. The specific feature of the realization of the dynamic contact problem was the combination of the iteration procedures for the refinement of the contact problem solution according to the Newton-Raphson algorithm with the iterations of the DDS method. As the test calculation results have shown, the use of the Schwarz contact algorithm and the HHT-α scheme in combination with the mass redistribution method on the contact boundary is a reliable, stable and sufficiently accurate way for the numerical integration of the contact problems at impact interaction. Finally, accuracy of the proposed method is verified by a conservation of momentum through three contact examples.

Contact interaction of machine elements with non–linear elastic intermediate layer

The work presents a variational formulation of a contact problem for elastic bodies with an intermediate nonlinear contact layer. The layer is introduced in order to model compliant gaskets, coatings and roughness between or on the surface of the machine parts. The contact conditions in this case incorporate not only the elastic displacements but the local deformations attributed to the contact layer. This additional deformation is determined at every point of the contact surface as a function of the acting normal tractions. This relation is in general nonlinear and can be accounted for by a special term in the complementary energy updating the original Kalker’s variational principle. As an outcome the problem acquires besides the structural nonlinearity (i.e. contact) as well the physical nonlinearity. This means that one needs to solve a system comprising both the inequalities and nonlinear equations. This is done by computing consecutive approximations from problem linearizations. The two particular variants of this procedure comprise the method of augmented gap and the method of variable compliance. Newton-Raphson iterations are also considered as an option. With this model and the numerical analysis method at hand one can solve certain inverse problems. The goal is to justify the the geometrical shape of the contacting bodies as well as the physical parameters of the contact layers. The objective criteria enforcing strength and durability in terms of the contact loads are proposed for the optimization. Furthermore deformation components arising from the applied external loads and the corresponding correction of the contact geometry are introduced as an additional design factor.

CONTACT PROBLEM FOR A TWO-LAYERED CYLINDER

Introduction. The investigation of the contact problems for cylindrical bodies is urgent due to the engineering contact strength analysis on shafts, cores and pipe-lines. In the present paper, a new contact problem of elastostatics on the interaction between a rigid band and an infinite two-layered cylinder, which consists of an internal continuous cylinder and an outer hollow one, with a frictionless contact between the cylinders, is studied. The outer cylindrical band of finite length is press fitted. By using a Fourier integral transformation, the problem is reduced to an integral equation with respect to the unknown contact pressure.Materials and Methods. Different combinations of linearly elastic materials of the composite cylinder are considered. Asymptotics of the symbol function of the integral equation kernel at zero and infinity is analyzed. This plays an important role for the application of the analytical solution methods. A key dimensionless geometric parameter is introduced, and a singular asymptotic technique is employed to solve the integral equation.Research Results. On the basis of the symbol function properties, a special easily factorable approximation being applicable in a wide variation range of the problem parameters is suggested. The Monte-Carlo method is used to determine the approximation parameters. The asymptotic formulas are derived both for the contact pressure, and for its integral characteristic. Calculations are made for different materials and for various relative thickness of the cylindrical layer including thin-walled layers.Discussion and Conclusions. The asymptotic solutions are effective for relatively wide bands when the contact zone length is bigger than the diameter of the composite cylinder. It is significant that the method is applicable also for those cases when the outer cylindrical layer is treated as a cylindrical shell. The asymptotic solutions can be recommended to engineers for the contact strength analysis of the elastic barrels with a flexible coating of another material.

METHOD OF DIMENSIONALITY REDUCTION IN CONTACT MECHANICS AND FRICTION: A USER’S HANDBOOK. III. VISCOELASTIC CONTACTS

Until recently the analysis of contacts in tribological systems usually required the solution of complicated boundary value problems of three-dimensional elasticity and was thus mathematically and numerically costly. With the development of the so-called Method of Dimensionality Reduction (MDR) large groups of contact problems have been, by sets of specific rules, exactly led back to the elementary systems whose study requires only simple algebraic operations and elementary calculus. The mapping rules for axisymmetric contact problems of elastic bodies have been presented and illustrated in the previously published parts of The User's Manual, I and II, in Facta Universitatis series Mechanical Engineering [5, 9]. The present paper is dedicated to axisymmetric contacts of viscoelastic materials. All the mapping rules of the method are given and illustrated by examples.

Modeling of normal contact of elastic bodies with surface relief taken into account

An approach to account the surface relief in normal contact problems for rough bodies on the basis of an additional displacement function for asperities is considered. The method and analytic expressions for calculating the additional displacement function for one-scale and two-scale wavy relief are presented. The influence of the microrelief geometric parameters, including the number of scales and asperities density, on additional displacements of the rough layer is analyzed.

Contact Problem for a Viscoelastic Half-Space and a Rough Rigid Cylinder under the Conditions of Hydrodynamic Lubrication

A model has been proposed for use in studying the combined effect of the roughness of a rigid punch and the viscous properties of a base separated by a thin lubricant layer exerted on the characteristics of contact interactions and the sliding friction force. The problem of the motion of a thin a lubricant layer between a fixed rigid cylinder with a regular relief, as well as the surface of a moving viscoelastic half-space, the rheological properties of which are described by an integral operator with an exponential creep kernel, has been considered. The pressure and thickness of the lubricating layer, as well as the deformation component of the frictional force depending on the sliding velocity, have been analyzed. A comparison of the results of solutions of contact problems for viscoelastic and elastic rough bodies in the presence of a lubricant has been presented.

Modeling Solutions of Axisymmetric Contact Problems Using Dimensionality Reduction Method with or without Adhesion

Axisymmetric contact problems of rigid bodies with elastomeric semi three-dimensional environment have been studied. It has been shown that three-dimensional contact problems with or without adhesion can be mapped with the dimensionality reduction method as one-dimensional problems with minimum error. The efficiency of the method has been illustrated by numerical calculations of practical problems. This also confirms the correctness of the analytical estimators.

Sliding and Crossing Dynamics in Extended Filippov Systems

In spatial contact problems of rigid bodies, Coulomb friction results in dynamical systems with codimension-2 discontinuity manifolds, which are outside the scope of piecewise smooth dynamical syst...

Domain decomposition algorithms for problems of thermomechanical contact between several elastic bodies

We consider a thermoelastic multibody contact problem for finite bodies with unilateral mechanical and imperfect thermal contact conditions. Using a penalty method, we obtain a weak formulation of this problem in the form of a system of linear and nonlinear variational equations in Hilbert space. To solve this variational system, we propose a class of iterative Robin type domain decomposition algorithms. In each iterative step of these algorithms one have to solve two linear variational equations for each of the bodies, which correspond to heat conduction problem with Newton boundary conditions on the possible contact areas and linear elasticity problem with additional volume forces and Robin boundary conditions respectively. The program implementation of proposed algorithms is made for plane thermoelastic contact problems with the use of linear and quadratic finite element approximations on triangles. The numerical analysis is performed for one-body and two-body thermoelastic contact problems.

Numerical solution of smooth and rough contact problems

A general procedure for the numerical solution of three-dimensional normal and tangential contact problems for non-conforming bodies of arbitrary shape is presented. The proposed procedure is based on the interpolation of known analytical solutions relating surface displacements of an elastic half-space subjected to pyramidal distributions of normal and tangential surface tractions. These formulas are used to interpolate unknown pressure distributions on the contact zone of two elastic bodies in contact. The proposed interpolation significantly reduces the computational burden of the numerical procedure required to interpolate the actual pressure distribution and to determine the initially unknown contact region. It amounts to expressing the contact pressure as a function of the elastic parameters of the two bodies, the distance between their surfaces and the relative displacement between two far-points pertaining to the bodies in contact. The procedure has been validated by comparison with classical contact problems and the results show excellent agreement with existing analytical and numerical solutions.

On the control of time discretized dynamic contact problems

We consider optimal control problems with distributed control that involve a time-stepping formulation of dynamic one body contact problems as constraints. We link the continuous and the time-stepping formulation by a nonconforming finite element discretization and derive existence of optimal solutions and strong stationarity conditions. We use this information for a steepest descent type optimization scheme based on the resulting adjoint scheme and implement its numerical application.