Normal Contact Problem Research Articles

Loaded tooth contact analysis for helical gears with surface waviness error

Meshing characteristics are crucial for the transmission performance of vibration noise and strength of gear systems. Gear transmission error has been acknowledged as a prominent excitation in vibration systems, but the surface waviness deviations that directly affect it in contact analysis are generally ignored. This study proposes a comprehensive model for quasi-static analysis of tooth surfaces with waviness error. The model reconstructs the actual tooth surface in reverse based on the gear Fourier measurement principle. Instead of solving the meshing equations to pre-assume contact position, the problem of normal contact between surfaces with deviations is solved by an innovative multiscale grid and a local iterative numerical algorithm. Combining the finite element numerical method and surface integral of the Boussinesq solution determines gear deformation and develops a loaded contact model. The validity of the model is verified through numerical examples. Furthermore, the proposed model is applied to discuss the tooth surfaces with different waviness amplitudes, frequencies, and distribution angles for the first time. It explains the interaction between local microscale and global scale introduced by waviness error on tooth surfaces and reveals the mechanism of the waviness error on meshing performance.

SELF-CONSISTENCY CONDITIONS IN STATIC THREE-BODY ELASTIC TANGENTIAL CONTACT

The contact problem for an elastic third-body particle between two elastic half-spaces is considered. The contact is assumed to consist of three Hertzian contact spots. The normal and tangential contact problems are analyzed analytically considering partial slip in the contacts and the influence of third-body weight. Self-consistency conditions between global equilibrium and the contact solution are formulated to give criteria, under which circumstances static slip and stationary sliding are possible states for the third-body particle. The sliding case is solved in detail.

Stochastic process model for interfacial gap of purely normal elastic rough surface contact

In purely normal elastic rough surface contact problems, Persson’s theory of contact shows that the evolution of the probability density function (PDF) of contact pressure with the magnification is governed by a diffusion equation. However, there is no partial differential equation describing the evolution of the PDF of the interfacial gap. In this study, we derive a convection–diffusion equation in terms of the PDF of the interfacial gap based on stochastic process theory, as well as the initial and boundary conditions. A finite difference method is developed to numerically solve the partial differential equation. The predicted PDF of the interfacial gap agrees well with that by Green’s Function Molecular Dynamics (GFMD) and other variants of Persson’s theory of contact at high load ranges. At low load ranges, the obvious deviation between the present work and GFMD is attributed to the overestimated mean interfacial gap and oversimplified magnification-dependent diffusion coefficient used in the present model. As one of its direct application, we show that the present work can effectively solve the adhesive contact problem under the DMT limit. The current study provides an alternative methodology for determining the PDF of the interfacial gap and a unified framework for solving the complementary problem of random contact pressure and random interfacial gap based on stochastic process theory.

Some remarks on the two-dimensional contact of dissimilar elastic solids under non-slipping boundary conditions

Spence's self-similarity approach is often preferred to an incremental method for solving plane, non-slipping normal contacts of dissimilar elastic solids under progressive loading. The number of plane contact problems solved in this way so far is nevertheless very limited, since explicit expressions especially for the pressure distribution and the tangential tractions within the contact area require the solution of complicated singular integral equations. In contrast, the application of the incremental method leads to much simpler integral equations, which are derived here and applied to solve some new non-slipping contact problems including the one characterized by an initial power-law gap profile function. The incremental method is also capable to determine the so-called interface mismatch eigenstrain. In Spence's method, however, the determination of its magnitude represents a mandatory step within the solution procedure. By means of a comprehensive comparison of both methods this work provides new insights into the solution of non-slipping plane normal contact problems including explicit analytical solutions for the interface mismatch eigenstrain.

A General Approximate Solution for the Slightly Non-Axisymmetric Normal Contact Problem of Layered and Graded Elastic Materials

We present a general approximate analytical solution for the normal contact of layered and functionally graded elastic materials for almost axisymmetric contact profiles. The solution only requires knowledge of the corresponding contact solution for indentation using a rigid cylindrical flat punch. It is based on the generalizations of Barber’s maximum normal force principle and Fabrikant’s approximation for the pressure distribution under an arbitrary flat punch in an inhomogeneous case. Executing an asymptotic procedure suggested recently for almost axisymmetric contacts of homogeneous elastic media results in a simple approximate solution to the inhomogeneous problem. The contact of elliptical paraboloids and indentation using a rigid pyramid with a square planform are considered in detail. For these problems, we compare our results to rigorous numerical solutions for a general (bonded or unbonded) single elastic layer based on the boundary element method. All comparisons show the quality and applicability of the suggested approximate solution. Based on our results, any compact axisymmetric or almost axisymmetric contact problem of layered or functionally graded elastic materials can be reduced asymptotically to the problem of indenting the material using a rigid cylindrical flat punch. The procedure can be used for different problems in tribology, e.g., within the framework of indentation testing or as a tool for the analysis of local features on a rough surface.

A Review of Wheel-Rail Contact Mechanics for Railway Vehicles

The dynamic behaviour of a railway vehicle is mainly depending on how the interaction of wheel and rail is taking place in running operation and what is the accuracy and efficiency of the applied contact models. Contact mechanics of railway vehicle is keyed with the history of the wheel-rail contact mechanics. For a designer of braking and traction control systems, it is essential to go through the fundamental knowledge of forces produced due to contact of wheel-rail and geometric constraints as well as moments to assess the wheel-rail wear and to evaluate the ride performance indices. This article gives a brief review to the evolution and present scenario of the previously published contact theories to get rid of the wheel-rail contact problems. The key theories which have most attention to solve normal contact problems are Hertzian contact theory as well as non-Hertzian contact theory, whereas, to tackle the problems related to tangential contact, linear as well as non-linear theories proposed by Kalker, theory proposed by Polach, FASTSIM, CONTACT etc are considered in this paper. This paper reviews the state-of-the-art of past studies based on experiments to study various contact models.

A new approach for solving three-dimensional elasto-plastic wheel-rail normal contact problems

A new approach for solving three-dimensional elasto-plastic wheel-rail normal contact problems

Analytic contact solutions of the Boussinesq and Cattaneo problems for an ellipsoidal power-law indenter

Based on Galin’s theorem for the indentation of an elastic half-space by a polynomial punch and Barber’s theorem for the determination of the contact area in elastic normal contact problems, an exact contact solution is developed for the frictionless indentation of an elastic half-space by an ellipsoidal power-law punch. Within the Cattaneo–Mindlin-approximation of tangential contacts, the tangential contact problem with friction can be reduced to the frictionless normal contact via the Ciavarella–Jäger principle, based on which also the tangential contact solution for the ellipsoidal power-law indenter is given. All known solutions for special cases (axisymmetric contact, elliptical Hertzian contact) are exactly recovered. A comparison of the pressure distribution obtained analytically for the 4-th power ellipsoidal indenter with a corresponding numerical solution based on the boundary element method shows no noticeable differences. The proposed solution procedure can also be used exactly for an arbitrary finite superposition of ellipsoidal power-law profiles, and, in an approximate sense, is applicable for arbitrary monotonous ellipsoidal profiles.

On normal compliance quasistatic viscoplastic contact problem with friction, damage and foundation memory property

The goal of this paper is to study a mathematical model which describes the quasistatic frictional contact between aviscoplastic body and a foundation. The contact is modeled with normal compliance, unilateral constraint, friction, damage andmemory term, which also includes foundation memory property. We present the classical formulation of the problem and derivevariational-hemivariational formulation. Finally we provide a unique weak solution of the problem.

FFT-BASED IMPLEMENTATION OF THE MDR TRANSFORMATIONS FOR HOMOGENEOUS AND POWER-LAW GRADED MATERIALS

It is shown how the Abel transform solution to the general axisymmetric normal contact problem for homogeneous and power-law graded elastic materials, which is paramount for the solution of different classes of tribological problems with the help of the method of dimensionality reduction (MDR), can be written in terms of explicit convolutions. These can be very efficiently evaluated with the 1D Fast Fourier Transform (FFT), which reduces the numerical complexity of the transformations from the order of N2 for the standard matrix-vector-multiplication (MVM) algorithm to the order of N. Convergence and performance of the proposed method are studied in detail. As an illustrative example a fretting wear simulation based on the new implementation is shown, the results of which are compared to the standard MVM implementation.

A framework for the analysis of fully coupled normal and tangential contact problems with complex interfaces

An extension to the interface finite element with eMbedded Profile for Joint Roughness (MPJR interface finite element) is herein proposed for solving the frictional contact problem between a rigid indenter of any complex shape and an elastic body under generic oblique load histories. The actual shape of the indenter is accounted for as a correction of the gap function. A regularised version of the Coulomb friction law is employed for modeling the tangential contact response, while a penalty approach is introduced in the normal contact direction. The development of the finite element (FE) formulation stemming from its variational formalism is thoroughly derived and the model is validated in relation to challenging scenarios for standard (alternative) finite element procedures and analytical methods, such as the contact with multi-scale rough profiles. The present framework enables the comprehensive investigation of the system response due to the occurrence of tangential tractions, which are at the origin of important phenomena such as wear and fretting fatigue, together with the analysis of the effects of coupling between normal and tangential contact tractions. This scenario is herein investigated in relation to challenging physical problems involving arbitrary loading histories.

Viscoelastic Effects during Tangential Contact Analyzed by a Novel Finite Element Approach with Embedded Interface Profiles

A computational approach that is based on interface finite elements with eMbedded Profiles for Joint Roughness (MPJR) is exploited in order to study the viscoelastic contact problems with any complex shape of the indenting profiles. The MPJR finite elements, previously developed for partial slip contact problems, are herein further generalized in order to deal with finite sliding displacements. The approach is applied to a case study concerning a periodic contact problem between a sinusoidal profile and a viscoelastic layer of finite thickness. In particular, the effect of using three different rheological models that are based on Prony series (with one, two, or three arms) to approximate the viscoelastic behaviour of a real polymer is investigated. The method allows for predicting the whole transient regime during the normal contact problem and the subsequent sliding scenario from full stick to full slip, and then up to gross sliding. The effects of the viscoelastic model approximation and of the sliding velocities are carefully investigated. The proposed approach aims at tackling a class of problems that are difficult to address with other methods, which include the possibility of analysing indenters of generic profile, the capability of simulating partial slip and gross slip due to finite slidings, and, finally, the possibility of simultaneously investigating dissipative phenomena, like viscoelastic dissipation and energy losses due to interface friction.

Solution of Contact Problems for Half-Space by Boundary Element Methods Based on Singular Solutions of Flamant and Boussinesq’s Problems

The paper considers normal contact problems formulated as follows: an indenter with negligible weight presses the surface half-space with a certain force, i.e., normal stress acts on the contact surface and tangential stress is zero. In particular, we consider two types of distributed load that correspond to the following cases: when half-space is subjected to frictionless flat rigid indenter, and when half-space is subject to frictionless cylindrical rigid indenter. The paper considers plane deformation. Problems are solved by boundary element methods (BEM), which are based on singular solutions of Flamant (BEMF) and Boussinesq (BEMB) problems. The stress–strain state of the half-plane, particularly the constructed contours (isolines) of stresses in half-plane, was studied. The results obtained by BEMF and BEMB are discussed and compared.

A simplified model for solving wheel-rail non-Hertzian normal contact problem under the influence of yaw angle

Wheelset yaw motion significantly affects wheel-rail non-Hertzian contact behaviour, but currently, few wheel-rail normal contact models consider this issue. In this paper, a simplified wheel-rail contact model is built to study the wheel-rail non-Hertzian normal contact problem considering the yaw angle. First, three basic assumptions for the approximate calculation of this problem are proposed and are validated by using results calculated by Kalker's variational method (KVM), and the approximate expression of the wheel-rail non-Hertzian contact problem considering the yaw angle is obtained based on these assumptions. Then, the approximate expression is applied to the MKP (Modified Kik–Piotrowski) contact model, and an improved MKP model that considers the wheelset yaw angle is proposed, i.e., the MKP-YAW model. Compared with the results calculated by KVM, the computational accuracy of the proposed MKP-YAW model in calculating the wheel-rail contact considering the yaw angle is similar to that of the MKP contact model in calculating the wheel-rail contact without considering the yaw angle.

Formulation of the pextension finite elements for the solution of normal contact problems

This work deals with normal contact problems. After a wide literature review, we look for the possibility of achieving a high-precision solution using the principle of minimum potential energy and the Hellinger-Reissner variational principle with penalty and augmented Lagrangian techniques. By positioning of the border of the contact elements, the whole surfaces of the eligible elements fall in contact or in gap regions. This reduces the error of the singularity in the border of the contact domain. Computations with $h$-, $p$- and $rp$-versions are performed. For the $rp$-version, the pre-fixed number of finite elements are moved so that small elements are placed in one or two element layers at the ends of the contact zone. A number of diagrams and tables showing the convergence of the solution (by increasing the number of polynomial degrees p) demonstrate the high efficiency of the proposed solution procedure.

Virtual Element Method for an Elliptic Hemivariational Inequality with Applications to Contact Mechanics

This paper is on the numerical solution of an elliptic hemivariational inequality by the virtual element method. We introduce an abstract framework of the numerical method and provide an error analysis. We then apply the virtual element method to solve two contact problems: a bilateral contact problem with friction and a frictionless normal compliance contact problem. Error estimates of their numerical solutions are derived, which are of optimal order for the linear virtual element method, under appropriate solution regularity assumptions. The discrete problem can be formulated as an optimization problem for a difference of two convex (DC) functions, and a convergent algorithm is introduced to solve it. Numerical examples are reported to show the performance of the proposed methods.

Quantifying uncertainties in contact mechanics of rough surfaces using the multilevel Monte Carlo method

We quantify the effect of uncertainties on quantities of interest related to contact mechanics of rough surfaces. Specifically, we consider the problem of frictionless non adhesive normal contact between two semi infinite linear elastic solids subject to uncertainties. These uncertainties may for example originate from an incomplete surface description. To account for surface uncertainties, we model a rough surface as a suitable Gaussian random field whose covariance function encodes the surface’s roughness, which is experimentally measurable. Then, we introduce the multilevel Monte Carlo method which is a computationally efficient sampling method for the computation of the expectation and higher statistical moments of uncertain system output’s, such as those derived from contact simulations. In particular, we consider two different quantities of interest, namely the contact area and the number of contact clusters, and show via numerical experiments that the multilevel Monte Carlo method offers significant computational gains compared to an approximation via a classic Monte Carlo sampling.

Comparison of non-Hertzian modeling approaches for wheel–rail rolling contact mechanics in the switch panel of a railway turnout

Due to the complicated wheel–rail contact relation of railway turnouts, it is necessary to select a reasonable rolling contact model to simulate the vehicle–turnout dynamics and wheel–rail damages. This paper mainly aims to evaluate the calculation accuracy and efficiency of different non-Hertzian modeling approaches in solving normal and tangential wheel–rail contact problems of railway turnouts. Four different non-Hertzian approaches, namely CONTACT, Kik–Piotrowski, Ayasse–Chollet, and Sichani methods are compared and analyzed. The above four models are built considering the relative motion of stock/switch rails. A wheel profile called LMA contacting with stock/switch rails (head width 35 mm) of CN60-1100-1:18 turnouts is selected as the object of analysis. The normal contact problems are evaluated by the wheel–rail contact areas, shapes, and normal contact pressures. The assessment of tangential contact problems is based on the creep curves, tangential contact stresses, and distribution of the stick/slip region. In addition, a contrast analysis is performed on the calculation efficiencies of the four approaches. It is found that the normal and tangential contact results calculated based on the Sichani method coincide well with those obtained according to CONTACT, and the calculation efficiency is about 262 times that of CONTACT. The conclusions can provide some guidance to the selection of wheel–rail rolling contact approach in the simulation of vehicle–turnout dynamics and wheel–rail damages.

Repeated loading model for elastic–plastic contact of geomaterial

A new nonlinear hysteretic model with considering the loading, unloading, and reloading processes is developed based on Drucker–Prager yield criterion and finite-element analysis. This model can be used for multiple repeated elastic–plastic normal direction contact problems between two identical spherical geomaterials. After examining the influence of material properties, strain hardening, and loading histories, we found that the hysteretic phenomena (represented by residual displacement and plastic work) become weak after the first cycle, and the subsequent cycles step into elastic shakedown state eventually. A critical number of cycles can be used to estimate the state of ratchetting, plastic shakedown, as well as elastic shakedown. It also found that the subsequent curves will be stiffer than the previous ones, especially when the yield strength is high and ratchetting effect is not strong. This new model can be used for a wide range of geomaterials under different loading levels, and it can also be extended to describe the constitutive behavior of spheres under earthquake as well as aftershocks.

Peculiarities of mathematical modeling of contact interaction of massive bodies and shells

The limits of using a mathematical model of contact interaction of elastic bodies, based on the Hertz’s solution in engineering practice are defined. Numerical investigation of the contact of a massive body and a shell has been conducted when the parameters of the latter vary. Three-dimensional finite-element models of contacting bodies have been developed in the case of different curvatures and thicknesses of the shell walls. A solution of the normal contact problem is given by the force method for various configuration models of the shell. Ratios of the geometric parameters of models the finite-element schemes of the contacting bodies are ere determined where the problem is described with sufficient accuracy by the Hertz’s solution.