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Edited by: B.H.V. Topping
Chapter 9

A Review of Boundary Element Analysis of Contact Mechanics

A.A. Becker

School of Mechanical, Materials, Manufacturing Engineering and Management, University of Nottingham, United Kingdom

Full Bibliographic Reference for this chapter
A.A. Becker, "A Review of Boundary Element Analysis of Contact Mechanics", in B.H.V. Topping, (Editor), "Progress in Civil and Structural Engineering Computing", Saxe-Coburg Publications, Stirlingshire, UK, Chapter 9, pp 209-236, 2003. doi:10.4203/csets.10.9
Keywords: boundary element method, contact, friction, inverse methods, creep.

The stress analysis of contact problems is a major concern in many engineering applications such as ball bearings, gears, rollers, mechanical seals and pressure vessel attachments. The numerical modelling of practical contact problems requires special attention because the actual contact area between the contacting bodies is usually unknown in advance and if friction is present, the behaviour may be dependent on load history [1]. The two widely used numerical techniques in the analysis of contact problems are the Finite Element (FE) and the Boundary Element (BE) methods.

The BE method, with its surface-only modelling capability in linear problems offers several advantages over the FE method in solving contact problems [2]. The contact stresses calculated by the BE methods are more accurate because there is a lesser degree of approximation imposed on the solution, and the surface tractions, which are fundamental in determining the contact pressures, are calculated to the same degree of accuracy as the displacements, which enables accurate coupling of the contact variables. In BE formulations, since the contact variables are directly coupled to satisfy equilibrium and compatibility relationships, there is no need to use any form of additional gap or interface elements across the contact interface. Another advantage of using the BE method in contact problems is the ease of re- meshing a local contact surface, rather than modifying a volume mesh.

In this paper, a review of BE formulations for contact problems is presented, where the contact variables are coupled directly to produce a set of simultaneous linear equations with a unique solution. Several types of contact interface conditions are covered including heat conduction, frictionless and frictional stick-slip. The review also covers three-dimensional contact problems, independent meshing of contact surfaces, non-linear material behaviour and inverse techniques.

The contact applications demonstrate the versatility of the BE technique in modelling industrial problems such as pressure vessels and three-dimensional geometries. The contact surfaces can be discretised using a matching node-on-node mesh or an independent mesh which allows the user to specify a coarse mesh on one contact surface and a fine mesh on the other surface [3]. Thermoelastic stresses can be easily incorporated into contact problems by coupling the temperatures and heat flux between the contact nodes. In three-dimensional contact stick-slip problems, robust iterative schemes are needed to identify the correct direction of slip along the contact plane [4]. The contact formulation is extended to model contact under creep conditions in which the contact solutions have to be marched in time to arrive at the final creep time [5]. The inverse BE method determines the surface contact stresses from stress information obtained inside the solution domain, and is shown to be an ideal companion to experimental stress measurement techniques, such as photoelasticity, because the interior domain can be modelled with un-connected points without the need for full inferior discretisation [6].

All of the contact applications demonstrate the versatility and suitability of the BE method for contact problems, particularly for linear three-dimensional problems with rapidly changing stress gradients and inverse problems. BE contact formulations enable the direct coupling of the contact variables on the surface through the direct incorporation of the equilibrium, compatibility and frictional slip relationships into the overall system of equations without the need for gap or interface elements. The BE method provides an alternative accurate technique for contact analysis.

Johnson, K.L., "Contact Mechanics", Cambridge University Press, Cambridge, 1985.
Becker, A.A., "The boundary element method in engineering", McGraw- Hill, London, 1992.
Olukoko, O.A., Becker, A.A. and Fenner, R.T., "A review of three alternative approaches to modelling frictional contact problems using the boundary element method", Proc. Royal Society of London, Series A, 444, 37-51, 1994. doi:10.1098/rspa.1994.0003
Leahy, J.G. and Becker, A.A. "A quadratic boundary element formulation for three-dimensional contact problems with friction", J. Strain Analysis, 34, 235-251, 1999. doi:10.1243/0309324991513786
Chandenduang, C. and Becker, A.A., "Boundary element formulation for two-dimensional creep problems using isoparametric quadratic elements", Computers & Structures, 81, 1611-1619, 2003. doi:10.1016/S0045-7949(03)00182-2
Chen, D., Becker, A.A., Jones, I.A., Hyde, T.H. and Wang, P. "Development of new inverse boundary element techniques in photoelasticity", J. Strain Analysis, 36, 253-264, 2001. doi:10.1243/0309324011514449

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