Keywords: boundary element method, film stiffness, contact, quadratic elements.

This paper presents a new BE contact formulation for modelling film stiffness between contact surfaces using a quadratic element formulation. An outline of the BE formulation is presented followed by a number of examples of film stiffness contact in compound cylinders and spheres.

The phenomenon of contact film stiffness can be observed in terms of the behaviour of a lubricant being applied between two contacting surfaces. The purpose of a lubricant in this context is to either prevent contact between the two surfaces by creating a barrier or film or enable sliding by effectively reducing the coefficient of friction between the two surfaces. The lubricant film is able to keep the two surfaces apart because the lubricant is being confined to a finite volume between the two surfaces at elevated pressures.

The Boundary Element (BE) method [1,2], with its surface-only modelling capability and the direct coupling of equilibrium and compatibility equations at the contact interface, is a very suitable computational method for contact problems. Many BE contact developments have been published, including load incrementation procedures (Man et al [3]), three-dimensional contact (Paris et al [4] and Leahy and Becker [5]), independent mesh discretisation (Olukoko et al [6]) and tangential loading (Hack and Becker [7]).

In this paper, a BE formulation for contact film stiffness has been developed and implemented into a BE contact formulation using quadratic elements. The equations of equilibrium and compatibility are directly incorporated in the final system of equations. Rather than using the Cartesian system conventionally used in BE matrices, a local axes system, based on the normal and tangential directions is implemented. The main advantage of the local axes system is that it can be easily extended to three-dimensional contact problems [5]. The stiffness behaviour of the squeeze film is represented by a spring stiffness in the normal direction, while the tangential stiffness is assumed negligible. The contact variables are rearranged in the final system of equations such that the total number of equations is exactly equal to the number of unknown variables.

The contact film stiffness formulation is incorporated into a quadratic BE computer program using local contact variables. Three examples are presented to test the accuracy of the new BE formulation. In all the cases, analytical solutions are also presented to verify the BE solutions. The first problem is a compound cylinder presented as an axisymmetric model having internal and external applied pressures. The second problem is an identical compound cylinder but instead presented as a plane strain model; also with internal and external applied pressures. The third problem is a compound sphere arrangement, presented as an axisymmetric model, with internal and external applied pressures. Isoparametric quadratic 3-node elements with 4 Gaussian integration points are used in all the examples. Contact surfaces with film stiffness and an interference fit are included in the analyses.

The examples demonstrate that the BE film stiffness contact solutions are in very good agreement with the corresponding analytical solutions, where the effect of the film stiffness can be clearly observed keeping the surfaces apart. This formulation can be used in the study of the mechanical effects of squeeze film type lubrication problems.

1

Brebbia, C.A., Telles, J.C.F. and Wrobel, L.C., "Boundary Element Techniques", Springer Verlag, Berlin, 1983.

2

Becker, A.A., "The boundary element method in engineering", McGraw-Hill, London, 1992.

3

Man, K.W., Aliabadi, M.H. and Rooke, D.P., "BEM frictional contact analysis: load incremental technique", Computers and Structures, 47, 893-905, 1993. doi:10.1016/0045-7949(93)90294-N

4

Paris, F., Foces, A. and Garrido, J.A., "Application of boundary element method to solve three-dimensional elastic contact problems without friction", Computers and Structures, 43, 19-30, 1992. doi:10.1016/0045-7949(92)90076-C

5

Leahy, J.G. and Becker, A.A., "The numerical treatment of local variables in 3D contact problems using the boundary element method", Computers and Structures, 71, 383-395, 1999. doi:10.1016/S0045-7949(98)00294-6

6

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 London, 444, 37-51, 1994. doi:10.1098/rspa.1994.0003

7

Hack, R.S. and Becker, A.A. , "Frictional contact analysis under tangential loading using a local axes boundary element formulation", Int J. Mechanical Sciences, 41, 419-436, 1999. doi:10.1016/S0020-7403(98)00075-7

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