## University of Wollongong Thesis Collection 1954-2016

1992

#### Degree Name

Doctor of Philosophy

#### Department

Department of Mathematics

#### Abstract

This thesis deals primarily with a class of elastostatic contact problems involving circular elastic cylinders, whose axes are parallel and whose boundaries touch initially along a line. The object of the study is to establish the nature of the contact region (a narrow rectangular strip) which develops when a pair of equal and opposite forces is applied compressing these cylinders together. Of prime interest is the pressure distribution which acts in this region for a given set of variables: the radii and elastic properties of the cylinders, combined with either their relative approach or the total compressive force. Specifically, the problems of incorporating the effects of curvature, sharp edges and surface layers, which are all neglected in the existing Hertz theory are investigated within the framework of linear elasticity. Results are discussed in light of their implications to the industrial process of roller coating.

The effects of curvature are accounted for by considering the two-dimensional symmetrical compression of a circular elastic cylinder by two identical arbitrary compressors. Two cases of boundary conditions are studied. In each case the general approach adopted involves a simple superposition of basic point force solutions for the circular boundary to derive the governing integral equation for the pressure distribution in the contact region. Complex variable methods are utilised to obtain the point force solutions, while the contact pressure distribution is derived using the properties of the finite Hilbert transform. The first boundary condition constitutes a prescribed horizontal circumferential displacement while the vertical displacement is left to take on whatever value is generated by the given deformation; in this case, the horizontal pressure distribution is sought. The second boundary condition pertains to the physical situation of complete adhesion wherein both the horizontal and vertical circumferential displacements are specified; in this case, both the horizontal and vertical pressures are determined. Analysis relating to the first boundary condition yields an exact formula for the pressure distribution for the following rigid compressors: (i) parallel plates, (ii) circular cylinders of equal radius to the central cylinder, and (iii) elliptical cylinders whose minor axes coincide with that of the central cylinder. Approximate expressions are given for the general cases of (ii) and (iii), and for the situations in which these compressors are elastic. Comparisons are drawn between some of the results and the Hertz pressure.

The problem of incorporating the effects of sharp edges and surface layers is dealt with in the context of three-dimensional linear elasticity. A numerical method of solution is developed which utilises the generalised Boussinesq stress-displacement relation for a layered elastic half-space, in conjunction with the flexibility method of structural analysis. A n iterative scheme is employed in which the systems of linear equations for the contact pressures so obtained are solved subject to the conditions of total equilibrium and the exclusion of tensile forces. Subsequently, the method is extended to account for the presence of sharp edges in roller contact by assuming at the outset the inverse square root singularity form for the pressure near the sharp edges. Numerical results are compared with both published theoretical and experimental data. The method is applied to specific roller coating systems, demonstrating how it may be utilised to identify the most influential factors which govern the intensity of the contact pressures for a given roll profile.

Finally, a brief examination of the contact between an elastic cylinder and a thin beam in bending is included. The deflection of the beam is deduced in accordance with elementary beam theory. This is followed by a treatment of the limiting case of contact with a rigid cylinder. For the elastic cylinder, the displacement solution corresponding to a point force acting on its circumferential boundary is determined using complex variable methods.

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