A Numerical Approach to the Contact of Nominally Flat Surfaces

Abstract

Machined surfaces can be described by heights and wavelengths of the surface asperities that show a statistical variation. Considering that a regular wavy surface with a sinusoidal profile is the crudest model for a rough surface, studying the contact of regular wavy surfaces is a good approximation for the contact of nominally flat surfaces. Such contact problems exhibit periodicity that can be simulated with the aid of computational techniques derived for contact mechanics in the frequency domain. The displacement calculation, which is a necessary step in the resolution of the contact problem, is mathematically a convolution product that can be calculated in the frequency domain with increased computational efficiency. The displacement induced by a unit surface load can be expressed in the frequency domain by the frequency response functions, which are counterparts of the space domain solutions to half-space fundamental problems such as the Boussinesq problem. The displacement induced by a periodic pressure distribution can be computed by executing the convolution product between the frequency response function and pressure on a single period. It should be noted that the convolution calculation in the spectral domain implies that the contributions of all neighbouring pressure periods are accounted for. The need to treat numerically only a single period results in remarkable computational efficiency, allowing for high density meshes that can capture the essential features of any textured real surface. The displacement calculation promotes the solution of the contact problem by an iterative approach. The advanced method is benchmarked against existing analytical solutions for the 3D contact of surfaces possessing two-dimensional waviness. This essentially deterministic model, supported by a direct numerical solution that can be obtained for samples of real rough surfaces, presents itself as a worthy alternative to the existing statistical models for rough contact interaction.

Symmetrically Distributed Eigenstrains In a Thin Sheet of Material

“…If this condition is met, the fourth state is that of a symmetrically loaded thin sheet. A suitable Green's function for the thin elastic layer compressed between two equal and collinear concentrated forces is readily available [11]. Considering that on the two boundaries the normal tractions 33  are piece-wise constant, integration of the before mentioned solution over a rectangular domain    …”

“…For the former, solutions derived in the linear theory of elasticity for the latter, also referred to as the Green's functions for the elastic and isotropic half-space, cannot be directly used. Based on the works of Diaconescu and Glovnea [9] and of Spinu and Glovnea, [10], these authors [11] proposed an appropriate Green's solution for the thin layer, by adding a corrective term to the Boussinesq half-space solution. This new result facilitates an extension of the inclusion problem solution, in which a thin sheet of material stands as the surrounding matrix.…”

“…To achieve this, the mirror image method is applied twice, with respect to each of the two boundaries of the thin sheet. Under the assumption that the eigenstrains are symmetrically distributed with respect to the sheet mid plane, the boundary relief can be achieved with the Green's functions derived in [11]. An increased computational efficiency is attained with the aid of the fast Fourier transform in performing the mathematical operations resulting from superposition.…”