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.
Various biomedical components, such as dental crowns and hip prostheses, data processing devices, and other numerous mechanical components that transmit load through a mechanical contact, may benefit from a tri-layer design. The coating may be optimized for wear protection and corrosion prevention, whereas the intermediate layer provides increased adhesion between the outer layer and the substrate, and confines the crack propagation. The solution to the contact problem involving tri-layered materials can be pursued numerically with the finite element or the boundary element methods, but semi-analytical techniques benefitting from the efficiency of the fast Fourier transform (FFT) technique have also been successfully applied. At the heart of the FFT-assisted approach lie the frequency response functions (FRFs), which are analytical solutions for fundamental problems of elasticity such as the Boussinesq and Cerruti problems, but expressed in the frequency domain. Considering recent efforts and results in application of FFT to convolution calculations in contact problems, the displacement arising in a tri-layer configuration is computed in the frequency domain, and the contact problem is subsequently solved in the space domain using a state-of-the-art algorithm based on the conjugate gradient method. The method relies on the FRFs derived in the literature for tri-layered materials, and the efficiency and accuracy of computations in the frequency domain is assured by using the Discrete Convolution Fast Fourier Technique (DCFFT) with influence coefficients derived from the FRFs. The computer program reproduces well-known results for bi-layered materials. Numerical simulations are performed for various configurations in which the elastic properties of the layers, as well as the frictional coefficient, are varied. By using the newly advanced simulation technique, design recommendations may be advanced for the optimal configuration of tri-layered materials under contact load.
Finding the distributions of eigenstresses induced by eigenstrains regardless of their type is a fundamental problem in mechanical engineering, described by complex mathematical models. Analytical solutions exist only for a small number of particular distributions of eigenstrains. This paper advances a numerical solution for the eigenstresses due to arbitrary distributions of eigenstrains in an infinite space. The imposed discretization transforms the continuous problem space into a set of adjacent cuboids, each characterized by a single value calculated analytically in a chosen point, usually the cuboid centre. In this manner, continuous functions are replaced in the mathematical model by sets of values calculated in discrete points, which, if the discretization is fine enough, replicate well the continuous distributions. The contribution of the uniform eigenstrains from a specific cuboid, to the eigenstresses in the calculation point, expressed analytically in the literature, is used as a starting point. To reduce the high computational requirements for superposition, state-of-the-art spectral methods for the acceleration of convolution products are applied. A Matlab computer program was developed to implement the newly advanced method. The case of a cuboid containing uniform dilatational eigenstrains was first simulated for validation purposes. Small deviations from the analytical solution can be observed near the inclusion boundary, but their magnitude decreases with finer meshes, suggesting it’s a discretization related error. The results were then extended by considering radially decreasing eigenstrains inside an ellipsoid.
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