FFT-Assisted Algorithms for 3d Line-Contact Problems

Abstract

The line-contact is a particular type of contact with a contact length much greater than its width. Such contact scenarios can be treated in the frame of a two-dimensional plane-strain problem if the contacting surfaces can be considered nominally smooth. However, surface irregularities inherent to any manufacturing technique lead to a discontinuous contact area that differs from the one derived on the basis of the smooth profile assumption. It is therefore tantalizing to pursue the solution of a line-contact problem using an intrinsically three-dimensional (3D) model, which can only be numerical due to lack of general analytical solutions in contact mechanics. Considering the geometry of the line-contact, a major challenge in its numerical modelling is that the expected contact area is orders of magnitude larger in one direction compared to the other. This may lead to an unreasonably large number of grids in the contact length direction, which translates to a prohibitive computational burden. An alternative approach, employed in this paper, is to treat the line-contact as non-periodic in the contact width direction, but periodic in the contact length direction, with a period equal to the window required to capture and replicate the surface specific texture. This periodicity encourages the contact problem solution by spectral methods based on the fast Fourier transform (FFT) algorithm. Based on this idea, two methods are derived in this paper from the existing Discrete Convolution Fast Fourier Transform (DCFFT) technique, which was previously developed for purely non-periodic contact problems. A first algorithm variant employs a special padding technique for pressure, whereas a second one mimics the contribution of multiple pressure periods by summation of the influence coefficients over a domain a few times larger than the target domain. Both techniques are validated against the existing analytical Hertz solution for the line-contact and a good agreement is found. The advanced methods seem well adapted to the simulation of contact problems that can be approximated as periodic in one direction and non-periodic in the other.

FFT-Assisted Solution for the Eigenstress Problem In an Elastic Half-Space

“…These summations replicate the elastic state in the half-space eigenstress problem. This author [13] advanced a numerical solution for the eigenstress problem in an infinite space, capable of finding the stresses induced by an arbitrary distribution of eigenstrains. To this end, superposition was applied to eigenstresses induced by cuboidal regions of uniform eigenstrains.…”

“…The model equations were then written taking into account the indexes of the centres of the elementary cuboids instead of continuous coordinates. For a 3D mesh with 1 2 3 ,, N N N grids along the three directions of a Cartesian coordinate system, the full space eigenstress tensor σ due to the eigenstrains tensor e, was expressed [13] as a 3D convolution product:…”

“…In [13], the 3D convolution operation (1) was performed via the DCFFT algorithm [7,8]. The latter is built on the convolution theorem, stating that a convolution in the time/space domain can be calculated, in the Fourier domain, as an element-by-element product between the transformed convolution factors.…”

“…. The index "1" is used to suggest that, although relation ( 5) generates a 3D stress array, only the first slice (a 2 2D array) is needed in the second term of the convolution (13), whereas in the first term, the index traverses all depth grids. Thus, ( 13) is only a 2D convolution product about the directions of 1…”