A detailed review of various fractal models used in tribology is presented. The analysis of the models is based on the use of the Cantor–Borodich (C_B) profile and its modifications. This profile and related models may be studied analytically and, therefore, they provide us with tools for rigorous analysis of fractal approaches to description of surface roughness and corresponding contact problems. In turn, this allows us to present a critical review of current fractal approaches to tribology. It will be demonstrated that fractal dimension alone cannot give a full description of surface roughness, however, some of these models may reflect the multilevel hierarchical structure of real surface roughness. This review helps to avoid the repetition of common erroneous statements about the use of fractal concepts in tribology.

This article addresses the plane strain problem of a bi-material system containing an elliptical cylindrical thermal inclusion. Using Eshelby ’s inclusion analysis method, we derive closed-form analytical solutions for the elastic field induced by the thermal inclusion. Inspired by Dundurs’ parameters, we introduce a new material parameter (ranging from -1 to 1) and five tensorially structured expressions to succinctly represent the analytical solution, facilitating its practical applications. For circular inclusion scenarios, the analytical solution simplifies significantly, and we derive explicit jump conditions for displacement, strain, and stress at the bonded interface of the bi-material. By adjusting the Young ’s moduli and Poisson ’s ratios of the bi-material, the solution can reduce to cases of a full or half-plane containing a thermal elliptical inclusion. The accuracy of the proposed solution is validated through consistency with previously published analytical results and by matching numerical solutions from the literature, confirming the correctness and reliability of the derived analytical expressions.

Effectively solving the governing equations for contact problems often involves complex mathematical theory, while the distribution of contact stress is highly random in practical engineering applications. This study proposes a novel algorithm based on the triangular load discrete element and the discrete convolution fast Fourier transform (DC-FFT) algorithm. This algorithm provides a high-precision and reliable method for efficiently solving the contact response of a solid under any load distribution. Compared to the commonly used uniform load element discrete method, the analytical solution of the triangular element is more complex. However, it better simulates the characteristics of contact load distribution, accounting for situations where the load at the contact edge increases from zero or decreases to zero. The stress component under the action of the triangular and uniform load elements is derived based on the "excitation-response" characteristics of the contact influence coefficient matrix. This information is used to optimize the solution method of the triangular load discrete element. By constructing the stress solution in the form of a discrete convolution, including the influence coefficient matrix, the stress superposition effect of a target node under the action of all elements can be further simplified and accelerated by using the DC-FFT algorithm for highly repetitive matrix calculations. Programming and calculation analysis show that the proposed algorithm based on the triangular load element is accurate and efficient.