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A novel analytical model is derived based on fractal geometry theory to characterize the permeability of power-law fluids through fractured porous media with fractal asperities on surfaces. The proposed model is expressed as a function of structure parameters of fractured porous media, fractal dimensions for porous matrix and for fracture as well as the index $n$ of power-law fluid behavior. The influences of these parameters on the permeability are analyzed in detail. It is found that the relationship between permeability and roughness exhibits nonlinear characteristics. A higher power-law fluid behavior index significantly affects the reduction rate of permeability with relative roughness. Moreover, permeability is sensitive to the perturbation due to the rate of aperture to length. The proposed fractal model can clearly reveal more physical mechanisms for power-law fluid in fractured porous media than the existing models. Compared with the experimental data available in the literature, the validity of model predictions is verified.

Lateral loading and interfacial slip of multi-asperity (i.e., rough) elastic contacts are studied for micro-slip contact conditions. The Mindlin micro-slip model for smooth surfaces is generalized to rough surface contacts using the Greenwood–Williamson (GW) approach, and the general relations of lateral contact force and related stiffness are obtained. The method extends previous approaches by incorporating micro-slip, allowing application to rough surfaces and providing simple expressions for experimental analysis by use of the Greenwood–Williams roughness parameters. As applications, the numerical results of micro-slip contacts on rough surfaces for Gaussian and exponential asperity height distributions, respectively, are obtained based on the general relations of the extended model, and are compared and discussed for low load cases.

The load–area relation of rough surfaces is of great interest in tribology. For elastic–plastic solids with strain hardening, an incremental model is adopted to analyze the contact of rough surfaces, in which the contact is modeled by accumulation of equivalent circular contacts with varying radius. For three typical rough surfaces with various material properties, comparisons with direct finite element calculations demonstrate the efficiency of this incremental contact model. An approximate linear relation between load and contact area is predicted by both methods up to a contact fraction of 15%. The influence of yield stress and strain-hardening exponent on the load-area proportionality is presented. This work gives a simple while effective method to calculate the load-area relation for rough contact of strain-hardening materials.