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The contact sport of rough surfaces

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Science  05 Jan 2018:
Vol. 359, Issue 6371, pp. 38
DOI: 10.1126/science.aaq1814

Describing the way two surfaces touch and make contact may seem simple, but it is not. Fully describing the elastic deformation of ideally smooth contacting bodies, under even low applied pressure, involves second-order partial differential equations and fourth-rank elastic constant tensors. For more realistic rough surfaces, the problem becomes a multiscale exercise in surface-height statistics, even before including complex phenomena such as adhesion, plasticity, and fracture. A recent research competition, the “Contact Mechanics Challenge” (1), was designed to test various approximate methods for solving this problem. A hypothetical rough surface was generated, and the community was invited to model contact with this surface with competing theories for the calculation of properties, including contact area and pressure. A supercomputer-generated numerical solution was kept secret until competition entries were received. The comparison of results (2) provides insights into the relative merits of competing models and even experimental approaches to the problem.

Understanding contact and friction fascinated da Vinci, Coulomb, Prandtl, and de Gennes, among others (3). In 1881, Hertz solved the problem of two smooth, elastic spheres coming into contact, launching the field of “contact mechanics” (4) and leading to the development of equations for contact area, contact pressure, stresses, strains, and deformations for smooth, loaded contact geometries. However, real surfaces are rarely smooth. Even finely polished metals retain peaks and valleys with wavelengths ranging from nanometers to micrometers, leading to a seemingly random distribution of asperity contacts separated by stress-free noncontact zones (see the figure). Solving such contact problems is fundamental to understanding how gears turn, tires roll, and geckos climb (5), as well as for engineering any device with moving or contacting parts. It also underpins theories of friction, adhesion, lubrication, and wear, which can be empirical and highly system-specific.

Just over 50 years ago, an elegant analysis of contacting rough surfaces assumed that surface summit heights, all with the same radii, had an approximately Gaussian distribution (6). This Greenwood-Williamson (GW) model is the most common approach to modeling rough surfaces and has since spawned a plethora of other models. These approaches involve both analytical extensions to the GW model as well as numerical approaches, with different degrees of complexity that account for effects such as plastic flow, adhesive forces, and anisotropic properties.

This wide variety of models not only makes it difficult for experimentalists and engineers to decide which approach to adopt but also throws into question their accuracy. Predictions from different models disagree strongly (as do their authors), and experimental validations are scant. In late 2015 (1), Müser formulated a contact problem and, along with Spencer and Tysoe, invited researchers to partake in the Contact Mechanics Challenge. Müser used a supercomputer to conduct brute-force numerical simulations of a rough surface contacting a flat plate with a moderate amount of adhesion. The topography and material properties were shared on the internet, but the solution was held secret. The scientific community was challenged to share results from their preferred models for properties including the total contact area, the contact spot distribution, stresses, and deformations.

The contacting surfaces like those used in the competition. The blue grid shows the deformations of the lowermost surface of an elastic solid (the left half is rendered more transparent) pressed against a rigid substrate with fractal roughness. The color range of the lower surface corresponds to height, enlarged ∼30 times versus the in-plane directions. [Adapted from (10).]

The response was impressive: 12 groups involving more than 30 authors from Austria, England, France, Germany, Italy, Iran, the Netherlands, Taiwan, and the United States submitted entries that ranged from analytical solutions based on GW-inspired models to more recent fractal-based theories (7) and numerical solutions. One group even three-dimensionally printed a model of the rough surface and measured the resulting contact properties with a frustrated total-internal-reflection method (8).

Müser and Greenwood compared and summarized the results (2). The multiscale approaches had the winning score, with an honorable mention for the experimental method, which matched the solution extremely well. Multiscale approaches based on Persson's theory (7) fitted much better than those based on GW for two reasons. First, ignoring small-scale roughness neglects high-pressure zones, whereas ignoring largescale roughness neglects substantial overall deformation as surfaces conform together. Second, with elastic solids, pushing down on one asperity creates a long-range elastic field that affects the neighbors by pushing them down as well. This effect renders summits near another summit less likely to make contact themselves, thus breaking up the contact area into smaller patches.

The importance of multiscale roughness has long been recognized (9), but the Contact Mechanics Challenge results signal that the time has come to standardize multiscale descriptions with elastic coupling, to develop next-generation models to account for behavior beyond the elastic limit, and to confront the next frontier in contact modeling—namely, describing rough surface contact during sliding. This last step would help fulfill da Vinci's dream of understanding what causes friction.

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