Adhesive (e.g. van der Waals) forces were not generally taken into account in contact mechanics until 1971, when Johnson, Kendall and Roberts (JKR) generalized Hertz’ solution for an elastic sphere using an energetic argument which we now recognize to be analogous to that used in linear elastic fracture mechanics. A significant result is that the load–displacement relation exhibits instabilities in which approaching bodies ‘jump in’ to contact, whereas separated bodies ‘jump out’ at a tensile ‘pull-off force’. The JKR approach has since been widely used in other geometries, but at small length scales or for stiffer materials it is found to be less accurate. In conformal contact problems, other instabilities can occur, characterized by the development of regular patterns of regions of large and small traction. All these instabilities result in differences between loading and unloading curves and consequent hysteretic energy losses. Adhesive contact mechanics has become increasingly important in recent years with the focus on soft materials (which generally permit larger areas of the interacting surfaces to come within the range of adhesive forces), nano-devices and the analysis of bio-systems. Applications are found in nature, such as insect attachment forces, in nano-manufacturing, and more generally in industrial systems involving rubber or polymer contacts. In this paper, we review the strengths and limitations of various methods for analysing contact problems involving adhesive tractions, with particular reference to the effect of the inevitable roughness of the contacting surfaces.

True contact between randomly rough solids consists of myriad individual microjunctions. While their total area controls the adhesive friction force of the interface, other macroscopic features, including viscoelastic friction, wear, stiffness, and electric resistance, also strongly depend on the size and shape of individual microjunctions. We show that, in rough elastomer contacts, the shape of microjunctions significantly varies as a function of the shear force applied to the interface. This process leads to a growth of anisotropy of the overall contact interface, which saturates in the macroscopic sliding regime. We show that smooth sphere-plane contacts have the same shear-induced anisotropic behavior as individual microjunctions, with a common scaling law over 4 orders of magnitude in the initial area. We discuss the physical origin of the observations in light of a fracture-based adhesive contact mechanics model, described in the companion article, which captures the smooth sphere-plane measurements. Our results shed light on a generic, overlooked source of anisotropy in rough elastic contacts, not taken into account in current rough contact mechanics models.

This paper gives a theoretical analysis for the fundamental problem of anisotropy induced by shear forces on an adhesive contact, discussing the experimental data of the companion Letter. We present a fracture mechanics model where two phenomenological mode-mixity functions are introduced to describe the weak coupling between modes I and II or I and III, which changes the effective toughness of the interface. The mode-mixity functions have been interpolated using the data of a single experiment and then used to predict the behavior of the whole set of experimental observations. The model extends an idea by Johnson and Greenwood, to solve purely mode I problems of adhesion in the presence of a nonaxisymmetric Hertzian geometry, to the case of elliptical contacts sheared along their major or minor axis. Equality between the stress intensity factors and their critical values is imposed solely at the major and minor axes. We successfully validate our model against experimental data. The model predicts that the punch geometry will affect both the shape and the overall decay of the sheared contact area.

The dynamical behavior of a single-degree-of-freedom system that experiences friction-induced vibrations is studied with particular interest on the possibility of the so-called hard effect of a subcritical Hopf bifurcation, using a velocity weakening–strengthening friction law. The bifurcation diagram of the system is numerically evaluated using as bifurcation parameter the velocity of the belt. Analytical results are provided using standard linear stability analysis and nonlinear stability analysis to large perturbations. The former permits to identify the lowest belt velocity \(({v_\mathrm{lw}})\) at which the full sliding solution is stable, the latter allows to estimate a priori the highest belt velocity at which large amplitude stick–slip vibrations exist. Together the two boundaries \([v_\mathrm{lw}, v_\mathrm{up}] \) define the range where two equilibrium solutions coexist, i.e., a stable full sliding solution and a stable stick–slip limit cycle. The model is used to fit recent experimental observations.