Adhesive contact of the Weierstrass profile

The Weierstrass series was considered in Ciavarella et al. (Ciavarella et al. 2000 Proc. R. Soc. Lond. A 456, 387-405. (doi: 10.1098/rspa.2000.0522)) to describe a linear contact problem between a rigid fractally rough surface and an elastic half-plane. In such cases, no applied mean pressure is sufficiently large to ensure full contact, and specifically there are not even any contact areas of finite dimension. Later, Gao & Bower (Gao & Bower 2006 Proc. R. Soc. A 462, 319-348. (doi: 10.1098/rspa.2005.1563)) introduced plasticity in the Weierstrass model, but concluded that the fractal limit continued to lead to what they considered unphysical predictions of the true contact size and number of contact spots, similar to the elastic case. In this paper, we deal with the contact problem between rough surfaces in the presence of adhesion with the assumption of a Johnson, Kendall and Roberts (JKR) regime. We find that, for fractal dimension D > 1.5, the presence of adhesion does not qualitatively modify the contact behaviour. However, for fractal dimension D < 1.5, a regularization of the contact area can be observed at a large magnification where the contact area consists of segments of finite size. Moreover, full contact can occur at all scales for D < 1.5 provided the mean contact pressure is larger than a certain value. We discuss, however, the implication of our assumption of a JKR regime.