Data from: 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.

魏尔斯特拉斯级数（Weierstrass series）曾被Ciavarella等人（Ciavarella et al. 2000 *Proc. R. Soc. Lond. A* 456, 387–405. (doi:10.1098/rspa.2000.0522)）用于描述刚性分形粗糙表面与弹性半平面（elastic half-plane）之间的线性接触问题。在此类场景中，不存在足够大的平均外加载荷压力以实现完全接触，且甚至不存在有限尺寸的接触区域。后续，Gao与Bower（Gao & Bower 2006 *Proc. R. Soc. A* 462, 319–348. (doi:10.1098/rspa.2005.1563)）在魏尔斯特拉斯模型中引入了塑性效应，但得出结论：分形极限（fractal limit）仍会导致他们认为不符合物理规律的真实接触面积与接触斑点（contact spots）数量的预测结果，与弹性情形类似。本文研究考虑粘附作用（adhesion）的粗糙表面接触问题，采用约翰逊-肯德尔-罗伯茨（Johnson, Kendall and Roberts, JKR）接触模型假设。研究发现，当分形维数（fractal dimension）D>1.5时，粘附的存在不会从定性层面改变接触行为。而当D<1.5时，在高放大倍率下可观察到接触区域的正则化现象，此时接触区域由有限尺寸的分段构成。此外，若平均接触压力大于某一临界值，当D<1.5时可在所有尺度上实现完全接触。不过，本文还就所采用的JKR接触模型假设的相关推论与适用边界展开了讨论。