Modeling and Analysis of Adhesion Hysteresis Between Rough Surfaces

如果断裂接触需要有限的力，则表面具有粘性或“粘性”。在原子尺度上，所有表面都通过无处不在的范德华相互作用相互作用，这种相互作用产生的单位面积的力比大气压力大几个数量级。这导致了小物体的强附着力，比如壁虎的刚毛和工程模拟物。这些相互作用的强度通常用粘附的内在功来描述，即每一亲密接触表面积的微观相互作用所获得的能量。而对于硬基板的粗糙度限制了该区域的最高突起，软固体是粘性的，因为它们可以变形以接触粗糙地形的很大一部分。在这种一致性接触的热力学观点中，破坏粘接接触成为一个断裂力学问题。从软接触中观察到的一个常见现象是，打破接触所需的力通常比压痕期间测量到的力要高得多。这一观察结果与假设接触遵循热力学平衡的理论的预期相矛盾。事实上，表面粗糙度将附着问题置于“摆动类”而不是“Serfaty类”——简而言之，取能量均匀化极限与解决相关的梯度流动演化问题并不一致。这导致了一个速率无关的接触线迟滞出现在一个快速振荡的势能景观和粘性演化之间的相互作用。本项目的主要目标如下。我们将从一阶近似模型开始，研究粗糙表面上的裂纹前钉住和脱屑效应，其中包括一个分数阶拉普拉斯算子来描述材料中的弹性相互作用。这使得这些模型既适用于解析处理，也适用于有效的数值处理。下一步是将这些模型扩展到更高阶。在这里，需要证明比较原理，以使这些高阶展开式符合严格的钉钉和脱钉效应的数学研究，同时也证明有效的数值方法。最后，我们将通过边界元方法对完全非线性模型进行数值分析。得到的钉住结果将再次与构造的亚解和超解进行比较，这些解给出了对新出现的接触线迟滞的确定性估计。

Surfaces are adhesive or "sticky" if breaking contact requires a finite force. At atomic scales, all surface interact via ubiquitous van-der-Waals interactions, that produce forces per unit area that are orders of magnitude larger than atmospheric pressure. This leads to strong adhesion of small objects, such as Gecko setae and engineered mimics. The strength of these interactions is commonly described by the intrinsic work of adhesion, i.e., the energy that is gained by microscopic interactions per surface area of intimate contact. While for hard substrates roughness limits this area to the highest protrusions, soft solids are sticky because they can deform to come into contact over a large portion of the rough topography. In this thermodynamic view of conforming contact, breaking adhesive contact becomes a fracture-mechanical problem. A common observation from soft contact is that the force needed to break the contact is typically much higher than the force measured during indentation. This observation contradicts expectations from theories that assume the contact follows thermodynamic equilibrium. Indeed, the surface roughness puts the adhesion problem in the "wiggly class" instead of the "Serfaty class" - in short, taking the energetic homogenization limit does not commute with solving the associated gradient flow evolution problem. This leads to a rate-independent contact-line hysteresis emerging from the interplay between a rapidly oscillating potential energy landscape and a viscous evolution. The main objectives in this project are as follows. We will study crack-front pinning and depinning effects on rough surfaces, starting with first-order approximate models, which include a fractional-order Laplacian to describe the elastic interaction in the material. This makes those models amenable to both analytic as well as efficient numerical treatment. The next step is then to extend these models to higher order. Here, comparison principles need to be proved, both to make these higher-order expansions amenable to rigorous mathematical study of pinning and depinning effects, but also to justify efficient numerical methods. Finally, we will examine the fully nonlinear models numerically by means of boundary-element methods. The pinning results obtained will be again compared to constructed sub- and super-solutions which give deterministic estimates on the emerging contact line hysteresis.