Putelat T, Dawes J H P, Champneys A R
Department of Engineering Mathematics, University of Bristol, Bristol BS8 1UB, UK.
Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK.
Proc Math Phys Eng Sci. 2017 Jul;473(2203):20160606. doi: 10.1098/rspa.2016.0606. Epub 2017 Jul 5.
Sliding frictional interfaces at a range of length scales are observed to generate travelling waves; these are considered relevant, for example, to both earthquake ground surface movements and the performance of mechanical brakes and dampers. We propose an explanation of the origins of these waves through the study of an idealized mechanical model: a thin elastic plate subject to uniform shear stress held in frictional contact with a rigid flat surface. We construct a nonlinear wave equation for the deformation of the plate, and couple it to a spinodal rate-and-state friction law which leads to a mathematically well-posed problem that is capable of capturing many effects not accessible in a Coulomb friction model. Our model sustains a rich variety of solutions, including periodic stick-slip wave trains, isolated slip and stick pulses, and detachment and attachment fronts. Analytical and numerical bifurcation analysis is used to show how these states are organized in a two-parameter state diagram. We discuss briefly the possible physical interpretation of each of these states, and remark also that our spinodal friction law, though more complicated than other classical rate-and-state laws, is required in order to capture the full richness of wave types.
在一系列长度尺度上的滑动摩擦界面会产生行波；例如，这些行波被认为与地震地面运动以及机械制动器和减震器的性能都相关。我们通过研究一个理想化的力学模型来解释这些波的起源：一块薄弹性板，受到均匀剪应力作用，并与一个刚性平面保持摩擦接触。我们为板的变形构建了一个非线性波动方程，并将其与一个旋节线速率-状态摩擦定律耦合，这导致了一个数学上适定的问题，能够捕捉许多库仑摩擦模型中无法实现的效应。我们的模型支持多种丰富的解，包括周期性的粘滑波列、孤立的滑动和粘着脉冲，以及分离和附着前沿。通过解析和数值分岔分析来展示这些状态在双参数状态图中的组织方式。我们简要讨论了这些状态各自可能的物理解释，并且还指出，为了捕捉波类型的全部丰富性，我们的旋节线摩擦定律虽然比其他经典的速率-状态定律更复杂，但却是必需的。
