Analytical and numerical 1D modelling of the nonlinear scattering at a rough-surface contact interface with clapping

The nonlinear scattering of a longitudinal wave induced by the clapping of a pair of rough surfaces under a compressive pre-stress is investigated using a model that combines nonlinear springs and unilateral contact. This novel approach captures both the strong nonlinear response induced by clapping (intermittent loss of contact) and the weaker nonlinear response induced by rough surface contact. An analytical solution is first derived for the case of linear springs and clapping, which provides a useful basis for understanding the interface response. It reveals the existence of a critical interface gap opening required to trigger the loss of contact, and leads to a judicious definition of a non-dimensional load ratio characterizing the onset of clapping. The accuracy of this solution is confirmed numerically using a finite difference model, which is then applied to the more general case of nonlinear springs with clapping. Using the analytical and finite difference methods, we obtain the evolution of the coefficients of reflection and transmission, as well as the second harmonic amplitude, which are evaluated as a function of the non-dimensional load ratio, by varying the incident wave amplitude, the static compression of the interface, and the frequency. The results show that a quadratic nonlinearity in the spring response has a relatively small influence on the reflection and transmission coefficients at the fundamental frequency, but a significant influence on the relation between the amplitude of the second harmonic amplitude and the load ratio. The presence of nonlinearity during the contact phase is shown to cause a smoother variation of second harmonic amplitude with the load ratio, and to a slight increase in the maximum amplitude that can be achieved at any given frequency.