Analytical and numerical modelling of wave scattering by a linear and nonlinear contact interface

A new analytical approach is presented for describing plane-wave scattering at a plane interface that is characterised by a linear or nonlinear traction law. The crucial step is a decomposition of the scattered field into two contributions that exhibit symmetries corresponding to Mode I or Mode II fields, in the terminology of fracture mechanics. As a consequence, the scattering problem for linear springs is reduced to determining two unknowns, corresponding to the normal and the tangential components of the relative displacement across the interface, thereby leading to an explicit analytical solution. By contrast, the conventional approach is formulated in terms of the four unknown amplitudes for the reflected and transmitted longitudinal and transverse waves, with the solution having to be determined numerically for particular values of the incident angle and material properties. Next, a perturbation approach is employed to derive analytical solutions for nonlinear springs. This approach leads to successive approximations that can also be solved analytically, thereby resulting in analytical formulae for scattering by weakly nonlinear springs. These analytical formulae are compared with computational results based on the finite element method, utilising parameter values that are representative of rough-surface contact. Excellent agreement with the computational results is demonstrated, thereby highlighting the practical value of the analytical formulae for parametric studies.