Wave transmission in 2D nonlinear granular-solid interfaces, including rotational and frictional effects

Abstract

We study the highly complex wave transmission at the interface between a two-dimensional (2D) hexagonally structured granular medium and a linearly elastic thin plate; we refer to this system as the “granular-solid interface”. By applying an impulsive excitation at the free end of the granular medium we study the nonlinear acoustics at the interface. A computational model is developed, where the thin plate under the plane-stress assumption is discretized by finite-elements (FEs), whereas the granular medium by discrete-elements (DEs). Apart from the highly discontinuous Hertzian granule-to-granule and granule-to-plate interactions, we also take into account rotational and frictional effects in the granules; these effects render the acoustics of the granular-solid interface strongly nonlinear and highly discontinuous. The interaction forces coupling the granular medium to the plate are computed by means of an algorithm of interrelated iterations and interpolations at successive time steps. Since frictional effects may yield numerical instabilities, our approach incorporates the continuous “Coulomb–tanh” friction model, whose efficacy is verified through convergence studies. By formulating appropriate theoretically predicted convergence criteria, we show that the stability of the algorithm depends on the time step, the mesh size of the FE model, and the frictional model parameters. Accordingly, convergence is ensured by introducing a self-adaptive time step scheme, which is informed by theoretical convergence criteria. An application of the algorithm for a specific granular-solid interface demonstrates its validity, accuracy and robustness. Wave transmission through the discrete–continuum interface is drastically delayed by the granular medium, which, inflicts significant “softening” to the nonlinear acoustics. Moreover, there is strong nonlinear wave dispersion and energy localization in the granular medium, resulting in highly reduced wave transmission to the plate. Moreover, these nonlinear acoustical features are tunable with the applied shock (or input energy). The model and results presented in this work apply to a broad class of nonlinear discrete–continuum interfaces, with broad applications, e.g., shock/blast mitigation, granular containers with flexible boundaries and acoustic non-reciprocity.