Shear to longitudinal mode conversion via second harmonic generation in a two-dimensional microscale granular crystal

TL;DR

This study explores shear to longitudinal mode conversion via second harmonic generation in a 2D microscale granular crystal, revealing resonant behaviors and emphasizing the importance of including rotational degrees of freedom.

Contribution

It provides a theoretical and computational analysis of second harmonic generation in a 2D granular medium, highlighting the role of rotations and shear interactions in nonlinear wave phenomena.

Findings

Second harmonic waves are longitudinal in all cases.

Resonant and antiresonant wave numbers are identified.

Rotations are essential for antiresonant behavior.

Abstract

Shear to longitudinal mode conversion via second harmonic generation is studied theoretically and computationally for plane waves in a two-dimensional, adhesive, hexagonally close-packed microscale granular medium. The model includes translational and rotational degrees of freedom, as well as normal and shear contact interactions. We consider fundamental frequency plane waves in all three linear modes, which have infinite spatial extent and travel in one of the high-symmetry crystal directions. The generated second harmonic waves are longitudinal for all cases. For the lower transverse-rotational mode, an analytical expression for the second harmonic amplitude, which is derived using a successive approximations approach, reveals the presence of particular resonant and antiresonant wave numbers, the latter of which is prohibited if rotations are not included in the model. By simulating a lattice with adhesive contact force laws, we study the effectiveness of the theoretical analysis for non-resonant, resonant, and antiresonant cases. This work is suitable for the analysis of microscale and statically compressed macroscale granular media, and should inspire future studies on nonlinear two- and three-dimensional granular systems in which interparticle shear coupling and particle rotations play a significant role.