  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.
