Left-handed shear and longitudinal elastic waves in 2D phononic crystals made of a solid matrix

Waves propagating in left -handed materials have unusual properties such as phase and group velocities of opposite signs and negative refraction index. Periodic lattices have been shown to exhibit such properties both for electromagnetic (photonic crystals) and in-fluid acoustic (phononic crystals) waves. This work addresses the question of the existence of left handed elastic waves in phononic crystals. Two-dimensional phononic crystals made of square lattices of cyl indrical cavities or inclusions in a solid matrix are considered. Dispersion curves are computed using plane wave expansion method for real wave vectors in the Brillouin zone and finite element method for complex wavenumbers along a specific propagation direction. From these results, the existence and symmetry of the lefthanded propagation mode in the phononic crystal is discussed and its relationship with lattice geometry and constitutive materials is analyzed.