We experimentally demonstrate the presence of negative refraction on acoustic passbands of two-dimensional phononic crystals for the in-plane and the anti-plane shear waves. We investigate the phenomenon on two geometrically identical two-phase crystals of different material properties, i.e., one stiff crystal (Aluminum matrix with PMMA inclusions), and another a soft crystal (PMMA matrix with Aluminum inclusions). We demonstrate that in the case of in-plane shear wave, the soft crystal does not show negative refraction in the first passband, however, the stiff crystal does show negative refraction for the first mode. In the case of the anti-plane shear wave tests, both crystals show only positive refraction in the first passband. However, in the soft crystal, negative refraction is present in the second passband. We also investigate and show that for the longitudinal mode (second acoustic passband), both stiff and soft crystals, exhibit negative refraction. The experimental results confirm the prediction of our theoretical model, which allows us to predict the behaviors of phononic crystals as we change the properties of their constituents.

In the second part of this dissertation, by the application of Noether’s theorem, conservation laws in linear elastodynamics are derived by invariance of the Lagrangean functional under a class of infinitesimal transformations. The work of Gupta and Markesncoff (2012), which provides a physical meaning to the dynamic J-integral as the variation of the Hamiltonian of the system due to an infinitesimal translation of the inhomogeneity if linear momentum is conserved in the domain, is extended here to the dynamic M- and L- integrals in terms of the ‘if’ conditions. We show that the variation of the Lagrangean is equal to the negative of the variation of the Hamiltonian under the above transformations for inhomogeneities, and hence provide a physical meaning to the dynamic J-, L- and M-integrals as dissipative mechanisms in elastodynamics. We prove that if linear momentum is conserved in the domain, the total energy loss of the system per unit scaling under the infinitesimal scaling transformation of the inhomogeneity is equal to the dynamic M-integral. Moreover, if linear and angular momenta are conserved, the total energy loss of the system per unit rotation under the infinitesimal rotational transformation is equal to the dynamic L-integral.