Dynamic Green’s functions for lattice systems with frontiers possessing contrasting periodicity and material properties

We consider anti-plane waves propagating in a lattice, formed from periodically placed masses interconnected by springs, that possesses a bi-periodic frontier with contrasting inertial and elastic properties. In particular, we study the response of such a system to the application of a point force, corresponding to Green’s function. This problem is studied via the application of the Z-transform which yields Green’s function in an exact form in terms of a contour integral. We demonstrate how this approach also provides explicit dispersion relations for waves along the lattice frontier. We study the influence of the lattice frontier’s composition the pass band structure linked to the existence of these waves. A low frequency homogenisation model is also developed that incorporates the effects due to the elastic and inertial properties of the frontier. We also show how our technique can be extended to study the dissimilar lattice systems with a bi-periodic interface. All analytical results are accompanied by numerical illustrations.