Continuum modelling of frequency dependent acoustic beam focussing and steering in hexagonal lattices

Abstract

Wave propagation in hexagonal lattices is characterised by a transition from isotropic to anisotropic regime as the frequency increases. This feature cannot be observed when the lattice material is modelled within the framework of classic Cauchy theory of elasticity, except when the real geometry of the microstructure is explicitly described. Homogenized equivalent continua can capture the onset of this anisotropy only if tensors involved in the constitutive law are at least of order six. This requirement is met in the case of Strain-Gradient Elasticity (SGE). In this work the SGE model is calibrated to quantitatively describe wave propagation within hexagonal lattices in a sufficiently large region of the dispersion diagram, by fitting the dispersion relations obtained from a Bloch-Floquet analysis on the unit cell. It is then shown that focussing and beam steering effects can be obtained by acting only on the local material orientation of the hexagonal lattice. This is achieved by computing the Poynting vector in the SGE model and optimising the distribution of material orientation.