Dispersion and localization of elastic waves in materials with microstructure

Abstract

This paper considers the interaction of elastic waves with materials with microstructure. The paper presents a mathematical model of elastic waves within a lattice system incorporating rotational motions and interaction between different lattice elements through elastic links. The waves are dispersive and the lattice system itself is heterogeneous, i.e. the elastic stiffness and/or mass are non-uniformly distributed. For such systems, one can identify stop bands, representing the intervals of frequencies of waves, which become evanescent and cannot propagate through the structure. Filtering properties of such lattices are studied in this paper. Defect modes are created by removing a periodic array of elastic links, which leads to localization within a macro-cell. Special attention is given to the evaluation of the effective group velocities and to the study of standing waves within the system. Analytical estimates are accompanied by numerical simulations and analysis of dispersion surfaces. We also consider an example showing the focusing and the creation of an image point by a flat elastic ‘lens’ formed from a finite micropolar lattice system.