Band-gap shift and defect-induced annihilation in prestressed elastic structures

Design of filters for electromagnetic, acoustic, and elastic waves involves structures possessing photonic/phononic band gaps for certain ranges of frequencies. Controlling the filtering properties implies the control over the position and the width of the band gaps in question. With reference to piecewise homogeneous elastic beams on elastic foundation, these are shown to be strongly affected by prestress (usually neglected in these analyses) that (i) shifts band gaps toward higher (lower) frequencies for tensile (compressive) prestress and (ii) may annihilate certain band gaps in structures with defects. The mechanism in which frequency is controlled by prestress is revealed by employing a Green's-function-based analysis of localized vibration of a concentrated mass, located at a generic position along the beam axis. For a mass perturbing the system, our analysis addresses the important issue of the so-called effective negative mass effect for frequencies within the stop bands of the unperturbed structure. We propose a constructive algorithm of controlling the stop bands and hence filtering properties and resonance modes for a class of elastic periodic structures via prestress incorporated into the model through the coefficients in the corresponding governing equations.