Effects of the strain gradients on the band structures of the elastic waves propagating in 1D phononic crystals: An analytical approach

In this paper, an analytical method is proposed for the analysis of the effects of the strain gradients on the frequency band structures and band-gaps of the elastic waves propagating in one-dimensional (1D) nano-sized phononic crystals (PCs). The considered 1D nano-sized PCs are made of periodic unit-cells containing a metal section and an epoxy section. The strain gradients are taken into account to obtain the governing equations by using the strain energy, the kinetic energy and the Hamilton's principle. An analytical solution based on the transfer matrix method is derived to compute the frequency dispersion curves or band structures of the elastic waves. The continuity conditions on the interface between the neighboring sections and the periodic boundary conditions for the unit-cell are satisfied using the proposed analytical method. Two band-gap characteristics, namely the starting frequency and the width of each band-gap, are defined and investigated. The effects of the small-scale parameters in the strain-gradient theory on the band structures and the band-gap characteristics are analyzed and discussed in detail. It is demonstrated that the strain gradients may significantly influence the frequency band structures and the band-gap characteristics of the elastic waves propagating in 1D nano-sized PCs.