Fractional order models for the homogenization and wave propagation analysis in periodic elastic beams

The advancement of manufacturing techniques has led to a rapid increase in the design and fabrication of periodic and architectured materials for a variety of dynamics and wave manipulation applications. Classical low-frequency homogenization techniques have been popular and invaluable tools to facilitate the numerical simulation of these systems, although their application is naturally limited to the long-wavelength regime. As a result, scattering dominated features, such as the occurrence of frequency band-gaps, cannot be captured by these models. Indeed, the first band-gap typically marks the approximate limit of validity of low-frequency homogenization approaches. Abandoning the use of this approach allows recovering the detailed dynamic behavior due to the geometric features of the periodic units, but it also comes at a significant increase in computational cost. In this study, we leverage the use of fractional operators to revisit the classical low-frequency homogenization approaches and explore their possible extension beyond the initial band-pass (low-frequency) regime. Remarkably, the resulting fractional-order homogenization approach is capable of representing the dynamic behavior of periodic structures within the first few frequency band-gaps. In particular, we apply the fractional approach to model elastic wave propagation in a bi-material periodic beam which serves as an idealization of a one-dimensional elastic metamaterial. The use of the fractional operator formalism allows casting the classical integer-order wave equation with spatially-variable coefficients, typical of an inhomogeneous beam, into a fractional-order differential equation with constant coefficients. Thus, the spatial heterogeneity of the periodic system is accounted for via the order of the fractional derivative. The fractional-order governing equation is obtained via variational principles, starting from fractional-order kinematic relations. It is found that the resulting fractional differential model of the heterogeneous system has, in its most general form, a complex valued and frequency-dependent order. The results from the fractional order model are compared with those obtained from the classical wave equation in order to assess the validity of the approach and its performance. The dynamic analysis is carried out at both band-gap and band-pass regimes showing a good agreement with results from traditional methodologies.