A Perturbation Approach for Analyzing Dispersion and Group Velocities in Two-Dimensional Nonlinear Periodic Lattices

Abstract

The paper investigates wave dispersion in two-dimensional, weakly nonlinear periodic lattices. A perturbation approach, originally developed for one-dimensional systems and extended herein, allows for closed-form determination of the effects nonlinearities have on dispersion and group velocity. These expressions are used to identify amplitude-dependent bandgaps, and wave directivity in the anisotropic setting. The predictions from the perturbation technique are verified by numerically integrating the lattice equations of motion. For small amplitude waves, excellent agreement is documented for dispersion relationships and directivity patterns. Further, numerical simulations demonstrate that the response in anisotropic nonlinear lattices is characterized by amplitude-dependent “dead zones.”

Multiple scales analysis of wave–wave interactions in a cubically nonlinear monoatomic chain

“…Various types of periodic media have been studied in the past, including strongly nonlinear contact in beaded systems [4], kink dynamics [21], and weakly coupled layered systems [25,26]. One of the most commonly studied systems in the last few decades is the one-dimensional, undamped spring-mass chain [13][14][15]17]. The dispersion relation of a linear chain relating the frequency and wavenumber is well known, and in one dimension the analysis may be confined to the first Brillouin Zone due to spatial periodicity [11].…”

“…Recent research has focused on wave propagation in nonlinear periodic media and its application to the design of novel metamaterials [1,3,7,8,[13][14][15]22]. Nonlinearities play an increasingly dominant role in the dynamics of small scale applications [5,9,10], such as periodic metamaterials with micro-or nanoscale unit cells.…”

Modeling of static deformations and dynamics of localized rotation in a chain of finite size particles

“…It is important to note, that linear analysis is rather standard approach to study localization effects, which decay with the distance. It is usually used in the study of dynamical effects in lattices [10,11,12], frequency filters [12,13], the tails of localized excitations, such as discrete breathers [14,15] and roto-breathers [16], for the analysis of their existence and properties.…”

Piezo-electromechanical smart materials with distributed arrays of piezoelectric transducers: Current and upcoming applications

“…For a comprehensive description of the models which we consider needed in this context we refer to [135]. If nonlinearities are to be included into the picture then the results described in [134,148,147] can be an useful starting point. Interesting multiple time scale analyses are conceivable to design wave modulations presenting energy transfer from one vibration mode to others and possibly its more effective dissipation (see the methods presented in [123,125] and references therein).…”