Dispersion and topological characteristics of permutative polyatomic phononic crystals

Abstract

This work presents a comprehensive mathematical treatment of phononic crystals (PCs) which comprise a finite lattice of repeated polyatomic unit cells. Wave dispersion in polyatomic lattices is susceptible to changes in the local arrangement of the monatoms (subcells) constituting the individual unit cell. We derive and interpret conditions leading to identical and contrasting band structures as well as the possibility of distinct eigenmodes as a result of cyclic and non-cyclic cellular permutations. Different modes associated with cyclic permutations yield topological invariance, which is assessed via the winding number of the complex eigenmode. Wave topology variations in the polyatomic PCs are quantified and conditions required to support edge modes in such lattices are established. Next, a transfer function analysis of finite polyatomic PCs is used to explain the formation of multiple Bragg band gaps as well as the emergence of truncation resonances within them. Anomalies arising from the truncation of the infinite lattice are further exploited to design mirror symmetrical edge modes in an extended lattice. We conclude with a generalized explanation of the band gap evolution mechanism based on the Bode plot analysis.

Highly Localized and Efficient Energy Harvesting in a Phononic Crystal Beam: Defect Placement and Experimental Validation

“…The passband resonance modes are in general larger than 0 dB in the displacement transmission. Although the considered PC beam is also truncated from an ideally infinite PC beam, we note that the defect modes are different from the truncation modes or surface modes in that wave will be localized around the mismatched unit cell at the frequency of the defect mode [29][30][31]. The insets in Figure 2a show the predicted output voltage around the two band gaps when an external load resistance of 500 kΩ is applied in the FEM simulation using the module "Piezoelectric Devices" in COMSOL MULTIPHYSICS.…”

“…(c) Full-field strain distribution at the frequency of the second defect mode of the defect and perfect PC beam. Crystals 2019, 9, x FOR PEER REVIEW 5 of 13mode[29][30][31]. The insets inFigure 2(a)show the predicted output voltage around the two band gaps when an external load resistance of 500 k is applied in the FEM simulation using the module "Piezoelectric Devices" in COMSOL MULTIPHYSICS.…”

Tunable topological interface states in one-dimensional inerter-based locally resonant lattices with damping

“…The analysis in the aforementioned works [38,40] is confined to the diatomic-like locally resonant lattices with a limited number of non-trivial gaps and topologically protected edge/interface states. In [41], the authors have demonstrated the existence of several edge states in 1D polyatomic lattices with different cyclic permutations of springs and masses in the chain. Complex band structures and topological properties were obtained for different lattice configurations having multiple bands associated with higher winding numbers.…”

Hybrid Bandgaps in Mass-coupled Bragg Atomic Chains: Generation and Switching

“…Being classic textbook models that can explain lattice vibrations in solid state physics, onedimensional atomic chains can capture the most fundamental properties of phononic crystals (PCs) (Kittel et al, 1996;Hofmann, 2015), which are known for having Bragg bandgaps that can suppress the propagation of mechanical waves (Deymier, 2013;Khelif and Adibi, 2015). Over the past decade, based on PCs that contain the simplest spring-mass systems in a unit cell, some important advances in the domains of nonlinear wave guides (Narisetti et al, 2010;Porubov and Andrianov, 2013;Ganesh and Gonella, 2015;Fang et al, 2016), topological edge states (Pal et al, 2018;Al Ba'ba'a et al, 2019), and diode-like acoustic structures (Vila et al, 2017;Attarzadeh et al, 2018) have been achieved. The atomic chains have also been used to explain nonlocal interactions of the panels of origami metamaterials (Pratapa et al, 2018).…”