Metastable Modular Metastructures for On-Demand Reconfiguration of Band Structures and Non-Reciprocal Wave Propagation

Metastable Modular Metastructures for On-Demand Reconfiguration of Band Structures and Non-Reciprocal Wave Propagation
Z. Wu1*, Y. Zheng1, 2 and K.W. Wang1 1 Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109-2125 2 State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi’an Jiaotong University, Xi’an 710049, P.R. China

• Corresponding author, email: wuzhen@umich.edu Abstract We present a novel approach to achieve adaptable band structures and non-reciprocal wave propagation by exploring and exploiting the concept of metastable modular metastructures. Through studying the dynamics of wave propagation in a chain composed of finite metastable modules, we provide experimental and analysis results on non-reciprocal wave propagation and unveil the underlying mechanisms in accomplishing such unidirectional energy transmission. Utilizing the property adaptation feature afforded via transitioning amongst metastable states, we uncovered an unprecedented bandgap reconfiguration characteristic, which enables the adaptivity of wave propagation within the metastructure. Overall, this investigation elucidates the rich dynamics attainable by periodicity, nonlinearity, asymmetry, and metastability, and creates a new class of adaptive structural and material systems capable of realizing tunable bandgaps and non-reciprocal wave transmissions. 1 Introduction Reciprocity of wave propagation is a fundamental principle [1] [2], describing the symmetry of wave transmission between two points in space. If wave can propagate from a source to a receiver, it is equally possible for the wave to travel in the opposite path, from the receiver to the source. Motivated by the concept of electrical diodes, directional flow of electrons in the presence of electric field, large amount of research attention have been devoted to explore the possibility of breaking the time-reversal symmetry and realizing one-way propagation in energy forms [3] [4] [5] [6] [7] [8] [9] [10]. Since linear structures alone cannot break the reciprocity in reflection-transmission if time reversal symmetry is preserved [11], considerable efforts have been devoted to realize non-reciprocal behavior in linear systems with additional symmetry breaking mechanisms. For instance, Fleury et al presented an acoustic circulator based on angular-momentum biasing through a circulating fluid [12]; Swinteck et al demonstrated bulk waves with unidirectional backscattering-immune topological states using superlattice with spatial and 2

temporal modulation of the stiffness [13]; Wang et al proposed an all-optical optical diode via a “moving” photonic crystal to control the flow of light [14]; and Thota et al realized reconfigurable one-way acoustic wave propagation via origami folding with spatial modulated lattices [15]. In parallel to advances in spatiotemporal modulated linear systems, another major contribution in achieving non-reciprocal wave propagation is through nonlinear systems. Liang et al coupled a nonlinear medium with a supperlattice and accomplished unidirectional acoustic wave propagation by exploiting second-harmonic generation (SHG) of the nonlinear medium together with frequency selectivity of the linear lattices [7]. Boechler et al utilized the combination of frequency filtering and asymmetrically excited bifurcations in a defected granular chain to obtain rectification ratios greater than 104 [8]. Popa and Cummer characterized an active acoustic metamaterial coupled to a nonlinear electronic circuit and demonstrated an isolation factor of >10 dB [16]. While many of these and other pioneer works pivoted primarily on the realization of unidirectional energy transmission, systems capable of on-demand tuning of non-reciprocal wave propagations, which are beneficial in many applications [17] [18], are yet to be addressed.

In complement but in contrast to previous contributions, in this research, we present a novel approach to accomplish non-reciprocal wave propagation with exceptional adaptivity by exploiting the concept of metastable modular metastructures, systems that exhibit coexisting stable states for the same topology. The proposed bottom-up metastructure concept is invested with direct pathways to facilitate global property adaptation by switching amongst the metastable states. Indeed, studies on metamaterials [19] [20], adaptive machines [21], and sensory adaptation systems [22] have all provided evidence that adaptivity may be induced by leveraging coexisting metastable states. To examine how the combination of nonlinearity, spatial asymmetry, periodicity and reconfigurability gives rise to unprecedented adaptive unidirectional wave propagation characteristics, the paper is organized as follows. In Sec II, the overall concept of the proposed unit module and metastructure is introduced. In Sec. III, the equations describing the nonlinear dyn

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