Low-frequency flexural wave attenuation in metamaterial sandwich beam with hourglass lattice truss core

Highlights

• A novel metamaterial sandwich beam is designed for flexural wave attenuation.

• A novel approach is developed to establish the model of metamaterial sandwich beam.

• The mass-loaded struts are represented by equivalent mass–spring local resonators.

• The feasibility of using the simplified CMB model to represent MSB is validated.

Abstract

Though lightweight sandwich structures have been extensively applied in practical engineering, it remains a challenge to control wave propagation and vibration in these structures in a low-frequency range. In this work, the band structure of flexural waves in a metamaterial sandwich beam (MSB) with hourglass lattice truss core is investigated using the transfer matrix method (TMM). The hourglass truss structure with lumped masses is modelled as a series of local resonators with determined equivalent stiffnesses and masses. A metamaterial dual-beam (MDB) model is then established to describe the MSB, and the MDB model is noted to be equivalent to the conventional metamaterial beam (CMB) model under base excitation. The MSB is further studied directly by the finite element method that confirmed the MSB can be represented by the CMB through the transmittance analysis and band structure analysis. Subsequently, parametric study is performed to investigate the effects of the material and structural parameters on the band structures of the MSB. This work provides a roadmap of modelling of lightweight lattice sandwich beams with complex core structures and presents guidelines for applying sandwich beams to control wave propagation.

Introduction

Metamaterials are artificial periodic structures that can generate band gaps that are frequency ranges within which the propagation of waves is forbidden [1], [2], [3], [4], [5], [6], [7]. Because of this intriguing phenomenon, extensive research interests have been attracted into exploring the theory and applications of metamaterials in wave attenuation and vibration control. There exist two basic band gap generation mechanism, i.e., Bragg scattering (BS) [8], [9], [10] and local resonance (LR)  [11], [12]. BS exists in phononic crystal and the band gap occurs in the frequency range where wavelengths are of the same order of magnitude as the lattice constant. Thus, a large lattice constant is required to attenuate low-frequency waves, which is often not realistic in the design of practical engineering structures. Different from phononic crystals, based on the local resonance mechanism, metamaterials could easily generate a low-frequency band gap. Therefore, enormous efforts have been devoted to studying and tailoring band gaps of metamaterials.

As well-known that lots of engineering structures, such as bridges [13], spacecraft arms [14] and building frames [15] can often be modelled as flexible beams, if the transverse dimension is much smaller than the longitudinal dimension. The terminology ‘beam’ implies that the structure is relatively thin and the transverse modes occur at lower frequencies than longitudinal modes. For this reason, in the low-frequency domain, the transverse vibration of a beam-like structure has a more significant effect on the structural safety and stability. Hence, significant attention has been given to study the transverse vibration of beams [16], [17], [18]. In recent years, some researchers focused on using metamaterials for low-frequency flexural wave attenuation of continuous systems. By using the transfer matrix method (TMM), Yu et al. [11], [19] researched the flexural wave attenuation of LR beams, in which the local resonators were made up of soft rubber rings and copper rings. Xiao et al. [20] investigated the band gap formation mechanism with an analytical method on basis of the periodic structure theory and spectral element method. In order to broaden the width of the band gap, periodic local resonators arrays with different resonant frequencies were designed and studied  [21], [22]. Another way to achieve a wide band gap is to construct a resonator with multiple degrees of freedom (DOF). Pai et al. [23], [24] proposed a two-DOF subsystem to be attached to the beam and plate for inducing two band gaps. The inertial force produced by the resonance of the absorber enhances the wave attenuation. Wang et al. [25], [26] theoretically and numerically studied flexural vibration of a metamaterial beam and plate with attached lateral local resonators. They found it could generate two band gaps to attenuate the flexural vibration, for which the formation mechanism is due to the transition from the flexural wave to longitudinal wave by a four-link-mechanism, which stimulates the lateral resonance to generate inertial force to counterbalance the shear force of the plate. Miranda Jr. et al. [27] studied flexural waves propagation in a metamaterial plate using Kirchhoff–Love theory, and found that by changing the arrays of multiple DOF resonators, the locally resonant band gaps could be effectively widened. By combining auxeticity and phononic crystals band gap properties, D’Alessandro et al. [28] obtained a tunable wide band gap in numerical and analytical models. On the other hand, some researchers installed piezoelectric shunts periodically on host structures as adjustable resonators to control vibration and wave propagation [29], [30], [31].

In addition to widening the band gaps, many researchers poured attention into achieving band gaps in low or ultra-low-frequency range. Since the band gap location of LR metamaterial beam mainly depends on the natural frequency of resonators, to achieve low-frequency band gap, one has to increase the mass or decrease the stiffness of the resonators. The concept of inertial amplification is proposed by Yilmaz et al. [32] to embed the amplification mechanism into the unit cell, which could effectively increase the inertia of resonators and reduce the resonance frequency. Assouar et al. [33] presented hybrid metamaterial plates that consisted of periodic stepped pillars and holes, in which the waves scattered simultaneously by the pillars and holes in certain frequency ranges generate wide and low band gaps. Zhou et al. [34] presented a novel resonator which combines a vertical spring with two oblique springs that provide negative stiffness in vertical direction, and found the band gap could be shifted into very low frequencies by tuning the stiffness of the oblique springs. D’Alessandro et al. [35] presented a novel 3D elastic periodic structure with a distributed set of local resonators to realize low-frequency band gaps. Fang et al. [36] unveiled the nonlinear chaotic mechanism in nonlinear acoustic metamaterials (NAMs) for achieving band gaps and chaotic bands in an ultra-low and ultra-broad frequency range.

In past decades, many researchers focused on the dynamics of sandwich structures [37], [38], [39], [40], [41], because sandwich structures with high strength and low weight are ideal solutions to realize lightweight characteristics in practical applications for bearing large bending load. In order to reduce the possibility of catastrophic accidents caused by vibration, it is necessary to control the vibration levels of the sandwich structures. Relative to the traditional active [42] and passive [43] vibration control methods, as well as Bragg scattering metamaterial [44], the metamaterial theory could generate and tune band gaps to attenuate the flexural wave in specific low-frequency ranges. Chen et al. [45], [46] investigated flexural vibration behaviour of a sandwich beam with local resonators embedded into foam cores analytically and experimentally. It is assumed that resonators were uniformly distributed along the sandwich beam with the volume averaging technique. Their proposed model, however, did not account for the effect of the periodicity of discrete resonators on band structures of the sandwich system. Based on this work, Sharma and Sun [47] used the phased array method to calculate propagation constants of a sandwich beam with resonators inserted into the core layer, concluding a more clear explanation for the effect of periodicity of resonators on flexural vibration of the sandwich beam. Furthermore, Chen and Huang [48] proposed a sandwich structure embedded with multiple resonators that can generate a plurality of band gaps with excellent wave attenuation characteristics. Nevertheless, widths of band gaps for the proposed single resonator or multiple resonators sandwich structures are still relatively narrow, which cannot completely satisfy the requirement of practical applications such as the blast- or impact-induced wave in which the frequency ranges tend to be quite broad. For this reason, Chen et al. [49] added the dissipative multiple resonators to the core layer of sandwich beams to achieve a wide wave absorption band efficiently.

Though some literature has been published on wave attenuation of sandwich structures, most of them inserted lumped mass–spring resonators into the idealized homogenized core [45], [46], [47], [48], [49]. Thus, the proposed systems are not realistic especially in terms of the implementation of the resonators. Other researchers proposed some practical meta-structures designed with complicated micro-structures and manufactured using 3D printing technology  [7], [35]. However, the proposed structures are relatively arbitrary. Though excellent dynamic properties are achieved in the proposed structures, the static properties which are also of great importance in the engineering field [50], [51], [52] can not be guaranteed. For this reason, based on the sandwich beam with hourglass truss structure whose static properties have already been experimentally studied and proved to outperform those with traditional lattice cores [53], [54], [55], such as the hourglass truss structure with relative density ranging from about 1.1% to 2.7% has 40%–60% higher shear strength and 26%–47% higher out-of-plane compressive strength than those of the pyramidal truss structure with similar relative density, as well as better bending strength. This paper proposes a novel metamaterial sandwich structure to realize wave attenuation in a low-frequency range. The hourglass truss structure is made up of eight struts. It is similar to a two-layer pyramidal truss structure but has a smaller slenderness ratio and a superior resistance to the buckling of the core layer. The oblique struts in the hourglass core sandwich beam are readily used as the springs for implementing the local resonators, making the proposed meta-structure relatively more realistic. Besides of the novel structure, from the methodology perspective, the model presented in this work simplifies the oblique struts as vertical springs to help reduce the difficulty in mathematical modelling.

Researchers have investigated the dynamic behaviour of such structures  [56], [57], however, designing the hourglass core sandwich beam with a metamaterial characteristic for vibration suppression has never been found in the existing literature. In this work, we develop a novel approach to establish the model of the metamaterial sandwich beam with hourglass core and investigate its band gap phenomenon. A lumped mass is added on the intersection of the eight oblique struts for the hourglass structure unit cell to assemble a resonator, in which the oblique struts provide elastic stiffness in vertical direction. The equivalent stiffness of the hourglass truss structure is obtained by using Hook’s law. The metamaterial sandwich beam (MSB) is then simplified into a metamaterial dual-beam (MDB), which can eventually be modelled as a conventional metamaterial beam (CMB) under base excitation. The feasibility of using the simplified CMB model to represent MSB is validated based on both transmittance and band structure analysis for the MSB using finite element method (FEM). The effects of material and structural parameters on the band structures of MSB are then studied.