Bandgaps in functionally graded phononic crystals containing graphene origami-enabled metamaterials
Graphical abstract
Introduction
Periodic structures made of two or more repetitively distributed materials are known as phononic crystals (PCs). These artificial composites were first proposed as analogous systems to photonic crystals and proved to exhibit a frequency-selecting property for elastic waves in the 1990s [1,2]. Over the last few decades, PCs have attracted great interest from various fields to explore practical applications such as sound insulation [3], seismic isolation [4,5], filters [6], waveguide transducers for sensing [7], and the design of novel acoustic devices like wave transistors [8], acoustic lenses [9], [10], [11], [12], wave detectors [13].
In order to make the PCs more versatile in applications, various smart materials including piezoelectric materials [14,15], piezomagnetic materials [16,17], thermal sensitive materials [18,19], functionally graded materials (FGMs) [20,21], etc., have been introduced into PCs to manipulate elastic waves. Based on previous studies, bandgaps result from the destructive interference of Bragg scattering and their existence requires a periodic variation of mechanical properties of materials in PCs [22,23], but it also makes the structures more fragile at interfaces between different materials. Consequently, FGMs are attracting more and more research interest in the last decade, as they can smoothen the transition between different materials and improve the bonding of adjacent components [24]. Besides, patterns of the graded distribution can also be designed to change the dispersion characteristics of elastic waves in PCs, which is one of the great advantages of functionally graded PCs. The first research on functionally graded laminated phononic crystals (FGLPCs) started with a simple laminated model investigated by Wu et al.[25]. The propagation of longitudinal waves (P-waves) was studied based on the plane wave expansion method and transfer matrix method. They found that the band structures in FGLPCs can be controlled by different functionally graded distributions and geometry parameters. Su et al.[26] discussed the effect of material properties on band structures of FGLPCs with continuous functionally graded distribution patterns. By transforming the model equation to Heun's equation, Bednarik et al.[27] analyzed the propagation of P-waves in FGMs under a trigonometric distribution. Recently, periodic thermal fields were considered to manipulate the band structures of P-waves in FGLPCs [18]. Besides, shear horizontal (SH) waves in periodic structures are also extensively investigated. Golub et al.[28] analyzed the transmission of SH waves in FGLPCs with both exact and approximate functionally graded models. Yan et al.[29] discussed the characteristics of bandgaps of SH waves in nanoscale FGLPCs with considering the imperfections in the structures. By introducing the thermal sensitive materials, Bian et al.[30,31] explored manipulating SH waves in functionally graded PCs with the applying thermal fields. Piezoelectric and piezomagnetic materials were also introduced into the manipulation of SH waves in functionally graded PCs [32,33]. For PCs with more complex wave motions, there were many excellent research works. Fomenko et al.[34] investigated the transmission and bandgaps of in-plane waves in laminated periodic structures composed of functionally graded interlayers by transfer matrix method. Sepehri et al.[35] studied the wave propagation properties of PCs made of FGMs with hexagonal, rectangular, and triangular unit cells using the FEM and Bloch theorem. Yan et al.[36] calculated the dispersion relations of in-plane waves propagating in the periodic grid structures with local resonators by spectral element method and Bloch theorem. Lu et al.[37,38] analyzed the vibration response and band gaps of functionally graded frame structures by using the spectral element method. In-plane and anti-plane waves in functionally graded PCs made of piezoelectric and piezomagnetic materials were also studied [39,40].
Poisson's ratio (PR) is one of the important physical parameters to describe the mechanical characteristics of solids. It is used to quantify the material deformation in directions perpendicular to the direction of external forces [41]. Although the theoretical limit of PR for isotropic material is from −1 to 0.5 [42], [43], [44], common solids, such as metals and functional oxides, have a positive Poisson's ratio of 0.1–0.3 [45]. In fact, most of nature materials’ PRs are usually between 0 (cork) and 0.5 (rubber) [46,47]. Until 1987, negative PR was observed in a novel foam structure by Lakes [48]. Inspired by his findings, many studies were conducted to investigate these unusual materials with negative PRs which are also termed auxetic metamaterials [43,[49], [50], [51]]. Man-made materials with negative PRs are usually obtained by designing specific internal geometries in base materials [52,53], which makes them more similar to structures but not materials. Recently, a class of novel composites, named graphene origami-enabled auxetic metamaterials (GOEAMs), has been designed [54,55]. Different from most mechanical metamaterials with negative PRs, GOEAMs do not rely on the topology of structures but achieve auxetic properties by adding origami graphene filler into a metal matrix. They can be used to design structures as composite materials. Moreover, with the addition of graphene origami fillers, the mechanical properties of the composites will outperform their conventional counterparts [54]. These novel composites may bring opportunities to renovate manipulation methods for elastic waves in periodic structures. Previous studies for bandgaps in PCs pay less attention to the effect of PR on elastic waves, as the propagation characteristics of elastic waves have no significant change in the positive region of PR. By introducing GOEAMs into PCs, ones can explore special propagation properties in materials with negative PRs and look for novel paths to manipulate elastic waves. Besides, compared to conventional materials, using GOEAMs to fabricate PCs can also improve mechanical properties, including indentation resistance, fracture toughness, shear resistance, etc.[56]. Nevertheless, to the best of our knowledge, there is a gap in the study of functionally graded PCs made of auxetic metamaterials and in the investigation of PR's effects on band structures.
Hence, this paper focuses on the dispersion characteristic of elastic waves in FGLPCs made of GOEAMs with different graded distributions of graphene origami in unit cells. First, the model of FGLPCs and the graded distributions for graphene content and hydrogen coverage are illustrated. The material properties of GOEAMs are determined by a genetic programming-assisted micromechanical model [57]. Then, the governing equations for each layer of laminated PCs are established, and the state space approach [58] is used to cast all equations into compact state equations. Based on the solutions of the state equations, the dispersion relations of elastic waves propagating along the thickness direction in FGLPCs are obtained under the framework of the method of reverberation-ray matrix (MRRM) [59], [60], [61]. In numerical examples, dispersion curves are depicted according to the dispersion relations to analyze the effect of graded distributions of graphene origami weight fraction (WGr) and hydrogen coverage (HGr) on bandgaps. Then, the influence of PR on bulk waves in GOEAMs and band structures of elastic waves in FGLPCs are investigated.
Section snippets
Methods
In this section, details of the FGLPC model will be illustrated and the process of obtaining dispersion relations of elastic waves in FGLPCs will be revealed. The FGLPC model description includes the explanation of relevant symbols, the introduction for GOEAM used in FGLPC, and the illustration of functionally graded distributions. In theoretical formulation, the dispersion relations of elastic waves will be obtained based on the state space approach and MRRM.
Numerical results
In this section, the effects of PR on the elastic waves propagating along the thickness direction of FGLPCs will be discussed through parametric studies for the WGr and HGr of graphene origami in GOEAMs. In unit cells, the material properties of GOEAMs in each layer are determined by Eqs. (1) to (3) combined with the graded distributions of WGr and HGr shown in Fig. 3. Material parameters of pure Cu and graphene at 300 K used in the following sections are: ECu = 65.79GPa, , ρCu
Conclusions
This paper investigates band structures in FGLPCs made of GOEAMs. WGr and HGr of graphene origami are distributed into functionally graded forms in unit cells of FGLPCs to reduce the difference of material properties between layers in FGLPCs and improve the bonding of structures. Introducing GOEAMs into FGLPCs can improve the indentation resistance, fracture toughness, and shear resistance of the structures due to the features of GOEAMs. Meanwhile, the PR of GOEAMs can be changed from positive
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
The work was fully supported by the Australian Research Council grant under the Discovery Project scheme (DP210103656) and China Scholarship Council. The authors are very grateful for the financial support.
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