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Diagrams of vibration amplitude ratios corresponding to the (a) acoustic, (b) fourth, (c) third, and (d) optic branches in Fig. 2(a) when α = 4, β = 2, α0 = 2, and β0 = 0.5 (color online)
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We have proposed an “exact” strain gradient (SG) continuum model to properly predict the dispersive characteristics of diatomic lattice metamaterials with local and nonlocal interactions. The key enhancement is proposing a wavelength-dependent Taylor expansion to obtain a satisfactory accuracy when the wavelength gets close to the lattice spacing....
Contexts in source publication
Context 1
... take the case with α = 4, β = 2, α 0 = 2, and β 0 = 0.5 to demonstrate relative motions of particles. Based on Eqs. (11), (13), and (14), we plot the vibration amplitude ratios in Fig. 3. Figure 3(a) gives the result for the acoustic branch in Fig. 2(a). At the extreme of ka 2π → 0, all amplitude ratios get close to 1, indicating that for an infinite wavelength, all particles in a unit cell move together like a rigid unit. Throughout the first Brillouin zone, A2 A1 0 suggests that particles M 2 and M 1 always keep the ...
Context 2
... on Eqs. (11), (13), and (14), we plot the vibration amplitude ratios in Fig. 3. Figure 3(a) gives the result for the acoustic branch in Fig. 2(a). At the extreme of ka 2π → 0, all amplitude ratios get close to 1, indicating that for an infinite wavelength, all particles in a unit cell move together like a rigid unit. ...
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Metamaterials are artificially engineered devices that go beyond the properties of conventional materials in nature. Metamaterials allow for the creation of negative refractive indexes; light trapping with epsilon-near-zero compounds; bandgap selection; superconductivity phenomena; non-Hermitian responses; and more generally, manipulation of the pr...
Citations
... Their properties are influenced by the intrinsic material properties as well as the geometric characteristics of the periodic unit cell. There have been intensive efforts which are mainly focused on various outstanding effective properties, such as negative Poisson's ratio [1], high specific strength [2], acoustic bandgaps [3][4][5], adjustable stiffness [6,7], negative effective modulus [8,9], zero Poisson's ratio [10], negative effective mass density [11] and so on. One important application lies in that lattice metamaterials work under vibration conditions in various fields, including wind turbines, mechanical engines, aerospace equipment and so on. ...
We examine the macroscopic moduli of lattices composed of multi-segment beams (MSBs), each of which includes more than one segment with independent geometric and material parameters, with a belief that a wider adjustable range can be obtained via introducing such MSB. To investigate the frequency dependence of effective moduli, we propose a dynamic stiffness formulation which is exactly calculated based on corresponding equations of motion. First, the MSB is taken as a multi-node finite element and its stiffness matrix is derived with the help of continuity requirements on inner joints. Then, macroscopic properties are obtained one by one by force analysis on the structural unit cell subjected to various external macrostress fields, resulting in exact analytical frequency-dependent expressions of effective moduli. The above formulation is verified by comparing with the dynamic finite element method (DFEM), and it shows the predicting ability in high-frequency domain. Compared to lattice with one-segment homogeneous truss, effects of geometric and material parameters of the truss segments on the macroscopic properties are investigated systematically. The adoption of MSBs is believed to be a feasible way of adjusting effective properties of lattices.
... Micro-scale models show local interactions and internal mechanisms [70][71][72]. As microscale models consider microstructural heterogeneity, they offer distinct advantages for investigating and optimizing lattice metamaterials [18, 73,74]. This section presents a comprehensive overview of computational methods for analyzing the mechanical behavior of lattice metamaterials. ...
... The Roton-like dispersion phenomenon shows that nonlocal connections can provide broadband acoustic backward waves and lead to negative refraction at the interface [25][26][27]. This mechanism allows us to design roton-like dispersion relations under ambient conditions that control the acoustic waves in an unusual way, resulting in lower frequency band gaps [28][29][30][31]. Wang et al. construct an accurate strain gradient (SG) continuum model to predict the dispersion properties of diatomic lattice metamaterials with local and nonlocal interactions throughout the first Brillouin zone [28]. ...
... This mechanism allows us to design roton-like dispersion relations under ambient conditions that control the acoustic waves in an unusual way, resulting in lower frequency band gaps [28][29][30][31]. Wang et al. construct an accurate strain gradient (SG) continuum model to predict the dispersion properties of diatomic lattice metamaterials with local and nonlocal interactions throughout the first Brillouin zone [28]. Ganesh and Gonella performed a comprehensive study of fluctuations in nonlinear chains and derived non-dispersive features during wave propagation [29]. ...
This paper presents a refined model for a mechanical diode based on a mass-spring system. The proposed model utilizes a bilinear spring to construct a frequency converter, which effectively disrupts the reciprocal transmission of acoustic waves. By employing a mass-spring-mass system as a filter, a nonlocal connection is introduced to generate an extremely low-frequency band gap (2–4 Hz), thereby achieving a mechanical diode with a lower operating frequency. The feasibility of these low-frequency mechanical diodes is demonstrated through comprehensive numerical simulations and experimental analyses. In addition, we evaluated the effect of bilinear springs and nonlocal connection parameters on the diode performance.
... They investigate its transient response using both discrete unit cell modelling and continuum modelling. Wang et al [37] proposed a strain gradient continuum model to predict the dispersive characteristics of diatomic lattice metamaterials. To the best of our knowledge, many continuous methods are used to study traditional local metamaterials. ...
Compared with classical metamaterials, nonlocal metamaterials have distributed long-range interactions. In this paper, a gradient continuum model is developed to properly predict the dispersive behaviour of a one-dimensional nonlocal metamaterial with long-range interactions. First, a discrete monoatomic model is reconstructed into a supercell model. Then, a Taylor expansion based on supercell model is applied to the continuous displacement field, resulting in a gradient continuum model. The dispersive relation of the gradient continuum model is obtained and compared with discrete supercell model to evaluate its suitability. The proposed gradient continuum model with the eighth-order truncation is found to be enough to capture the dispersion behaviours all over the first Brillouin zone. The results indicate that the proposed gradient continuum model can predict the dispersion behaviour of the one-dimensional nonlocal system very well. Furthermore, the gradient continuous model of two mass-in-mass system with long-range interactions are verified.
... Recently, Wang et al. 18 proposed an effective SG continuum model by adopting a wavelength-dependent Taylor expansion, and the corresponding results agree well with discrete results throughout the whole Brillouin zone. ...
... The distance between two same particles is a. The EOMs for each type of particle are as follows: 18,22 The dispersive equation of the diatomic mass-in-mass lattice is ...
Based on the symbiotic organisms search (SOS) optimization algorithm, a robust strain gradient (SG) continuum model has been proposed
to accurately capture the broadband dispersion relations of one-dimensional acoustic metamaterials. Under the continuous assumption, an
unavoidable key step is the Taylor expansion of displacements, which directly influences the accuracy of the corresponding continuum theory.
When the wavelength becomes comparable to the periodic characteristic size, the coefficients of Taylor expansions need necessary adjustments
due to the discreteness of the microstructure. Thus, the continuum theories still face critical challenges in predicting the broadband dispersion
feature. This remains widely open so far. In this study, we attempt to adopt the SOS optimization to determine the optimal Taylor expansion
coefficients to guarantee the dispersion diagrams causing the minimal error throughout the first Brillouin zone. The robustness of the
SOS-based SG continuum model is demonstrated with three benchmark examples, i.e., the monoatomic, diatomic, and mass-in-mass lattices.
Such an attempt of constructing continuum models with the help of optimization tools may shed some new light on continuum mechanics of
structure media.
Dispersion relations govern wave behaviors, and tailoring them is a grand challenge in wave manipulation. We demonstrate the inverse design of phononic dispersion using nonlocal interactions on one-dimensional spring-mass chains. For both single-band and double-band cases, we can achieve any valid dispersion curves with analytical precision. We further employ our method to design phononic crystals with multiple ordinary (roton or maxon) and higher-order (undulation) critical points and investigate their wave packet dynamics.
The micropolar (MP) and strain gradient (SG) continua have been generally adopted to investigate the relations between the macroscopic elastic constants and the microstructural geometric parameters. Owing to the fact that the microrotation in the MP theory can be expressed in terms of the displacement gradient components, we may regard the MP theory as a particular incomplete SG theory called the MPSG theory, compared with the existing SG theories which are deemed complete since all the SGs are included. Taking the triangular lattice comprising zigzag beams as an example, it is found that as the angle of the zigzag beams increases, the bending of the beams plays a more important role in the total strain energy, and the difference between the results by the two theories gradually decreases. Finally, the models are verified with the pure bending and simple shear of lattices by comparing with the results obtained by the finite element method (FEM)-based structure analyses.
In this article, an exact strain gradient (SG) continuum model to capture the broadband bandgap characteristics of the prestressed diatomic lattice (PDL) with local/nonlocal interactions has been proposed. By considering the periodicity of wave motion, the wavelength-dependent Taylor expansion is established, leading to a desirable prediction when the wavelength is close to the lattice scale. For PDL, the dispersion curves of the proposed continuum model are always consistent with those of discrete model in the first Brillouin zone. The proposed SG continuum model is used to investigate the effects of structural parameters, orders of SG continua and various nonlocal interactions on bandgap properties of PDL.
• Highlights
• Continuum modelling of prestressed acoustic diatomic lattices.
• Perfect agreements achieved between dispersion diagrams by discrete and proposed continuum models.
• Effects of prestress, nonlocal interactions, mass and stiffness ratios on band gap structures analyzed.
• Proper strain gradient (SG) orders determined to guarantee a satisfactory accuracy.
The Roton-like dispersions support the “return flow” of acoustic waves, such phenomena were only observed quantum system. In this paper, we aim to investigate the nonlinear roton-like dispersion in a mechanical metamaterial with both nonlinear chains and nonlinear resonators with nonlocal connection, both theoretical and numerical method are used to analyze the system, some new phenomena such as amplitude-dependent roton-like behaviors are observed. This work opens a new way for designing an extremely low-frequency vibration isolators with a stable configuration.
