Abstract
A new model for producing band gaps for flexural elastic wave propagation in a periodic microbeam structure is developed using an extended transfer matrix method and a non-classical Bernoulli–Euler beam model that incorporates the strain gradient, couple stress and velocity gradient effects. The band gaps predicted by the new model depend on the three microstructure-dependent material parameters of each constituent material, the beam thickness, the unit cell length and the volume fraction. A parametric study is conducted to quantitatively illustrate these factors. The numerical results reveal that the first band gap frequency range increases with the increases of the three microstructure-dependent material parameters, respectively. In addition, the band gap size predicted by the current model is always larger than that predicted by the classical model, and the difference is large for very thin beams. Furthermore, both the unit cell length and volume fraction have significant effects on the band gap.
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Abbreviations
- a :
-
Unit cell length of a two-phase periodic composite beam
- \({a}_{\mathrm{1}}\) :
-
Length of material I in the unit cell
- \({a}_{\mathrm{2}}\) :
-
Length of material II in the unit cell
- b :
-
Width of the beam
- f :
-
Wave frequency
- h :
-
Thickness of the beam
- i :
-
Imaginary unit
- k :
-
Wave number
- l :
-
Material length scale parameter
- \(l_{m}\) :
-
Couple stress coefficient
- \(l_{s}\) :
-
Strain gradient coefficient
- \(l_{v}\) :
-
Velocity gradient coefficient
- \(m_{{0}}, m_{\mathrm{2}}\) :
-
Mass inertia
- q :
-
Bloch wave number in the x-direction
- w :
-
Deflection on the beam centroidal axis
- A :
-
Cross-sectional area of the beam
- \(C_{\mathrm{0}}, C_{\mathrm{2}}, D, S\) :
-
Convenient material parameters
- E :
-
Young’s modulus
- I :
-
Second moment of cross-sectional area of the beam
- \({{\varvec{I}}}\) :
-
6 by 6 identity matrix
- \({{\varvec{T}}}\) :
-
Transfer matrix
- \(V_{f}^{\mathrm{(I)}}\) :
-
Volume fraction of material I
- W :
-
Coefficient for a harmonic propagating wave
- \({\omega }\) :
-
Angular frequency of the wave
- \({\mu }\) :
-
Shear modulus
- \({\rho }\) :
-
Mass density of the beam material
- \({\nu }\) :
-
Poisson’s ratio
- \(^{\mathrm{(I)}}\) :
-
Item for material I
- \(^{\mathrm{(II)}}\) :
-
Item for material II
References
Liu L, Hussein MI. Wave motion in periodic flexural beams and characterization of the transition between Bragg scattering and local resonance. J Appl Mech. 2012;79:011003-1–17.
Liu J, Yu D, Zhang Z, Shen H, Wen J. Flexural wave bandgap property of a periodic pipe with axial load and hydro-pressure. Acta Mech Solida Sin. 2019;32:173–85.
An X, Lai C, Fan H, Zhang C. 3D acoustic metamaterial-based mechanical metalattice structures for low-frequency and broadband vibration attenuation. Int J Solids Struct. 2020;191–192:293–306.
Zhao P, Zhang K, Deng Z. Elastic Wave Propagation in Lattice Metamaterials with Koch Fractal. Acta Mech Solida Sin. 2020;33:600–11.
Olsson RH, Elkady I. Microfabricated phononic crystal devices and applications. Meas Sci Technol. 2009;20:012002-1–13.
Yan Z, Zhang C, Wang Y. Attenuation and localization of bending waves in a periodic/disordered fourfold composite beam. J Sound Vib. 2009;327:109–20.
Trainiti G, Rimoli JJ, Ruzzene M. Wave propagation in periodically undulated beams and plates. Int J Solids Struct. 2015;75:260–76.
Lam DCC, Yang F, Chong ACM, Wang J, Tong P. Experiments and theory in strain gradient elasticity. J Mech Phys Solids. 2003;51:1477–508.
McFarland AW, Colton JS. Role of material microstructure in plate stiffness with relevance to microcantilever sensors. J Micromech Microeng. 2005;15:1060–7.
Maranganti R, Sharma P. Length scales at which classical elasticity breaks down for various materials. Phys Rev Lett. 2007;98:195504-1–4.
Zhang GY, Gao X-L, Bishop JE, Fang HE. Band gaps for elastic wave propagation in a periodic composite beam structure incorporating microstructure and surface energy effects. Compos Struct. 2018;189:263–72.
Gao RZ, Zhang GY, Ioppolo T, Gao X-L. Elastic wave propagation in a periodic composite beam structure: a new model for band gaps incorporating surface energy, transverse shear and rotational inertia effects. J Micromech Molecular Phys. 2018;3:1840005-1–22.
Mindlin RD. Micro-structure in linear elasticity. Arch Rat Mech Anal. 1964;16:51–78.
Li Y, Wei P, Zhou Y. Band gaps of elastic waves in 1-D phononic crystal with dipolar gradient elasticity. Acta Mech. 2016;227:1005–23.
Gourgiotis PA, Georgiadis HG. Plane-strain crack problems in microstructured solids governed by dipolar gradient elasticity. J Mech Phys Solids. 2009;57:1898–920.
Yang F, Chong ACM, Lam DCC, Tong P. Couple stress based strain gradient theory for elasticity. Int J Solids Struct. 2002;39:2731–43.
Park SK, Gao X-L. Variational formulation of a modified couple stress theory and its application to a simple shear problem. Z Angew Math Phys. 2008;59:904–17.
Zhang GY, Gao X-L, Ding SR. Band gaps for wave propagation in 2-D periodic composite structures incorporating microstructure effects. Acta Mech. 2018;229:4199–214.
Zhang GY, Gao X-L. Elastic wave propagation in a periodic composite plate structure: band gaps incorporating microstructure, surface energy and foundation effects. J Mech Mater Struct. 2019;14:219–36.
Zhang GY, Gao X-L. Elastic wave propagation in 3-D periodic composites: band gaps incorporating microstructure effects. Compos Struct. 2018;204:920–32.
Zhang GY, Gao X-L, Guo ZY. A non-classical model for an orthotropic Kirchhoff plate embedded in a viscoelastic medium. Acta Mech. 2017;228:3811–25.
Zhang GY, Gao X-L. Band gaps for wave propagation in 2-D periodic three-phase composites with coated star-shaped inclusions and an orthotropic matrix. Compos Part B: Eng. 2020;182:107319-1–13.
Dontsov EV, Tokmashev RD, Guzina BB. A physical perspective of the length scales in gradient elasticity through the prism of wave dispersion. Int J Solids Struct. 2013;50:3674–84.
Zhang GY, Gao X-L. A new Bernoulli–Euler beam model based on a reformulated strain gradient elasticity theory. Math Mech Solids. 2020;25:630–43.
Mindlin RD. Influence of couple-stresses on stress concentrations. Exp Mech. 1963;3:1–7.
Park SK, Gao X-L. Bernoulli–Euler beam model based on a modified couple stress theory. J Micromech Microeng. 2006;16:2355–9.
Papargyri-Beskou S, Polyzos D, Beskos DE. Wave dispersion in gradient elastic solids and structures: a unified treatment. Int J Solids Struct. 2009;46:3751–9.
Yu D, Liu Y, Zhao H, Wang G, Qiu J. Flexural vibration band gaps in Euler-Bernoulli beams with locally resonant structures with two degrees of freedom. Phys Rev B. 2006;73:064301-1–5.
Kittel C. Introduction to solid state physics. 8th ed. New York: Wiley; 2004.
Davis JR. ASM specialty handbook: stainless steels. Materials Park, OH: ASM International; 1994.
Chen Y, Wang L. Periodic co-continuous acoustic metamaterials with overlapping locally resonant and Bragg band gaps. Appl Phys Lett. 2014;105:191907-1–5.
Bao B, Lallart M, Guyomar D. Manipulating elastic waves through piezoelectric metamaterial with nonlinear electrical switched Dual-connected topologies. Int J Mech Sci. 2020;172:105423-1–17.
Bao B, Lallart M, Guyomar D. Structural design of a piezoelectric meta-structure with nonlinear electrical Bi-link networks for elastic wave control. Int J Mech Sci. 2020;181:105730-1–19.
Bao B, Wang Q. Elastic wave manipulation in piezoelectric beam meta-structure using electronic negative capacitance dual-adjacent/staggered connections. Compos Struct. 2019;210:567–80.
Wang L. Size-dependent vibration characteristics of fluid-conveying microtubes. J Fluids Struct. 2010;26:675–84.
Wen J, Wang G, Yu D, Zhao H, Liu Y. Theoretical and experimental investigation of flexural wave propagation in straight beams with periodic structures: Application to a vibration isolation structure. J Appl Phys. 2005;97:114907-1–4.
Zhou W, Chen Z, Chen Y, Chen W, Lim CW, Reddy JN. Mathematical modelling of phononic nanoplate and its size-dependent dispersion and topological properties. Appl Math Model. 2020;88:774–90.
Zhou W, Chen W, Lim CW. Surface effect on the propagation of flexural waves in periodic nano-beam and the size-dependent topological properties. Compos Struct. 2019;216:427–35.
Acknowledgements
The work reported here is funded by the National Natural Science Foundation of China [grant numbers 12002086, 11872149 and 11472079] and the Fundamental Research Funds for the Central Universities [grant number 2242020R10027]. These supports are gratefully acknowledged.
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Zhang, G., Zheng, C., Qiu, X. et al. Microstructure-dependent Band Gaps for Elastic Wave Propagation in a Periodic Microbeam Structure. Acta Mech. Solida Sin. 34, 527–538 (2021). https://doi.org/10.1007/s10338-021-00217-z
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DOI: https://doi.org/10.1007/s10338-021-00217-z