周期结构中弹性波的色散关系与振动局部化问题研究
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摘要
周期结构广泛应用于工程中的诸多领域,这种结构具有特殊的力学性质。表现为当弹性波在周期性结构中传播时,会产生弹性波频率的通带和禁带特性。在通带范围内其振动会遍及整个结构,在禁带范围内振动就会受到“抑制”。通过分析色散关系,可以计算出结构带隙的起始、截止频率。声子晶体是一种具有弹性波禁带的周期性结构功能材料。随着声子概念的提出,以及其本身所具有的优良的减振降噪功能,周期结构的研究引起了各国学者的广泛关注。
建立适合的模型,使之能够有效的描述物理现象,是分析、解决问题的关键步骤。本文分别运用连续介质力学理论、晶格动力学理论(单弹簧振子模型)以及多弹簧振子模型,对一维周期结构(声子晶体)中的弹性波色散特性及带隙结构进行了分析、比较,在低频振动与波动问题中,连续性模型与离散粒子模型非常接近,然而当发生高频率振动或短波长波动时,两种模型却存在较大差异,说明了经典的连续介质理论的局限性。
作为算例,分析了一定组分比条件下钨/天然橡胶声子晶体的剪切波与扭转波的传播特性,得到了一、二带隙起始、截止频率,研究了三种典型激振力频率作用下结构内部质点位移,从中可以发现稳态幅值的差别。通过分析了结构内周期数目对结构振动频谱的影响,以及验证周期结构中的禁带与通带特性,可以形成对周期结构中带隙——这种特殊力学的性能更加深刻、全面的理解。
若存在失谐(缺陷),弹性波或振动就会限制在局部区域,形成局部振荡,即产生弹性波的局部化现象。根据弹性波传递矩阵的表达式,计算了周期结构中的局部化因子,讨论了谐和周期结构中的材料组分比对局部化因子的影响。
Periodic structures are extensively used in many engineering fields; there are a lot of particular mechanic characteristics in these structures. Such as when elastic waves propagate in them, pass band and stop band will appear. The modes of vibration can extend all over the whole structure in pass band, but it will disappear in stop band. The limiting frequencies of starting and stopping band gaps are derived by analyzing the dispersion relation.
Phonon crystal is a type of functional materials, which has the elastic band gap. As the conception of phonon crystal is introduced that has outstanding function of damping and denoising, extensive attentions are paid on.
Establishing proper models to describe the physical phenomena validly is the key step for solving problems. In this paper, based on the theory of continuum mechanics, lattice dynamics (single spring-oscillator), and multiple spring-oscillator model, the properties of elastic wave in 1D periodic structure are investigated, the dispersion relation and band gap structure of elastic waves in are analyzed and compared, continuum models and discrete particle models are similar when vibration frequency is low and wavelength is long. However, as vibration frequency is high and wavelength is short, there is large difference between the two models, which are continuum models are invalid in this case.
As an example, the shear and torsion wave characteristics of tungsten/natural rubber phonon crystal in certain components are analysis, the first and second starting、stopping frequencies are derived, the displacement of internal particle under three typical exciting force are studied, from which, the difference of steady-state amplitude is founded.
With the effect that the number of unit cells to the vibration being analyzed and the properties of forbidden band and passing band being verified, the band gap in periodic structures, which is a special mechanic performance, is profound and overall comprehended.
Propagation of elastic waves through disordered periodic structures will be limited in local region; the localization phenomenon of elastic wave will appear. By employing the expression of elastic wave transfer matrix, the localization factor of this system is calculated; the influence of material component ratio on
建立适合的模型,使之能够有效的描述物理现象,是分析、解决问题的关键步骤。本文分别运用连续介质力学理论、晶格动力学理论(单弹簧振子模型)以及多弹簧振子模型,对一维周期结构(声子晶体)中的弹性波色散特性及带隙结构进行了分析、比较,在低频振动与波动问题中,连续性模型与离散粒子模型非常接近,然而当发生高频率振动或短波长波动时,两种模型却存在较大差异,说明了经典的连续介质理论的局限性。
作为算例,分析了一定组分比条件下钨/天然橡胶声子晶体的剪切波与扭转波的传播特性,得到了一、二带隙起始、截止频率,研究了三种典型激振力频率作用下结构内部质点位移,从中可以发现稳态幅值的差别。通过分析了结构内周期数目对结构振动频谱的影响,以及验证周期结构中的禁带与通带特性,可以形成对周期结构中带隙——这种特殊力学的性能更加深刻、全面的理解。
若存在失谐(缺陷),弹性波或振动就会限制在局部区域,形成局部振荡,即产生弹性波的局部化现象。根据弹性波传递矩阵的表达式,计算了周期结构中的局部化因子,讨论了谐和周期结构中的材料组分比对局部化因子的影响。
Periodic structures are extensively used in many engineering fields; there are a lot of particular mechanic characteristics in these structures. Such as when elastic waves propagate in them, pass band and stop band will appear. The modes of vibration can extend all over the whole structure in pass band, but it will disappear in stop band. The limiting frequencies of starting and stopping band gaps are derived by analyzing the dispersion relation.
Phonon crystal is a type of functional materials, which has the elastic band gap. As the conception of phonon crystal is introduced that has outstanding function of damping and denoising, extensive attentions are paid on.
Establishing proper models to describe the physical phenomena validly is the key step for solving problems. In this paper, based on the theory of continuum mechanics, lattice dynamics (single spring-oscillator), and multiple spring-oscillator model, the properties of elastic wave in 1D periodic structure are investigated, the dispersion relation and band gap structure of elastic waves in are analyzed and compared, continuum models and discrete particle models are similar when vibration frequency is low and wavelength is long. However, as vibration frequency is high and wavelength is short, there is large difference between the two models, which are continuum models are invalid in this case.
As an example, the shear and torsion wave characteristics of tungsten/natural rubber phonon crystal in certain components are analysis, the first and second starting、stopping frequencies are derived, the displacement of internal particle under three typical exciting force are studied, from which, the difference of steady-state amplitude is founded.
With the effect that the number of unit cells to the vibration being analyzed and the properties of forbidden band and passing band being verified, the band gap in periodic structures, which is a special mechanic performance, is profound and overall comprehended.
Propagation of elastic waves through disordered periodic structures will be limited in local region; the localization phenomenon of elastic wave will appear. By employing the expression of elastic wave transfer matrix, the localization factor of this system is calculated; the influence of material component ratio on
引文
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2 Kushwaha M S, Halevi P, Dobrzynski L, Djafarirouhani B. Acoustic Band-structure of Periodic Elastic Composites. Physical Review Letters. 1993, 71(13) : 2022～2025
3 Mingrong Shen, Wenwu Cao. Accoustic Bandgap Formation in a Periodic Structure with Multilayer Unit Cells. Journal Physics D: Applied Physics. 2000, 33: 1150～1154
4 F.Kobayashi, S.Biwa, N.Ohno. Wave Transmission Characteristics in Periodic Media of Finite Length: Multilayers and Fiber Arrays. International Journal of Solids and Structures. 2004, 41: 7361～7375
5 A.S.J.Suiker, A.V.Metrikine, R.de Borst. Comparison of Wave Propagation Characteristics of the Cosserat Continuum Model and Corresponding Discrete Lattice Models. International Journal of Solids and Structures. 2001, 38: 1563～1583
6 Andrei V. Metrikine, Harm Askes. One-Dimensional Dynamically Consistent Consistent Gradient Elasticity Models Derived from a Discrete Microstructure Part1: Generic Formulation. European Journal of Mechanics A/Solids. 2002, 21: 557～572
7 N.C.Ghosh, S.Nath, L.Debnath. Propagation of Waves in Micropolar Solid-Solid Semispaces in the Presence of a Compressional Wave Source in the Upper Solid Substratum. Mathematical and Computer Modelling. 2001, 34: 557～563
8 R. Bladh, M.P. Castanier, C. Pierre. Component-Mode-Based Reduced Order Modeling Techniques for Mistuned Bladed Disks—Part II: Application. ASME Journal of Engineering for Gas Turbines and Power. 2001, 123: 100～108
9 J.H. Kuang, B.W. Huang. The Effect of Blade Crack on Mode Localization in Rotating Bladed Disks. Journal of Sound and Vibration. 1999, 227(1): 85～103
10 李凤明. 结构中弹性波与振动局部化问题的研究: [学位论文]. 哈尔滨工业大学, 2003
11 Kolsky H. Stress Waves in Solids. Clarendon Press, Oxford, 1953
12 Achenbach J D. Wave Propagation in Elastic Solids. North Holland. Amsterdam, 1973
13 Eringen A C, Suhubi E S. Elastodynamic. Vol. 2 Linear Theory. Academic Press, New York, 1975
14 鲍亦兴, 毛昭宙著, 刘殿魁, 苏先樾译. 弹性波的衍射与动应力集中. 北京: 科学出版社, 1993: 3～25, 72～131
15 Alexander F. Vakakis, Cetin Cetinkaya. Dispersion of Stress Waves in One-Dimensional Semi-Infinite, Weakly Coupled Layered Systems. International Journal of Solids. 1996, 33(28): 4195～4213
16 A. M. EI-Naggar, M. M. Saliem. Wave Propagation in Layered Media under Initial Strsses. Applied Mathematics and Computation. 1996, 74: 95～117
17 Graham A. Rogerson, Kevin J. Sandiford. The Effect of Finite Primary Deformations on Harmonic Waves in Layered Elastic Media. International Journal of Solids and Structures. 2000, 37: 2059～2087
18 Christine Valle, Jianmin Qu, Laurence J. Jacobs. Guided Circumferential Waves in Layered Cylinders. International Journal of Engineering Science. 1999, 37: 1369～1387
19 S. M. Howard, Y. H. Pao. Analysis and Experiment on Stress Waves in Planar Trusses. ASME Journal of Engineering Mechanics. 1998, 128: 854～891
20 Yin-Hing Pao, Xian-Yue Su, Jia-Yong Tian. Reverberation Matrix Method For Propagation of Sound in A Multilayered Liquid. Journal of Sound and Vibration. 2000, 230(4): 743～760
21 孙雁, 刘正兴. Hamilton 体系及弹性波在层状介质中的传播问题. 地球物理学报. 2002, 45(4): 569～574
22 Zhenghua Qian, Feng Jin, Zikun Wang, Kikuo Kishimoto. Dispersion Relation for SH-wave Propagation in Periodic Piezoelectric Composite Layered Structures. International Journal of Engineering Science. 2004, 42: 673～689
23 Thompson W T. Transmission of Elastic Waves Through a Stratified Solid Medium. Journal of Applied Physics. 1950, 21: 89
24 Haskell N A. The Dispersion of Surface Waves on Multi-layered Media. Bulletin of the Seismological. Society of America Acoustic. 1953, 43: 17
25 F. Gilbert, G. E. Backus. Propagator Matrices in Elastic Wave and Vibration Problem. Geophysics. 1966, 31: 326
26 E. Cosserat, F. Cosserat. Théorie des Corps Deformables. Herman et fils. 1909:12～63
27 A. C. Eringen, E. S. Suhubi. Mechanics of Micromorphic Continuum. Springer-Verlag. 1967, 3: 139～152
28 A. C. Eringen, E. S. Suhubi. Nonlinear Theory of Simple Micro-elastic Solids-I. Journal of Engineering Science. 1964, 2: 189～203
29 A. C. Eringen. Linear Theory of Nonlocal Elasticity and Dispersion of Plane Waves. International Journal of Engineering Science. 1972, 10: 425～435
30 A. C. Eringen. Nonlocal Polar Elastic Continua. International Journal of Engineering Science. 1972, 10: 1～16
31 R.D. Mindlin. Micro-structure in Linear Elasticity. Archive for Rational Mechanics and Analysis. 1964, 16: 51～78
32 Ching S. Chang, Jian Gao. Second-gradient Constitutive Theory for Granular Material with Random Packing Structure. International Journal of Solids and Structures. 1995, 32(16): 2279～2293
33 A.S.J. Suiker, Chang C. S, De Borst. Micro-mechanical Modeling of Granular Material-Part1: Derivation of a Second-gradient Micro-polar Constitutive Theory. Acta Mechanics. 2001, 149: 161～180
34 A.S.J. Suiker, Chang C. S, De Borst. Micro-mechanical Modeling of Granular Material-Part2: Plane Wave Propagation in infinite media. Acta Mechanics. 2001, 149: 181～201
35 C.S. Chang, Gao Jian. Wave Propagation in Granular Rod Using High-gradient Theory. Journal of Engineering Mechanics. 1997, 12: 52～59
36 Taiji Adachi, Youshihiro Tomita, Masao Tanaka. Computational Simulation of Deformation Behavior of 2D-Lattice Continuum. Journal of Mechanics Science. 1998, 40(9): 857～866
37 N. C. Ghosh, S. Nath, L. Debnath. Propagation of Waves in Micropolar Solid-Solid Semispaces in the Presence of a Compressional Wave Source in the Upper Solid Substratum. Mathematical and Computer Modeling. 2001, 34:557～563
38 L. E. Fernndez, G. Ayala. Constitutive Modeling of Discontinuities by Means of Discrete and Continuum Approximations and Damage Models. International Journal of Solids and Structures. 2004, 41: 1453～1471
39 E.C. Aifantis. Gradient Deformation Models at NANO, Micro, and Macro Scales. Journal of Engineering Materials Technology. 1999, 121: 189～202
40 Gero Friesecke, Richard D. James. A Scheme for the Passage from Atomic to Continuum Theory for Thin Films, Nanotubes and Nanorods. Journal of the Mechanics and Physics of Solids. 2000, 48: 1519～1540
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