Bloch analysis was originally developed by Bloch to study the electron behavior in crystalline solids. His method has been adapted to study the elastic wave propagation in periodic structures. The absence of a rigorous mathematical analysis of the approach, as applied to periodic structures, has resulted in mistreatment of internal forces and misapplication to nonlinear media. In a previous article (Farzbod and Leamy, 2009, “The Treatment of Forces in Bloch Analysis,” J. Sound Vib., 325(3), pp. 545–551), we clarified the treatment of internal forces. In this article, we borrow the insight from the previous work to detail a mathematical basis for Bloch analysis and thereby shed important light on the proper application of the technique. For example, we conclusively show that translational invariance is not a proper justification for invoking the existence of a “propagation constant,” and that in nonlinear media, this results in a flawed analysis. We also provide a simple, two-dimensional example, illustrating what the role stiffness symmetry has on the search for a band gap behavior along the edges of the irreducible Brillouin zone. This complements other treatments that have recently appeared addressing the same issue.
Issue Section:
Research Papers
1.
Gibson
, L. J.
, and Ashby
, M. F.
, 1997, Cellular Solids: Structure and Properties
, 2nd ed., Cambridge University Press
, Cambridge, UK
.2.
Brillouin
, L.
, 1946, Wave Propagation in Periodic Structures
, Dover
, New York
.3.
Cherchi
, M.
, 2005, “Bloch Analysis of Finite Periodic Microring Chains
,” Appl. Phys. B: Lasers Opt.
0946-2171, 80
(1
), pp. 109
–113
.4.
Tanaka
, Y.
, Tomoyasu
, Y.
, and Tamura
, S. -i
, 2000, “Band Structure of Acoustic Waves in Phononic Lattices: Two-Dimensional Composites With Large Acoustic Mismatch
,” Phys. Rev. B
0556-2805, 62
(11
), pp. 7387
–7392
.5.
Miyashita
, T.
, 2005, “Sonic Crystals and Sonic Wave-Guides
,” Meas. Sci. Technol.
0957-0233, 16
(5
), pp. R47
–R63
.6.
Heckl
, M. A.
, 1964, “Investigations on the Vibrations of Grillages and Other Simple Beam Structures
,” J. Acoust. Soc. Am.
0001-4966, 36
(7
), pp. 1335
–1343
.7.
Mead
, D. J.
, 1970, “Free Wave Propagation in Periodically Supported, Infinite Beams
,” J. Sound Vib.
0022-460X, 11
(2
), pp. 181
–197
.8.
Mead
, D. J.
, 1973, “A General Theory of Harmonic Wave-Propagation in Linear Periodic Systems With Multiple Coupling
,” J. Sound Vib.
0022-460X, 27
(2
), pp. 235
–260
.9.
Sengupta
, G.
, 1970, “Natural Flexural Waves and Normal Modes of Periodically-Supported Beams and Plates
,” J. Sound Vib.
0022-460X, 13
(1
), pp. 89
–101
.10.
Ruzzene
, M.
, and Baz
, A.
, 2000, “Control of Wave Propagation in Periodic Composite Rods Using Shape Memory Inserts
,” ASME J. Vibr. Acoust.
0739-3717, 122
(2
), pp. 151
–159
.11.
Pany
, C.
, and Parthan
, S.
, 2003, “Axial Wave Propagation in Infinitely Long Periodic Curved Panels
,” ASME J. Vibr. Acoust.
0739-3717, 125
(1
), pp. 24
–30
.12.
Treyssede
, F.
, 2008, “Elastic Waves in Helical Waveguides
,” Wave Motion
0165-2125, 45
(4
), pp. 457
–470
.13.
Romeo
, F.
, and Paolone
, A.
, 2007, “Wave Propagation in Three-Coupled Periodic Structures
,” J. Sound Vib.
0022-460X, 301
(3–5
), pp. 635
–648
.14.
Kohrs
, T.
, and Petersson
, B. A. T.
, 2007, “Wave Propagation in Light Weight Profiles With Truss-Like Cores: Wavenumber Content, Forced Response and Influence of Periodicity Perturbations
,” J. Sound Vib.
0022-460X, 304
(3–5
), pp. 691
–721
.15.
Duhamel
, D.
, Mace
, B. R.
, and Brennan
, M. J.
, 2006, “Finite Element Analysis of the Vibrations of Waveguides and Periodic Structures
,” J. Sound Vib.
0022-460X, 294
(1–2
), pp. 205
–220
.16.
Houillon
, L.
, Ichchou
, M. N.
, and Jezequel
, L.
, 2005, “Wave Motion in Thin-Walled Structures
,” J. Sound Vib.
0022-460X, 281
(3–5
), pp. 483
–507
.17.
Saeed
, H. M.
, and Vestroni
, F.
, 1998, “Simulation of Combined Systems by Periodic Structures: The Wave Transfer Matrix Approach
,” J. Sound Vib.
0022-460X, 213
(1
), pp. 55
–74
.18.
Baz
, A.
, 2001, “Active Control of Periodic Structures
,” ASME J. Vibr. Acoust.
0739-3717, 123
(4
), pp. 472
–479
.19.
El-Raheb
, M.
, 1997, “Frequency Response of a Two-Dimensional Trusslike Periodic Panel
,” J. Acoust. Soc. Am.
0001-4966, 101
(6
), pp. 3457
–3465
.20.
Mead
, D. J.
, 1996, “Wave Propagation in Continuous Periodic Structures: Research Contributions From Southampton, 1964–1995
,” J. Sound Vib.
0022-460X, 190
(3
), pp. 495
–524
.21.
Orris
, R. M.
, and Petyt
, M.
, 1974, “Finite-Element Study of Harmonic Wave-Propagation in Periodic Structures
,” J. Sound Vib.
0022-460X, 33
(2
), pp. 223
–236
.22.
Phani
, A. S.
, Woodhouse
, J.
, and Fleck
, N. A.
, 2006, “Wave Propagation in Two-Dimensional Periodic Lattices
,” J. Acoust. Soc. Am.
0001-4966, 119
(4
), pp. 1995
–2005
.23.
Askar
, A.
, 1973, “Dispersion Relation and Wave Solution for Anharmonic Lattices and Korteweg Delaunays Vries Continua
,” Proc. R. Soc. London, Ser. A
0950-1207, 334
(1596
), pp. 83
–94
.24.
Chakraborty
, G.
, and Mallik
, A. K.
, 2001, “Dynamics of a Weakly Non-Linear Periodic Chain
,” Int. J. Non-Linear Mech.
0020-7462, 36
(2
), pp. 375
–389
.25.
Farzbod
, F.
, and Leamy
, M. J.
, 2009, “The Treatment of Forces in Bloch Analysis
,” J. Sound Vib.
0022-460X, 325
(3
), pp. 545
–551
.26.
Harrison
, J. M.
, Kuchment
, P.
, Sobolev
, A.
, and Winn
, B.
, 2007, “On Occurrence of Spectral Edges for Periodic Operators Inside the Brillouin Zone
,” J. Phys. A: Math. Theor.
1751-8113, 40
(27
), pp. 7597
–7618
.27.
Adams
, S. D. M.
, Craster
, R. V.
, and Guenneau
, S.
, 2008, “Bloch Waves in Periodic Multi-Layered Acoustic Waveguides
,” Proc. R. Soc. London, Ser. A
0950-1207, 464
(2098
), pp. 2669
–2692
.28.
Patterson
, J. D.
, and Bailey
, B. C.
, 2007, Solid-State Physics
, Springer
, New York
.29.
Bloch
, F.
, 1928, “Über die Quantenmechanik der Elektronen in Kristallgittern
,” Z. Phys.
0044-3328, 52
, pp. 555
–600
.30.
Langley
, R. S.
, 1993, “A Note on the Force Boundary-Conditions for 2-Dimensional Periodic Structures With Corner Freedoms
,” J. Sound Vib.
0022-460X, 167
(2
), pp. 377
–381
.31.
Lowdin
, P. O.
, 1998, Linear Algebra for Quantum Theory
, Wiley-Interscience
, USA
.32.
Griffiths
, D. J.
, 2004, Introduction to Quantum Mechanics
, 2nd ed., Benjamin Cummings
, Upper Saddle River, NJ
.33.
Farzbod
, F.
, 2010, “Analysis of Bloch Formalism in Undamped and Damped Periodic Structures
,” Ph.D. thesis, Georgia Institute of Technology, Atlanta, GA.34.
Johnson
, S. G.
, 2008, September 26, private communication.35.
Wang
, Y.
, Li
, F.
, Wang
, Y.
, Kishimoto
, K.
, and Huang
, W.
, 2009, “Tuning of Band Gaps for a Two-Dimensional Piezoelectric Phononic Crystal With a Rectangular Lattice
,” Acta Mech. Sin.
0459-1879, 25
(1
), pp. 65
–71
.36.
Wang
, Y. -Z.
, Li
, F. -M.
, Kishimoto
, K.
, Wang
, Y. -S.
, and Huang
, W. -H.
, 2009, “Elastic Wave Band Gaps in Magnetoelectroelastic Phononic Crystals
,” Wave Motion
0165-2125, 46
(1
), pp. 47
–56
.37.
Yilmaz
, C.
, Hulbert
, G. M.
, and Kikuchi
, N.
, 2007, “Phononic Band Gaps Induced by Inertial Amplification in Periodic Media
,” Phys. Rev. B
0556-2805, 76
(5
), p. 054309
.38.
Jensen
, J. S.
, 2003, “Phononic Band Gaps and Vibrations in One- and Two-Dimensional Mass-Spring Structures
,” J. Sound Vib.
0022-460X, 266
(5
), pp. 1053
–1078
.39.
Gonella
, S.
, and Ruzzene
, M.
, 2008, “Analysis of In-Plane Wave Propagation in Hexagonal and Re-Entrant Lattices
,” J. Sound Vib.
0022-460X, 312
(1–2
), pp. 125
–139
.40.
Kushwaha
, M. S.
, Halevi
, P.
, Dobrzynski
, L.
, and Djafari-Rouhani
, B.
, 1993, “Acoustic Band Structure of Periodic Elastic Composites
,” Phys. Rev. Lett.
0031-9007, 71
(13
), pp. 2022
–2025
.41.
Vasseur
, J. O.
, Deymier
, P. A.
, Djafari-Rouhani
, B.
, Pennec
, Y.
, and Hladky-Hennion
, A. -C.
, 2008, “Absolute Forbidden Bands and Waveguiding in Two-Dimensional Phononic Crystal Plates
,” Phys. Rev. B
0556-2805, 77
(8
), p. 085415
.42.
Meirovitch
, L.
, 2002, Fundamentals of Vibrations
, 1st ed., McGraw-Hill
, New York
.Copyright © 2011
by American Society of Mechanical Engineers
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