對於任何形式的波而言，是無法僅藉由單一均質性的材料所控制，必須具有如梯
度分佈、材料間介面以及曲線排列等某種程度的非均質性存在，才能被加以控制。在
早期聲子晶體的研究與探討主要是集中於能隙的應用與計算，如濾波器、波導或共振
腔的設計。而近年來研究發現在其傳導區擁有異常之頻散現象，發現聲子晶體具有負
折射特性，使得該方面之研究更成為最近學術界的熱門話題。
而具有負蒲松比特性之彈性材料，又稱膨脹材料，而膨脹材料多為孔隙材料所構
成，其組成皆為彈性體的周期性排列，分析對象為彈性波，學者將其視為聲子晶體的
一種，透過有限元素分析與布洛赫定理分析此類型週期結構的波傳行為。本文以膨脹
材料結合聲子晶體為主軸，使用有限元素軟體分析負蒲松比對於能帶結構以及波傳行
為的影響，選用背景基材為膨脹材料，散射體為鋼材所構成的二維聲子晶體，並在兩
種幾何形式的散射體下，計算聲子晶體之能帶結構圖與等頻圖來分析。而膨脹材料對
於能隙頻率影響非常關鍵，可以藉由調變蒲松比來改變其能隙頻率。最後，針對由矩
形柱週期排列而成的聲子晶體上，所對應到的自我準直現象。並且設計特殊佈局及超
常介面操控彈性波。最後討論二維以及三維局部共振型聲子晶體之低頻率能隙的特性，
藉由局部共振型聲子晶體本身特性，用較小的尺寸獲得更低的能隙，並且使用負蒲松
比材料替換包覆軟材可以大幅降低其鋼性，達到更低的能隙。

In this thesis, the wave propagation in phononic crystal composed of auxetic star-shaped
honeycomb matrix with negative Poisson’s ratio is presented. Two types of inclusions with
circular and rectangular cross sections are considered. The band structures of the phononic
crystals are obtained by the finite element method. The band structure of the phononic crystal
is affected significant by the auxeticity of the star-shaped honeycomb. Some other interesting
findings are also presented, such as the negative refraction and the self-collimation. The
present study demonstrates the potential applications of the star-shaped honeycomb in
phononic crystals, such as vibration isolation and the elastic waveguide. The results reveal
that the phononic crystals composed of auxetic materials can have a great potential for the
design of novel acoustic devices. Furthermore, the low frequency band gap of local resonant
phononic crystals with auxetic coating material are also investigated. Low frequency waves,
in the range 3–15 Hz for earthquakes and up to hundreds of Hz for vibrations generated by
machine tool, cause a large amount of damage or inconvenience. The first band gap of local
resonant phononic crystals with auxetic coating material can applied for this region of
frequency.

[1] Yablonovitch, Eli, and T. J. Gmitter. "Photonic band structure: The face-centered-cubic
case." Physical Review Letters 63.18 (1989): 1950.
[2] Yablonovitch, Eli. "Inhibited spontaneous emission in solid-state physics and
electronics." Physical review letters 58.20 (1987): 2059.
[3] John, Sajeev. "Strong localization of photons in certain disordered dielectric
superlattices." Physical review letters 58.23 (1987): 2486.
[4] M. Sigalas and E. Economou,"Elastic and acoustic wave band structure", Journal of
sound and vibration, Vol. 158, No. 2, pp. 377-382. (1992).
[5] Kushwaha, Manvir S., et al. "Acoustic band structure of periodic elastic composites."
Physical review letters 71.13 (1993): 2022.
[6] Brillouin, Leon. Wave propagation in periodic structures: electric filters and crystal
lattices. Courier Corporation, 2003.
[7] L. Cremer and H. O. Leilich, “Zur theorie der Biegekettenleiter (On theory of flexural
periodic systems)” Archiv der Elektrischen Ubertragung, Vol. 7, pp.261 (1953)
[8] Mead, D. M. "Wave propagation in continuous periodic structures: research
contributions from Southampton, 1964–1995." Journal of sound and vibration 190.3
(1996): 495-524.
[9] Mester, S. S., and Haym Benaroya. "Periodic and near-periodic structures." Shock and
Vibration 2.1 (1995): 69-95.
[10] Bousfia, A., et al. "Omnidirectional phononic reflection and selective transmission in
one-dimensional acoustic layered structures." Surface science 482 (2001): 1175-1180.
[11] Manzanares-Martínez, Betsabe, et al. "Experimental evidence of omnidirectional
elastic bandgap in finite one-dimensional phononic systems." Applied physics letters
85.1 (2004): 154-156.
[12] Kushwaha, M. S., and P. Halevi. "Band‐gap engineering in periodic elastic
composites." Applied Physics Letters 64.9 (1994): 1085-1087.
[13] Kushwaha, Manvir S., et al. "Theory of acoustic band structure of periodic elastic
composites." Physical Review B 49.4 (1994): 2313.
77
[14] Kushwaha, Manvir S. "Stop-bands for periodic metallic rods: Sculptures that can filter
the noise." Applied Physics Letters 70.24 (1997): 3218-3220.
[15] R. Martinezsala, J. Sancho, J. V. Sanchez, V. Gomez, J. Llinares and F.
Meseguer,"SOUND-ATTENUATION BY SCULPTURE", Nature, Vol. 378, No. 6554,
pp. 241-241. (1995).
[16] Thomas, Edwin L., Taras Gorishnyy, and Martin Maldovan. "Phononics: Colloidal
crystals go hypersonic." Nature materials 5.10 (2006): 773-774.
[17] Wen, Jihong, et al. "Theoretical and experimental investigation of flexural wave
propagation in straight beams with periodic structures: Application to a vibration
isolation structure." Journal of Applied Physics 97.11 (2005): 114907.
[18] Sigalas, M. M. "Elastic wave band gaps and defect states in two-dimensional
composites." The Journal of the Acoustical Society of America 101.3 (1997): 1256-
1261.
[19] Wu, Fugen, et al. "Point defect states in two-dimensional phononic crystals." Physics
Letters A 292.3 (2001): 198-202.
[20] Miyashita, Toyokatsu. "Experimentally study of a sharp bending wave-guide
constructed in a sonic-crystal slab of an array of short aluminum rods in air."
Ultrasonics Symposium, 2004 IEEE. Vol. 2. IEEE, 2004.
[21] Zhang, Xin, et al. "Defect states in 2D acoustic band-gap materials with bend-shaped
linear defects." Solid State Communications 130.1 (2004): 67-71.
[22] Veselago, Viktor G. "The electrodynamics of substances with simultaneously negative
values of and μ." Soviet physics uspekhi 10.4 (1968): 509.
[23] Pendry, John Brian. "Negative refraction makes a perfect lens." Physical review letters
85.18 (2000): 3966.
[24] Liu, Wei, and Xianyue Su. "Collimation and enhancement of elastic transverse waves
in two-dimensional solid phononic crystals." Physics Letters A 374.29 (2010): 2968-
2971.
[25] Cicek, Ahmet, Olgun Adem Kaya, and Bulent Ulug. "Impacts of uniaxial elongation
on the bandstructures of two-dimensional sonic crystals and associated applications."
Applied Acoustics 73.1 (2012): 28-36.
[26] Cicek, Ahmet, Olgun Adem Kaya, and Bulent Ulug. "Wide-band all-angle acoustic
78
self-collimation by rectangular sonic crystals with elliptical bases." Journal of Physics
D: Applied Physics 44.20 (2011): 205104.
[27] Soliveres, Ester, et al. "Self collimation of ultrasound in a three-dimensional sonic
crystal." Applied Physics Letters 94.16 (2009): 164101.
[28] Cebrecos, A., et al. "Formation of collimated sound beams by three-dimensional sonic
crystals." Journal of Applied Physics 111.10 (2012): 104910.
[29] Morvan, B., et al. "Ultra-directional source of longitudinal acoustic waves based on a
two-dimensional solid/solid phononic crystal." Journal of Applied Physics 116.21
(2014): 214901.
[30] Lin, Sz-Chin Steven, et al. "Gradient-index phononic crystals." Physical Review B
79.9 (2009): 094302.
[31] Liu, Zhengyou, et al. "Locally resonant sonic materials." science 289.5485 (2000):
1734-1736.
[32] James, Richard, et al. "Sonic bands, bandgaps, and defect states in layered structures—
theory and experiment." The Journal of the Acoustical Society of America 97.4 (1995):
2041-2047.
[33] Liu, Zhengyou, Che Ting Chan, and Ping Sheng. "Three-component elastic wave
band-gap material." Physical Review B 65.16 (2002): 165116.
[34] Gang, Wang, et al. "Accurate evaluation of lowest band gaps in ternary locally resonant
phononic crystals." Chinese Physics 15.8 (2006): 1843.
[35] Zhang, Xin, et al. "Large two-dimensional band gaps in three-component phononic
crystals." Physics Letters A 317.1 (2003): 144-149.
[36] Zhang, Shu, and Jianchun Cheng. "Existence of broad acoustic bandgaps in threecomponent
composite." Physical Review B 68.24 (2003): 245101.
[37] Larabi, H., et al. "Multicoaxial cylindrical inclusions in locally resonant phononic
crystals." Physical Review E 75.6 (2007): 066601.
[38] Wu, Ying, Yun Lai, and Zhao-Qing Zhang. "Elastic metamaterials with simultaneously
negative effective shear modulus and mass density." Physical Review Letters 107.10
(2011): 105506.
[39] Fok, L., and X. Zhang. "Negative acoustic index metamaterial." Physical Review B
83.21 (2011): 214304.
79
[40] N. J. Hoff, “Structural Problems of Future Aircraft” Third Anglo-American
Aeronautical Conference. The Royal Aeronautical Society, London, p.77 (1951)
[41] Ren, Xin, et al. "Experiments and parametric studies on 3D metallic auxetic
metamaterials with tuneable mechanical properties." Smart Materials and Structures
24.9 (2015): 095016.
[42] Ren, Xin, et al. "A simple auxetic tubular structure with tuneable mechanical
properties." Smart Materials and Structures 25.6 (2016): 065012.
[43] Bertoldi, Katia, et al. "Negative Poisson's ratio behavior induced by an elastic
instability." Advanced Materials 22.3 (2010): 361-366.
[44] Shan, Sicong, et al. "Harnessing multiple folding mechanisms in soft periodic
structures for tunable control of elastic waves." Advanced Functional Materials 24.31
(2014): 4935-4942.
[45] Phani, A. Srikantha, J. Woodhouse, and N. A. Fleck. "Wave propagation in twodimensional
periodic lattices." The Journal of the Acoustical Society of America 119.4
(2006): 1995-2005.
[46] Karasudhi, Pisidhi. Foundations of solid mechanics. Vol. 3. Springer Science &
Business Media, 2012.
[47] Love, Augustus Edward Hough. A treatise on the mathematical theory of elasticity.
Cambridge university press, 2013.
[48] Simmons, Gene. Single crystal elastic constants and calculated aggregate properties.
SOUTHERN METHODIST UNIV DALLAS TEX, 1965.
[49] Milstein, Frederick, and K. Huang. "Existence of a negative Poisson ratio in fcc
crystals." Physical Review B 19.4 (1979): 2030.
[50] Baughman, Ray H., et al. "Negative Poisson's ratios as a common feature of cubic
metals." Nature 392.6674 (1998): 362-365.
[51] Gibson, Lorna J., et al. "The mechanics of two-dimensional cellular materials."
Proceedings of the Royal Society of London A: Mathematical, Physical and
Engineering Sciences. Vol. 382. No. 1782. The Royal Society, 1982.
[52] Robert, F. "An isotropic three-dimensional structure with Poisson's ratio--1." Journal
of Elasticity 15 (1985): 427-430.
[53] Lakes, Roderic. "Foam structures with a negative Poisson's ratio." Science 235 (1987):
80
1038-1041.
[54] Evans, Ken E. "Auxetic polymers: a new range of materials." Endeavour 15.4 (1991):
170-174.
[55] Prall, D., and R. S. Lakes. "Properties of a chiral honeycomb with a Poisson's ratio
of—1." International Journal of Mechanical Sciences 39.3 (1997): 305-314.
[56] Murray, Gabriel J., and Farhan Gandhi. "Auxetic honeycombs with lossy polymeric
infills for high damping structural materials." Journal of Intelligent Material Systems
and Structures 24.9 (2013): 1090-1104.
[57] Mousanezhad, Davood, et al. "Hierarchical honeycomb auxetic metamaterials."
Scientific reports 5 (2015).
[58] Saxena, Krishna Kumar, Raj Das, and Emilio P. Calius. "Three Decades of Auxetics
Research− Materials with Negative Poisson's Ratio: A Review." Advanced
Engineering Materials 18.11 (2016): 1847-1870.
[59] Schenk, Mark, and Simon D. Guest. "Geometry of Miura-folded metamaterials."
Proceedings of the National Academy of Sciences 110.9 (2013): 3276-3281.
[60] Yasuda, H., and Jinkyu Yang. "Reentrant origami-based metamaterials with negative
Poisson’s ratio and bistability." Physical review letters 114.18 (2015): 185502.
[61] Lv, Cheng, et al. "Origami based mechanical metamaterials." Scientific reports 4
(2014): 5979.
[62] Scarpa, Fabrizio, et al. "Kirigami auxetic pyramidal core: mechanical properties and
wave propagation analysis in damped lattice." Journal of Vibration and Acoustics
135.4 (2013): 041001.
[63] Virk, K., et al. "SILICOMB PEEK Kirigami cellular structures: mechanical response
and energy dissipation through zero and negative stiffness." Smart Materials and
Structures 22.8 (2013): 084014.
[64] Zirbel, Shannon A., et al. "Accommodating thickness in origami-based deployable
arrays." Journal of Mechanical Design 135.11 (2013): 111005.
[65] Ruzzene, Massimo, Fabrizio Scarpa, and Francesco Soranna. "Wave beaming effects
in two-dimensional cellular structures." Smart materials and structures 12.3 (2003):
363.
[66] Gonella, Stefano, and Massimo Ruzzene. "Analysis of in-plane wave propagation in
81
hexagonal and re-entrant lattices." Journal of Sound and Vibration 312.1 (2008): 125-
139.
[67] Ruzzene, Massimo, and Panos Tsopelas. "Control of wave propagation in sandwich
plate rows with periodic honeycomb core." Journal of engineering mechanics 129.9
(2003): 975-986.
[68] Ruzzene, Massimo, and Fabrizio Scarpa. "Control of wave propagation in sandwich
beams with auxetic core." Journal of intelligent material systems and structures 14.7
(2003): 443-453.
[69] Liebold‐Ribeiro, Yvonne, and Carolin Körner. "Phononic band gaps in periodic
cellular materials." Advanced Engineering Materials 16.3 (2014): 328-334.
[70] Zhang, Yan, et al. "Regulating elastic bandgaps in two‐dimensional lattice grid by
periodical cut‐off operations." physica status solidi (b) 253.3 (2016): 566-572.
[71] Meng, J., et al. "Band gap analysis of star-shaped honeycombs with varied Poisson’s
ratio." Smart Materials and Structures 24.9 (2015): 095011.
[72] Lim, Teik-Cheng. Auxetic materials and structures. Springer, 2014.
[73] Berryman, James G. "Long-wavelength propagation in composite elastic media I.
Spherical inclusions." J. Acoust. Soc. Am 68 (1980): 6.
[74] Hudson, John Arthur. The excitation and propagation of elastic waves. CUP Archive,
1980.
[75] Achenbach, Jan. Wave propagation in elastic solids. Vol. 16. Elsevier, 2012.
[76] Lim, Teik‐Cheng, Philip Cheang, and Fabrizio Scarpa. "Wave motion in auxetic
solids." physica status solidi (b) 251.2 (2014): 388-396.
[77] Chen, Y. J., et al. "Elasticity of anti-tetrachiral anisotropic lattices." International
Journal of Solids and Structures 50.6 (2013): 996-1004.
[78] Ma, Tian-Xue, et al. "Elastic band structures of two-dimensional solid phononic crystal
with negative Poisson's ratios." Physica B: Condensed Matter 407.21 (2012): 4186-
4192.
[79] Liu, Geng-Ting, et al. "Elastic Dispersion Analysis of Two Dimensional Phononic
Crystals Composed of Auxetic Materials." ASME 2015 International Mechanical
Engineering Congress and Exposition. American Society of Mechanical Engineers,
2015.
82
[80] 周偉迪(2016)。可調式膨脹星形蜂巢狀結構之波傳分析。國立成功大學機系研究
所碩士論文，未出版，台灣台南。
[81] Zhang, Xiangdong, and Zhengyou Liu. "Negative refraction of acoustic waves in twodimensional
phononic crystals." Applied Physics Letters 85.2 (2004): 341-343.
[82] Torres, M., and FR Montero de Espinosa. "Ultrasonic band gaps and negative
refraction." Ultrasonics 42.1 (2004): 787-790.
[83] Qiu, Chunyin, Xiangdong Zhang, and Zhengyou Liu. "Far-field imaging of acoustic
waves by a two-dimensional sonic crystal." Physical Review B 71.5 (2005): 054302.
[84] Li, Jing, Zhengyou Liu, and Chunyin Qiu. "Negative refraction imaging of acoustic
waves by a two-dimensional three-component phononic crystal." Physical Review B
73.5 (2006): 054302.
[85] Veselago, Viktor G. "The electrodynamics of substances with simultaneously negative
values of and μ." Soviet physics uspekhi 10.4 (1968): 509.
[86] Tian, Ye, et al. "Broadband manipulation of acoustic wavefronts by pentamode
metasurface." Applied Physics Letters 107.22 (2015): 221906.
[87] Yu, Nanfang, et al. "Light propagation with phase discontinuities: generalized laws of
reflection and refraction." science 334.6054 (2011): 333-337.
[88] Gang, Wang, et al. "Accurate evaluation of lowest band gaps in ternary locally resonant
phononic crystals." Chinese Physics 15.8 (2006): 1843.