The review is devoted to introduce the recent development of the study in mathematical theory and methods of mechanics of quasicrystals, respectively. The mechanics of quasicrystalline materials includes elasticity, plasticity, defects, dynamics, fracture etc. In the article some relevant measured data are collected for some important quasicrystal systems, which are necessary for understanding physics and applications of the materials. It is very interesting that the mathe-m atical theory and solving methods of the mechanics of quasicrystals have developed rapidly in recent years, which is strongly supported by the experiments and applications. The theoretical development strongly enhances the under standing in-depth the physics including mechanics of the materials. The mathematical theory and computational meth o ds provide a basis to the applications of quasicrystals as functional and structural materials in practice as well. More recently the quasicrystals in soft matter are observed, which challenge the study of based on the quasicrystals of binary and ternary alloys and greatly enlarge the scope of the materials and have aroused a great deal attention of researchers, an introduction about this new phase and its mathematical theory is also given in the review.
D. Shechtman, I. Blech, D. Gratias and J. W. Cahn, “Metallic Phase with Long-Range Orientational Order and No Translational Symmetry,” Physical Review Letters, Vol. 53, No. 20, 1984, pp. 1951-1953. doi:10.1103/PhysRevLett.53.1951
D. Levine and P. J. Steinhardt, “Quasicrystal: A New Class Ordered Structure,” Physical Review Letters, Vol. 53, No. 26, 1984, pp. 2477-2480. doi:10.1103/PhysRevLett.53.2477
H. Q. Ye, D. Wang and K. H. Kuo, “Five-Fold Symmetry in Real and Reciprocal Space,” Ultramicrossopy, Vol. 16, No, 2, 1985, pp. 273-277. doi:10.1016/0304-3991(85)90083-X
Z. Zhang, H. Q. Ye and K. H. Kuo, “A New Icosahedralphase with m35 Symmetry,” Philosophical Magazine A, Vol. 52, No. 6, 1985, pp. L49-L52. doi:10.1080/01418618508242135
L. D. Landau and E. M. Lifshitz, “Theoretical Physics V: Statistical Physics,” 3rd Edition, Pergamon Press, Oxford, 1980.
P. W. Anderson, “Basic Notations of Condensed Matter Physics,” Benjamin-Cummings, Menlo Park, 1984.
A. Einstein, “Die PlackschenTheorie der Strahlung und die Theorie der spezifischenWaerme,” Annalen der Physik, Vol. 22, No. 2, 1907, pp. 180-190.
P. Debye, “Die Eigentuemlichkeit der spezifischen Waer menbeitiefen Temperaturen,” Arch de Genéve, Vol. 33, No. 4, 1912, pp. 256-258.
M Born and Th. von Kármán, “ZurTheorie der Spezifis chen Waermen,” Physikalische Zeitschrift, Vol. 14, No. 1, 1913, pp. 15-19. M. Born and K. Huang, “Dynamic Theory of Crystal Lattices,” Clarendon Press, Oxford, 1954.
B. Blinc and A. P. Lavanyuk, “Incommensurate Phases in Dielectrics I, II,” North Holland, Amsterdam, 1986.
R. Penrose, “The Role of Aesthetics in Pure and Applied Mathematical Research,” Bulletin of the Institute of Mathematics and Its Applications, Vol. 10, No. 2, 1974, pp. 266-271.
P. Bak, “Phenomenological Theory of Icosahedral In commensurate (“Quaisiperiodic”) Order in Mn-Al Alloys,” Physical Review Letters, Vol. 54, No. 8, 1985, pp. 1517-1519. doi:10.1103/PhysRevLett.54.1517
P. Bak, “Symmetry, Stability and Elastic Properties of Icosahedral Incommensurate Crystals,” Physical Review B, Vol. 32, No. 9, 1985, pp. 5764-5772. doi:10.1103/PhysRevB.32.5764
D. Levine, T. C. Lubensky, S. Ostlund, S. Ramaswamy, P. J. Steinhardt and J. Toner, “Elasticity and Dislocations in Pentagonal and Icosahedral Quasicrystals,” Physical Re view Letters, Vol. 54, No. 8, 1985, pp. 1520-1523. doi:10.1103/PhysRevLett.54.1520
T. C. Lubensky, S. Ramaswamy and J. Toner, “Hydro dynamics of Icosahedral Quasicrystals,” Physical Review B, Vol. 32, No. 11, 1985, pp. 7444-7452. doi:10.1103/PhysRevB.32.7444
T. C. Lubensky, S. Ramaswamy and J. Toner, “Dislocation Motion in Quasicrystals and Implications for Macroscopic Properties,” Physical Review B, Vol. 33, No. 11, 1986, pp. 7715-7719. doi:10.1103/PhysRevB.33.7715
T. C. Lubensky, J. E. S. Socolar, P. J. Steinhardt, P. A. Bancel and P. A. Heiney, “Distortion and Peak Broadening in Quasicrystal Diffraction Patterns,” Physical Review Letters, Vol. 57, No. 12, 1986, pp. 1440-1443. doi:10.1103/PhysRevLett.57.1440
T. C. Lubensky, “Symmetry, Elasticity and Hydrodynamics in Quasiperiodic Structures,” In: M. V. Jaric, Ed., Introduction to Quasicrystals, Academic Press, New York, 1988, pp. 199-280. doi:10.1016/B978-0-12-040601-2.50011-1
P. A. Kalugin, A. Kitaev and L. S. Levitov, “6-Dimensional Properties of Al0.86Mn0.14alloy,” Journal de Physique letters, Vol. 46, No. 13, 1985, pp. 601-607. doi:10.1051/jphyslet:019850046013060100
S. M. Torian and D. Mermin, “Mean-Field Theory of Quasicrystalline Order,” Physical Review Letters, Vol. 54, No. 14, 1985, pp. 1524-1527. doi:10.1103/PhysRevLett.54.1524
M. V. Jaric, “Long-Range Icosahedral Orientational Order and Quasicrystals,” Physics Review Letters, Vol. 55, No. 6, 1985, pp. 607-610. doi:10.1103/PhysRevLett.55.607
M. Duneau and A. Katz, “Quasiperiodic Patterns,” Physical Review Letters, Vol. 54, No. 25, 1985, pp. 2688-2691. doi:10.1103/PhysRevLett.55.607
J. E. S. Socolar, T. C. Lubensky and P. J. Steinhardt, “Phonons, Phasons, and Dislocations in Quasicrystals,” Physical Review B, Vol. 34, No. 5, 1986, pp. 3345-3360.
F. Gahler and J. Rhyner, “Equivalence of the Generalised Grid and Projection Methods for the Construction of Quasiperiodictilings,” Journal of Physics A: Mathematical and General, Vol. 19, No. 2, pp. 267-277. doi:10.1088/0305-4470/19/2/020
D. H. Ding, W. G. Yang, R. H. Wang and C. Z. Hu, “Generalized Elasticity Theory of Quasicrystals,” Physical Review B, Vol. 48, No. 10, 1993, pp. 7003-7010. doi:10.1103/PhysRevB.48.7003
T. Janssen, “The Symmetry Operations for N-Dimensional Periodic and Quasi-Periodic Structures,” Zeits-chrift fuerKristall, Vol. 198, No. 1, 1992, pp. 17-32.
W. G. Yang, D. H. Ding, R. H. Wang and C. Z. Hu, “Group-Theoretical Derivation of the Numbers of Independent Physical Constants of Quasicrystals,” Physical review B, Vol. 49, No. 18, 1994, pp. 12656-12661. doi:10.1103/PhysRevB.49.12656
C. Z. Hu, R. H. Wang and D. H. Ding, “Piezoelectric Effects in Quasicrystals,” Physical Review B, Vol. 56, No. 5, 1997, pp. 2463-2468. doi:10.1103/PhysRevB.49.12656
G. A. M. Reynolds, B. Golding, A. R. Kortan et al., “Iso tropic Elasticity of the Al-Cu-Li Quasicrystal,” Physical Review B, Vol. 41, No. 2, 1997, pp. 1194-1195. doi:10.1103/PhysRevB.41.1194
P. S. Spoor, J. D. Maynard and A. R. Kortan, “Elastic Isotropy and Anisotropy in Quasicrystalline and Cubic AlCuLi,” Physical Review Letters, Vol. 75, No. 19, 1995, pp. 3462-3465. doi:10.1103/PhysRevLett.75.3462
K. Tanaka, Y. Mitarai and M. Koiwa, “Elastic Constants of Al-based Icosahedral Quasicrystals,” Philosophical Magazine A, Vol. 73, No. 6, 1996, pp. 1715-1723. doi:10.1080/01418619608243008
J.-Y. Duquesne and B. Perrin, “Elastic Wave Interaction in Icosahedral AlPdMn,” Physica B: Condensed Matter, Vol. 316-317, 2002, pp. 317-320. doi:10.1016/S0921-4526(02)00496-9
K. Foster, R. G. Leisure, A. Shaklee, J. Y. Kim and K. F. Kelton, “Elastic Moduli of a Ti-Zr-Ni Icosahedral Quasi crystal and a 1/1 bcc Crystal Approximant,” Physical Re view B, Vol. 59, No. 17, 1999, pp. 11132-11135. doi:10.1103/PhysRevB.59.11132
J. Schreuer, W. Steurer, T. A. Lograsso and D. Wu, “Elastic Properties of Icosahedral i-Cd84Yb16 and Hexagonal h-Cd51Yb14,” Philosophical Magazine Letters, Vol. 84, No. 10, 2004, pp. 643-653. doi:10.1080/09500830512331329132
R. Sterzel, C. Hinkel, A. Haas et al., “Ultrasonic Measurements on FCI Zn-Mg-Y Single Crystals,” Europhysics Letters, Vol. 49, No. 6, 2004, pp. 742-747. doi:10.1209/epl/i2000-00213-1
A. Letoublon, M. De Boissieu, M. Boudard, et al., “Phason Elastic Constants of the Icosahedral Al-Pd-Mn Phase Derived from Diffuse Scattering Measurements,” Philosophical Magazine Letters, Vol. 81, No. 4, 2001, pp. 273-283. doi:10.1080/09500830010029409
M. De Boissieu, S. Francoual, Y. Kaneko, et al., “Diffuse Scattering and Phason Fluctuations in the Zn-Mg-Sc Ico-sahedral Quasicrystal and Its Zn-Sc Periodic Approximant,” Physics Review Letters, Vol. 95, No. 10, 2005, p. 105503. doi:10.1103/PhysRevLett.95.105503
K. Edagawa and Y. GI So, “Experimental Evaluation of Phonon-Phason Coupling in Icosahedral Quasicrystals,” Philosophical Magazine, Vol. 87, No. 1, 2007, pp. 77-95. doi:10.1080/14786430600891352
M. A. Chernikov, H. R. Ott, A. Bianchi, A. Miglion and T. W. Darling, “Elastic Moduli of a Single Quasicrystal of Decagonal Al-Ni-Co: Evidence for Transverse Elastic Isotropy,” Physics Review Letters, Vol. 80, No. 2, 1998, pp. 321-324. doi:10.1103/PhysRevLett.80.321
H. C. Jeong and P. J. Steinhardt, “Finite-Temperature Elasticity Phase Transition in Decagonal Quasicrystals,” Physical Review B, Vol. 48, No. 13, 1993, pp. 9394-9403.
A. E. H. Love, “A Treatise on the Mathematical Theory of Elasticity,” 4th Edition, Dover, New York, 1954.
R. Courant and D. Hilbert, “Mathematical Physics Method,” Interscience, New York, 1954.
T. Y. Fan, “Mathematical Theory of Elasticity of Quasi crystals and Its Applications,” Beijing Institute Technology Press, Beijing, 1999.
T. Y. Fan, “Mathematical Theory of Elasticity of Quasi crystals and Its Applications,” Science Press, Beijing/ Springer-Verlag, Heidelberg, 2010.
Y. Z. Peng and T. Y. Fan, “Elastic Theory of 1D Qua-siperiodic Stacking of 2D Crystals,” Journal of Physics: Condensed Matter, Vol. 12, No. 45, 2000, pp. 9381-9387.
G. T. Liu, T. Y. Fan and R. P. Guo, “Governing Equations and General Solutions of Plane Elasticity of One Dimensional Quasicrystals,” International Journal of So lids and Structures, Vol. 41, No. 14, 2004, pp. 3949-3959.
W. Q. Chen, Y. L. Ma and H. J. Ding, “On Three-Dimensional Elastic Problems of One-Dimensional Hexagonal Quasicrystal Bodies,” Mechanics Research Communications, Vol. 31, No. 6, 2004, pp. 633-641. doi:10.1016/j.mechrescom.2004.03.007
X. Wang, “The General Solution of One-Dimensional Hexagonal Quasicrystal,” Journal of Applied Mathematics and Mechanics, Vol. 33, No. 4, 2006, pp. 576-580.
Y. Gao, Y. T. Zhao and B. S. Zhao, “Boundary Value Problems of Holomorphic Vector Functions in One-Dimensional Hexagonal Quasicrystals,” Physica B: Condensed Matter, Vol. 394, No. 1, 2007, pp. 56-61. doi:10.1016/j.physb.2007.02.007
X. F. Li and T. Y. Fan, “New Method for Solving Elastic ity Problems of Some Plana Quasicrystals and Solutions,” Chinese Physics Letters, Vol. 15, No. 4, 1998, pp. 278 280. doi:10.1088/0256-307X/15/4/016
X. F. Li, T. Y. Fan and Y. F. Sun, “A Decagonal Qua-sicrystal with a Griffith Crack,” Philosophical Magazine A, Vol. 79, No. 8, 1999, pp. 1943-1952.
X. F. Li, X. Y. Duan, T. Y. Fan and Y. F. Sun, “Elastic Field for a Straight Dislocation in a Decagonal Quasi crystal,” Journal of Physics: Condensed Matter, Vol. 11, No. 3, 1999, pp. 703-711. doi:10.1088/0953-8984/11/3/009
Y. C. Guo and T. Y. Fan, “ModeⅡ Griffith Crack in Decagonal Quasicrystals,” Journal of Applied Mathematics and Mechanics, Vol. 22, No. 10, 2001, pp. 1311 1317.
W. M. Zhou and T. Y. Fan, “Axisymmetric Elasticity Problem of Cubic Quasicrystal,” Chinese Physics B, Vol. 9, No. 4, 2000, pp. 294-303. doi:10.1088/1009-1963/9/4/009
W. M. Zhou and T. Y. Fan, “Plane Elasticity and Crack Problem of Octagonal Quasicrystals,” Chinese Physics, Vol. 10, No. 8, 2001, pp. 743-747. doi:10.1088/1009-1963/10/8/315
T. Y. Fan and L. H. Guo, “Final Governing Equation of Plane Elasticity of Icosahedral Quasicrystals,” Physics Letter A, Vol. 341, No. 4, pp. 235-239.
L. H. Li and T. Y. Fan, “Final Governing Equation of Plane Elasticity of Icosahedral Quasicrystals-Stress Potential Method,” Chinese Physics Letters, Vol. 24, No. 9, 2006, pp. 2519-2521.
P. De and R. A. Pelcovits, “Linear Elasticity Theory of Pentagonal Quasicrystals,” Physical Review B, Vol. 35, No. 16, 1987, pp. 8609-8620. doi:10.1103/PhysRevB.35.8609
D. H. Ding, R. H. Wang, W. G. Yang and C. Z. Hu, “General Expressions for the Elastic Displacement Fields Induced by Dislocations in Quasicrystals,” Journal of Physics: Condensed Matter, Vol. 7, No. 28, 1995, pp. 5423-5436. doi:10.1088/0953-8984/7/28/003
D. H. Ding, R. H. Wang, W. G. Yang, C. Z. Hu and Y. L. Qin, “Elasticity Theory of Straight Dislocations in Quasicrystals,” Philosophical Magazine Letters, Vol. 72, No. 5, 1975, pp. 352-359.
S. H. Yang and D. H. Ding, “Crystal Dislocation The ory,” Science Press, Beijing, 1998.
W. G. Yang, M. Feuerbacher, N. Tamura, D. H. Ding, R. H. Wang and K. Urban, “Atomtic Model of Dislocation in Icosahedral Quasicrystals,” Philosophical Magazine A, Vol. 77, No. 6, 1998, pp. 1481-1497. doi:10.1080/01418619808214265
A. Y. Zhu, T. Y. Fan and L. H. Guo, “A Straight Dislocation in an Icosahedral Quasicrystal,” Journal of Physics: Condensed Matter, Vol. 19, No. 23, 2007, pp. 236216. doi:10.1088/0953-8984/19/23/236216
A. Y. Zhu and T. Y. Fan, “Elastic Analysis of a Griffith Crack in Icosahedral Al-Pd-Mn Quasicrystal,” International Journal of Modern Physics B, Vol. 23, No. 16, 2009, pp. 1-16. doi:10.1142/S0217979209052510
I. N. Sneddon, “Fourier Transforms,” McGrow-Hill, New York, 1952.
G. T. Liu and T. Y. Fan, “The Complex Method of the Plane Elasticity in 2D Quasicrystals Point Group 10mm Ten-Fold Rotation Symmetry Notch Problems,” Science in China E, Vol. 46, No. 4, 2003, pp. 326-336.
L. H. Li and T. Y. Fan, “Complex Function Method for Solving Notch Problem of Point Group 10 and Two-Dimensional Quasicrystal Based on the Stress Potential Function,” Journal of Physics: Condensed Matter, Vol. 18, No. 47, 2006, pp. 10631-10641. doi:10.1088/0953-8984/18/47/009
L. H. Li and T. Y. Fan, “Complex Function Method for Notch Problem of Plane Elasticity of Icosahedral Quasi crystals,” Science in China G, Vol. 51, No. 6, 2008, pp. 773-780.
T. Y. Fan, T. Z. Yang, L. H. Li and W. Li, “The Strict Theory of Complex Variable Function Method of Sextuple Harmonic Equation and Applications,” Journal of Mathematical Physics, Vol. 51, No. 5, 2010, Article ID: 053519. doi:10.1063/1.3421668
I. N. Muskhelishvil, “Some Basic Problems of Mathematical Theory of Elasticity,” Nordhoff, Groringen, 1956.
L. H. Guo and T. Y. Fan, “Solvability on Boundary Value Problems of Elasticity of Three-Dimensional Qua-sicrystals,” Applied Mathematics and Mechanics, Vol. 28, No. 8, 2007, pp. 1061-1070. doi:10.1007/s10483-007-0808-y
T. Y. Fan, X. F. Li and Y. F. Sun, “A Moving Screw Dislocation in One-Dimensional Hexagonal Quasicrystal,” Acta Physica Sinica (Overseas Edition), Vol. 8, No. 3, 1999, pp. 288-295.
T. Y. Fan, “A Study on Special Heat of One-Dimensional Hexagonal Quasicrystals,” Journal of Physics: Condensed Matter, Vol. 11, No. 45, 1999, pp. L513-L517. doi:10.1088/0953-8984/11/45/101
C. L. Li and Y. Y. Liu, “Phason-Strain Influences on Low-Temperature Specific Heat of the Decagonal Al-Ni Co Quasicrystal,” Chinese Physics Letters, Vol. 18, No. 4, 2001, pp. 570-572. doi:10.1088/0256-307X/18/4/335
C. L. Li and Y. Y. Liu, “Low-Temperature Lattice Excitation of Icosahedral Al-Mn-Pd Quasicrystals,” Physical review B, Vol. 63, No. 6, 2001, p. 064203. doi:10.1103/PhysRevB.63.064203
S. B. Rocal and V. L. Lorman, “Anisotropy of Acoustic Phonon Properties of an Icosahedral Quasicrystal at High Temperature Due to Phonon-Phason Coupling,” Physical Review B, Vol. 62, No. 2, 2000, pp. 874-879. doi:10.1103/PhysRevB.62.874
S. B. Rochal and V. L. Lorman, “Minimal Model of the Phonon-Phason Dynamics on Icosahedral Quasicrystals and Its Application for the Problem of Internal Friction in the i-AIPdMn Alloys,” Physical Review B, Vol. 66, No. 14, 2002, Article ID: 144204. doi:10.1103/PhysRevB.66.144204
T. Y. Fan, X. F. Wang, W. Li and A. Y. Zhu, “Elasto-Hydrodynamics of Quasicrystals,” Philosophical Magazine, Vol. 89, No. 6, 2009, pp. 501-512. doi:10.1080/14786430802562157
A. Y. Zhu and T. Y. Fan, “Dynamic Crack Propagation in a Decagonal Al-Ni-Co Quasicrystal,” Journal of Physics: Condensed Matter, Vol. 20, No. 29, 2008, pp. 295217. doi:10.1088/0953-8984/20/29/295217
X. F. Wang, T. Y. Fan and A. Y. Zhu, “Dynamic Behaviour of the Icosahedral Al-Pd-Mnquasicrystal with a Griffith Crack,” Chinese Physics B, Vol. 18, No. 2, 2009, pp. 709-714. doi:10.1088/1674-1056/18/2/050
X. F. Wang and T. Y. Fan, “Study on the Dynamics of the Double Cantilever-Beam Specimen of Decagonal Al-Ni Co Quasicrystals,” Applied Mathematics and Computation, Vol. 211, No. 2, 2009, pp. 336-346. doi:10.1016/j.amc.2009.01.044
M. Feuerbacher and K. Urban, “Platic Behaviour of Quasicrystalline Materials,” In: H. R. Trebin, Ed., Quasi crystals, Wiley Press, Berlin, 2003, pp. 432-461.
D. Calliard, “Dislocation Mechanism and Plasticity of Quasicrystals: TEM Observations in Icosahedral Al-Pd-Mn,” Materials Science Forum, Vol. 509, No. 1, 2006, pp. 49-56. doi:10.4028/www.scientific.net/MSF.509.49
B. Geyer, M. Bartisch, M. Feuerbacher, K. Urban and U. Messerschmidt, “Plastic Deformation of Icosahedral Al Pd-Mn Single Quasicrystals, I. Experimental Results,” Philosophical Magazine A, Vol. 80, No. 5, 2000, pp. 1151-1164. doi:10.1080/01418610008212107
U. Messerschmidt, M. Bartisch, B. Geyer, M. Feuerbacherz and K. Urban, “Plastic Deformation of Icosahedral Al-Pd-Mn Single Quasicrystals, II, Interpretation of Experimental Results,” Philosophical Magazine A, Vol. 80, No. 5, 2000, pp. 1165-1181. doi:10.1080/01418610008212108
K. Urban and M. Wollgarten, “Dislocation and Plasticity of Quasicrstals,” Materials Science Forum, Vol. 150-151, No. 2, 1994, pp. 315-322. doi:10.4028/www.scientific.net/MSF.150-151.315
M. Wollgten, M. Bartschs, U. Messerschmidt, M. Feuerbacher, M. Rosenfeld, M. Beyss and K. Urban, “In-Situ Observation of Dislocation Motion in Icosahedral Al Pd-Mn Single Quasicrystals,” Philosophical Magazine Letters, Vol. 71, No. 2, 1995, pp. 99-105. doi:10.1080/09500839508241001
M. Feuerbacher, B. Bartsch, B. Grushk, U. Messersch-midt and K. Urban, “Plastic Deformation of Decagonal Al-Ni-Co Quasicrystals,” Philosophical Magazine Letters, Vol. 76, No. 6, 1997, pp. 396-375. doi:10.1080/095008397178788
U. Messerschmidty, M. Bartschy, M. Feuerbacherz, B. Geyery and K. Urbanz, “Friction Mechanism Ofdislocation Motion in Icosahedral Al-Pd-Mn Single Quasicrystals,” Philosophical Magazine A, Vol. 79, No. 9 1999, pp. 2123-2135.
P. Guyot and G. Canova, “The Plasticity of Icosahedral Quasicrystals,” Philosophical Magazine A, Vol. 79, No. 11, 1999, pp. 2815-2822. doi:10.1080/01418619908212026
M. Feuerbacher, P. Schall, Y. Estrin and Y. Brecht, “A Constitutive Model for Quasicrystal Plasticity,” Philosophical Magazine Letters, Vol. 81, No. 7, 2011, pp. 473-482. doi:10.1080/09500830110049983
T. Y. Fan and L. Fan, “Plastic Fracture of Quasicrystals,” Philosophical Magazine, Vol. 88, No. 4, 2008, pp. 523-535. doi:10.1080/14786430801894536
D. S. Dugdale, “Yielding of Steel Sheets Containing Slits,” Journal of the Mechanics and Physics of Solids, Vol. 8, No. 2, 1960, pp. 105-108. doi:10.1016/0022-5096(60)90013-2
G. I. Barenblatt, “The Mathematical Theory of Equilibrium of Crack in Brittle Fracture,” Advances in Applied Mechanics, Vol. 7, 1962. pp. 55-129. doi:10.1016/S0065-2156(08)70121-2
W. Li and T. Y. Fan, “Study on Plastic Analysis of Crack in Three-Dimensional Quasicrystal Materials,” Philosophical Magazine, Vol. 72, No. 6, 2009, pp. 2823-2831. doi:10.1080/14786430903082014
T. Y. Fan, H.-R. Trebin, U Messerschmidt and Y W Mai, “Plastic Flow Coupled with a Griffith Crack in some One and Two-Dimensional Quasicrystals,” Journal of Physics: Condensed Matter, Vol. 16, No. 29, 2004, pp. 5229-5240. doi:10.1088/0953-8984/16/29/014
B. A. Bilby, A. H. Cottrell and K. H. Swinden, “The Sp read of Plastic Yield from a Notch,” Proceedings of the Royal Society A, Vol. 272, No. 2, 1963, pp. 304-314.
B. A. Bilby, A. H. Cottrell, E. Smith and K. H. Swinden. “Plastic Yealding from Sharp Notches,” Proceedings of the Royal Society A, Vol. 279, No. 1, 1964, pp. 1-9.
N. I. Muskhelishvili, “Singular Integral Equations,” Groningen, Nordhoff, 1956.
T. Y. Fan and L. Fan, “Relation between Eshelby Integral and Generalized BCS and Generalized DB Models for Some One and Two-Dimensional Quasicrystals,” Chinese Physics B, Vol. 20, No. 3, 2011, Article ID: 036102. doi:10.1088/1674-1056/20/3/036102
J. D. Eshelby, “The Continuum Theory of Dislocations in Crystals,” In: F. Seits, Ed., Solid State Physics, Academic Press, New York, 1956, pp. 79-144.
T. Y. Fan and Y. W. Mai, “Partition Function and State Equation of Point Group 12mm Dodecagonal Quasi crystals,” European Physical Journal B, Vol. 31, No. 1, 2003, pp. 25-27. doi:10.1140/epjb/e2003-00004-0
J. Wang and T. Y. Fan, “Analytic Study on Specific Heat of Icosahedral Al-Pd-Mnquasicrystals,” Modern Physics Letters B, Vol. 22, No. 17, 2008, pp. 1651-1659. doi:10.1142/S021798490801611X
M. Casdagli, “Symbolic Dynamics for the Renormaliza tion Map of a Quasiperiodic Schroedinger Equation,” Communications in Mathematical Physics, Vol. 107, No. 2, 1986, pp. 295-318. doi:10.1007/BF01209396
A. Suetoe, “The Spectrum of a Quasiperiodic Schroedinger Operator,” Communications in Mathematical Physics, Vol. 111, No. 3, 1987, pp. 409-415. doi:10.1007/BF01238906
S. Kotani, “Jacobi Matrices with Random Potentials Taking Finitely Many Values,” Reviews in Mathematical Physics, Vol. 1, No. 1, 1989, pp. 129-133. doi:10.1142/S0129055X89000067
J. Bellissard, B. Iochum, E. Scoppola and D. Testart, “Spectral Properties of One Dimensional Quasi-Crystals,” Communications in Mathematical Physics, Vol. 125, No. 3, 1989, pp. 527-543. doi:10.1007/BF01218415
A. Bovier and J.-M. Ghez, “Spectrum Properties of One-Dimensional Schroedinger Operators with Potentials Generated by Substitutions,” Communications in Mathematical Physics, Vol. 158, No. 1, 1993, pp. 45-66. doi:10.1007/BF02097231
Q. H. Liu, B. Tan, Z. X. Wen and J. Wu, “Measure Zero Spectrum of a Class of Schroedinger Operators,” Journal of Statistical Physics , Vol. 106, No. 3, 2002, pp. 681-691. doi:10.1023/A:1013718624572
D. Lenz, “Singular Spectrum of Lebesgue Measure Zero for One-Dimensional Quasicrystals,” Communications in Mathematical Physics, Vol. 227, No. 1, 2002, pp. 119-130. doi:10.1007/s002200200624
A. Furman, “On the Multiplicative Ergodic Theorem for Uniquely Ergodic Systems,” Annales de l’Institut Henri Poincare (B) Probability and Statistics, Vol. 33, No. 6, 1997, pp. 97-815. doi:10.1016/S0246-0203(97)80113-6
D. Damanik and D. Lenz, “A Condition of Boshernitzan and Uniform Convergence in the Multiplicative Ergodic Theorem,” Duke Mathematical Journal, Vol. 133, No. 1, 2006, pp. 95-123. doi:10.1215/S0012-7094-06-13314-8
Q. H. Liu and Y. H. Qu, “Uniform Convergence of Schroedinger Cocycles over Simple Toeplitz Subshift,” Annales Henri Poincare, Vol. 12, No. 1, 2011, pp. 153-172. doi:10.1007/s00023-010-0075-y
Q. H. Liu and Y. H. Qu, “Uniform Convergence of Schroedinger Cocycles over bounded Toeplitz Subshift,” Annales Henri Poincare, Vol. 13, No. 6, 2012, pp. 1483 1550. doi:10.1007/s00023-011-0157-5
Q. H. Liu and Z. Y. Wen, “Hausdorff Dimension of Spectrum of One-Dimensional Schroedinger Operator with Sturmian Potentials,” Potential Analysis, Vol. 20, No. 1, 2004, pp. 33-59. doi:10.1023/A:1025537823884
D. Damanik, M. Embree, A. Gorodetski and S. Tcheremchantsev, “The Fractal Dimension of the Spectrum of the Fibonacci Hamiltonian,” Communications in Mathematical Physics, Vol. 280, No. 2, 2008, pp. 499-516. doi:10.1007/s00220-008-0451-3
Q. H. Liu, J. Peyrière and Z. Y. Wen, “Dimension of the Spectrum of One-Dimensional Discrete Schrodinger Operators with Sturmian Potentials,” Comptes Randus Mathematique, Vol. 345, No. 12, 2007, pp. 667-672. doi:10.1016/j.crma.2007.10.048
S. Fan, Q. H. Liu and Z. Y. Wen, “Gibbs-Like Measure for Spectrum of a Class of Quasi-Crystals,” Ergodic Theory Dynam. Systems, Vol. 31, No. 6, 2011, pp. 1669-1695. doi:10.1017/S0143385710000635
S. Fischer, A. Exner, K. Zielske, J. Perlich, S. Deloudi, W. Steuer, P. Linder and S. Foestor, “Colloidal Quasicrystals with 12-Fold and 18-Fold Symmery,” Proceedings of the National Academy of Sciences of the United States, Vol. 108, No. 5, 2011, pp. 1810-1814. doi:10.1073/pnas.1008695108
A. R. Denton and H. Loewen, “Stability of Colloidal Quasicrystals,” Physical Review Letters, Vol. 81, No. 2, 1998, pp. 469-472. doi:10.1103/PhysRevLett.81.469
X. B. Zheng, G. Ungar, Y. S. Liu, P. Virgil, E. Andrés. Dulcey and K. J. Hobbs, “Supermolecular Dentritic Liquid Quasicrystals,” Nature, Vol. 428, No. 6979, 2004, pp. 157-160.
K. Takano, “A Mesoscopicarchimedian Tiling Having a New Complexity in an ABC Star Polymer,” Journal of Polymer Science Part B: Polymer Physics, Vol. 43, No. 18, 2005, pp. 2427-2432.doi:10.1002/polb.20537
K. Hayashida, T. Dotera, A. Takano, Y. Matsushita, “Polymeric Quasicrystal: Mesoscopicquasicrystalline Tiling in ABC Star Polymers,” Physical review Letters, Vol. 98, No. 19, 2007, Article ID: 195502. doi:10.1103/PhysRevLett.98.195502
V. D. Talapin, “Quasicrystalline Order in Self-Assembled Binary Nanoparticle Superlattices,” Nature, Vol. 461, 2009, pp. 964-967. doi:10.1038/nature08439
C. Z. Hu, D. H. Ding, W. G. Yang and R. H. Wang, “Possible Two-Dimensional Quasicrystals Structures with a Six-Dimensional Embedding Space,” Physical Review B, Vol. 49, No. 14, 1994, pp. 9423-9427. doi:10.1103/PhysRevB.49.9423
T. Y. Fan, “The Elasticity and Hydrodynamics of Quasi crystals with 7-, 14-, 9 and 18-Fold Symmetries in Sol ids,” 2012. http://arxiv.org/abs/1210.069
T. Y. Fan, “The Elasto-/ Hydro-Dynamics of Quasicrystals with 12 and 18-Fold Symmetries in some Soft Matters,” 2012. http://arxiv.org/abs/1210.1667.
T. Y. Fan, “Poisson Brackets Method and Some Applications to Quasicrystals, Liquid Crystals and Soft Matters Study,” 2012, Acta Mechanica Sinica, Unpublished.
T. Y. Fan, “Equations of Motion of some Soft Matter and Mathematical Solutions,” Chinese Physics Letters, 2013, Unpublished.
E. Radi and P. M. Mariano, “Dynamic Steady-Stae Crack Propagation in Quasicrystals,” Mathematical Methods in the Applied Sciences, Vol. 34, No. 1, 2011, pp. 1-23. doi:10.1002/mma.1325
H. K. Akmaz and U. Akinci, “Dynamic Plane Elasticity Problems of 2D Quasicrystals,” Physics Letters A, Vol. 373, No. 12-13, 2009, pp. 1105-1110.
G. Altay and M. C. Doekmeci, “On the Fundamental Equation of Piezoelasticity of Quasicrystal Media,” International Journal of Solids and Structures, Vol. 49, No. 23-24, 2012, pp. 3255-3262. doi:10.1016/j.ijsolstr.2012.06.016
X. F. Li, “General Solution of Elasto-Hydrodynamics of Two-Dimensional Quasicrystals,” Philosophical Maga zine Letters, Vol. 91, No. 4, 2011, pp. 313-320. doi:10.1080/09500839.2011.559179
Y. Gao and B. S. Zhao, “General Solutions of Three Dimensional Problem of Two Dimensional Quasicrystals,” Applied Mathematical Modelling, Vol. 33, No. 8, 2009, pp. 3382-3391. doi:10.1016/j.apm.2008.11.001