准晶数学弹性理论和某些有关研究的进展(上)

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doi:10.6052/1000-0992-11-168

范天佑^,

北京理工大学物理学院, 北京 100081

基金项目:国家自然科学基金项目(10372016,10672022)资助

详细信息

通讯作者:
范天佑

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- 被引次数:0
出版历程
- 收稿日期:2011-12-01
- 修回日期:2012-07-12
- 刊出日期:2012-09-25

DEVELOPMENT ON MATHEMATICAL THEORY OF ELASTICITY OF QUASICRYSTALS AND SOME RELEVANT TOPICS

FAN Tianyou^,

School of Physics, Beijing Institute of Technology, Beijing 100081, China

Funds:The project was supported by the National Natural Science Foundation of China (10372016, 10672022).

More Information

Corresponding author:FAN Tianyou

摘要

摘要:本文对固体准晶力学性能和准晶数学弹性, 塑性, 断裂以及有关研究的进展作了评论, 尤其对材料常数和塑性变形行为的测量, 一维、二维、三维准晶弹性理论, 动力学、非线性、缺陷理论、准晶弹性新型偏微分方程的推导和精确分析解, 复分析方法, 变分原理和有限元方法, 有限差分方法这些宏观问题和它们的数学方法进行了分析, 同时对准晶晶格动力学问题的数学理论也作了初步讨论. 近来在软物质中发现了12 次和18次对称准晶, 意义重大, 这里也做了初步介绍. 文中重点讨论此领域最近这些年来中国科学工作者的工作.
- 准晶/
- 弹性/
- 塑性/
- 缺陷/
- 动力学/
- 精确解
Abstract:This paper gives an introduction to the development on mathematical theory of elasticity of quasicrystals and some relevant topics, in which the physical framework of elasticity is discussed first, the measuremental data of material constants for quasicrystals in solid phase are one of the most important basis to the theory and applications and are listed in detail. As the fundamental points, the analytic methods and exact solutions for various boundary value or boundary-initial value problems are summarized, the Chinese researchers contributed the effort in this respect. Apart from elasticity, dynamics and plasticity of the novel material are also discussed. At last the paper offers a simple description on a possible theoretical treatment, from the view point of elasticity and hydrodynamics, to quasicrystals in soft matter phase observed very recently.

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参考文献 (65)

1 Shechtman D, Blech I, Gratias D, et al. Metallic phase with long-range orientational order and no translational symmetry. Phys. Rev. Lett., 1984, 53(20): 1951-1953

2 Ye H Q, Wang D, Kuo K H. Five-fold symmetry in real and reciprocal space. Ultramicrossopy, 1985, 16(2): 273-277

3 Zhang Z, Ye H Q, Kuo K H. A new icosahedral phase with m35 symmetry. Phil. Mag. A, 1985, 52(6): L49-L52

4 Landau L D, Lifshitz M E. Theoretical Physics V: Statistical Physics, 3rd Edition, Pergamon Press, Oxford. 1980

5 Anderson P W. Basic Notations of Condensed Matter Physics, Menlo Park: Benjamin-Cummings, 1984

6 Einstein A. Die plackschen theorie der strahlung und die theorie der spezifischen waerme. Ann. d. Phys., 1907,22(4): 180-190

7 Debye P. Die Eigentuemlichkeit der spezifischen waermen bei tiefen temperaturen. Arch de Gen eve, 1912, 33(4):256-258

8 Born M, von Kármán Th. Zur theorie der spezifischen waermen. Physikalische Zeitschrift, 1913, 14(1): 15-19 Born M, Huang K. Dynamic Theory of Crystal Lattices, Clarendon Press, Oxford. 1954

9 Blinc B, Lavanyuk A P. Incommensurate Phases in Dielectrics I, II,. Amsterdam: North Holand, 1986

10 Penrose R. The role of aesthetics in pure and applied mathematical research. Bull. Inst. Math. Appl., 1974,10: 266-271

11 Bak P. Phenomenological theory of icosahedral incommensurate (“quaisiperiodic”) order in Mn-Al alloys. Phys. Rev. Lett., 1985, 54(8): 1517-1519

12 Bak P. Symmetry, stability and elastic properties of icosahedral incommensurate crystals. Phys. Rev. B, 1985,32(9): 5764-5772

13 Levine D, Lubensky T C, Ostlund S, et al. Elasticity and dislocations in pentagonal and icosahedral quasicrystals. Phys. Rev. Lett., 1985, 54(8): 1520-1523

14 Lubensky T C, Ramaswamy S, Toner J. Hydrodynamics of icosahedral quasicrystals. Phys. Rev. B, 1985, 32(11):7444-7452

15 Lubensky T C, Ramaswamy S, Toner J. Dislocation motion in quasicrystals and implications for macroscopic properties. Phys. Rev. B, 1986, 33(11): 7715-7719

16 Lubensky T C, Socolar J E S, Steinhardt P J, et al. Distortion and peak broadening in quasicrystal diffraction patterns. Phys. Rev. Lett., 1986, 57(12): 1440-1443

17 Lubensky T C. Introduction to Quasicrystals, Boston: Academic Press, 1988

18 Kalugin P A, Kitaev A, Levitov L S. 6-dimensional properties of Al0:86Mn0:14 alloy. J. Phys. Lett., 1985, 46(13):601-607

19 Torian S M, Mermin D. Mean-field theory of quasicrystalline order. Phys. Rev. Lett., 1985, 54(14): 1524-1527

20 Jaric M V. Long-range icosahedral orientational order and quasicrystals. Phys. Rev. Lett., 1985, 55(6): 607-610

21 Duneau M, Katz A. Quasiperiodic patterns. Phys. Rev. Lett., 1985, 54(25): 2688-2691

22 Socolar J E S, Lubensky T C, Steinhardt P J. Phonons, phasons, and dislocations in quasicrystals. Phys. Rev. B,1986, 34(5): 3345-3360

23 Gahler F, Rhyner J. Equivalence of the generalised grid and projection methods for the construction of quasiperiodic tilings. J. Phys. A: Math. Gen., 1986, 19(2): 267-277

24 Ding D H, Yang W G, Wang R H, et al. Generalized elasticity theory of quasicrystals. Phys. Rev. B, 1993, 48(10):7003-7010

25 Janssen T. The symmetry operations for n-dimensional periodic and quasi-periodic structures. Zeitschrift fuer Kristall, 1992, 198(1): 17-32

26 Yang W G, Ding D H, Wang R H, et al. Phys. Rev. B,1994, 49: 12616-12621

27 Hu C Z, Wang R H, Ding D H. Rezoelectric effects in quasicrystals. Phys. Rev. B, 1997, 56: 2463-2468

28 Hu C Z, Wang R H, Ding D H. Symmetry groups physical property tensors, elastricity and disolcations in quasicrystals. Rep Prog Phys, 2000, 63(1): 1-39

29 Reynolds G A M, Golding B, Kortan A R, et al. Isotropic elasticity of the Al-Cu-Li quasicrystal. Phys. Rev. B,1990, 41(2): 1194-1195

30 Spoor P S, Maynard J D, Kortan A R. Elastic isotropy and anisotropy in quasicrystalline and cubic AlCuLi. Phys. Rev. Lett., 1995, 75(19): 3462-3465

31 Tanaka K, Mitarai, Koiwa M. Elastic constants of Albased icosahedral quasicrystals. Phil. Mag. A, 1996,73(6): 1715-1723

32 Duquesne J Y, Perrin B. Elastic wave interaction in icosahedral AlPdMn. Physics B, 2002. 316: 317-320

33 Foster K, Leisure R G, Shaklee A, et al. Elastic moduli of a Ti-Zr-Ni icosahedral quasicrystal and a 1/1 bcc crystal approximant. Phys. Rev. B, 1999, 59(17): 11132-11135

34 Schreuer J, Steurer W, Lograsso T A, et al. Elastic properties of icosahedral i-Cd84Yb16 and hexagonal h-Cd51Yb14. Phil. Mag. Lett., 2004, 84(10): 643-653

35 Sterzel R, Hinkel C, Haas A, et al. Ultrasonic measurements on FCI Zn-Mg-Y single crystals. Europhys. Lett.,2000, 49(6): 742-747

36 Letoublon A, de Boissieu M, Boudard M, et al. Phason elastic constants of the icosahedral Al-Pd-Mn phase derived from diffuse scattering measurements. Phil. Mag. Lett., 2001, 81(4): 273-283

37 de Boissieu M, Francoual S, Kaneko Y, et al. Diffuse scattering and phason fluctuations in the Zn-Mg-Sc icosahedral quasicrystal and its Zn-Sc periodic approximant. Phys. Rev. Lett., 2005, 95(10): 105503/1-4

38 Edagawa K, So GI Y. Experimental evaluation of phononphason coupling in icosahedral quasicrystals. Phil. Mag.,2007, 87(1): 77-95

39 Chernikov M A, Ott H R, Bianchi A, et al. Elastic moduli of a single quasicrystal of decagonal Al-Ni-Co: evidence for transverse elastic isotropy. Phys. Rev. Lett., 1998,80(2): 321-324

40 Jeong H C, Steinhardt P J. Finite-temperature elasticity phase transition in decagonal quasicrystals. Phys. Rev. B, 1993, 48(13): 9394-9403

41 Love A E H. A Treatise on the Mathematical Theory of Elasticity, 4th Edition, New York: Dover. 1944

42 Courant R, Hilbert D. Mathematical Physics Method, New York: Interscience, 1954

43 范天佑. 准晶数学弹性理论及应用. 北京: 北京理工大学出版社, 1999

44 Fan T Y. Mathematical Theory of Elasticity of Quasicrystals and Its Applications. Beijing: Science Press/Heidelberg: Springer-Verlag, 2010

45 Peng Y Z, Fan T Y. Elastic theory of 1D quasiperiodic stacking of 2D crystals. J. Phys.: Condens. Matter, 2000,12(45): 9381-9387

46 Liu G T, Fan T Y, Guo R P. Governing equations and general solutions of plane elasticity of one-dimensional quasicrystals. Int. J. Solid and Structures, 2004, 41(14):3949-3959

47 Chen W Q, Ma Y L, Ding H J. On three-dimensional elastic problems of one-dimensional hexagonal quasicrystal bodies. Mech. Res. Commun., 2004, 31(5): 633-641

48 Wang X. The general solution of one-dimensional hexagonal quasicrystal. Appl. Math. Mech., 2006, 33(4): 576-580

49 Gao Y, Zhao Y T, Zhao B S. Boundary value problems of holomorphic vector functions in one-dimensional hexagonal quasicrystals. Physica B, 2007, 394(1): 56-61

50 Li X F, Fan T Y. New method for solving elasticity problems of some planar quasicrystals and solutions. Chin. Phys. Letter., 1998, 15(4): 278-280

51 Li X F, Fan T Y, Sun Y F. A decagonal quasicrystal with a Griffith crack. Phil Mag A, 1999, 79(8): 1943-1952

52 Li X F, Fan T Y, Sun Y F. Elastic field for a straight dislocation in a decagonal quasicrystal. J Phys.: Condens Matter, 1999, 11(3): 703-711

53 Guo Y C, Fan T Y. Mode II Griffith crack in decagonal quasicrystals. Appl. Math. Mech., English Edition. 2001,22(10): 1311-1317

54 Zhou W M, Fan T Y. Axisymmetric elasticity problem of cubic quasicrystal. Chinese Physics B, 2000, 9(4): 294-303

55 Zhou W M, Fan T Y. Plane elasticity and crack problem of octagonal quasicrystals. Chin. Phys., 2001, 10(6):277-284

56 Fan T Y, Guo L H. Final governing equation of plane elasticity of icosahedral quasicrystals. Phys. Lett. A, 2005,341(5): 235-239

57 Li L H, Fan T Y. Final governing equation of plane elasticity of icosahedral quasicrystals-stress potential method. Chin. Phys. Lett., 2006, 24(9): 2519-2521

58 De P, Pelcovits R A. Linear elasticity theory of pentagonal quasicrystals. Phys. Rev. B, 1987, 35(16): 8609-8620

59 Ding D H, Wang R H, Yang W G, et al. General for the elastic displacement fields induced by dislocations in quasicrystals. J. Phys: Condens Matter, 1995, 7(28): 5423-5436

60 Ding D H, Wang R H, Yang W G, et al. Elasticity theory of straight dislocations in quasicrystals. Phil. Mag. Lett.,1995, 72(2): 352-359

61 杨顺华, 丁棣华. 晶体位错理论, 第2卷. 北京: 科学出版社,1998

62 Yang W G, Ding D H, Feuerbacher U, et al. Atomtic model of dislocation in Al-Pd-Mn icosahedral quasicrystals. Phil. Mag. A, 1998, 77(6): 1481-1497

63 Zhu A Y, Fan T Y, Guo L H. A straight dislocation in an icosahedral quasicrystal. J. Phys.: Condens. Matter,2007, 19(23): 236216

64 Zhu A Y, Fan T Y. Elastic analysis of a Griffith crack in icosahedral Al-Pd-Mn quasicrystal. Int. J. Mod. Phys. B, 2009, 23(10): 1-16

65 Sneddon I N. Fourier Transforms. New York: McGrow-Hill, 1952

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