Elasticity theory, fracture mechanics, and some relevant thermal properties of quasi-crystalline materials

[+] Author and Article Information
Tian-You Fan

Department of Physics, Beijing Institute of Technology, PO Box 327, Beijing 100081, PR China; tianyoufan@yahoo.com

Yiu-Wing Mai

School of Aerospace, Mechanical and Mechatronic Engineerinig, University of Sydney, Sydney, NSW 2006, Australia; mai@aeromech.usyd.edu.au

Appl. Mech. Rev 57(5), 325-343 (Dec 21, 2004) (19 pages) doi:10.1115/1.1763591 History: Online December 21, 2004
Copyright © 2004 by ASME
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Shechtman  D, Blech  I, Gratins  D, and Cahn  JW (1984), Metallic phase with long-range orientational order and no translational symmetry, Phys. Rev. Lett. 53, 1951–1953.
Ye  HQ, Wang  D, and Kuo  KH (1985), Five-fold symmetry in real and reciprocal space, Ultramicroscopy 16, 273–278.
Zhang  Z, Ye  HQ, and Kuo  KH (1985), A new icosahedron phase with m35 symmetry, Philos. Mag. A 52, L49–L52.
Levine  D, and Steinhardt  PJ (1984), Quasi-crystals: A new class of ordered structure, Phys. Rev. Lett. 53, 2477–2450.
Fung  Y, Lu  G, and Witers  RI (1989), An incommensurate structure with cubic point group symmetry in rapidly solidified V-Ni-Si alloy, J. Phys.: Condens. Matter 1, 3695–3770.
Bendesky  L (1985), Quasicrystal with one-dimensional translational symmetry and a ten-fold rotation axis, Phys. Rev. Lett. 55, 1461–1463.
Chattopadhyay  K, Ranganathan  S, Subbanna  GN, and Thangaraj  N (1985), Electron microscopy of quasicrystals in rapidly solidified Al-14%Mn alloys, Scr. Metall. 19, 767–771.
Fung  KK, Yang  CY, Zhou  YQ, Zhao  JG, Zhan  WS, and Shen  BG (1986), Icosahedrally decagonal in rapidly cooled Al-14-at.%-Fe alloy, Phys. Rev. Lett. 56, 2060–2063.
Urban  K, Myer  J, Rapp  M, Wikens  M, Casanady  A, and Filder  J (1986), Studies on aperiodic crystals in Al-Mn and Al-V alloys by means of transmission electron-microscopy, J. Phys. Colloq. 47(C3), 465–475.
Wang  N, Chen  H, and Kuo  KH (1987), Two-dimensional quasi-crystal with eight-fold rotational symmetry, Phys. Rev. Lett. 59, 1010–1013.
Li  XZ, Yu  RC, Kuo  KH, and Hiraga  K (1996), Two-dimensional quasi-crystal with five-fold rotational symmetry and superlattice, Philos. Mag. Lett. 37, 255–261.
Merlin  R, Bajema  K, Clarke  R, Juang  FY, and Bhattacharya  PK (1985), Quasiperiodic GaAs-AlAs heterostructures, Phys. Rev. Lett. 55, 1768–1770.
Hu  A, Tien  C, Li  XJ, Wang  YH, and Feng  D (1986), X-ray diffraction pattern of quasi-periodic (Fibonacci) Nb-Cu superlattice, Phys. Lett. A 119, 313–314.
Feng  D, Hu  A, Chen  K, and Xiong  S (1987), Research on quasi-periodic superlattice, Mater. Sci. Forum 22–24, 489–498.
Teranchi  H, Noda  Y, Kamigaki  K, Matsunaka  S, Nakayama  M, Kato  H, Sano  N, and Yamada  Y (1988), X-ray diffraction patterns of configuration Fibonacci lattice, J. Phys. Soc. Jpn. 17, 2416–2424.
Chen  KJ, Mao  GM, Feng  D, Yan  Y, Du  JF, Li  ZF, Chen  HH, and Fritzsche  H (1987), Quasiperiodic a-Si: H/a-SiNx:H multiplayer structures, J. Non-Cryst. Solids 97, 341–344.
Yang  WG, Gui  JN, and Wang  RH (1996), Some-new stable one-dimensional quasi-crystals Al65Cu20Fe10Mn5 alloy, Philos. Mag. Lett. 74, 357–366.
Penrose  H (1974), The role of arethetics in pure and applied mathematical research, Bull Inst Math Appl, 10, 266–271. (On Penrose tiling, can also refer to Gandner  M (1977), Extraordinary non-periodic tiling that enriches the theory of tiles, Sci. Am. 236, 110–119; and Senehal M (1994), Quasi-crystals and Geometry, Cambridge Univ Press, Cambridge).
Bak  P (1985), Symmetry, stability and elastic properties of icosahedron incommensurate crystals, Phys. Rev. B 32, 3764–3772.
Bak  P (1985), Phenomenological theory of icosahedron incommensurate (quasiperiodic) order in Mn-Al alloys, Phys. Rev. Lett. 54, 1517–1519.
Socolar  JES, Lubensky  TC, and Steinhart  PJ (1986), Phonons, phasons and dislocations in quasi-crystals, Phys. Rev. B 34, 3345–3360.
Landau LD, and Lifshitz EM (1958), Statistical Physics, 2nd Edition (English translation), New York, Pergamon Press.
Hu  CZ, Yang  WG, Wang  RH, and Ding  DH (1997), Symmetry and physical properties of quasicrystals (in Chinese), Prog. Phys. 17, 3345–375.
Ding  DH, Wang  RH, Yang  WG, and Hu  CZ (1998), Elasticity, plasticity and dislocations of quasi-crystals (in Chinese), Prog. Phys. 18, 223–260.
Ding  DH, Yang  WG, Hu  CZ, and Wang  RH (1993), Generalized elasticity theory of quasi-crystals, Phys. Rev. B 48, 7003–7010.
Wang  RH, Yang  WG, Hu  CZ, and Ding  DH (1997), Point and space groups and elastic behaviours of one-dimensional quasi-crystals, J. Phys.: Condens. Matter 9, 2411–2422.
Hu  CZ, Yang  WG, Wang  RH, and Ding  DH (1996), Point groups and elastic properties of two-dimensional quasicrystals, Acta Crystallogr. 52, 251–256.
Fan TY (1999), Mathematical Theory of Elasticity of Quasi-crystals and Its Applications (Chinese), Beijing Institute of Technology Press, Beijing.
Li  XF, and Fan  TY (1998), New method for solving elasticity problems of some planar quasi-crystals and solutions, Chin. Phys. Lett. 15, 278–280.
Guo  YC, and Fan  TY (2001), A model II Griffith crack in decagonal quasi-crystal, Appl. Math. Mech. 22, 1311–1317.
Ding  DH, Wang  RH, Yang  WG, and Hu  CZ (1995), General expressions for the elastic displacement fields induced by dislocations in quasi-crystals, J. Phys.: Condens. Matter 7, 5423–5436.
De  P, and Pelcovits  RA (1987), Linear elasticity theory of pentagonal quasicrystals, Phys. Rev. B 36, 9304–9307 (Note: The structure was seen as pentagonal quasicrystal by the original authors, but which should be considered as decagonal quasicrystal according to the point of view at present—noted by the citer).
Li  XF, Duan  XY, Fan  TY, and Sun  YF (1999), Elastic field for a straight dislocation in a decagonal quasi-crystal, J. Phys.: Condens. Matter 11, 703–711.
Li  XF, Fan  TY, and Sun  YF (1999), A decagonal quasi-crystal with a Griffith crack, Philos. Mag. A 79, 1942–1953.
Li  XF, and Fan  TY (2002), Elastic analysis of a mode-II crack in a decagonal quasi-crystal, Chin. Phys. 11, 266–271.
Zhou WM (2000), Dislocation, crack and contact problems of two-and three-dimensional quasi-crystals (Chinese), Dissertation, Beijing Institute of Technology, Beijing.
Zhou  WM, and Fan  TY (2001), Plane elasticity and crack problem of octagonal quasi-crystals, Chin. Phys. 10, 277–284.
Titchmarsh EC (1937), Introduction to the Theory of Fourier Integrals, Clarenden Press, Oxford.
Busbridge  IW (1938), Dual integral equations, Proc. London Math. Soc. 44, 115–132.
Mong  XM, Tong  PY, and Wu  YQ (1994), Mechanical properties of quasicrystal Al65Cu20Co15 (in Chinese), Acta Mech. Sin. 30, 61–64.
Muskhelishvili, NI (1956), Some Basic Problems of Mathematical Theory of Elasticity, English translation, by JRMP Radok, Nordhoff, Gloringen.
Liu  GT, and Fan  TY (2003), Complex method of the plane elasticity in 2D quasi-crystal with point group 10 mm ten-fold symmetry and notch problems, Sci. China, Ser. E: Technol. Sci. 46, 326–336.
Peng  YZ, and Fan  TY (2000), Elastic theory of 1D quasi-periodic stacking of 2D crystals, J. Phys.: Condens. Matter 12, 9381–9388.
Peng  YZ, and Fan  TY (2001), Crack and indentation problems for one-dimensional hexagonal quasi-crystal, Eur. Phys. J. B 21, 39–44.
Peng  YZ, and Fan  TY (2002), Perturbation theory of 2D decagonal quasi-crystals, Physica B 311, 324–330.
Zhou  WM, and Fan  TY (2000), Axisymmetry elasticity problem of cubic quasi-crystal, Chin. Phys. 9, 295–303.
Zhou  WM, and Fan  TY (2002), Axisymmetric contact problem of cubic quasi-crystalline materials, Acta Mechanica Solida Sinica, 15, 68–73.
Peng  YZ, and Fan  TY (2000), Perturbation method solving elastic problems of icosahedral quasicrystals containing a circular crack, Chin. Phys. 9, 762–766.
Bachteler  J, and Trebin  HR (1998), Elastic Green’s function of icosahedral quasi-crystals, Eur. Phys. J. B 4, 299–306.
Ricker  M, Bachteler  J, and Trebin  H-R (2001), Elastic theory of icosahedral quasi-crystals: application to straight dislocation, Eur. Phys. J. B 23, 351–363.
Bachteler J (1999), Kontinuumstheorie der Versezungen in ikosaedrische Quasikristallen, Shaker Verlage, Aachen.
Ricker M (2000), Zur Kontinuumstheorie der Versezungen in ikosaedrischen Quasikristallen, Diplomarbeit, Institut fuer Theoretische und Angewandte Physik, Univ Stuttgart.
Lubensky  TC, Ramaswamy  S, and Toner  J (1985), Hydrodynamics of icosahedral quasi-crystals, Phys. Rev. B 32, 7444–7452.
Trebin  H-R, Mikulla  R, and Roth  J (1993), Motion of dislocations in two-dimensional quasi-crystals, J Noncrystalline Solids, 153–154, 272–276.
Fan  TY (1999), A study on specific heat of one-dimensional hexagonal quasicrystal, J. Phys.: Condens. Matter 11, L513–L517.
Fan  TY, Li  XF, and Sun  YF (1999), A moving screw dislocation in one-dimensional quasicrystal, Acta Phys. Sin. (Overseas Ed.) 8, 288–295.
Eshelby  JD (1949), Uniformly moving dislocation, Proc. Phys. Soc., London, Sect. A 62, 407–414.
Franck  FC (1949), On the equation of motion crystal dislocation, Proc. Phys. Soc., London, Sect. A 62, 131–134.
Yoffe  EH (1950), Moving Griffith crack, Philos. Mag. 42, 739–750.
Fan  TY, and Mai  Y-W (2003), Partition function and state equation of the point group 12 mm two-dimensional dodecagonal quasicrystal, Eur. Phys. J. B 31, 25–27.
Debye  P (1912), Die Eigentuemlichkeit der specifischen Waerme bei tiefen Temperaturen, Arch de Gene’ve, 33, 256–258.
Fan TY, and Mai Y-W (2003), Specific heat of point group 10 mm two-dimensional quasi-crystal, Eur. Phys. J. B (submitted).
Bianchi  AD, Bommeli  P, Felder  E, Kenzelmann  M, Chernikov  MA, Degiorgi  L, Ott  HR, and Edagawa  K (1998), Low-temperature thermal and optical properties of single grained decagonal Al-Ni-Co quasicrystal, Phys. Rev. B 58, 3046–305.
Chernikov  MA, Ott  HR, Bianchi  A, Migliori  A, and Darling  TW (1998), Elastic moduli of a single quasicrystal of decagonal Al-Ni-Co: evidence for transverse elastic isotropy, Phys. Rev. Lett. 80, 321–324.
Jeong  HC, and Steinhardt  PJ (1993), Finite-temperature elasticity phase transition in decagonal quasicrystals, Phys. Rev. B 48, 9394–9403.
Fan TY (2001), Phenomenological theory of lattice dynamics of quasi-crystals and its applications, GAMM Jahrestagung 2001, Zuerich, Feb. 12–15.
Rosenfeld  R, Feurbacher  M, Baufeld  B, Bartsch  M, Wollgartten  M, Hanke  G, Beyss  M, and Messerschmidt  U (1995), Study of plastically deformed icosahedral Al-Pd-Mn single quasi-crystals by transmission electron microscopy, Philos. Mag. Lett. 72, 375–384.
Bartsch  M, Geyer  B, Haeussler  D, Feuerbacher  M, Urban  K, and Messesrschmidt  U (2000), Plastic properties of icosahedral Al-Pd-Mn single quasicrystals, Mater. Sci. Eng. 294–296, 761–764.
Wollgarten  M, Bartsch  M, Messerschmidt  U, Feuerbach  M, Rosenffeld  R, Beyss  M, and Urban  K (1995), In-situ observation of dislocation motion in icosahedral Al-Pd-Mn single quasi-crystals, Philos. Mag. Lett. 71, 99–105.
Wang RH (1996), Micro-mechanism of plasticity of quasi-crystals (in Chinese), Proc. Annual Conf of Hubei/Wuhan Phys Soc, 56–60.
Eshelby JD (1956), The continuum theory of dislocations in crystals, Solid-State Physics, 3 F Seitz et al. (ed), Academic Press, New York.
Rice JR (1967), Brown Univ ARPA SD-86 Report E39.
Rice  JR (1968), A path independent integral and the approximate analysis of strain concentration by notches and cracks, ASME J. Appl. Mech. 35, 379–386.
Wu XF (1999), Mathematical and numerical simulation of elasticity of quasi-crystals (in Chinese) Dissertation, Beijing Institute of Technology, Beijing.
Oden JT, and Reddy JN (1976), An Introduction to the Mathematical Theory of Finite Element, Springer-Verlag, Berlin.


Grahic Jump Location
Phonon displacement versus radial distance
Grahic Jump Location
Phason displacement versus radial distance
Grahic Jump Location
Strain energy density of hollow cylinder versus radial distance: a) distribution curve, and b) distribution along thickness
Grahic Jump Location
The mesh of phason field before and after deformation: a) before deformation, and b) after deformation
Grahic Jump Location
FEM numerical results of normalized phonon stress near the crack tip versus normalized distance
Grahic Jump Location
The pattern of diffraction and the stereographic structure of an icosahedral quasi-crystal
Grahic Jump Location
Penrose tiling of a 2D quasi-crystal with five-fold symmetry
Grahic Jump Location
A hollow cylinder of octagonal quasi-crystal subjected to a pressure
Grahic Jump Location
Phonon stress versus radial distance
Grahic Jump Location
Phason stress versus radial distance



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