# Elasto-Dynamics of Quasicrystals

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## Abstract

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## 1. Introduction

**u**and

**w**, in which the former

**u**called the phonon field is similar to that in crystals; the latter

**w**is called the phason field. They were first discussed by Bak [12,13] and Socalar et al. [14]. So the total displacement field in a quasicrystal can be expressed by $\overline{u}={u}^{\parallel}\oplus {u}^{\perp}=u\oplus w$, in which

**u**is in the parallel space, or the physical space;

**w**is in the complement space, or the perpendicular space; which is an internal space and $\oplus $ denotes the direct sum. On the basis of the above physical framework and the extended methodology in mathematical physics from classical elasticity, the independent elastic constants for different symmetries of quasicrystals can be determined [16,17,18,19,20,21]. Then the mathematical elasticity theory of quasicrystals has been developed rapidly. In 2004, Fan and Mai give a review based on the static elasticity theory of various quasicrystals [22]. A monograph is devoted to the development of a mathematical elasticity theory of quasicrystals and its applications [23]. First, Levine and Lubensky et al. carried out extensive work in terms of the elasticity and dislocations in pentagonal and icosahedral quasicrystals, resulting in many solutions for the dislocations [24]. Then Li presented two solutions for a Griffith crack and a straight dislocation embedded in a decagonal quasicrystal [25,26]. Chen provided a three-dimensional elastic problem of one-dimensional hexagonal quasicrystal, and gave a general solution for this problem [27]. Meanwhile, Liu et al. obtained the general solutions and the governing equations for plane elasticity of one-dimensional quasicrystals [28]. Based on the stress potential function, Li used a complex function method to accomplish notch problem of two-dimensional quasicrystals [29]. Wang and Gao obtained some solutions for some defect problems of one-, two-dimensional quasicrystal [30,31,32]. Coddens discussed the elasticity and dynamics of the phason in quasicrystals [33]. Wang et al. discussed the phonon- and phason-type inclusions in icosahedral quasicrystals [34]. Guo et al. also discussed an elliptical inclusion in hexagonal quasicrystals [35]. More recently, the phonon-phason elasticity of QCs has attracted a lot of attention too. For example, Radi and Mariano have described linear elasticity of QCs, and obtained some meaningful results for the straight cracks and dislocations in two-dimensional quasicrytals [36,37,38]. An important reference concerning fundamental aspects of generalized elasticity and dislocation theory deriving the generalized three-dimensional elastic Green tensor and all the dislocation key-formulas including an application to dislocation loops for arbitrary quasicrytals is presented by Lazar and Agiasofitou [39]. Li provided fundamental solutions for thermo-elasticity based on half infinite plane cracks embedded in one-dimensional hexagonal QCs and some solutions for one-dimensional hexagonal QCs [40,41,42]. Of course, dislocation and plate theory of QCs have been developed by some researchers, for example, Sladek et al. [43], Li and Chai [44]. Because of the lack of data of constitutive equations about plastic deformation of quasicrystals, the related researches are fairly complicated. The existence of phason degrees of freedom leads to the essential difference between crystals and quasicrystals. Wollgarten et al. used a dislocation mechanism to study plastic deformation of quasicrystals [45]. A model of the plastic deformation of icosahedral quasicrystals was provided by Feuerbacher et al. [46] and refined by Messerschmidt et al. [47]. A mass of studies have been executed on the deformed Al–Pd–Mn single quasicrystals through a series of experimental observations [48,49,50,51].

**w**is similar to phonon field

**u**in quasicrystals, represents long-wavelength propagation. This idea originated from Bak [12,13] who indicated that the phason describes particular structure fluctuations (or structure disorders) in quasicrystals, and it can be viewed as a six-dimensional space description; (2) The phason field

**w**is diffusion rather than wave propagation, so phasons play different roles to phonons in the hydrodynamics of quasicrystals. This idea can be found in Lubensky et al. [52], Francoual et al. [53] and Socolar et al. [54], etc. They claimed that the phason field

**w**represents diffusion rather than wave propagation; thus, phasons play different roles than phonons in the hydrodynamics of QCs. The phason modes denote the relative motion of the density waves and the phason field

**w**is not sensitive to spatial translations, such that the phasons are not oscillatory, instead, they are diffusive with very large diffusion times. Furthermore, some researchers explained that the motion of the phasons is atomic jumps. Following the above two models, Mikulla and Trebin et al. [55], Takeuchi et al. [56], Rösch [57] and co-workers carried out extensive work in terms of analytic and numerical methods, this results in many solutions in dislocation and crack dynamics for different quasicrystal systems. Similar to [58,59], Fan and his co-workers obtained some analytical solutions for some 1D and 2D quasicrystals [60,61,62,63]. More recently, Tupholme studied an anti-plane shear crack moving in one-dimensional hexagonal quasicrystals based on the reference [64,65]. Li gave a general solution for elasto-dynamics of two dimensional quasicrystals [66]. In contrast to the generalized elasto-dynamics, this kind of study may be named elasto-/hydro-dynamics of quasicrystals, because the equations of motion of phonons are elasto-dynamics equations, while the equations of motion of phasons are diffusion equations originated from the hydrodynamics; (3) There is a recent and promising model introduced by Agiasofitou and Lazar for describing the dynamics of quasicrystals, in which the authors clarify that an elasto-dynamic model of wave-telegraph type can be used [67]. Based on this model, phonons are represented by waves, and phasons by waves damped in time and propagating with finite velocity. Therefore, the equations of motion for the phonon fields are of wave type and for the phason fields are of telegraph type. The proposed model constitutes a unified theory in the sense that already established models in the literature can be recovered as asymptotic cases of it. Moreover, the same authors have investigated and compared three models of dynamics: the elasto-dynamic model of wave type and the elasto-hydrodynamic model and the elasto-dynamic model of wave-telegraph type [68]. For these models, they derived the equations of motion of dislocations for arbitrary quasicrystals. To the best of the author’s knowledge these are the main-current models in the literature for the description of the dynamics of quasicrystals.

## 2. Elasto-Dynamics of Quasicrystals Followed Bak’s Argument

#### 2.1. Example 1: Basic Equations for Elasto-Dynamics of Anti-Plane Elasticity of Some Quasicrystals

#### 2.2. Example 2: Elasto-Dynamics of Hexagonal Quasicrystals

#### 2.3. Example 3: Elasto-Dynamics of Plane Elasticity of Dodecagonal Quasicrystals

#### 2.4. Example 4: The Approximate Form of Elasto-Dynamics of Two-Dimensional Elasticity (or Simplified Three-Dimensional Elasticity) of Icosahedral Quasicrystals

#### 2.5. A Moving Screw Dislocation in Anti-Plane Elasticity of Some Quasicrystals

**V**. Meanwhile, suppose that the dislocation has the Burgers vector is $(0,0,{\mathrm{b}}_{3}^{\left|\right|},0,{\mathrm{b}}_{3}^{\perp})$. The dislocation conditions can be repressed by

#### 2.6. A Mode III Moving Griffith Crack in Anti-Plane Elasticity

#### 2.7. The Moving Dugdale Model for Plane Elasticity of Dodecagonal Quasicrystals

**V**in the direction $O{x}_{1}$. We introduce the Galileo transformation for convenience

## 3. Elasto-/Hydro-Dynamics of Quasicrystals Based on the Argument of Lubensky et al.

#### 3.1. Application 1 of Elasto-/Hydro-Dynamics-Approximate Solution for a Moving Screw Dislocation in Anti-Plane Elasticity

#### 3.2. Application 2 of Elasto-/Hydro-Dynamics—Dynamic Propagating Crack and Solutions of Two-Dimensional Decagonal Quasicrystals

#### 3.3. Application 3 of Elasto-/Hydro-Dynamics-Dynamic Crack Propagation of Icosahedral Quasicrystals

#### 3.3.1. A Cracked Specimen of Icosahedral Quasicrystals of Plane Problem Based on FDM

#### 3.3.2. A Cracked Specimen of Anti-Plane Problem in Icosahedral QCs Based on FDM

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Shechtman, D.; Blech, I.; Gratias, D.; Cahn, J.W. Metallic phase with long-range orientational order and no translational symmetry. Phys. Rev. Lett.
**1984**, 53, 1951–1953. [Google Scholar] [CrossRef] - Ohashi, W.; Spaepen, F. Stable Ga-Mg-Zn quasi-periodic crystals with pentagonal dodecahedral solidification morphology. Nature
**1987**, 330, 555–556. [Google Scholar] [CrossRef] - Wang, N.; Chen, H.; Kuo, K.H. Two-dimensional quasicrystal with eightfold rotational symmetry. Phys. Rev. Lett.
**1987**, 59, 1010–1013. [Google Scholar] [CrossRef] [PubMed] - Janot, C. The structure of quasicrystals. J. Non-Cryst. Solids
**1993**, 156–158, 852–864. [Google Scholar] [CrossRef] - Ishimasa, T.; Nissen, H.U. New ordered state between crystalline and am or phous in Ni-Cr particles. Phys. Rev. Lett.
**1985**, 55, 511–513. [Google Scholar] [CrossRef] [PubMed] - Feng, Y.C.; Lu, G. Experimental evidence for and a projection model of a cubic quasicrystal. J. Phys. Condens. Matter
**1990**, 2, 9749–9755. [Google Scholar] [CrossRef] - Chen, H.; Li, X.Z.; Zhang, Z.; Kuo, K.H. One-dimensional quasicrystals with twolvefold rotational symmetry. Phys. Rev. Lett.
**1998**, 60, 1645–1648. [Google Scholar] [CrossRef] [PubMed] - Bohsung, J.; Trebin, H.R. Disclinations in quasicrystals. Phys. Rev. Lett.
**1987**, 58, 1204–1207. [Google Scholar] [CrossRef] [PubMed] - Ebert, P.H.; Feuerbacher, M.; Tamura, N. Evidence for a cluster-based on structure of Al-Pd-Mn single quasicrystals. Phys. Rev. Lett.
**1996**, 77, 3827–3830. [Google Scholar] [CrossRef] [PubMed] - Li, C.L.; Liu, Y.Y. Low-temperature lattice excitation of icosahedral Al-Mn-Pd quasicrystals. Phys. Rev. B
**2001**, 63, 064203. [Google Scholar] [CrossRef] - Rochal, S.B.; Lorman, V.L. Anisotropy of acoustic-phonon properties of an icosahedral quasicrystal at high temperature due to phonon-phason coupling. Phys. Rev. B
**2000**, 62, 849–874. [Google Scholar] [CrossRef] - Bak, P. Phenomenological theory of icosahedral in commensurate (quasiperiodic) order in Mn-Al alloys. Phys. Rev. Lett.
**1985**, 54, 1517–1519. [Google Scholar] [CrossRef] [PubMed] - Bak, P. Symmetry, stability and elastic properties of icosahedral in commensurate crystals. Phys. Rev. B
**1985**, 32, 5764–5772. [Google Scholar] [CrossRef] - Socolar, J.E.S.; Lubensky, T.C.; Steinhardt, P.J. Phonons, phasons and dislocations in quasicrystals. Phys. Rev. B
**1986**, 34, 3345–3360. [Google Scholar] [CrossRef] - Landau, L.D.; Lifshitz, E.M. Statistical Physics; Pergamon Press: Elmsford, NY, USA, 1958. [Google Scholar]
- Edagawa, K. Phonon-phason coupling in decagonal quasicrystals. Philos. Mag.
**2007**, 87, 2789–2798. [Google Scholar] [CrossRef] - Cheminkov, M.A.; Ott, H.R.; Bianchi, A.; Migliori, A.; Darling, T.W. Elastic moduli of a single quasicrystal of decagonal Al-Ni-Co: Evidence for transverse elastic isotropy. Phys. Rev. Lett.
**1998**, 80, 321–324. [Google Scholar] - Tanaka, K.; Mitarai, Y.; Koiwa, M. Elastic constants of Al-based icosahedral quasicrystals. Philos. Mag. A
**1996**, 73, 1715–1723. [Google Scholar] [CrossRef] - Ding, D.H.; Yang, W.G.; Hu, C.Z. Generalized elasticity theory of quasicrystals. Phys. Rev. B
**1993**, 48, 7003–7010. [Google Scholar] [CrossRef] - Hu, C.Z.; Wang, R.H.; Ding, D.H. Symmetry groups, physical property tensors, elasticity and dislocations in quasicrystals. Rep. Prog. Phys.
**2000**, 63, 1–39. [Google Scholar] [CrossRef] - Jeong, H.C.; Steinhardt, P.J. Finite-temperature elasticity phase transition in decagonal quasicrystals. Phys. Rev. B
**1993**, 48, 9394–9403. [Google Scholar] [CrossRef] - Fan, T.Y.; Mai, Y.W. Elasticity theory, fracture mechanics and some relevant thermal properties of quasicrystalline materials. Appl. Mech. Rev.
**2004**, 57, 325–344. [Google Scholar] [CrossRef] - Fan, T.Y. Mathematical Theory of Elasticity of Quasicrystals and Its Applications; Springer: Heideberg, Germany, 2010. [Google Scholar]
- Levine, D.; Lubensky, T.C.; Ostlund, S. Elasticity and dislocations in pentagonal and icosahedral quasicrystals. Phys. Rev. Lett.
**1985**, 54, 1520–1523. [Google Scholar] [CrossRef] [PubMed] - Li, X.F.; Duan, X.Y.; Fan, T.Y. Elastic field for a straight dislocation in a decagonal quasicrystal. J. Phys. Condens. Matter
**1999**, 11, 703–711. [Google Scholar] [CrossRef] - Li, X.F.; Fan, T.Y.; Sun, Y.F. A decagonal quasicrystal with a Griffith crack. Philos. Mag. A
**1999**, 79, 1943–1952. [Google Scholar] - 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, 633–641. [Google Scholar] [CrossRef] - Liu, G.T.; Fan, T.Y.; Guo, R.P. Governing equations and general solutions of plane elasticity of one-dimensional quasicrystals. Int. J. Solids Struct.
**2004**, 41, 3949–3959. [Google Scholar] [CrossRef] - Li, L.H.; Fan, T.Y. Complex function method for solving notch problem of point 10 two-dimensional quasicrystal based on the stress potential function. J. Phys. Condens. Matter
**2006**, 18, 10631–10641. [Google Scholar] [CrossRef] - Wang, X.; Pan, E. Analytical solutions for some defect problems in 1D hexagonal and 2D octagonal quasicrystals. Pramana—J. Phys.
**2008**, 70, 911–933. [Google Scholar] [CrossRef] - Gao, Y.; Xu, S.P.; Zhao, B.S. Boundary conditions for plate bending in one dimensional hexagonal quasicrystals. J. Elast.
**2007**, 86, 221–233. [Google Scholar] [CrossRef] - Gao, Y.; Ricoeur, A. The refined theory of one-dimensional quasi-crystals inthick plate structures. J. Appl. Mech.
**2011**, 78, 031021. [Google Scholar] [CrossRef] - Coddens, G. On the problem of the relation between phason elasticity and phason dynamics in quasicrystals. Eur. Phys. J. B
**2004**, 54, 37–65. [Google Scholar] [CrossRef] - Wang, J.B.; Mancini, L.; Wang, R.H.; Gastaldi, J. Phonon- and phason-type spherical inclusions in icosahedral quasicrystals. J. Phys. Condens. Matter
**2003**, 15, L363–L370. [Google Scholar] [CrossRef] - Guo, J.H.; Zhang, Z.Y.; Xing, Y.M. Antiplane analysis for an elliptical inclusion in 1D hexagonal piezoelectric quasicrystal composites. Philos. Mag.
**2016**, 96, 349–369. [Google Scholar] [CrossRef] - Radi, E.; Mariano, P.M. Stationary straight cracks in quasicrystals. Int. J. Fract.
**2010**, 166, 102–120. [Google Scholar] [CrossRef] - Radi, E.; Mariano, P.M. Steady-state propagation of dislocations in quasi-crystals. Proc. R. Soc. A Math. Phys.
**2011**, 467, 3490–3508. [Google Scholar] [CrossRef] - Mariano, P.M.; Planas, J. Phason self-actions in quasicrystals. Physica D
**2013**, 249, 24946–24957. [Google Scholar] [CrossRef] - Lazar, M.; Agiasofitou, E. Fundamentals in generalized elasticity and dislocation theory of quasicrystals: Green tensor, dislocation key-formulas and dislocation loops. Philos. Mag.
**2014**, 94, 4080–4101. [Google Scholar] [CrossRef] - Li, X.Y. Fundamental solutions of penny-shaped and half infinite plane cracks embedded in an infinite space of one dimensional hexagonal quasi-crystal under thermal loading. Proc. R. Soc. A Math. Phys.
**2013**, 469, 20130023. [Google Scholar] [CrossRef] - Li, X.Y. Three-dimensional thermo-elastic general solutions of one-dimensional hexagonal quasi-crystal and fundamental solutions. Phys. Lett. A
**2012**, 376, 2004–2009. [Google Scholar] [CrossRef] - Li, X.Y. Elastic field in an infinite medium of one-dimensional hexagonal quasicrystalwith a planar crack. Int. J. Solids Struct.
**2014**, 51, 1442–1455. [Google Scholar] [CrossRef] - Sladek, J.; Sladek, V.; Pan, E. Bending analyses of 1D orthorhombic quasicrystal plates. Int. J. Solids Struct.
**2013**, 50, 3975–3983. [Google Scholar] [CrossRef] - Li, W.; Chai, Y.Z. Anti-plane problem analysis for icosahedral quasicrystals under shear loadings. Chin. Phys. B
**2014**, 23, 116201. [Google Scholar] [CrossRef] - Wollgarten, M.; Beyss, M.; Urban, K.; Liebertz, H.; Koster, U. Direct evidence for plastic deformation of quasicrystals by means of a dislocationmechanism. Phys. Rev. Lett.
**1993**, 71, 549–552. [Google Scholar] [CrossRef] [PubMed] - Feuerbacher, M.; Bartsch, M.; Grushko, B.; Messerschmidt, U.; Urban, K. Plastic deformation of decagonal Al-Ni-Co quasicrystals. Philos. Mag. Lett.
**1997**, 76, 369–376. [Google Scholar] [CrossRef] - Messerschmidt, U.; Bartsch, M.; Feuerbacher, M.; Geyer, B.; Urban, K. Friction mechanism of dislocation motion in icosahedralAl-Pd-Mn quasicrystals. Philos. Mag. A
**1999**, 79, 2123–2135. [Google Scholar] [CrossRef] - Schall, P.; Feuerbacher, M.; Bartsch, M.; Messerschmidt, U.; Urban, K. Dislocation density evolution upon plastic deformation of Al-Pd-Mn single quasicrystals. Philos. Mag. Lett.
**1999**, 79, 785–796. [Google Scholar] [CrossRef] - Geyer, B.; Bartsch, M.; Feuerbacher, M.; Urban, K.; Messerschmidt, U. Plastic deformation of icosahedral Al-Pd-Mn single quasicrystals I. Experimental results. Philos. Mag. A
**2000**, 80, 1151–1163. [Google Scholar] [CrossRef] - Rosenfeld, R.; Feuerbacher, M. Study of plastically deformed icosahedral Al-Pd-Mn single quasicrystals by transmission electron microscopy. Philos. Mag. Lett.
**1995**, 72, 375–384. [Google Scholar] [CrossRef] - Caillard, D.; Vanderschaeve, G.; Bresson, L.; Gratias, D. Transmission electron microscopy study of dislocations and extended defects in as-grown icosahedral Al-Pd-Mn single grains. Philos. Mag. A
**2000**, 80, 237–253. [Google Scholar] [CrossRef] - Lubensky, T.C.; Ramaswamy, S.; Toner, J. Hydrodynamics of icosahedral quasicrystals. Phys. Rev. B
**1985**, 32, 7444–7452. [Google Scholar] [CrossRef] - Francoual, S.; Livet, F.; de Boissieu, M.; Yakhou, F.; Bley, F.; Letoublon, A.; Caudron, R.; Gastaldi, J. Dynamics of Phason fluctuation in i-Al-Pd-Mn quasicrystals. Phys. Rev. Lett.
**200**