Abstract
In view of the same model of fivefold and tenfold symmetry quasicrystals, with the same boundary and the same initial conditions, we have obtained a lot of results by phonon–phason dynamics and hydrodynamics and have performed a detailed comparative analysis. The quantitative results on mass density, viscosity velocities, phonon displacements, phason displacements, phonon stresses, phason stresses, and viscosity stresses and their time–space variations help us to understand the motion of solid quasicrystals. The analysis for octagonal and dodecagonal quasicrystals can be easily extended to other two-dimensional quasicrystals and three-dimensional icosahedral quasicrystals. These results reveal that the phonon field is dominated; moreover, the coupling between phonon and phason fields is important for the studied dynamic process, and these two elementary excitations are the main figures in the dynamic process. In contrast, the effects of the compressibility and viscosity are very weak, and they make almost no contribution to the dynamic process. For this dynamic process, the hydrodynamics and phonon–phason dynamics are equivalent, and the hydrodynamics can be simplified by phonon–phason dynamics for solid quasicrystals.
Similar content being viewed by others
References
Penrose, R.: The role of aesthetics in pure and applied mathematical research. Bull. Inst. Math. Appl. 10, 266–271 (1974)
Gardner, M.: Extraordinary nonperiodic tiling that enriches the theory of tiles. Sci. Amer. 236, 110–119 (1977)
Shechtman, D., Blech, I., Gratias, D., Cahn, J.W.: Metallic phase with long-range orientational order and no translational symmetry. Phys. Rev. Lett. 53, 1951 (1984)
Ye, H.Q., Wang, D.N., Kuo, K.H.: Fivefold symmetry in real and reciprocal spaces. Ultramicroscopy 16, 273–277 (1985)
Zhang, Z., Ye, H.Q., Kuo, K.H.: A new icosahedral phase with m35 symmetry. Philos. Mag. A 52, L49–L52 (1985)
Bindi, L., Steinhardt, P.J., Yao, N., Lu, P.J.: Natural quasicrystals. Science 324, 1306–1309 (2009)
Lubensky, T.C., Ramaswamy, S., Toner, J.: Hydrodynamics of icosahedral quasicrystals. Phys. Rev. B 32, 7444 (1985)
Lubensky, T.C.: Symmetry, elasticity and hydrodynamics of quasicrystals. In: Introduction to Quasicrystals, pp. 199–280, Academic Press, Boston (1988)
Rochal, S.B., Lorman, V.L.: Minimal model of the phonon-phason dynamics in icosahedral quasicrystals and its application to the problem of internal friction in the i-AlPdMn alloy. Phys. Rev. B 66, 144204 (2002)
Fan, T.Y., Wang, X.F., Li, W., Zhu, A.Y.: Elasto-hydrodynamics of quasicrystals. Philos. Mag. 89, 501–512 (2009)
Martin, P.C., Parodi, O., Pershan, P.S.: Unified hydrodynamic theory for crystals, liquid crystals, and normal fluids. Phys. Rev. A 6, 2401 (1972)
Fleming, P.D., Cohen, C.: Hydrodynamics of solids. Phys. Rev. B 13, 500 (1976)
Ding, D.H., Yang, W., Hu, C., Wang, R.: Generalized elasticity theory of quasicrystals. Phys. Rev. B 48, 7003 (1993)
Hu, C., Wang, R., Ding, D.H.: Symmetry groups, physical property tensors, elasticity and dislocations in quasicrystals. Rep. Prog. Phys. 63, 1 (2000)
Fan, T.Y.: Mathematical Theory of Elasticity of Quasicrystals and Its Applications, 2nd edn. Springer, Heidelberg (2016)
Dzyaloshinskii, I.E., Volovick, G.E.: Poisson brackets in condensed matter physics. Ann. Phys. 125, 67–97 (1980)
Fan, T.Y.: Poisson bracket method and its applications to quasicrystals, liquid crystals and a class of soft matter. Chin. J. Theor. Appl. Mech. 45, 548–559 (2013) (in Chinese)
Jeong, H.C., Steinhardt, P.J.: Finite-temperature elasticity phase transition in decagonal quasicrystals. Phys. Rev. B 48, 9394 (1993)
Shmuely, M., Alterman, Z.S.: Crack propagation analysis by finite differences. J. Appl. Mech. 40, 902–908 (1973)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Cheng, H., Fan, T.Y. & Wei, H. Phonon–phason dynamics and hydrodynamics of fivefold and tenfold symmetry quasicrystals. Acta Mech 228, 1363–1372 (2017). https://doi.org/10.1007/s00707-016-1779-y
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00707-016-1779-y