Wave propagation of phonon and phason displacement modes in quasicrystals: Determination of wave parameters

Abstract

The paper presents the study of wave propagation in quasicrystals. Our interest is in the computation of the wavenumber (k n ) and group speed (c g ) of the phonon and phason displacement modes of one, two, and three dimensional quasicrystals. These wave parameter expressions are derived and computed using the elasto-hydrodynamic equations for quasicrystals. For the computation of the wavenumber and group speeds, we use Fourier transform approximation of the phonon and the phason displacement modes. The characteristic equations obtained are a polynomial equation of the wavenumber (k n ), with frequency as a parameter. The corresponding group speeds (c g ) for different frequencies are then computed from the wavenumber k n . The variation of wavenumber and group speeds with frequency is plotted for the 1-D quasicrystal, 2-D decagonal Al-Ni-Co quasicrystals, and 3-D icosahedral Al-Pd-Mn and Zn-Mg-Sc quasicrystals. From the wavenumber and group speeds plots, we obtain the cut-off frequencies for different spatial wavenumber g m . The results show that for 1-D, 2-D, and 3-D quasicrystals, the phonon displacement modes are nondispersive for low values of g m and becomes dispersive for increasing values of g m . The cut-off frequencies are not observed for very low values of g m , whereas the cut-off frequency starts to appear with increasing g m . The group speeds of the phason displacement modes are orders of magnitude lower than that of the phonon displacement modes, showing that the phason modes do not propagate, and they are essentially the diffusive modes. The group speeds of the phason modes are also not influenced by g m . The group speeds for the 2-D quasicrystal at 35 kHz is also simulated numerically using Galerkin spectral finite element methods in frequency domain and is compared with the results obtained using wave propagation analysis. The effect of the phonon and phason elastic constants on the group speeds is studied using 3-D icosahedral Al-Pd-Mn and Zn-Mg-Sc quasicrystals. It is also shown that the phason elastic constants and the coupling coefficient do not affect the group speeds of the phonon displacement modes. V C 2015 AIP Publishing LLC.

Scattering of SH wave by a crack in a functionally graded one dimensional hexagonal piezoelectric quasicrystals

“…The portions of 𝐴 1 (𝑠), 𝐴 2 (𝑠) and 𝐴 3 (𝑠) that contribute to the singular behaviours are found from the integration by part of Equations (24)(25)(26) in the forms:…”

“…Waksmanski et al [24][25] studied the free vibration and harmonic response of 1D QCs multilayer plate under load using the transfer matrix method. Wave propagation in QCs beams is studied by Chellappan [26] using Fourier transform approximation. Wang et al [27] obtained approximate analytical solutions of dislocation dynamics in 3D icosahedral QCs.…”

Axial guided wave characteristics in functionally graded one-dimensional hexagonal piezoelectric quasi-crystal cylinders

“…The bending analysis in a one-dimensional orthorhombic quasi-crystal plate under static and transient dynamic loads was investigated by Sladek et al [19]. The wave propagation in quasi-crystal structures was investigated by Chellappan et al [20]. The plate bending analysis in orthorhombic quasi-crystals under static and transient dynamic loads was investigated by Chiang et al [21].…”