Rayleigh Waves Propagating in the Functionally Graded One-Dimensional Hexagonal Quasicrystal Half-Space
Abstract
:1. Introduction
2. Mathematics and Formulation
- (1)
- z-FGQC half-space
- (2)
- x-FGQC half-space
3. Analytical Integration Expression of Laguerre Polynomial Method
4. Numerical Results
4.1. Method Validation
4.2. Convergence and Computational Efficiency Analysis
4.3. Phonon–Phason Coupling Effect
4.4. Inhomogeneity of Material
4.5. The Influence of Quasiperiodic Direction on Wave Characteristics
5. Conclusions
- (1)
- Compared with the classical Laguerre polynomial method, the improvement in computational efficiency of the analytical Laguerre polynomial method is more than 99%.
- (2)
- The energy penetration depth of the phason modes for Rayleigh waves is greater than that of the phonon modes.
- (3)
- The variation in the graded function has a considerable influence on the phase velocity and displacement distribution. Therefore, the performance of FGQC half-spaces can be adjustable by changing the graded function.
- (4)
- The phase velocity of Rayleigh waves is higher when the directions of quasiperiodicity and wave propagation coincide.
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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C11 | C12 | C13 | C22 | C23 | C33 | C44 | C55 | |
---|---|---|---|---|---|---|---|---|
QC1 | 23.433 | 5.741 | 6.663 | 23.433 | 6.663 | 23.222 | 7.019 | 7.019 |
QC2 | 20 | 10 | 10 | 20 | 10 | 15 | 5 | 5 |
C66 | ρ | R1 | R2 | R3 | K1 | K2 | ||
QC1 | 8.846 | 4.186 | 8.846 | 8.846 | 0.8846 | 12.2 | 2.4 | |
QC2 | 5 | 5.07 | 0.5 | 0.5 | 0.5 | 5 | 2 |
C11 | C12 | C44 | ρ | E | σ |
---|---|---|---|---|---|
10.77 | 5.55 | 2.61 | 2.7 | 7 | 0.34 |
Mode | k | M = 2 | M = 4 | M = 8 | M = 16 | M = 32 |
---|---|---|---|---|---|---|
The first phonon mode | 0.5 | 2.922 | 2.913 | 2.912 | 2.912 | 2.912 |
1 | 3.010 | 2.993 | 2.989 | 2.989 | 2.989 | |
2 | 3.183 | 3.139 | 3.122 | 3.121 | 3.121 | |
The first phason mode | 0.5 | 2.482 | 2.114 | 2.006 | 2.005 | 2.005 |
1 | 2.175 | 2.037 | 1.996 | 1.994 | 1.994 | |
2 | 2.088 | 2.008 | 1.990 | 1.988 | 1.988 |
Method | M = 8 | M = 9 | M = 10 | M = 11 |
---|---|---|---|---|
CLPA | 217.391 | 258.969 | 376.578 | 448.281 |
ALPA | 0.360 | 0.469 | 0.812 | 0.875 |
Save | 99.83% | 99.82% | 99.78% | 99.80% |
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Zhang, B.; Tu, H.; Li, L.; Yu, J.; Dai, J. Rayleigh Waves Propagating in the Functionally Graded One-Dimensional Hexagonal Quasicrystal Half-Space. Crystals 2023, 13, 1205. https://doi.org/10.3390/cryst13081205
Zhang B, Tu H, Li L, Yu J, Dai J. Rayleigh Waves Propagating in the Functionally Graded One-Dimensional Hexagonal Quasicrystal Half-Space. Crystals. 2023; 13(8):1205. https://doi.org/10.3390/cryst13081205
Chicago/Turabian StyleZhang, Bo, Honghang Tu, Liangjuan Li, Jiangong Yu, and Jun Dai. 2023. "Rayleigh Waves Propagating in the Functionally Graded One-Dimensional Hexagonal Quasicrystal Half-Space" Crystals 13, no. 8: 1205. https://doi.org/10.3390/cryst13081205
APA StyleZhang, B., Tu, H., Li, L., Yu, J., & Dai, J. (2023). Rayleigh Waves Propagating in the Functionally Graded One-Dimensional Hexagonal Quasicrystal Half-Space. Crystals, 13(8), 1205. https://doi.org/10.3390/cryst13081205