Dispersion and Attenuation Characteristics of Fractional-Order Thermoelastic Guided Waves in Functionally Graded Piezoelectric Hollow Cylinders
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摘要: 基于分数阶热电弹性理论和Legendre多项式方法，构建了功能梯度空心圆柱中导波传播的数学模型. 讨论了分数阶次、压电效应、径厚比等对导波传播，特别是对其衰减的影响规律. 数值结果表明，压电效应对衰减的影响主要集中在截止频率和突变频率附近，并使得突变频率发生前移；分数阶对热波模态相速度和衰减的影响较大，且热波相速度存在模态交叉，在交叉频率点附近分数阶对相速度的影响相反；热波衰减随着分数阶增大而逐渐减小；第一阶纵向模态衰减受到了压电效应的抑制，其余模态衰减都显著增大，并且电开路受到的影响要比电短路状态大.Abstract: Based on the fractional-order thermo-electric-elastic theory and the Legendre polynomial series method, a mathematical model for guided wave propagation in functionally graded hollow cylinders was established. The effects of the fractional order, the piezoelectric effect, and the radius-thickness ratio on the wave propagation, especially on its attenuation, were discussed. The numerical analysis results indicate that, the piezoelectric effect on attenuation mainly concentrates near the cutoff frequency and the mutation frequency, and causes the mutation frequency to shift forward. The fractional order has a great impact on the phase velocity and attenuation of the thermal wave mode, and has an opposite impact on the phase velocity around the crossover frequency point where the crossover mode occurs with the thermal wave velocity. But the thermal wave attenuation gradually decreases with the fractional order. Meanwhile, the 1st longitudinal mode attenuation is suppressed by the piezoelectric effect. However, the attenuation of other modes significantly increases, and the impact of the electrical open circuit is greater than that of the electrical short circuit.
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图 2 色散曲线的实部收敛分析(N=3)
Figure 2. The real part convergence analysis of the dispersion curve (N=3)
图 3 色散曲线的虚部收敛分析(N=3)
注 为了解释图中的颜色，读者可以参考本文的电子网页版本，后同.
Figure 3. The imaginary part convergence analysis of the dispersion curve (N=3)
图 4 轴对称模态中第一热波的色散曲线(N=0)
Figure 4. Dispersion curves of the axisymmetric mode thermal wave (N=0)
图 5 轴对称模态中准弹性波的相速度曲线(N=0)
Figure 5. Phase velocity curves of the quasi-elastic wave in the axisymmetric mode (N=0)
图 6 轴对称模态中准弹性波的衰减曲线(N=0)
Figure 6. Attenuation curves of the quasi-elastic wave in the axisymmetric mode (N=0)
图 7 非轴对称模态中准弹性波的相速度曲线(N=3)
Figure 7. Phase velocity curves of the quasi-elastic wave in the non-axisymmetric mode (N=3)
图 8 非轴对称模态中准弹性波的衰减曲线(N=3)
Figure 8. Attenuation curves of the quasi-elastic wave in the non-axisymmetric mode (N=3)
图 9 不同α下第一热波的色散曲线
Figure 9. Dispersion curves of the 1st thermal wave with different α values
图 10 不同α下准弹性波的相速度曲线
Figure 10. Phase velocity curves of the quasi-elastic wave in the FGPM hollow cylinder with different α values
图 11 不同α下准弹性波的衰减曲线
Figure 11. Attenuation curves of the quasi-elastic wave with different α values
图 12 不同μ下第一热波的色散曲线
Figure 12. Dispersion curves of the 1st thermal wave with different μ values
图 13 不同μ下准弹性波的相速度曲线
Figure 13. Phase velocity curves of the quasi-elastic wave with different μ values
图 14 不同μ下准弹性波的衰减曲线
Figure 14. Attenuation curves of the quasi-elastic wave with different μ values
图 15 不同梯度下第一热波的色散曲线
Figure 15. Dispersion curves of the 1st thermal wave with different gradients
图 16 不同梯度下准弹性波的相速度曲线
Figure 16. Phase velocity curves of the quasi-elastic wave with different gradients
图 17 不同梯度下准弹性波的衰减曲线
Figure 17. Attenuation curves of the quasi-elastic wave with different gradients
property C11/(N·m-2) C12/(N·m-2) C13/(N·m-2) C22/(N·m-2) C23/(N·m-2) C33/(N·m-2) PZT-4 1.39×1011 7.78×1010 7.43×1010 1.39×1011 7.43×1010 1.15×1011 property C44/(N·m-2) C55/(N·m-2) C66/(N·m-2) e15/(C·m-2) e24/(C·m-2) e31/(C·m-2) PZT-4 2.56×1010 2.56×1010 3.06×1010 12.7 12.7 -5.2 property e32/(C·m-2) e33/(C·m-2) ε11/ (C2·N-1·m-2) ε22/(C2·N-1·m-2) ε33/(C2·N-1·m-2) ρ/(kg·m-3) PZT-4 -5.2 15.1 6.46×10-9 6.46×10-9 5.62×10-9 7.5×103 
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material C11/(N·m-2) C12/(N·m-2) C13/(N·m-2) C33/(N·m-2) C44/(N·m-2) C66/(N·m-2) steel 2.692 3×1011 1.153 8×1011 1.153 8×1011 2.692 3×1011 7.692×1010 7.692×1010 cobalt 3.071×1011 1.65×1011 1.027×1011 3.581×1011 7.55×1010 7.105×1010 material ρ/(kg·m-3) Ce/(J·kg-1·K-1) β1/(N·K-1·m-2) β3/(N·K-1·m-2) K1/(W·m-1·K-1) K3/(W·m-1·K-1) steel 7.85×103 477 6.0×106 6.0×106 52 52 cobalt 8.836×103 427 7.04×106 6.9×106 69 69
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property CdSe PZT-5A property CdSe PZT-5A C11/(N·m-2) 7.41×1010 1.39×1011 ε11/(C2·N-1·m-2) 8.26×10-11 6.00×10-9 C12/(N·m-2) 4.52×1010 7.78×1010 ε22/(C2·N-1·m-2) 8.26×10-11 6.00×10-9 C22/(N·m-2) 7.41×1010 1.39×1011 ε33/(C2·N-1·m-2) 9.03×10-11 5.47×10-9 C13/(N·m-2) 3.93×1010 7.54×1010 K1/(W·m-1·K-1) 9 1.5 C23/(N·m-2) 3.93×1010 7.54×1010 K2/(W·m-1·K-1) 9 1.5 C33/(N·m-2) 8.36×1010 1.13×1011 K3/(W·m-1·K-1) 9 1.5 C44/(N·m-2) 1.32×1010 2.56×1010 β1/(N·K-1·m-2) 6.21×105 1.52×106 C55/(N·m-2) 1.32×1010 2.56×1010 β2/(N·K-1·m-2) 6.21×105 1.52×106 C66/(N·m-2) 1.445×1010 3.06×1010 β3/(N·K-1·m-2) 5.51×105 1.53×106 e31/(C·m-2) -0.16 -6.98 P1/(C·K-1·m-2) 0 0 e32/(C·m-2) -0.16 -6.98 P2/(C·K-1·m-2) 0 0 e33/(C·m-2) 0.347 13.8 P3/(C·K-1·m-2) -2.94×10-6 -4.52×10-4 e15/(C·m-2) -0.138 13.4 Ce/(J·kg-1·K-1) 260 420 e24/(C·m-2) -0.138 13.4 ρ/(kg·m-3) 5.504×103 7.75×103
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