Wave Propagation in Functionally Graded Piezoelectric Nanoshells
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摘要: 基于非局部应变梯度理论，探究了含孔隙的功能梯度压电陶瓷纳米壳中波传播特性. 利用Hamilton原理和一阶剪切理论推导了控制方程. 结合非局部应变梯度理论和谐波解得到了尺度依赖的特征方程. 数值讨论了尺度参数、波数、梯度指数、壳厚、孔隙率及电压对波传播特性的影响. 研究表明：非局部参数和应变梯度参数对波传播频率的影响与波数密切相关，在一定范围内波数越大，尺度参数对频率的影响越大；另外，孔隙和梯度指数对频率具有耦合作用.Abstract: The waves propagation characteristics in porous functionally graded piezoelectric nanoshells were investigated based on the nonlocal strain gradient theory. The governing equations were developed under Hamilton's principle and the 1st-order shear theory. The scale-dependent characteristic equations were obtained through combination of the nonlocal strain gradient theory and the harmonic solutions. The effects of the scale parameter, the wave number, the gradient index, the thickness, the porosity and the voltage on the wave propagation characteristics were discussed numerically. The results show that, the influences of the nonlocal parameter and the strain gradient parameter on the wave propagation frequency are closely related to the wave number, and the larger the wave number is in a certain range, the greater the influence of scale parameters on the frequency will be. In addition, the porosity and the gradient index have a coupling effect on the frequency.
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图 2 退化验证(η=1 nm2)
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 2. Comparison of frequency dispersion results (η=1 nm2)
图 3 基于不同波数，变化的η/μ比值对FGPM纳米壳波传播频率的影响
Figure 3. Frequencies vs. η/μ values for different wave numbers of the FGPM nanoshell
图 4 基于不同波数，变化的尺度参数和功能梯度指数对FGPM纳米壳波传播频率的影响
Figure 4. Frequencies vs. scale parameters and FG indexes for different wave numbers of the FGPM nanoshell
图 5 不同壳厚下FGPM纳米壳的频率与功能梯度指数的关系
Figure 5. Frequencies vs. FG indexes for different thicknesses of the FGPM nanoshell
图 6 不同孔隙率下FGPM纳米壳的频率与功能梯度指数的关系
Figure 6. Frequencies vs. FG indexes for different porosities of the FGPM nanoshell
图 7 不同外电压作用下FGPM纳米壳的频率与功能梯度指数的关系
Figure 7. Frequencies vs. FG indexes for different electric voltages of the FGPM nanoshell
表 1 FGPM的材料特性数值
Table 1. Material properties of the FGPM
material unit PZT-4 PZT-5H elastic modulus GPa c11=139, c12=77.8, c13=74,
c22=139, c23=74, c33=115,
c44=25.6, c55=25.6, c66=30.6c11=126, c12=79.1, c13=83.9,
c22=126, c23=83.9, c33=117,
c44=23, c55=23, c66=23.5piezoelectric modulus C/m2 e31=－5.2, e32=－5.2, e33=15.1,
e15=12.7, e24=12.7e31=－6.5, e32=－6.5, e33=23.3,
e15=17, e24=17dielectric modulus C/(V·m) s11=5.841×10-9, s33=7.124×10-9 s11=1.505×10-8, s33=1.302×10-8 mass density kg/m3 ρ=7 500 ρ=7 500 
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表 2 前三阶固有频率(f1, f2, f3)与纵向波数k和周向波数n的关系
Table 2. The 1st 3 orders of frequencies vs. longitudinal and circumferential wave numbers
n k/m-1 f1/THz f2/THz f3/THz 1 1×108 0.727 8 1.407 4 2.375 5 5×108 2.881 4 3.119 8 3.219 8 1×109 6.448 6 7.129 6 7.514 7 10 1×108 0.773 5 1.330 0 2.242 7 5×108 2.929 7 3.175 6 3.275 2 1×109 6.457 7 7.140 8 7.526 7 100 1×108 2.973 8 3.235 4 3.336 7 5×108 4.301 9 4.838 0 4.916 2 1×109 7.327 1 8.172 2 8.646 4
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