# 环形厚度线性变化薄板中Lamb波传播轨迹研究The Propagation Trajectory of Lamb Waves on Thin Plates with Circular Thickness of Linear Variation

Abstract: In this paper, the propagation trajectory of Lamb waves on thin plates with circular thickness of linear variation is studied by using laser ultrasonic technology. Based on the thermoelastic mech-anism, three-dimensional finite element model of Lamb wave excited by laser line source and laser point source is established, respectively. The Lamb wave excited by laser line source propagates into the area of thickness variation, and focuses at the bottom boundary. The sound wave propagates along the bottom boundary; part is strong, while the other parts are relatively weak. The Lamb wave generated by laser point source propagates along the radius into area of thickness variation. The acoustic accumulates continuously at the bottom boundary and the amplitude in-creases gradually. Moreover, the Lamb wave propagates along two opposite directions of the circle with the same amplitude and frequency. The results of this paper can be used to detect and evalu-ate thin plates with circular thickness of linear variation.

1. 引言

Lamb波是由H. Lamb求解波动方程式时，采用自由边界条件得到的一种特殊的波，主要存在于声波波长和板厚具有数量级相同的板中。与横波、纵波相比，其能量集中、衰减小，可以非接触式激发，在薄板材料的无损检测中具有广泛的运用 [1]。由于激光超声的非接触激发和检测，同时激发出多种形式的超声波，可以对薄板进行快速的全方位扫描，对形状复杂、小尺寸的物体进行检测的优点，其广泛用于材料结构和质量的无损检测。

2. 理论模型

(a) (b)

Figure 1. Schematic diagram of laser irradiation on annular thin plate with linear changes (a), Cone radius R1 = 5 mm, R0 = 1 mm (b)

(a) 对称模态的Lamb波 (b) 反对称模态的Lamb波

Figure 2. Schematic diagram of symmetry and antisymmetry Lamb wave mode

$-k\frac{\partial T\left(x,y,z,t\right)}{\partial n}={I}_{0}A\left(T\right)f\left(x\right)g\left(t\right),$ (1)

$-k\frac{\partial T\left(x,y,z,t\right)}{\partial n}={I}_{0}A\left(T\right)f\left(x,y\right)g\left(t\right),$ (2)

$g\left(t\right)=\frac{8{t}^{3}}{{t}_{0}^{4}}\mathrm{exp}\left(\frac{-2{t}^{2}}{{t}_{0}^{2}}\right),$ (3)

$f\left(x\right)=\frac{1}{\sqrt{2\text{π}}}\frac{2}{{R}_{G}}\mathrm{exp}\left(\frac{-2{\left(x-{x}_{0}\right)}^{2}}{{R}_{G}^{2}}\right),$ (4)

$f\left(x,y\right)=\frac{1}{\sqrt{2\text{π}}}\frac{2}{{R}_{G}}\mathrm{exp}\left(\frac{-2{\left(x-{x}_{0}\right)}^{2}}{{R}_{G}^{2}}-\frac{-2{\left(y-{y}_{0}\right)}^{2}}{{R}_{G}^{2}}\right),$ (5)

Figure 3. The meshing of thin plates with circular thickness of linear variation

Table 1. Parameters of aluminum

${\frac{\partial {V}_{P}\left(r\right)}{\partial r}|}_{{r}_{0}}=\frac{{V}_{P}\left({r}_{0}\right)}{\partial {r}_{0}},$ (6)

${\frac{\partial \Omega \left(r\right)}{\partial r}|}_{{r}_{0}}=0,$ (7)

3. 仿真结果与讨论

3.1. 激光线源激发Lamb波传播轨迹研究

(a) t = 0 μs (b) t = 1 μs (b) t = 2 μs (d) t = 3 μs (e) t = 4 μs (f) t = 5 μs

Figure 4. The displacement field of Lamb wave at t = 0 μs, 1 μs, 2 μs, 3 μs, 4 μs, 5 μs

(a) (b)

Figure 5. Diagram of three-dimensional point (a) and displacement curve (b)

Figure 6. The B-scan of the displacement excited by line source on R0 = 1 mm

3.2. 激光线源激发Lamb波传播轨迹研究

(a) t = 0 μs (b) t = 1 μs (c) t = 2 μs (d) t = 3 μs (e) t = 4 μs (f) t = 5 μs

Figure 7. The displacement field of Lamb wave at t = 0 μs, 1 μs, 2 μs, 3 μs, 4 μs, 5 μs

Figure 8. The B-scan of the displacement excited by point source on R0 = 1 mm

4. 总结

[1] Rose, J.L., 何存富, 吴斌, 王秀彦. 固体中的超声波[M]. 北京: 科学出版社, 2004.

[2] Silva, M., Gouyon, R. and Lepoutre, F. (2003) Hidden Corrosion Detection in Aircraft Aluminum Structures Using Laser Ultrasonics and Wavelet Transform Signal Analysis. Ultrasonics, 41, 301-305. https://doi.org/10.1016/S0041-624X(02)00455-9

[3] 张慧, 刘玉振, 于露, 曾周末. 复合板缺陷的空耦Lamb波扫描仿真与成像研究[J]. 仪器仪表学报, 2019, 40(1): 150-157.

[4] 丁红星, 沈中华, 李加, 祝雪丰, 倪晓武. 复合兰姆波声子晶体中超宽部分禁带[J]. 物理学报, 2012(19): 394-401.

[5] Krylov, V. and Tilman, F. (2004) Acoustic “Black Holes” for Flexural Waves as Effective Vibration Dampers. Journal of Sound and Vibration, 274, 605-619. https://doi.org/10.1016/j.jsv.2003.05.010

[6] Yan, S.L., Lomonosov, A.M., Han, B., Zhang, H.C., Shen, Z.H. and Ni, X.W. (2015) Investigation of Wedge Waves Using Digital Shearing Speckle Interferometry. International Journal of Thermophysics, 36, 1074-1080. https://doi.org/10.1007/s10765-014-1737-7

[7] 李睿奇, 祝雪丰, 梁彬, 李勇，程建春. 声黑洞结构[C]//中国声学学会. 中国声学学会第九届青年学术会议论文集. 2011: 55-56.

[8] 季宏丽, 黄薇, 裘进浩, 成利. 声学黑洞结构应用中的力学问题[J]. 力学进展, 2017, 47(1):337-388.

[9] 李海勤, 孔宪仁, 刘源. 接触非线性对声黑洞梁减振效果的影响[J]. 力学学报, 2019, 51(4): 1189-1201.

[10] Jia, J., Shen, Z., Han, Q. and Jing, X.P. (2017) Design of Wedge Structure with Non-Dispersive Wedge Wave Propagation. Applied Optics, 56, 8564. https://doi.org/10.1364/AO.56.008564

Top