Scattering of SH wave by multiple semi-cylindrical depressions in an elastic strip
-
摘要: 对稳态SH（shear horizontal）导波在表面含有多个半圆柱形凹陷的弹性带形介质内的散射问题进行了研究，并给出了解析解。首先，运用导波展开法构造平面SH导波；然后，利用累次镜像法构造出满足带形域上、下两条直边界应力自由条件的散射波；最后，根据凹陷边沿的切应力为零的条件得到定解方程。通过算例分析了累次镜像法的精度、凹陷边沿的动应力集中和上、下边界位移幅值的变化情况。数值结果表明：只有一个凹陷时，中高频率的入射波和小厚度的带形域会引起凹陷边沿更高的动应力集中，上边界位移幅值的最大值会出现在凹陷的迎波面附近；当有两个凹陷时，大多数情况下，第二个凹陷对第一个凹陷边沿的动应力集中起放大作用，并且在理想弹性带形介质内，两凹陷之间的影响在相距无穷远时也会存在。Abstract: The scattering of steady-state shear horizontal (SH) guided waves from the elastic strip media with multiple semi-cylindrical depressions on the surface was studied and the analytical solution was given. Firstly, the guided wave expansion method was used to construct SH guided waves. Then, the scattered waves satisfying the zero-stress boundary conditions of the upper and lower surfaces in the strip were constructed by employing the repeated image method. Finally, according to the condition that the shear stress of the edge of the depression is zero, the definite solution equation was derived. The accuracy of repeated image method, the dynamic stress concentration around a depression and the displacement amplitude at the upper and lower boundaries were analyzed by examples. The numerical results show that when there is only one depression, the incident waves with middle and high frequency and the strip with small thickness cause higher dynamic stress concentration around the depression, and the maximum displacement amplitude of the upper boundary occurs near the incident surface of the depression. When there are two depressions, in most cases, the second depression amplifies the dynamic stress concentration around the first depression. And in the ideal elastic strip, even if the two depressions are infinitely far apart, the influence between them still exists.
-
图 8 弹性带形域上边界存在两半圆柱形凹陷
Figure 8. Two semi-cylindrical depressions on the upper boundary of the elastic strip
图 9 下边界
$\tau _{y{\textit z}}^*$ 的变化Figure 9. Variation of
$\tau _{y{\textit z}}^*$ in the lower boundary
图 10 下边界一点处的
$\tau _{y{\textit z}}^*$ 随P的变化规律Figure 10. Variation of
$\tau _{y{\textit z}}^*$ at a certain point in the lower boundary with P
图 11 凹陷边沿动应力集中系数随镜像次数的变化规律
Figure 11. Variation of dynamic stress concentration factor around the depression with P
图 13 动应力集中系数随带形域无量纲厚度的变化 (g=1, m=0)
Figure 13. Variation of dynamic stress concentration factor with dimensionless thickness (g=1, m=0)
图 14 不同k*时动应力集中系数随角度θ变化 (g=1, m=0)
Figure 14. Variation of dynamic stress concentration factor with θ at different k* (g=1, m=0)
图 15 不同h*时动应力集中系数随角度θ的变化 (g=1, m=0)
Figure 15. Variation of dynamic stress concentration facor with θ at different h* (g=1, m=0)
图 16 凹陷边沿最大动应力集中随k*的变化
Figure 16. Variation of maximum dynamic stress concentration factor with k* around the depression
图 17 1号凹陷边沿动应力集中系数的最大值随两凹陷之间量纲距离a*的变化 (m=0, r*=1)
Figure 17. Variation of maximum dynamic stress concentration factor around the first depression with a* (m=0, r*=1)
图 18 表面位移幅值随k*的变化 (g=1, m=0, h* =10.0)
Figure 18. Variation of surface displacement amplitude with k* (g=1, m=0, h* =10.0)
-
[1] TRIFUNAC M D. Scattering of plane sh waves by a semi-cylindrical canyon [J]. Earthquake Engineering and Structural Dynamics, 1972, 1(3): 267–281. DOI: 10.1002/eqe.4290010307. [2] WONG H L, TRIFUNAC M D. Scattering of plane SH waves by a semi-elliptical canyon [J]. Earthquake Engineering and Structural Dynamics, 1974, 3(2): 157–169. DOI: 10.1002/eqe.4290030205. [3] LIU D K, GAI B Z, TAO G Y. Applications of the method of complex functions to dynamic stress concentrations [J]. Wave Motion, 1982, 4(3): 293–304. DOI: 10.1016/0165-2125(82)90025-7. [4] LIU D K, HAN F. Scattering of plane SH-wave by cylindrical canyon of arbitrary shape [J]. Soil Dynamics and Earthquake Engineering, 1991, 10(5): 249–255. DOI: 10.1016/0267-7261(91)90018-U. [5] 许贻燕, 韩峰. 平面SH波在相邻多个半圆形凹陷地形上的散射 [J]. 地震工程与工程振动, 1992, 12(2): 12–18. DOI: 10.13197/j.eeev.1992.02.002.XU Y Y, HAN F. Scattering of SH-waves by multiple semi-cylindrical canyons [J]. Earthquake Engineering and Engineering Vibration, 1992, 12(2): 12–18. DOI: 10.13197/j.eeev.1992.02.002. [6] 刘刚, 李宏亮, 刘殿魁, 等. SH波对浅埋裂纹的半圆形凹陷地形的散射 [J]. 爆炸与冲击, 2007, 27(2): 171–178. DOI: 10.11883/1001-1455(2007)02-0171-08.LIU G, LI H L, LIU D K, et al. Scattering of a semi-cylindrical canyon and a crack with incident SH waves [J]. Explosion and Shock Waves, 2007, 27(2): 171–178. DOI: 10.11883/1001-1455(2007)02-0171-08. [7] 齐辉, 蔡立明, 潘向南, 等. 弹性直角域中半圆形凹陷的SH波散射的稳态解 [J]. 天津大学学报(自然科学与工程技术版), 2014, 47(12): 1065–1071. DOI: 10.11784/tdxbz201403015.QI H, CAI L M, PAN X N, et al. Steady state solution of SH wave scattering of a semi-circular cylindrical canyon in an elastic quarter space [J]. Journal of Tianjin University (Science and Technology), 2014, 47(12): 1065–1071. DOI: 10.11784/tdxbz201403015. [8] CHANG K H, TSAUR D H, WANG J H. Scattering of SH waves by a circular sectorial canyon [J]. Geophysical Journal International, 2013, 195(1): 532–543. DOI: 10.1093/gji/ggt236. [9] SHYU W S, TENG T J, YEH C S. Surface motion of two canyons for incident SH waves by hybrid method [J]. Procedia Engineering, 2014, 79: 533–539. DOI: 10.1016/j.proeng.2014.06.376. [10] BA Z N, LIANG J W. Dynamic response analysis of periodic alluvial valleys under incident plane SH-waves [J]. Journal of Earthquake Engineering, 2017, 21(4): 531–550. DOI: 10.1080/13632469.2016.1178192. [11] ACHENBACH J D. Wave propagation in elastic solids [M]. Amsterdam: North-Holland, 1973: 202−261. [12] LU Y C. Guided antiplane shear wave propagation in layers reinforced by periodically spaced cylinders [J]. The Journal of the Acoustical Society of America, 1996, 99(4): 1937–1943. DOI: 10.1121/1.415377. [13] HAYIR A, BAKIRTAS I. A note on a plate having a circular cavity excited by plane harmonic SH waves [J]. Journal of Sound and Vibration, 2004, 271(1-2): 241–255. DOI: 10.1016/S0022-460X(03)00751-X. [14] 齐辉, 折勇, 赵嘉喜. 带形域内圆柱形夹杂对SH型导波的散射 [J]. 振动与冲击, 2009, 28(5): 142–145. DOI: 10.3969/j.issn.1000-3835.2009.05.032.QI H, SHI Y, ZHAO J X. Scattering of SH waves from a circular inclusion in an infinite strip region [J]. Journal of Vibration and Shock, 2009, 28(5): 142–145. DOI: 10.3969/j.issn.1000-3835.2009.05.032. [15] QI H, ZHANG X M. Scattering of SH guided wave by a circular inclusion in an infinite piezoelectric material strip [J]. Waves in Random and Complex Media, 2019, 29(1): 93–110. DOI: 10.1080/17455030.2017.1413262. [16] QI H, XIANG M, GUO J. The dynamic stress analysis of an infinite piezoelectric material strip with a circular cavity [J]. Mechanics of Advanced Materials and Structures, 2020. DOI: 10.1080/15376494.2019.1709676. [17] QI H, XIANG M, GUO J. Scattering of a shear horizontal wave by a circular cavity in a piezoelectric bi-material strip based on guided wave theory [J]. Mathematics and Mechanics of Solids, 2020, 25(4): 968–985. DOI: 10.1177/1081286519897353. -
下载:
点击查看大图
计量
- 文章访问数: 3038
- HTML全文浏览量: 1402
- PDF下载量: 41
- 被引次数: 0