Scattering of SH wave by multiple semi-cylindrical depressions in an elastic strip
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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.
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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)
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