摘要
本文在弹性力学动力范畴内,研究了带有覆盖层的圆孔和圆形夹杂对SH波的散射问题。利用复变函数法、波函数展开法以及移动坐标技术求解了具有覆盖层的圆孔和圆柱形夹杂在SH波作用下孔边与夹杂周边动应力集中问题的解。主要内容如下：
(1)采用大圆弧假设原理：即半空间水平表面采用一个半径很大的圆弧来近似代替。这一假设解决了在半无限空间表面无法进行波函数展开的难题,进而可以用大圆弧的内聚波来替代在半空间表面的散射波。
(2)在复平面上,借助Helmholtz定理预先构造出地下圆形结构对SH波散射的波函数一般形式的解,再通过边界条件来求解控制方程。其中采用复数Fourier-Hankel级数展开把控制方程的问题转化为无穷代数方程组,通过截断该方程组的有限项,求得波函数中未知系数,从而完成该问题的求解。
(3)结合两个具体算例讨论了地表覆盖层对浅埋圆孔和圆柱形夹杂动应力集中的影响以及圆孔和夹杂对地表位移的影响,讨论了在不同影响因素下地下结构周边动应力集中系数和地表位移幅值的变化趋势和规律。分析结果显示,半空间中地表覆盖层参数的变化对入射SH波的散射影响较大,可明显改变结构周边动应力系数及地表位移值的分布。入射波的频率、入射波的角度以及浅埋圆孔和夹杂间的距离对结构周边动应力系数具有较大的影响。入射波的波数、地下结构的埋深和覆盖层的参数使地表位移发生较大的改变。
The scattering of SH-wave by subsurface circular cavity and circular inclusions in a layered elastic half-space was researched on the category of elasticity. The solution of the dynamic stress concentration of SH-wave by a subsurface circular cavity and circular inclusions in a layered elastic half-space was solved based on the complex function method, wave functions expansion method and moving coordinate. The main contents are as follow:
(1) The principle of orthodrome assumption was used. The principle assumes that the horizontal surface in infinite half-space is replaced by circular arc with large radius. Under this assumption, wave function can be expanded on the horizontal surface in infinite half-space. On the basis of wave function expansion, scattering waves on the surface of half-space can be substituted by cohesion waves.
(2) On complex plane, general solution of the wave function which expresses the scattering of circular structure underground under SH-wave was given using the theory of Helmholtz, and then the solution of controlling equation was finished by boundary conditions. The solution of controlling equation was converted into infinite linear algebraic equations by expansion of complex Fourier-Hankel series. Unknown coefficient in wave function was obtained after the finite terms in equations were truncated, and the solution of this problem was completed.
(3) The ground motion and the dynamic stress concentration factors around the circular inclusion and circular cavity were discussed according to two numerical examples. The variation law of dynamic stress concentration coefficient and surface displacement was also analyzed under different factors. The results show that the parameters of surface overburden layer in half-space have great influence on the scattering of SH-wave, which can change the distribution of dynamic stress concentration coefficient and surface displacement around the circular inclusion and circular cavity. The frequency and angle of incident wave and the distance between circular inclusion and cavity badly affect the dynamic stress concentration coefficient. The wave number, buried depth of the structure underground and the parameters of surface overburden layer all can change the surface displacement obviously.
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