波函数展开法在弹性波散射中的应用
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摘要
弹性波的散射与动应力集中问题是近几十年提出的课题,它在理论上,特别是工程应用上有重要的意义。本文选取了工程中常见的几种典型情况为研究对象,得到了它们对于入射SH波的散射作用封闭解析解。
采用分区思想和傅里叶级数展开方法,利用波函数展开技术并引入辅助函数,求解并得出了任意圆弧形隧洞对平面SH波的散射的封闭解析解。并在收敛性和误差分析的基础上,讨论了不同频率下隧洞的形状以及埋深对地表位移和隧洞环向应力幅值的影响。结果表明入射角度、无量纲频率、隧洞埋置深度和隧洞形状均对地表位移形式有影响,劣弧形隧洞与半圆形隧洞以及优弧形隧洞有较大不同。
波函数形式采用傅里叶级数展开为余弦函数形式,利用波函数展开技术,给出了水平地表圆形非匀质凸起出平面动力相应的闭合级数解答并对水平地表圆形非均匀凸起出平面运动特征进行详细分析。计算结果表明,位移应力残量在包括角点的整个边界上非常小可以忽略,计算精度很高,同时对于高频也能有很好的解答。结果表明,凸起区域介质软硬程度对其出平面地表运动影响非常大,软质凸起表面的地表位移较匀质凸起情况有显著放大,地表运动波动效应明显增强,而硬质凸起则减弱地表运动。
同样通过将位移应力函数用余弦函数形式表达,对公式和求解过程重新推导,给出弹性半空间带同心半圆形隧洞和全圆隧洞的半圆形凸起地形对平面SH波二维散射的改进精确级数展开解析解。计算结果表明位移应力残量在包括角点的整个边界上可以忽略,计算精度很高,同时对于较高频率和较高孔洞与凸起地形半径比也能有很好的解答。最后,还比较了无隧洞凸起地形以及含全圆或半圆隧洞凸起地形对地表位移结果的影响。
Diffraction of Elastic Waves and Dynamics Stress Concentrations has great academic significance and distinct referenced value in practical project..Based on this as engineering background, an analytical solution of incident SH waves by some kind of representative site in a half-space is presented.
A closed-form solution for two-dimensional scattering of plane SH waves by a tunnel of circular-arc cross-section in a half-space are presented by using the region division and auxiliary functions methods as well as the wave functions expansion.Based on convergence and error analyse, the effects of the frequency of the incident waves,the shape and position of tunnel on displacement amplitude of ground surface and hoop stress amplitude on the wall of tunnel are illustrated.The results indicate:The incidence angles and the frequency of the incident waves,as well as the shape and position of tunnel,all have influence on displacement amplitude of ground surface.
By expressing all displacement and stress waves as cosine functions,a closed-form solution for two-dimensional scattering of plane SH waves by a semi-cylindrical heterogeneous hill are presented using the wave functions expansion.For the solution to be correct,the computed stress and displacement residual amplitudes were showen to be numerically negligible everywhere,including those at both rims of the hill.Displacements at higher frequencies are also computed. The results indicate the effects of rigidity of the hilly region on the surface ground displacements of the hill and its adjacent region are siginificantly strong.
Using the improved accurate closed-form wave function analytic solution,the solution of two-dimensional scattering and diffraction of plane SH waves by a semi-cylindrical hill with a semi-cylindrical or cylindrical concentric tunnel inside an elastic half-space was reformulated.Similarly, stress and displacement residual amplitudes were showen to be numerically negligible everywhere. Displacements at higher frequencies are also computed,as well as great radius of tunnel.In the end, the effects of the frequency of the incident waves,the shape and radius of tunnel on displacement amplitude of ground surface are illustrated.
采用分区思想和傅里叶级数展开方法,利用波函数展开技术并引入辅助函数,求解并得出了任意圆弧形隧洞对平面SH波的散射的封闭解析解。并在收敛性和误差分析的基础上,讨论了不同频率下隧洞的形状以及埋深对地表位移和隧洞环向应力幅值的影响。结果表明入射角度、无量纲频率、隧洞埋置深度和隧洞形状均对地表位移形式有影响,劣弧形隧洞与半圆形隧洞以及优弧形隧洞有较大不同。
波函数形式采用傅里叶级数展开为余弦函数形式,利用波函数展开技术,给出了水平地表圆形非匀质凸起出平面动力相应的闭合级数解答并对水平地表圆形非均匀凸起出平面运动特征进行详细分析。计算结果表明,位移应力残量在包括角点的整个边界上非常小可以忽略,计算精度很高,同时对于高频也能有很好的解答。结果表明,凸起区域介质软硬程度对其出平面地表运动影响非常大,软质凸起表面的地表位移较匀质凸起情况有显著放大,地表运动波动效应明显增强,而硬质凸起则减弱地表运动。
同样通过将位移应力函数用余弦函数形式表达,对公式和求解过程重新推导,给出弹性半空间带同心半圆形隧洞和全圆隧洞的半圆形凸起地形对平面SH波二维散射的改进精确级数展开解析解。计算结果表明位移应力残量在包括角点的整个边界上可以忽略,计算精度很高,同时对于较高频率和较高孔洞与凸起地形半径比也能有很好的解答。最后,还比较了无隧洞凸起地形以及含全圆或半圆隧洞凸起地形对地表位移结果的影响。
Diffraction of Elastic Waves and Dynamics Stress Concentrations has great academic significance and distinct referenced value in practical project..Based on this as engineering background, an analytical solution of incident SH waves by some kind of representative site in a half-space is presented.
A closed-form solution for two-dimensional scattering of plane SH waves by a tunnel of circular-arc cross-section in a half-space are presented by using the region division and auxiliary functions methods as well as the wave functions expansion.Based on convergence and error analyse, the effects of the frequency of the incident waves,the shape and position of tunnel on displacement amplitude of ground surface and hoop stress amplitude on the wall of tunnel are illustrated.The results indicate:The incidence angles and the frequency of the incident waves,as well as the shape and position of tunnel,all have influence on displacement amplitude of ground surface.
By expressing all displacement and stress waves as cosine functions,a closed-form solution for two-dimensional scattering of plane SH waves by a semi-cylindrical heterogeneous hill are presented using the wave functions expansion.For the solution to be correct,the computed stress and displacement residual amplitudes were showen to be numerically negligible everywhere,including those at both rims of the hill.Displacements at higher frequencies are also computed. The results indicate the effects of rigidity of the hilly region on the surface ground displacements of the hill and its adjacent region are siginificantly strong.
Using the improved accurate closed-form wave function analytic solution,the solution of two-dimensional scattering and diffraction of plane SH waves by a semi-cylindrical hill with a semi-cylindrical or cylindrical concentric tunnel inside an elastic half-space was reformulated.Similarly, stress and displacement residual amplitudes were showen to be numerically negligible everywhere. Displacements at higher frequencies are also computed,as well as great radius of tunnel.In the end, the effects of the frequency of the incident waves,the shape and radius of tunnel on displacement amplitude of ground surface are illustrated.
引文
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