半空间饱和土中圆形壳结构对平面P波的散射
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摘要
在Biot波动理论的基础上,引入更符合工程实际的半透水边界条件,给出了半空间均质饱和土中半渗透圆柱形壳结构对平面P波散射的级数解。具体做法是利用波函数展开法,将入射波和散射波的势函数展开成Fourier-Bessel函数的无穷级数的形式,采用大半径凸圆弧来近似半空间表面,根据问题的连续性条件和边界条件,确定波函数展开式中的未知系数,进而得到问题的求解。计算结果表明:半空间饱和土和圆柱形壳界面处的径向应力与壳和土体的相对渗透性、入射角等有关,壳和饱和土体的相对渗透性越小,饱和土体和壳结构底部界面处径向应力所受影响越大。
Based on Biot’s theory, a series solution for the scattering of an elastic wave by a cylindrical shell in a homogeneous saturated half-space was investigated theoretically with the assumption that the shell material was also a saturated medium. The wave potentials were expanded in Fourier-Bessel series using the wave function expansion method. A convex approximation was used to replace the straight boundary of the half-space. In terms of continuity conditions and boundary conditions of the problem, the unknown coefficients in potentials were determined. Numerical results show that radial stress amplitudes at the interface of the saturated soil and shell depend on the relative permeability of the saturated soil and shell and the angle of incidence. The radial stress amplitudes are significantly affected at the interface of the shell bed and soil when the relative permeability is low.
Based on Biot’s theory, a series solution for the scattering of an elastic wave by a cylindrical shell in a homogeneous saturated half-space was investigated theoretically with the assumption that the shell material was also a saturated medium. The wave potentials were expanded in Fourier-Bessel series using the wave function expansion method. A convex approximation was used to replace the straight boundary of the half-space. In terms of continuity conditions and boundary conditions of the problem, the unknown coefficients in potentials were determined. Numerical results show that radial stress amplitudes at the interface of the saturated soil and shell depend on the relative permeability of the saturated soil and shell and the angle of incidence. The radial stress amplitudes are significantly affected at the interface of the shell bed and soil when the relative permeability is low.
引文
[1]Pao Y H,Mow C C.Diffraction of elastic waves and dynamic stress concentrations[M].New York:Crane and Russak,1973.
[2]Lee V W,Trifunac M D.Response of tunnels to incident SH-waves[J].Journal of Engineering Mechanics,ASCE,1979,105(4):643―659.
[3]Cao H,Lee V W.Scattering and diffraction of plane P waves by circular cylindrical canyons with variable depth-to-width ratio[J].Soil Dynamics and Earthquake Engineering,1990,9(3):141―150.
[4]Lee V W,Karl J.On deformations near a circular underground cavity subjected to incident plane P waves[J].European Earthquake Engineering,1993,7(1):29―36.
[5]Luo J E,Debarros F C P.Dynamic displacements and stresses in the vicinity of a cylindrical cavity embedded in a half-space[J].Earthquake Engineering and Structural Dynamics,1994,23(33):321―340.
[6]Davis C A,Lee V W,Bardet J P.Transverse response of underground cavities and pipes to incident SV waves[J].Earthquake Engineering and Structural Dynamics,2001,30:383―410.
[7]Kattis S E,Beskos D E,Cheng A H D.2D dynamic response of unlined and lined tunnels in poroelastic soil to harmonic body waves[J].Earthquake Engineering and Structural Dynamics,2003,32(1):97―110.
[8]胡亚元,王立忠,陈云敏.饱和土中平面应变波在圆柱体上的散射和折射[J].地震学报,1998,20(3):300―307.Hu Yayuan,Wang Lizhong,Chen Yunmin.Scattering and refracting of plane strain wave by a cylindrical inclusion in fluid-saturated soils[J].Acta Seismologica Sinica,1998,20(3):300―307.(in Chinese)
[9]Liu Ganbin,Xie Kanghe,Yao Hailin.Scattering of elastic wave by a partially sealed cylindrical shell embedded in poroelastic medium[J].International Journal for Numerical and Analytical Methods in Geomechanics,2005,29(11):1109―1126.
[10]Esmaeili M,Vahdani S,Noorzad A.Dynamic response of lined circular tunnel to plane harmonic waves[J].Tunnelling and Underground Space Technology,2006,21(5):511―519.
[11]Karinski Y S,Shershnev V V,Yankelevsky D Z.Analytical solution of the harmonic waves diffraction by a cylindrical lined cavity in poroelastic saturated medium[J].International Journal for Numerical and Analytical Methods in Geomechanics,2007,31(5):667―689.
[12]Biot M A.Theory of propagation of elastic waves in a fluid-saturated porous solid,I,Low-frequency range[J].Journal of the Acoustical Society of America,1956,28(2):168―178.
[13]Philippacopoulos A J.Wave in a partially saturated medium due to surface loads[J].Journal of Engineering Mechanics,ASCE,1988,114(10):1740―1759.
[14]李伟华,赵成刚.具有饱和土沉积层的充水河谷对平面波的散射[J].地球物理学报,2006,49(1):212―224.Li Weihua,Zhao Chenggang.Scattering of plane waves by circular-arc alluvial valleys with saturated soil deposits and water[J].Chinese Journal of Geophysics,2006,49(1):212―224.(in Chinese)
[2]Lee V W,Trifunac M D.Response of tunnels to incident SH-waves[J].Journal of Engineering Mechanics,ASCE,1979,105(4):643―659.
[3]Cao H,Lee V W.Scattering and diffraction of plane P waves by circular cylindrical canyons with variable depth-to-width ratio[J].Soil Dynamics and Earthquake Engineering,1990,9(3):141―150.
[4]Lee V W,Karl J.On deformations near a circular underground cavity subjected to incident plane P waves[J].European Earthquake Engineering,1993,7(1):29―36.
[5]Luo J E,Debarros F C P.Dynamic displacements and stresses in the vicinity of a cylindrical cavity embedded in a half-space[J].Earthquake Engineering and Structural Dynamics,1994,23(33):321―340.
[6]Davis C A,Lee V W,Bardet J P.Transverse response of underground cavities and pipes to incident SV waves[J].Earthquake Engineering and Structural Dynamics,2001,30:383―410.
[7]Kattis S E,Beskos D E,Cheng A H D.2D dynamic response of unlined and lined tunnels in poroelastic soil to harmonic body waves[J].Earthquake Engineering and Structural Dynamics,2003,32(1):97―110.
[8]胡亚元,王立忠,陈云敏.饱和土中平面应变波在圆柱体上的散射和折射[J].地震学报,1998,20(3):300―307.Hu Yayuan,Wang Lizhong,Chen Yunmin.Scattering and refracting of plane strain wave by a cylindrical inclusion in fluid-saturated soils[J].Acta Seismologica Sinica,1998,20(3):300―307.(in Chinese)
[9]Liu Ganbin,Xie Kanghe,Yao Hailin.Scattering of elastic wave by a partially sealed cylindrical shell embedded in poroelastic medium[J].International Journal for Numerical and Analytical Methods in Geomechanics,2005,29(11):1109―1126.
[10]Esmaeili M,Vahdani S,Noorzad A.Dynamic response of lined circular tunnel to plane harmonic waves[J].Tunnelling and Underground Space Technology,2006,21(5):511―519.
[11]Karinski Y S,Shershnev V V,Yankelevsky D Z.Analytical solution of the harmonic waves diffraction by a cylindrical lined cavity in poroelastic saturated medium[J].International Journal for Numerical and Analytical Methods in Geomechanics,2007,31(5):667―689.
[12]Biot M A.Theory of propagation of elastic waves in a fluid-saturated porous solid,I,Low-frequency range[J].Journal of the Acoustical Society of America,1956,28(2):168―178.
[13]Philippacopoulos A J.Wave in a partially saturated medium due to surface loads[J].Journal of Engineering Mechanics,ASCE,1988,114(10):1740―1759.
[14]李伟华,赵成刚.具有饱和土沉积层的充水河谷对平面波的散射[J].地球物理学报,2006,49(1):212―224.Li Weihua,Zhao Chenggang.Scattering of plane waves by circular-arc alluvial valleys with saturated soil deposits and water[J].Chinese Journal of Geophysics,2006,49(1):212―224.(in Chinese)
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