半空间饱和土中圆形衬砌对弹性压缩波的散射
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摘要
借助于Biot波动理论和弹性波的传播理论,采用复变函数和多级坐标法,对半空间饱和土中圆形衬砌结构对弹性稳态压缩波的散射问题进行求解和分析。利用一个半径很大的圆弧来逼近半空间直边界,将待解问题转化为稳态弹性压缩波在一个大圆孔和一个弹性衬砌结构的散射问题。通过引入势函数,将饱和土的Biot波动方程和衬砌的弹性波动方程解耦成Helmholtz方程,借助复变函数级数展开便可以预先写出该组Helmholtz方程的通解。然后,通过引用复变量,把饱和土和衬砌结构中的应力、位移及孔压用设定的势函数表示出来,再利用半空间饱和土和衬砌结构的连续性条件和近似直边界的圆弧边界和衬砌内边界的边界条件求解出该组势函数的特解。最后,利用势函数的特解,得到饱和土中的位移,应力和孔压及衬砌结构的位移和应力;变换不同的参数求解衬砌结构内外边界的动应力和孔压的集中系数,通过对算例结果的分析得出一系列有益的结论。
In terms of Biot's dynamic theory and elastic wave theory,the method of functions of complex variable and multi-polar coordinate is used to solve the problem of scattering around a circular lining in saturated poroelastic half-space under harmonic plane dilatational waves.Here,a circular cavity with large radius is used to replace the straight boundary of the saturated poroelastic half space.The equations of the Biot wave motion for poroelastic and the equations of the isotropic elastic wave motion for lining are decoupled to Helmholtz equations by introducing potential functions.The general solution of Helmholtz equations is given by means of the complex series expansion technology.Utilizing the general solution of the potential functions,the expressions of the displacements,stresses and pore pressures of saturated soil and those of the displacements and stresses of lining structure can be obtained by using the boundary conditions and continuous conditons.Then the variations of the coefficients of dynamic stress concentration and the pore water pressures concentration on boundaries of the lining structure are discussed for different parameter conditions.The results of the given numerical example indicate that the method is useful and efficient to the scattering of lining in poroelastic half-space.
In terms of Biot's dynamic theory and elastic wave theory,the method of functions of complex variable and multi-polar coordinate is used to solve the problem of scattering around a circular lining in saturated poroelastic half-space under harmonic plane dilatational waves.Here,a circular cavity with large radius is used to replace the straight boundary of the saturated poroelastic half space.The equations of the Biot wave motion for poroelastic and the equations of the isotropic elastic wave motion for lining are decoupled to Helmholtz equations by introducing potential functions.The general solution of Helmholtz equations is given by means of the complex series expansion technology.Utilizing the general solution of the potential functions,the expressions of the displacements,stresses and pore pressures of saturated soil and those of the displacements and stresses of lining structure can be obtained by using the boundary conditions and continuous conditons.Then the variations of the coefficients of dynamic stress concentration and the pore water pressures concentration on boundaries of the lining structure are discussed for different parameter conditions.The results of the given numerical example indicate that the method is useful and efficient to the scattering of lining in poroelastic half-space.
引文
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[5]BIOT M A.Theory of propagation of elastic waves in a fluid-saturated porous solid.I:low frequency range[J].The Journal of the Acoustical Society of America,1956,28(2):168-178.
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[7]WANG J H,ZHOU X L,LU J F.Dynamic stress concentration around elliptic cavities in saturated poroelastic soil under harmonic plane waves[J].Inter-national Journal of solids and Structures,2005,42:4 295-4 310.
[8]周香莲,周光明,王建华.饱和土中圆形衬砌结构对弹性波的散射[J].岩石力学与工程学报.2005,24(9):1 572-1 576.ZHOU Xiang-lian,ZHOU Guang-ming,WANG Jian-hua.Scattering of elastic wave by circular cavity with lining in saturated soil[J].Chinese Journal of Rock Mechanics and Engineering,2005,24(9):1 572-1 576.
[9]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):293-304.
[10]ZIMMERMAN C.Scattering of plane compressional waves by spherical inclusions in a poroelastic medium[J].The Journal of the Acoustical Society of America,1993,94(1):527-536.
[11]DERESIEWICZ H.The effect of boundaries on wave propagation in a liquid-filled porous solid:I.Reflection of plane wave a free plane boundary(non-dissipative case)[J].Bulletin of the Seismological Society of America,1960,50(4):599-607.
[12]DAVIS C A,LEE V W,BARDET J P.Transverse reponse of underground cavities and pipes to incident SV waves[J].Earthquake Engineering and Structural Dynamics,2001,30:383-410.
[13]鲍亦兴,毛昭宙.弹性波的衍射与动应力集中[M].北京:科学出版社,1993.PAO Yi-xing,MAO Zhao-yu.Scattering and Dynamics Stresses Concentration of Elastic Wave[M].bei Jing:The Science Press,1993.
[2]EL-Akily N,DATTA S K.Response of a circular cylindrical shell to disturbances in a half–space-numerical results[J].Earthquake Engineering and Structural Dynamics,1981,9:477-487.
[3]DATTA S K,SHAH A H,WONG KC.Dynamics stress and displacements in buried pipe[J].Journal of Engineering Mechanics,ASCE,1984,110(10):1451-1466.
[4]史文谱,刘殿魁,林宏,等.半无限空间中稳态P波在衬砌周围的散射[J].地震工程与工程振动,2002,22(3):19-26.SHI Wen-pu,LIU Dian-kui,LIN Hong,ZHANG Yao-liang.Scattering of steady P-wave around a circular lining in half space[J].Earthquake Engineering and Engineering Vibration,2002,22(3):19-26.
[5]BIOT M A.Theory of propagation of elastic waves in a fluid-saturated porous solid.I:low frequency range[J].The Journal of the Acoustical Society of America,1956,28(2):168-178.
[6]BIOT M A.Mechanics of deformation and acoustic propagation in porous media[J].Journal of Applied Physics,1962,33(4):1 482-1 498.
[7]WANG J H,ZHOU X L,LU J F.Dynamic stress concentration around elliptic cavities in saturated poroelastic soil under harmonic plane waves[J].Inter-national Journal of solids and Structures,2005,42:4 295-4 310.
[8]周香莲,周光明,王建华.饱和土中圆形衬砌结构对弹性波的散射[J].岩石力学与工程学报.2005,24(9):1 572-1 576.ZHOU Xiang-lian,ZHOU Guang-ming,WANG Jian-hua.Scattering of elastic wave by circular cavity with lining in saturated soil[J].Chinese Journal of Rock Mechanics and Engineering,2005,24(9):1 572-1 576.
[9]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):293-304.
[10]ZIMMERMAN C.Scattering of plane compressional waves by spherical inclusions in a poroelastic medium[J].The Journal of the Acoustical Society of America,1993,94(1):527-536.
[11]DERESIEWICZ H.The effect of boundaries on wave propagation in a liquid-filled porous solid:I.Reflection of plane wave a free plane boundary(non-dissipative case)[J].Bulletin of the Seismological Society of America,1960,50(4):599-607.
[12]DAVIS C A,LEE V W,BARDET J P.Transverse reponse of underground cavities and pipes to incident SV waves[J].Earthquake Engineering and Structural Dynamics,2001,30:383-410.
[13]鲍亦兴,毛昭宙.弹性波的衍射与动应力集中[M].北京:科学出版社,1993.PAO Yi-xing,MAO Zhao-yu.Scattering and Dynamics Stresses Concentration of Elastic Wave[M].bei Jing:The Science Press,1993.
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