饱和土中深埋圆柱形衬砌洞室对瞬态平面波的散射
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摘要
基于Biot饱和多孔介质动力学理论,运用Laplace变换和波函数展开法,根据饱和土体与衬砌结构交界面的连续条件和衬砌结构内边界上的应力自由条件,得到饱和土中深埋圆柱形衬砌洞室对瞬态平面P波和SV波散射问题的解答,该解答可以退化成为饱和土中深埋圆柱形空穴或弹性夹塞物的情形,并很容易转换成为对稳态波散射的解.通过与已有的相关问题的解析解答进行对比,验证了该解答的正确性.同时利用Laplace逆变换的数值方法,给出了饱和土和衬砌中应力和位移场在时域内的数值解,通过算例,分析了衬砌厚度、刚度对衬砌内边界处应力集中因子的影响.
On the basis of Biot dynamic theory for saturated porous media,this work finds the analytical solutions for two-dimensional scattering and diffraction of transient plane P waves and SV waves by deeply buried cylindrical lined cavity in saturated soil.This process employs the Laplace transform and wave function expansion method according to the continuous conditions of saturated soil and lining structure interface and the stress-free condition on the surface of lining structure.The solutions can be degenerated to the case of deeply buried cylindrical hole in saturated soil or in the elastic medium and under the condition of steady wave incident.The solutions also are proofed by comparison with the existing related solutions.Numerical results are given and the effects of the lining thickness and stiffness on the stress concentration factor of the inside boundary of lining are analyzed.
On the basis of Biot dynamic theory for saturated porous media,this work finds the analytical solutions for two-dimensional scattering and diffraction of transient plane P waves and SV waves by deeply buried cylindrical lined cavity in saturated soil.This process employs the Laplace transform and wave function expansion method according to the continuous conditions of saturated soil and lining structure interface and the stress-free condition on the surface of lining structure.The solutions can be degenerated to the case of deeply buried cylindrical hole in saturated soil or in the elastic medium and under the condition of steady wave incident.The solutions also are proofed by comparison with the existing related solutions.Numerical results are given and the effects of the lining thickness and stiffness on the stress concentration factor of the inside boundary of lining are analyzed.
引文
[1]Pao Y H,Mow C C.Diffraction of Elastic Waves and Dynamic Stress Concentrations.New York:Crane and Russak,1973.
[2]徐长节,吴世明,蔡袁强等.波在饱和土中柱状空穴界面上的反射.计算力学学报,1997,14(增刊):291-294.Xu C J,Wu S M,Cai Y Q,et.al.Refracting of plane waves by cylindrical cavity in saturated soil.Chinese Journal of Computational Mechanics(in Chinese),1997,14(Suppl.):291-294.
[3]胡亚元,王立忠,陈云敏等.饱和土中平面应变波在圆柱体上的散射和折射.地震学报,1998,20(3):300-307.Hu Y Y,Wang L Z,Chen Y M,et.al.Scattering and refracting of plane strain waves by cylinder in saturated soil.Acta Seismological Sinica(in Chinese),1998,20(3):300-307.
[4]陆建飞,王建华.饱和土中的任意形状孔洞对弹性波的散射.力学学报,2002,34(6):904-913.Lu J F,Wang J H.The scattering of elastic waves by holes of arbitrary shapes in saturated soil.Acta Mechanica Sinica(in Chinese),2002,34(6):904-913.
[5]Kattis S E,Beskos D E,Cheng A H D.2Ddynamic response of unlined and lined tunnels in poroelastic soil to harmonic body waves.Earthquake Engineering and Structural Dynamics,2003,32(1):97-110.
[6]李伟华,赵成刚.饱和土半空间中圆柱形孔洞对平面P波的散射.岩土力学,2004,25(12):1867-1872.Li W H,Zhao C G.An analytical solution for diffraction of plane P-waves by cylindrical cavity in a fluid-saturated porous media semi-space.Rock and Soil Mechanics(in Chinese),2004,25(12):1867-1872.
[7]李伟华,赵成刚.平面SV波在饱和土半空间中圆柱形孔洞周边的散射.地震工程与工程振动,2008,28(6):1-7.Li W H,Zhao C G.An analytical solution for the scattering of plane SV-waves around cylindrical cavity in a fluid-saturated porous media half space.Journal of Earthquake Engineering and Engineering Vibration(in Chinese),2008,28(6):1-7.
[8]李伟华,申齐豪,赵成刚.饱和土半空间中地下圆形衬砌洞室对平面SV波的散射.防灾减灾工程学报,2009,29(2):172-178.Li W H,Shen Q H,Zhao C G.An analytical solution for scattering and diffraction of plane-SV-waves by a cylindrical lined cavity in half-space of saturated soil.Journal of Disaster Prevention and Mitigation Engineering(in Chinese),2009,29(2):172-178.
[9]尤红兵.层状饱和土半空间中凹陷地形或隧洞对弹性波的散射[博士论文].天津:天津大学,2005.You H B.Elastic wave scattering by a canyon or tunnel in layered saturated half space[Ph.D.](in Chinese).Tianjin:Tianjin University,2005.
[10]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.岩石力学与工程学报,2005,24(9):1572-1576.Zhou X L,Zhou G M,Wang J H.Scattering of elastic wave by circular cavity with lining in saturated soil.Chinese Journal of Rock Mechanics and Engineering(in Chinese),2005,24(9):1572-1576.
[11]徐平,夏唐代,韩同春.饱和弹性土体中圆柱壳体对弹性波的散射.地震学报,2006,28(2):183-189.Xu P,Xia T D,Han T C.Scattering of elastic wave by a cylindrical shell embedded in saturated soils.Acta Seismologica Sinica(in Chinese),2006,28(2):183-189.
[12]王建华,姜领发,周香莲.稳态P波对半无限饱和土中的圆形孔洞的散射.南昌工程学院学报,2005,24(1):8-19.Wang J H,Jiang L F,Zhou X L.Scattering of steady P-wave around a circular cavity in half space saturated soil.Journal of Nanchang Institute of Technology(in Chinese),2005,24(1):8-19.
[13]Gatmiri B,Eslami H.Scattering of Harmonic waves by a circular cavity in a porous medium:Complex functions theory approac.International Journal of Geomechanics,2007,7(5):371-381.
[14]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.International Journal for Numerical and Analytical Methods in Geomechanics,2007,31(5):667-689.
[15]Hasheminejad S M,Kazemirad S.Dynamic response of an eccentrically lined circular tunnel in poroelastic soil under seismic excitation.Soil Dynamics and Earthquake Engineering,2008,28(4):277-292.
[16]丁光亚,蔡袁强,徐长节.半空间饱和土中圆形壳结构对平面P波的散射.工程力学,2008,25(12):35-41.Ding G Y,Cai Y Q,Xu Z J.Scattering of P waves by a cylindrical shell in a saturated half-space.Engineering Mechanics(in Chinese),2008,25(12):35-41.
[17]姜领发,陈善雄,于忠久.饱和土中任意形状衬砌对稳态压缩波的散射.岩土力学,2009,30(10):3063-3070.Jiang L F,Chen S X,Yu Z J.Scattering around a liner of arbitrary shape in saturated soil under dilatational waves.Rock and Soil Mechanics(in Chinese),2009,30(10):3063-3070.
[18]Schanz M.Poroelastodynamics:Linear models,analytical solutions,and numerical methods.Applied Mechanics Reviews,2009,62(3):030803,doi:10.1115/1.3090831.
[19]高盟,高广运,王滢等.饱和土与衬砌动力相互作用的圆柱形孔洞内源问题解答.固体力学学报,2009,30(5):481-488.Gao M,Gao G Y,Wang Y,et al.A solution on the internal source problem of a cylindrical cavity cosidering the dynamic interaction between lining and saturated soil.Chinese Journal of Solid Mechanics(in Chinese),2009,30(5):481-488.
[20]Biot M A.Theory of propagation of elastic waves in fluid-saturated porous solid.I.Low-frequency range.The Journal of the Acoustical Society of America,1956,28(2):168-178.
[21]Deresiewicz H,Skalak R.On uniqueness in dynamic poroelasticity.Bulletin of the Seismological Society of America,1963,53(4):783-788.
[22]Hajra S,Mukhopadhyay A.Reflection and refraction of seismic waves incident obliquely at the boundary of a liquid-saturated porous solid.Bulletin of the Seismological Society of America,1982,72(5):1509-1533.
[23]Durbin F.Numerical inversion of Laplace transforms:an efficient improvement to Dubner and Abate's method.The Computer Journal,1974,17(4):371-376.
[2]徐长节,吴世明,蔡袁强等.波在饱和土中柱状空穴界面上的反射.计算力学学报,1997,14(增刊):291-294.Xu C J,Wu S M,Cai Y Q,et.al.Refracting of plane waves by cylindrical cavity in saturated soil.Chinese Journal of Computational Mechanics(in Chinese),1997,14(Suppl.):291-294.
[3]胡亚元,王立忠,陈云敏等.饱和土中平面应变波在圆柱体上的散射和折射.地震学报,1998,20(3):300-307.Hu Y Y,Wang L Z,Chen Y M,et.al.Scattering and refracting of plane strain waves by cylinder in saturated soil.Acta Seismological Sinica(in Chinese),1998,20(3):300-307.
[4]陆建飞,王建华.饱和土中的任意形状孔洞对弹性波的散射.力学学报,2002,34(6):904-913.Lu J F,Wang J H.The scattering of elastic waves by holes of arbitrary shapes in saturated soil.Acta Mechanica Sinica(in Chinese),2002,34(6):904-913.
[5]Kattis S E,Beskos D E,Cheng A H D.2Ddynamic response of unlined and lined tunnels in poroelastic soil to harmonic body waves.Earthquake Engineering and Structural Dynamics,2003,32(1):97-110.
[6]李伟华,赵成刚.饱和土半空间中圆柱形孔洞对平面P波的散射.岩土力学,2004,25(12):1867-1872.Li W H,Zhao C G.An analytical solution for diffraction of plane P-waves by cylindrical cavity in a fluid-saturated porous media semi-space.Rock and Soil Mechanics(in Chinese),2004,25(12):1867-1872.
[7]李伟华,赵成刚.平面SV波在饱和土半空间中圆柱形孔洞周边的散射.地震工程与工程振动,2008,28(6):1-7.Li W H,Zhao C G.An analytical solution for the scattering of plane SV-waves around cylindrical cavity in a fluid-saturated porous media half space.Journal of Earthquake Engineering and Engineering Vibration(in Chinese),2008,28(6):1-7.
[8]李伟华,申齐豪,赵成刚.饱和土半空间中地下圆形衬砌洞室对平面SV波的散射.防灾减灾工程学报,2009,29(2):172-178.Li W H,Shen Q H,Zhao C G.An analytical solution for scattering and diffraction of plane-SV-waves by a cylindrical lined cavity in half-space of saturated soil.Journal of Disaster Prevention and Mitigation Engineering(in Chinese),2009,29(2):172-178.
[9]尤红兵.层状饱和土半空间中凹陷地形或隧洞对弹性波的散射[博士论文].天津:天津大学,2005.You H B.Elastic wave scattering by a canyon or tunnel in layered saturated half space[Ph.D.](in Chinese).Tianjin:Tianjin University,2005.
[10]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.岩石力学与工程学报,2005,24(9):1572-1576.Zhou X L,Zhou G M,Wang J H.Scattering of elastic wave by circular cavity with lining in saturated soil.Chinese Journal of Rock Mechanics and Engineering(in Chinese),2005,24(9):1572-1576.
[11]徐平,夏唐代,韩同春.饱和弹性土体中圆柱壳体对弹性波的散射.地震学报,2006,28(2):183-189.Xu P,Xia T D,Han T C.Scattering of elastic wave by a cylindrical shell embedded in saturated soils.Acta Seismologica Sinica(in Chinese),2006,28(2):183-189.
[12]王建华,姜领发,周香莲.稳态P波对半无限饱和土中的圆形孔洞的散射.南昌工程学院学报,2005,24(1):8-19.Wang J H,Jiang L F,Zhou X L.Scattering of steady P-wave around a circular cavity in half space saturated soil.Journal of Nanchang Institute of Technology(in Chinese),2005,24(1):8-19.
[13]Gatmiri B,Eslami H.Scattering of Harmonic waves by a circular cavity in a porous medium:Complex functions theory approac.International Journal of Geomechanics,2007,7(5):371-381.
[14]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.International Journal for Numerical and Analytical Methods in Geomechanics,2007,31(5):667-689.
[15]Hasheminejad S M,Kazemirad S.Dynamic response of an eccentrically lined circular tunnel in poroelastic soil under seismic excitation.Soil Dynamics and Earthquake Engineering,2008,28(4):277-292.
[16]丁光亚,蔡袁强,徐长节.半空间饱和土中圆形壳结构对平面P波的散射.工程力学,2008,25(12):35-41.Ding G Y,Cai Y Q,Xu Z J.Scattering of P waves by a cylindrical shell in a saturated half-space.Engineering Mechanics(in Chinese),2008,25(12):35-41.
[17]姜领发,陈善雄,于忠久.饱和土中任意形状衬砌对稳态压缩波的散射.岩土力学,2009,30(10):3063-3070.Jiang L F,Chen S X,Yu Z J.Scattering around a liner of arbitrary shape in saturated soil under dilatational waves.Rock and Soil Mechanics(in Chinese),2009,30(10):3063-3070.
[18]Schanz M.Poroelastodynamics:Linear models,analytical solutions,and numerical methods.Applied Mechanics Reviews,2009,62(3):030803,doi:10.1115/1.3090831.
[19]高盟,高广运,王滢等.饱和土与衬砌动力相互作用的圆柱形孔洞内源问题解答.固体力学学报,2009,30(5):481-488.Gao M,Gao G Y,Wang Y,et al.A solution on the internal source problem of a cylindrical cavity cosidering the dynamic interaction between lining and saturated soil.Chinese Journal of Solid Mechanics(in Chinese),2009,30(5):481-488.
[20]Biot M A.Theory of propagation of elastic waves in fluid-saturated porous solid.I.Low-frequency range.The Journal of the Acoustical Society of America,1956,28(2):168-178.
[21]Deresiewicz H,Skalak R.On uniqueness in dynamic poroelasticity.Bulletin of the Seismological Society of America,1963,53(4):783-788.
[22]Hajra S,Mukhopadhyay A.Reflection and refraction of seismic waves incident obliquely at the boundary of a liquid-saturated porous solid.Bulletin of the Seismological Society of America,1982,72(5):1509-1533.
[23]Durbin F.Numerical inversion of Laplace transforms:an efficient improvement to Dubner and Abate's method.The Computer Journal,1974,17(4):371-376.
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