排水状态下饱和土中基桩动力特性分析
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摘要
饱和砂土在地震作用下会喷砂冒水,一些饱和土中打桩也会冒水,但目前都无法计算排水。根据Biot弹性两相饱和介质控制方程的固相与流相Green函数频域解析解,通过Sommerfeld积分,化为轴对称坐标下的形式,结合基桩的竖向振动方程与排水状态的Lamb积分公式,计算得到桩顶作用集中力时,符合Darcy定律的排水状态下饱和土中圆柱基桩的动刚度、饱和土孔压、排水量等重要参数,讨论了诸如桩径比等参数在排水状态下对桩的动力特性的影响。结论可供饱和土中桩基的动力响应与桩土相互作用的动力特性研究借鉴。
Sand liquefaction and water drainage will occur at a saturated sandy soil site under earthquakes.Similar phenomena will also happen during pile driving.However,the drainage can not be calculated so far.Based on the solutions to the Green’s functions of the solid and fluid phases in frequency domain pertaining to the Biot's governing differential equations of a two-phase saturated medium,the Sommerfeld's integral and the transformation of axisymmetrical coordinate,dynamic equations and Lamb’s integral are combined to calculate the forces acting on a single pile driven under drainage.According to the Darcy’s law,the corresponding parameters such as dynamic stiffness of a cylinder pile,pore pressure and flux in the saturated soil are calculated when the pile top is subjected to concentrated forces.The effects of some parameters such as pile slenderness on the dynamic properties of piles are revealed.The results may benefit further researches on the dynamic response of piles in saturated soil and the dynamic properties of pile-soil interaction.
Sand liquefaction and water drainage will occur at a saturated sandy soil site under earthquakes.Similar phenomena will also happen during pile driving.However,the drainage can not be calculated so far.Based on the solutions to the Green’s functions of the solid and fluid phases in frequency domain pertaining to the Biot's governing differential equations of a two-phase saturated medium,the Sommerfeld's integral and the transformation of axisymmetrical coordinate,dynamic equations and Lamb’s integral are combined to calculate the forces acting on a single pile driven under drainage.According to the Darcy’s law,the corresponding parameters such as dynamic stiffness of a cylinder pile,pore pressure and flux in the saturated soil are calculated when the pile top is subjected to concentrated forces.The effects of some parameters such as pile slenderness on the dynamic properties of piles are revealed.The results may benefit further researches on the dynamic response of piles in saturated soil and the dynamic properties of pile-soil interaction.
引文
[1]TAJIMI H.Dynamic analysis of a structure embedded in anelastic stratum[C]//Proceeding of the 4th World Conferencein Earthquake Engineering,Santiago,Chile.1969:53–69.
[2]MATTES N S,POULOS H G.Settlement of singlecompressible pile[J].Journal of the Soil Mechanics andFoundations Division,ASCE,1900,95(1):189–208.
[3]POULOS H G.Behavior of laterally loaded piles:II.Pilegroups[J].Journal of the Soil Mechanics and FoundationsDivision,ASCE,1971,97(5):733–751.
[4]NOVAK M.Vertical vibration of floating piles[J].Journal ofEngineering Mechanics Division,ASCE,1977,103(1):153–168.
[5]NOVAK M.Piles under dynamic loads[C]//Proc 2nd Int Confon Recent Advances in Geotechnical Earthquake Engineeringand Soil Dynamics,St Louis,Missouri.1991,22(3):250–273.
[6]KAVVADS M,GAZETAS G.Kinematic seismic response andbending of free-head piles in layered soil[J].Géotechnique,1993,43(2):207–222.
[7]MATLOCK H.Correlations for design of laterally loaded pilesin soft clay[C]//Proc 2nd Annual Offshore TechnologyConference,Houston,Texas.1970,OTC 1204:577–594.
[8]REESE L C,COX W R,KOOP ED.Field testing and analysisof laterally loaded piles in stiff clay[C]//Proc 7th AnnualOffshore Technology Conference,Houston,Texas.1975.OTC 2312:671–690.
[9]王惠初,武冬青,田平.黏土中横向静载桩P–Y曲线的一种新的统一法[J].河海大学学报,1991,19(1):9–17.(WANG Hui-chu,WU Dong-qing,TIAN Ping.A new unitedmethod of P–Y curves for laterally statically loaded Piles inclay[J].Journal of Hohai University,1991,19(1):9–17.(inChinese))
[10]程泽坤.关于P–Y曲线法分析横向力作用下高桩结构物的讨论[J].港工技术,1997,3:53–57.(CHENG Ze-kun.Discussion on P–Y curve method for high pile structuresunder lateral loads[J].Port Engineering Technology,1997,3:53–57.(in Chinese))
[11]杨国平,张志明.对大应变条件下横向受力桩P–Y曲线的研究[J].水运工程,2002(7):40–45.(YANG Guo-ping,ZHANG Zhi-ming.Researches on P–Y curve of lateralbearing piles under large displacement[J].Port&WaterwayEngineering,2002(7):40–45.(in Chinese))
[12]ISMAEL N.Behavior of laterally loaded bored piles incemented sands[J].Journal of Geotechnical Engineering,1990,116(11):1678–1699.
[13]OKAHARA M.Horizontal loading tests of group piles and itsstimulation analysis[J].Journal of Structural Engineering,JSCE,1993,39:1371–1384.(in Japanese)
[14]ROLLINS K M,PETERSON K T,WEAVER T J.Lateralload behavior of full-scale pile group in clay[J].Journal ofGeotechnical and Geoenvironmental Engineering,ASCE,1998,124(6):468–478.
[15]KUHLEMEYER R L.Static and dynamic laterally loadedfloating piles[J].Journal of Geotechnical EngineeringDivision,ASCE,1979,105(2):289–304.
[16]ROESSET J M.Dynamic stiffness of pile groups[C]//Analysis and Design of Pile Foundations,ASCE,New York.1984:263–286.
[17]RAJAPAKSE R K N D.A note on the elastodynamic loadtransfer problem[J].International Journal of Solids andStructures,1988,24(9):963–972.
[18]RAJAPAKSE R K N D,SHAH A H.On the lateral harmonicmotion of an elastic bar embedded in an elastic half-space[J].International Journal of Solids and Structures,1987,23(2):287–303.
[19]SEN R,DAVIES T G,BANERJEE P K.Dynamic analysis ofpiles and pile groups embedded in homogeneous soils[J].Earthquake Engineering and Structural Dynamics,1985,13(1):53–65.
[20]MAMOON S M,BANERJEE P K.Time domain analysis ofdynamically loaded single piles[J].Journal of EngineeringMechanics,ASCE,1992,118(1):140–160.
[21]LIU W M,NOVAK M.Dynamic response of single pilesembedded in transversely isotropic layered media[J].Earthquake Engineering Structural Dynamics,1994,23(11):1239–1257.
[22]胡昌斌,王奎华,谢康和.考虑桩土相互作用效应的桩顶纵向振动时域响应分析[J].计算力学学报,2004,21(4):392–399.(HU Chang-bin,WANG Kui-hua,XIE Kang-he.Time domain analysis of vertical dynamic response of a pileconsidering the effect of pile-soil interaction[J].ChineseJournal of Computational Mechanics,2004,21(4):392–399.(in Chinese))
[23]程昌均,胡育佳.受轴力作用的桩基在水平振动时的抗力[J].上海大学学报(自然科学版),2005,11(3):275–281.(CHENG Chang-jun,HU Yu-jia.Resistance of pile againsthorizontal vibration under axial force[J].Journal of ShanghaiUniversity(Natural Science),2005,11(3):275–281.(inChinese))
[24]胡育佳,程昌钧,杨骁.水平振动下桩基的非线性动力学特性[J].应用数学和力学,2005,26(6):645–652.(HUYu-jia,CHENG Chang-jun,YANG Xiao.Nonlineardynamical characteristics of piles under horizontalvibration[J].Applied Mathematics Mechanics,2005,26(6):645–652.(in Chinese))
[25]ELLISON R D,D’APPOIONIA E,THIERS G R.Load-deformation mechanism for bored Piles[J].Journal of SoilMechanics and Foundation Division,1971,97(4):661–678.
[26]FARUQUE M O,DESAI C S.3-D material and geometricnonlinear analysis of piles[C]//Proc 2nd InternationalConference on Numerical Methods in Offshore Piling,University of Texas at Austin,Texas.1982:553–575.
[27]SHAW R P,MAKRIS N.Green’functions for Helmholtz andLaplace equations in heterogeneous media[J].EngineeringAnalysis with Boundary Elements,1992,10(2):179–183.
[28]LIYANAPATHIRANA D S,POULOS H G.Pseudostaticapproach for seismic analysis of piles in liquefying soil[J].Journal Geotechnical Geoenvironmental Engineering,ASCE,2005,131(12):1480–1487.
[29]ZENG X,RAJAPAKSE R K N D.Dynamic axial loadtransfer from elastic bar to poroelastic medium[J].Journal ofEngineering Mechanics,ASCE,1999,125(9):1048–1055.
[30]JIN B,ZHOU D,ZHONG Z.Lateral dynamic compliance ofpile embedded in poroelastic half space[J].Soil Dynamicsand Earthquake Engineering,2001,21(6):519–525.
[31]陆建飞.频域内半空间饱和土中水平受荷桩的动力分析[J].岩石力学与工程学报,2002,21(4):577–581.(LUJian-fei.Dynamic frequency domain analysis on laterallyloaded pile in saturated soil[J].Chinese Journal of RockMechanics and Engineering,2002,21(4):577–581.(inChinese))
[32]GAO S W,WANG J H,ZHOU X L.Solution of a rigid diskon saturated soil considering consolidation and rheology[J].Journal of Zhejiang University Science,2005,6A(3):222–228.
[33]李强,王奎华,谢康和.饱和土中端承桩纵向振动特性研究[J].力学学报,2004,36(4):435–442.(LI Qiang,WANG Kui-hua,XIE Kang-he,Vertical vibration of an endbearing pile embedded in saturated soil[J].Acta MechnicaSinica,2004,36(4):435–442.(in Chinese))
[34]BUTTERFIELD R,BANERJEE P K The elastic analysis ofcompressible pile and pile groups[J].Géotechnique,1971,21(1):43–60.
[35]CHAPEL F,TSAKALIDIS C.Computation of the Greenfunction of elastodynamic for a layered half-space through aHankel transformation:applications to foundation vibrationand seismology[M]//Rotterdam:Balkema.NumericalMethods in Geomechanics.1985:1311–1318.
[36]BESKOS D E.Boundary element methods in dynamicanalysis:Part II(1986-1996)[J].Applied Mechanics Reviews,1997,50:149–197.
[37]DOMINGUEZ J.Boundary element approach for dynamicporoelastic problems[J].International Journal for NumericalMethods in Engineering,1992,35(2):307–324.
[38]丁伯阳,袁金华,潘晓东.饱和土动力学问题Green函数计算的抽象集成与OOP实现[J].中国科学G辑,2009,39(2):284–292.(DING Bo-yang,YUAN Jin-hua,PANXiao-dong.Saturated soil dynamics calculation of Greenfunctions of the abstract to achieve integrated and OOP[J].Sci China Ser G:Phys Mech Astron,2009,39(2):284–292.(in Chinese))
[39]张建民.地震液化后地基侧向变形对桩基础的影响[C]//第八届土力学及岩土工程学术会议论文集,北京:万国学术出版社,1999:577–580.(ZHANG Jian-min.Effect ofseismic liquefaction-induced lateral deformation on pilefoundation[C]//Proc 8 th Academic Conference on SoilMechanics and Geotechnical Engineering,Beijing:International Academic Publishers,1999:577–580.(inChinese))
[40]CUBRINOVSKI M,KOKUSHO T.ISHIHARA K.Interpretation from large-scale shake table tests on pilessubjected to spreading of liquefied soils[C]//11thInternational Conference on Soil Dynamics and EarthquakeEngineering,Berkeley.2004.
[41]HARADA N,TOWHATA I,TSUNODA S,et al.Development of new drain method for protection of existingpile foundation from liquefaction effects[C]//Proc 11thInternational Conference on Soil Dynamics and EarthquakeEngineering and 3rd International Conference on EarthquakeGeotechnical Engineering,Berkeley,2004:498–505.
[42]CHENG A H D,BADMUS T,BESKOS D E,Integralequation for dynamic poroelasticity in frequency domain withBEM solution[J].1991,Journal of Engineering Mechanics,ASCE,117(5):1136–1157.
[43]DING B Y,YUAN J H.Dynamic Green’s functions of atwo-phase saturated medium subjected to concentratedforce[J].International Journal of Solid Structures,2011,48:2288–2303.
[44]丁伯阳,党改红,袁金华.伴有排水的两相饱和介质动力问题的Lamb积分公式[J].应用数学和力学,2010,31(9):1066–1074.(DING Bo-yang,DANG Gai-hong,YUANJin-hua.Lamb’s integral formulas of two-phase saturatedmedium for soil dynamic problems with drainage[J].AppliedMathematics and Mechanics,2010,31(9):1066–1074.(inChinese))
[45]BIOT M A.Mechanics of deformation and acousticpropagation in porous media[J].Journal of Applied Physics,1962,33(4):1482–1498.
[46]丁伯阳,宋新初,袁金华.关于两相饱和介质中流相Green函数的解析解[J].地球物理学报,2009,52(7):1858–1866.(DING Bo-yang,SONG Xin-chu,YUAN Jin-hua.The solution of Green function on fluid phase in two-phasesaturated medium[J].Chinese Journal of Geophysics,2009,52(7):1858–1866.(in Chinese))
[47]丁伯阳,陈军,潘晓东.饱和多孔介质中Lamb问题的Green函数解答[J].力学学报,2011,43(3):533–541.(DING Bo-yang,CHEN Jun,PAN Xiao-dong.The solutionsof Green's function for Lamb’s problem in a two-phasesaturated medium[J].Chinese Journal of Theoretical andApplied Mechanics,2011,43(3):533–541.(in Chinese))
[48]DING B Y,CHEN J.Solutions of Green’s function forLamb’s problem of a two-phase saturated medium[J].Theoretical&Applied Mechanics Letters,2011,1:052003.
[2]MATTES N S,POULOS H G.Settlement of singlecompressible pile[J].Journal of the Soil Mechanics andFoundations Division,ASCE,1900,95(1):189–208.
[3]POULOS H G.Behavior of laterally loaded piles:II.Pilegroups[J].Journal of the Soil Mechanics and FoundationsDivision,ASCE,1971,97(5):733–751.
[4]NOVAK M.Vertical vibration of floating piles[J].Journal ofEngineering Mechanics Division,ASCE,1977,103(1):153–168.
[5]NOVAK M.Piles under dynamic loads[C]//Proc 2nd Int Confon Recent Advances in Geotechnical Earthquake Engineeringand Soil Dynamics,St Louis,Missouri.1991,22(3):250–273.
[6]KAVVADS M,GAZETAS G.Kinematic seismic response andbending of free-head piles in layered soil[J].Géotechnique,1993,43(2):207–222.
[7]MATLOCK H.Correlations for design of laterally loaded pilesin soft clay[C]//Proc 2nd Annual Offshore TechnologyConference,Houston,Texas.1970,OTC 1204:577–594.
[8]REESE L C,COX W R,KOOP ED.Field testing and analysisof laterally loaded piles in stiff clay[C]//Proc 7th AnnualOffshore Technology Conference,Houston,Texas.1975.OTC 2312:671–690.
[9]王惠初,武冬青,田平.黏土中横向静载桩P–Y曲线的一种新的统一法[J].河海大学学报,1991,19(1):9–17.(WANG Hui-chu,WU Dong-qing,TIAN Ping.A new unitedmethod of P–Y curves for laterally statically loaded Piles inclay[J].Journal of Hohai University,1991,19(1):9–17.(inChinese))
[10]程泽坤.关于P–Y曲线法分析横向力作用下高桩结构物的讨论[J].港工技术,1997,3:53–57.(CHENG Ze-kun.Discussion on P–Y curve method for high pile structuresunder lateral loads[J].Port Engineering Technology,1997,3:53–57.(in Chinese))
[11]杨国平,张志明.对大应变条件下横向受力桩P–Y曲线的研究[J].水运工程,2002(7):40–45.(YANG Guo-ping,ZHANG Zhi-ming.Researches on P–Y curve of lateralbearing piles under large displacement[J].Port&WaterwayEngineering,2002(7):40–45.(in Chinese))
[12]ISMAEL N.Behavior of laterally loaded bored piles incemented sands[J].Journal of Geotechnical Engineering,1990,116(11):1678–1699.
[13]OKAHARA M.Horizontal loading tests of group piles and itsstimulation analysis[J].Journal of Structural Engineering,JSCE,1993,39:1371–1384.(in Japanese)
[14]ROLLINS K M,PETERSON K T,WEAVER T J.Lateralload behavior of full-scale pile group in clay[J].Journal ofGeotechnical and Geoenvironmental Engineering,ASCE,1998,124(6):468–478.
[15]KUHLEMEYER R L.Static and dynamic laterally loadedfloating piles[J].Journal of Geotechnical EngineeringDivision,ASCE,1979,105(2):289–304.
[16]ROESSET J M.Dynamic stiffness of pile groups[C]//Analysis and Design of Pile Foundations,ASCE,New York.1984:263–286.
[17]RAJAPAKSE R K N D.A note on the elastodynamic loadtransfer problem[J].International Journal of Solids andStructures,1988,24(9):963–972.
[18]RAJAPAKSE R K N D,SHAH A H.On the lateral harmonicmotion of an elastic bar embedded in an elastic half-space[J].International Journal of Solids and Structures,1987,23(2):287–303.
[19]SEN R,DAVIES T G,BANERJEE P K.Dynamic analysis ofpiles and pile groups embedded in homogeneous soils[J].Earthquake Engineering and Structural Dynamics,1985,13(1):53–65.
[20]MAMOON S M,BANERJEE P K.Time domain analysis ofdynamically loaded single piles[J].Journal of EngineeringMechanics,ASCE,1992,118(1):140–160.
[21]LIU W M,NOVAK M.Dynamic response of single pilesembedded in transversely isotropic layered media[J].Earthquake Engineering Structural Dynamics,1994,23(11):1239–1257.
[22]胡昌斌,王奎华,谢康和.考虑桩土相互作用效应的桩顶纵向振动时域响应分析[J].计算力学学报,2004,21(4):392–399.(HU Chang-bin,WANG Kui-hua,XIE Kang-he.Time domain analysis of vertical dynamic response of a pileconsidering the effect of pile-soil interaction[J].ChineseJournal of Computational Mechanics,2004,21(4):392–399.(in Chinese))
[23]程昌均,胡育佳.受轴力作用的桩基在水平振动时的抗力[J].上海大学学报(自然科学版),2005,11(3):275–281.(CHENG Chang-jun,HU Yu-jia.Resistance of pile againsthorizontal vibration under axial force[J].Journal of ShanghaiUniversity(Natural Science),2005,11(3):275–281.(inChinese))
[24]胡育佳,程昌钧,杨骁.水平振动下桩基的非线性动力学特性[J].应用数学和力学,2005,26(6):645–652.(HUYu-jia,CHENG Chang-jun,YANG Xiao.Nonlineardynamical characteristics of piles under horizontalvibration[J].Applied Mathematics Mechanics,2005,26(6):645–652.(in Chinese))
[25]ELLISON R D,D’APPOIONIA E,THIERS G R.Load-deformation mechanism for bored Piles[J].Journal of SoilMechanics and Foundation Division,1971,97(4):661–678.
[26]FARUQUE M O,DESAI C S.3-D material and geometricnonlinear analysis of piles[C]//Proc 2nd InternationalConference on Numerical Methods in Offshore Piling,University of Texas at Austin,Texas.1982:553–575.
[27]SHAW R P,MAKRIS N.Green’functions for Helmholtz andLaplace equations in heterogeneous media[J].EngineeringAnalysis with Boundary Elements,1992,10(2):179–183.
[28]LIYANAPATHIRANA D S,POULOS H G.Pseudostaticapproach for seismic analysis of piles in liquefying soil[J].Journal Geotechnical Geoenvironmental Engineering,ASCE,2005,131(12):1480–1487.
[29]ZENG X,RAJAPAKSE R K N D.Dynamic axial loadtransfer from elastic bar to poroelastic medium[J].Journal ofEngineering Mechanics,ASCE,1999,125(9):1048–1055.
[30]JIN B,ZHOU D,ZHONG Z.Lateral dynamic compliance ofpile embedded in poroelastic half space[J].Soil Dynamicsand Earthquake Engineering,2001,21(6):519–525.
[31]陆建飞.频域内半空间饱和土中水平受荷桩的动力分析[J].岩石力学与工程学报,2002,21(4):577–581.(LUJian-fei.Dynamic frequency domain analysis on laterallyloaded pile in saturated soil[J].Chinese Journal of RockMechanics and Engineering,2002,21(4):577–581.(inChinese))
[32]GAO S W,WANG J H,ZHOU X L.Solution of a rigid diskon saturated soil considering consolidation and rheology[J].Journal of Zhejiang University Science,2005,6A(3):222–228.
[33]李强,王奎华,谢康和.饱和土中端承桩纵向振动特性研究[J].力学学报,2004,36(4):435–442.(LI Qiang,WANG Kui-hua,XIE Kang-he,Vertical vibration of an endbearing pile embedded in saturated soil[J].Acta MechnicaSinica,2004,36(4):435–442.(in Chinese))
[34]BUTTERFIELD R,BANERJEE P K The elastic analysis ofcompressible pile and pile groups[J].Géotechnique,1971,21(1):43–60.
[35]CHAPEL F,TSAKALIDIS C.Computation of the Greenfunction of elastodynamic for a layered half-space through aHankel transformation:applications to foundation vibrationand seismology[M]//Rotterdam:Balkema.NumericalMethods in Geomechanics.1985:1311–1318.
[36]BESKOS D E.Boundary element methods in dynamicanalysis:Part II(1986-1996)[J].Applied Mechanics Reviews,1997,50:149–197.
[37]DOMINGUEZ J.Boundary element approach for dynamicporoelastic problems[J].International Journal for NumericalMethods in Engineering,1992,35(2):307–324.
[38]丁伯阳,袁金华,潘晓东.饱和土动力学问题Green函数计算的抽象集成与OOP实现[J].中国科学G辑,2009,39(2):284–292.(DING Bo-yang,YUAN Jin-hua,PANXiao-dong.Saturated soil dynamics calculation of Greenfunctions of the abstract to achieve integrated and OOP[J].Sci China Ser G:Phys Mech Astron,2009,39(2):284–292.(in Chinese))
[39]张建民.地震液化后地基侧向变形对桩基础的影响[C]//第八届土力学及岩土工程学术会议论文集,北京:万国学术出版社,1999:577–580.(ZHANG Jian-min.Effect ofseismic liquefaction-induced lateral deformation on pilefoundation[C]//Proc 8 th Academic Conference on SoilMechanics and Geotechnical Engineering,Beijing:International Academic Publishers,1999:577–580.(inChinese))
[40]CUBRINOVSKI M,KOKUSHO T.ISHIHARA K.Interpretation from large-scale shake table tests on pilessubjected to spreading of liquefied soils[C]//11thInternational Conference on Soil Dynamics and EarthquakeEngineering,Berkeley.2004.
[41]HARADA N,TOWHATA I,TSUNODA S,et al.Development of new drain method for protection of existingpile foundation from liquefaction effects[C]//Proc 11thInternational Conference on Soil Dynamics and EarthquakeEngineering and 3rd International Conference on EarthquakeGeotechnical Engineering,Berkeley,2004:498–505.
[42]CHENG A H D,BADMUS T,BESKOS D E,Integralequation for dynamic poroelasticity in frequency domain withBEM solution[J].1991,Journal of Engineering Mechanics,ASCE,117(5):1136–1157.
[43]DING B Y,YUAN J H.Dynamic Green’s functions of atwo-phase saturated medium subjected to concentratedforce[J].International Journal of Solid Structures,2011,48:2288–2303.
[44]丁伯阳,党改红,袁金华.伴有排水的两相饱和介质动力问题的Lamb积分公式[J].应用数学和力学,2010,31(9):1066–1074.(DING Bo-yang,DANG Gai-hong,YUANJin-hua.Lamb’s integral formulas of two-phase saturatedmedium for soil dynamic problems with drainage[J].AppliedMathematics and Mechanics,2010,31(9):1066–1074.(inChinese))
[45]BIOT M A.Mechanics of deformation and acousticpropagation in porous media[J].Journal of Applied Physics,1962,33(4):1482–1498.
[46]丁伯阳,宋新初,袁金华.关于两相饱和介质中流相Green函数的解析解[J].地球物理学报,2009,52(7):1858–1866.(DING Bo-yang,SONG Xin-chu,YUAN Jin-hua.The solution of Green function on fluid phase in two-phasesaturated medium[J].Chinese Journal of Geophysics,2009,52(7):1858–1866.(in Chinese))
[47]丁伯阳,陈军,潘晓东.饱和多孔介质中Lamb问题的Green函数解答[J].力学学报,2011,43(3):533–541.(DING Bo-yang,CHEN Jun,PAN Xiao-dong.The solutionsof Green's function for Lamb’s problem in a two-phasesaturated medium[J].Chinese Journal of Theoretical andApplied Mechanics,2011,43(3):533–541.(in Chinese))
[48]DING B Y,CHEN J.Solutions of Green’s function forLamb’s problem of a two-phase saturated medium[J].Theoretical&Applied Mechanics Letters,2011,1:052003.
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