竖向力下圆形隧道振动反应的频域Green函数解答
摘要
饱和土体的隧道在竖向荷载作用下的隧道底面振动位移反应关系着饱和砂土的震陷和液化等。本文利用前人研究的基础上,通过矩阵转换、Manolis(1983a)积分方法、Bessel以及Hankel函数的性质,得到了二维频域Green函数,当锐化成单项介质时与Doyle(1966)以及Cruse和Rizzo(1968)的基本解一致;接着前人得到的三维Lamb积分方程,进而利用Manolis方法得出二维Lamb积分方程。以饱和土中圆形隧道为例,将得到的Green函数代入二维Lamb方程,得到机动车在竖向力振动引起的圆形隧道振动位移图象,文中的计算结果,对于以后研究饱和多孔介质动力学流相问题提供基础。
引文
[1]Biot,M,A.Theory of elastic waves in a fluid-saturated porous solid,Ⅰ.Low frequeney range[J].Acoust,Soc.Am.1956,28:168-178.
[2]Biot,M,A.Theoryofelasticwavesinafluid-saturatedporoussolid,Ⅱ.Highfrequeneyrange[J].Acoust,Soc.Am.1956,28:168-178.
[3]Biot,M,A.Mechanics of deformation and acoustic propagation in porous dispative media[J].Acoust.Soc.Am.1962,34:1482-1498.
[4]Biot,M,A.Generalized theroy of acoustic propagation in porous dissipative media[J].Acoust.Soc.Am.1962,34:1254-1264.
[5]Burridge,R.and Vargas,C.A.,The Fundamental Solution in DynamicPoroelasticity,Geophys[J].J.R.Astr.Soc,1979,58:61-90.
[6]Norris AN.Radiation from a point source and scattering theory in a fluid-saturated porous solid[J].Acoust SocAm,1985,77(6):2012-2023.
[7]Philippaopoulos AJ.Lamb's problem for fhud-saturated porous media[J].Bull Seism Soc Am,1988,78(2):908923.
[8]Philippaopoulos AJ.Wave in partially saturated medium due to surface loads[J].Engrg Mech,ASCE,1988,114(10):1740-1759.
[9]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.
[10]Manoils G,Beskos DE.Dynamic response of lined tunnels by an isoparametric boundary element method.Comp.Meth.Appl.Mech.Engng,1983,36:291-307.
[11]DoyleTM.Integration of the Laplace Trasformed Equation of classical Elastokinetics[J].MathAppl,1966.
[12]Cruse T A,Rizzo F J.A direct formulation and numerical solution of the general elastodynamic problem[J].Journal of Mathmatical Analysis and Applications,1968,22(1):244-259.(下转第243页)
[13]丁伯阳,宋新初,袁金华.饱和土隧道内集中荷载作用下振动位移反应的Green函数解答[J].工程力学,2009,30(3):153-157.
[14]刘银斌,李幼铭,吴如山.横向各向同性多孔介质中的地震波传播[J].地球物理学报,1994,37(4):499-514.
[15]Chen J.Time Domain fundamental solution to Boit’s complete equations of dynamic poroelasticity PartⅠ[J].Int.J.Solid Struct,1994,31(10):1447-1490.
[16]Chen J.Time Domain fundamental solution to Boit’s complete equations of dynamic poroelasticity PartⅡ[J].Int.J.Solid Struct,1994,31(2):169～202.
[17]Bonnet G.Basic singular solutions for a poroelastic medium in the dynamic range[J].J.Acoust.Soc.Amer.November,1987,82:1758-1762.
[18]丁伯阳,丁翠红等.集中力作用下两相饱和介质二维位移场Green函数[J].应用数学和力学,2004,25(8):869-874.
[19]Gradshteyn,I.S.,Ryzhik,I.M.,1980.Table of Integrals,Series and Products.Academic,Orlando.
[20]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].应用数学与力学,2010,31(9):1065-1074.
[2]Biot,M,A.Theoryofelasticwavesinafluid-saturatedporoussolid,Ⅱ.Highfrequeneyrange[J].Acoust,Soc.Am.1956,28:168-178.
[3]Biot,M,A.Mechanics of deformation and acoustic propagation in porous dispative media[J].Acoust.Soc.Am.1962,34:1482-1498.
[4]Biot,M,A.Generalized theroy of acoustic propagation in porous dissipative media[J].Acoust.Soc.Am.1962,34:1254-1264.
[5]Burridge,R.and Vargas,C.A.,The Fundamental Solution in DynamicPoroelasticity,Geophys[J].J.R.Astr.Soc,1979,58:61-90.
[6]Norris AN.Radiation from a point source and scattering theory in a fluid-saturated porous solid[J].Acoust SocAm,1985,77(6):2012-2023.
[7]Philippaopoulos AJ.Lamb's problem for fhud-saturated porous media[J].Bull Seism Soc Am,1988,78(2):908923.
[8]Philippaopoulos AJ.Wave in partially saturated medium due to surface loads[J].Engrg Mech,ASCE,1988,114(10):1740-1759.
[9]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.
[10]Manoils G,Beskos DE.Dynamic response of lined tunnels by an isoparametric boundary element method.Comp.Meth.Appl.Mech.Engng,1983,36:291-307.
[11]DoyleTM.Integration of the Laplace Trasformed Equation of classical Elastokinetics[J].MathAppl,1966.
[12]Cruse T A,Rizzo F J.A direct formulation and numerical solution of the general elastodynamic problem[J].Journal of Mathmatical Analysis and Applications,1968,22(1):244-259.(下转第243页)
[13]丁伯阳,宋新初,袁金华.饱和土隧道内集中荷载作用下振动位移反应的Green函数解答[J].工程力学,2009,30(3):153-157.
[14]刘银斌,李幼铭,吴如山.横向各向同性多孔介质中的地震波传播[J].地球物理学报,1994,37(4):499-514.
[15]Chen J.Time Domain fundamental solution to Boit’s complete equations of dynamic poroelasticity PartⅠ[J].Int.J.Solid Struct,1994,31(10):1447-1490.
[16]Chen J.Time Domain fundamental solution to Boit’s complete equations of dynamic poroelasticity PartⅡ[J].Int.J.Solid Struct,1994,31(2):169～202.
[17]Bonnet G.Basic singular solutions for a poroelastic medium in the dynamic range[J].J.Acoust.Soc.Amer.November,1987,82:1758-1762.
[18]丁伯阳,丁翠红等.集中力作用下两相饱和介质二维位移场Green函数[J].应用数学和力学,2004,25(8):869-874.
[19]Gradshteyn,I.S.,Ryzhik,I.M.,1980.Table of Integrals,Series and Products.Academic,Orlando.
[20]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].应用数学与力学,2010,31(9):1065-1074.
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