稳态P波对半无限饱和土中的圆形孔洞的散射
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摘要
借助在非耗散情况下的Biot波动理论,采用复变函数及多级坐标的方法对半无限饱和土中P波在一个圆形空洞周围的散射和动应力集中的问题提出了一种近似求解分析方法.具体做法是利用一个半径很大的圆来逼近半空间的直边界,将半空间直边界问题转化为曲面边界问题.借助Helmholtz定理预先写出问题波函数的一般形式解,再利用边界条件并借助复数傅立叶级数展开把问题化为求解波函数中未知系数的无穷线性代数方程组,通过变换不同的条件组合,得出半空间的圆形孔洞周围的动应力及孔压集中系数的数值解的分布和变化情况.由算例可知:该方法对研究与P波有关的散射问题是可行的.
In terms of Biot's dynamic theory, the multi-polar coordinate and complex function are used to put forward an approximate analysis method for scattering and dynamic stress concentration of steady P-wave around a circular cavity in half space saturated soil . The circular cavity with large radius is used to replace the straight boundary of the half space of saturated soil. Hence the problem in this paper will be translated into the scattering and dynamic stress concentration of steady P-wave by two cavities. By using the theory of Helmholtz , the general solution of the Biot's wave function is given and a system of infinite linear algebraic equations of the problem studied in this paper can be given by means of the complex series expansion technology and the boundary conditions of the solid matrix and the fluid. Then the variation of the coefficient of dynamic stress concentration on boundaries of the cavity is discussed with different parameter conditions .The results of the given example indicate that the method presented in this paper is useful and efficient to the scattering and dynamic stress concentration of P- wave in which the straight boundary exists.
In terms of Biot's dynamic theory, the multi-polar coordinate and complex function are used to put forward an approximate analysis method for scattering and dynamic stress concentration of steady P-wave around a circular cavity in half space saturated soil . The circular cavity with large radius is used to replace the straight boundary of the half space of saturated soil. Hence the problem in this paper will be translated into the scattering and dynamic stress concentration of steady P-wave by two cavities. By using the theory of Helmholtz , the general solution of the Biot's wave function is given and a system of infinite linear algebraic equations of the problem studied in this paper can be given by means of the complex series expansion technology and the boundary conditions of the solid matrix and the fluid. Then the variation of the coefficient of dynamic stress concentration on boundaries of the cavity is discussed with different parameter conditions .The results of the given example indicate that the method presented in this paper is useful and efficient to the scattering and dynamic stress concentration of P- wave in which the straight boundary exists.
引文
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[13]陆建飞,王建华.饱和土中的任意形状空洞对弹性波的散射[J].力学学报,2002,34(6):904-913.
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[15]HCAO,VWLEE.ScatteringandDiffractiontoP-WavebyCirlindricalCanyonswithVarisbleDepth-to-widthRatio[J].SoilDynamicsandEarthquakeEngineering,1990,9(3):141-150.
[16]Biot,MA.GeneralTheoryofthreeDimensionalConsolidation[J].J.Appl.Phys,1941,(12):155-164.
[17]Biot,MA.TheoryofPropagationofElasticWavesinaFluid SaturatedPorousSolid,I,lowFrenquecyrang[J].J.Acoust.Soc.Am,1956,28(2),168-178.
[18]Biot,MA.MechanicsofDeformationandAcousticPorousMedia[J].J.Appl.Phy,1962,33(4):1482-1498.
[19]梁建文,严林隽,VincentWlee.圆弧形层状沉积谷地对入射平面P波的散射解析解[J].地震学报,2001,23(2):167-184.
[2]SavinG.孔附近的应力集中[M].北京:科学出版社,1965.
[3]鲍亦兴,毛昭宙.弹性波衍射与动应力集中[M].北京:科学出版社,1993.
[4]PAOYih Hsing,MOWChao chow.DiffractionofElasticWavesandDynamicStressConcentrations[M].Crane,Russak＆CompanyInc.inUS,1973.
[5]DattaSK,WongKC,ShahAH.DynamicStressandDisplacementAroundCylindricalCavitiesofArbitraryShapes[J].ASMEJAppl.Mech,1984.51:798-803.
[6]ManolisGD,BeskosDE.BoundaryElementsMethodsinElastodynamics[M].UnwinHyman,London,1988.
[7]PaoYH,Varathatajuluv.PrincipleConditionandIntegralFormulasfortheScatteringofElasticWaves[J].JAcoust.Soc.Am.,1976,59:1361-1371.
[8]GautesenAK,VchenbachJD,McMakenD.SurfaceWaveRaysinElastodynamicDiffractionbyCracks[J].JofAcoust.Soc.Am.,1978,63:1828-1831.
[9]刘殿魁,盖秉政,陶贵源.论孔附近的动应力集中[J].力学学报特刊,1981:65-77.
[10]ZimmermanC.ScatteringofPlaneCompressionalWavesbySphericalInclusionsinaPoroelasticMedium[J].JAcoust.Soc.Am,1993,94(1):527-536.
[11]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[J].地震学报,1998,20(3):300-307.
[12]刘殿魁,许贻燕.各向异性介质中SH波与多个半圆形凹陷地形的相互作用[J].力学学报,1993,1(25):93-102.
[13]陆建飞,王建华.饱和土中的任意形状空洞对弹性波的散射[J].力学学报,2002,34(6):904-913.
[14]HDeresiewicz.TheEffectofBoundariesonWavePropagationinaLiquid-FilledPorousSolid:I.ReflectionofPlaneWaveaFreePlaneBoundary(non-dissipativecase)[J].BulletinoftheSeismologicalSocietyofAmerica,1960,50(4):599-607.
[15]HCAO,VWLEE.ScatteringandDiffractiontoP-WavebyCirlindricalCanyonswithVarisbleDepth-to-widthRatio[J].SoilDynamicsandEarthquakeEngineering,1990,9(3):141-150.
[16]Biot,MA.GeneralTheoryofthreeDimensionalConsolidation[J].J.Appl.Phys,1941,(12):155-164.
[17]Biot,MA.TheoryofPropagationofElasticWavesinaFluid SaturatedPorousSolid,I,lowFrenquecyrang[J].J.Acoust.Soc.Am,1956,28(2),168-178.
[18]Biot,MA.MechanicsofDeformationandAcousticPorousMedia[J].J.Appl.Phy,1962,33(4):1482-1498.
[19]梁建文,严林隽,VincentWlee.圆弧形层状沉积谷地对入射平面P波的散射解析解[J].地震学报,2001,23(2):167-184.
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