饱和土中结构对稳态体波散射问题的研究
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摘要
本文以Biot波动理论和弹性波理论为基础,借助复变函数和多级坐标变换及保角映射等方法,并且运用大圆弧来代替半空间饱和土直边界的近似处理手段,对具有不同的工程背景的饱和土中结构在稳态弹性平面入射体波作用下所产生的散射和集中的问题进行了较系统的研究,得出了一些有益结论。研究的内容包括以下几个方面:
(1)利用复变函数和多级坐标并借助以曲代直的大圆弧边界处理方法,通过引入位移势函数,解耦Biot波动理论在不同模型中的控制方程和运动方程,研究半空间饱和土介质中的圆形孔洞对弹性体波在有无流体耗散情况下的散射及动应力、孔压集中问题。求解过程为:首先通过引入位移势函数将半空间饱和土中在Biot模型下的波动方程解耦成三个Helmholtz方程;并利用分离变量法求解该组方程,得到用复数表示的三个势函数的通解;之后根据饱和土中的控制方程及复势函数特性,得到介质应力、位移和孔压和势函数的关系式,并将这些关系式转化为复平面极坐标形式;在引进一个大圆弧近似代替半空间直边界基础上,利用应力边界条件、位移边界条件或混合边界条件求解出复势函数级数展开式中的未知系数值,进而求得复势函数的特解,最后利用复势函数的特解求解出孔洞周边的动应力和孔压的集中系数,讨论半空间饱和土中的孔洞周围的散射和集中系数对直边界散射、孔洞边界透水性、孔洞埋深、饱和土的孔隙率、土骨架的密度、剪切模量及土骨架的泊松比等不同的参数条件组合下的分布和变化情况。从而得出一些较普遍的规律。
(2)借助Biot多孔介质波动理论和弹性介质中的波动理论,引入位移势函数来解
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 concentration coefficient of harmonic plane body wave around a circular cavity or the structure in a saturated soil half-space. Here, a circular cavity with large radius is used to replace the straight boundary of saturated soil half-space. The studies of the paper consist of the following parts.
(1) With the dissipation condition, the steady state Biot’s dynamic field equations of saturated solid are uncoupled into Helmholtz equations via given potential functions. Here, the complex function method based on the potential function and multi-polar coordinate method are used. A circular-arc with large radius is used to replace the straight boundary of the saturated soil half-space. The stresses and pore water pressures are obtained by the potential functions with certain boundary conditions of the saturated soil. Then the variations of the coefficients of dynamic stress concentration and the pore pressures concentration on boundaries of the cavity are discussed with different parameter conditions
(2) Based on Biot’s dynamic theory and elastic wave theory, the potential functions is used to decoupled the equations the saturated soil and the elastic lining with the complex function method and the multi-polar coordinate method. Considering the conditions between the saturated soil and the lining, the coefficients in the potentials can be determines by using the boundary condition of the lining and the straight boundary of the saturated soil half-space replaced by a circular-arc with large radius. Utilizing the solutions of the potential functions, the expressions of the displacements, stresses and pore pressures of saturated around the lining structure can be obtained. Some numerical results are given by the different parameter conditions and some conclusions are obtained.
(3) By using a circular-arc with large radius to replaced the straight boundary of the saturated half-space and the circular-arc alluvial valley, the scattering of saturated circular-arc alluvial valley and straight boundary of the saturated half-space can be studied. The coefficients of dynamic stresses concentration and
(1)利用复变函数和多级坐标并借助以曲代直的大圆弧边界处理方法,通过引入位移势函数,解耦Biot波动理论在不同模型中的控制方程和运动方程,研究半空间饱和土介质中的圆形孔洞对弹性体波在有无流体耗散情况下的散射及动应力、孔压集中问题。求解过程为:首先通过引入位移势函数将半空间饱和土中在Biot模型下的波动方程解耦成三个Helmholtz方程;并利用分离变量法求解该组方程,得到用复数表示的三个势函数的通解;之后根据饱和土中的控制方程及复势函数特性,得到介质应力、位移和孔压和势函数的关系式,并将这些关系式转化为复平面极坐标形式;在引进一个大圆弧近似代替半空间直边界基础上,利用应力边界条件、位移边界条件或混合边界条件求解出复势函数级数展开式中的未知系数值,进而求得复势函数的特解,最后利用复势函数的特解求解出孔洞周边的动应力和孔压的集中系数,讨论半空间饱和土中的孔洞周围的散射和集中系数对直边界散射、孔洞边界透水性、孔洞埋深、饱和土的孔隙率、土骨架的密度、剪切模量及土骨架的泊松比等不同的参数条件组合下的分布和变化情况。从而得出一些较普遍的规律。
(2)借助Biot多孔介质波动理论和弹性介质中的波动理论,引入位移势函数来解
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 concentration coefficient of harmonic plane body wave around a circular cavity or the structure in a saturated soil half-space. Here, a circular cavity with large radius is used to replace the straight boundary of saturated soil half-space. The studies of the paper consist of the following parts.
(1) With the dissipation condition, the steady state Biot’s dynamic field equations of saturated solid are uncoupled into Helmholtz equations via given potential functions. Here, the complex function method based on the potential function and multi-polar coordinate method are used. A circular-arc with large radius is used to replace the straight boundary of the saturated soil half-space. The stresses and pore water pressures are obtained by the potential functions with certain boundary conditions of the saturated soil. Then the variations of the coefficients of dynamic stress concentration and the pore pressures concentration on boundaries of the cavity are discussed with different parameter conditions
(2) Based on Biot’s dynamic theory and elastic wave theory, the potential functions is used to decoupled the equations the saturated soil and the elastic lining with the complex function method and the multi-polar coordinate method. Considering the conditions between the saturated soil and the lining, the coefficients in the potentials can be determines by using the boundary condition of the lining and the straight boundary of the saturated soil half-space replaced by a circular-arc with large radius. Utilizing the solutions of the potential functions, the expressions of the displacements, stresses and pore pressures of saturated around the lining structure can be obtained. Some numerical results are given by the different parameter conditions and some conclusions are obtained.
(3) By using a circular-arc with large radius to replaced the straight boundary of the saturated half-space and the circular-arc alluvial valley, the scattering of saturated circular-arc alluvial valley and straight boundary of the saturated half-space can be studied. The coefficients of dynamic stresses concentration and
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