Diffraction of plane P waves by a canyon of arbitrary shape in poroelastic half-space (Ⅰ): Formulation

This paper presents an indirect boundary integration equation method for diffraction of plane P waves by a two-dimensional canyon of arbitrary shape in poroelastic half-space. The Green's functions of compressional and shear wave sources in poroelastic half-space are derived based on Biot's theory. The scattered waves are constructed using the fictitious wave sources close to the boundary of the canyon, and magnitude of the fictitious wave sources are determined by the boundary conditions. The precision of the method is verified by the satisfaction extent of boundary conditions, the comparison between the degenerated solutions of single-phased half-space and the well-known solutions, and the numerical stability of the method.     

平面SV波在饱和半空间中沉积谷地周围的散射

采用一种特殊的间接边界积分方程法,求解了平面SV波在饱和半空间中任意形状沉积谷地周围的二维散射问题。结合饱和半空间中膨胀波源和剪切波源格林函数,由分布在沉积和半空间交界面附近两虚拟波源面上的波源分别构造沉积内外的散射波场,由交界面连续条件建立方程并求解确定虚拟波源密度,总波场反应即可由自由波场和散射波场叠加而得。然后通过边界条件验算、退化解答与现有结果的比较以及稳定性检验,验证了方法的计算精度。通过一组典型算例,研究了平面SV波在饱和半空间中沉积谷地周围散射的基本规律,详细给出了不同参数情况沉积谷地附近地表位移幅值和孔隙水压,着重分析了入射SV波频率和角度、边界渗透条件、沉积孔隙率等因素对场地反应的影响,得出了一些有益的结论。     

3D Diffraction of obliquely incident SH waves by twin infinitely long cylindrical cavities in layered poroelastic half-space

This paper studies three-dimensional diffraction of obliquely incident plane SH waves by twin infinitely long cylindrical cavities in layered poroelastic half-space using indirect boundary element method. The approach is validated by comparison with the literature, and the effects of cavity interval, incident frequency, and boundary drainage condition on the diffraction are studied through numerical examples. It is shown that, the interaction between two cavities is significant and surface displacement peaks become large when two cavities are close, and the surface displacement may be significantly amplified by twin cavities, and the influence range with large amplification can be as wide as 40 times of the cavity radius. Surface displacements in dry poroelastic case and saturated poroelastic cases with drained and undrained boundaries are evidently different under certain circumstances, and the differences may be much larger than those in the free-field response.     

Diffraction of plane SV waves by a shallow circular-arc canyon in a saturated poroelastic half-space

An analytical solution for the scattering and diffraction of incident plane SV waves by a shallow circular-arc canyon in a saturated poroelastic half-space is derived by the wave function expansion method. The solution is utilized to analyze the dependence of the computed surface motions on the incident frequencies, incident angles, porosity, boundary drainage and Poisson's ratio. It is shown that, depending on the incident angles, the surface displacement amplitudes around a canyon in a dry poroelastic half-space and saturated poroelastic half-space can be very different. The surface displacement amplitudes of an undrained saturated poroelastic half-space are close to those of a drained saturated poroelastic half-space. For low porosity, the surface displacement amplitudes of a saturated poroelastic half-space are almost identical to those of a dry poroelastic half-space, and drainage condition has little influence on the surface displacement amplitudes. But for high porosity, the effect of drainage condition becomes significant, and for the same porosity, the displacement amplitudes of an undrained saturated half-space will be larger than those of a drained saturated half-space. Poisson's ratio is also an important factor affecting the surface displacement amplitudes around the canyon, both in drained and undrained conditions, but leads to larger effects for an undrained saturated half-space than for a drained saturated half-space. Large pore pressures are found around the canyon and their amplitudes depend on the incident angles and frequencies. Below the surface, the amplitudes of pore pressures are less than they are at the surface, especially for high frequencies.     

Dynamic displacements and stresses in the vicinity of a cylindrical cavity embedded in a half-space

The two-dimensional response of a viscoelastic half-space containing a buried, unlined, infinitely long cylindrical cavity of circular cross-section subjected to harmonic plane SH, P, SV and Rayleigh waves is obtained by an indirect boundary integral method based on the two-dimensional Green's functions for a viscoelastic half-space. An extensive critical review of the existing numerical results obtained by other techniques is presented together with some new numerical results describing the motion on the ground surface and the motion and stresses on the wall of the cavity for P, SV, SH and Rayleigh waves.     

平面SV波在层状半空间中沉积谷地周围的散射

采用间接边界元方法求解了入射平面SV波在层状半空间中沉积谷地周围的散射问题.问题的解答包含自由场和散射场两部分.自由场可由直接刚度法求得,散射场由层状半空间中斜线荷载动力格林函数来模拟.文中以入射平面SV波在基岩上单一土层中沉积谷地周围散射为例研究了土层和沉积谷地周围的影响.结果表明,由于考虑了土层的动力特性,平面SV...     

Diffraction of plane SH waves by a semi-circular cavity in half-space

This paper presents a closed-form solution for diffraction of plane SH waves by a semi-circular cavity in half-space by using wave function expansion method. Accuracy of the solution is checked by the displacement residual and stress residual along the boundaries. Numerical results show that there are notable differences for response amplitudes between a semi-circular cavity and a whole-circular cavity in a half-space.     

平面P波在饱和半空间中洞室周围的散射(I):解析解

利用波函数展开法给出了入射平面P波在饱和半空间中圆形洞室周围散射问题的一个解析解。半空间假定为无粘性流体饱和介质,满足Biot理论。采用一种基于实验数据的孔隙率和模量之间的线性关系来确定Biot模型中的介质参数。解答考虑了透水边界和非透水边界两种情况。对边界条件进行了数值检验,结果表明,随着级数截断项数的增大,边界残量衰减很快。解答为进一步研究入射波频率和角度、边界渗透条件、孔隙率、泊松比等参数对散射的影响奠定了基础。     

SV波在饱和土体自由表面的反射

基于饱和两相介质弹性波动方程分析SV波在饱和土体自由表面的反射问题,引入波动方程的势函数解答,求解出二维问题中SV波入射情况下饱和土体自由场的位移、速度、加速度和应力响应。在饱和土体自由场响应解析解基础上,建立SV波入射下饱和土体自由场静、动力有限元模型。建模中考虑了如下几方面因素:(1)在不同分析步,对土体单元赋予不同材料本构。通过*model change命令进行单元生死设定,从而实现在初始应力场平衡的静力状态下采用DuncanChang本构模型,而地震波动输入时采用Davidenkov动力本构模型;(2)采用多孔介质黏弹性人工边界条件,在人工边界上分别施加固相和液相介质的弹簧和阻尼来模拟饱和土体中能量的传播;(3)将地震波转化为作用在人工边界上的等效地震荷载,施加到人工边界节点上;(4)土体单元采用4结点平面应变孔压单元(CPE4P)。有限元计算与解析解比较结果表明:SV波在垂直入射和掠入射时,竖向位移响应为零;在45°左右入射时,水平位移响应最大;60°左右入射时,竖向位移响应最大。这些结论与解析解吻合较好,本文模型为建立土-结构动力相互作用模型打下良好的基础。     

层状半空间中洞室对平面SH波的放大作用

利用间接边界元法,求解了弹性层状半空间中洞室对入射平面SH波的放大作用问题,并以基岩上单一土层中洞室对入射平面SH波的放大作用为例进行了数值计算分析。本文模型的特点之一是考虑了层状场地的动力特性,因而更接近于实际工程;特点之二是计算精度非常高。研究表明,层状半空间中洞室对波的放大作用与均匀半空间中情况有着本质的差别;层状半空间中洞室附近地表动力响应由土层动力特性和洞室对波的散射二者共同决定。土层动力特性不仅影响洞室附近地表位移的幅值,还会影响地表位移的频谱。在土层的前几阶共振频率附近,随着基岩与土层剪切波速比的增大,土层的影响随之增大,而随着土层厚度的增加,土层的影响随之减小,并逐渐趋于均匀半空间情况。     

2D dynamic response of unlined and lined tunnels in poroelastic soil to harmonic body waves

The problem of harmonic wave diffraction by tunnels in an infinite poroelastic saturated soil obeying Biot's theory is studied numerically under conditions of plane strain and the effect of poroelasticity on the response is assessed through some parametric studies. The method is based on the theory of Mei and Foda, which considers the total field to be approximated by the superposition of an elastodynamic problem with modified elastic constants and mass density for the whole domain and a diffusion problem for the pore fluid pressure confined to a boundary layer at the free boundaries. Both problems are solved numerically by the boundary element method in the frequency domain. Results dealing with the response of a circular tunnel with and without an elastic concrete liner in an infinite poroelastic medium to incident harmonic P and SV plane waves are provided and compared against analytical ones as well as to those corresponding to linear elastic soil behaviour. Copyright © 2002 John Wiley & Sons, Ltd.     

弹性层状半空间中凸起地形对入射平面SH波的放大作用

对Wolf理论进行拓展,使之可解决凸起地形对波的散射问题,进而利用间接边界元法,求解了弹性层状半空间中凸起地形对入射平面SH波的放大作用问题。本文模型的显著特点之一是考虑了层状半空间的动力特性以及层状半空间和凸起地形的阻尼;特点之二是计算精度高。文中以基岩上单一土层中半圆凸起地形对入射平面SH波的放大作用为例进行了数值计算分析。研究表明,基岩上单一土层中凸起地形对入射平面SH波放大作用和均匀半空间中凸起地形有着本质的差别;土层动力特性不仅影响凸起地形地表位移的幅值,还会影响地表位移的频谱;阻尼会显著降低凸起地形对高频波的放大作用。     

P波入射时含圆柱形空腔板弹性波散射与动应力集中

采用波函数展开法给出了板的横截面处孔洞在平面P波入射下动应力集中问题的解。本文使用大圆弧假定法来满足板中的自由表面的边界条件。数值结果表明,上表面和孔洞间的距离与下表面和孔洞间的距离都对结果有影响。如果孔洞半径相对板的厚度很小,解近似等于孔洞全空间解。所给出的解是半解析解。因此能被用来验证诸如BEM,FEM,FDM等数值法。     

平面P波在饱和半空间中洞室周围的散射(II):数值结果

本文通过数值计算研究了入射平面P波在饱和半空间中洞室周围散射问题,分析了入射波频率和角度、边界渗透条件、孔隙率、泊松比等参数对散射的影响。研究表明,平面P波入射情况下,饱和半空间和单相(干土)半空间中洞室附近地表位移幅值的差别很大,干土情况的水平位移幅值相对较大,饱和情况的竖向地表位移幅值相对较大;由于波在洞室附近的干涉,饱和情况与干土情况的地表位移出现相位漂移。随着孔隙率的增大,洞室附近水平地表位移幅值逐渐减小,竖向地表位移幅值则逐渐增大;当孔隙率较低时,边界渗透条件对地表位移幅值的影响很小,而当孔隙率较大时,边界渗透条件的影响则不可忽视,不透水情况下,水平和竖向地表位移幅值的峰值均相对较大;随着入射频率的增加,孔隙率的影响逐渐增大,而且不透水情况下孔隙率的影响相对较大。随着泊松比的增大,洞室附近水平地表位移幅值逐渐降低,竖向地表位移幅值则逐渐增大;泊松比较小时,边界渗透条件对位移幅值的影响较大,泊松比较大时,边界渗透条件对位移幅值的影响则较小;随着入射频率的增加,泊松比的影响逐渐增大。当孔隙率较小时,半空间地表和洞室表面孔隙水压幅值较小,但空间变化比较剧烈,随着孔隙率的增大,孔隙水压逐渐增大但空间变化逐渐平缓;随着入射频率的增加,孔隙水压幅值逐渐增大,且孔隙水压的空间变化逐渐变得复杂。     

弹性层状半空间中沉积谷地对入射平面SH波的放大作用

对Wolf理论进行拓展,使之可解决沉积谷地对波的散射问题,进而利用间接边界元法,求解了弹性层状半空间中沉积谷地对入射平面SH波的放大作用问题。通过自由场反应分析,求得假想边界上各点的位移和各单元的应力反应。施加虚拟均布荷载,求得位移和应力的格林函数。根据应力和位移连续边界条件确定虚拟分布荷载,将自由场反应和虚拟分布荷载产生的反应叠加起来,即得到问题的解答。最后,以基岩上单一土层中沉积谷地对入射平面SH波的放大作用为例进行了数值计算分析,结果表明,层状半空间中沉积谷地和均匀半空间中沉积谷地对波的放大作用存在显著差别。文中分析了造成差别的原因,并讨论了覆盖层厚度和刚度对放大作用的影响。     

Two-dimensional scattering and diffraction of P- and SV-waves around a semi-circular canyon in an elastic half-space: An analytic solution via a stress-free wave function

A well-defined boundary-valued problem of wave scattering and diffraction in elastic half-space should have closed-form analytic solutions. This two-dimensional (2-D) scattering around a semi-circular canyon in elastic half-space subjected to seismic plane and cylindrical waves has long been a challenging boundary-value problem. In all cases, the diffracted waves will consist of both longitudinal (P-) and shear (S-) rotational waves. Together at the half-space surface, these in-plane longitudinal P- and shear SV-waves are not orthogonal over the infinite half-space flat-plane boundary. Thus, to simultaneously satisfy both the zero normal and shear stresses at the flat-plane boundary, some approximation of the geometry and/or wave functions often has to be made, or in some cases, relaxed (disregarded). This paper re-examines this two-dimensional (2-D) boundary-value problem from an applied mathematics points of view and redefines the proper form of the orthogonal cylindrical-wave functions for both the longitudinal P- and shear SV-waves so that they can together simultaneously satisfy the zero-stress boundary conditions at the half-space surface. With the zero-stress boundary conditions satisfied at the half-space surface, the most difficult part of the problem will be solved, and the remaining boundary conditions at the finite-canyon surface are then comparatively less complicated to solve. This is now a closed-form analytic solution of the 2-D boundary-valued problem satisfying the half-space zero-stress boundary conditions exactly.     

Diffraction of plane P waves by a canyon of arbitrary shape in poroelastic half-space （Ⅱ）： Numerical results and discussion

This paper investigates in detail the nature of diffraction of plane P waves around a canyon in poroelastic half-space, and studies the effects of incident frequency, drainage condition, porosity, etc, on the diffraction of waves. It is shown that the surface displacement amplitudes of the drained case are close to those of the undrained case, however, the surface displacement amplitudes of the dry case are very different from those of the saturated (either drained or undrained) cases. There are large phase...     

Scattering of SV waves by a canyon in a fluid-saturated, poroelastic layered half-space, modeled using the indirect boundary element method

The scattering of SV waves by a canyon in a fluid-saturated, poroelastic layered half-space is modeled using the indirect boundary element method in the frequency domain. The free-field responses are calculated to determine the displacements and stresses at the surface of the canyon, and fictitious distributed loads are then applied at the surface of the canyon in the free field to calculate the Green's functions for displacements and stresses. The amplitudes of the fictitious distributed loads are determined from the boundary conditions, and the displacements arising from the waves in the free field and from the fictitious distributed loads are summed to obtain the solution. The effects of fluid saturation, boundary conditions, porosity, and soil layers on the surface displacement amplitudes and phase shifts are discussed, and some useful conclusions are obtained. It is shown that the surface displacement amplitudes due to saturation and boundary conditions, different porosities, or the presence of a soil layer can be very dissimilar, and large phase shifts can be observed. The resulting wavelengths for an undrained saturated poroelastic medium are slightly longer than those for a drained saturated poroelastic medium; and are longer for a drained saturated poroelastic medium than those for a dry poroelastic medium. As porosity increases, the wavelengths become longer; and a layered half-space produces longer wavelengths than a homogeneous half-space.     

饱和土中深埋圆柱形衬砌洞室对瞬态平面波的散射

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.     

Time harmonic point load and dynamic contact problem of contacting fluid and poroelastic half-spaces

The dynamic response of contacting fluid and fluid-saturated poroelastic half- spaces to a time-harmonic vertical point force or a point pore pressure is investigated. The solutions are formulated using the boundary conditions at the fluid-porous medium interface. The point load solutions are then used to solve the dynamic problem of the vertical vibration of a rigid disc (both permeable and impermeable discs are included) on the surface of the poroelastic half-space. The contact problems are solved by integrating the point force and point pore pressure solutions over the contact area with unknown discontinuous force and pore pressure distributions, which are determined from the boundary conditions. The solutions are expressed in terms of dual integral equations, which are converted to Fredholm integral equations of the second kind and solved numerically. Selected numerical results for the vertical dynamic compliance coefficient for the cases with or without fluid overlying the poroelastic half-space are presented to show the effects of the fluid. The influence of the permeability condition of the disc on the compliance of the poroelastic half-space is investigated. The displacement, vertical stress, pore pressure in the poroelastic half-space and water pressure in the fluid half-space are also examined for different poroelastic materials and frequencies of excitation. The present results are helpful in the study of the dynamic response of foundations on the seabed under seawater.