Elastic waves at a corrugated interface between two dissimilar fibre‐reinforced elastic half‐spaces

The reflection and transmission phenomena of elastic waves incident at a corrugated interface between two dissimilar fibre‐reinforced elastic half‐spaces have been analysed. Using Rayleigh method of approximation, the expressions of the reflection and transmission coefficients are obtained in closed form for the plane interface as well as for the first order approximation of the periodic interface ζ = d cos px. All these reflection and transmission coefficients of regular and irregular waves are found to be the functions of angle of incidence and elastic parameters of the media. Moreover, the coefficients of irregularly reflected and transmitted waves are found to be proportional to the amplitude of the corrugated interface and are functions of the frequency of the incident wave. Numerical computations have been performed for a specific model to compute these coefficients and results obtained are shown graphically. The results of Singh and Singh (Sadhana 2004; 29 :249–257) and Ben‐Menahem and Singh (Seismic Waves and Sources. Springer: New York) have been derived from our analysis as particular cases. Copyright © 2006 John Wiley & Sons, Ltd.     

Dynamic stiffness for rectangular rigid foundations on a semi-infinite elastic medium

The present paper deals with the dynamic steady-state force-displacement relationships (complex stiffness) for rectangular rigid foundations resting on a semi-infinite medium, consisting of homogenous, isotropic, linear elastic materials. The foundations are considered to be excited under harmonic vertical and rocking vibration. This gives mixed boundary value problems which cannot be easily solved by analytical approaches. Therefore, a numerical method is proposed here. The method is based on quite, simple equations, and is straightforward in computation, compared with other methods. Although the proposed method gives just approximate solutions, it is satisfactory for engineering practices, and the soluations become highly accurate for a small value of ωB/V_s. The results obtained by the proposed method are compared with those of other methods to evaluate the accuracy of the results. The effects of length/width ratio and the area of the contact plane of the foundations are also discussed.     

基于离散颗粒模型的岩体结构面动-静态刚度系数对比的数值模拟研究

岩体工程计算分析中结构面刚度系数是至关重要的力学参数,计算分析的精度和可靠程度与这个参数密不可分,结构面刚度系数取值仍然是一个难点。岩体中应力波传播至结构面处将会发生反射和透射现象,利用应力波透射系数可反演结构面动态刚度系数。本文从细观力学角度运用颗粒离散元方法,开发分段线性接触模型及应力波吸收边界模型,开展宏观岩体中应力波传播的模拟,结合准静态压缩试验模拟,研究了较为平直的岩体结构面分别在不同正应力条件下的动、静态刚度系数的变化特征。模拟结果表明:(1)利用C++语言开发的分段线性接触模型很好地实现了结构面非线性变形特征的模拟;(2)基于颗粒离散元方法能够准确地反映岩体中应力波传播规律,应力波通过不同刚度结构面的透射系数与理论解一致;(3)在离散颗粒模型中加入黏滞吸收边界条件很好地实现了在有限尺寸模型中远场应力波传播模拟;(4)在岩体模型中结构面接触部位运用分段线性接触模型,通过模拟应力波传播与单轴压缩试验分别获得了一致性较好的结构面动、静态刚度系数,结构面动/静态刚度系数之比值约为1.0。本文对岩体中结构面刚度的测试和取值具有重要的指导意义。     

Generalization of two-dimensional discontinuous deformation analysis for forward modelling

This paper provides the theoretical basis for a forward numerical model of rock block systems. This method was developed to compute stress, strain, sliding and opening of rock blocks; rigid body movement and deformation occur simultaneously. Input data consist of block geometry, loading forces, the deformability constants E and v, and the restraint or boundary conditions of the block system. Output data give the movements, deformations, stresses and strains of each block, and the sliding and detachment or rejoining of blocks. The forces acting on each block, from external loading or contact with other blocks, satisfy the equilibrium equations. Equilibrium is also achieved between external forces and the block stresses. Furthermore, this analysis fulfills constraints of no tension between blocks and no penetration of one block into another. To acheive equilibrium and satisfy these constraints, simultaneous equations are solved repeatdly with partial changes of the coefficients each time the contact constraints are chosen.     

Transmission of Elastic P-waves across Single Fractures with a Nonlinear Normal Deformational Behavior

Summary This paper presents a theoretical study on normally incident elastic P-wave transmission across single dry fractures with a nonlinear normal deformational behavior. The effects of nonlinear fracture normal behavior on P-wave transmission are examined without the mixture of fracture shear behavior. The linear displacement discontinuity model for wave propagation across fractures is extended to a nonlinear model – the hyperbolic elastic model (BB model). Numeric solutions of magnitudes of transmission (|T _non|) and reflection (|R _non|) coefficients, for normally incident P-wave transmission across the nonlinear deformable fractures, are obtained and related to the closure behavior of fractures. Parametric studies are conducted to acquire an insight into the effects of the nonlinear fracture normal deformation on P-wave transmission, in terms of initial normal stiffness and the ratio of current maximum closure to maximum allowable closure of the fractures, as well as the incident wave amplitude and frequency. Comparisons between the linear and nonlinear models are presented. It is shown that, |T _lin| and |R _lin| for the linear model are special solutions of |T _non| and |R _non| for the nonlinear model, when the incident wave amplitude is so low that the current maximum closure of fracture incurred during the wave transmission is much smaller, relative to the maximum allowable closure. In addition, the nonlinear fracture behavior gives rise to a phenomenon of higher harmonics during the wave transmission across the fracture. The higher harmonics contribute to the increase of |T _non| from |T _lin|.     

Exact solutions for one‐dimensional consolidation of single‐layer unsaturated soil

Based on the Fredlund consolidation theory of unsaturated soil, exact solutions of the governing equations for one‐dimensional consolidation of single‐layer unsaturated soil are presented, in which the water permeability and air transmission are assumed to be constants. The general solution of two coupled homogeneous governing equations is first obtained. This general solution is expressed in terms of two functions psi₁ and ψ₂, where ψ₁ and ψ₂, respectively, satisfy two second‐order partial differential equations, which are in the same form. Using the method of separation of variables, the two partial differential equations are solved and exact solutions for three typical homogeneous boundary conditions are obtained. To obtain exact solutions of nonhomogeneous governing equations with three typical nonhomogeneous boundary conditions, the nonhomogeneous boundary conditions are first transformed into homogeneous boundary conditions. Then according to the method of undetermined coefficients and exact solutions of homogenous governing equations, the series form exact solutions are put forward. The validity of the proposed exact solutions is verified against other analytical solutions in the literature. Copyright © 2011 John Wiley & Sons, Ltd.     

斜入射线弹性节理应力波传播特征的数值模拟

依据弹性波理论,应力波斜入射线弹性节理时会发生波场分解。根据入射、透射及反射各波形的不同到时,运用离散元软件UDEC模拟应力波在含倾斜节理岩体中的传播并计算其透射、反射系数,并分析其波型转换规律。应力波斜入射单节理时,模拟得到的透射、反射系数随节理刚度、入射角度的变化规律,与已有的理论解是吻合的。应力波斜入射一组平行节理时,随着节理间距的增大,其同类波的透射系数Tpp、Tss先增大后减小,最后趋于稳定值;节理条数越多,Tpp、Tss越小。此外,不同条数的节理,透射系数达到最大值的临界节理间距值基本一致,但趋于稳定时的节理间距值随节理条数的增加而逐渐增大。     

Analytical solutions for the consolidation of unsaturated foundation with prefabricated vertical drain

Based on the consolidation theory raised by Fredlund, the solutions for the equal-strain consolidation of unsaturated foundation with the prefabricated vertical drain considering smear effect and drain resistance are analytically formulated in this paper. Firstly, governing equations for excess pore pressures (i.e., excess pore-air and pore-water pressures) under the equal-strain hypothesis are derived with the introduction of radial boundary conditions. Afterwards, the obtained coupled equations are solved by applying general integration, decoupling process, and Fourier sine series expansion. The smear coefficients and factors of drain resistance corresponding to air and water phases are both captured explicitly in the final solutions. Furthermore, the degenerated solutions are employed to verify the reliability of the current solutions. Finally, a parametric study is conducted to study the consolidation characteristics of the proposed foundation model against modeling sizes (S and N), smear coefficients (α_a and α_w), and drain resistance factors (G_a and G_w).     

Effect of dry sandiness parameter and initial stress on the scattering of plane SH wave

The present paper is incorporated into a mathematical model of transmission and reflection of shear waves through the initially stressed dry sandy medium sandwiched between two initially stressed distinct orthotropic half-spaces. The formulae of transmission and reflection coefficients have been deduced for the propagation of SH waves in the described model subjected to certain boundary conditions applying Snell’s law and Cramer’s method. These coefficients are observed as a function of wave number, phase velocity, initial stress, rigidity, and dry sandiness coefficients. The concepts of energy partition, phase shift, conservation of energy, critical angle, and slowness section are introduced. Graphical approach has been carried out to accomplish a relation between reflection/refraction coefficient, phase shift, and energy ratio with an angle of incidence and wave number on the propagation of shear wave. The study reveals that sandiness parameter, initial stress, and wave number have a cogent respond to the scattering of shear wave that has been illustrated graphically.     

Ultrashallow seismic experiment on a trenched section of the Chelunpu fault zone, Taiwan

Ultrashallow P-wave seismic reflection experiments were conducted at a model test site and in a trenched shallow fault zone along the Chelunpu fault line. The field layout was designed to have the shallowest undistorted reflection from about 1 m depth with 0.5 m vertical resolution. The smallest group interval tested in this study was 0.05 m with a 0.25 ms sample interval, which can avoid spatial aliasing of ground roll if the target is very shallow and the velocities are low. Data processing was designed to be simple but consistent. As the ultrashallow reflections may be contaminated with high-amplitude coherent noise in many aspects, first break muting and surgical muting were performed on each file as detailed as possible, and f–k filtering was applied mainly for the purpose of attenuating the aliasing energy and back-scattered noise. Data acquired in this study show that the low P-wave velocities (< 200 m/s) and high dominant frequencies (120–200 Hz) of near-surface layers may have a potential vertical resolution of 0.4 m or even better.Comparing the test profile with the ground-penetrating radar (GPR) control profile of the same test site and correlating the results obtained from the study site with those of the geologic cross-section of the trench, this experiment demonstrates the possibility of using seismic methods in investigating shallow structures at depths of less than a few meters with vertical resolution comparable to the GPR technique.     

强夯加固地基的土体竖向位移计算方法研究

根据动力分析中应力边界与速度边界之间的关系,将应力边界时程转化为速度边界时程,提出计算土体竖向位移的新方法,并进行了简化和多角度对比性研究。在钱家欢加卸载模型应力和竖向位移计算的基础上,提出将正弦荷载形式和三角形荷载形式分别引入,推导出竖向位移的两种简化计算方法。两种方法概念清楚、物理意义明确,形式简单且相同,区别仅在于系数不同。工程实例表明,两种方法计算出的竖向位移均接近于实测位移,但正弦荷载形式下的位移计算方法更能反映实际情况。     

On the stability of fissured slopes subject to seismic action

A set of analytical solutions achieved by the upper bound theorem of limit analysis and the pseudo‐static approach is presented for the assessment of the stability of homogeneous c, ϕ slopes manifesting vertical cracks and subject to seismic action. Rotational failure mechanisms are considered for slopes with cracks of either known or unknown depth and location. A validation exercise was carried out based on numerical limit analyses and displacement‐based finite‐element analyses with strength reduction technique. Charts providing the stability factor for fissured slopes subject to both horizontal and vertical accelerations for any combination of c, ϕ and slope inclination are provided. The effect of the direction of the vertical acceleration on slope stability is specifically analysed. Yield seismic coefficients are also provided. When the presence of cracks within the slope can be ascertained with reasonable confidence, maps showing the zones within the slope where they have no destabilising effect are provided. Finally, Newmark's method was employed to assess the effect of cracks on earthquake induced displacements. To this end, displacement coefficients are provided in chart form as a function of the slope characteristics. Two examples of slopes subjected to known earthquakes are illustrated. © 2016 The Authors. International Journal for Numerical and Analytical Methods in Geomechanics published by John Wiley & Sons Ltd.     

Shear-wave velocity structure of Africa from Rayleigh-wave analysis

The 3D S-velocity structure beneath Africa is shown by means of a 2D S-velocity mapping for depths raging from zero to 500 km, determined by the regionalization and inversion of Rayleigh-wave dispersion. The traces of 94 earthquakes, occurred from 1990 to 2009 in the study area, have been used to obtain the Rayleigh-wave dispersion. These earthquakes were registered by 61 seismic stations located on Africa and the surrounding area. The dispersion curves were obtained for periods between 5 and 300 s, by digital filtering with a combination of MFT and TVF filtering techniques. After that, all seismic events (and some stations) were grouped to obtain a dispersion curve for each source-station path. These dispersion curves were regionalized and after inverted according to generalized inversion theory, to obtain shear-wave velocity models for rectangular blocks with a size of 5° × 5°. The 3D S-velocity structure obtained through this procedure is shown in the 2D S-velocity maps plotted for several depths. These results agree well with the geology and other geophysical results previously obtained. The obtained S-velocity models suggest the existence of lateral and vertical heterogeneity. The zones with consolidated and old structures (as cratons) present greater S-velocity values than the other younger zones. Nevertheless, in the depth range from 20 to 40 km, the different Moho depths present in the study area generate the principal variation of S-velocity. A similar behaviour is found for the depth range from 60 to 230 km, in which the lithosphere–asthenosphere boundary generates the principal variations of S-velocity. Finally, it should be highlighted a new and interesting feature obtained in this study: the definition of the base of the asthenosphere, for depths ranging from 160 to 280 km, in the whole African continent.     

P波在弹性介质与饱和冻土介质分界面上的透反射问题研究

基于弹性波在冻结饱和多孔介质与单相弹性介质中的传播理论,研究了平面P波在饱和冻土介质与单相弹性介质分界面上的透反射问题。利用Helmholtz矢量分解定理,根据分界面上的边界条件,获得了平面P波从单相弹性介质入射到饱和冻土介质分界面上透反射振幅比的理论表达式。通过数值计算,分析了在不同入射频率、胶结参数、孔隙率、饱和度和接触参数下,弹性波的透反射振幅比随入射角变化的关系。研究结果表明：P波从单相弹性介质垂直入射到饱和冻土介质分界面上时只有反射P波和3种透射P波产生,当掠入射时只产生反射而没有透射现象发生。入射频率、胶结参数、孔隙率、饱和度以及接触参数等参数对反射波和透射波的振幅比影响显著。     

不同饱和度土层分界面上剪切波的反射与透射

基于多相孔隙介质弹性理论,给出了非饱和土中不同弹性波的传播方程。根据分界面上的边界条件,建立了各势函数波幅值之间的关系式,讨论了入射剪切波在不同饱和度土层分界面上的反射与透射问题。在无限空间非饱和土体中存在3种压缩波和1种剪切波,因此,当剪切波传播到不同饱和度的非饱和土层分界面上将分别在上、下土层激发产生4种反射波和4种透射波。推导出不同反射波和透射波的振幅比例系数和能量比例系数的理论表达式,并且在此基础上进行数值分析。在数值算例中分别研究了各反射波与透射波的能量比例系数（即能量反射率和能量透射率）受入射频率、入射角度以及上、下土层土体饱和度变化的影响情况。计算结果表明：各能量反射率和能量透射率不仅与入射角和入射频率有关,而且其受上、下土层饱和度变化的影响也同样不能忽视。     

Numerical study of boundary conditions for solute transport through a porous medium

A transition region may be defined as a region of rapid change in medium properties about the interface between two porous media or at the interface between a porous medium and a reservoir. Modelling the transition region between different porous media can assist in the selection of the most appropriate boundary conditions for the standard advection–dispersion equation (ADE). An advantage of modelling the transition region is that it removes the need for explicitly defining boundary conditions, though boundary conditions may be recovered as limiting cases. As the width of a transition region is reduced, the solution of the transition region model (TR model) becomes equivalent to the solution of the standard ADE model with correct boundary conditions. In this paper numerical simulations using the TR model are employed to select the most appropriate boundary conditions for the standard ADE under a variety of configurations and conditions. It is shown that at the inlet boundary between a reservoir and porous medium, continuity of solute mass flux should be used as the boundary condition. At the boundary interface between two porous media both continuity of solute concentration and solute mass flux should be used. Finally, in a finite porous medium where the solute is allowed to advect freely from the exit point, both continuity of solute concentration and solute mass flux should be used as the outlet boundary condition. The findings made here are discussed with reference to a detailed review of previous relevant theoretical and experimental observations. Copyright © 2001 John Wiley & Sons, Ltd.     

关于薄层与单界面模型弹性反射透射系数的讨论

在均匀各向同性水平层状介质假设下,采用位移与位移位两种不同的平面波表示方式,推导了层状介质的传播矩阵;并对层状模型逐步简化,依次给出了薄层、单阻抗差界面纵波入射时的反射、透射系数公式,指出了位移与位移位函数这两种不同形式下各公式的异同和适用条件,对于薄层地震反演精度的提高具有一定的指导作用。      

CASM: a unified state parameter model for clay and sand

The purpose of this paper is to present a simple, unified critical state constitutive model for both clay and sand. The model, called CASM (Clay And Sand Model), is formulated in terms of the state parameter that is defined as the vertical distance between current state (v, p′) and the critical state line in v–ln p′ space. The paper first shows that the standard Cam-clay models (i.e. the original and modified Cam-clay models) can be reformulated in terms of the state parameter. Although the standard Cam-clay models prove to be successful in modelling normally consolidated clays, it is well known that they cannot predict many important features of the behavior of sands and overconsolidated clays. By adopting a general stress ratio-state parameter relation to describe the state boundary surface of soils, it is shown that a simple, unified constitutive model (CASM) can be developed for both clay and sand. It is also demonstrated that the standard Cam-clay yield surfaces can be either recovered or approximated as special cases of the yield locus assumed in CASM. The main feature of the proposed model is that a single set of yield and plastic potential functions has been used to model the behaviour of clay and sand under both drained and undrained loading conditions. In addition, it is shown that the behaviour of overconsolidated clays can also be satisfactorily modelled. Simplicity is a major advantage of the present state parameter model, as only two new material constants need to be introduced when compared with the standard Cam-clay models. © 1998 John Wiley & Sons, Ltd.     

Multi‐genetic origin of the continental Moho: insights from Lithoprobe

This paper reviews four hypotheses for the origin of the continental crust–mantle boundary and discusses seismic parameters with which these hypotheses might be tested. The relict Moho hypothesis posits that the oceanic Moho is preserved during continental assembly; the magmatic underplating hypothesis posits formation of a new Moho by episodic emplacement of sill‐like intrusive bodies; the metamorphic (or metasomatic) front hypothesis posits that the Moho is overprinted by a phase transformation; and the regional décollement hypothesis posits that the Moho behaves as a structural detachment. These hypotheses are not mutually exclusive, and examples from the Canadian Lithoprobe program suggest that all four may be applicable in different regions of North America. Comparison of seismic images from a fossil subduction zone with modern subduction at Cascadia suggests that serpentinization of the forearc mantle, a previously unrecognized mechanism for overprinting and erasing the reflection Moho, may have occurred in the Proterozoic.