Reflection and refraction of SH‐waves at a corrugated interface between two monoclinic elastic half‐spaces

A problem of reflection and transmission of shear wave incident upon a corrugated interface between two monoclinic solid half‐spaces have been investigated. Rayleigh's method of approximation is used to investigate the reflection and transmission coefficients for first and second approximation of the corrugation. For a special interface, closed‐form expressions of these coefficients for the first order approximation of the corrugation are obtained. It is found that these coefficients are functions of amplitude of corrugation, elastic parameters of the media, frequency of the incident wave and angle of incidence. The numerical computations are performed for a specific model and the results obtained are presented graphically. It is found that the reflection and transmission coefficients are strongly influenced by the corrugation and the elastic properties of the media. Results of some earlier workers in this field have been reduced as particular case from the present formulation. Copyright © 2004 John Wiley & Sons, Ltd.     

Response of shear wave from a corrugated interface between elastic solid/viscoelastic half‐spaces

The propagation of plane shear wave at a corrugation interface between elastic solid and viscoelastic liquid half‐spaces has been investigated. Adopting Rayleigh's method of approximation, the reflection and refraction coefficients corresponding to reflected and refracted SH‐waves have been derived for the first order of approximation of the corrugated interface. These coefficients are derived in the closed form for a periodic interface. Numerical computation has been performed for a particular model and the results are depicted graphically. The results of Asano (Bull. Earthquake Res. Inst. 1960; 38 (2):177–197) and Kielczynski and Pajewski (J. Acoust. Soc. Am. 1987; 81 (3):599–605) are recovered from this study. Copyright © 2010 John Wiley & Sons, Ltd.     

Plane SH‐waves at a corrugated interface between two dissimilar perfectly conducting self‐reinforced elastic half‐spaces

In this paper, we have attempted a problem of reflection and refraction of plane harmonic SH‐wave at a corrugated interface between two different perfectly conducting self‐reinforced elastic half‐spaces. Rayleigh's method is employed to find out the expressions of reflection and refraction coefficients for first‐ and second‐order approximation of the corrugation. The expressions of these coefficients show that they depend on the properties of half‐spaces, angle of incidence, frequency of the incident wave and are strongly influenced by the corrugation of the interface. Numerical computations are performed for a particular model having special type of interface and the variation of these coefficients are depicted graphically against the angles of incidence, frequency parameter, corrugation parameter at different values of reinforcement parameters. Results of some earlier works are reduced as a particular case of this formulation. Copyright © 2005 John Wiley & Sons, Ltd.     

Reflection of<Emphasis Type="Italic">P</Emphasis> and<Emphasis Type="Italic">SV</Emphasis> waves at the free surface of a monoclinic elastic half-space

The propagation of plane waves in an anisotropic elastic medium possessing monoclinic symmetry is discussed. The expressions for the phase velocity ofqP andqSV waves propagating in the plane of elastic symmetry are obtained in terms of the direction cosines of the propagation vector. It is shown that, in general,qP waves are not longitudinal andqSV waves are not transverse. Pure longitudinal and pure transverse waves can propagate only in certain specific directions. Closed-form expressions for the reflection coefficients ofqP andqSV waves incident at the free surface of a homogeneous monoclinic elastic half-space are obtained. These expressions are used for studying numerically the variation of the reflection coefficients with the angle of incidence. The present analysis corrects some fundamental errors appearing in recent papers on the subject.     

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。     

The study of reflection/transmission phenomena in a corrugated interface between two magnetoelastic transversely isotropic media

In this paper, an integrated approach has been carried out with an intent to study the reflection and transmission phenomena of plane SH-type wave on a corrugated interface separating two magnetoelastic transversely isotropic half-space. Closed form expressions for reflection and transmission coefficients have been derived for both plane and corrugated surface. Rayleigh’s method of approximation have been incorporated to deduce equations for the first- and second-order approximation of corrugation. Analytical solutions for both the half-spaces have been worked out. All possible cases have been apprehended specially for anisotropic and an isotropic medium. The effects of magnetoelastic coupling parameter, angle at which the wave crosses the magnetic field, corrugation amplitude, frequency factor and wave number have been explained by collaborating with graphical analysis.     

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。     

Analytical solutions for dynamic pressures of coupling fluid–porous medium–solid due to SV wave incidence

This paper presents the results of theoretical investigation on the dynamic coupling of an ideal fluid‐porous medium‐elastic half‐space system subjected to SV waves to study the effect of sediment on the seismic response of dams for reservoirs that are deposited with a significant amount of sediment after a long period of operation. The effects of the porous medium and the incident wave angle on dynamic pressures in the overlying ideal fluid are analyzed, and the reflection and transmission coefficients of the wave at the material interfaces are derived using an analytical solution in terms of displacement potentials. The numerical test of modeling shows that the dynamic pressures significantly depend on the properties of porous medium. The fully saturated porous medium reduces the response peaks slightly, while the partially saturated porous medium causes a considerable increase in the resonance peaks. Copyright © 2009 John Wiley & Sons, Ltd.     

Review of the methodology for computation of transmission and reflection coefficients for the surface waves propagated through a vertical contact

In this paper, a review of the methodology for the computation of transmission and reflection coefficients for the surface waves, propagated through a vertical contact, will be presented and discussed. The surface-wave transmission and reflection coefficients will be obtained using a simple method. This method will be very briefly described and compared with a more sophisticate method, obtaining the Rayleigh-wave transmission coefficients for the ocean-continent boundary, existing between the Atlantic Ocean and the Iberian Peninsula. This simple method will be shown as efficient as other more sophisticate methods, computing a theoretical curve (the theoretical Rayleigh-wave transmission coefficients) as close as possible to the observed data (the observed Rayleigh-wave transmission coefficients, previously measured for the boundary considered). Thus, the validity and reliability of this simple method, proposed in this paper, will be proven. This simple method should be a useful tool, to study the vertical contacts between two structures (as faults or continent-ocean boundaries), for many areas of the Earth, if the elastic properties of the media in both sides of the contact (and the density distribution with depth) are well known in advance.     

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。     

Poroelastic model for pile–soil interaction in a half‐space porous medium due to seismic waves

In this paper, frequency domain dynamic response of a pile embedded in a half‐space porous medium and subjected to P, SV seismic waves is investigated. According to the fictitious pile methodology, the problem is decomposed into an extended poroelastic half‐space and a fictitious pile. The extended porous half‐space is described by Biot's theory, while the fictitious pile is treated as a bar and a beam and described by the conventional 1‐D structure vibration theory. Using the Hankel transformation method, the fundamental solutions for a half‐space porous medium subjected to a vertical or a horizontal circular patch load are established. Based on the obtained fundamental solutions and free wave fields, the second kind of Fredholm integral equations describing the vertical and the horizontal interaction between the pile and the poroelastic half‐space are established. Solution of the integral equations yields the dynamic response of the pile to plane P, SV waves. Numerical results show the parameters of the porous medium, the pile and incident waves have direct influences on the dynamic response of the pile–half‐space system. Significant differences between conventional single‐phase elastic model and the poroelastic model for the surrounding medium of the pile are found. Copyright © 2007 John Wiley & Sons, Ltd.     

Self‐similar solution of a plane‐strain fracture driven by a power‐law fluid

This paper analyses the problem of a hydraulically driven fracture, propagating in an impermeable, linear elastic medium. The fracture is driven by injection of an incompressible, viscous fluid with power‐law rheology and behaviour index n?0. The opening of the fracture and the internal fluid pressure are related through the elastic singular integral equation, and the flow of fluid inside the crack is modelled using the lubrication theory. Under the additional assumptions of negligible toughness and no lag between the fluid front and the crack tip, the problem is reduced to self‐similar form. A solution that describes the crack length evolution, the fracture opening, the net fluid pressure and the fluid flow rate inside the crack is presented. This self‐similar solution is obtained by expanding the fracture opening in a series of Gegenbauer polynomials, with the series coefficients calculated using a numerical minimization procedure. The influence of the fluid index n in the crack propagation is also analysed. Copyright © 2002 John Wiley & Sons, Ltd.     

Visco‐elastic load transfer models for axially loaded piles

Viscoelastic or creep behaviour can have a significant influence on the load transfer (t–z) response at the pile–soil interface, and thus on the pile load settlement relationship. Many experimental and theoretical models for pile load transfer behaviour have been presented. However, none of these has led to a closed‐form expression which captures both non‐linearity and viscoelastic behaviour of the soil. In this paper, non‐linear viscoelastic shaft and base load transfer (t–z) models are presented, based on integration of a generalized viscoelastic stress–strain model for the soil. The resulting shaft model is verified through published field and laboratory test data. With these models, the previous closed‐form solutions evolved for a pile in a non‐homogeneous media have been readily extended to account for visco‐elastic response. For 1‐step loading case, the closed‐form predictions have been verified extensively with previous more rigorous numerical analysis, and with the new GASPILE program analysis. Parametric studies on two kinds of commonly encountered loading: step loading, ramp (linear increase followed by sustained) loading have been performed. Two examples of the prediction of the effects of creep on the load settlement relationship by the solutions and the program GASPILE, have been presented. Copyright © 2000 John Wiley & Sons, Ltd.     

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|.     

Finite element modelling of thick plates on two‐parameter elastic foundation

This paper is intended to give some information about how to build a model necessary for bending analysis of rectangular and circular plates resting on a two‐parameter elastic foundation, subjected to combined loading and permitting various types of boundary conditions. The formulation of the problem takes into account the shear deformation of the plate and the surrounding interaction effect outside the plate. The numerical model based on an 18‐node zero‐thickness isoparametric interface element interacting with a thick Reissner–Mindlin plate element with three degrees of freedom at each of the nine nodes, which enforce C⁰ continuity requirements for the displacements and rotations of the midsurface, is proposed. Stiffness matrices of a special interface element are superimposed on the global stiffness matrix to represent the stiffening elastic foundation under and beyond the plate. Some numerical examples are given to illustrate the advantages of the method presented. Copyright © 2001 John Wiley & Sons, Ltd.     

Analysis of Blast Wave Interaction with a Rock Joint

The interaction between rock joints and blast waves is crucial in rock engineering when rock mass is suffered from artificial or accidental explosions, bursts or weapon attacks. Based on the conservation of momentum at the wave fronts and the displacement discontinuity method, quantitative analysis for the interaction between obliquely incident P- or S-blast wave and a linear elastic rock joint is carried out in the present study, so as to deduce a wave propagation equation. For some special cases, such as normal or tangential incidence, rigid or weak joint, the analytical solution of the stress wave interaction with a rock joint is obtained by simplifying the wave propagation equation. By verification, it is found that the transmission and reflection coefficients from the wave propagation equation agree very well with the existing results. Parametric studies are then conducted to evaluate the effects of the joint stiffness and incident waves on wave transmission and reflection. The wave propagation equation derived in the present study can be straightforwardly extended for different incident waveforms and nonlinear rock joints to calculate the transmitted and reflected waves without mathematical methods such as the Fourier and inverse Fourier transforms.     

Quartz‐in‐garnet and Ti‐in‐quartz thermobarometry: Methodology and first application to a quartzofeldspathic gneiss from eastern Papua New Guinea

Mineral inclusions are ubiquitous in metamorphic rocks and elastic models for host‐inclusion pairs have become frequently used tools for investigating pressure–temperature (P–T) conditions of mineral entrapment. Inclusions can retain remnant pressures () that are relatable to their entrapment P–T conditions using an isotropic elastic model and P–T–V equations of state for host and inclusion minerals. Elastic models are used to constrain P–T curves, known as isomekes, which represent the possible inclusion entrapment conditions. However, isomekes require a temperature estimate for use as a thermobarometer. Previous studies obtained temperature estimates from thermometric methods external of the host‐inclusion system. In this study, we present the first P–T estimates of quartz inclusion entrapment by integrating the quartz‐in‐garnet elastic model with titanium concentration measurements of inclusions and a Ti‐in‐quartz solubility model (QuiG‐TiQ). QuiG‐TiQ was used to determine entrapment P–T conditions of quartz inclusions in garnet from a quartzofeldspathic gneiss from Goodenough Island, part of the (ultra)high‐pressure terrane of Papua New Guinea. Raman spectroscopic measurements of the 128, 206, and 464 cm^?1 bands of quartz were used to calculate inclusion pressures using hydrostatic pressure calibrations (), a volume strain calculation (), and elastic tensor calculation (), that account for deviatoric stress. values calculated from the 128, 206, and 464 cm^?1 bands’ hydrostatic calibrations are significantly different from one another with values of 1.8 ± 0.1, 2.0 ± 0.1, and 2.5 ± 0.1 kbar, respectively. We quantified elastic anisotropy using the 128, 206 and 464 cm^?1 Raman band frequencies of quartz inclusions and stRAinMAN software (Angel, Murri, Mihailova, & Alvaro, 2019,  234 :129–140). The amount of elastic anisotropy in quartz inclusions varied by ~230%. A subset of inclusions with nearly isotropic strains gives an average and of 2.5 ± 0.2 and 2.6 ± 0.2 kbar, respectively. Depending on the sign and magnitude, inclusions with large anisotropic strains respectively overestimate or underestimate inclusion pressures and are significantly different (<3.8 kbar) from the inclusions that have nearly isotropic strains. Titanium concentrations were measured in quartz inclusions exposed at the surface of the garnet. The average Ti‐in‐quartz isopleth (19 ± 1 ppm [2σ]) intersects the average QuiG isomeke at 10.2 ± 0.3 kbar and 601 ± 6°C, which are interpreted as the P–T conditions of quartzofeldspathic gneiss garnet growth and entrapment of quartz inclusions. The P–T intersection point of QuiG and Ti‐in‐quartz univariant curves represents mechanical and chemical equilibrium during crystallization of garnet, quartz, and rutile. These three minerals are common in many bulk rock compositions that crystallize over a wide range of P–T conditions thus permitting application of QuiG‐TiQ to many metamorphic rocks.     

弹性层系的反射系数正演

应用层系位移和应力递推公式，针对固体多层介质，导出了纵、横波反射、透射系数公式，并对新疆某油田模型进行了试算，探讨了反射系数与入射角之间的变化规律。结果表明，该区含气砂岩易于检测。通过zoeppritz方程和垂直入射的反射系数解析式分别对单界面和本文模型进行了检验，结果是完全一致的。     

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.     

The Reissner–Sagoci problem for a transversely isotropic half‐space

A transversely isotropic linear elastic half‐space, z?0, with the isotropy axis parallel to the z‐axis is considered. The purpose of the paper is to determine displacements and stresses fields in the interior of the half‐space when a rigid circular disk of radius a completely bonded to the surface of the half‐space is rotated through a constant angle θ₀. The region of the surface lying out with the circle r?a, is free from stresses. This problem is a type of Reissner–Sagoci mixed boundary value problems. Using cylindrical co‐ordinate system and applying Hankel integral transform in the radial direction, the problem may be changed to a system of dual integral equations. The solution of the dual integral equations is obtained by an approach analogous to Sneddon's (J. Appl. Phys. 1947; 18 :130–132), so that the circumferential displacement and stress fields inside the medium are obtained analytically. The same problem has already been approached by Hanson and Puja (J. Appl. Mech. 1997; 64 :692–694) by the use of integrating the point force potential functions. It is analytically proved that the present solution, although of a quite different form, is equivalent to that given by Hanson and Puja. To illustrate the solution, a few plots are provided. The displacements and the stresses in a soil deposit due to a rotationally symmetric force or boundary displacement may be obtained using the results of this paper. Copyright © 2006 John Wiley & Sons, Ltd.