P-and SV-wave dispersion and attenuation in saturated microcracked porous rock with aligned penny-shaped fractures

P-and SV-wave dispersion and attenuation have been extensively investigated in saturated poroelastic media with aligned fractures.However,there are few existing models that incorporate the multiple wave attenuation mechanisms from the microscopic scale to the macroscopic scale.Hence,in this work,we developed a unified model to incorporate the wave attenuation mechanisms at different scales,which includes the microscopic squirt flow between the microcracks and pores,the mesoscopic wave-induced fluid flow between fractures and background(FB-WIFF),and the macroscopic Biot's global flow and elastic scattering(ES)from the fractures.Using Tang's modified Biot's theory and the mixed-boundary conditions,we derived the exact frequency-dependent solutions of the scattering problem for a single penny-shaped fracture with oblique incident P-and SV-waves.We then developed theoretical models for a set of aligned fractures and randomly oriented fractures using the Foldy approximation.The results indicated that microcrack squirt flow considerably influences the dispersion and attenuation of P-and SV-wave velocities.The coupling effects of microcrack squirt flow with the FB-WIFF and ES of fractures cause much higher velocity dispersion and attenuation for P waves than for SV waves.Randomly oriented fractures substantially reduce the attenuation caused by the FB-WIFF and ES,particularly for the ES attenuation of SV waves.Through a comparison with existing models in the limiting cases and previous experimental measurements,we validated our model.

SCATTERING OF SH-WAVES BY AN INTERACTING INTERFACE LINEAR CRACK AND A CIRCULAR CAVITY NEAR BIMATERIAL INTERFACE

An analytical method is developed for scattering of SH-waves and dynamic stressconcentration by an interacting interface crack and a circular cavity near bimaterial interface.Asuitable Green’s function is contructed,which is the fundamental solution of the displacement fieldfor an elastic half space with a circular cavity impacted by an out-plane harmonic line source loadingat the horizontal surface.First,the bimaterial media is divided into two parts along the horizontalinterface,one is an elastic half space with a circular cavity and the other is a complete half space.Then the problem is solved according to the procedure of combination and by the Green’s functionmethod.The horizontal surfaces of the two half spaces are loaded with undetermined anti-plane forcesin order to satisfy continuity conditions at the linking section,or with some forces to recover cracks bymeans of crack-division technique.A series of Fredholm integral equations of first kind for determiningthe unknown forces can be set up through continuity conditions as expressed in terms of the Green’sfunction.Moreover,some expressions are given in this paper,such as dynamic stress intensity factor(DSIF)at the tip of the interface crack and dynamic stress concentration factor(DSCF)around thecircular cavity edge.Numerical examples are provided to show the influences of the wave numbers,the geometrical location of the interface crack and the circular cavity,and parameter combinations ofdifferent media upon DSIF and DSCF.

Scattering of SH-wave by multiple circular cavities in half space

In this paper,an analytic method is developed to address steady SH-wave scattering and perform dynamic analysis of multiple circular cavities in half space.The scattered wave function used for scattering of SH-waves by multiple circular cavities,which automatically satisfies the stress-free condition at the horizontal surface,is constructed by applying the symmetry of the SH-wave scattering and the method of multi-polar coordinates system.Applying this scattered wave function and method of moving coordinates,the original problem can be transformed to the problem of SH-wave scattering by multiple circular cavities in the full space.Finally,the solution of the problem can be reduced to a series of algebraic equations and solved numerically by truncating the infinite algebraic equations to the finite ones.Numerical examples are provided for case with two cavities to show the effect of wave number,and the distances between the centers of the cavities and from the centers to the ground surface on the dynamic stress concentration around the cavity impacted by incident steady SH-wave.

Antiplane response of two scalene triangular hills and a semi-cylindrical canyon by incident SH-waves

Antiplane response of two scalene triangular hills and a semi-cylindrical canyon by SH-waves is studied using wave function expansion and complex function method. Firstly, the analytical model is divided into three parts, and the displacement solutions of wave fields are constructed based on boundary conditions in the three regions. Three domains are then conjoined to satisfy the ＂conjunction＂ condition at shared boundary. In addition, combined with the zero-stress condition of semi-cylindrical canyon, a series of infinite algebraic equations for the problem are derived. Finally, numerical examples are provided and the influence of different parameters on ground motion is discussed.

SCATTERING FAR FIELD SOLUTION OF SH-WAVE BY MOVABLE RIGID CYLINDRICAL INTERFACE INCLUSION

The Green's function is used to solve the scattering far fieldsolution of SH-wave by a mov- able rigid cylindrical interfaceinclusion in a linear elastic body. First, a suitable Green'sfunction is devel- oped, which is the fundamental displacementsolution of an elastic half space with a movable rigid half-cylin-drical inclusion impacted by out-of-plane harmonic line source loadedat any point of its horizontal surface.

An IBEM solution to the scattering of plane SH-waves by a lined tunnel in elastic wedge space

The indirect boundary element method （IBEM） is developed to solve the scattering of plane SH-waves by a lined tunnel in elastic wedge space. According to the theory of single-layer potential, the scattered-wave field can be constructed by applying virtual uniform loads on the surface of lined tunnel and the nearby wedge surface. The densities of virtual loads can be solved by establishing equations through the continuity conditions on the interface and zero-traction conditions on free surfaces. The total wave field is obtained by the superposition of free field and scattered-wave field in elastic wedge space. Numerical results indicate that the IBEM can solve the diffraction of elastic wave in elastic wedge space accurately and effi- ciently. The wave motion feature strongly depends on the wedge angle, the angle of incidence, incident frequency, the location of lined tunnel, and material parameters. The waves interference and amplification effect around the tunnel in wedge space is more significant, causing the dynamic stress concentration factor on rigid tunnel and the displacement amplitude of flexible tunnel up to 50.0 and 17.0, respectively, more than double that of the case of half-space. Hence, considerable attention should be paid to seismic resistant or anti-explosion design of the tunnel built on a slope or hillside.

Ground motion duration effect on responses of hydraulic shallow-buried tunnel under SV-waves excitations

Although intensive research of the influence of ground motion duration on structural cumulative damage has been carried out, the influence of dynamic responses in underground tunnels remains a heated debate. This study attempts to highlight the importance of the ground motion duration effect on hydraulic tunnels subjected to deep-focus earthquakes. In the study, a set of 18 recorded accelerograms with a wide-range of durations were employed. A spectrally equivalent method serves to distinguish the effect of duration from other ground motion features, and then the seismic input model was simulated using SV-wave excitation based on a viscous-spring boundary, which was verified by the time-domain waves analysis method. The nonlinear analysis results demonstrate that the risk of collapse of the hydraulic tunnel is higher under long-duration ground motion than that of short-duration ground motion of the same seismic intensity. In a low intensity earthquake, the ground motion duration has little effect on the damage energy consumption of a hydraulic tunnel lining, but in a high intensity earthquake, dissipation of the damage energy and damage index of concrete shows a nonlinear growth trend accompanied by the increase of ground motion duration, which has a great influence on the deformation and stress of hydraulic tunnels, and correlation analysis shows that the correlation coefficient is greater than 0.8. Therefore, the duration of ground motion should be taken into consideration except for its intensity and frequency content in the design of hydraulic tunnel, and evaluation of seismic risk.

Diffraction of SH-waves by topographic features in a layered transversely isotropic half-space

Abstract： The scattering of plane SH-waves by topographic features in a layered transversely isotropic （TI） half-space is investigated by using an indirect boundary element method （IBEM）. Firstly, the anti-plane dynamic stiffness matrix of the layered TI half-space is established and the free fields are solved by using the direct stiffness method. Then, Green＇s functions are derived for uniformly distributed loads acting on an inclined line in a layered TI half-space and the scattered fields are constructed with the deduced Green＇s functions. Finally, the free fields are added to the scattered ones to obtain the global dynamic responses. The method is verified by comparing results with the published isotropic ones. Both the steady-state and transient dynamic responses are evaluated and discussed. Numerical results in the frequency domain show that surface motions for the TI media can be significantly different from those for the isotropic case, which are strongly dependent on the anisotropy property, incident angle and incident frequency. Results in the time domain show that the material anisotropy has important effects on the maximum duration and maximum amplitudes of the time histories.

Dynamic soil-tunnel interaction in layered half-space for incident P-and SV-waves

The dynamic soil-tunnel interaction is studied by the model of a rigid tunnel embedded in layered half-space, which is simplified as a single soil layer on elastic bedrock to the excitation of P- and SV-waves. The indirect boundary element method is used, combined with the Green＇ s function of distributed loads acting on inclined lines. It is shown that the dynamic characteristics of soil-tunnel interaction in layered half-space are different much from that in homoge- neous half-space, and that the mechanism of soil-tunnel interaction is also different much from that of soil-founda- tion-superstructure interaction. For oblique incidence, the tunnel response for in-plane incident SV-waves is com- pletely different from that for incident SH-waves, while the tunnel response for vertically incident SV-wave is very similar to that of vertically incident SH-wave.

Dynamic stress concentration and scattering of SH-wave by interface elliptic cylindrical cavity

In this paper,the dynamic stress concentration and scattering of SH-waves by bi-material structures that possess an interface elliptic cavity are investigated.First,by using the complex function method,the Green's function is constructed.This yields the solution of the displacement field for an elastic half space with a semi-elliptic canyon impacted by an anti-plane harmonic line source loading on the horizontal surface.Then,the problem is divided into an upper and lower half space along the horizontal interface,regarded as a harmony model.In order to satisfy the integral continuity condition, the unknown anti-plane forces are applied to the interface.The integral equations with unknown forces can be established through the continuity condition,and after transformation,the algebraic equations are solved numerically.Finally,the distribution of the dynamic stress concentration factor(DSCF)around the elliptic cavity is given and the effect of different parameters on DSCF is discussed.

Time-history responses on the surface by regularly distributed enormous embedded cavities:Incident SH-waves

The time-history responses of the surface were obtained for a linear elastic half-plane including regularly distributed enormous embedded circular cavities subjected to propagating obliquely incident plane SH-waves. An advanced numerical approach named half-plane time-domain boundary element method（BEM）, which only located the meshes around the cavities, was used to create the model. By establishing the modified boundary integral equation（BIE）independently for each cavity and forming the matrices, the final coupled equation was solved step-by-step in the timedomain to obtain the boundary values. The responses were developed for a half-plane with 512 cavities. The amplification patterns were also obtained to illustrate the frequencydomain responses for some cases. According to the results,the presence of enormous cavities affects the scattering and diffraction of the waves arrived to the surface. The introduced method can be recommended for geotechnical/mechanical engineers to model structures in the fields of earthquake engineering and composite materials.

Investigation of SH-Wave Fundamental Modes in Piezoelectromagnetic Plate: Electrically Closed and Magnetically Closed Boundary Conditions

It is theoretically considered the propagation (first evidence) of new dispersive shear-horizontal (SH) acoustic waves in the piezoelectromagnetic (magnetoelectroelastic) composite plates. The studied two-phase composites (BaTiO3-CoFe2O4 and PZT-5H-Terfenol-D) possess the piezoelectric phase (BaTiO3, PZT-5H) and the piezomagnetic phase (CoFe2O4, Terfenol-D). The mechanical, electrical, and magnetic boundary conditions applied to both the upper and lower free surfaces of the plate are as follows: the mechanically free, electrically closed, and magnetically closed surfaces. As a result, the fundamental modes of two new dispersive SH-waves recently discovered in book [Zakharenko, A.A. (2012) ISBN: 978-3-659-30943-4] were numerically calculated. It was found that for large values of normalized plate thickness kd (k and d are the wavenumber and plate half-thickness, respectively) the velocities of both the new dispersive SH-waves can approach the nondispersive SH-SAW velocity of the piezoelectric exchange surface Melkumyan (PEESM) wave. It was also discussed that for small values of kd, the experimental study of the new dispersive SH-waves can be preferable in comparison with the nondispersive PEESM wave. The obtained results can be constructive for creation of various technical devices based on (non)dispersive SH-waves and two-phase smart materials. The new dispersive SH-waves propagating in the plates can be also employed for nondestructive testing and evaluation. Also, it is obvious that the plates can be used in technical devices instead of the corresponding bulk samples for further miniaturization.

SCATTERING OF CIRCULAR CAVITY IN RIGHT-ANGLE PLANAR SPACE TO STEADY SH-WAVE

Complex function method and multi-polar coordinate transformation technology are used here to study scattering of circular cavity in right-angle planar space to SH-wave with out-of-plane loading on the horizontal straight boundary. At first, Green function of right-angle planar space which has no circular cavity is constructed; then the scattering solution which satisfies the free stress conditions of the two right-angle boundaries with the circular cavity existing in the space is formulated. Therefore, the total displacement field can be constructed using overlapping principle. An infinite algebraic equations of unknown coefficients existing in the scattering solution field can be gained using multi-polar coordinate and the free stress condition at the boundary of the circular cavity. It can be solved by using limit items in the infinite series which can give a high computation precision. An example is given to illustrate the variations of the tangential stress at the boundary of the circular cavity due to different dimensionless wave numbers, the location of the circular cavity, the loading center and the distributing range of the out-of-plane loading. The results show the efficiency and effectiveness of the mothod introduced here.

SCATTERING OF PLANE SH-WAVE BY A CYLINDRICAL HILL OF ARBITRARY SHAPE

The problems of scattering of plane SH-wave by a cylindrical hill of arbitrary shape is studied based on the methods of conjunction and division of solution zone. The scattering wave function is given by using the complex variable and conformal mapping methods. The conjunction boundary conditions are satisfied. Furthermore appling orthogonal function expanding technique, the problems can finally be summarized into the solution of a series of infinite algebraic equations. At last, numerical results of surface displacements of a cylindrical arc hill and of a semi-ellipse hill are obtained. And those computational results are compared with the results of finite element method (FEM).

Global SH-wave propagation in a 2D whole Moon model using the parallel hybrid PSM/FDM method

We present numerical modeling of SH-wave propagation for the recently proposed whole Moon model and try to improve our understanding of lunar seismic wave propagation. We use a hybrid PSM/FDM method on staggered grids to solve the wave equations and implement the calculation on a parallel PC cluster to improve the computing efficiency. Features of global SH-wave propagation are firstly discussed for a 100-km shallow and900-km deep moonquakes, respectively. Effects of frequency range and lateral variation of crust thickness are then investigated with various models. Our synthetic waveforms are finally compared with observed Apollo data to show the features of wave propagation that were produced by our model and those not reproduced by our models. Our numerical modeling show that the low-velocity upper crust plays significant role in the development of reverberating wave trains. Increasing frequency enhances the strength and duration of the reverberations.Surface multiples dominate wavefields for shallow event.Core–mantle reflections can be clearly identified for deep event at low frequency. The layered whole Moon model and the low-velocity upper crust produce the reverberating wave trains following each phases consistent with observation. However, more realistic Moon model should be considered in order to explain the strong and slow decay scattering between various phases shown on observation data.

Scattering of SH-wave from interface cylindrical elastic inclusion with a semicircular disconnected curve

Scattering of SH wave from an interface cylindrical elastic inclusion with a semicircular disconnected curve is investigated. The solution of dynamic stress concentration factor is given using the Green＇s function and the method of complex variable functions. First, the space is divided into upper and lower parts along the interface. In the lower half space, a suitable Green＇s function for the problem is constructed. It is an essential solution of the displacement field for an elastic half space with a semi-cylindrical hill of cylindrical elastic inclusion while bearing out-plane harmonic line source load at the horizontal surface. Thus, the semicircular disconnected curve can be constructed when the two parts are bonded and continuous on the interface loading the undetermined anti-plane forces on the horizontal surfaces. Also, the expressions of displacement and stress fields are obtained in this situation. Finally, examples and results of dynamic stress concentration factor are given. Influences of the cylindrical inclusion and the difference parameters of the two mediators are discussed.

Surface Effects on the Scatter of SH-Wave by a Shallow Buried the Elliptical Hole

The methods of complex function, multi-polar coordinate system, and conformal mapping are used to solve dynamic stress concentration factor. The surface elasticity theory is applied to obtain the stress boundary conditions on the surface. The effects of frequency and the ration of the major and minor axis of the ellipse on the dynamic stress concentration factor around the elliptical nano-hole are discussed in detail. When the size of elliptical hole shrinks to nanometers, the numerical results show that the surface effect has a significant effect on the scattering of SH-wave.

Global SH-wavefield calculation for a two-dimensional whole-Earth model with the parallel hybrid PSM/FDM algorithm

We present a parallel hybrid algorithm based on pseudospectral method （PSM） and finite difference method （FDM） for two-dimensional （2-D） global SH- wavefield simulation. The whole-Earth model is taken as a cross section of spherical Earth, and corresponding wave equations are defined in 2-D cylindrical coordinates. Spatial derivatives in the wave equations are approximated with efficient and high accuracy PSM in the lateral and high-order FDM in the radial direction on staggered grids. This algorithm allows us to divide the whole-Earth into sub-domains in radial direction and implement efficient parallel computing on PC cluster, while retains high accuracy and efficiency of PSM in lateral direction. A transformation of moment tensor between 3-D spherical Earth and our 2-D model was proposed to give corre- sponding moment tensor components used in 2-D modeling. Comparison of modeling results with those obtained by direct solution method shows very good accuracy of our algorithm. We also demonstrate its feasibility with a lateral heterogeneous whole-Earth model with localized velocity perturbation.

Scattering of SH-waves on triangular hill joined bysemi-cylindrical canyon

Scattering of SH-waves on the triangular hill joined by semi-cylindrical canyon in half-space is studied using the method of complex function and moving coordinates. The model being studied is divided into two domains. The wave functions satisfying the required condition at each wedge are constructed in each equation. The equations are solved with Fourier expansion. Numerical results are provided to discuss the influence of scattering of SH-waves.

SCATTERING OF PLANE SH-WAVES ON SEMI-CANYON TOPOGRAPHY OF ARBITRARY SHAPE WITH LINGING IN ANISOTROPIC MEDIA

The purpose of this paper is to use the conforma mapping method[1]to analyzeand evaluate the ground displacement and scattering of incident SH-waves, on thesurface of semi-canyon topography of arbitrary shape with lining in anisotropic media.The problem to be solved can be reduced to the solution of an infinite algebraicequation set by using the method of full-space expansion of Fourier progression Usingthe mapping function and scattering theory to solve problems due to semi-canyon topography with lining is just like mapping the semi-cylindrical canyon of arbitraryshape into a cylindrical canyon in full-space.Moreover,it is far practical inengineering practice.From the computational examples,it is obvious that the variation of displacement amplitudes on the surface near the canyon topography is rather sharp. especially when the freqencies of incident SH-waves increase.