Acceleration of near-field scattering from an inhomogeneous spherical shell

The three dimensional scattering of near-field, from a point source, is studied for acceleration in the time domain. The perturbation method is applied to define the acceleration for the first order scattering from a weak inhomogeneity in a homogeneous surrounding. A body force, arising from the interaction between the primary waves and the inhomogeneity, acts as the source generating the scattered motion. The acceleration of scattered waves is related to the velocity and density fluctuations of the inhomogeneity. No restrictions are placed on the inhomogeneity size or locations of the source and receiver. Decoupling of scattered motion enables the identification of different phases. Integral expressions are derived for the scattering acceleration due to the incidence of near-field wave (from an impulsive point force) at a radially inhomogeneous volume element. These integrals are solved further for scattering from an inhomogeneous spherical shell. The accelerations for back scattering are obtained as a special case. These accelerations are simple analytically solvable expressions in closed form. Only spherical asymmetry ofP wave velocity inhomogeneity can affect the scatteredS acceleration. ScatteredP acceleration is affected by the gradient ofS wave velocity inhomogeneity. The back scattering of near-field from a spherical shell, is independent of radial inhomogeneity ofP wave velocity. Inhomogeneity with smoothly perturbedS wave velocity does not back-scatter any acceleration. Accelerations are computed numerically for scattering from a part of inhomogeneous spherical shell. Hypothetical models are considered to study the effects of the distances of spherical shell from source, receiver, its thickness and its position relative to the direction of impulsive force.     

Scattering of a spherical pulse from a small inhomogeneity: Dilatation and rotation

Perturbations in elastic constants and density distinguish a volume inhomogeneity from its homogeneous surroundings. The equation of motion for the first order scattering is studied in the perturbed medium. The scattered waves are generated by the interaction between the primary waves and the inhomogeneity. First order scattering theory is modified to include the source term generating the primary waves. The body force equivalent to the scattering source is presented in a convenient form involving the perturbations in wave velocities and gradient of density perturbation. A procedure is presented to study the scattering of a spherical pulse from a small inhomogeneity, in time domain. The size of inhomogeneity is assumed small as compared to its distance from source and receiver. No restrictions are placed on the positions of source, receiver and inhomogeneity. The dilatation and rotations are calculated for a pulse scattered from an arbitrary point in a spherical volume. The aggregate of the scattered phases from all the points of the inhomogeneity, reaching at a fixed receiver, gives the amount of scattering from the inhomogeneity. The interaction of bothP andS waves with inhomogeneity are considered. Dilatation and rotations for scattering are obtained as integral expressions over the solid angle of inhomogeneity. These expressions are computed numerically, for hypothetical models. The effects of source (unit force) orientations, velocity and density perturbations, and size of inhomogeneity, on the scattered phases, are discussed.     

Dynamic response of an elastic and viscoelastic full-space to a spherical source

The dynamic response due to a spherical source of radius a embedded in an elastic and viscoelastic full-space is investigated at a distance R from the source. Previous solutions to the elastic case are extended to incorporate realistic source pressure functions. The elastic solution is then cast in a scale independent form in order to generalize the application. The results show that the near-field of the spherical source may be defined by R/a < 5. For this region the particle velocity and displacement decrease as R^?2, and the risetime decreases as R^?1. However. in the far-field region (R/a > 5) the particle velocity and displacement decrease as R^?1, and the risetime is independent of R. A non-constant Q model is developed to model viscoelastic attenuation and a complete analytical solution for wave propagation is obtained by cascading the separate mechanisms of geometric attenuation and viscoelastic attenuation. A comparison of our analytical model with the results of dynamic finite element modelling shows excellent agreement. This suggests that the method of cascading the separate transfer functions is a valid approach for wave propagation in viscoelastic media.     

Torsional wave propagation in non‐homogeneous layer between non‐homogeneous half‐spaces

The study of surface wave in a layered medium has their possible application in geophysical prospecting. In the present work, dispersion equation for torsional wave in an inhomogeneous isotropic layer between inhomogeneous isotropic half‐spaces has been derived. Two cases are discussed separately for torsional wave propagation in inhomogeneous layer between homogeneous and non‐homogeneous half‐spaces, respectively. Further, two possible modes for torsional wave propagation are obtained in case of inhomogeneous layer sandwiched between non‐homogeneous half‐spaces. Closed form solutions for displacement in the layer and half‐spaces are obtained in each case. The study reveals that the layer width, layer inhomogeneity, frequency of inhomogeneity, as well as inhomogeneity in the half‐space has significant effect on the propagation of torsional surface waves. Displacement and implicit dispersion equation for torsional wave velocities are expressed in terms of Heun functions and their derivatives. Effects of inhomogeneity on torsional wave velocity are also discussed graphically by plotting the dimensionless phase velocity against dimensionless and scaled wave number for different values of inhomogeneity parameter. Copyright © 2012 John Wiley & Sons, Ltd.     

The method of fundamental solution for 3‐D wave scattering in a fluid‐saturated poroelastic infinite domain

By using a complete set of poroelastodynamic spherical wave potentials (SWPs) representing a fast compressional wave P_I, a slow compressional wave P_II, and a shear wave S with 3 vectorial potentials (not all are independent), a solution scheme based on the method of fundamental solution (MFS) is devised to solve 3‐D wave scattering and dynamic stress concentration problems due to inhomogeneous inclusions and cavities embedded in an infinite poroelastic domain. The method is verified by comparing the result with the elastic analytical solution, which is a degenerated case, as well as with poroelastic solution obtained using other numerical methods. The accuracy and stability of the SWP‐MFS are also demonstrated. The displacement, hoop stress, and fluid pore pressure around spherical cavity and poroelastic inclusion with permeable and impermeable boundary are investigated for incident plane P_I and SV waves. The scattering characteristics are examined for a range of material properties, such as porosity and shear modulus contrast, over a range of frequency. Compared with other boundary‐based numerical strategy, such as the boundary element method and the indirect boundary integral equation method, the current SWP‐MFS is a meshless method that does not need elements to approximate the geometry and is free from the treatment of singularities. The SWP‐MFS is a highly accurate and efficient solution methodology for wave scattering problems of arbitrary geometry, particularly when a part of the domain extends to infinity.     

2D rotation of rigid inclusions in confined bulk simple shear flow: a numerical study

We have used incompressible Navier–Stokes in 2D finite element modelling to investigate rigid inclusion rotation under confined bulk simple shear flow. Confinement is defined as the ratio (S) between the channel width (H) and the inclusion's least axis (e₂)(S=H/e₂). The numerical results show that (i) inclusion rotation is strongly influenced by S and, when the confinement is effective, aspect ratio (R) and shape also play an important role. (ii) Back rotation is limited because inclusions reach a stable equilibrium orientation (_se). (iii) There is also an unstable equilibrium orientation (_ue), which defines an antithetic rotation field with _se, and both _se and _ue depend on S, and inclusion R and shape.     

2-D rotation behavior of a rigid ellipse in confined viscous simple shear: numerical experiments using FEM

The rotation behavior of rigid elliptical inclusions adherent to the viscous matrix in simple shear flow is investigated using a 2-D finite element numerical model. Several simulations were performed using different ratios (S) between shear zone width and inclusion's least principal axis. A computational strategy was devised to calculate pressure and viscous forces exerted on the inclusion and deduce its angular velocity. For large S values, results agree remarkably well with theoretical predictions, while for small S values results deviate significantly from theory but are in agreement with previous analogue experiments. The numerical model provided detailed and coherent information about the physical parameters involved in the process (e.g., pressure, strain rate and vorticity distributions within the model).     

Simultaneous inversion of the aftershock data of the 1993 Killari earthquake in Peninsular India and its seismotectonic implications

The aftershock sequence of the September 30th, 1993 Killari earthquake in the Latur district of Maharashtra state, India, recorded by 41 temporary seismograph stations are used for estimating 3-D velocity structure in the epicentral area. The local earthquake tomography (LET) method of Thurber (1983) is used. About 1500P and 1200S wave travel-times are inverted. TheP andS wave velocities as well asV _P/V_Sratio vary more rapidly in the vertical as well as in the horizontal directions in the source region compared to the adjacent areas. The main shock hypocentre is located at the junction of a high velocity and a low velocity zone, representing a fault zone at 6–7 km depth. The estimated average errors ofP velocity andV _P/V_Sratio are ±0.07 km/s and ±0.016, respectively. The best resolution ofP and S-wave velocities is obtained in the aftershock zone. The 3-D velocity structure and precise locations of the aftershocks suggest a ‘stationary concept’ of the Killari earthquake sequence.     

Near-field ground motion during earthquake preparation process

Wave number discretization method is applied to study the near-field of seismic sources embedded in a cracked elastic solid. Near-field solutions are obtained for horizontal and vertical line forces. Effects of modifications in cracks of focal region on ground motion, in the near-field, are studied numerically for different

? values of crack density

? saturation of cracks

? width of cracks, and

? regimes of connection between cracks.

An earthquake process is assumed to be going through five major stages. These stages represent continuous accumulation of stress, interconnections between cracks leading to eventual failure and drainage of fluid from cracks after the major shock. Variations in the velocity ratio of waves noted from accelerograms verify the process of preparation of an earthquake.P wave contribution to vertical acceleration is negligible when the source is a vertical line force andS wave contributes only a little to horizontal acceleration when the source is a horizontal line force.     

Wave propagation in an inhomogeneous cross‐anisotropic medium

Analytical solutions for wave velocities and wave vectors are yielded for a continuously inhomogeneous cross‐anisotropic medium, in which Young's moduli (E, E′) and shear modulus (G′) varied exponentially as depth increased. However, for the rest moduli in cross‐anisotropic materials, ν and ν′ remained constant regardless of depth. We assume that cross‐anisotropy planes are parallel to the horizontal surface. The generalized Hooke's law, strain–displacement relationships, and equilibrium equations are integrated to constitute governing equations. In these equations, displacement components are fundamental variables and, hence, the solutions of three quasi‐wave velocities, V_P, V_SV, and V_SH, and the wave vectors, $\mathop{\mathop{l}\limits^{\rightharpoonup}}\nolimits_{P}$ $\mathop{\mathop{l}\limits^{\rightharpoonup}}\nolimits_{\mathit{SV}}$, and $\mathop{\mathop{l}\limits^{\rightharpoonup}}\nolimits_{{\mathit{SH}}}$, can be generated for the inhomogeneous cross‐anisotropic media. The proposed solutions and those obtained by Daley and Hron, and Levin correlate well with each other when the inhomogeneity parameter, k, is 0. Additionally, parametric study results indicate that the magnitudes and directions of wave velocity are markedly affected by (1) the inhomogeneous parameter, k; (2) the type and degree of geomaterial anisotropy (E/E′, G′/E′, and ν/ν′); and (3) the phase angle, θ. Consequently, one must consider the influence of inhomogeneous characteristic when investigating the behaviors of wave propagation in a cross‐anisotropic medium. Copyright © 2009 John Wiley & Sons, Ltd.     

岩石非均质度对其力学性能的影响研究

根据非均质岩石参数赋值方法具有随机性的特点,以随机参数的变异系数定义岩石非均质度,推导了Weibull分布参数赋值方法的非均质度计算方法,并与形状参数进行了对比分析;推导了基于矿物细胞元参数赋值方法的岩石非均质度的计算方法;利用上述2种赋值方法研究了岩石非均质度对其力学性能的影响。研究结果表明,利用变异系数定义岩石非均质度是合理的,可用于描述不同非均质岩石模型的非均质程度;非均质岩石的力学特性由细观单元力学特性和其非均质特性共同决定,岩石的非均质性对其极限强度具有弱化影响;对于不同的非均质岩石模型,可建立相同的线性函数关系表示非均质岩石极限强度受其非均质度的影响。     

Relating discrete element method parameters to rock properties using classical and micropolar elasticity theories

Micro–macro relations for discrete element method (DEM) media are derived using both classical and micropolar elasticity theories. The DEM media are classified into two main categories: dense packing, and loose packing. For both categories, relations for Young modulus (E), Poisson's ratio (ν) to represent static behaviors, and wave velocities (P‐wave and S‐wave) to represent dynamic behaviors are derived using the internal DEM parameters (k_n, k_s) and compared with values obtained from static and dynamic numerical tests. Whereas the dynamic behaviors for the two categories and the static behaviors for the dense packing match the analytical relations, the static behavior for the loose packing does not. Micropolar elasticity theory is also used to study the behaviors of the DEM media, where it is shown that if element rotation is included, DEM media behave according to linear elasticity theory. However, if element rotation is constrained, asymmetrical stresses arise in the DEM media, and a new expression is derived for the S‐wave, which allows it, under certain conditions, to travel faster than the P‐wave. Copyright © 2011 John Wiley & Sons, Ltd.     

基于大直径霍普金森压杆数值试验的非均匀介质动态破坏过程分析

基于细观损伤力学基础而开发的动态岩石破裂过程分析系统RFPA2D代替大直径SHPB试验技术对非均匀介质的动态破坏过程和动态性能进行数值模拟,分析了加载波形和介质的非均匀性对数值试样的动态性能,如应力-时间曲线、应变-时间曲线、应力-应变曲线和应变率-时间曲线的影响。分析表明,大直径SHPB弥散效应对试验结果的影响较大,选择合适的加载波形可以减小SHPB装置中应力波的弥散效应,得到较准确的试验结果,其中三角形波加载可以有效降低大直径SHPB动态测试中的应力波弥散,是岩石等非均匀材料SHPB动态测试的较理想加载波形;在相同加载条件下,岩石的非均匀性对波的弥散效应影响不大,但非均匀岩石的试验曲线比均匀岩石的曲线在波峰后都出现较大的振荡,这主要是由于不同均质度的岩石其单元具有不同的破坏分布造成的,不是波形弥散造成的。     

Compressional and shear wave velocities at high temperature and confining pressure in basalts from the Faeroe Islands

Compressional (V_P) and shear (V_S) wave velocities and the dependent elastic constants have been determined by the pulse transmission technique to 6 kb confining pressure at room temperature and to 700° C at 6 kb confining pressure for eleven basalts from the Faeroe Islands. The Faeroe basalts investigated are tholeiitic, they clearly lie within the tholeiitic area, and display a pronounced trend of iron enrichment from rocks with an M/M + F ratio of 0.5 to rocks with an M/M + F ratio of about 0.25. The mean V_P and V_S for eleven specimens are 5.57 km/sec and 3.18 km/sec, respectively. Velocity—density relations for the basalts might be more appropriately described by non-linear solutions than by linear relations commonly used for basalts. In general, V_P and V_S remain unaffected by temperature up to 300° C. At higher temperature the changes in wave velocities are influenced by metamorphic processes and are, therefore, somewhat erratic. In zeolite-bearing specimens an abrupt velocity decrease around 350°C is observed, which correlates well with a drastic compaction of bulk volume. Additional experiments on cold-pressed zeolite powder clearly indicate that the sharp velocity decrease in the basalts is related to dehydration of zeolite minerals. Partial-melting processes, which occur within vesicules and pore-spaces at distinctly higher temperatures have no additional effect on wave velocity. Comparison with field data reveals that, without exception, the velocities at 0.5 kb confining pressure display the same range that has been commonly noted in refraction data for Layer 2. There are no significant differences in wave velocities and the pressure—temperature dependence in samples recovered from the upper, middle, and lower basalt series in the Faeroe Islands.     

The shape of pebbles

A theory of pebble erosion is presented, based on the assumption that the rate of erosion at a point on the surface is a function Vof the curvature there. It is proved that for physically reasonable functions V,the sphere is the only shape of pebble which can maintain its proportions as it wears away. An argument is given which leads to a particular form for the function Vand a few qualitative consequences of this form are indicated. The surface of the pebble at time tmay be described using spherical polar coordinates θ, Φ by the radius function r (θ, Φ, t). This function is given by a highly nonlinear partial differential equation. However, in the case of the erosion of a deformed sphere, when terms which are of second order or higher in the deformation are neglected, the equation becomes linear and is a version of the diffusion equation. The stability of the spherical shape against deformations of the various harmonic types is then easily analyzed.     

抗转动对颗粒材料组构特性的影响研究

颗粒材料大多由不规则形状的颗粒组成,如砂土、谷物等,抵抗转动是不规则形状颗粒的固有特性。已有研究表明,颗粒抗转动特性对其宏观力学特性有显著影响。因此,在颗粒材料的细观数值模拟中或采用非圆颗粒,或在圆颗粒离散元模拟中采用考虑抗转动的接触模型。采用有限元-离散元耦合方法（FDEM）和离散元方法（DEM）分别对椭球形状颗粒和具有抗转动能力的圆球颗粒进行三轴剪切数值模拟,指出了采用抗转动接触模型考虑颗粒形状影响的局限性,并基于颗粒的局部排布结构揭示了形状影响的细观来源。峰值内摩擦角和剪胀均随着转动摩擦系数和形状偏离圆球程度而单调增加,但颗粒形状对它们的影响呈现出明显的收敛趋势。细观组构分析也表明,虽然颗粒形状和转动摩擦都能显著增强组构各向异性,但是组构各向异性的演化模式有明显的区别。造成以上结果的差异在于其抵抗转动的影响机制不同。转动摩擦是通过限制颗粒转动,增强了颗粒间的稳定承载能力,而非圆颗粒是通过咬合作用形成稳定的局部排列结构。由于椭球颗粒腹部比端部能够传递更大的接触力,颗粒受剪切后发生转动,颗粒长轴倾向于正交大主应力方向,呈现交错排列,颗粒间相互锁定。     

Reorientation of early biotites during schistosity formation in andalusite schists, Mount Franks area, Broken Hill, N.S.W.

Study of schistosity formation in andalusite—grade mica schists from part of the Pre-cambrian Willyama Complex in New South Wales is facilitated by the coarse grained nature of the rocks, and by the presence of deformation indicators provided by pre-S₂ biotite. S₂, the dominant schistosity, is generally domainal. It is defined by the alternation of mica (M) and quartz + mica (QM) domains. M domains are dominated by biotite aggregates with a very marked shape (but not a crystallographic) orientation parallel to S₂, and by S₂ muscovite laths. QM domains are dominated by kinked biotite grains, biotites aligned across S₂, and biotite lozenges and grains in which (001) traces are aligned oblique to S₂. Biotite grains in these domains are less elongate than those in M domains. Individual biotite grains have been reoriented by kinking and corrosion; some homo- geneous glide on (001) may also have taken place. The variation in these deformation effects indicates that M domains represent zones of high strain with respect to the QM domains. M domains have undergone a history of shortening, rotation, diffusive mass transfer, volume reduction and syntectonic crystallization of muscovite. QM domains have also undergone syntectonic crystallization of muscovite, but their history is marked by less rotation, shortening, mass transfer and volume reduction than that of M domains. The greater activity of mass transfer mechanisms in M domains suggests that they are strain dependent, and proceed more easily in more highly deformed grains. Metamorphic driving forces associated with chemical reaction may play a part in the generation of these mass transfer mechanisms.     

Manifestation of Singlets of the Basic Spheroidal Mode of the Earth 0S2 in Geomagnetic Variations

The results of instrumental observations have shown that the spectra of geomagnetic variations exhibit a quasi-harmonic component, the frequency of which is close to the basic spheroidal mode of the Earth ₀S₂. In periods 15 days long after three large earthquakes and in the absence of strong magnetic disturbances, the fine structure of this mode in geomagnetic variations is identified in the form of singlets. The frequencies of these singlets are similar to the theoretical values calculated using an ideal elastic isotropic model of the Earth of spherical symmetric form without its rotation.

     

The average molecular weight and shape of the “polymeric acids” found in Black Trona Water from the Green River Basin

Macromolecular organic material, called “polymeric acids”, has been isolated from Black Trona Water by exhaustive dialysis and characterized as the sodium salt in 0.10 M sodium carbonate, pH 10, by several physico-chemical methods. Analysis by gel filtration chromatography on Sepharose-CL 6B indicates that the “polymeric acids” are polydisperse and composed of species of relatively high molecular weight ( 4 × 10⁵, using proteins as standards). With this method, the range of molecular weights appears to be rather narrow. If “polymeric acids” are transferred from sodium carbonate, pH 10, into distilled water, selfassociation occurs and all species elute in the void volume. The weight-average molecular weight determined in 0.10 M sodium carbonate, pH 10, by the light scattering method is 1.7 × 10⁵. Sedimentation velocity analysis at 20°C with the analytical ultracentrifuge gives a value for S_20,w of 5.4 and the shape of the Schlieren patterns suggest a polydisperse sample with a relatively narrow range of sizes. Analysis of the molecular weight distribution by a sedimentation equilibrium method indicates that the range of molecular weights is 8 × 10⁴ to 2.1 × 10⁵. The partial specific volume ( ) of “polymeric acids” is 0.874 ml/g. Viscosity measurements yield a value for [η] of 2.5 ml/g, which indicates that the “polymeric acids” are compact (spherical or ellipsoidal) in shape.     

Locating a point on a spherical surface relative to a spherical polygon of arbitrary shape

An algorithm for determining if any given point,P, on the surface of a sphere is located inside, outside, or along the border of an arbitrary spherical polygon,S, is described. The polygon is described by specifying coordinates of its vertices, and coordinates of some pointX which is known to lie withinS. The algorithm is based on the principle that an arc joiningX andP will cross the border ofS an odd number of times ifP lies outsideS, and an even number of times ifP lies withinS. The algorithm has been implemented as a set of FORTRAN subroutines, and a listing is provided. The algorithm and subroutine package can be used with spherical polygons containing holes, or with composited spherical polygons.