Differential geometric approach to the stress aspect of a fault: Airy stress function surface and curvatures

We considered the two-dimensional stress aspect of a fault from the viewpoint of differential geometry. For this analysis, we concentrated on the curvatures of the Airy stress function surface. We found the following: (i) Because the principal stresses are the principal curvatures of the stress function surface, the first and the second invariant quantities in the elasticity correspond to invariant quantities in differential geometry; specifically, the mean and Gaussian curvatures, respectively; (ii) Coulomb’s failure criterion shows that the coefficient of friction is the physical expression of the geometric energy of the stress function surface; (iii) The differential geometric expression of the Goursat formula shows that the fault (dislocation) type (strike-slip or dip-slip) corresponds to the stress function surface type (elliptic or hyperbolic). Finally, we discuss the need to use non-biharmonic stress tensor theory to describe the stress aspect of multi-faults or an earthquake source zone.     

Asymptotic Elastodynamic Green Function in the Kiss Singularity in Homogeneous Anisotropic Solids

The far-field asymptotic formula is derived for the elastodynamic Green function in the kiss singularity in homogeneous anisotropic solids. In contrast to standard asymptotics in regular directions the derived formula is more complex and expressed in the form of a 1-D integral. This integral is specified for the kiss singularity along the symmetry axis in transverse isotropy and along the fourfold symmetry axes in tetragonal and cubic symmetries. The shape of the slowness surface in the singularity is regular in transverse isotropy and the amplitude of the Green function is expressed by means of the Gaussian curvature of this surface in the singularity. However, the shape of the slowness surface is irregular and the Gaussian curvature is not defined in the singularity in tetragonal or cubic symmetries. In this case, the amplitude of the Green function is expressed by means of the generalized Gaussian curvature.     

A method of reconstructing complex stratigraphic surfaces with multitype fault constraints

The construction of complex stratigraphic surfaces is widely employed in many fields, such as petroleum exploration, geological modeling, and geological structure analysis. It also serves as an important foundation for data visualization and visual analysis in these fields. The existing surface construction methods have several deficiencies and face various difficulties, such as the presence of multitype faults and roughness of resulting surfaces. In this paper, a surface modeling method that uses geometric partial differential equations (PDEs) is introduced for the construction of stratigraphic surfaces. It effectively solves the problem of surface roughness caused by the irregularity of stratigraphic data distribution. To cope with the presence of multitype complex faults, a two-way projection algorithm between threedimensional space and a two-dimensional plane is proposed. Using this algorithm, a unified method based on geometric PDEs is developed for dealing with multitype faults. Moreover, the corresponding geometric PDE is derived, and an algorithm based on an evolutionary solution is developed. The algorithm proposed for constructing spatial surfaces with real data verifies its computational efficiency and its ability to handle irregular data distribution. In particular, it can reconstruct faulty surfaces, especially those with overthrust faults.     

基于重力梯度张量曲率的边界识别

重力梯度张量曲率目前广泛用于重力数据的处理和解释中.为了拓宽重力梯度张量曲率的应用,本文回顾了重力梯度张量曲率的定义,从等位面的曲率定义出发,讨论了正确计算曲率的测量参考系及局部旋转的相关理论,并以单个球体和棱柱体为例来说明曲率的正确计算方式.然后,在正确计算重力梯度张量曲率的基础上,将重力梯度张量曲率应用到重力数据的边界识别中,通过理论模型和实际数据详细分析和比较了各种曲率在重力边界识别中的应用效果.结果表明:基于等位面的局部旋转坐标系是各种曲率正确计算的先决条件,纠正了曲率计算的误区;在边界识别中,局部坐标系下所计算的高斯曲率进行边界识别能够较好的圈定地下地质体的边界.     

Relation of the Wave-Propagation Metric Tensor to the Curvatures of the Slowness and Ray-Velocity Surfaces

The contravariant components of the wave-propagation metric tensor equal half the second-order partial derivatives of the selected eigenvalue of the Christoffel matrix with respect to the slowness-vector components. The relations of the wave-propagation metric tensor to the curvature matrix and Gaussian curvature of the slowness surface and to the curvature matrix and Gaussian curvature of the ray-velocity surface are demonstrated with the help of ray-centred coordinates.     

The effects of couple-stresses on the propagation of waves from cylindrical cavity

Summary In the present paper, the wave propagation from a cylindrical cavity in an infinite medium exhibiting couple-stress is considered. The variations of Hoop-stress on the cavity surface are investigated for different couple-stress material moduli and wave lengths. Significant differences are found when compared with those based on classical theory of elasticity.     

Numerical modelling of aeolian erosion over rough surfaces

The presence of non‐erodible roughness elements on erodible surfaces has the effect of absorbing part of the wind shear stress and thus protecting the erodible surface from wind erosion. This paper examines the shear stress distribution over roughness arrays of varying density, representing the progress of erosion on a bed of erodible and non‐erodible particles. Three‐dimensional numerical simulations, simulating wind flow over a bed of particles covered by roughness elements, were conducted in order to investigate the effect of roughness elements on the shear stress near the surface. The results of these simulations confirm that the erosion of soil by wind is strongly attenuated by the presence of roughness elements on the surface and depends on the geometric properties of the roughness elements. Based on the new numerical results obtained, a refinement of existing theoretical approaches is developed to describe the dependence of the friction velocity upon roughness frontal area and real exposed cover rate. The new formulation proposed will allow a more accurate evaluation of shear stress partitioning as a function of topographic changes. Copyright © 2010 John Wiley & Sons, Ltd.     

Effect of different meander curvatures on spatial variation of coherent turbulent flow structure inside ingoing multi-bend river meanders

In this paper, the effect of different curvatures on the spatial variation of coherent flow structure inside two physical models with both strongly curved and mild multi-bend meanders is investigated. Three dimensional flow velocities at three sequential meanders were measured using an Acoustic Doppler Velocity meter (Micro-ADV). Three dimensions of flow velocity are classified into two major classes and eight different bursting events. The contribution probability and transition probability of each zone is calculated from experimental data. The results indicated that the effect of curvature in sequential bends was important particularly for strongly curved bends. The contribution probability of the events for strongly curved meanders with relative curvature (R_c/B) of 2.6 were found to be higher than for mild curved meanders with relative curvature (R_c/B) of 4.43. The minimum contribution probability was found in external inward interaction event. In addition, analysis of bursting events showed that the highest values of transition probabilities occurred in the stable organizations for both models. The influences of different curvatures on distributions of the Reynolds shear stress, the turbulent kinetic energy, the streamwise velocity and the vertical velocity were also shown to be in good agreement with eroded bed. The above results can be useful for finding meandering patterns inside rivers and also in river training works.     

Migration of growth axial surfaces and its implications for multiphase tectono-sedimentary evolution of the Zhangwu fault depression,southern Songliao Basin,NE China

The normal fault-bend folding theory uses active axial surfaces, inactive axial surfaces, and growth axial surfaces to describe the geometric relationship between faults and deformation of a hanging wall. The dip of a growth axial surface is related to the fault slip rate and the basin sedimentation rate: higher fault slip rates result in smaller dips of growth axial surfaces, whereas higher basin sedimentation rates produce larger dips of growth axial surfaces. Moreover, the growth axial surface will be a straight line if both the fault slip and sedimentation rates remain relatively unchanged, but will become curved if both rates are variable. Therefore, the characteristics of growth axial surfaces can provide clear information on the evolution of faulting and deposition. By studying the seismic profiles of the Zhangwu fault depression of the southern Songliao Basin, we show that the migration of growth axial surfaces and unconformities can be used as indicators of basin development. Multiphase tectonic activity will not only produce unconformities but also result in migration of growth axial surfaces or inactive axial surfaces. Therefore, the normal fault-bend folding theory, particularly with regard to the evolution of growth axial surfaces, can be applied to the interpretation of geometric and kinematic evolutions of half-grabens and the exploration of related tectono-sedimentary processes.     

Fractal generation of surface area of porous media

Many natural porous geological rock formations, as well as engineered porous structures, have fractal properties, i.e., they are self-similar over several length scales. While there have been many experimental and theoretical studies on how to quantify a fractal porous medium and on how to determine its fractal dimension, the numerical generation of a fractal pore structure with predefined statistical and scaling properties is somewhat scarcer. In the present paper a new numerical method for generating a three-dimensional porous medium with any desired probability density function (PDF) and autocorrelation function (ACF) is presented. The well-known Turning Bands Method (TBM) is modified to generate three-dimensional synthetic isotropic and anisotropic porous media with a Gaussian PDF and exponential-decay ACF. Porous media with other PDF's and ACF's are constructed with a nonlinear, iterative PDF and ACF transformation, whereby the arbitrary PDF is converted to an equivalent Gaussian PDF which is then simulated with the classical TBM. Employing a new method for the estimation of the surface area for a given porosity, the fractal dimensions of the surface area of the synthetic porous media generated in this way are then measured by classical fractal perimeter/area relationships. Different 3D porous media are simulated by varying the porosity and the correlation structure of the random field. The performance of the simulations is evaluated by checking the ensemble statistics, the mean, variance and ACF of the simulated random field. For a porous medium with Gaussian PDF, an average fractal dimension of approximately 2.76 is obtained which is in the range of values of actually measured fractal dimensions of molecular surfaces. For a porous medium with a non-Gaussian quadratic PDF the calculated fractal dimension appears to be consistently higher and averages 2.82. The results also show that the fractal dimension is neither strongly dependent of the porosity nor of the degree of anisotropy assumed.     

Continuous wavelet-like filter for a spherical surface and its application to localized admittance function on Mars

We have developed a 2D isotropic continuous wavelet-like transform for a spherical surface. The transform is simply defined as the surface convolution between the original field and a kernel, based on the zeroth-order Bessel function with a spherical correction. This spherical correction violates the geometric similarity for the various scales of the kernels, which becomes more apparent at longer wavelengths. We found numerically that this transform is practically equivalent to a Gaussian bandpass filter in the spherical harmonic domain. We have applied this wavelet-like transform on the recently acquired Martian gravity and topography fields. Using a ratio constructed locally from these two fields, we have constructed a map describing the lateral variations of the localized admittance function on Mars.     

Fractal generation of surface area of porous media

Many natural porous geological rock formations, as well as engineered porous structures, have fractal properties, i.e., they are self-similar over several length scales. While there have been many experimental and theoretical studies on how to quantify a fractal porous medium and on how to determine its fractal dimension, the numerical generation of a fractal pore structure with predefined statistical and scaling properties is somewhat scarcer. In the present paper a new numerical method for generating a three-dimensional porous medium with any desired probability density function (PDF) and autocorrelation function (ACF) is presented. The well-known Turning Bands Method (TBM) is modified to generate three-dimensional synthetic isotropic and anisotropic porous media with a Gaussian PDF and exponential-decay ACF. Porous media with other PDF's and ACF's are constructed with a nonlinear, iterative PDF and ACF transformation, whereby the arbitrary PDF is converted to an equivalent Gaussian PDF which is then simulated with the classical TBM. Employing a new method for the estimation of the surface area for a given porosity, the fractal dimensions of the surface area of the synthetic porous media generated in this way are then measured by classical fractal perimeter/area relationships. Different 3D porous media are simulated by varying the porosity and the correlation structure of the random field. The performance of the simulations is evaluated by checking the ensemble statistics, the mean, variance and ACF of the simulated random field. For a porous medium with Gaussian PDF, an average fractal dimension of approximately 2.76 is obtained which is in the range of values of actually measured fractal dimensions of molecular surfaces. For a porous medium with a non-Gaussian quadratic PDF the calculated fractal dimension appears to be consistently higher and averages 2.82. The results also show that the fractal dimension is neither strongly dependent of the porosity nor of the degree of anisotropy assumed.     

Theory and experimental study for sliding isolators with variable curvature

Because a conventional isolation system with constant isolation frequency is usually a long‐period dynamic system, its seismic response is likely to be amplified in earthquakes with strong long‐period wave components, such as near‐fault ground motions. Seismic isolators with variable mechanical properties may provide a promising solution to alleviate this problem. To this end, in this work sliding isolators with variable curvature (SIVC) were studied experimentally. An SIVC isolator is similar to a friction pendulum system (FPS) isolator, except that its sliding surface has variable curvature rather being spherical. As a result, the SIVC's isolation stiffness that is proportional to the curvature becomes a function of the isolator displacement. By appropriately designing the geometry of the sliding surface, the SIVC is able to possess favorable hysteretic behavior. In order to prove the applicability of the SIVC concept, several prototype SIVC isolators, whose sliding surfaces are defined by a sixth‐order polynomial function, were fabricated and tested in this study. A cyclic element test on the prototype SIVC isolators and a shaking table test on an SIVC isolated steel frame were all conducted. The results of both tests have verified that the prototype SIVC isolators do indeed have the hysteretic property of variable stiffness as prescribed by the derived formulas in this study. Moreover, it is also demonstrated that the proposed SIVC is able to effectively reduce the isolator drift in a near‐fault earthquake with strong long‐period components, as compared with that of an FPS system with the same friction coefficient. Copyright © 2011 John Wiley & Sons, Ltd.     

The fractal geometry of the soil—covered landscape

The soil-covered landscape surface can be idealized from two viewpoints. The intuitive view is of a smooth, absolutely continuous surface with continuous contour lines and measurable in integral dimensions. The alternative view emphasizes the roughness, a surface of little regularity and at the limit of no contours, the appropriate measure being that of fractional Hausdorff dimension. Regularity is a local property and both idealizations need to stop far short of the limit to avoid awkward consequences. The dichotomy of viewpoint can be matched in the theory of Gaussian random fields. These, if they are smooth, are very smooth but if they are irregular they are highly irregular (erratic); there is no middle ground. This Belayev dichotomy is defined and both modes applied to the soil-covered landscape. On the one hand, if the landscape is subject to a general diffusive type degradation or more generally a Davisian downwasting regime then the curvature of the landscape surface is progressively straightened and the distribution of gradient (increments) along a typical traverse will eventually adopt a Gaussian form. Then from the irregular viewpoint the surface is ultimately well represented by a fractional Brownian surface of low Hausdorff dimension (2·0 < dim < 2·3). The Hausdorff dimension is directly related to the entropy of the landscape and as degradation proceeds both quantities decrease in value. On the other hand, if the surface is regarded as smooth and well represented by an absolutely continuous Gaussian field then the mean value of the number of upcrossings of a level or the extent of an excursion set will also be Gaussian. This analysis is restricted to one dimension; the number of times a profile curve crosses or the amount of time it spends above any given level. Predictions from both viewpoints are substantially corroborated in a map analysis of 15 sites on varied terrains in Southern England and the map analysis checked against one based upon digital tape data for one of the sites.     

A NEW DERIVATION OF THE WAVEFRONT CURVATURE TRANSFORMATION AT AN INTERFACE BETWEEN TWO INHOMOGENEOUS MEDIA*

The principles for ray-tracing and wavefront curvature calculations in a three-dimensional medium are reviewed. A new derivation of the transformation of the wavefront curvature matrix at an interface between two inhomogeneous media is given. The derivation is based on a Taylor series expansion of the ray refraction equation at the interface between two inhomogeneous media, and only elementary geometric arguments are used. The wave-front curvature transformation at the interface is obtained by neglecting all terms in the direction of the surface normal. With proper definition of the variables, the derivation is also valid for a reflected wave-front. A simplified transformation rule is derived for a reflected wave of the same type as the incident wave.     

Approximation of interfacial properties in multiphase porous medium systems

We investigate the estimation of interfacial areas, curvatures, and common curve lengths in multiphase porous medium systems. Algorithms are developed to obtain estimates of these quantities based upon a variety of potential data sources and estimation approaches. The accuracy of the derived approximations are evaluated as a function of the data type and resolution of the data. The methods advanced improve upon standard approaches now in use and show excellent accuracy at resolutions on the order of five lattice points per minimum radius of curvature of the object being resolved. Finally, we suggest a promising class of extensions that could lead to further improvements in the accuracy of such methods.     

垂直地震力作用下弹性半空间上扁壳基础的解析解

用布希涅斯克定义的弹性半空间内的垂直位移包括两项积分,除了积分号前面系数的差别之外,第一项积分是单层位势而第二项积分为双层位势。若扁壳基础是正高斯曲率的几何曲面,则壳底与半空间表面间的挤压强度就是半空间表面作用的分布垂直荷载。当越过边界时,双层势位的函数值和单层势位的法向导数值发生跳跃。利用这些性质,本文得出布希涅斯克积分的反演公式,从而避开要求解偏微分—积分方程组的巨大数学困难而易于得出解析解。以椭园抛物面扁壳为例说明本文方法的应用。     

Spherical Harmonic Analysis of Gravitational Curvatures and Its Implications for Future Satellite Missions

In this study we assume that a gravitational curvature tensor, i.e. a tensor of third-order directional derivatives of the Earth’s gravitational potential, is observable at satellite altitudes. Such a tensor is composed of ten different components, i.e. gravitational curvatures, which may be combined into vertical–vertical–vertical, vertical–vertical–horizontal, vertical–horizontal–horizontal and horizontal–horizontal-horizontal gravitational curvatures. Firstly, we study spectral properties of the gravitational curvatures. Secondly, we derive new quadrature formulas for the spherical harmonic analysis of the four gravitational curvatures and provide their corresponding analytical error models. Thirdly, requirements for an instrument that would eventually observe gravitational curvatures by differential accelerometry are investigated. The results reveal that measuring third-order directional derivatives of the gravitational potential imposes very high requirements on the accuracy of deployed accelerometers which are beyond the limits of currently available sensors. For example, for orbital parameters and performance similar to those of the GOCE mission, observing third-order directional derivatives requires accelerometers with the noise level of \({\sim}10^{-17}\,\hbox {m}\,\hbox {s}^{-2}\) Hz\(^{-1/2}\).     

The propagation of surface waves in elastic mediums with slightly curved boundaries of sinusoidal type

Summary The effect of slightly curved boundaries (free surfaces and interfaces) of the elastic mediums on the components of displacement of a particle in a medium due to the propagation of the surface waves has been investigated in this paper. It has been found that along with the usual displacement components for stratified boundaries (designated here as primary components), there exist secondary displacement components arising from the presence of curvature in the boundaries. They are constituted of different harmonic components with their amplitudes proportional to the parameters which measure the extent of curvature of the boundaries. The wave numbers of these harmonic constituents are related to those of the primary components in a definite way decided by the shapes of the boundaries.