Static deformation of two welded monoclinic elastic half-spaces due to a long inclined strike-slip fault

Static deformation of two monoclinic elastic half-spaces in welded contact due to a long inclined strike-slip fault situated in one of the half-spaces is studied analytically and numerically. Closed-form algebraic expressions for the displacement at any point of the medium are obtained. The variation of the displacement at the interface with the horizontal distance from the fault is studied. The effect of anisotropy on the displacement field is examined. It is found that while the anisotropy of the source half-space has a significant effect on the displacement at the interface, the anisotropy of the other half-space has only a marginal effect.     

Static deformation of two welded half-spaces due to dip-slip faulting

Closed-form expressions for the displacements and stresses at any point of either of two elastic half-spaces in welded contact caused by a dip-slip line source obtained earlier are integrated analytically to derive the elastic residual field due to a long dip-slip fault of finite width. The results are valid for an arbitrary dip of the fault. The variation of the displacement field with the distance from the fault as well as with the distance from the interface is studied numerically. It is found that the displacement field is heavily dependent on the dip angle. Contour maps showing the residual elastic field in the two half-spaces caused by a vertical dip-slip fault located in one of the half-spaces are also obtained.     

2-D deformation of two welded half-spaces due to a blind dip-slip fault

The solution of two-dimensional problem of an interface breaking long inclined dip-slip fault in two welded half-spaces is well known. The purpose of this note is to obtain the corresponding solution for a blind fault. The solution is valid for arbitrary values of the fault-depth and the dip angle. Graphs showing the variation of the displacement field with the distance from the fault, for different values of fault depth and dip angle are presented. Contour maps showing the stress field around a long dip-slip fault are also obtained     

Static deformation of two welded half-spaces due to very long strike-slip dislocations

Closed form analytic expressions for displacement and stresses at any point of either of the two homogeneous, isotropic, perfectly elastic half-spaces in welded contact due to very long strike-slip dislocations are obtained. Both cases of vertical and horizontal strike-slip dislocations are discussed in detail. Variation of the displacement with horizontal distance from the fault and with vertical distance from the interface for a vertical strike-slip fault is studied numerically.     

2-D deformation analysis of a half-space due to a long dip-slip fault at finite depth

Closed form analytical expressions of stresses and displacements at any field point due to a very long dip-slip fault of finite width buried in a homogeneous, isotropic elastic half-space, are presented. Airy stress function is used to derive the expressions of stresses and displacements which depend on the dip angle and depth of the upper edge of the fault. The effect of dip angle and depth of the upper edge of the fault on stresses and displacements is studied numerically and the results obtained are presented graphically. Contour maps for stresses and displacements are also presented. The results of Rani and Singh (1992b) and Freund and Barnett (1976) have been reproduced.     

Quasi-static deformation due to two-dimensional seismic sources embedded in an elastic half-space in welded contact with a poroelastic half-space

The Biot linearized theory of fluid saturated porous materials is used to study the plane strain deformation of a two-phase medium consisting of a homogeneous, isotropic, poroelastic half-space in welded contact with a homogeneous, isotropic, perfectly elastic half-space caused by a two-dimensional source in the elastic half-space. The integral expressions for the displacements and stresses in the two half-spaces in welded contact are obtained from the corresponding expressions for an unbounded elastic medium by applying suitable boundary conditions at the interface. The case of a long dip-slip fault is discussed in detail. The integrals for this source are solved analytically for two limiting cases: (i) undrained conditions in the high frequency limit, and (ii) steady state drained conditions as the frequency approaches zero. It has been verified that the solution for the drained case (ω → 0) coincides with the known elastic solution. The drained and undrained displacements and stresses are compared graphically. Diffusion of the pore pressure with time is also studied.     

Seismic plastic deformation in the free-field

An analytical solution for the free-field seismic elastoplastic deformation of granular dry soil is developed. This solution is possible since the stress field solution from pseudostatic analysis is known at any acceleration level. A plastic potential function with two plastic material parameters is suggested to model the soil. This model can describe both the volume dilation and contraction of soil under shearing. Closed-form deformation solutions are derived for two special cases and the results from a numerical evaluation of the solution are compared with those measured from shaking table tests. From the deformation solution the shape of the horizontal plastic displacement distribution with depth is identical to that for the elastic deformation since both depend only on the distribution of the elastic shear modulus. Copyright © 1999 John Wiley & Sons, Ltd.     

On the formation and growth of faults: an experimental study

By using an experimental geometry, which includes a slot loaded to produce a mode 3 shear stress concentration, shear fractures have been grown in two fine-grained rock types. The deformation is accomplished by initially generating an array of oblique, mainly tensile, cracks at the tip of the slot, which are subsequently linked to form a rupture zone. This suggests a more general model for the formation of brittle shear rupture. A through-going shear surface will form if a damage zone is concentrated enough in the appropriate orientation. This may occur at the edges of an existing shear strain discontinuity, such as a fault, particularly in a mode 3 displacement field. Thus under some conditions faults may grow or link up as true macroscopic shear ruptures. This may be part of a solution to the apparent paradox of the existence of long faults. We suggest features that would indicate this sequence of formation in the field and review some geological examples.     

Changes in structural style of normal faults due to failure mode transition: First results from excavated scale models

The effects of failure mode transition from tensile to shear on structural style and fault zone architecture have long been recognized but are not well studied in 3D, although the two modes are both common in the upper crust of Earth and terrestrial planets, and are associated with large differences in transport properties. We present a simple method to study this in physical scale models of normal faults, using a cohesive powder embedded in cohesionless sand. By varying the overburden thickness, the failure mode changes from tensile to hybrid and finally to shear. Hardening and excavating the cohesive layer allows post mortem investigation of 3D structures at high resolution. We recognize two end member structural domains that differ strongly in their attributes. In the tensile domain faults are strongly dilatant with steep open fissures and sharp changes in strike at segment boundaries and branch points. In the shear domain fault dips are shallower and fault planes develop striations; map-view fault traces undulate with smaller changes in strike at branches. These attributes may be recognized in subsurface fault maps and could provide a way to better predict fault zone structure in the subsurface.     

A note on 2-D lithospheric deformation due to a blind strike-slip fault

Analytical solution for the problem of a surface-breaking long strike-slip fault in an elastic layer overlying an elastic half-space is well known. The purpose of this note is to obtain the corresponding solution for a blind fault. Since the solution is valid for arbitrary values of the fault-depth and the dip angle, the effects of these two important fault parameters can be studied numerically. The variation of the parallel displacement and shear stress with the distance from the fault is studied numerically for different values of the fault-depth and dip angle.     

巴西劈裂试验应力场解析解应力函数解法

巴西圆盘试验是国际岩石力学学会推荐的用于测量岩石抗拉强度的间接实验,其理论基础是圆盘在荷载作用下应力场的分布。本文以平面应力状态为基础,根据弹性理论中的Airy应力函数和线弹性叠加原理,采用与Н. И. Мусхелишьили所用的复变函数方法完全不同的方法,给出了巴西圆盘内部任意一点应力的直角坐标形式的解析表达式。将得到的应力场解析解和Н.И. Мусхелишьили 采用复变函数方法给出的经典解对比发现,两者完全吻合一致,从而表明本文所采用的方法是合理正确的。     

Quasi-static deformation of a layered half-space by a long strike-slip fault

Theoretical expressions for the surface displacement and shear stress caused by a long strike-slip dislocation in an elastic layer overlying an elastic half-space are derived and the correspondence principle is used to obtain the quasi-static response when the half-space is Maxwell-viscoelastic. Variation of the surface displacement and shear stress with horizontal distance is studied for various times and vertical extents of the fault. It is seen that the quasi-static response differs significantly from the corresponding elastic response.     

The relationship between the geometry of normal faults and that of the sedimentary layers in their hanging walls

We derive an analytical expression that relates the shape of a fault in cross-section to the shape of the bedding horizons in its hanging wall block. The expression assumes that the hanging wall deforms by simple shear and that the footwall remains undeformed throughout. Although this paper concentrates on normal faults, the expression is equally valid and applicable to thrust faults. The direction of simple shear in the hanging wall block is arbitrary and has a dramatic effect on the predicted fault or bedding geometry. There is no reason to believe that the simple shear occurs on vertical planes, as is commonly assumed in graphical approaches to this problem, and ignoring the presence of inclined simple shear is likely to lead to considerable underestimates of the amount of extension across normal faults and in the amount of shortening across thrusts. Similar though more complicated expressions can be obtained when compaction within the hanging wall block is taken into account. For a planar normal fault such compaction may result in the development of a hanging wall syncline.     

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

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

唐山地震强余震分布特征和迁移规律

引言从1976年7月28日在唐山发生7.8级强震起,到同年12月底止,该震区已发生5.5级（Ms）以上强余震39次。它们集中分布在唐山断裂附近及其两端偏在压缩区的一侧。与其它大震的余震分布迥然不同,具有独特的形式。它究竟由何种因素支配?是一个值得探讨的问题。正确地回答这个问题,不但有助于弄清11月15日在宁河发生的6.9级强震的性质,而且对于预测今后在唐山地区的余震活动也有帮助。由于单纯运用弹性力学理论解决的问题比较简单,所以作者引用一些破裂和光弹试验的成果进行综合分析研究。     

Quasistatic propagation of a normal fault: A fracture mechanics model

We have carried out boundary element calculations to simulate quasistatic propagation of a normal fault in the earth's crust under a horizontal tensile loading. Byerlee's frictional law is employed to describe the mechanical behavior of the fault surface. We hypothesize that in order for a normal fault to grow quasistatically, the mixed-mode effective shear stress intensity factor must exceed a threshold value (fracture toughness), a crustal material property. We suggest that the fault grows in a direction of local maximum shear stress. The direction of fault propagation thus depends on the ratio of tensile and shear stress intensity factors. A listric normal fault is likely to form in crustal material with a small shear fracture toughness. A listric normal fault is also more likely to form in crustal material with a high degree of plasticity.The propagation trajectory of an incrementally growing normal fault is examined. As the normal fault extends to a greater depth, the shear stress intensity factor drops, owing to an increase in fault surface friction. The equilibrium depth to which a normal fault will grow is controlled by the far field loading and the fracture mechanical property of the crustal material. The decrease of shear stress intensity with fault length also stabilizes the fault growth.     

竖向掏土纠倾孔周塑性区计算研究

采用桩筏基础的高层建筑倾斜后,常采用竖向孔掏土进行地应力解除实现纠倾,工程设计与施工依据经验进行,缺乏理论指导。将一定深度处的竖向孔孔周应力重分布问题简化为平面应变问题,采用摩尔?库仑强度条件推导了塑性区半径的解析解。该解析解表明,塑性半径随土体抗剪参数的降低而增大,随掏土孔半径的增大而增大,随地应力的增大而增大。结合解析解的表达式,探讨了存在地下水、注水、抽水、循环掏土扰动等因素对塑性区半径的影响。最后用数值分析方法对该解析解进行模拟验证,表明公式的正确性与适用性。该研究成果对纠倾工程的设计与施工具有积极的指导作用。     

En echelon crack-arrays in potash salt rock

Summary There are two types of fracture patterns in the yield pillars of the potash mines of Saskatchewan. The individual members of both patterns are tensile (extension) fractures that propagate parallel with the maximum principal stress trajectory (perpendicular to the minimum principal stress). The difference between the two patterns lies in the arrangement of the member fractures. In theen echelon tensile crack-array, the macroscopic fracture consists of individual tensile cracks that are slightly offset from each other. They have only a small overlap and the child crack seems to form randomly on either side of its parent. Consequently, the en echelon tensile crack-array inherits the axial orientation of its members. In contrast, the tensile cracks of anen echelon shear crack-array, have a larger overlap and their lateral displacement from each other is biased in one direction. Therefore, the crack-array is no longer axial but inclined 20–25 degrees from the maximum principal stress direction. With increasing stress, the shear crack-array often collapses, forming theenvelope orhourglass structures of the potash mines.     

堆石料强度变形特性与非线性弹性本构模型研究

在大型三轴试验研究的基础上,提出了堆石料应力-应变关系、侧向应变与轴向应变关系的指数函数表达式,以及抗剪强度包线在q-p平面上的幂函数表达式。继而推导得到了一个反映堆石料强度与变形特性的非线性弹性本构模型。通过对实体工程的数值模拟,将Duncan E-B模型与新建模型进行了对比分析,验证了该模型的可行性。     

塔中地区构造应力场数值模拟研究

构造应力场研究对于油气运聚分析及储层特征描述具有重要的理论和实际意义。本文根据塔中隆起断裂特征及其演化史,结合钻井资料,选择中下奥陶统碳酸盐岩为主要研究对象,应用大型有限元分析软件ANSYS,在处理断层问题时运用非连续的接触算法,模拟了中奥陶世末、中泥盆世末、三叠纪末、新近纪末的构造应力场,研究了该区不同时期的古应力分布特征。研究表明,最大主压应力受东部车尔臣-星星峡走滑断裂影响显著,在隆起区沿断裂带呈条带状低值分布,在不同时期塔中Ⅰ号断裂带无一例外地位于最大主压应力的最低区,反映了塔中断裂带尤其是塔中Ⅰ号断裂带是油气运移的有利指向区。最小主压应力沿断裂带呈条带状高值分布,在断裂的上盘、断裂走向发生变化和断裂的倾末端是张应力高值区,为张裂缝发育的重要构造部位。最小主压应力和最大剪应力受车尔臣-星星峡走滑断裂的影响明显减弱。