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     

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.     

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.     

Deformation of a layered half-space due to a very long tensile fault

The problem of the coseismic deformation of an earth model consisting of an elastic layer of uniform thickness overlying an elastic half-space due to a very long tensile fault in the layer is solved analytically. Integral expressions for the surface displacements are obtained for a vertical tensile fault and a horizontal tensile fault. The integrals involved are evaluated approximately by using Sneddon’s method of replacing the integrand by a finite sum of exponential terms. Detailed numerical results showing the variation of the displacements with epicentral distance for various source locations in the layer are presented graphically. The displacement field in the layered half-space is compared with the corresponding field in a uniform half-space to demonstrate the effect of the internal boundary. Relaxed rigidity method is used for computing the postseismic deformation of an earth model consisting of an elastic layer of uniform thickness overlying a viscoelastic half-space.     

Deformation of two welded elastic half-spaces due to a long inclined tensile fault

The calculation of the deformation caused by shear and tensile faults is necessary for the investigation of seismic and volcanic sources. The solution of the two-dimensional problem of a long inclined shear fault in two welded half-spaces is well known. The purpose of this note is to present the corresponding solution for a tensile fault. Closed-form analytical expressions for the Airy stress function for a tensile line source in two welded half-spaces are first obtained. These expressions are then integrated analytically to derive the Airy stress function for a long tensile fault of arbitrary dip and finite width. Closed-form analytical expressions for the displacements and stresses follow immediately from the Airy stress function. These expressions are suitable for computing the displacement and stress fields around a long inclined tensile fault near an internal boundary.     

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.     

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.     

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.     

蒙山断裂地应力特征及其稳定性分析

蒙山断裂带是沂沭断裂（郯庐断裂带山东段）西侧的一条北西向断裂,控制着长清-临沂中强地震带,该蒙山断裂带的地应力状态对研究山东中部地震活动的危险性有重要的意义。在蒙山断裂附近开展了2个深孔的水压致裂地应力测量,得到最大水平主应力SH为7.0～17.0 MPa,最小水平主应力Sh为5.0～11.0 MPa。测量结果显示,300 m以上的三向主应力的相对大小表现为SH＞Sv≥Sh（Sv为垂直主应力）;400～450 m范围内,三向主应力相对大小表现为SH＞Sv＞Sh;最大水平主应力的方向为NE-NEE,与利用其他资料得到的结果一致;实测资料得到侧压力系数K的平均值,ZK134孔KHmax=1.54（KHmax为最大水平侧压系数）,ZK8孔KHmax=1.38。根据Byerlee定律判定蒙山断裂处于相对稳定的地应力状态,2个孔的? m(最大剪应力与平均主应之比)在2010－2011年间几乎没有变化,亦表明该断层处在一个稳定的状态。     

唐山地区3～4 km深部地应力测量及断层稳定性分析

地应力大小和方向是干热岩开发中注采井网部署、水力压裂设计和诱发地震评估等方面的重要基础数据。本文利用非弹性应变恢复(ASR)地应力测试方法,实测获取了唐山市乐亭县马头营干热岩勘探区3～4 km深度范围的地应力状态。研究结果表明:(1)地应力量值随深度增加而加大,3139～3934 m深度范围内水平最小主应力介于59.0～90.7 MPa之间,水平最大主应力介于103.7～123.6 MPa之间。水平最大主应力方向介于N83°～114°E之间。(2)三向主应力总体表现为σ_H>σ_v>σ_h,表明研究区3～4 km深度构造应力占主导地位,该应力状态有利于走滑断层活动。(3)利用摩尔-库伦准则对邻区断层的稳定性进行分析,结果表明研究区3～4 km深度范围内的断层总体处于稳定的应力环境。(4)干热岩注水开发与断层稳定性分析表明,在统一的区域地应力场作用下,研究区3900～4000 m干热岩注水开发过程中,当地面持续注入压力达到或超过约28 MPa时,可能引起场区内断层的滑动失稳,导致中小地震的发生,在干热岩开发利用中需注...     

陡倾断层上下盘开挖引起地表变形的数值模拟分析

考虑地下开挖区与断层相对位置关系,采用简化数值模型对陡倾断层上下盘开挖引起的地表变形特征进行了数值模拟。结果表明,当断层位于地下开挖引起地表变形的压缩区且开挖区位于下盘时,地表出现不连续变形的可能性小;当断层位于地下开挖引起地表变形的压缩区且开挖区位于上盘时,随下向开挖进行地表断层处裂缝有减小趋势;而当断层位于地下开挖引起的地表变形拉张区时,断层出露处水平拉张明显,表现为拉张裂缝,随下向开挖深度增加、规模增大,不论开挖区位于上盘还是下盘,极可能导致地表出现正断层式的错动．     

Deformation of an elastic layer coupling in different ways to a base due to a very long vertical strike-slip dislocation

The closed-form analytic expressions for the displacement and stresses at any point of an elastic layer lying over a base due to a very long vertical strike-slip dislocation are obtained. The interface between the layer and the base is assumed to be either ‘smooth-rigid’ or ‘rough-rigid’ or ‘welded’. The variations of displacement and stresses with the horizontal distance from the fault for different types of coupling of the layer with the base have been studied. It is found that the displacement for ‘welded interface’ lies between the displacements due to ‘smooth rigid’ and ‘rough-rigid’ interfaces for different positions of the observer and different values of the ratio of rigidities of the layer and half-space.     

An improved return‐map stress update algorithm for finite deformation analysis of general isotropic elastoplastic geomaterials

This paper develops a novel return mapping algorithm for the numerical integration of general isotropic finite strain elastoplastic constitutive models for geomaterials. The constitutive formulation is founded on multiplicative decomposition of the deformation gradient. The logarithmic strain measure as well as the exponential approximation of the plastic flow rule is utilized to restore the standard infinitesimal format return mapping algorithm. Central to the algorithm is the exploitation of a set of three mutually orthogonal unit base tensors for the representation of constitutive relations and the corresponding integration of the rate form of the constitutive equations. The base tensors constitute a local cylindrical coordinate system in the principal space, which allows to formulate the return mapping algorithm in the three‐dimensional space and reduce the dimension of the problem to be analyzed from six down to three. With the proposed approach, direct determination of the principal axes and the transformation procedure between the general space and the principal space, as required in traditional spectral decomposition, are avoided. Furthermore, the matrices that are involved in the inversion evaluation take simple forms, leading to extremely easy inverse computation. As a result, the consistent tangent operator can be streamlined into a form simpler and more compact than those by conventional integration methods. Following the formulation of the integration procedure, a numerical experiment is performed to assess the accuracy and efficiency of the proposed algorithm. Copyright © 2014 John Wiley & Sons, Ltd.     

Characterisation of deformation and flow mechanics around porphyroclasts in a calcite marble ultramylonite by means of EBSD analysis

This paper studies the flow heterogeneity around porphyroclasts associated with greenschist facies deformation of a calcite marble shear zone. Microstructural data from electron backscatter diffraction analyses (EBSD) are used to constrain the flow mechanics of this dominantly non-coaxial type of deformation. The microstructure of the undisturbed ultramylonite (grain-size range 5–100 μm, mean 40 μm) is interpreted to represent steady-state (time-independent) flow conditions with flow planes parallel to the shear zone boundary. Single calcite porphyroclasts (grain-size 1–3 mm) caused flow perturbation in the fine-grained marble ultramylonite. It is the shape, in particular, of these rigid porphyroclasts that controls their rotational behaviour during deformation and, therefore, the development of specific flow fabrics. The flow planes around elongated-rhomboidal, stable porphyroclasts change the orientation to become roughly parallel to the porphyroclast margin, whereas the geometry of flow planes around nearly equant, rotating porphyroclasts describes a δ-type flow pattern. We infer that to some extent decoupling at the clast–matrix interface has occurred to guarantee a stable orientation of elongated porphyroclasts, but was not sufficient to reduce the rotation rate of equant clasts to zero. According to the flow deflection, the general crystallographic preferred orientation (CPO) with its single c-axis maximum perpendicular to the flow plane is rotated about an axis which is (sub)parallel to the kinematic rotation axis of the shear zone. Ultramylonite microstructures, CPOs and misorientation data are best explained by the dual operation of grain-size-insensitive (dislocation creep with recovery and recrystallization) and grain-size-sensitive (diffusion creep) mechanisms. The limited grain-size reduction around porphyroclasts suggests that the grain-size-insensitive mechanisms controlled rheology.     

Steady‐state analytical solutions of flow and deformation coupling due to a point sink within a finite fluid‐saturated poroelastic layer

An exact steady‐state closed‐form solution is presented for coupled flow and deformation of an axisymmetric isotropic homogeneous fluid‐saturated poroelastic layer with a finite radius due to a point sink. The hydromechanical behavior of the poroelastic layer is governed by Biot's consolidation theory. Boundary conditions on the lateral surface are specifically chosen to match the appropriate finite Hankel transforms and simplify the transforms of the governing equations. Ordinary differential equations in the transformed domain are solved, and then the analytical solutions in the physical space for the pore pressure and the displacements are finally obtained by using finite Hankel inversions. The analytical solutions at some special locations such as the top and bottom surfaces, lateral surface, and the symmetrical axis are given and analyzed. And a case study for the consolidation of a water‐saturated soft clay layer due to pumping is conducted. The analytical solution is verified against the finite element solution. Meanwhile, an analysis of coupled hydromechanical behavior is carried out herein. The presented analytical solution is an exact solution to the practical poroelastic problem within an axisymmetric finite layer. It can provide us a better understanding of the poroelastic behavior of the finite layer due to fluid extraction. Besides, it can be applied to calibrate numerical schemes of axisymmetric poroelasticity within finite domains. Copyright © 2017 John Wiley & Sons, Ltd.