Elastic and inelastic triggering of earthquakes in the North Anatolian Fault zone

Deformation models used to explain the triggering mechanism often assume pure elastic behaviour for the crust and upper mantle. In reality however, the mantle and possibly the lower crust behave viscoelastically, particularly over longer time scales. Consequently, the stress field of an earthquake is in general time-dependent. In addition, if the elastic stress increase were enough to trigger a later earthquake, this triggered event should occur instantaneously and not many years after the triggering event. Hence, it is appropriate to include inelastic behaviour when analysing stress transfer and earthquake interaction.In this work, we analyse a sequence of 10 magnitude M_s > 6.5 events along the North Anatolian Fault between 1939 and 1999 to study the evolution of the regional Coulomb stress field. We investigate the triggering of these events by stress transfer, taking viscoelastic relaxation into account. We evaluate the contribution of elastic stress changes, of post-seismic viscoelastic relaxation in the lower crust and mantle, and of steady tectonic loading to the total Coulomb stress field. We analyse the evolution of stress in the region under study, as well as on the rupture surfaces of the considered events and their epicentres. We study the state of the Coulomb stress field before the 1999 İzmit and Düzce earthquakes, as well as in the Marmara Sea region.In general, the Coulomb stress failure criterion offers a plausible explanation for the location of these events. However, we show that using a purely elastic model disregards an important part of the actual stress increase/decrease. In several cases, post-seismic relaxation effects are important and greater in magnitude than the stress changes due to steady tectonic loading. Consequently, viscoelastic relaxation should be considered in any study dealing with Coulomb stress changes.According to our study, and assuming that an important part of the rupture surface must be stressed for an earthquake to occur, the most likely value for the viscosity of the lower crust or mantle in this region is 5 · 10¹⁷–10¹⁸ Pa · s. Our results cannot rule out the possibility of other time-dependent processes involved in the triggering of the 1999 Düzce event. However, the stress increase due to viscoelastic relaxation brought 22% of the 1999 Düzce rupture area over the threshold value of Δσ_c ≥ 0.01 MPa (0.1 bar), and took the whole surface closer to failure by an average of 0.2 MPa. Finally, we argue that the Marmara Sea region is currently being loaded with positive Coulomb stresses at a much faster rate than would arise exclusively from steady tectonic loading on the North Anatolian Fault.     

Co and postseismic characteristics of Indian sub-continent in response to the 2004 Sumatra earthquake

The M_w 9.3 Sumatra earthquake of December 26, 2004 caused extensive coseismic displacements globally, measurements of which were made essentially using modern geodetic techniques. This earthquake induced considerable perturbation in stress distribution as far as ∼8000 km away from the epicenteral region, which is tending to relax to its normal rates as seen from postseismic transient deformation. The monitoring of crustal displacements from strategically located sites using GPS provides coseismic as well as postseismic deformation that facilitates the understanding of the fault geometry, elastic thickness, postseismic relaxation mechanisms, rheology and earthquake recurrence time interval.We investigated coseismic and postseismic GPS derived displacements in Indian region together with the GPS data collected from Andaman and Sumatra region. It is found that while EW displacements are significantly large in peninsular India, those in the region to the north of Central India Tectonic Zone (CITZ) are relatively small. We could delineate the postseismic transients from position time series and interpreted them in terms of viscoelastic relaxation. It is inferred that the postseismic deformation is characterized by a power-law viscoelastic flow in the mantle. In Indian peninsula region, the timescale parameter of the exponential decay (τ = 250 days) would require an extremely low viscosity for the upper mantle. Relying on the prevailing coseismic and postseismic displacement fields, the present study also reflects upon the contemporary litho-tectonics of the Indian sub-continent.     

Finite element analyses of layered visco‐elastic system under vertical circular loading

Analyses for the response of a linear visco‐elastic system subjected to axi‐symmetric vertical circular loading are presented. Hankel transforms with respect to the radial spatial coordinate are used to reduce the three‐dimensional problem to that involving only a single spatial dimension, which is then discretized using the finite element method. Three techniques are employed to handle the time factor in the visco‐elastic material: (i) direct time integration; (ii) Fourier transforms; and (iii) Laplace transforms. These methods are compared and evaluated through their numerical results. Copyright © 2007 John Wiley & Sons, Ltd.     

2008年汶川大地震孕震、同震及震后变形和应力演化全过程的数值模拟

2008年MS 8.0级汶川大地震发生在具有复杂的地质构造背景、 强烈的地表起伏、 不均匀的弹性和黏性结构的龙门山断裂带上.由于震前地震活动性不够强烈且地表构造变形较小,龙门山断裂带的地震危险性在汶川地震之前被低估.从数值模拟的角度,建立黏弹性有限元模型,考虑了初始地形、 重力、构造加载、 黏弹性松弛等因素对2008年...     

爆炸荷载作用下深埋圆形隧洞的饱和黏弹性土-衬砌系统动力响应

假定土骨架服从标准线性固体黏弹性本构关系,研究了深埋圆形隧洞的饱和黏弹性土-弹性衬砌耦合系统在轴对称爆炸作用下的瞬态动力响应。首先,基于饱和土的Biot模型和衬砌的弹性理论,通过引入势函数和Laplace变换,利用弹性衬砌和饱和黏弹性土界面处的连续性条件以及边界条件,得到饱和黏弹性土体和弹性衬砌位移、应力和孔隙水压力等在Laplace变换域中的解析解。其次,利用Laplace数值Crump逆变换得到耦合系统在时间域的动力响应,数值分析了不同土体模型下土体-衬砌耦合系统的径向位移和环向应力以及土体孔隙水压力等。结果表明：对不同土体模型的土体-衬砌耦合系统,其在爆炸载荷作用下的动力响应性态基本一致,但动力响应的振动周期和幅值等具有明显的差异。同时,对于饱和黏弹性土-弹性衬砌系统,土体黏性参数对土体径向位移和孔隙水压力有明显的影响,但对土体环向应力影响较小。     

Late Quaternary glacio- and hydro-isostasy, on a layered earth

Isostatic response of the Earth to changes in Quaternary Times of ice and water loads is partly elastic, and partly involves viscous mantle flow. The relaxation spectrum of the Earth, critical for estimation of the mantle flow component, is estimated from published determinations of Fennoscandian and Laurentide rebound, and of the nontidal acceleration of the Earth's rotation. The spectrum is consistent with an asthenosphere viscosity around 10²¹P, and a viscosity around 10²³P below 400 km depth. Calculation of relaxation effects is done by convoluting the load history with the response function in spherical harmonics for global effects, and in rectangular or cylindrical transforms for smaller regional effects. Broad-scale deformation of the globe, resulting from the last deglaciation and sea level rise, is calculated to have involved an average depression of ocean basins of about 8 m, and mean upward movement of continents of about 16 m, relative to the center of the Earth, in the last 7000 yr. Deflection in the ocean margin “hinge zone” varies with continental shelf geometry and rigidity of the underlying lithosphere: predictions are made for different model cases. The computational methods is checked by predicting Fennoscandian and Laurentide postglacial warping, from published estimates of icecap histories, with good results. The depth variations of shorelines formed around 17,000 BP (e.g., North America, 90–130 m; Australia, 130–170 m), are largely explainable in terms of combined elastic and relaxation isostasy. Differences between Holocene eustatic records from oceanic islands (Micronesia, Bermuda), and continental coasts (eastern North America, Australia), are largely but not entirely explained in the same terms.     

Coupling Rheological Behavior of Rock Mass Supported with Rockbolts Based on the Point Load Model

Rockbolts can be regarded as a permanent support structure for underground engineering. However, the coupled rheological mechanism between rockbolts and rock mass remain poorly understood. In this research, the rock mass supported with discretely mechanically or frictionally coupled rockbolts with the point load model was investigated. At first, the elastic solutions of the coupling model were solved. Subsequently, the viscoelastic analytical solutions were acquired by applying the Laplace inverse transforms. Finally, the effect of viscosity coefficients and supporting parameters on the coupling model rheological behavior were explored. The results indicated that the angle of rockbolts support has a greater influence on the radial stress and tangential stress of the rock mass in the elastic state, but has little influence on the rheological state. Moreover, the roof, the interface between the left wall and right wall should be supported strengthen; the viscosity coefficient has little influence on the radial stress and tangential stress in the rheological state and has a negative correlation with the radial displacement. In order to limit the displacement in the rheological state of the rock mass, we choose the rockbolts with a large viscosity coefficient.

     

A semi-discrete procedure for dynamic response analysis of saturated soils

Biot's equations of wave propagation through fluid-saturated porous elastic media are discretized spatially using the finite element method in conjunction with Galerkin's procedure. Laplace transformation of the discretized equations is used to suppress the time variable. Introducing Laplace transforms of constituent velocities at nodal points as additional variables, the quadratic set of equations in the Laplace transform parameter is reduced to a linear form. The solution in the Laplace transform space is inverted, term by term, to get the complete time history of the solid and fluid displacements and velocities. Since the solution is exact in the time domain, the error in the calculated response is entirely due to the spatial approximation. The procedure is applied to one-dimensional wave propagation in a linear elastic material and in a fluid-saturated elastic soil layer with ‘weak’, ‘strong’ as well as ‘moderate’ coupling. With refinement of the spatial mesh, convergence to the exact solution is established. The procedure can provide a useful benchmark for validation of approximate temporal discretization schemes and estimation of errors due to spatial discretization.     

A method of time-dependent analysis using elastic solutions for nonlinear materials

The formulation of viscoelastic solutions from elastic equations using the ‘correspondence principle’ and an inverse Laplace transform has been discussed extensively in the literature. Because this method has been developed, many time-dependent solutions can be obtained from closed form elastic solutions and conditions have been delineated in which the ‘quasi-elastic’ approximation of the viscoelastic solution is within acceptable tolerance. This communication shows the feasibility of the application of these methods to formulate approximate nonlinear viscoelastic solutions with nonlinear stress-strain materials, and for want of a specific nonlinear model to demonstrate this, the hyperbolic model was selected. The ‘power law’ is used to model the relaxation modulus of the viscoelastic materials. There are five related development that are discussed here using a simple numerical example to illustrate each of them and they are: (1) a linear elastic solution, (2) a linear viscoelastic solution, (3) a nonlinear elastic solution, (4) a nonlinear viscoelastic solution and finally, (5) a ‘regression’ approximation of the nonlinear viscoelastic solution which is suggested by the series form of the elastic solution. All of these are related to one another and each provides an acceptably accurate solution of the problem it addresses. The latter is of particular practical interest since it can be used to provide answers to problems involving nonlinear viscoelastic materials while requiring only very small calculation times. The problem used as an example is the calculation of the displacement of a circular hole in an infinite plate made of a material with a nonlinear time-dependent stress-strain relationship. The nonlinear elastic form of the solution was developed by matching results from nonlinear finite element analysis.     

The effects of a variation of the radial viscosity profile on mantle evolution

The present paper describes a set of numerical experiments on the mantle's thermal evolution with an infinite Prandtl number fluid in a compressible spherical shell heated mainly from within. We used the anelastic liquid approximation with Earth-like material parameters. The usual variable-viscosity approach in mantle-convection models is the assumption of a temperature dependence only. The resulting thermal boundary layers are included in our model also, but an additional viscosity profile of the interior mantle was derived: The Birch–Murnaghan equation was employed to derive the Grüneisen parameter and other physical quantities as a function of depth from observational values provided by PREM. We computed the melting temperature and a new mantle viscosity profile, called eta3, using the Grüneisen parameter, Lindemann's law and some solid-state physics considerations. The new features of eta3 are a high-viscosity transition layer with rather high viscosity gradients at its boundaries, a second low-viscosity layer beginning under the 660-km discontinuity, and a strong viscosity increase in the central parts of the lower mantle. The rheology is Newtonian but it is supplemented by a viscoplastic yield stress, σ_y. A viscosity-level parameter, r_n, and σ_y have been varied. For a medium-sized Rayleigh-number–yield-stress area, eta3 generates a stable, plate-tectonic behavior near the surface and simultaneously thin sheet-like downwellings in the depth. Outside this area, three other types of solution were found. Not only the planforms but also the evolution of the Rayleigh number, the reciprocal Urey number, the Nusselt number, the surface heat flow, etc., have been studied. We repeated this investigation with two very different basic viscosity profiles, etaKL5a and etaKM, of other authors. A comparison reveals that eta3 facilitates the generation of surface plates and thin sheet-like downwellings in the depth considerably more than etaKL5a or even etaKM. The presence of two internal low-viscosity layers is obviously conducive for plateness and thin sheet-like downwellings. For an infinite yield stress, the thin cold sheet-like downwellings are reticularly connected. However, the distribution of the downwellings is more Earth-like if a realistic yield stress is added.     

Fractional order constitutive model of geomaterials under the condition of triaxial test

Fractional calculus has been successfully applied to characterize the rheological property of viscoelastic materials; however, geomaterials were seldom involved in fractional order constitutive models (FOCM), and the topic of first loading and then unloading is rarely discussed through fractional calculus. In this paper, mechanical properties are considered as a ‘spectrum’, both ends of which are elasticity and viscosity, and the fractional order can be utilized to describe such properties quantitatively. In addition to conditions such as creep, stress‐relaxation, and constant‐strain‐rate loading, stress‐strain relationship under the condition of first loading and then unloading was also derived using FOCM. FOCM is then adopted to simulate triaxial tests of geomaterials under corresponding conditions. A comparison of test and numerical results demonstrates that FOCM can reasonably describe the mechanical characteristics of geomaterials.Copyright © 2012 John Wiley & Sons, Ltd.     

全球海面变化的两极冰盖模型

在Clark等(1976)提出的在粘弹性地球上全球海面变化的数值模型和笔者(1992)提出的一极冰盖质点模型的基础上,考虑冰盖在两极的实际分布,建立两极冰盖质点分布的数理模型。计算结果表明,一极模型中海面变化的的极值区在S极,而两极模型中海面变化极值区在南半球的中纬度地区。     

PROPAGATION OF TORSIONAL SURFACE WAVES IN VISCOELASTIC MEDIUM

It is well known that an elastic homogeneous half-space does not allow torsional surface waves to propagate. The present paper attempts to find out the possibility of propagation of such waves in a viscoelastic half-space. The study reveals that although the homogeneous elastic half-space does not allow torsional surface waves to propagate, a viscoelastic half-space does so. The wave is damped due to the viscoelastic parameter. It has also been found that as the viscoelastic parameter decreases, the medium becomes elastic and the torsional surface waves ceases to propagate.     

弹性地基中厚板弯曲问题的无网格LRPIM分析

无网格局部径向点插值方法（LRPIM）的形状函数具有Kronecker delta函数的特性,便于施加位移边界条件,不需要进行特殊地处理。采用局部加权残值法推导了双参数弹性地基中厚板的离散系统方程,利用无网格LRPIM对四边简支和四边固支以及筏式双参数弹性地基中厚板的弯曲问题进行了分析和计算。算例表明,用无网格LRPIM分析弹性地基中厚板问题具有非常灵活和易于实现等优点。     

循环荷载作用下黏弹性地基一维固结性状研究

针对单层黏弹性地基模型,对循环荷载作傅立叶级数展开,由固结方程求得循环荷载作用下不透水（透水）边界饱和软黏土一维固结解析解。通过不同渗透系数、黏滞系数和弹性模量以及黏弹性与弹性不同地基模型比较等对有效应力的影响,进行了固结性状分析。结果表明、在循环荷载作用下地基中各点的有效应力并不随荷载的变化而同步变化,而是按一定规律滞后发展;有效应力最终达到一个稳定状态,每一个加载卸载循环下有效应力幅值在一个振荡周期的平均值变化趋近于0;软黏土的流变特性对固结的影响主要发生在中、后期。     

东西构造带形成机制和有关问题的讨论

<正> 早在二十年代初期,李四光教授在研究中国和东亚各种类型构造体系的时候,就明确指出地壳表面存在着沿一定纬度分布的东西向构造带。以后又进一步讨论了它们长期发展的历史和它们在全球的规模。他从不同类型构造体系在地壳上分布和排列的规律,探索构造运动的起源,认为这些构造体系所表现的方向性显然与地球的旋转轴有一定的联系,它们的成生与地球自转角速度的变化密切相关,当地球自转角速度发生变化时,它的离心惯性力也随着发生变化,离心惯性力的水平分力推动着地壳的定向水平运动——经向和纬向水平运动。     

Parameter identification of viscoelastic materials

Two parameter identification procedures for linear viscoelastic materials are presented. One is the method using the incremental constitutive relation for linear viscoelastic materials, and the other is the method using the elastic–viscoelastic correspondence principle. Part of back analysis in both methods is formulated based on the boundary control concept (Ichikawa Y, Ohkami T. A parameter identification procedure as a dual boundary control problem for linear elastic materials. Soils and Foundations 1992;32(2):35–44). Two numerical examples are presented to compare the efficiency of both methods.     

原油粘度变化对水驱油开发动态影响的数学模拟方法

为了研究注水开发油田原油粘度升高对开发效果的影响, 通过对实际油藏原油粘度统计, 回归出了原油粘度增长模型.在三维三相黑油渗流模型的基础上, 建立了一个原油粘度随含水和压力变化的油藏渗流数学模型, 并采用有限差分方法建立了相应的数值模型, 采用超松弛法对该模型进行了求解, 用Fortran90语言开发了一个新的数值模拟器.应用该模拟器模拟了不同的原油粘度变化规律对水驱效果的影响, 并与常规模拟器的结果进行了对比.结果表明: 初始水油粘度比为1∶10、含水达到98%时, 粘度增长指数由0增加到0.02, 对应的原油采出程度由44.80%降低到34.29%.目前商业软件中忽略了原油粘度随含水升高而增加的因素, 使得预测的采收率明显偏高.      

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.     

Dynamic Study on Fracture Problems in Viscoelastic Sedimentary Rocks Using the Numerical Manifold Method

The viscoelastic deformation behavior of a sedimentary rock under different loading rates is numerically modeled and investigated by the numerical manifold method (NMM). By incorporating a modified 3-element viscoelastic constitutive mode in the NMM, crack initiation and propagation criteria, and crack identification and evolution techniques, the effects of the loading rates on the cracking behavior of a sedimentary rock, such as crack open displacement, crack sliding displacement, crack initiation, crack propagation and final failure mode, are successfully modeled. The numerical results reveal that under a high loading rate (>1,000 MPa/s), due to the viscoelastic property of the sedimentary rock, not only the structural behavior deviates from that of elastic model, but also different cracking processes and final failure modes are obtained.