Non-hypersingular boundary integral equations for two-dimensional non-planar crack analysis

We derive a set of non-hypersingular boundary integral equations, both elastodynamic and elastostatic, for the analysis of arbitrarily shaped 2-D anti-plane and in-plane cracks located in an infinite homogeneous isotropic medium, rendered in a unified nomenclature for all cases. The hypersingularities that appear in the usual formulations for the dynamic cases, existent both at the source point and at the wavefront, are removed by way of a regularization technique based on integration by parts. The equations for the in-plane cases are presented in terms of a local Cartesian coordinate system, one of the axes of which is always held locally tangential to the crack trace. The expressions for the elastic field at any point on the model plane are also given.
Our formulations are shown to yield accurate numerical results, as long as appropriate stabilization measures are taken in the numerical scheme. The numerical applicability of our method to non-planar crack problems is illustrated by simulations of dynamic growth of a hackly crack which has small off-plane side-branches. The results imply that the branching of a crack brings about a significant decrease in the crack-tip stress concentration level and consequently may play an essential role in the arrest of earthquake rupturing.     

An efficient method to compute the dynamic response of a fluid-filled crack

The vibration of a fluid-filled crack is considered to be one of the most plausible source mechanisms for the long-period events and volcanic tremors occurring around volcanoes. As a tool for the quantitative interpretation of source process of such volcanic seismic signals, we propose a method to numerically simulate the dynamic response of a fluid-filled crack. In this method, we formulate the motions of the fluid inside and the elastic solid outside of the crack, using boundary integrals in the frequency domain and solve the dynamic interactions between the fluid and the elastic solid using the point collocation method. The present method is more efficient compared with the time-domain finite difference method, which has been used in simulations of a fluid-filled crack and enables us to study the dynamics of a fluid-filled crack over a wide range of physical parameters. The method also allows us direct calculation of the attenuation quality factor of the crack resonance, which is an indispensable parameter for estimating the properties of the fluid inside the crack. The method is also designed to be flexible to many applications, which may be encountered in volcano seismology, and thus, extensions of the method to more complicated problems are promising.     

Bilateral propagation of a spontaneous two-dimensional anti-plane shear crack under the influence of cohesion

Summary. We consider the problem of the bilateral extension of a two-dimensional anti-plane shear crack that initiates spontaneously at a point and extends under the influence of cohesive forces at the edges. An approximation to the stresses in the regions beyond the edges of the crack has been found that simplifies the calculation. The exact stresses in these regions are also found iteratively. In the cases of uniformly propagating cracks, the estimates of cohesive forces obtained from the approximation are close to the exact values for high crack speeds but are significantly different for low crack speeds. It is also found that if healing is initiated due to the encounter of one end of a uniformly propagating crack with an unbreakable barrier, the static stress drop in the torn region is constant but may either overshoot or undershoot the dynamical stress drop. In these cases, the final static slip distribution is obtained by freezing the dynamic solution along a characteristic line through the location of the barrier. We find that the crack length cannot be unambiguously derived from the far field spectral properties.     

Dynamic nucleation process of shallow earthquake faulting in a fault zone

Two distinct phases are commonly observed at the initial part of seismograms of large shallow earthquakes: low-frequency and low-amplitude waves following the onset of a P wave ( P ₁) are interrupted by the arrival of the second impulsive phase P₂ enriched with high-frequency components. This observation suggests that a large shallow earthquake involves two qualitatively different stages of rupture at its nucleation.
We propose a theoretical model that can naturally explain the above nucleation behaviour. The model is 2-D and the deformation is assumed to be anti-plane. A key clement in our model is the assumption of a zone in which numbers of pre-existing cracks are densely distributed; this cracked zone is a model for the fault zone. Dynamic crack growth nucleated in such a zone is intensely affected by the crack interactions, which exert two conflicting effects: one tends to accelerate the crack growth, and the other tends to decelerate it. The accelerating and decelerating effects are generally ascribable to coplanar and non-coplanar crack interactions, respectively. We rigorously treat the multiple interactions among the cracks, using the boundary integral equation method (BIEM), and assume the critical stress fracture criterion for the analysis of spontaneous crack propagation.
Our analysis shows that a dynamic rupture nucleated in the cracked zone begins to grow slowly due to the relative predominance of non-coplanar interactions. This process radiates the P₁ phase. If the crack continues to grow, coalescence with adjacent coplanar cracks occurs after a short time. Then, coplanar interactions suddenly begin to prevail and crack growth is accelerated; the P₂ phase is emitted in this process. It is interpreted that the two distinct phases appear in the process of the transition from non-coplanar to coplanar interaction predominance.     

Quasi-static crack models and the frictional stress threshold criterion for slip arrest

Summary. Two-dimensional crack problems in elastic homogeneous isotropic media are considered which describe rupture over a fault surface characterized by non-uniform stress drop. Solutions can be found in which the stress field is finite at the crack tips and the rupture surface is not assigned a priori , but is part of the solution. These crack models are found to be consistent with the frictional stress threshold criterion for slip arrest over pre-existing fault surfaces. A crack is found to stop when its contribution to the stress field is opposite to the stress drop at the crack tips. The quasi-static propagation of a crack up to the arrest configuration is studied in terms of the minimum energy principle. The crack spontaneously propagates in such a way as to make the value of the stress intensity factor at one tip equal to the value at the other tip. Furthermore a tip propagating in a region with higher friction is found to move more slowly than the other tip propagating in a region with lower friction. Simple criteria for fracture arrest are derived, in terms of a properly averaged stress drop. Piecewise constant stress drop profiles are explicitly considered yielding a variety of solutions which can be applied to modelling asperities or barriers over a fault plane. The evaluation of the amount of the energy released during the quasi-static crack propagation shows that stopping phases cannot be efficiently radiated if the crack comes to rest in a low friction region.     

Dynamic non-planar crack rupture by a finite volume method

Modelling dynamic rupture for complex geometrical fault structures is performed through a finite volume method. After transformations for building up the partial differential system following explicit conservative law, we design an unstructured bi-dimensional time-domain numerical formulation of the crack problem. As a result, arbitrary non-planar faults can be explicitly represented without extra computational cost. On these complex surfaces, boundary conditions are set on stress fluxes and not on stress values. Prescribed rupture velocity gives accurate solutions with respect to analytical ones depending on the mesh refinement, while solutions for spontaneous propagation are analysed through numerical means. An example of non-planar spontaneous fault growth in heterogeneous media demonstrates the good behaviour of the proposed algorithm as well as specific difficulties of such numerical modelling.     

Evaluation of stress and displacement fields due to an elliptical plane shear crack

Summary. Following the classic work of Eshelby, the slip and stress distributions due to an elliptical plane shear crack are evaluated. The relation between average (or maximum) slip on the crack and the (constant) static stress drop, for faults of different aspect ratios, is found. The slip vector is not parallel to the applied stress but makes a small angle to it, except when the stress is applied along the major or minor axis of the ellipse. The stress -distribution around the crack shows that in addition to the expected stress concentration along the crack edge, there are broad regions of stress increase off the crack plane for circular and elliptical cracks, similar to those known to exist for in-plane but not for antiplane shear cracks. Whether the stress- intensity factor at the end of one axis is greater or less than that at the end of the other axis ( k_a ≶ k_b ), depends on the condition: √ b/a ≶ (1 − v ) where a and b are the semi-axes of the ellipse, k_a and k_b are the stress-intensity factors at the end of the a- and b -axes and v is Poisson's ratio. The total stress-intensity factor varies smoothly along the edge of the ellipse from one axis to the other and it is found that this variation is rather small.     

Unilateral extension of a two-dimensional shear crack under the influence of cohesive forces

Summary. We consider the problem of the unilateral extension of a two-dimensional anti-plane crack that initiates spontaneously at a point. The crack extends under the influence of cohesive resistance at the edge and dynamical friction along the crack walls. The stresses in the region beyond the edge of the crack are approximated so that they are exactly equal to the cohesive stresses near the edge of the crack, and are zero on the wavefront. An exact method of solving such problems is also given and can be used to determine the validity of the approximation. We find that the crack will not grow if the cohesion exceeds some critical value; this is consistent with an earlier result obtained by Knopoff, Mouton & Burridge for a similar one-dimensional model of crack propagation.     

Hydraulic crack propagation in a porous medium

We develop a model for the propagation of a fluid-filled crack in a porous medium. The problem is motivated by the mechanism whereby drainage networks may form in partially molten rock below the Earth's lithosphere. Other applications include the propagation of hydraulic fractures in jointed rocks and in oil drilling operations, and the formation of dessication cracks in soils. Motivated by the lithospheric problem, we study a situation in which gravity acts in the direction of crack propagation. The model couples the elastoliydrodynamic equations of crack propagation with a pore pressure field in the porous rock, which drives the fluid flow which supplies the crack. The effect of the pore flow is to include a diffusional term in the evolution equation for the crack width, thus allowing a crack initiated at the base of the lithosphere to propagate down into the asthenosphere. Asymptotic and numerical solutions are presented for this crack evolution. However, the predicted drainage of melt into this crack is tiny compared with the upward percolative melt migration, and the predicted width of cracks (millimetres) is much too small to allow propagation of melt into the lithosphere without freezing. As a mechanism to explain magma fracturing in the lithosphere, the process described here therefore requires further refinement.     

A crack model of creep processes on deep sections of faults

A crack model in antiplane shear configuration is shown representing creep processes interpreted in terms of 'viscous' deformation of a narrow plastic layer, characterized by inhomogeneous rheological properties, embedded within a homogeneous elastic medium. The evolution in time of slip and stress over the crack plane is studied through a truncated expansion in Chebyshev polynomials, and convergence is proved to be fast in the simple examples considered. Finite-stress solutions are found which are compatible with constitutive relations of elasto-plastic materials and furthermore these allow us to simulate creep propagation and stress transfer between locked and unlocked fault segments. This model provides a simple interpretation of the shallow depth of the seismogenic layer observed in several areas of the world and lends itself to modelling creep processes during either post-seismic rebound or pre-seismic stress buildup. Stress transfer is accomplished mostly by the slow extension of the creeping section. During a seismic cycle it is envisaged that different regimes dominate over deep, intermediate and shallow sections of faults: (i) slow pre-seismic stress build-up accompanied by creep and stress migration toward intermediate depths; (ii) brittle fracture over shallow and intermediate sections of faults; (iii) post-seismic rebound over intermediate and deep sections of faults. The present crack model, while providing finite-stress solutions, allows a better understanding of how stress may accommodate at different depths over a fault plane during a seismic cycle.     

Quasi-static analysis of strike fault growth in layered media

We study the effects of structural inhomogeneity on the quasi-static growth of strike-slip faults. A layered medium is considered, made up of an upper layer bounded by a free surface and welded to a lower half-space with different elastic property. Mode III crack is employed as a mathematical model of strike-slip fault, which is nucleated in the lower half-space and then propagates towards the interface. We adopt FEM-β, newly proposed analysis method for failure, to simulate the quasi-statistic crack growth governed by the stress distribution in layered media. Our results show that along planar traces across interfaces a compliant upper layer has significant effects on promoting/suppressing crack growth before/after its extension into the layer and vice versa for a rigid one. This proposes a possibility that surface breaks due to strike-slip faulting could be arrested by deposit layers at the topmost part of the Earth's crust.     

High-frequency radiation from crack (stress drop) models of earthquake faulting

We present a theory for the radiation of high-frequency waves by earthquake faults. We model the fault as a planar region in which the stress drops to the kinematic friction during slip. This model is entirely equivalent to a shear crack. For two-dimensional fault models we show that the high frequencies originate from the stress and slip velocity concentrations in the vicinity of the fault's edges. These stress concentrations radiate when the crack expands with accelerated motion. The most efficient generation of high-frequency waves occurs when the rupture velocity changes abruptly. In this case, the displacement spectrum has an ω^-2 behaviour at high frequencies. The excitation is proportional to the intensity of the stress concentration near the crack tips and to the change in the focusing factor due to rupture velocity. We extend these two-dimensional results to more general three-dimensional fault models in the case when the rupture velocity changes simultaneously on the rupture front. Results are similar to those described for two-dimensional faults. We apply the theory to the case of a circular fault that grows at constant velocity and stops suddenly. The present theory is in excellent agreement with a numerical solution of the same problem.
Our results provide upper bounds to the high-frequency radiation from more realistic models in which rupture velocity does not change suddenly. The ω^-2 is the minimum possible decay at high frequencies for any crack model of the source.     

经济全球化背景下的产业空间重构

探讨在全球化背景下影响产业空间重构的成本因素变化,通过对企业迁移区位和政府产业转移园区区位的分析,揭示广东产业空间重构的趋势与可能的窄间格局,并且从绩效的角度对比了2003和2006年两个时间的广东省域经济增长格局,证明原有的珠三角和外围地区的"核心一边缘"结构已经被打破,产业转移不仅导致了珠三角外围地区经济的快速增长,也同时促使"后厂"的空间拓展和珠三角产业空间的转型.     

Dynamics of a one-dimensional crack with variable friction

Summary. For a one-dimensional model for which the stress drop is a function of the relative displacement, the absence of an initial instantaneous stress drop from the static level inhibits the formation of a spontaneous, growing crack.     

深圳市土地供给与经济增长关系研究

利用连续的动态资料，对深圳土地供给稀缺性、土地供给与经济发展的相关性、土地利用弹性系数、土地供给对经济增长贡献率及其变化进行了定量分析。结果表明，深圳建设用地的快速扩张支撑了经济总量和城市建设的高速发展；经济增长与土地供给呈现一致波动性，并表现出滞后性；土地投入对经济增长的贡献具有“隐性贡献”的特点。     

城市用地与人口的异速增长和相关经验研究

由于城市土地利用变化涉及的因素多，使获取动态研究研究所需的资料十分困难，所以，用少数几个主要因素定量地表达其变化就显得十分重要。从前人的成果，即以人口表示的城市位序-规模法则和建成区面积表示的位序-规模法则出发，绽绎出城市的用地规模和人口数量呈异速增长。这意味着，如果把整个城市看成是一个生命有机体，那么作为反映城市特征的城市用地规模和城市人口这两个重要变量，就是城市这个有机体的两个器官，他们的增长率是成比例的。还通过这个关系建立了城市建成区面积与市场人口和经济发展水平的数学模式。对我国部分城市的经验研究在一定程度上分别证明了这两经验关系。     

华南地区龙眼种植的温度风险评估

建立了龙眼温度适宜度模型和风险动态评估模型,对华南地区龙眼温度适宜性和风险性及其时空差异进行了计算、评价和预测。首先,分析龙眼各生育期适宜度,结果表明:影响华南地区龙眼生产的关键问题不是冷害,而是冬春季节的热害;未来各生育期中除果实发育成熟期和抽梢期无明显变动外,其他生育期的温度适宜度都有下降趋势。其次,根据华南地区龙眼不同减产率的温度适宜度的概率分布,将华南地区龙眼温度风险大致划分高、中、低三个区,并对不同时段华南地区龙眼温度风险进行了对比分析,结果表明:在空间上,华南地区龙眼温度风险性的地域分异呈现纬度地带性规律;在时间上,其风险性随时间推移有增加的趋势。最后,分析了龙眼生育期温度风险对气候变暖的响应。     

中国沿海地区产业结构水平的横向对比与纵向测度模型及应用——基于福建省的视角

优化、高效的区域产业结构是区域社会经济发展的必备条件.本研究基于传统偏离-份额分析法(SSM)以及动态偏离-份额分析法对区域产业结构进行横向对比与纵向测度,从福建省的视角进行实证分析,旨在为区域产业结构优化升级、经济发展方式转变提供科学指导.研究表明:1)中国沿海地区的产业结构份额和产业竞争份额大部分大于全国平均水平,说明沿海地区相对于其他地区具有产业结构的优势;2)福建的经济增长主要是依靠全国经济增长的作用,而不是靠产业结构和产业竞争力的优势;3)相对于其他沿海地区,福建省的发展速度相对落后;4)福建省应利用区位优势,承接台湾地区的产业转移,促进福建省的产业结构优化升级.     

Seismic radiation from dynamic coalescence, and the reconstruction of dynamic source parameters on a planar fault

The dynamic coalescence of two mode II cracks on a planar fault is simulated here using the elastodynamic boundary integral equation method. We focus on the complexity of the resultant slip rate and seismic radiation in the crack coalescence model (CCM) and on the reconstruction of a single crack model (SCM) that can reproduce the CCM waveforms from heterogeneous source parameters rather than coalescence. Simulation results reveal that localized higher slip rates are generated by coalescence as a result of stress interaction between the approaching crack tips. The synthesized seismic radiation exhibits a distinct coalescence phase that has striking similarities to stopping phases in the radiation and propagation properties. The corresponding SCM yields a singular increase in the stress drop distribution, which is accompanied by a sudden decrease in it across the point of coalescence in the CCM. This implies that the generation of high-frequency radiation is more efficient from coalescence than from stopping, although both phenomena exhibit the same strong  ω⁻²  -type displacement spectra.     

广州市荔湾区大坦沙岩溶地面塌陷成因及其稳定性评价

广州市荔湾区大坦沙是广州地区岩溶地面塌陷主要发生地区之一,2007-11-07-2008-01-29共发生7起岩溶地面塌陷,造成多间房屋倒塌和开裂,经济损失巨大.通过参与现场地质调查,之后收集、整理该地区地下水的动态监测资料,结合地面塌陷发育的基本特征,从区域地质环境条件分析其地面塌陷的形成原因,采用定性、半定量的评价方法,构建岩溶地面塌陷稳定性判别因子的量化指标体系,对其地面塌陷的稳定性进行综合预测.结果认为:(1)大坦沙地区岩溶地面塌陷具有一因多效现象,地下水位下降是引发该时段系列塌陷的主导因素,次要