A branching model for crack propagation

Summary A branching model for crack propagation is proposed, a branch corresponding to an existing microfissure or flaw in the material, and the propagation of the crack to the coalescence of such branches. Increase in external stress increases the probability that a given branch will link into more than a specified number of further branches. Such increases can continue until a critical state is reached when the mean number of branches linking into a given branch is equal to unity; beyond this point, the system becomes unstable, and any slight movement is likely to lead to catastrophic rupture. The distribution of the sums of the lengths of the branches linked together in a cracking episode is investigated, and shown to lead, in the critical case, to a Gutenberg-Richter type relation with parameterb=0.75. Departures from this value are attributed to the influence of the distribution of the lengths of preexisting fissures, this distribution varying with the strength of the material and its stress history. Some difficulties with the theoretical model of Scholz are raised, and it is suggested that a more complete analysis of Scholz's model should lead to results qualitatively similar to those obtained for the branching model.     

A statistical model of the geomagnetic field

Summary A statistical model of the geomagnetic field is derived, based on the assumption of an axial geocentric dipole field of strengthH _e at the equator perturbed by randomly directed components of constant magnitudeh. The model fits the dispersions found from an analysis of the 1945 field, and the ratioh/H _e obtained for this field and from the palaeomagnetic data both average to about 0.4. The model predicts that during reversal of the dipole field, the field intensity falls to between 0.2 and 0.4 of the steady field intensity, and this agrees with estimates made from the palaeomagnetic observations.     

A statistical model for vesuvius and its volcanological implications

The last 300 years of Vesuvius history are reconstructed as a chronological succession of 4 phenomenological states: i) repose, ii) persistent activity, iii) intermediate eruption and iv) final eruption. It turns out that the times of permanence in each state are distributed according to the same exponential law. Vesuvius activity is then described by a Markov chain of these 4 states, with transition probabilities determined from the previous phenomenological analysis. The model reproduces the Vesuvius activity between 1694 and 1872 and possibly also in the 1872–1944 period. It turns out that, at least between 1694 and 1872, the volcano was behaving like a quasistationary system with 4 equilibrium states, perturbed by a stochastic noise responsible for occasional transitions from an equilibrium state to another. Major physical or structural changes of the volcanic system around 1872 and possibly in the whole subsequent period, are clearly shown by the statistical analysis.     

A statistical damage model with implications for precursory seismicity

Acoustic emissions prior to rupture indicate precursory damage. Laboratory studies of frictional sliding on model faults feature accelerating rates of acoustic emissions prior to rupture. Precursory seismic emissions are not generally observed prior to earthquakes. To address the problem of precursory damage, we consider failure in a fiber-bundle model. We observe a clearly defined nucleation phase followed by a catastrophic rupture. The fibers are hypothesized to represent asperities on a fault. Two limiting behaviors are the equal load sharing p = 0 (stress from a failed fiber is transferred equally to all surviving fibers) and the local load sharing p = 1 (stress from a failed fiber is transferred to adjacent fibers). We show that precursory damage in the nucleation phase is greatly reduced in the local-load sharing limit. The local transfer of stress from an asperity concentrates nucleation, restricting precursory acoustic emissions (seismic activity).     

A forest floor model for heat and moisture including a litter layer

Forest soils are often covered with a litter that influences the rate of mass and energy transfer between the soil and the air above, thereby modifying the temperature and moisture fields in the soil. The presence of a litter should therefore be accounted for in forest SVAT models, especially when long-term simulations are to be performed. A heat and moisture litter model has been developed by adding two dynamical equations to a force-restore type soil model. The experimental data used for the model validation was collected in a pine forest canopy in the South-West of France, that was part of the Euroflux network. The model is tested and validated over a two-year period. It is shown to provide a fairly good simulation of soil and litter moisture, soil and litter temperature and turbulent fluxes measured above the forest floor. It is also shown that simulations without the litter layer are unable to reproduce all these variables simultaneously. We then perform a sensitivity analysis to the parameters whose values are either uncertain or likely to be variable in time and space, such as the litter thickness, the rainfall fraction intercepted by the litter or the maximum value of the surface resistance. A threshold value of the litter moisture used in the surface resistance parameterisation turns out to be the most critical parameter. Further work is needed to investigate the possible relationships between the various parameters describing the litter, but the present litter model can already be used in combination with other forest SVAT models.     

A gradient plasticity model for lüders band propagation

At the onset of plastic deformation some materials exhibit non-monotonic behaviour in that after initially yielding the flow stress decreases with continuing strain, passing through a minimum. Strain softening destabilizes homogeneous configurations and results in the formation of bands of localized deformation. Within these bands the macro and microscales of the deformation overlap and accordingly terms have to be included in the evolution equation for the plastic strain to provide the necessary information on the material's behaviour at the next smaller scale. In the model chosen here the evolution equation has the form of a reaction diffusion equation, whereby physically the diffusion term accounts for the nonlocal interaction between dislocations on neighbouring slip planes. The model predicts the band propagation velocity, the width of the propagation front and the strain profile.     

A two-dimensional horizontal wave propagation and mud mass transport model

The present study offers a two-dimensional horizontal wave propagation and morphodynamic model for muddy coasts. The model can be applied on a general three-dimensional bathymetry of a soft muddy coast to calculate wave damping, fluid mud mass transport and resulting bathymetry change under wave actions. The wave propagation model is based on time-dependent mild slope equations including the wave energy dissipation due to the wave-mud interaction of bottom mud layers as well as the combined effects of the wave refraction, diffraction and breaking. The constitutive equations of the visco-elastic–plastic model are adopted for the rheological behavior of fluid mud. The mass transport velocity within the fluid mud layer is calculated combining the Stokes’ drift, the mean Eulerian velocity and the gravity-driven mud flow. The results of the numerical model are compared against a series of conducted wave basin experiments, wave flume experiments and field observations. Comparisons between the computed results with both the field and laboratory data reveal the capability of the proposed model to predict the wave transformation and mud mass transport.     

A physically constrained wavelet-aided statistical model for multi-decadal groundwater dynamics predictions

A physically constrained wavelet-aided statistical model (PCWASM) is presented to analyse and predict monthly groundwater dynamics on multi-decadal or longer time scales. The approach retains the simplicity of regression modelling but is constrained by temporal scales of processes responsible for groundwater level variation, including aquifer recharge and pumping. The methodology integrates statistical correlations enhanced with wavelet analysis into established principles of groundwater hydraulics including convolution, superposition and the Cooper–Jacob solution. The systematic approach includes (1) identification of hydrologic trends and correlations using cross-correlation and multi-time scale wavelet analyses; (2) integrating temperature-based evapotranspiration and groundwater pumping stresses and (3) assessing model prediction performances using fixed-block k-fold cross-validation and split calibration-validation methods. The approach is applied at three hydrogeologicaly distinct sites in North Florida in the United States using over 40 years of monthly groundwater levels. The systematic approach identifies two patterns of cross-correlations between groundwater levels and historical rainfall, indicating low-frequency variabilities are critical for long-term predictions. The models performed well for predicting monthly groundwater levels from 7 to 22 years with less than 2.1 ft (0.7 m) errors. Further evaluation by the moving-block bootstrap regression indicates the PCWASM can be a reliable tool for long-term groundwater level predictions. This study provides a parsimonious approach to predict multi-decadal groundwater dynamics with the ability to discern impacts of pumping and climate change on aquifer levels. The PCWASM is computationally efficient and can be implemented using publicly available datasets. Thus, it should provide a versatile tool for managers and researchers for predicting multi-decadal monthly groundwater levels under changing climatic and pumping impacts over a long time period.     

A thin-layer interface model for wave propagation through filled rock joints

The present study essentially employs a thin-layer interface model for filled rock joints to analyze wave propagation across the jointed rock masses. The thin-layer interface model treats the rough-surfaced joint and the filling material as a continuum medium with a finite thickness. The filling medium is sandwiched between the adjacent rock materials. By back analysis, the relation between the normal stress and the closure of the filled joint are derived, where the effect of joint deformation process on the wave propagation through the joint is analyzed. Analytical solutions and laboratory tests are compared to evaluate the validity of the thin-layer interface model for filled rock joints with linear and nonlinear mechanical properties. The advantages and the disadvantages of the present approach are also discussed.     

包含Biot机制和分数阶黏弹性机制的波传播模型

含流体孔隙介质中的波能量耗散通常由多种力学机制造成.传统Biot理论中的能量耗散仅仅考虑了固流两相相对运动引起的摩擦耗散,无法准确预测波在孔隙介质中低频段出现的高频散与强衰减现象.为了建立一个能准确预测地震波频段高频散与强衰减现象的动力学模型,我们在Biot理论的基础上引入黏弹性机制,并利用分数阶导数刻画黏弹性本构关系,最终获得了一种新的孔隙介质波传播模型.与传统的Biot模型相比,新模型考虑了含流体孔隙介质中固体骨架的内耗散,对波能量耗散的刻画更为精准.通过数值算例,我们研究了分数阶导数的阶数参数对快P波和S波频散和衰减的影响,并通过来自不同地区且具有不同物理性质的几组流体饱和岩芯实验数据,对比研究了新模型的有效性.结果表明,文章提出的新模型能更准确地预测快P波和S波在低频段出现的高频散和强衰减现象.     

用传输函数构建的大气重力波传播理论模式

本文根据考虑大气热传导和黏滞的重力波复色散关系,采用传输函数的概念,基于重力波的线性理论,构建了用于研究对流层内重力波激发源与电离层响应之间的传输函数数值模式.在相空间中讨论了传输函数振幅的分布特性,并以地面单位脉冲源为例,分析了从地面到300 km高空的响应,得到了物理量的时空分布特征.结果表明：（1）对内重力波的传播而言,大气相当于一个滤波器,只有波动周期在15～30 min,水平波长在200～450 km之间的重力波扰动才最容易到达300 km电离层高度;（2）电离层的响应主要在与地面的激发源之间相隔较远的水平距离上发生;（3）黏滞和热传导系数在低层对上传重力波的影响较小,随着高度的增加它们对重力波的影响越来越大;（4）在低层计算的波动频率与Row理论的计算结果比较一致,然而到了高层却相差较大.     

砾岩储层地震波传播方程:三重孔隙结构模型

针对砾岩储层的砂、砾、泥三重孔隙结构特征,本文分析砾岩孔隙区域、砂岩孔隙区域以及泥岩孔隙区域相互之间的孔隙流体流动机制,将静态的砾岩骨架本构方程与动态的孔隙流体运动方程联立,提出了复杂砾岩储层的弹性波传播理论方程.采用实测砾岩储层参数,在算例中与双重孔隙介质理论进行对比分析,验证了本文理论方程的合理性;基于三重孔隙介质模型,分析不同储层环境下纵波的传播特征,结果显示：随流体黏滞系数增大,在衰减-频率轴坐标系中,砾与砂、砂与泥孔隙区域间局域流导致的两个衰减峰向低频端移动,而Biot全局流导致的衰减峰向高频端移动;嵌入体尺寸及背景相介质渗透率的变化,主要影响纵波速度频散曲线沿频率轴左、右平移,不影响波速低频、高频极限幅值;嵌入体含量及孔隙度的变化改变了岩石干骨架的弹性、密度参数,不仅影响速度频散曲线沿频率轴平移,而且影响其上、下限幅值;砾包砂包泥三重孔隙介质模型所预测的衰减曲线中,低频段“第一个衰减峰”主要由砾岩孔隙区域与砂岩孔隙区域之间的局域流导致,中间频段“第二个衰减峰”主要由砂岩孔隙区域与泥岩孔隙区域之间的局域流导致,超声频段“第三个衰减峰”由Biot全局流导致.对慢纵波传播特征的分析显示,砂岩骨架（局部孔隙度较大）内部的宏观孔隙流体流动造成的耗散明显强于砾岩与泥岩骨架.

     

The propagation of a single shear crack: A displacement discontinuity model

Numerical studies using the displacement discontinuity method show that a single shear crack under compression propagates in its own direction, because such propagation results in the maximum release of strain energy. The methods of linear elastic fracture mechanics may not be used for such a closed crack, and the stress intensity factors are meaningless in that case. Laboratory observations of propagation by means of kinks at an angle of approximately 70° to the crack may be due to heterogeneities, to the effect of a preexisting crack, to end effects, to microcracking, or to some combination of these factors. Such kinks may thus be local phenomena which cannot release most of the strain energy, and are not incompatible with our numerical results which are based on a global energy balance.     

A wave propagation model with the Biot and the fractional viscoelastic mechanisms

Energy loss in porous media containing fluids is typically caused by a variety of dynamic mechanisms.In the Biot theory,energy loss only includes the frictional dissipation between the solid phase and the fluid phase,resulting in underestimation of the dispersion and attenuation of the waves in the low frequency range.To develop a dynamic model that can predict the high dispersion and strong attenuation of waves at the seismic band,we introduce viscoelasticity into the Biot model and use fractional derivatives to describe the viscoelastic mechanism,and finally propose a new wave propagation model.Unlike the Biot model,the proposed model includes the intrinsic dissipation of the solid frame.We investigate the effects of the fractional order parameters on the dispersion and attenuation of the P-and S-waves using several numerical experiments.Furthermore,we use several groups of experimental data from different fluid-saturated rocks to testify the validity of the new model.The results demonstrate that the new model provides more accurate predictions of high dispersion and strong attenuation of different waves in the low frequency range.     

Multiparticle statistical model of the solar wind

A multiparticle statistical approach to plasma (gas) modeling is presented, in which the fact that the macroscopic parameters are measured with finite resolution scales is taken into account [Minkova, 2004; 2005; 2007]. This approach is based on the Liouville theorem formulated for a stationary open system in the approximation of detailed dynamic balance with its surroundings. When the finite resolution scales of measuring instruments are taken into account, the plasma (gas) is described by multiparticle distribution functions. The latter are used to derive the probability distribution functions of fluctuations and the average values of macroscopic parameters. The multiparticle statistical approach allows a stationary solar wind model to be constructed under a number of simplifying assumptions. Its results for the average values of macroscopic parameters coincide with those of the two-particle kinetic model [Vasenin et al., 2003] and agree with inecliptic observational data.     

Key principles of constructing probabilistic statistical ionosphere models for the radiowave propagation problems

It is demonstrated that it is required to create probabilistic statistical models of the ionosphere for calculating radio propagation in a wide frequency range. This, in fact, presents a new type of ionospheric modeling. These models are classified into pure statistical and deterministic-stochastic. We describe the key principles of building such models, present some examples of their construction, and discuss some difficulties arising from them.     

A general Lagrangian tracking methodology for riverine flow and transport

This article presents a new public domain tool for generalized Lagrangian particle tracking in rivers. The approach can be applied with a variety of two- and three-dimensional flow solvers. Particle advection by the flow is incorporated using flow fields from the chosen solver assuming particles follow the Reynolds-averaged flow, although some other simple passive and active particle behaviors are also treated. Turbulence effects are treated using a random walk algorithm with spatial step lengths randomly chosen from Gaussian distributions characterized by the diffusivity from the flow solver. Our work extends this concept to a general framework that is solver and coordinate system independent to allow easy comparisons between differing flow treatments. To better treat problems where detailed information is required in specific regions, the approach includes novel cloning and colligation algorithms which enhance local resolution at modest computational expense. We also provide tools for computing local concentrations and total exposure over a user-specified time interval. Several examples of predictions are provided to illustrate applications of the technique, including examination of the role of curvature-driven secondary flows, storage in lateral separation eddies, treatment of larval drift, treatment of fuel spill dispersion, river-floodplain connections, and sedimentation in floodplain ponds by tie channel connections. We also demonstrate that the model can reproduce analytically derived concentration profiles for simple diffusivities. These examples show that the Lagrangian particle tracking approach and the extensions proposed here are broadly applicable and viable for treating difficult river problems with multiple temporal and spatial scales. The examples also illustrate the utility of the cloning/colligation extensions and show how these can decrease the computational effort required on problems where high local resolution is required. Enhancement of the tools and even broader applicability can be achieved through the inclusion of multiple particle populations and particle–particle interactions.     

A comparison between two statistical and a physically-based model in snow water equivalent mapping

Snow water equivalent (SWE) estimates at the end of the winter season have been compared for the 2002–2006 period in a 200 km² mountainous area in Switzerland, using three different models. The first model, ALPINE3D, is a physically based process-oriented model, which solves the snowpack energy and mass balance equations. The other two models, SWE-SEM and HS-SWE, are statistical algorithms interpolating snow data on a grid. While SWE-SEM interpolates local estimates of SWE, HS-SWE converts interpolated snow depth maps into maps of SWE using a regionally-calibrated conversion model. We discuss similarities and differences among the models’ results, both in terms of total volume, and spatial distribution of SWE. The comparison shows a general good agreement of the results of the three models, with a mean difference in the total volumes between the two statistical models of ∼8%, and between the physical model and the statistical ones of ∼−3% to −10%.     

Simple two-dimensional model of propagation of magma-filled cracks

The direction of propagation of magma-filled cracks was theoretically examined for a two-dimensional model. The analytical result indicates that magma-filled cracks, which have magmatic pressures larger than a critical value, propagate in parallel with the maximum principal direction of far-field stress. This may give one of theoretical grounds to the dike method by which the regional stress field is estimated.