The spacetime gravitational field of a deformable body

The high-resolution analysis of orbit perturbations of terrestrial artificial satellites has documented that the eigengravitation of a massive body like the Earth changes in time, namely with periodic and aperiodic constituents. For the space-time variation of the gravitational field the action of internal and external volume as well as surface forces on a deformable massive body are responsible. Free of any assumption on the symmetry of the constitution of the deformable body we review the incremental spatial (“Eulerian”) and material (“Lagrangean”) gravitational field equations, in particular the source terms (two constituents: the divergence of the displacement field as well as the projection of the displacement field onto the gradient of the reference mass density function) and the `jump conditions' at the boundary surface of the body as well as at internal interfaces both in linear approximation. A spherical harmonic expansion in terms of multipoles of the incremental Eulerian gravitational potential is presented. Three types of spherical multipoles are identified, namely the dilatation multipoles, the transport displacement multipoles and those multipoles which are generated by mass condensation onto the boundary reference surface or internal interfaces. The degree-one term has been identified as non-zero, thus as a “dipole moment” being responsible for the varying position of the deformable body's mass centre. Finally, for those deformable bodies which enjoy a spherically symmetric constitution, emphasis is on the functional relation between Green functions, namely between Fourier-/ Laplace-transformed volume versus surface Love-Shida functions (h(r),l(r) versus h ^′(r),l ^′(r)) and Love functions k(r) versus k ^′(r). The functional relation is numerically tested for an active tidal force/potential and an active loading force/potential, proving an excellent agreement with experimental results. Received: December 1995 / Accepted: 1 February 1997     

Thermoelastic stress due to a rectangular heat source in a semi-infinite medium. Presentation of an analytical solution

The thermoelastic response due to a time-dependent rectangular heat source in a semi-infinite medium is analyzed. The problem originates from studies of nuclear waste repositories in rock. Canisters containing heat-emitting nuclear waste are deposited over a large rectangular area deep below the ground surface. The solution for a time-dependent heat source is obtained from the corresponding instantaneous heat source by superposition. The thermoelastic problem for the instantaneous rectangular heat source in a infinite surrounding is solved exactly. An important step is the introduction of so-called quadrantal heat sources. The solution for the rectangle is obtained from four quadrantal solutions. The solution for the quadrantal heat source depends on the three dimelasionless coordinates only. Time occurs in the scale factors only. The condition of zero normal and shear stresses at the ground surface is fulfilled by using a mirror heat source and a boundary solution. The boundary solution accounts for the residual normal stress at the ground surface. Using a Hertzian potential, a surprisingly simple solution is obtained. The final analytical solution is quite tractable considering the complexity of the initial problem. The solution may be used to test numerical models for coupled thermoelastic processes. It may also be used in more detailed numerical simulations of the process near the heat sources as boundary conditions to account for the three-dimensional global process.     

Experimental assessment and prediction of temporal scour depth around a spur dike

Spur dikes are river training structures that have been extensively used worldwide for towards enhancing flood control and the stability of embankments and riverbanks.However,scour around spur dikes can be a major problem affecting their stability and hydraulic performance.The precise computation of temporal scour depth at spur dikes is very important for the design of economical and safe spur dikes.This study focuses on experimentally assessing the temporal variation of scour depth around a vertical wall spur dike and identifying the parameters,which mostly influence spur dike performance for a channel bed surface comprised of sand-gravel mixtures.In the current study,the authors did physical experiments in a flume based study to obtain new data,aimed at deriving a new predictive model for spur dike scour and comparing its performance to others found in the literature.It was found that the dimensionless temporal scour depth variation increases with an increase in(i)the threshold velocity ratio,(ii)the densimetric Froude number of the bed surface sediment mixture,(iii)the flow shallowness(defined as the ratio of the approach flow depth,y,to the spur dike’s transverse length,l),and(iv)the flow depth-particle size ratio.It is also concluded that the temporal scour depth variation in the sediment mixture is influenced by the non-uniformity of sediment and decreases with an increase in the non-uniformity of the sediment mixture.A new mathematical model is derived for the estimation of temporal scour depths in sand-gravel sediment mixtures.The proposed equation has been calibrated and validated with the experimental data,demonstrating a good predictive capacity for the estimation of temporal scour depth evolution.     

“Transverse Action” – A model benchmark exercise for numerical analysis of the Callovo-Oxfordian claystone hydromechanical response to excavation operations

The Callovo-Oxfordian claystone (COx) is considered as a potential geological formation to host an industrial radioactive waste repository in France. A detailed understanding of the thermo-hydro-mechanical (THM) behavior of the COx is a key issue for design of different repository structures and the safety calculations of the project. More particularly, numerical modeling of induced fracture networks around drifts excavated at the main level of the Andra’s Meuse/Haut-Marne Underground Research Laboratory (M/HM URL) and short and long term behavior of the COx around these drifts are of great interest. Several constitutive models have been developed/used in the framework of the R&D and simulation programs of Andra. A model benchmark exercise has been launched since 2012 to provide an overall view of the developed models regarding the in situ observations. In this view, two series of test cases, respectively at material point scale and at drift excavation scale are defined. Different kinds of constitutive models based on the elasto-visco-plasticity concept, continuum damage mechanics, the rigid block spring method and two-scale computational homogenized model (CHM) are used within this exercise. The obtained results show that accounting for material anisotropy and strain localization treatment techniques can improve the obtained results when elasto-visco-plastic models are used. Damage mechanics based approaches and methods accounting for discontinuities through discrete elements provide also interesting insights especially when fracturing processes must be modeled. However, more efforts are necessary to improve the robustness of these kinds of approaches in the complex context of COx response to excavation works.     

Kinematic dynamo action in a sphere: Effects of periodic time-dependent flows on solutions with axial dipole symmetry

Choosing a simple class of flows, with characteristics that may be present in the Earth's core, we study the ability to generate a magnetic field when the flow is permitted to oscillate periodically in time. The flow characteristics are parameterised by D, representing a differential rotation, M, a meridional circulation, and C, a component characterising convective rolls. The dynamo action of all solutions with fixed parameters (steady flows) is known from earlier studies. Dynamo action is sensitive to these flow parameters and fails spectacularly for much of the parameter space where magnetic flux is concentrated into small regions, leading to high diffusion. In addition, steady flows generate only steady or regularly reversing oscillatory fields and cannot therefore reproduce irregular geomagnetic-type reversal behaviour. Oscillations of the flow are introduced by varying the flow parameters in time, defining a closed orbit in the space ( D,?M ). When the frequency of the oscillation is small, the net growth rate of the magnetic field over one period approaches the average of the growth rates for steady flows along the orbit. At increased frequency time-dependence appears to smooth out flux concentrations, often enhancing dynamo action. Dynamo action can be impaired, however, when flux concentrations of opposite signs occur close together as smoothing destroys the flux by cancellation. It is possible to produce geomagnetic-type reversals by making the orbit stray into a region where the steady flows generate oscillatory fields. In this case, however, dynamo action was not found to be enhanced by the time-dependence. A novel approach is being taken to solve the time-dependent eigenvalue problem where, by combining Floquet theory with a matrix-free Krylov-subspace method, we can avoid large memory requirements for storing the matrix required by the standard approach.     

地震的"序列归属"问题与ETAS模型——以唐山序列为例

历史上发生过强震地区的余震活动可能持续较长时间,而余震序列在何时可被看作正常的"背景地震活动",即"序列归属"问题在地球动力学和地震物理中有重要意义.时-空"传染型余震序列"(ETAS)模型可分离"背景"地震和"丛集"地震,并用概率形式表示作为相应事件的可能性,为考察此问题提供了可能.本文以1976年唐山M_S7.8地震序列为例,对唐山地区1970年以来的M_L4.0以上地震进行了时-空ETAS模型拟合,并以2010年以来发生的3次M_S4.0以上地震为例讨论了它们的"序列归属"问题.研究结果显示,3次M_S4.0以上地震的背景地震概率分别为0.72、0.88和0.76,表明它们作为1976年唐山M_S7.8的余震的可能性较低,更可能为背景地震.     

2010年玉树MS7.1地震前的中长期加速矩释放(AMR)问题

2010年4月14日青海玉树MS7.1地震前的加速矩释放(AMR)现象的研究,对理解这次地震的孕震过程、对于时间相依的地震危险性分析(或中长期地震预测)具有重要意义.鉴于以往AMR研究中的争论,本文不刻意选取AMR分析的时空尺度,而是在已知发震时刻和震中位置情况下,对T-R-MC三维空间中矩释放指数m值的分布进行分析,...     

Probabilistic aftershock hazard assessment I: numerical testing of methodological features

Probabilistic aftershock hazard assessment (PAHA, Wiemer, Geoph Res Lett 27:3405–3408, 2000), provided for California within the frame of the STEP project, is based on a methodology, two features of which are addressed in detail: (1) the parameters of Omori’s law and (2) application of attenuation relations in evaluating peak ground velocity exceedence probability. Concerning the first point, we perform a simple parametric study. We assume the generalized Omori’s law by Shcherbakov et al. (Geoph Res Lett 31:L11613, 2004), in which characteristic time c scales with aftershock magnitude. The study shows that, among all the parameters, the hazard is most sensitive to the choice of m* (controlling the overall aftershock productivity) and least sensitive to the scaling of c. We also conclude that the hazard is mainly due to very early (less than 1 day) aftershocks. As regards the second point, we employ various attenuation relations from different tectonic areas to study their effect on the hazard analysis. We conclude that the resulting variations are relatively large, comparable to those obtained for varying m*.     

Probabilistic aftershock hazard assessment II: application of strong ground motion simulations

Probabilistic aftershock hazard assessment (PAHA) has been introduced by Wiemer (Geophys Res Lett 27:3405–3408, 2000). The method, in its original form, utilizes attenuation relations in evaluating peak ground velocity (PGV) exceedence probability. We substitute the attenuation relations together with their uncertainties by strong ground motion simulations for a set of scenarios. The main advantage of such an approach is that the simulations account for specific details of the aftershock source effects (faulting style, slip distribution, position of the nucleation point, etc.). Mean PGVs and their standard deviations are retrieved from the simulation results obtained by the new hybrid k-squared source model, and they are used for the PAHA analysis at a station under study. The model chosen for the testing purposes is inspired by the Izmit A25 aftershock (M _w  = 5.8) that occurred 26 days after the mainshock. The PAHA maps are compared with (1) those obtained by the use of attenuation relations and (2) the peak values of ten selected strong-motion recordings written by the aftershock at epicentral distances <50 km. We conclude that, although the overall hazard decay with increasing fault distance is similar, the PAHA maps obtained by the use of simulations exhibit remanent radiation pattern effect and prolongation in the strike direction due to the directivity effect pronounced for some of the scenarios. As regard the comparison with real data, we conclude that the PAHA maps agree with observed peak values due to appropriate attenuation model adopted in the analysis.