Some problems of rotation and equilibrium of transversely isotropic elastic bodies

Summary The present discussion is concerned with the rotation of a few problems of aeolotropic elastic solids. The symmetry of the material considered in the discussion is transverse-isotropy. The problems solved are: (i) rotation of a circular cylinder with axle-hole, (ii) rotation of a circular disc with axle-hole, (iii) a cylindrical tube under external and internal pressures. Complete cylinder and complete disc are the particular cases of the above.     

Mechanical problems in geodynamics and work done in China

The subject of geodynamics concerns the dynamics of the global motion of the earth, of the motion in the earth's interior and its interaction with surface features, together with mechanical processes in the deformation and rupture of geological structures. A brief historical review is given from Sir I. Newton, Maclauwin, Jacobi, Poincaré, Poisson, Lamé, Darwin, Rayleigh, and Love, who dealt with the homogeneous sphere, to Leibenson, Takeuchi, and Meinesz, concentrating on layered spheres, and after the advent of plate tectonics, on the use of numerical simulation, to the analyses of tectonic features, earthquake mechanism, the application of nonlinear dynamics. By discussing the forward and inverse mechanical problems, and the questions facing the inverse problems of searching for the structural parameters, driving forces, etc. are raised in more detail. Thereafter, some works accomplished in China on the global and regional stress fields analyses; tectonic features' analyses, mantle flow studies; experimental studies of rocks and their constitutive relations are presented. Finally, the interdisciplinary nature of the subject is emphasized, and the main mechanical problems that need special attention are then proposed.     

The topical problems of identifying the results of the observations in recent geodynamics

The paper addresses the problems of identifying the results of deformational observations on the Earth, which arise from the new measurement technologies and lead to the ambiguity relationships such as the “spatial size of the anomaly—density of the observation sites” and the “duration of the anomalies—degree of temporal detail of the measurements.” It is found that many “paradoxical” conclusions concerning the rates and scales of the recent geodynamical processes are removed if the parameters of the measurement system match the properties of the studied object and if the relative character of the observational means is taken into account. It is shown that the time variation in the uniaxial deviatoric stress leads to the variation in the volumetric strain and, consequently, to the variation in gravity. The ambiguity in determining the ground displacement vectors by SAR interferometry is demonstrated. It is concluded that the autonomous use of the interferometry data leads to the significant distortions of the results, and these data should be necessarily used in combination with the ground-based geodetic observations.     

Recent applications of sliding block theory to geotechnical design

The sliding block theory was proposed by Newmark for determining the permanent displacement of embankments and dams under earthquake loading. This paper highlights recent applications of sliding block theory to different geotechnical structures. The equations to determine seismic factor of safety, yield acceleration and permanent displacement are given for rock block, soil slope, landfill cover, geosynthetic-reinforced soil retaining wall, and composite breakwater. The presented equations for seismic stability degenerate to that of static stability in the absence of earthquake. The permanent displacement for various structures can be obtained from that of a horizontal sliding block through a correction factor. A simplified procedure is included for the permanent displacement under vertical acceleration. The sliding block approach is rational for design under high seismic load.     

地球物理学及地球动力学研究与计算数学

地球物理学是20世纪,特别是在其中叶以后迅速发展起来的一门边缘科学.它以物理学、数学和信息科学为依托,并与地质学、地球化学密切结合;以研究和探索地球内部介质的属性、结构变异和深部物质与能量的交换、深层过程和动力学响应.这在促进社会与经济的发展和科学与技术的进步中占有重要地位,因为大量资源的需求和自然灾害的防范在人类生活和生存空间以及可持续发展中乃核心所在.在研究和探索这一系列的地球科学问题的进程中,首先必须通过地表进行高精度的地球物理探测、观测和高分辨率的数据采集,进而利用不同的数学方法反演计算地球内部介质的精细结构和其物理—力学过程与动力学响应.基于此,本文主要讨论了：1）地球物理学和地球动力学的发展趋势及特点;2）地球物理学的发展对计算数学的需求;3）地球动力学和数值模拟与计算地球物理学.最后对这些问题和基本导向进行了讨论.     

Introduction to geodynamics for high- and ultrahigh-pressure metamorphism

Since the first workshop on ultrahigh-pressure (UHP) metamorphism at Stanford in 1994 and the special issue of The Island Arc `Ultrahigh-Pressure Metamorphism and Tectonics' published in December 1995, many symposia and special sessions specifically with regard to the UHP metamorphic terrane have been held. While we are still wondering how exhumation of UHP rocks from mantle depths to the surface takes place, the finding of possible records from the mantle transition zone at 300–400 km depths is astonishing. The study of the UHP regime has expanded to include input from mineral physics, experimental geochemistry and kinetics in addition to the petrochemical and tectonic study of a variety of HP–UHP rocks. It was with this theme that the second workshop for the task group III-6 `Ultrahigh-Pressure Metamorphism and Geodynamics in Collision-type Orogenic Belts' of the International Lithosphere Program was held during the International Geological Congress in Beijing, 1996. The Symposium 8–9 `Dynamic Metamorphic Rocks and High- and Ultrahigh-Pressure Metamorphism' (Cong Bolin & J. G. Liou conveners) had more than 25 presentations in two oral sessions and 70 papers in one poster session. This second special issue of The Island Arc includes nine papers from this symposium and a few related contributions to the geodynamics of HP–UHP metamorphism and tectonics. It is our hope that The Island Arc will continue to publish a special issue on this increasingly recognized subject that is essential to our understanding of continental collision, mantle dynamics and geochemical + fluid cycles.     

Some unintended features of elastic finite-difference models

While finite-difference methods have been used extensively for many years to model wave propagation in elastic media, some of the more subtle effects observable in such models are very inadequately documented in the geophysical literature, especially in regard to their practical numerical consequences. In addition to the intended travelling waves, and the undesirable exponential instability revealed by the von Neumann test, typical second-order-time finite-difference equations also support drifting linear solutions, as can be verified, both theoretically and by numerical experiment. The necessity of these solutions, and their relationship to the incompleteness of the set of travelling-wave eigenfunctions of the finite-difference operator, can be exposed by a matrix-based analysis, and exact expressions for them can be obtained by using standard algebraic techniques. A further peculiarity of the finite-difference formulation is numerical anisotropy, which emerges in a grid of more than one spatial dimension, even when the modelled medium is intended to be isotropic. This anisotropy can be explained and quantified in terms of the exact eigenfunction solutions to the finite-difference equation, which, it is found, can be obtained in a simple, closed form, for a typical modern 3D staggered scheme.     

Transformations of spherical harmonics and applications to geodesy and satellite theory

am ¶rt; mauau uu u u aamuam u a. m num u m a¶rt;a muu nmuaa u ¶rt;uau nmu¶rt;uu.

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Invited Paper, 9^th Brazilian Congress in Cartography, Curitiba, Paraná, Brazil, February 4–9, 1979.     

A complex variable technique to study notch-effects in geodynamics

Summary The techniques of Complex Variables are used to obtain a solution for a general notch problem having a load distributed over any portion of its free surface. This solution is very general in that it can be used for many different shapes of notches, including protrusions, by simply varying a few parameters. The same techniques are then used to solve the problem of a notched plate acted on by loads at infinity. This solution is of a very general nature. One particular solution holds for the problem studied in an earlier publication.     

Recent geodynamics of dangerous faults

The analysis of the existing information concerning the present-day deformation activity of the fault zones in seismically active and aseismic regions suggests that the notions of an active fault and a dangerous fault should be distinguished. It is shown that a fault which is active for an expert in geotectonics will not be considered dangerous by an expert in geotechnical monitoring of buildings. The definition is given according to which a dangerous fault is understood as a zone of linear destruction which accommodates the contemporary short-period (a few months and years) pulsed and/or alternating motions with strain rates above 5 × 10^–5 per annum and earthquakes with M ≥ 5. A technique is developed for identifying the dangerous faults based on monitoring the recent ground surface displacements in accordance with a special protocol which ensures an increased degree of detail in time and space. Based on the idea of the probable accumulation of dangerous strains during the operating cycle of the objects, the criteria for assessing their geodynamical risks are formulated.     

Propagator matrices in elastic wave and vibration problems

The boundary value problems most frequently encountered in studies of elastic wave propagation in stratified media can be formulated in terms of a finite number of linear, first order and ordinary differential equations with variable coefficients. Volterra (1887) has shown that solutions to such a system of equations are conveniently represented by the product integral, or propagator, of the matrix of coefficients. In this paper we summarize some of the better known properties of propagators plus numerica methods for their computation. When the dispersion relation is somem ^th order minor of the integral matrix it is possible to deal withm ^th minor propagators so that the dispersion relation is a single element of them ^th minor integral matrix. In this way one of the major sources of loss of numerical accuracy in computing the dispersion relation is avoided. Propagator equations forSH and forP-SV waves are given for both isotropic and transversely isotropic media. In addition, the second minor propagator equations forP-SV waves are given. Matrix polynomial approximations to the propagators, obtained from the method of mean coefficients by the Cayley-Hamilton theorem and the Lagrange-Sylvester, interpolation formula, are derived.     

Some problems of marine toxicity studies

Toxicity tests are necessarily carried out on animals in the laboratory, but are these test organisms healthy? Dr Perkins suggests some of the pitfalls and describes some of the precautions that must be observed to make sure the subjects of toxicity tests are not already unhealthy and dying because of the way they were collected or kept in the laboratory.     

Some aspects of stochastic reservoir theory

This paper examines the problem of the reservoir size-reliability-draft relationship. It compares the mathematicians' analytical efforts with the simulation techniques of the engineers and hydrologists, and suggests that the two methods should complement each other. It examines the various models for generating gamma Markov sequences and suggests a new one which is analytically more suitable to the problem in question. An alternative to the traditional formulation of the problem in terms of probability of failure is suggested, and a method based on random-walk techniques is employed to give close approximations. Finally, the effects of various parameters of the inflow distribution on the reservoir size to give the same reliability for a constant draft ratio are discussed.     

The interaction of elastic waves with a solid-liquid interface,with applications to the core-mantle boundary

The paper discusses basic ideas and principles underlying methods, which have proved useful in the interpretation of diffraction and scattering phenomena by a smooth or slightly rough solid-liquid interface. Generally, the wave interaction may be formulated as an excitation problem; it involves (1) finding an equivalent dislocation or source distribution on the interface, and (2) evaluating the excited wave field. These steps are taken through perturbation theory and/or adopting the appropriate generalization of ray theory. In this context an explicit form of Green's function is also given. The methods have been applied to the core-mantle boundary, with a view toward the interpretation of recent data: (a) diffracted P and S waves around the core (in particular, their attenuation); (b) scattered short-period core phases (in particular, precursors to PKP and PKKP). Other types of wave interaction, and implications for models of the core-mantle boundary structure, are briefly mentioned.Paper presented at the EGS/ESC workshop on Generation and propagation of seismic waves in Neustadt, Federal Republic of Germany, August 1978.     

Some problems in digital heat-flow measurement

The most accurate method of digitizing a resistance requires that a bridge be brought to null balance by means of digital steps. Satisfactory circuits of reasonable complexity require irregular electrical dissipation in the sensors, and a relatively wide frequency bandwidth in the detector. This combination places considerably greater demands on sensor quality than do conventional electromechanical recorders, and this has been strikingly confirmed during the initial use at sea of a newly developed digital resistance recorder with a sensitivity of 10^?5. Mounted on a coring tool, with miniature outrigger sensors to measure geothermal gradients in the pelagic sediment, the recorder produces sensor readings with considerable scatter even when the sensor probes are buried in sediment. These contrast sharply to the steady reading of the calibration arm. An additional symptom with some sensors is the shunning of certain numbers in the low decades of the digital readings. The reading scatter varies sharply among different sensor types and can be satisfactorily explained by the production of a high level of white noise when the sensors are biased with excitation current. No adequate explanation has been found for the preferential reading of certain numbers, and it is necessary to speculate about obscure contact effects in those sensors.     

Spectral-element simulations of elastic wave propagation in exploration and geotechnical applications

We apply the spectral-element method (SEM), a high-order finite-element method (FEM) to simulate seismic wave propagation in complex media for exploration and geotechnical problems. The SEM accurately treats geometrical complexities through its flexible FEM mesh and accurately interpolates wavefields through high-order Lagrange polynomials. It has been a numerical solver used extensively in earthquake seismology. We demonstrate the applicability of SEM for selected 2D exploration and geotechnical velocity models with an open-source SEM software package SPECFEM2D. The first scenario involves a marine survey for a salt dome with the presence of major internal discontinuities, and the second example simulates seismic wave propagation for an open-pit mine with complex surface topography. Wavefield snapshots, synthetic seismograms, and peak particle velocity maps are presented to illustrate the promising use of SEM for industrial problems.