Some applications of the Mohr diagram for three-dimensional strain

The Mohr diagram for strain is rarely used in its full form, as a representation of three-dimensional strain. Recent attention has focused on various uses of the Mohr circle to express two-dimensional strain tensors. This contribution redescribes the Mohr diagram for three-dimensional strain and illustrates some new applications. The Mohr diagram for any strain ellipsoid provides an immediate method for ellipsoid shape classification. However, its greatest new potential is considered to be in the representation of strain ellipses as sections of ellipsoids.Any plane section of a strain ellipsoid can be plotted on the ellipsoid's Mohr diagram: it is here called a ‘Mohr locus’ because it is constructed as a locus of points representing the sheaf of lines which can be considered to define the plane. Mohr loci for sectional ellipses have a variety of forms, according to their orientation in the strain ellipsoid. Generally oblique sections are represented by loops bounded by the three principal circles. Their most leftward and rightward points are the plane's principal axes. Any Mohr locus can be transformed into a Mohr circle for the sectional ellipse.Mohr diagrams with Mohr loci have considerable potential as a graphical method of deriving best-fit strain ellipsoids from natural strain data. This is illustrated in three examples.     

Ellipsoid estimation in coal reflectance anisotropy

The procedure of fitting an ellipsoid to vitrinite reflectance anisotropy is described. Several authors on this subject use incorrect fitting formulae. The correct formulae are given and a least- squares procedure is developed to give confidence regions for the principal reflectances and their orientations. A FORTRAN program is offered that computes the principal reflectances using the correct formulae. A test is constructed for whether the principal reflectance orientations are the same at two locations. Because of the ellipsoid shape, the reflectance anisotropy can be compared to the strain ellipsoid. As an example, the methods are used to compare two coal blocks from an openpit mine in Alberta, Canada. The test shows that the principal reflectances of the two blocks have different orientations, indicating that the orientation of the principal strain axes is different at the two sites.     

Analysis of triaxial ellipsoids: Their shapes, plane sections, and plane projections

A method is presented for the determination of a triaxial ellipsoid (such as a strain ellipsoid)from three nonparallel plane sections of the ellipsoid. The sections need be neither orthogonal nor central sections of the ellipsoid. Measurement errors are used to adjust the observed plane ellipses so that they are exact sections of the nearest true ellipsoid, whose dimensions and orientation are then found by solution of a system of six linear equations. A solution of the inverse problem is also presented: given a triaxial ellipsoid with known orientation, to determine the shape and orientation of the ellipse on a plane section. The problem is solved by expanding the equation of an ellipsoid with rotated coordinates, then setting one dimension to zero. Also, a method is presented for the projection of a triaxial ellipsoid onto a plane surface. This is solved by taking the derivative of the ellipsoid equation in the direction of the normal to the plane surface.     

Rapid extraction of central vacancy by image-analysis of Fry plots

The Fry method, based on the relative movement of different material points, typically grain centers, with reference to each other graphically yields a point distribution that displays the finite strain ellipse as a central vacancy. The diffused nature of the central vacancy induces subjectivity in strain estimation, particularly, if the point population when undeformed lacked an isotropic anticlustered distribution. Most existing methods use analytical and/or iterative approaches for improving the sharpness of the central vacancy and positioning the best-fit strain ellipse in a Fry plot. We provide an image-analysis method that is independent of any iteration or analytical solution. It is also an efficient technique for extraction of the central vacancy without any subjectivity. The method is more direct, simple and easy-to-use than most existing techniques.The image-analysis method uses Gaussian blur filter for distinction between the areas of largest and smallest pixel intensities in a Fry plot image. It then applies the optimal threshold value and an inversion filter for extraction of the sharp central vacancy. The method also searches for the best-fit strain ellipse through the extracted central vacancy and displays axial ratio and orientation of the ellipse in a separate window. The validity of the method is tested using several computer-simulated and natural examples.     

Determination of fabric and strain ellipsoids from measured sectional ellipses—implementation and applications

Geologists examine fabrics to constrain models of formation or of deformation of rocks, and it is often convenient to summarise the results by a fabric ellipsoid. As fabric data are commonly collected on planar sections through the rock, estimating a fabric ellipsoid from sectional ellipses, often with arbitrary orientations, is an important task. An algebraic method to calculate such an ellipsoid, presented in an earlier paper, has been implemented with the program ellipsoid. It is used here on examples that illustrate questions and issues that arise when collecting, selecting and processing sectional fabric data, and when assessing the results. The quality of fit of the ellipsoid to the data is assessed in all cases. Examples include a case in which the average sizes of markers on individual sections can be used in the determination of the ellipsoid, and other cases in which such sizes are not useful; a case in which sectional ellipses are not obtained from closed markers; and a case in which data scatter and insufficient coverage of section orientations lead to a hyperboloid instead of an ellipsoid.     

Slope reliability analysis accounting for spatial variation

A practical and transparent procedure is described for implementing a generalized limit equilibrium method via cell-object-oriented constrained optimization in the spreadsheet platform. The formulation allows switching among the Spencer, Bishop simplified and wedge methods on the same template by specifying different side-force inclination and different constraints of optimization. Search for the critical circular or non-circular slip surface is possible. The deterministic procedure is extended probabilistically by implementing the first-order reliability method via constrained optimization of the equivalent dispersion ellipsoid in the original space of the random variables. This procedure is illustrated for an embankment on soft ground, and for a clay slope in southern Norway, both involving spatially correlated soil properties. The effects of autocorrelation distance on the results of reliability analysis are studied. Shear strength anisotropy is modelled via user-created simple function codes in the programming environment of the spreadsheet. The meaning of probability of failure is discussed.     

非饱和土热湿耦合传输问题的数值解——理论及一维问题的解

本文介绍了求解非饱和土壤中热量和水分耦合传输问题的一种数值方法——积分有限差分方法(IFDM)。基于菲利普-迪弗瑞斯(Philip-De Vries)多孔介质热湿耦合流动模型,采用积分有限差分方法,编制了求解热湿传输问题的计算程序 HM1,该程序可用于求解各类工程中遇到的多孔介质一维传热传湿问题。文中还给出了计算结果与实测结果相比较的实例。     

A study on selection of probability distributions for at-site flood frequency analysis in Australia

The most direct method of design flood estimation is at-site flood frequency analysis, which relies on a relatively long period of recorded streamflow data at a given site. Selection of an appropriate probability distribution and associated parameter estimation procedure is of prime importance in at-site flood frequency analysis. The choice of the probability distribution for a given application is generally made arbitrarily as there is no sound physical basis to justify the selection. In this study, an attempt is made to investigate the suitability of as many as fifteen different probability distributions and three parameter estimation methods based on a large Australian annual maximum flood data set. A total of four goodness-of-fit tests are adopted, i.e., the Akaike information criterion, the Bayesian information criterion, Anderson–Darling test, and Kolmogorov–Smirnov test, to identify the best-fit probability distributions. Furthermore, the L-moments ratio diagram is used to make a visual assessment of the alternative distributions. It has been found that a single distribution cannot be specified as the best-fit distribution for all the Australian states as it was recommended in the Australian rainfall and runoff 1987. The log-Pearson 3, generalized extreme value, and generalized Pareto distributions have been identified as the top three best-fit distributions. It is thus recommended that these three distributions should be compared as a minimum in practical applications when making the final selection of the best-fit probability distribution in a given application in Australia.     

Non-steady homogeneous deformations: Computational techniques using Lie theory,and application to ellipsoidal markers in naturally deformed rocks

The dynamic theory of deformable ellipsoidal inclusions in slow viscous flows was worked out by J.D. Eshelby in the 1950s, and further developed and applied by various authors. We describe three approaches to computing Eshelby's ellipsoid dynamics and other homogeneous deformations. The most sophisticated of our methods uses differential-geometric techniques on Lie groups. This Lie group method is faster and more precise than earlier methods, and perfectly preserves certain geometric properties of the ellipsoids, including volume. We apply our method to the analysis of naturally deformed clasts from the Gem Lake shear zone in the Sierra Nevada mountains of California, USA. This application demonstrates how, given three-dimensional strain data, we can solve simultaneously for best-fit bulk kinematics of the shear zone, as well as relative viscosities of clasts and matrix rocks.     

Selection of threshold values in geochemical data using probability graphs

A method of choosing threshold values between anomalous and background geochemical data, based on partitioning a cumulative probability plot of the data is described. The procedure is somewhat arbitrary but provides a fundamental grouping of data values. Several practical examples of real data sets that range in complexity from a single population to four populations are discussed in detail to illustrate the procedure.The method is not restricted to the choice of thresholds between anomalous and background populations but is much more general in nature. It can be applied to any polymodal distribution containing adequate values and populations with appropriate density distribution. As a rule such distributions for geochemical data closely approach a lognormal model. Two examples of the more general application of the method are described.     

滑面正应力近似方法对三维极限平衡法解的影响

采用楔形体与椭球滑面算例,研究了三维极限平衡法中滑面正应力近似方法对其计算结果的影响。使用线性插值近似滑面修正应力时,插值节点和网格数目的改变未对安全系数造成明显影响,但明显改变了修正应力的计算结果,网格交界处的修正应力易出现突变。采用移动最小二乘近似滑面修正应力后,安全系数基本不变,并且相对于线性插值,能在求解更少未知量的情况下获得更有规律的修正应力分布。算例表明,在采用线性插值和移动最小二乘近似计算滑面修正应力的情况下,安全系数基本不随滑面正应力的改变而变化,这与二维极限平衡法的相关研究结论基本相同,显示三维极限平衡法满足了平衡条件和合理性条件。     

椭球体的视电阻率特征

椭球体是最具代表性的典型三维地电体之一,它三个轴的长短经过变化后可以和多种形状的地电体相似。因此,实现了椭球体电阻率的求解,就可以模拟绝大多数三维地电体的电阻率曲线。从拉普拉斯方程的求解出发,得到了均匀空间中存在的水平椭球体异常表达式,并用简单加倍的方法得到了均匀半空间中水平椭球体的视电阻率公式,结合倾斜旋转椭球体的视电阻率公式的规律,得到了倾斜椭球体的视电阻率公式,再进行一系列简化得到板状体。最后分析对比了均匀电场中旋转椭球体和板状体的视电阻率曲线的形态和规律。结论表明:旋转椭球体与板状体的视电阻率曲线的走向大体相似,且板状体的曲线相比旋转椭球体较缓和。对于低阻情况,在异常体顶部附近的视电阻率异常为零,异常体的倾向一侧有极小值,反倾向一侧有极大值,曲线随倾角α的增大而逐渐变陡。对于高阻情况,视电阻率的极大值出现在异常体顶部附近,两侧有极小值,倾向一侧的曲线较缓和而反倾向一侧的曲线较陡峭,随着倾角α的增大,视电阻率异常变大。     

Determination of strain ellipsoid according to two-dimensional data on three or more intersection planes

The eigenvalue method (Gendzwill and Stauffer, 1981) is adopted to determine three-dimensional deformation of geological bodies from two-dimensional data on three or more nonparallel sections. A least-squares method is employed to fit an overdetermined system for uniquely determining an ellipsoid. The inverse problem is treated more directly. Calculated examples show that the errors are caused mainly from inaccuracy of the procedures used to obtain two-dimensional data.     

The New “York” Regression: Application of an Improved Statistical Method to Geochemistry

York's (1969) method of regression, determining the best-fit line to data with errors in both variables using a least-squares solution, has become an integral part of isotope geochemistry. Although other methods agree with York's best-fit line (e.g., maximum likelihood), there is little agreement on the standard-error estimates for slope and intercept values. The reasons for this are differing levels of approximation used to compute the standard error, doubts concerning procedures for determining a confidence interval once the standard error has been estimated, and a typographical error in the original publication. This paper examines York's method of regression and standard errors of the parameters of a best-fit line. A very accurate method for determining the standard error in slope and intercept values is introduced, which eliminates the need to multiply the standard-error estimate by the goodness-of-fit parameter known as MSWD. In addition, a derivation of a fixed-intercept method of regression is introduced, and interpretations of MSWD and use of the t-adjustment in confidence intervals are discussed. The accuracy of the standard-error computations is determined by comparing the results to slope and intercept statistics generated from several thousand Monte Carlo regressions using synthetic ⁴⁰Ar/³⁹Ar inverse isochron data.     

Steady-state solutions for the motion of a globular cluster in an inhomogeneous, rotating elliptical galaxy

Steady-state solutions for the motion of a passively gravitating globular cluster (GC) inside an inhomogeneous, rotating, ellipsoidal elliptical galaxy (EG) are considered. It is assumed that an EG with a halo is comprised of a triaxial ellipsoid consisting of two layers. The first is formed by an inner, uniform ellipsoid representing the luminous part of the galaxy, while the second corresponds to the space between an inner and outer ellipsoid, which is uniformly filled with dark matter. The triaxial ellipsoids are taken to be homothetic and to have a common center; the space between them is called a homeoid. The outer boundary of the homeoid is the boundary of the galaxy halo. The densities of the luminous part of the EG and the homeoid are different. This picture of an EG is in agreement with our current understanding of galactic structure. The motion of the GC occurs outside the luminous part of the EG, but inside the homeoid, which is treated like a perturbing body. Steady-state solutions (libration points) are found for the GC, and its Lyapunov stability determined. The elliptical galaxies NGC 4472 (M49), NGC 4636, and NGC 4374, which contain a large number of GCs, are used as examples. Analysis of these galaxies shows that the exact expression for the potential of the luminous part of the EG must be used to find the libration points and study their stability, rather than an approximate expression for this potential.     

Measuring fracture orientation at exposed rock faces by using a non-reflector total station

Measurements of fracture orientation are usually taken by using a compass-inclinometer device on exposed rock faces. The drawbacks when using this method is that it is time-consuming if many fractures are measured and that measurement might be impossible if the rock face cannot be safely reached physically. To improve field mapping of rock fractures, a method for applying a non-reflector total station to measuring fracture orientation is presented in this paper. A non-reflector total station is a geodetic device that captures three-dimensional co-ordinates of target points without using a reflector. Therefore, physical touching the rock surfaces is no longer required. To determine a fracture orientation, co-ordinates of a set of points on the exposed fracture surface are captured at a distance from the rock face. The best-fit plane of the exposed fracture surface is defined by the co-ordinates of the target points, and the orientation (e.g. dip angle and dip direction) of the fracture surface is determined as that of its best-fit plane. This paper presents the technical procedure and a portable system designed for the field mapping of fracture orientation. Results of a case study performed at an exposed rock face are also included.     

Statistical regression applied to borehole strain measurements data analysis

Summary In mining and geotechnical engineering, it is usually necessary to carry out field measurements in order to obtain information. Parameters are often measured indirectly and calculated based on certain relationships to the measured quantities. More often, the number of measurements taken is greater than the minimum required, in order to increase the reliability of results. However, some data points are less reliable than others for reasons such as measurement errors; a solution which best fits the measurement data is obtained accordingly. As a result, there is a residual or a difference between the individual quantities measured and those predicted from the best-fit solution. This brings about a question of how big a residual is acceptable for a solution to be reliable. It is also important to know whether the data point with the largest residual is the most erroneous, whether those data points with large residuals should be deleted and how many of them should be deleted. Standard deviation may provide a measure of the data divergence but it is questionable if this parameter can be used as a measure of the reliability of solution. In order to solve these problems, the author has done extensive study in this area, especially as part of geotechnical data analysis. In this paper, the statistical multiple regression method is introduced to analyse the measurement data. The method is applied to the analysis ofin situ stress measurement and can be easily adopted to analyse data from other field measurements and laboratory tests. An example is included which illustrates the analysis procedure and shows the advantages of the method.     

An internally consistent integration method for critical state models

A new procedure to integrate critical state models including Cam–Clay and modified Cam–Clay is proposed here. The proposed procedure makes use of the linearity of the virgin isotropic compression curve and the parallel anisotropic consolidation lines in e–ln p space which are basic features of the formulation of critical state models. Using this algorithm, a unique final stress state may be found as a function of a single unknown for elastoplastic loading. The key equations are given in this article for the Cam–Clay and modified Cam–Clay models. The use of the Newton–Raphson iterative method to minimize residuals and obtain a converged solution is described here. This new algorithm may be applied using the assumptions of linear elasticity or non‐linear elasticity within a given loading step. The new algorithm proposed here is internally consistent and has computational advantages over the current numerical integration procedures. Numerical examples are presented to show the performance of the algorithm as compared to other integration algorithms. Published in 2005 by John Wiley & Sons, Ltd.     

卧式椭球盐岩储气库理论解及设计参数分析

我国的层状盐岩储层具有岩层厚度薄、夹层含量多、杂质含量高的特点,盐层厚度较薄时立式盐岩储气库有时难以获得较为经济的溶腔体积,通过提高储气库水平方向尺寸而建造的卧式储气库则可以达到经济体积的要求。从力学稳定性角度探讨卧式储气库的可行性,提出了卧式椭球储气库的力学简化模型。在前期研究的基础上使用弹性理论方法,推导了卧式椭球上关键点处以椭球形状比s（水平半轴与竖直半轴之比）、材料泊松比 、腔体内压 和远场应力（ 、 、 ）为函数的应力解。对理论解和数值解进行了对比,结果表明二者误差平均在3%左右。使用推导得到的理论公式,结合DP屈服准则,计算了当椭球形状比一定时储气库的最小运营内压值,得到了椭球形状比与储气库最小运营内压的关系曲线,并对卧式储气库运营内压的稳压范围进行了简单讨论。