Average properties of ellipsoidal fabrics: Implications for two- and three-dimensional methods of strain analysis

Many rocks contain ellipsoidal objects (such as pebbles or reduction zones) which display a variety of shapes and orientations. In deformed rocks such objects may be used for strain analysis by using the concept of an average ellipsoid (here called the “fabric ellipsoid”). Two fabric ellipsoids are defined which are the results of two different algebraic averaging processes. During deformation of ellipsoidal distributions, the fabric ellipsoids change as if they were themselves material ellipsoids and are therefore of fundamental importance in strain analysis.In most studies to date, such 3-D fabric ellipsoids have been obtained from 2-D average ellipses determined on section planes cut through the rock sample. Previous work has assumed that the average ellipses will approximate to section through a single fabric ellipsoid. I show here that this is not the case as sectioning introduces a systematic bias into the section ellipse data. This bias is distinct from the statistical errors (due to finite sample size and measurement errors) discussed in other work and must be considered in any method of strain analysis using section planes.     

Determination of the strain ellipsoid from measurements on any three sections

An ellipsoid is defined by, and may be re-constructed from, any three sections through it. In the field, calculation of the strain ellipsoid from general sections (two-dimensional strain ellipses determined from measured strain markers) is complicated by the fact that, due to experimental error and/or strain inhomogeneity, the three ellipses may not come from the same ellipsoid. The ellipses must first be adjusted to make them compatible. A method is suggested by which an adjustment ellipse is determined analytically for each of the three sections. Application of these adjustment ellipses makes the three sections compatible, and the strain ellipsoid may be determined. The principal axes of the ellipsoid are derived from the ellipsoid matrix by eigenvector analysis. Examples are given of practical applications of this method.     

Analysis of three-dimensional, homogeneous, finite strain using ellipsoidal objects

It has been suggested (Oertel, 1971, 1972;Owens, 1974; Shimamoto and Ikeda, 1976) that some methods for analysis of finite homogeneous strain from deformed ellipsoidal objects (Ramsay, 1967; Dunnet, 1969a; Elliott, 1970; Dunnet and Siddans, 1971; Matthews et al., 1974) require sections to be cut in principal planes of the finite strain ellipsoid. A mathematical model is presented which enables the homogeneous deformation of a randomly oriented ellipsoid to be investigated. In particular the elliptical shapes that result on any three mutually perpendicular sections through the ellipsoid, in the deformed state, can be computed, together with the corresponding strain ellipses. The resulting ellipses can be unstrained in the section planes by applying the corresponding reciprocal strain ellipses. It is shown that these restored ellipses are identical with the elliptical shapes that result on planes through the original ellipsoid when the planes are parallel to the unstrained orientation of the section planes.The model is extended to investigate the finite homogeneous deformation of a suite of 100 randomly oriented ellipsoids of constant initial axial ratio. The pattern of elliptical shapes that result on any three mutually perpendicular section planes, in the deformed state, is computed. From this data the two-dimensional strain states in the section planes are estimated by a variety of methods. These are combined to recalculate the three-dimensional finite strain that was imposed on the system. It is thus possible to compare the results of the two- and three-dimensional analyses obtained by the various methods. It is found that providing all six independent combinations of the two-dimensional strain data are used to compute a best finite strain ellipsoid, the methods of Dunnet (1969a), Matthews et al. (1974) and Shimamoto and Ikeda (1976) provide accurate estimates of the three-dimensional finite strain state.It is concluded that measurement of the two-dimensional data on section planes parallel to the principal planes of the finite strain ellipsoid is not necessary and that all six independent combinations of the two-dimensional strain data should always be made and used to compute a best finite strain ellipsoid.     

利用惯量椭圆进行岩石有限应变分析

提出了利用惯量椭圆对任意形状矿物颗粒进行描述的方法,并利用惯量椭圆理论,计算了岩石薄片中任意形状矿物颗粒的惯量椭圆,通过颗粒面积对椭圆参数进行标准化,得到每一矿物颗粒的等效应变椭圆。等效应变椭圆能够有效地反映对应矿物颗粒的优选方位以及变形特征,进而利用椭圆的矩阵参数形式对等效应变椭圆进行统计分析,获得岩石的有限应变椭圆;同时给出了相应的数值计算方法,编制了软件Straindesk,并得到了成功的应用。该方法克服了先前应变测量中的局限性,方便实现计算机的自动分析,具有较强的有效性和广泛的适用性。     

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.     

Texture and fabric studies across the Kishorn Nappe, near Kyle of Lochalsh, Western Scotland

The progressive development of mylonitic fabrics in a series of Torridonian sandstones and shales has been studied along traverses across the Kishorn Nappe. The fabrics developed have been investigated using the following techniques.

1. 1. Optical examination of thin sections.

2. 2. Measurements of the anisotropy of magnetic susceptibility.

3. 3. X-ray texture goniometry.

The results are used in support of a proposed deformation history of the area and the relative advantages of the techniques used are discussed.The early deformation was well lubricated with layer-parallel sliding and little internal deformation of the rocks, except for development, in the east, of a layer-parallel penetrative fabric with an extension direction to the ESE. This deformation produced a westward facing isoclinal anticline and a recumbent syncline in the Torridonian rocks which became at least partly decoupled from the basement.The important phases of fabric development post date this folding. In the west the sandstones developed a spaced, pressure solution cleavage, but in the east the grain shape fabric has been produced by both dislocation and diffusion processes. The shales reveal more details of the deformation episodes than do the sandstones and thus show different fabric intensities and orientations when measured by magnetic and X-ray techniques.The magnetic anisotropy technique of fabric analysis gives a rapid method of mapping the deformation domains formed by different deformation mechanisms and intensities. However, the rocks carry several magnetic components and these have different anisotropy tensors and different responses to deformation, also, measurements made at high fields (5 kOe) give magnitudes and orientations of the magnetic anisotropy tensor which are different from those made at low fields. It is concluded that it is not possible to relate variations in the magnitude and shape of the magnetic anisotropy ellipsoid quantitatively to the deformation.Chlorite and muscovite fabrics measured by X-ray techniques show variations in intensity and orientation similar to those of the magnetic anisotropy ellipsoid due to paramagnetic minerals. However, the data demonstrate the difficulty of correlating this fabric intensity with deformation intensity where there has been a change in deformation mechanisms with time and space.     

Density distribution techniques and strained length methods for determination of finite strains

For a homogeneously deformed rock composed initially of an isotropic distribution of object shapes, finite strain may be determined from the correlation between the orientations of either two-dimensional or one-dimensional sample cuts and the frequencies with which they intersect marker objects. Mimran previously published an incorrect method for planar samples under the heading ‘density distribution technique’. Methods are described by which the three-dimensional strain may be directly determined from six general samples, either linear or planar. Construction of two-dimensional ellipses as an intermediate step is unnecessary and enforces practical difficulties.These methods may be simplified by use of samples parallel to known principal axes or planes of the finite strain. In this case the same large errors may arise from slight misorientation of samples as with other methods of strain measurement. A new quick method is proposed, combining linear and planar measurements of frequencies of intersected objects, which is thought to be the first method to circumvent a large part of the error from this error source. For example, if true X:Z ratio is 9 : 1, and orientations in the XZ plane are misjudged by 8°, normal methods give 38% error where the new method gives, with care, an error of 1.9%. For methods of strain measurement such as are described here the concept of strain ellipsoid is unnecessarily limiting, and should be abandoned.     

Quantifying Three-Dimensional Silicate Fabrics in Cumulates Using Cumulative Distribution Functions

We present a new method for quantifying three-dimensional silicatefabrics and the associated uncertainties from grain orientationdata on three orthogonal sections. Our technique is appliedto the orientation of crystallographic features and, hence,yields a fabric related to the lattice-preferred orientation,although the method could be applied to shape-preferred orientationsor strain analysis based on passive linear markers. The orientationdata for each section are represented by their cumulative distributionfunction, and an iterative procedure is used to find the symmetricsecond-rank strain tensor that will simultaneously satisfy thecumulative distribution functions observed on each section.For samples with well-developed fabrics, this technique providesa much closer match to the sectional data than do previous techniquesbased on eigenparameter analysis of two-dimensional orientationdata. Robust uncertainty estimates are derived from a non-parametricbootstrap resampling scheme. The method is applied to two cumulates:one with a well-developed fabric and the other with a weak fabric,from the Stillwater complex, Montana. The silicate petrofabricorientations obtained for these samples compare favorably withindependent direct estimates of the volume fabric from electronbackscatter diffraction and magnetic techniques. KEY WORDS: cumulates; fabrics; quantitative; Stillwater complex; textural analysis     

Role of particle size in till-fabric characteristics: systematic variation in till fabric from Vestari-Hagafellsjökull, Iceland

Till-fabric analysis has often been used to interpret glacier flow directions and subglacial dynamics using vector-based statistics, but recent data suggest that such analysis may also effectively indicate former glacier dynamics. The results of a fabric investigation of subglacially strained till deposited during a surge of Vestari-Hagafellsjökull, Iceland, are presented. Till fabric was collected at four sites within a limited area where ice-flow direction during deposition was known from subglacial bedforms at the site. Analysis was carried out on elongate grains (axial ratio <1.5:1) at seven size fractions, with the a-axis length ranging from 0.25 to 32 mm. The largest grains tend to be parallel to ice flow, whereas smaller grains reflect a mix of parallel and transverse orientations. The implications of these data for the role and validity of till-fabric analysis are discussed, with reference to vector analyses and the compilation of fabric shape envelopes. It is noted that, in contrast to laboratory experimental data, neither March nor Jeffery mechanisms explain the fabric configurations reported. It is concluded that a standardized approach is necessary for collecting fabric data, and in many situations analysis of data populations at a range of particle sizes is desirable.     

Quantification of microcrack anisotropy in quartzite — a comparison between experimentally undeformed and deformed samples

In this paper, microcrack patterns in a quartzite are quantified using fractal geometry based methods. Since the quartzite does not show a mesoscopic foliation, the fabric was recognized using anisotropy of magnetic susceptibility (AMS) analysis. Microcracks were investigated in thin sections prepared along the three principal planes of the AMS ellipsoid. Point load tests were performed on cores drilled parallel as well as perpendicular to the magnetic foliation. After experimental deformation, thin sections were prepared in two orientations — (a) parallel to the plane of failure (i.e., parallel to the direction of loading), (b) perpendicular to the plane of failure (i.e., perpendicular to the direction of loading), and microcrack patterns in these sections were investigated. The box-counting method of fractal analysis was first applied to microcracks traced from SEM images from each thin section of the experimentally undeformed as well as deformed samples to establish the fractal nature of the microcrack pattern. It was found that in thin sections perpendicular to the direction of loading, the box (fractal) dimension tends to marginally increase. This is inferred as a manifestation of the increase in complexity of the pattern. The software AMOCADO, which is based on the modified Cantor Dust method of fractal analysis, was applied to microcrack pattern from each thin section in order to quantify the pattern anisotropy. It is noted that the anisotropy significantly reduces in sections perpendicular to the loading direction. SEM data are presented to demonstrate that this reduction in anisotropy is on account of generation and/or growth of new cracks in random orientations. It is envisaged that the approach adopted in this investigation maybe useful in rock mechanics and mineral-resource applications in future.     

Strain analysis using deformed pebbles: The influence of initial pebble shape

Several methods exist for the determination of the finite strain ellipsoid from deformed pebble shapes. These methods are critically evaluated and others are proposed on the basis of calculations which predict both the sectional and three-dimensional shape of pebbles in simple deformed simulated conglomerates. In many cases it is found preferable to use an average pebble shape to estimate the tectonic strain and that the harmonic mean of the ratios of axial lengths yields an average pebble shape which is closest to the strain ellipsoid shape.     

磁化率各向异性与剪切带

岩石组构记录了地壳形成与演化的关键信息,提取这些信息对分析和恢复地球动力学过程具有重要意义.磁化率各向异性(AMS)是一种重要的岩石组构方法,可以有效地揭示岩石的应变特征,分析其地球动力学过程,是研究构造变形性质以及应力作用方式的有效手段.本文在梳理AMS的研究历史、主要成果和最新进展的基础上,系统阐述了AMS的基本原...     

Experimental shear zones and magnetic fabrics

Magnetic fabric analysis has been used as a non-destructive means of detecting petrofabric development during experimentally produced multi-stage, transpressive deformations in ‘shear zones’. Artificial, magnetic-bearing silicate sands and calcite sands, bonded with Portland cement, were deformed at room temperature and at 100 and 150 MPa confining pressure. The slip-rate for the shear zone walls was 0.73 × 10⁻⁴ mm s⁻¹ and the maximum shear strains were about 0.38, across zones that were initially about 5 mm thick. The magnetic fabric ellipsoid rapidly spins so that the maximum and intermediate susceptibilities tend to become parallel to the shear zone walls throughout the sheared zone. The ellipsoid becomes increasingly oblate with progressive deformation. However, in all cases, the anisotropy is strongly influenced by the pre-deformation magnetic fabric. During deformation the cement gel collapses so that cataclasis of the mineral grains is suppressed. In the quartz-feldspar aggregates the magnetite's alignment is accommodated by particulate flow (intergranular displacements) of the grains. In the calcite aggregates stronger magnetic fabrics develop due to plastic deformation of calcite grains as well as particulate flow. However, the calcite grain fabrics are somewhat linear (L ≥ S) whereas the magnetic fabrics are planar (S >L). The preferred dimensional orientations of magnetite are weak and it is possible that the magnetic fabrics are due to intragranular rearrangements of magnetic domains.The transpressive shear zones are much more efficient than axial-symmetric shortening in the increase of anisotropy of the magnetic fabrics, especially in the case of the calcite aggregates. This suggests that flow laws derived for axial-symmetric shortening experiments may not be appropriate for non-coaxial strain histories such as those of shear zones.     

Strain determination from the measurement of pebble shapes: A modification

A previously described method of combining measurements of numerous pebble cross-sections in three orthogonal planes into a single average ellipsoid was found to be in need of improvement. The method of reconciling redundancies and of distributing the errors can be left unchanged. The original measurements of principal half-axes, however, rather than being normalized as the ratios of larger half-axes divided by the smaller, should be normalized so that they produce ellipses of standard area. The coefficients of ellipses so normalized will produce distributions with a pronounced central tendency even in those cases that produce markedly skewed distributions of coefficients for ellipses normalized by the former method. Equations of the previous paper that must be amended are given in their new form.     

Comparative study of striations and basal till clast fabrics, Malpeque—Bedeque region, Prince Edward Island, Canada

Coastal erosion has resulted in exposure of a succession of basal till deposits and striation localities along the Malpeque and Bedeque Bays, PEI, Canada, permitting repetitive determinations of clast fabrics and investigation of recently exhumed striations. Striations indicate initial eastward ice flow, followed by a second glacial event with flow towards the south and southwest from the Gulf of St. Lawrence. The degree of local inconsistency and the variations with striation assemblages indicate that the orientations of individual striations cannot be regarded as 'absolute' indicators of ice flow directions. Diamictons identified as basal tills using sedimentological criteria have clast fabrics typical of basal till deposition. Local deviations of fabric trend within individual outcrops are common. A comparative analysis of the data obtained from striation and clast fabric orientations indicates that the ice flow directions independently suggested by each are in general agreement. Differences are present between the orientations of striations and clast fabrics at individual sites. The orientations of the fabrics lie within 20° of the modal striation orientations in only 28% of the cases, and the median deviation between the suggested ice flow directions was 29°. In 40%, of the cases, the azimuth deviation between the two methods exceeded 40°, and the maximum deviation recorded was 84°. At many sites, the deviations are so severe as to potentially result in the misinterpretation of the modal ice flow direction, based on the use of either striation orientation or clast fabric in isolation. Variations in glacial flow can readily produce striations and clast fabrics which record different phases of glacial activity associated with the same glacial event at an individual site.     

Study of ductile shear zones (Galicia, Spain) using texture goniometry and magnetic fabric methods

A joint magnetic and mica fabric study has been carried out on rocks from recognised shear zones in Galicia (Spain). The two main aims of the study are to determine strain gradients in a foliated material without any significant strain markers and to obtain a quantitative relationship between the magnetic susceptibility anisotropy ellipsoid and the mica fabric axes. The intensity parameters of the fabric on profiles intersecting the studied areas indicate the dominant influence of two major shear zones on the fabric development. The regional fabric shows less significant variations across minor (less developed) shear zones.The empirical relationship between magnetic fabric and strain reported by Rathore (1979, 1980) and Rathore and Henry (1982) is found to hold in the magnetic and mica fabrics comparison. The relationship found in this study is:     

Two-dimensional analysis of shape-fabric using projections of digitized lines in a plane

A method of two-dimensional shape-fabric analysis is presented which is based on the projection of lines or ellipses on the x-axis. If a set of lines or ellipses displays a preferred orientation, the average length of projection of the lines or ellipses on the x-axis changes with direction in the x-y plane. The proposed method involves use of Fortran programs which evaluate the average length of projection of lines (e.g., outlines of ooides, pressure solution seams) while the fabric is rotated by small increments in a counterclockwise sense about the origin of the x - y plane. Data acquisition is by digitizing the lines on an enlarged photograph. The sets of x-y coordinates that are thus obtained serve as input for the programs. The basic equations of the method are explained and application to a sedimentary fabric is demonstrated.The proposed method is both fast and sensitive. It is especially useful if the particle outlines are closely spherical or if the fabric is only weakly developed. Yet, the method is not restricted to weak fabrics only. Also, if the fabric is due to deformation this method yields the axes of the two-dimensional strain ellipse.     

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.     

总结大测程重力仪的分段格值整理重力资料的几种情况

对于大测程重力仪,当分段标定的格值相对变化大于1/1400时,应使用分段格值对重力资料进行整理.文中总结了如何使用重力仪的分段格值计算在不同测程段上观测点重力值可能出现的几种情况.     

Adaption of polygonal strain markers

Two methods are presented whereby finite-strain data may be determined from naturally occurring irregular strain markers (polygons) which are of unknown pre-deformation shape and distribution, without assumptions as to the orientation of the finite-strain ellipse. The first method describes “construction” of ellipses within the polygons, these ellipses providing the basis for analysis by already developed techniques. The second method is a simple extension of Wellman's method, which graphically establishes a strain ellipse from angle and line data.