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.     

Tests of computerised strain analysis methods by the analysis of simulated deformation of natural unstrained sedimentary fabrics

Axial ratio—fluctuation measurements made on markers (oncolites, oolites, quartz grains, and accretionary lapilli) on bedding-perpendicular sections of unstrained sedimentary rocks have been homogeneously strained mathematically by set amounts, using published transformation equations, to provide a simulation of natural tectonic deformation. The resultant data have been used as test input to the strain analysis computer programs of Dunnet and Siddans (1971) and Matthews et al. (1974), which were devised to assess tectonic strain in assemblages of strained elliptical markers, with the use of limited specific assumptions concerning the nature of the pre-tectonic fabrics of the markers. The tests have shown that even slight deviations in the initial fabrics from the assumptions involved in the use of the strain analysis methods may lead to significant errors in the strain determinations. Errors are particularly large when fabrics which were imbricate in the unstrained state are used with the mistaken assumption that they were bedding-symmetric.     

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.     

Comparison of methods of algebraic strain estimation from Rf/ɸ data: A unified theory of 2D strain analysis

A unified development of the subject of the algebraic strain analysis methods using Rf/ɸ data is outlined, embodying the main features the theories of Shimamoto and Ikeda, Mulchrone et al. and Yamaji. It is shown that the theories yields an identical strain ellipse from the same data set. However, error estimation in that of Shimamoto and Ikeda is difficult owing to the distortion of its parameter space: Resolution of their method depends on the choice of a reference orientation in the plane where strain markers are observed. In this respect, the remaining two theories have advantages. The hyperbolic vector mean method was developed in the Minkowski 3-space, thereby linked seamlessly with the visualizing methods of Rf/ɸ data, optimal strain and its confidence region. In addition, the residuals of the optimal strain ellipse determined by this method have clear physical meanings concerning logarithmic strains needed to transform a unit circle to given ellipses.     

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.     

Strain analysis using quartz deformation bands

Quartz deformation bands are kink bands in quartz crystals. A deformation band develops as a region of localized crystal-plastic deformation with boundaries perpendicular to the slip plane and slip direction, which usually is along an -axis in the basal plane. Under cross-polarized light, the difference in crystallographic orientation between a deformation band and its host is indicated by a difference in extinction positions. The displacement between the c axis in a deformation band and the c axis in the host represents the angular shear of the deformation band in the direction of the c axis in the host grain. Assuming the deformation is homogeneous at the grain scale, the angular shear of the grain (the gauge) is calculated by multiplying the angular shear of the deformation band by the ratio of the sheared part to the whole grain. Using the strain-gauge method for three-dimensional infinitesimal strain analysis, a minimum number of five grains measured on universal stage is needed to solve for the deviatoric strain components of the aggregate if the strain is homogeneous in the aggregate. Data from more than five grains are used to find the best-fit strain components by a least-squares method. The principal strains and their orientations are found from these strain components by calculating the eigenvalues and eigenvectors. A 3-D strain ellipsoid also is obtained from strain ellipses in three perpendicular planes determined from the two-dimensional flat-stage measurements by the Wellman method. Both the strain-gauge method and the Wellman method are tested by using synthetic data sets and applied to a naturally deformed sample. Both methods give similar results; the established Wellman method thus confirms the strain-gauge calculation.     

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.     

A comparative error analysis of manual versus automated methods of data acquisition for algebraic strain estimation

This paper presents a statistical analysis of the algebraic strain estimation algorithm of Shimamoto and Ikeda [Shimamoto, T., Ikeda, Y., 1976. A simple algebraic method for strain estimation from deformed eillipsoidal objects: 1. Basic theory. Tectonophysics 36, 315–337]. It is argued that the error in their strain estimation procedure can be quantified using an expected discrepancy measure. An analysis of this measure demonstrates that the error is inversely proportional to the number of clasts used. The paper also examines the role of measurement error, in particular that incurred under (i) a moment based and (ii) manual data acquisition methods. Detailed analysis of these two acquisition methods shows that in both cases, the effect of measurement error on the expected discrepancy is small relative to the effect of the sample size (number of objects). Given their relative speed advantage, this result favours the use of automated measurement methods even if they incur more measurement error on individual objects. Validation of these results is carried out by means of a simulation study, as well as by reference to studies appearing in previous literature. The results are also applied to obtain an upper bound on the error of strain estimation for various studies published in the literature on strain analysis.     

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.     

均匀离散分布的平面应变过程和Fry法应变测量

本文运用 Apple Macitonsh 计算机对无应变的均匀离散点分布进行系列平面应变模拟,应变叠加模式分别采用具普遍意义的成岩压实+顺层缩短(LPS)+压溶作用和成岩压实+简单剪切+压溶作用。对各应变阶段的变形点分布进行相应的 Fry 法应变测量,并配合低变形砂岩样品的 Fry 法应变分析实例.证实 Fry 法应变测量方法为一非常有用的应变测量方法,其结果不仅能较好地揭示全岩总应变特征,而且能揭示出许多应变叠加的信息,Fry 法揭示的全岩有限应变椭球主面的方位也较为真实可靠。而 Fry 法运用于应变分布不均匀的劈理化岩石中时,能揭示不同变形域的应变特征,从而达到应变分解的目的。     

A comparison of methods used in measuring finite strains from ellipsoidal objects

The available methods for measuring strains in two and three dimensions from passive ellipsoidal objects are compared in an attempt to determine the most useful and precise procedure for the structural geologist. A comparison of data collection techniques showed that the use of thin orthogonally cut slabs or mylar overlays used with a monocomparater provides the most reproducible data. Lisle's theta-curve and Shimamoto and Ikeda's algebraic methods provided the most precise, and probably most accurate, two-dimensional data while Miller and Oertel's procedure and possibly Dunnet's PHASE 5 program gave the best three-dimensional results.An examination of errors encountered in strain analyses suggests that all of the available methods give accurate orientations of the finite strain ellipsoid. However, the magnitudes of strain ratios show large variations that are dependent on the sample size and procedure used. Shimamoto and Ikeda's method again proved to be the most precise, giving reproducible results with as few as ten elliptical objects.The samples used in the above comparison are part of a larger analysis of strains occurring in southeastern Maine. Structural elements observed in four selected areas of Avalonian belt rocks and the strain data collected suggest that the region has undergone at least three non-coaxial deformations with D₁ >D₂ >D₃.     

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。     

Removal of finite deformation using strain trajectories

A technique is described for removing the effects of finite deformation, given the principal values and orientations of strain at a number of points throughout a deformed body.Using the principal orientations, strain trajectories are constructed for the deformed state. The body is divided into finite elements bounded by these trajectories. Each element is then unstrained without changing its orientation or position. This process creates artificial voids and overlaps, which are minimized by imparting rigid translations and rotations to the elements according to a least squares method.The result is the pattern of strain trajectories for the undeformed state. It is shown that the trajectories for the deformed and undeformed states may be used as reference coordinates in order to map the change in shape of any body as it passes from the deformed to the undeformed state or vice versa. The technique is tested using models of a folded layer and a shear zone. It is suggested that the technique is versatile enough to allow for errors in original strain data. Although the technique has so far been applied to two-dimensional deformations, a similar method should be usable in three dimensions.     

Analysis of the Variability of a Two-Dimensional Finite Strain Estimate

Finite strain estimation is a widely used technique for the study of rock deformation in structural geology. One particular algorithm proposed by Shimamoto and Ikeda uses the ‘average shape matrix’ of deformed markers. This paper provides a detailed error analysis for resulting strain estimates in two dimensions. When the number of markers exceeds 100, estimators of components of the strain tensor are shown to have an approximately Gaussian distribution with variances that increase with their mean. Equal variance estimators are obtained by applying a log transform for the elongation and an arcsin transformation for the orientation estimates. Confidence interval formulae for strain tensor components are proposed. Lithology specific constants arising in these formulae are estimated from undeformed samples. The results are validated by application to simulated data as well as observational data from thin sections of sandstone sampled from SE Ireland.     

Normalized center-to-center strain analysis of packed aggregates

The resolution of conventional techniques of center-to-center strain analysis is limited by the degree of original anticlustering of centers on the analyzed plane. However, the three-dimensional anticlustering of packed objects does not result in equivalent anticlustering on two-dimensional planes through these aggregates. Size variations due to imperfect sorting further decrease the anticlustering of natural aggregates. For the Fry all-object-object separations method, these problems are manifested in vague point-density distributions and ambiguously defined strain ellipses.Normalization of center-to-center distances allows more precise determination of small initial and tectonic anisotropies in packed aggregates. On planes through packed aggregates, object spacing is a function of object size, shape and the distance between object margins. Dividing the center-to-center distance between two objects by the sum of their average radii eliminates variations due to object size and sorting. Analyses of synthetic aggregates of packed spheres and statically recrystallized iron show that normalized Fry diagrams form better-defined vacancy fields and sharper rims of maximum point density regardless of the original sorting and anticlustering in the aggregate. Normalized strain analyses of deformed aggregates also show greatly increased resolution, with variable initial and tectonic ellipticity resulting in a wider ring of high point-density.     

Pure shear to simple shear-dominated transpression during the Eburnean major D2 deformation,Daléma sedimentary basin,eastern Senegal

Field studies in the Palaeoproterozoïc Daléma basin, Kédougou-Kéniéba Inlier, reveal that the main tectonic feature comprises alternating large shear zones relatively well-separated by weakly deformed surrounding rock domains. Analysis of the various structures in relation to this major D₂ phase of Eburnean deformation indicates partitioning of sinistral transpressive deformation between domains of dominant transcurrent and dominant compressive deformation. Foliation is mostly oblique to subvertical and trending 0–30° N, but locally is subhorizontal in some thrust-motion shear zones. Foliation planes of shear zones contain a superimposed subhorizontal stretching lineation which in places cross-cuts a steeply plunging stretching lineation which is clearly expressed in the metasedimentary rocks of weakly deformed surrounding domains. In the weakly deformed domains, the subhorizontal lineation is absent, whereas the oblique to subvertical lineation is more fully developed. Finite strain analyses of samples from surrounding both weakly deformed and shearing domains, using finite strain ratio and the Fry method, indicate flattened ellipsoid fabrics. However, the orientation of the long axis (X) of the finite strain ellipsoid is horizontal in the shear zones and oblique within the weakly deformed domains. Exceptionally, samples from some thrust zones indicate a finite strain ellipsoid in triaxial constriction fabrics with a subhorizontal long axis (X). In addition, the analysis of the strain orientation starting from semi-ductile and brittle structures indicates that a WNE–ESE (130° N to 110° N) orientation of strain shortening axis occurred during the Eburnean D₂ deformation.     

The sectional strain ellipse during progressive coaxial deformations

The two-dimensional strain history on a sheet which is inclined to the principal axes of the strain ellipsoid is considered. Even if the strain history in three dimensions is coaxial, the two-dimensional progressive deformation on the surface of the sheet is in general of a non-coaxial type. It is shown in this paper that the degree and sense of two-dimensional non-coaxiality is governed by the strain path followed during three-dimensional coaxial deformation. The general relationship is defined between the gradient of the strain path on the Flinn strain ellipsoid diagram and the nature of the two-dimensional strain increments. For most strain paths an asymmetrical arrangement of structures in the oblique sheet is to be expected. Hence, en échelon folds, transected folds and extension veins with curved fibres could be produced during three-dimensional coaxial deformation. Only if the strain path is of a rather special type will the deformation be coaxial in a two-dimensional as well as a three-dimensional sense.     

The Interrelationship of Micro-Meso and Macroscopic Structures on the Western Limb of the Hazara Kashmir Syntaxis, Pakistan

Detailed micro-meso to macroscopic structural analyses reveal two deformation phases in the western limb of the Hazara-Kashmir Syntaxis(HKS). Bulk top to NW shearing transformed initially symmetrical NNE-SSW trending meso to macroscopic folds from asymmetric to overturned ones without changing their trend. Sigmoidal en-echelon tension gashes developed during this deformation,that were oblique to bedding parallel worm burrows and bedding planes themselves. Strain analyses of deformed elliptical ooids using the R_f/φ method constrain the internal strain patterns of the NNE-SSW structures. The principal stretching axis(S_3) defined by deformed elliptical ooids is oriented N27°E at right angles to WNW-ESE shortening. The deformed elliptical ooids in sub-vertical bedding vertical planes contain ooids that plunge ~70° SE due to NW-directed tectonic transport. Finite strain ratios are1.45(R_(xy)) parallel to bedding plane and 1.46(R_(yz)) for the vertical plane. From these 2D strain values, we derive an oblate strain ellipsoidal in 3D using the Flinn and Hsu/Nadai techniques. Strains calculated from deformed elliptical ooids average-18.10% parallel to bedding and-18.47% in the vertical plane.However, a balanced cross-section through the study area indicates a minimum of~-28% shortening.Consequently, regional shortening was only partially accommodated by internal deformation.     

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.