Strain determination from the measurement of pebble shapes

Cross-sections of pebbles or other near-ellipsoidal bodies in rocks can be measured in three planes, conveniently chosen orthogonal to each other, but otherwise oriented arbitrarily. The three sets of measurements, referred to a single Cartesian coordinate system, can in principle be used to calculate the average pebble shape. Some of these measurements, however, are necessarily redundant and, generally, not compatible without adjustments for the distribution of an error. The calculation of the average ellipsoid is developed, as is that of the strain which transforms an original sphere into such an ellipsoid. This calculation involves the distribution by a weighted least-squares method of the error due to redundancy and leads to the assignment of an estimated error to each of the second-rank tensor components of the strain. The assumption is made that, after being referred to the common coordinate system, the measured values for each tensor component have a normal distribution. This assumption may not always be correct but could be tested before this method is used.     

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.     

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.     

Cataclastic deformation of geological materials in matrices of differing composition: II. Boudinage

Spheres and clylinders of various rock types were embedded in a matrix of crushed rock and the combined samples were then deformed by applying uniaxial compressive loads of up to 4.5 MN. Under these conditions, large confining pressures are built up in the centre of the samples; thus the granular matrix is compressed into relatively hard rock and the objects experience flattening and stretching strains of up to 30%. The rock types used for the objects ranged from weak sandstone and shale to very strong quartzite; matrix materials were crushed adamellite, marble and a marble—salt mixture. The experiments were designed to investigate the relative deformation of the objects and the composite samples and, in particular, the effect of the ductility contrasts (or complete differences) between the objects and the matrix.Spheres and cylinders with a length/diameter ratio of unity are called “pebbles” in the paper. They experience a homogeneous flattening during compression and the ductility contrast controls the load at which yielding occurs and the relative rates at which a pebble and the surrounding matrix deform. The changes in shape of pebble and sample are related linearly and the slope of the straight line graph gives a quantitative estimate of the ductility contrast ; in this way a table of ductility contrasts for the various rock types has been constructed and the relative responses of different pebbles in a granular matrix are nicely illustrated in an artificial deformed conglomerate.Cataclasis is the dominant mode of deformation and much of the large finite strain induced into the objects occurs at surprisingly low applied loads. This suggests that deformed pebbles in natural rock need not necessarily have deformed as a result of tectonic pressures but could have changed shape during diagenesis of the host rocks.     

Cataclastic deformation of geological materials in matrices of differing composition: I. Pebbles and conglomerates

Spheres and clylinders of various rock types were embedded in a matrix of crushed rock and the combined samples were then deformed by applying uniaxial compressive loads of up to 4.5 MN. Under these conditions, large confining pressures are built up in the centre of the samples; thus the granular matrix is compressed into relatively hard rock and the objects experience flattening and stretching strains of up to 30%. The rock types used for the objects ranged from weak sandstone and shale to very strong quartzite; matrix materials were crushed adamellite, marble and a marble—salt mixture. The experiments were designed to investigate the relative deformation of the objects and the composite samples and, in particular, the effect of the ductility contrasts (or complete differences) between the objects and the matrix.Spheres and cylinders with a length/diameter ratio of unity are called “pebbles” in the paper. They experience a homogeneous flattening during compression and the ductility contrast controls the load at which yielding occurs and the relative rates at which a pebble and the surrounding matrix deform. The changes in shape of pebble and sample are related linearly and the slope of the straight line graph gives a quantitative estimate of the ductility contrast ; in this way a table of ductility contrasts for the various rock types has been constructed and the relative responses of different pebbles in a granular matrix are nicely illustrated in an artificial deformed conglomerate.Cataclasis is the dominant mode of deformation and much of the large finite strain induced into the objects occurs at surprisingly low applied loads. This suggests that deformed pebbles in natural rock need not necessarily have deformed as a result of tectonic pressures but could have changed shape during diagenesis of the host rocks.     

在变形岩石中确定应变椭球体方法的探讨

本文研究了四十年代以来确定变形岩石应变椭球体的方法,并结合实例给出一种简明、精度较高的从三个或更多任意截面上的二维应变测量数据,计算三维应变椭球体之主轴大小和方向的算法。     

Tectonic strain of a deformed conglomerate determined from a single pebble

Individual rounded pebbles of schist or foliated gneiss included in a conglomerate can each be used as strain markers when the conglomerate has been deformed subsequently. The shape, orientation and the attitude of the earlier schistosity within a single pebble allow one to determine the strain ratio assuming passive behaviour during deformation. The method may also be applicable to certain individual lava pillows containing paleo-horizontal “lava-level” markers.     

Does AMS data from micaceous quartzite provide information about shape of the strain ellipsoid?

Anisotropy of magnetic susceptibility (AMS) in micaceous quartzites with mean susceptibility (K _m) >50 × 10⁻⁶ SI units is known to be on account of the orientation distribution of the para/ferromagnetic minerals (e.g. micas, magnetite), which comprise the minor phase in the rocks. However, the strain in such deformed micaceous quartzites is dominantly accommodated by the quartz grains, which are the major phase in them. The objective of this paper is to explore the extent to which AMS data from micaceous quartzites provide information about the shape of the strain ellipsoid. AMS analysis of 3 quartzite blocks is performed, and the shape of the AMS ellipsoid is recorded to be oblate. From AMS data, the three principal planes of the AMS ellipsoid are identified in each block and thin sections are prepared along them. Quartz grain shape (aspect ratio, R _q), intensity of quartz and mica shape preferred orientation (κ_q and κ_mi, respectively) and 2D strain (E) recorded by quartz are measured in each section. R _q, κ_q, κ_mi and E are all noted to be minimum in the section parallel to the magnetic foliation plane as compared to the other two sections. This indicates that the quartz grains have oblate shapes in 3D and accommodated flattening strain, which is similar to the shape of the AMS ellipsoid. The role of mica in causing Zener drag and pinning of quartz grain boundaries is discussed. It is concluded that during progressive deformation, migration of pinned grain boundaries is inhibited. This causes enhanced recrystallization at the grain boundaries adjacent to the pinned ones, thus guiding the shape modification of quartz grains. A strong correlation is demonstrated between κ_q and κ_mi as well as κ_mi and E. It is inferred that fabric evolution of quartz was controlled by mica. Hence, the shape of the AMS ellipsoid, which is on account of mica, provides information about shape of the strain ellipsoid.     

岩石变形变质过程中体积变化的估算方法

简林地介绍了估算岩石体积变化的５种方法：在可能求得岩石有限应变椭球体真实大小的情况下计算岩石体变的方法；在平面应变状态下求岩石体变的方法；非平面应变状态下岩石体变的估算方法；利用岩石中有关元素的含量变化求岩石体变的方法；利用岩石有关矿物的含量变化球岩石体变的方法，并对这些估算方法的运用条件及误差产生的原因进行了分析。     

Pulsating oblate and prolate three-dimensional strains

The progressive ductile deformation of competent spherical inclusions is modeled analytically. Results of this study may help to understand better the limitations connected to geological field methods using competent inclusions for strain analyses. Parameters studied and quantified here are the strain magnitude, the progressive change in inclusion shape, the orientation of the finite strain axes, the frequency of pulsation, and the coupling between the strain ellipticity and viscosity contrast. Competent inclusions develop pulsating apparent strains if the host material is subjected to a component of simple shear and provided time or strain rate is sufficient to complete the strain cycle. The disparity between the strain magnitude inferred from competent viscous inclusions and that undergone by the host rock, increases for larger viscosity between them. The pulsation of the inclusion may suggest zero strain after a strain cycle has been completed, even though strain in the host rock is extremely large. The inclusion will develop pulsating oblate strains if a shortening rate is superposed normal to the plane of pulsation. Conversely, pulsating prolate strains occur if an extension rate is superposed instead of shortening. Stretching lineations outlined by deformed competent inclusions within shear zones beneath collapsing nappe sheets may even point perpendicular to the direction of nappe transport. This finding offers an explanation for the occurrence of mutually perpendicular pebble elongations in nearby locations within the Bygdin conglomerate beneath the Jotun nappe, Norwegian Caledonides.     

惯量投影椭球在构造变形分析中的意义初探

本文讨论了一种新的数学工具,即惯量投影椭球（和椭圆）,在具有任意形状构造变形体的变形描述与应变分析中的意义。论证了在均匀的递进变形过程中,构造标志体惯量投影椭球的变形与其自身的变形保持一致,两者遵循同样的均匀变形方程。在进行构造变形描述和分析时,构造标志体形状与其惯量投影椭球是等效的。惯量投影椭球具有有限应变椭球相同的性质,可以用来描述任意形状构造标志体的变形,并可以替代具有任意形状构造标志体进行变形分析。先前适用于椭圆形标志体变形分析的方法均可应用于具有任意形状的构造标志体。这将为我们对具有任意形状的构造标志体变形的描述和分析提供方便和有用的数学工具。     

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.     

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 shape of pebbles

A theory of pebble erosion is presented, based on the assumption that the rate of erosion at a point on the surface is a function Vof the curvature there. It is proved that for physically reasonable functions V,the sphere is the only shape of pebble which can maintain its proportions as it wears away. An argument is given which leads to a particular form for the function Vand a few qualitative consequences of this form are indicated. The surface of the pebble at time tmay be described using spherical polar coordinates θ, Φ by the radius function r (θ, Φ, t). This function is given by a highly nonlinear partial differential equation. However, in the case of the erosion of a deformed sphere, when terms which are of second order or higher in the deformation are neglected, the equation becomes linear and is a version of the diffusion equation. The stability of the spherical shape against deformations of the various harmonic types is then easily analyzed.     

Quantitative finite strain and kinematic flow analyses along the Zagros transpression zone, Iran

The Ghouri area in southwest Iran exposes a cross section through the Zagros orogenic belt. The area provides an opportunity to investigate quantitative finite strain (Rs), kinematic vorticity number (W_k), proportions of pure shear and simple shear components, sense of shear indicators, steeply plunging lineations, and other moderate to steeply plunging stretching lineations in a transpressional zone. Based on a classical strain analysis of deformed microfossils with oblate strain ellipsoid shape, the Zagros orogenic belt is classified as a pure-shear dominated zone of transpression, but asymmetry of shear-sense indicators suggests that a significant component of simple shear was involved along the deformation zone boundaries. The long axes of the microfossils and stretched pebbles of a deformed conglomerate were used to indicate the stretching direction in this zone. The stretching lineations have a steep to moderate plunge but a constant strain magnitude. Characteristics of dextral inclined transpressional kinematics in the Zagros continental collision zone were quantified and indicate an estimated k-value < 1, an angle between the maximum horizontal axis of the instantaneous strain ellipsoid and the zone boundary (θ = 32°), asymmetrical dextral shear-sense indicators, and an angle of relative plate motion (α = 25°).     

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.     

Strain distribution in a fold in the West Norwegian Caledonides

Strain has been measured from clasts within a deformed conglomerate layer at 17 localities around an asymmetric fold in the Rundemanen Formation in the Bergen Arc System, West Norwegian Caledonides. Strain is very high and a marked gradient in strain ellipsoid shape exists. To either side of the fold, strain within the conglomerate bed is of the extreme flattening type. In the fold, especially on the lower fold closure, the strain is constrictional. Mathematical models of perturbations of flow in glacial ice have produced folds of the same geometry as this fold, with a strikingly similar pattern of finite strain. The fold geometry and strain pattern, as well as other field observations, suggest that the fold developed passively, as the result of a perturbation of flow in a shear zone, where the strain was accommodated by simple shear accompanied by extension along Y.     

The shape of pebbles

A theory of pebble erosion is presented, based on the assumption that the rate of erosion at a point on the surface is a function Vof the curvature there. It is proved that for physically reasonable functions V,the sphere is the only shape of pebble which can maintain its proportions as it wears away. An argument is given which leads to a particular form for the function Vand a few qualitative consequences of this form are indicated. The surface of the pebble at time tmay be described using spherical polar coordinates θ, Φ by the radius function r (θ, Φ, t). This function is given by a highly nonlinear partial differential equation. However, in the case of the erosion of a deformed sphere, when terms which are of second order or higher in the deformation are neglected, the equation becomes linear and is a version of the diffusion equation. The stability of the spherical shape against deformations of the various harmonic types is then easily analyzed.     

The relation between microfabric and strain in a progressively deformed quartzite sequence from Central Australia

In the Ormiston Nappe Complex, west of Alice Springs, central Australia, a deformed zone up to 0.7 km thick is developed in the sedimentary Heavitree Quartzite. The deformed zone is adjacent to a major thrust fault and is defined by mylonitic foliation, which is parallel to the thrust plane and by isoclinal folds. Recognition of original detrital quartz grains allows strain ellipsoids to be measured across the zone. The strain generally plots in the flattening field and many specimens show pure flattening strain. The mylonitic foliation is an axial-plane structure to the folds and is parallel to the XY-plane of the strain ellipsoid. A quartz elongation lineation may be present within the foliation and is parallel to the principal extension direction (the X-axis) of the strain ellipsoid.Strain is accommodated principally by intracrystalline plastic deformation of the quartz grains. In detail the strain is not homogeneous and may vary even between adjacent grains of the same specimen. Quartz optic axis fabrics reflect this strain inhomogeneity. If the strain ellipsoid is an oblate spheroid, c-axes lie in small-circle girdles about the principal shortening axis (the Z-axis). With general triaxial strain, the c-axes lie in a great-circle girdle or girdles which intersect the foliation parallel to the intermediate strain axis (the Y-axis) and lie symmetrically about the Z-axis. A random population of grains from a specimen often shows a composite c-axis pattern between these two types.With approach to the thrust there is an increase in the amount of strain within the specimens. The increasing strain correlates with an increase in the degree of c-axis preferred orientation of the deformed detrital grains and in the amount of new strain-free grains present in the deformed quartzite. Adjacent to the thrust the quartzite is completely composed of polygonal new grains. The new grains probably formed under syntectonic conditions caused by movement along the thrust. The bulk of the new grains developed by increasing misorientation between the subareas of an initially polygonized old grain. There is no evidence of any marked host control on new-grain orientation, but new grain c-axis plots are generally similar to the corresponding old-grain plots from the same specimen.