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.     

岩石的磁性组构及其在岩石变形分析中的应用

文本总结了对采自巴彦乌拉山韧性剪切带的部分标本进行岩石组构数据测定的分析结果,并将其与用常规应变分析方法所得结果予以比较,表明了岩石磁化率各向异性椭球体能够反映总的应变椭球体,证明岩石磁化率各向异性技术做为一种岩石组构因素的研究是有发展前景的。岩石磁性组构资料可应用于变形岩石的应变分析,特别是在通常的岩石组构分析技术太费时间和粗糙的情况下更显示了其优越性。     

衡山岩体西缘韧性剪切带磁性组构研究

本文对采自衡山花岗岩体西缘韧性剪切带的标本进行了岩石磁性组构研究,并将其与常规应变分析方法所得结果予以比较,表明了岩石磁化率各向异性椭球体与应变椭球体之间有一定的对应关系,岩石磁化率各向异性技术可望成为构造地质学的一种新手段。岩石磁性组构资料可应用于变形岩石的应变分析,特别是在没有宏观应变标志或者通常的岩石组构分析方法太费时间的情况下更显出其优越性。     

湖北青峰韧性剪切带构造岩的磁性组构研究

通过对青峰韧性剪切带中糜棱岩的磁性组构研究,同时与常规主应变分析方法所测结果比较,表明岩石磁化率各向异性椭球体与应变椭球体之间有一定的对应关系。磁性组构的特征为构造岩变形机制,以及断裂带的运动学、动力学等的研究提供了一种比较可靠、准确、方便的方法。     

Multi-laminate non-coaxial modelling of anisotropic sand behavior through damage formulation

A modified multi-laminate model, to predict non-coaxiality in anisotropic sand, is proposed in this paper. The model can easily be extended to other geo-materials only with implementing some minor provisions. To consider anisotropy of sand, two ellipsoids are utilized to summarize shear and compressive stiffness of material in different directions. Damage concept is used to take into account degradation of material through loading procedure. Ellipsoid of rigidity factors is being changed in both size and dimension, under applied strain path. Variation of ellipsoids results in change of stiffness distribution over different planes. In other words, fabric evolution in material is considered through variation of ellipsoids of rigidity factors. A simple rule is proposed for shear stress-strain relationship in loading-unloading and reloading, which captures most of the natural characteristics of sand behavior. In multi-laminate models, depending on stiffness distribution over sampling planes, stress and strain are not coaxial essentially. To achieve better results, non-coaxiality of shear stress and strain on sampling planes is considered by applying vector field concept. Shear stress in different directions of a sampling plane is considered as a vector field. This field is obtained from strain field, considering shear stiffness in various orientations. The model parameters are calibrated using uniaxial compressive test data in different directions, with respect to bedding plane on an anisotropic sand sample. To investigate capability of the model to predict non-coaxiality, results of the model are compared to experimental results obtained from pure principal stress rotation. Ultimately, good accuracy is observed in results.     

白银厂矿区劈理成因的磁组构分析

将磁性组构方法用于甘肃白银厂矿区的构造分析.结果表明该区岩石的磁性组构与岩石组构具有同一分布规律.岩石的磁线理与岩石的拉长线理L~a近于一致,磁面理与第一期劈理S_1基本平行,并垂直于岩石的磁化率椭球的最小主轴,也垂直于应变椭球的最短主轴方位,故推断第一期劈理S_1的成因主要属压扁成因.同时受到后期的多次变形与褶皱迭加影响.S_1轴面的优选面状分布发生变位不大,表明该区多次构造变形主要来自早期的同一主应力场作用所致.矿区岩石的磁面理与应变椭球拉长轴面或第一期劈理面S_1存在某些的角度差,表明该区岩石的劈理面由于受后期多次构造变形影响,可能发生剪切运动或旋转机制.     

Magnetic fabric and modeled strain distribution in the head of a nested granite diapir,the Melechov pluton,Bohemian Massif

The Melechov pluton, Bohemian Massif, is interpreted as a mid-crustal nested granitic diapir with an apical part exposed at the present-day erosion level. The diapir head exhibits a concentric structure defined by lithologic zoning and by the anisotropy of magnetic susceptibility (AMS). In concert with theoretical models, outward-dipping margin-parallel magnetic foliations are associated with oblate shapes of the susceptibility ellipsoids and higher degree of anisotropy, passing inward into weaker triaxial to prolate fabric. By contrast, magnetic fabric in an inner granite unit is in places oriented at a high angle to internal contacts and is interpreted as recording an internal diapir circulation. We use inverse modeling to calculate strain variations across the diapir from the AMS data. The magnetic fabric parameters and calculated strains are in agreement with strain distribution in heads of model Newtonian diapirs traveling a distance of two body radii and suggest granitic magma ascent as a crystal-poor suspension followed by crystallization of fabric markers and their response to strain near the final emplacement level. The intrusive fabric thus formed late but, though generally weak, was still capable of recording incremental strain gradient in the granite diapir.     

Oolite deformation in Windgällen, Canton Uri, Switzerland

Detailed analysis of the oolite deformation in Windgällen revealed that the finite-strain states vary significantly and systematically over the major recumbent structures. The strain is lowest at the fold cores and highest along the limbs, which have suffered extensions of up to 150%. These high strains are interpreted as the result of strong overfolding during the development of the nappe-like structure. The maximum ratio of the strain ellipsoids computed from the oolite shapes is 14.6:1, though individual oolites having axial ratios as high as 30 : 1 have been encountered. The intermediate axes of the strain ellipsoids throughout the area have been extended between 3 and 64% and the strain ellipsoids are of the flattening type with K being less than unity. The pre-deformation fabric of the oolites is very weak and their initial axial ratios in any two-dimensional section is generally less than 2:1. Slaty cleavage is coincident with the plane of maximum compression in the rock. The relationship of cleavage, folds and oolite deformation suggest that cleavage develops progressively during deformation and forms at the same time as the folds associated with it.     

Do reduction spots predate finite strain? A magnetic diagnosis of Cambrian slates in North Wales

The purple Cambrian slates of North Wales contain green-colored, irregular patches drawn out along cleavage and the more familiar ellipsoidal reduction spots that are parallel to cleavage. Because parallelism of pre-strain ellipsoids could only be produced by extremely high strain, we reject the hypothesis that these are strain markers. Here, we use magnetic methods to resolve this argument. The magnetic analysis reveals that the magnetic fabric of purple slate differs from the green variety of the slate only in the oxidation state, and indicates that the reduction process postdates cleavage. This suggests that the shape of reduction spots reflects the anisotropy of diffusion during reduction, not finite strain and alignment of an initially ellipsoidal object in the slates.     

Particle shape effect on macro‐ and micro‐behaviors of monodisperse ellipsoids

The microscopic and macroscopic behaviors of assemblages of monodisperse ellipsoids with different particle shapes were studied using the discrete element method. Four samples were created with 1170 identical prolate ellipsoids. The samples were compressed isotropically to 100 kPa. Then triaxial compression tests were carried out to very large strains until the ultimate state was reached. This paper presents typical macroscopic result including stress–strain relationship and volumetric behavior. In addition, the fabric of the samples was examined at the initial state, at the peak shear strength state, and at the ultimate state. We studied the evolution of three vector‐typed micromechanical arguments with strain including the particle orientation, branch vector, and normal contact force. The normal contact force (micromechanical argument) was found to have a direct relationship with the principal stress ratio (macroscopic parameter). The angles between these vectors were also investigated. The maximum angle between vectors is related to particle shape. The results indicate that the distributions and the maximum values of these angles do not change with loading. Copyright © 2008 John Wiley & Sons, Ltd.     

Magnetic fabric of Pleistocene continental clays from the hanging-wall of an active low-angle normal fault (Altotiberina Fault,Italy)

Anisotropy of magnetic susceptibility (AMS) represents a valuable proxy able to detect subtle strain effects in very weakly deformed sediments. In compressive tectonic settings, the magnetic lineation is commonly parallel to fold axes, thrust faults, and local bedding strike, while in extensional regimes, it is perpendicular to normal faults and parallel to bedding dip directions. The Altotiberina Fault (ATF) in the northern Apennines (Italy) is a Plio-Quaternary NNW–SSE low-angle normal fault; the sedimentary basin (Tiber basin) at its hanging-wall is infilled with a syn-tectonic, sandy-clayey continental succession. We measured the AMS of apparently undeformed sandy clays sampled at 12 sites within the Tiber basin. The anisotropy parameters suggest that a primary sedimentary fabric has been overprinted by an incipient tectonic fabric. The magnetic lineation is well developed at all sites, and at the sites from the western sector of the basin it is oriented sub-perpendicular to the trend of the ATF, suggesting that it may be related to extensional strain. Conversely, the magnetic lineation of the sites from the eastern sector has a prevailing N–S direction. The occurrence of triaxial to prolate AMS ellipsoids and sub-horizontal magnetic lineations suggests that a maximum horizontal shortening along an E–W direction occurred at these sites. The presence of compressive AMS features at the hanging-wall of the ATF can be explained by the presence of gently N–S-trending local folds (hardly visible in the field) formed by either passive accommodation above an undulated fault plane, or rollover mechanism along antithetic faults. The long-lasting debate on the extensional versus compressive Plio-Quaternary tectonics of the Apennines orogenic belt should now be revised taking into account the importance of compressive structures related to local effects.     

Pan-African strike–slip tectonics in eastern Cameroon—Magnetic fabrics (AMS) and structure in the Lom basin and its gneissic basement

Field, microstructural, and anisotropy of magnetic susceptibility (AMS) or magnetic fabric studies were applied to identify the sequence and character of the Pan-African structures in the basement of Eastern Cameroon at both sides of the regional scale Bétaré-Oya Shear Zone (BOSZ). The NE-SW trending BOSZ separates older gneisses and migmatites towards SE (domain I) from the younger rocks of the Lom meta-volcano-sedimentary basin towards NW (domain II). In domain I, early, ductile compressional deformation occurred in two events, D1 and D2, under relatively high T conditions. During subsequent cooling, strain partitioned between the competent basement gneisses with only mild compression and the bordering shear zone (BOSZ) with intense simple shear-wrenching (D3). Strain in the less competent rocks of domain II is dominated by simple shear, strike-slip wrenching (D3), with an earlier stage of compressional deformation preserved only in some low strain pods.Magnetic fabrics (AMS) document a progressive change from oblate ellipsoids towards prolate ellipsoids in domain I, when proceeding from the south towards the BOSZ. Foliations are mostly steep but define a girdle with a pole plunging gently towards WSW. The magnetic lineations also plunge mostly towards WSW at shallow angles. These fabrics indicate a compression approximately normal to the BOSZ, which is also the SE margin of the Lom Basin. In the Lom metasediments (domain II), AMS ellipsoids are typically oblate. Foliations trend NE-SW with mostly steep dips. Magnetic lineations plunge gently NE or SW. This fabric with foliations mostly steep and subparallel with the major BOSZ, combined with generally subhorizontal lineations implies the BOSZ as a Pan-African strike–slip shear zone with a subordinate component of compression.At a larger scale, the area is part of a continent-scale shear zone, separating external Pan-African domains of compression along the northern margin of the Congo craton from internal domains dominated by high-angle strike–slip and transpressional deformation. Together with published data, the present study thus demonstrates that transpression is a regional phenomenon in the Pan-African orogen of central and eastern Cameroon.     

岩石磁组构在断裂变形性状与期次研究中的应用——以广西博白-合浦断裂为例

岩石磁化率椭球体的三个轴与应变椭球体的三个轴方向相平行,并具有一定的共构关系.变形岩石的磁组构参数P、T、F、L以及磁化率椭球体主张量方向等可以用来定量地表征岩石变形的性状及期次.本文通过实测和计算27个样品的磁组构参数,研究了博白-合浦断裂带的变形性状与期次,结果表明:博白-合浦断裂带大致经历了三期构造变形作用,不同时期具不同性质的构造变形.变形性状分别表现为韧性、韧-脆性及脆性变形,应变行为分别表现为平面、非平面和线性应变.     

Structural geometry and kinematics of the Ailao Shan shear zone: insights from integrated structural,microstructural, and fabric studies of the Yao Shan complex,Yunnan, Southwest China

ABSTRACT

The Yao Shan complex, a massif near the southern segment of the Ailao Shan–Red River (ASRR) shear zone, bears important information on the structural framework of the massif and the kinematics of ductile shearing along the ASRR shear zone. In this contribution, structural, microstructural, quartz c-axis fabric, magnetic fabric, and geochronologic data are used to determine the structural framework of the Yao Shan massif and its tectonic implications for the ASRR shear zone. The Yao Shan complex is characterized by an overall linear A-type antiform that contains a core of high-grade metamorphic rocks with Palaeoproterozoic to Mesozoic protoliths and a mantle of Permo-Triassic low-grade rocks. Both the high-grade metamorphic core and low-grade Permo-Triassic rocks have experienced progressive ductile shearing. Anisotropy of magnetic susceptibility (AMS) results from 17 samples collected along the Xinjie–Pingbian section across the complex show that magnetic lineation (K_max) and foliation (K_max–K_int) are generally subparallel to the corresponding structural elements in the sheared rocks. The shape parameter E values of the magnetic ellipsoids are indicative of dominantly oblate and plane strain, but vary with protolith type and degree of strain among the various rock types. In agreement with the field and microstructural observations, the corrected degree of anisotropy (Pj) values reflect high shear strain in the core rocks and relatively low shear strain in the low-grade strata. A kinematic analysis based on structural and magnetic fabric data shows that both left- and right-lateral shear occurred during the deformation of the Yao Shan complex. Therefore, instead of being an element of the ASRR shear zone, the Yao Shan complex constitutes a crustal-scale inharmonic A-type fold with a fold axis parallel to the stretching lineation. Geochronologic data reveal that the folding occurred coevally with ductile shearing of the middle to lower crust between ca. 30 and 21 Ma.     

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.     

Pivoting analyses of the selective entrainment of sediments by shape and size with application to gravel threshold

Measured variations of pivoting angles with grain size, shape (‘reliability’ and angularity) and imbrication are employed in analyses of grain threshold to examine how these factors influence selective grain entrainment and sorting. With a bed of uniform grain sizes, as employed experimentally to establish the standard threshold curves such as that of Shields, the threshold condition depends on grain shape and fabric. The analysis demonstrates quantitatively that there should be a series of nearly-parallel threshold curves depending on grain pivoting angles. For a given grain size, the order of increasing flow strength required for entrainment is spheres, smooth ellipsoids (depending on their ‘reliability’), angular grains, and imbricated ellipsoids (depending on their imbrication angles). The relative threshold values for these different grain shapes and fabric are predicted according to their respective pivoting angles, but remain to be directly tested by actual threshold measurements. The pivoting angle of a grain also depends on the ratio of its size to those it rests upon. This dependence permits an evaluation of selective entrainment by size of grains from a bed of mixed sizes, the condition generally found in natural sediments. The pivoting model predicts systematic departures from the standard threshold curves for uniform grain sizes. Such departures have been found in recent studies of gravel threshold in rivers and offshore tidal currents. The pivoting model is compared with those threshold data with reasonable agreement. However, more controlled measurements are required for a satisfactory test of the model. It is concluded that variations in pivoting angles for grain entrainment are significant to the processes of selective sorting by grain size and shape.     

西昆仑山前晚新生代沉积岩磁组构及构造意义

西昆仑山前晚新生代沉积岩磁组构特征表明，沉积岩原生磁组构受后期构造活动改变。磁组构测试结果表明晚新生代沉积岩生较明显变形，岩石磁化率椭球体指示磁面理较发育，反映岩石受压扁型变形为主。磁化率椭球体最小轴方向为NW，指示该区最大主压应力为NW，与区域构造分析结果相一致。     

Comparison of strain and magnetic fabrics in Dalradian rocks from the southwest Highlands of Scotland

The magnetic fabrics of 235 samples from 31 localities in Argyllshire, Scotland were determined to study the development of the Caledonian tectonic fabric in the southwest Highlands of Scotland. The regional fabric indicates a strong NE-SW compressional foliation due to the primary deformational phases, which in parts has been overprinted by secondary deformations. A detailed comparison of the anisotropy data and the available strain data shows that the two fabric ellipsoids are co-axial, and that their axial mean ratios seem to be related by an empirical power relationship of the type:

$\text{Xi}\text{Xj}\text{=}\text{li}\text{lj}{}^{\mathrm{a}}$

(for i = 1,2,3; j = 1,2,3 and i ≠ j) where χ_i and χ_j are orthogonal principal axes and l_i and l_j are the corresponding orthogonal principal strain axes. The exponent a for the sites from Scotland is 0.088 ± 0.017 compared with 0.142 ± 0.001 and 0.145 ± 0.005 found in the Caledonian slates of the English Lake District and the Welsh slate belt.     

Comparison of Fry strain ellipse and AMS ellipsoid trends to tectonic fabric trends in very low-strain sandstone of the Appalachian fold–thrust belt

In moderately to highly strained sandstones, both the long axis of the bedding-parallel finite-strain ellipse, as calculated by the normalized Fry method, and the projection of the long axis of the AMS ellipsoid on the plane of bedding, align well with local “structural grain” (trends of cleavage, folds, and faults). This relationship implies that results of both 2D Fry and AMS analyses represent the local layer-parallel tectonic strain component. Do both methods provide comparable results for very low-strain sandstone (e.g., <5%)? To address this question, Fry and AMS analyses were conducted in very low-strain sandstone from two localities in the Appalachian foreland fold–thrust belt: near Rosendale in New York and the Lackawanna synclinorium of Pennsylvania. We compared the map projections of both bedding-parallel Fry ellipses and AMS ellipsoids to the local structural grain. In both study areas, projections of the long axis of Fry strain ellipses do not cluster in a direction parallel to structural grain, whereas the projection of the long axes of AMS ellipsoids do cluster closely to structural grain. This observation implies that in very low-strain sandstone, AMS analysis provides a more sensitive “quick” indicator of tectonic fabric than does normalized Fry analysis.     

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: