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.     

古地磁岩心定向及其在地应力测量上的应用

在实验室,将需测试的一段岩心柱标上标志线,制备成古地磁测试和差应变分析测试的标准试样。岩样经一定步骤的磁清洗、相应的磁性参数测量和数据处理,可恢复岩样在地下的原始方位。用差应变分析(DSA)法确定最大主应力相对标志线的方位,与古地磁测试结果结合,即可确定出该地区的最大主应力方向。      

Strain investigations on calcite marbles using neutron time-of-flight diffraction

In order to describe and explain the effect of bowing of marble facade panels, neutron time-of-flight diffraction was applied to determine residual macro- and microstrain on the calcite mineral phase. The measurements were combined with investigations of the crystallographic preferred orientation (texture) measurements by neutron diffraction, macroscopic measuring of the bowing on marble building stones, as well as microfabric analyses. Three samples were investigated to explain the bowing effect: a fresh broken sample, a good conditioned facade panel and a strongly deformed facade panel. Residual intracrystalline strain was detected in all investigated samples, which differed in the degree of bowing. For the first time, the preferred orientation and the residual strain were found to be related. The results show that different strain magnitudes are reflected by residual strains, which differ significantly in magnitude and direction. Furthermore, different Bragg peak widths have been detected as an indication of microscopic strain. The observed residual strain values in the samples are related with the grain shape and texture properties.     

The analysis of deformed belemnites

Deformed belemnites are important strain markers because they allow calculation of actual length changes in deformed rocks. They have been studied in the deformed Lower Lias sequence of part of the French Alps by measuring the orientation of about 100 belemnites at each locality, recording the amount of extension of as many as possible, and locating the sector of arc on the bedding plane occupied by extended belemnites. For comparison, studies on a similar but undeformed Lower Lias sequence in southern England show that dispersed belemnites have no strong initial preferred orientation, and that a proportion of undeformed belemnites show cross-fractures that would allow extension of the rostra from the onset of deformation.Graphs of frequency vs orientation for belemnites at locations deformed by irrotational strain lead directly to an estimate of two-dimensional strain ratio and orientation. Non-random initial distributions and rotational strain produce more complex graphs, which are treated only qualitatively. Extension-orientation graphs show that many deformed belemnites lie within the sector of extension of the strain ellipse. These graphs provide estimates of 1 + e₁ and 1 + e₂, which can be cross-checked using the orientation of the lines of no finite longitudinal strain derived from measuring the maximum sector of arc occupied by extended belemnites.Examples from the French Alps show how the rotational and irrotational strain of initially random samples may be distinguished, and how the more complex case of deformation of non-random samples differs from these. The direction of rotation of the strain ellipse can be derived from the geometry of the graphs of frequency and elongation.     

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.     

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.     

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     

The initiation and development of metamorphic foliation in the Otago Schist, Part 1: competitive oriented growth of white mica

The 3D shape, size and orientation data for white mica grains sampled along two transects of increasing metamorphic grade in the Otago Schist, New Zealand, reveal that metamorphic foliation, as defined by mica shape‐preferred orientation (SPO), developed rapidly at sub‐greenschist facies conditions early in the deformation history. The onset of penetrative strain metamorphism is marked by the rapid elimination of poorly oriented large clastic mica in favour of numerous new smaller grains of contrasting composition, higher aspect ratios and a strong preferred orientation. The metamorphic mica is blade shaped with long axes defining the linear aspect of the foliation and intermediate axes a partial girdle about the lineation. Once initiated, foliation progressively intensified by an increase in the aspect ratio, size and alignment of grains, although highest grade samples within the chlorite zone record a decrease in aspect ratio and reduction in SPO strength despite continued increase in grain size. These trends are interpreted in terms of progressive competitive anisotropic growth of blade‐shaped grains so that the fastest growth directions and blade lengths tend to parallel the extension direction during deformation. The competitive nature of mica growth is indicated by the progressive increase in size and resultant decrease in number of metamorphic mica with increasing grade, from c. 1000 relatively small mica grains per square millimetre of thin section at lower grades, to c. 100 relatively large grains per square millimetre in higher grade samples. Reversal of SPO intensity and grain aspect ratio trends in higher grade samples may reflect a reduction in the strain rate or reduction in the deviatoric component of the stress field.     

Random porosity fields and their influence on the stability of granular media

It is well established that the mechanical behavior of granular media is strongly influenced by the media's microstructure. In this work, the influence of the microstructure is studied by integrating advances in the areas of geostatistics and computational plasticity, by spatially varying the porosity on samples of sand. In particular, geostatistical tools are used to characterize and simulate random porosity fields that are then fed into a nonlinear finite element model. The underlying effective mechanical response of the granular medium is governed by a newly developed elastoplastic model for sands, which readily incorporates spatial variability in the porosity field at the meso‐scale. The objective of this study is to assess the influence of heterogeneities in the porosity field on the stability of sand samples. One hundred and fifty isotropic and anisotropic samples of dense sand are failed under plane‐strain compression tests using Monte Carlo techniques. Results from parametric studies indicate that the axial strength of a specimen is affected by both the degree and orientation of anisotropy in heterogeneous porosity values with anisotropy orientation having a dominant effect, especially when the bands of high porosity are aligned with the natural orientation of shear banding in the specimen. Copyright © 2007 John Wiley & Sons, Ltd.     

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。     

Clay orientation in some Upper Carboniferous mudrocks, its relationship to quartz content and some inferences about fissility, porosity and compactional history

The degree of preferred orientation of clay minerals in a series of fifteen fine-grained sediments has been precisely measured by transmitted X-ray goniometry. This varies systematically with quartz content. By contrast it is apparently unrelated to fissility since the one fissile sample (a ‘paper shale’) occurs at the mid-point of the sample series ranked in order of degree of preferred orientation. In this series fissility corresponds with the presence of fine-scale lamination. Estimates of compaction strain using the theory of March (1932) and assuming random initial clay orientation are consistent with the view that these samples started life with porosities similar to those of present-day muddy sediments. On this basis it is concluded that preferred orientation in clay-rich sediments results almost entirely from compaction strain. The degree to which this can be attained is limited by the presence of nonplaty particles (such as quartz grains) which prevent planar fabric development in their immediate vicinity.     

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.     

Regional scale strain variations in Banded Iron Formations of Eastern India: results from anisotropy of magnetic susceptibility studies

Anisotropy of magnetic susceptibility (AMS) data are used as a tool to determine strain variations in different parts of the Banded Iron Formations (BIFs) of the Bonai Synclinorium, eastern India. AMS data of 88 cylindrical cores drilled from 29 samples collected from the limb and hinge parts of mesoscopic scale folds as well as different parts of the entire synclinorium are presented. It is found that the samples from limbs of small-scale folds and also from limbs of the regional scale synclinorium have higher degrees of anisotropy than the hinges. This is inferred to indicate that the limbs accommodated higher strain than the hinges. AMS orientation data are analysed in conjunction with field data. It is concluded that the magnetic fabric developed in the limbs as well as hinges of the BIFs of the study area is related to deformation and is not a manifestation of sedimentary fabric.     

On the strength of transversely isotropic rocks

Accurate prediction of strength in rocks with distinct bedding planes requires knowledge of the bedding plane orientation relative to the load direction. Thermal softening adds complexity to the problem since it is known to have significant influence on the strength and strain localization properties of rocks. In this paper, we use a recently proposed thermoplastic constitutive model appropriate for rocks exhibiting transverse isotropy in both the elastic and plastic responses to predict their strength and strain localization properties. Recognizing that laboratory‐derived strengths can be influenced by material and geometric inhomogeneities of the rock samples, we consider both stress‐point and boundary‐value problem simulations of rock strength behavior. Both plane strain and 3D loading conditions are considered. Results of the simulations of the strength of a natural Tournemire shale and a synthetic transversely isotropic rock suggest that the mechanical model can reproduce the general U‐shaped variation of rock strength with bedding plane orientation quite well. We show that this variation could depend on many factors, including the stress loading condition (plane strain versus 3D), degree of anisotropy, temperature, shear‐induced dilation versus shear‐induced compaction, specimen imperfections, and boundary restraints.     

Static recrystallization and preferred orientation of phyllosilicates: Michigamme Formation,northern Michigan,USA

The Michigamme Formation of the Marquette District in Michigan's Upper Peninsula comprises a sequence of cleaved rocks of increasing metamorphic grade. Because metamorphism in the area occurred after cleavage formation, the rocks provide an opportunity to study preferred orientation development of phyllosilicates under conditions of static recrystallization.X-ray texture goniometry on samples from the greenschist-facies zone that were collected at varying distances from the bounding biotite-in and garnet-in isograds, shows that: (1) the preferred orientation of phyllosilicates is always parallel to the mesoscopic cleavage, and (2) the degree of preferred orientation of phyllosilicates improves as a function of increasing metamorphic grade (from <4 to >9 m.r.d.). Scanning electron microscopy on these samples shows that: (1) the length/width ratio increases with increasing grade, and (2) grain shapes are better defined with increasing grade.Previous work on slates showed mechanical processes dominate at very low-grade metamorphism, whereas chemical processes are favored at higher grades. The Michigamme samples show that improvement of preferred orientation occurrred by grain dissolution and crystallization. Noncleavage-parallel phyllosilicate grains were preferentially dissolved, probably facilitated by internal strain energy from mineral defects, aided by chemical energy, whereas cleavage-parallel phyllosilicates were hosts for new growth along their basal planes. These results show that significant fabric strengthening can be achieved by grain dissolution and crystallization in the absence of tectonic stress.     

Application of strain analysis to estimate pressure solution processes in regional shear zones

This paper considers the basic principles of the strain analysis method based on the analysis of antitaxial regeneration fibrous fringes around linear rigid inclusions in a low-viscosity rock matrix. This method has been developed for pressure shadows composed of fibrous minerals, whose orientation is controlled by the major elongation direction rather than the orientation of rigid inclusions. This approach is applicable only for rocks exposed to uniform coaxial straining. The strain ellipse is calculated in two ways: for three variably oriented strain markers, it is calculated using Mohr’s circles, and for numerous strain markers by average body ellipse. The strain ellipsoid is calculated using the parameters of a few strain ellipses calculated with three and more non-parallel planes. This paper provides the data on the method testing in reference sites of Dora–Pil’ ore field in the Upper Indigirka district and Vangash area in the Yenisei Range. Regeneration fibrous fringes around fragments of fern fossils and linear rutile metacrystals were used as markers. The results of strain analysis obtained for the reference sites in the Upper Indigirka district made it possible to describe the signs of variable strain stages of developing strike-slip zones making up the Adycha–Taryn Fault Zone. Sublatitudinal ore-bearing strike-slip zones are characterized by a subvertical orientation of the elongation axes X of elongated strain ellipsoids, which are subperpendicular to quartz–carbonate veins and slope kink zones. NW-trending strike-slip zones are characterized by subhorizontal orientation of the Z shortening axes of flattened strain ellipsoids, which are subparallel to the normals of quartz–carbonate veins and veinlets. The results of strain analysis obtained for reference sites in the Vangash area made it possible to describe the thrust strain environment following the metamorphism stage and to reveal specific features in the formation of the strain textures of ore-bearing rocks based on their rheological properties.     

橄榄石集合体的简单剪切形变实验研究

本文利用高精度Paterson气体介质变形装置对富铁橄榄石集合体(Mg_(0.5),Fe_(0.5))_2SiO_4进行了高温简单剪切变形试验.试验在温度1473K和围压300MPa的条件下进行,差应力为64～153MPa,应变率为10~-5～10~-3s~-1.一共进行了三组试验,试件的剪切变形量分别为89%,131%和200%,通过对变形后试件的反射光学显微结构分析,得到富铁橄榄石集合体动态重结晶的微观机制,由结晶各向异性分析给出晶格最优取向和波速各向异性的分布及随应变的增加而表现出的演化特征.     

Strain determination using a density-distribution technique and its application to deformed Upper Cretaceous Dorset chalks

A simple and highly accurate finite-strain measurement technique is presented. The principle behind the technique is that the longer the dimension of a strain indicator in a certain orientation, the higher the probability of its being intersected by a plane perpendicular to this orientation. Strain indicators have to be counted in any three mutually perpendicular orientations and their populations are analytically transferred into the parameters describing the strain ellipses in the three orientations. Strain indicators might be of any shape, but they must have had predeformational random distribution and random orientation.The geological implications of the technique are discussed. The technique is applied to deformed Upper Cretaceous chalk from Dorset (southern England) using microfossils as strain indicators.     

Constitutive analysis of the mechanical anisotropy of Opalinus Clay

This paper aims to analyse the anisotropic features of behaviour of Opalinus Clay using the theory of plastic multi-mechanisms. The results of triaxial tests conducted with different load levels and directions showed that the mechanical behaviour of this shale is cross-anisotropic. The stiffer samples are those in which the loading direction is parallel to the bedding plane. This indicates that the preconsolidation stress depends on the orientation of the load with respect to the fabric of Opalinus Clay. It is proposed to interpret the observed cross-anisotropy with an elastoplastic model based on four plastic strain mechanisms that may be successively mobilised depending on the loading direction. The predicted stress–strain responses vary according to the directions of the space as a result of the hardening process, depending on the number of plastic strain mechanisms that have been mobilised. The numerical predictions show overall good agreement with the experimental data in terms of deviatoric stress versus axial strain, demonstrating that multi-mechanism plasticity is a suitable constitutive tool for the interpretation of the mechanical anisotropy of this shale.     

Shear-zone deformation and bulk strain in granite-greenstone terrain of the Western Superior Province,Canada

The later Archaean history of the granite-greenstone terrain of the western Superior Province in Manitoba was marked by the development of numerous shear zones with varying orientation and sense of movement. Data are presented on the orientation, width and displacement of 15 shear zones in a 61,000 km² area east of Lake Winnipeg. By making simplifying assumptions, estimates are made of the orientation of the maximum compressive stress (N-S) and the bulk shortening strain caused by the shear zone displacements (39–45%).The estimates of shear zone displacement strain are compared with estimates of bulk strain obtained by analysing the granite-greenstone shape pattern over a large area of the western Superior Province, again making simplifying assumptions. As expected, the contribution of the more ductile greenstones to the bulk strain is proportionately greater. A value of around 40% is indicated for the bulk shortening strain.Estimates of mean shear strain obtained both from shear zone displacement/width measurements and from strain profiles show a wide range in values from around 10 in some narrow zones to around 1 in very wide zones.