Deformation of particles developed around rigid and deformable nuclei

The deformation of particles developed around rigid and deformable nuclei e.g., oolites, is investigated theoretically by considering the motion of the viscous fluid within the particles during deformation. It is postulated that the deformed shapes of these particles are influenced by the development of stagnation points around the nuclei. The shape changes of these particles are studied experimentally using an ethyl cellulose—benzyl alcohol solution to represent both the matrix and the particles developed around rigid or highly viscous nuclei. The models were subjected to pure-shear, simple-shear and irrotational non-plane strain deformation and the experimental model deformation is compared with naturally deformed particles. The deformed shapes of particles with nuclei of different competencies differ greatly from that of particles without nuclei; the differences depending on the viscosity difference of the two media and the relative size of the nuclei as compared to the size of the particles enclosing it. For both pure-shear and simple-shear deformation, the shapes of these particles would approximate to that of constrictional (prolate) ellipsoids. Irrotational non-plane strain deformation could, depending on the deformation paths, result in these particles having shapes which are more flattened (oblate) or more constrictional (prolate) than homogeneous particles subjected to the same deformation.     

Adaption of polygonal strain markers

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

Kinematic analyses of orogen-parallel L-tectonites from Pelling-Munsiari thrust of Sikkim Himalayan fold thrust belt: Insights from multiple,incremental strain markers

Fault rocks associated with the Pelling thrust (PT) in the Sikkim Himalayan fold thrust belt (FTB) change from SL tectonites to local, transport-parallel L-tectonites that are exposed in discontinuous klippen south of the PT zone. By estimating the incremental kinematic vorticity number (W_k) from quartz c-axes fabric, oblique fabric, and subgrains, we reconstruct a first-order, kinematic path of these L-tectonites. Quartz c-axes fabric suggests that the deformation initiated as pure-shear dominated (∼56–96%) that progressively became simple-shear dominated (∼29–54%), as recorded by the oblique fabric and subgrains in the L-tectonites. These rocks record a non-steady deformation where the kinematic vorticity varied spatially and temporally within the klippen.The L-tectonites record ∼30% greater pure-shear than the PT fault rocks outside the klippen, and the greatest pure-shear dominated flow among the published vorticity data from major fault rocks of the Himalayan FTB. The relative decrease in the transport-parallel simple-shear component within the klippen, and associated relative increase of transport-perpendicular, pure-shear component, support the presence of a sub-PT lateral ramp in the Sikkim Himalayan FTB. This study demonstrates the influence of structural architecture for fault systems for controlling spatial and temporal variations of deformation fabrics and kinematic path of deforming thrust wedges.     

Strain analysis of a shear zone in a granodiorite

A ductile shear zone in a late Precambrian granodiorite, from the Rouergue (southwest part of the French Massif Central) has been studied.A single episode of deformation is responsible for the formation of a foliation and a well-defined lineation which are localized into an elongated zone, a few decimeters wide.The strain features can be attributed to a simple-shear mechanism (Ramsay and Graham, 1970), so that the main parameters of the deformation are defined.At stages of increasing deformation, the quartz isotropic sub-fabric of the undeformed host rock is progressively transformed into an anisotropic fabric composed of a single oblique girdle while the subgrain size progressively decrease and the dislocation density remains constant. It is suggested that the gliding planes of quartz are the basal plane (0001) and a predominant prismatic plane 101̄0 the slip directions may be a for both glide-planes.The results obtained in this investigation provide a basis for a high voltage electron microscope (H.V.E.M.) study which shows that the fabrics development may be related to dislocation processes. The difference of strain rates in the host rock and in the shear zone is calculated from the dislocation microstructures.     

鄂东南铁山不对称鱼嘴状石香肠构造基质层中的应变分析

利用惯量椭圆法对鄂东南铁山不对称鱼嘴状石香肠构造基质层中不同位置的方解石进行二维有限应变测量。运动学涡度以及相关参数的等值线分布图表明:不对称鱼嘴状石香肠构造基质层中的应变分布是不均匀的。基质层中,靠近断开的能干层处的剪切变形中纯剪切占优势,伸长方向趋向于平行剪切方向;而靠近连续的能干层处的剪切变形中简单剪切占优势,伸长方向趋向于垂直剪切方向。     

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.     

Mohr circles for strain,simplified

The Mohr circle is a well known representation of two-dimensional strain. It is commonly used to illustrate strain algebra, or derive a strain ellipse from particular strain data. The established sign convention is positive (anticlockwise) shear strain on the upward ordinate, after Brace. However, in practice, confusion may easily arise from inconsistent conventions in standard text books. A Mohr-circle convention is proposed here which represents clockwise shear strain on the upward γ ordinate (i.e. upside-down). It simplifies the Mohr circle by representing both single and double angles in their natural sense. Single angles may be traced from rock to Mohr circle directly which is not the case for the Brace circle. Deformed brachiopods and stretched belemnites are used to illustrate this simplified Mohr circle. The pole to the Mohr circle is a useful addition which allows strain data to be represented in their true orientation. The pole method only works on the γ clockwise-up Mohr circle. The Mohr circle is useful to demonstrate simple-shear geometry and algebra. Successive Mohr circles with poles may be used to map strain across a heterogeneous simple-shear zone.     

Transpression

Transpression is considered as a wrench or transcurrent shear accompanied by horizontal shortening across, and vertical lengthening along, the shear plane. A model for the strain in transpression is derived, from which the shape and orientation of the finite strain ellipsoid, and the stretch and rotation of lines can be determined. Shortening across the zone of transpression leads to oblate finite strain ellipsoids (k<1).By considering the superposition of small increments of strain various model deformation paths are computed. These are used to interpret the development of structures, such as en-échelon folds, in transpression zones. The incremental strain ellipsoid allows prediction of the orientation of the principal stresses and hence brittle structures within such zones. The model is also applied to bends and terminations of shear zones and used to interpret the observed patterns of folds and fractures in these.     

Theoretical and natural strain patterns in ductile simple shear zones

A simple empirical model representing the variation of shear strain throughout a simple shear zone allows us to determine the evolution of finite strain as well as the progressive shape changes of passive markers. Theoretical strain patterns (intensity and orientation of finite strain trajectories, deformed shapes of initially planar, equidimensional and non-equidimensional passive markers) compare remarkably well with patterns observed in natural and experimental zones of ductile simple shear (intensity and orientation of schistosity, shape changes of markers, foliation developed by deformation of markers).The deformed shapes of initially equidimensional and non-equidimensional passive markers is controlled by a coefficient P, the product of

1. (1) the ratio between marker size and shear zone thickness

2. (2) the shear gradient across the zone.

For small values of P (approximately P < 2), the original markers change nearly into ellipses, while large values of P lead to “ retort” shaped markers.This theoretical study also allows us to predict, throughout a simple shear zone, various relationships between the principal finite strain trajectory, planar passive markers and foliations developed by deformation of initially equidimensional passive markers.     

东天山大型韧性剪切带基本特征与金矿预测

东天山大型韧性剪切带即秋格明塔什-黄山韧性剪切带,规模宏大,东西长逾600km,宽5～20km。变形标志明显,分四期变形。应变测量、岩组分析、剪切位移量计算、同位素测年等定量分析表明：①以平面单剪应变为主,稍晚向伸长应变转化;②存在逆冲剪切和脆-韧性变形转换;③剪切位移量在75km以上;④变形时代在300～250Ma之间,高峰期在285～265Ma,属海西中晚期。它是康-黄断裂带的重要组成体,控制着康古尔塔格金矿带的空间展布。据现有大中型金矿研究建立了区域脆-韧性变形转换带结构模式和金矿预测模式,可有效指导找矿勘探。     

含软弱夹层库岸复杂滑坡体形成机制

黄土坡复杂滑坡体形成机制研究对滑坡稳定性分析非常重要。通过地层对比并考虑软弱夹层的发育特征建立了巴东城区黄土坡斜坡原岩结构模型,斜坡前缘巴东组第三段上亚段岩体中软弱夹层发育密集,可与黄土坡滑坡深部多级滑带对应。数值模拟计算表明,斜坡在形成演化过程中斜坡前缘巴东组第三段上亚段岩体出现了明显的剪应变集中分布,变形深度与斜坡临江I号崩滑堆积体厚度基本一致,可见斜坡岩体结构和斜坡演化过程的应力应变特征控制了临江I号崩滑堆积体的形成。三峡水库蓄水后会加剧滑坡体的剪切变形,泥化夹层的存在容易诱发深层滑坡。      

Strain analysis of a Northern Apennine shear zone using deformed marble breccias

The Alpi Apuane region of the Northern Apennines appears to have been deformed within a large-scale, low-angle shear zone with an overthrust sense of movement. The presence of mineral stretching lineations, folds progressively rotated into the X strain direction, and schistosities which intersect the nappe boundaries at small angles suggest that a component of shear strain occurred during the deformation. The strain ratios and orientations on two-dimensional sections have been determined from deformed marble breccias, reduction spots, and oncalites. Data from three or more non-perpendicular, non-principal sections have been combined to determine the finite strain ellipsoids at 33 sites within the shear zone.The finite strains have been separated into components of simple shear (γ), longitudinal strain (λ), and volume change (Δ). Algebraic expressions have been derived and graphs constructed which enable components of γ, γ and Δ, and γ and λ to be determined directly from a knowledge of strain ratio (R) within the shear zone and the angle (θ) between the principal strain direction and the shear zone boundary. The Alpi Apuane data indicate that neither simple shear alone, nor simple shear with volume change can satisfactorily explain the observed strains. Consideration of simple shear plus longitudinal strain leads to a general relationship in which the value of shear increases, and the values of longitudinal strain change along a SW-NE profile across the zone. Integration of the resulting shear strain-distance curves gives a minimum displacement of 4 km within the shear zone. Combination of the finite strains with the total time of deformation known from K/Ar studies leads to average strain rates from 1.4 to 9.6 × 10⁻¹⁵ sec⁻¹.A characteristic flat-ramp-flat geometry initially formed the boundaries of what was later to develop into the overthrust shear zone, and deformation of the underlying crystalline basement is believed to have occurred by ductile shearing. Estimates of 21% crustal shortening for the region suggest that the crustal thickness prior to deformation was approximately 20 km in this part of the Northern Apennines.     

Numerical modeling of heterogeneous flow fields around rigid objects with special reference to particle paths, strain shadows and foliation drag

With the help of two-dimensional numerical models this paper investigates three aspects of heterogeneous deformation around rigid objects: (1) the nature of particle paths; (2) the development of strain shadow zones; and (3) the drag patterns of passive markers. In simple shear, spherical objects develop typically a concentric vortex motion, showing particle paths with an eye (double-bulge)-shaped separatrix. The separatrix has no finite dimension along the central line, parallel to the shear direction. Under a combination of pure shear and simple shear, the particle paths assume a pattern with a bow-tie shaped separatrix. With increase in the ratio of pure shear to simple shear (S_r), the separatrix around the object shrinks in size. The axial ratio of the object (R) is another important factor that controls the geometry of particle paths. When R<1.5, the loci of a particle close to the object form an elliptical shell with the long axis lying along the central line. With increase in axial ratio R, the loci form a doublet elliptical shell structure. Objects with R>3 do not show closed particle paths, but give rise to elliptical or circular spiral particle paths.

The development of strain shadow zones against equant rigid bodies depends strongly on the strain ratio S_r. When S_r=0 (simple shear), they develop opposite to the extensional faces of the object, forming a typical σ-type tail. The structure has a tendency to die out with an increase in the pure shear component of the bulk deformation (S_r). The initial angle of the long axis of the object with the shear direction (φ) and the axial ratio of the object (R) determine the development of strain shadow zones near inequant rigid objects. Objects with large R and φ between 60 and 120° form pronounced zones of low finite strain, giving rise to strain shadow structures. A geometrical classification of diverse drag patterns of passive markers around rigid objects is presented along with their conditions of formation.     

偏应力比和应力主轴偏转角同时变化下饱和砂土变形特性研究

利用空心圆柱扭剪仪对饱和砂土进行了应力主轴固定和偏应力比增大（即定向剪切）、偏应力比不变和应力主轴单调旋转（即纯应力主轴单调旋转）、偏应力比和应力主轴偏转角同时增加、偏应力比和应力主轴偏转角分段增加4个系列的排水剪切试验,着重分析不同应力路径下饱和砂土的变形特性及主应力和主应变增量方向变化规律。试验结果表明,纯应力主轴单调旋转下,主应力增量方向在45°～135°范围内变化,主应变增量方向逐渐偏向主应力方向;偏应力比和应力主轴偏转角同时增加下,砂土变形不断增大,当主应力增量方向 45°时,主应变增量方向与主应力增量方向基本一致,但当 45°时,主应变增量方向逐渐偏离主应力增量方向。当应力状态在偏应力比 0.75、应力主轴偏转角 45°范围内时,体应变、最大剪应变与应力路径无关,且后期纯应力主轴旋转下砂土变形不受前期加载历史的影响。     

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.     

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°).     

Evolution of a calcite marble shear zone complex on Thassos Island, Greece: microstructural and textural fabrics and their kinematic significance

The deformation history of a monophase calcite marble shear zone complex on Thassos Island, Northern Greece, is reconstructed by detailed geometric studies of the textural and microstructural patterns relative to a fixed reference system (shear zone boundary, SZB). Strain localization within the massive marble complex is linked to decreasing P–T conditions during the exhumation process of the metamorphic core complex. Solvus thermometry indicates that temperatures of 300–350°C prevailed during part of the shear zone deformation history. The coarse-grained marble protolith outside the shear zone is characterized by symmetrically oriented twin sets due to early coaxial deformation. A component of heterogeneous non-coaxial deformation is first recorded within the adjacent protomylonite. Enhanced strain weakening by dynamic recrystallization promoted strong localization of plastic deformation in the ultramylonite of the calcite shear zone, where high strain was accommodated by non-coaxial flow. This study demonstrates that both a pure shear and a simple shear strain path can result in similar crystallographic preferred orientations (single c-axis maximum perpendicular to the SZB) by different dominant deformation mechanisms. Separated a-axis pole figures (+a- and −a-axis) show different density distributions with orthorhombic texture symmetry in the protolith marble and monoclinic symmetry in the ultramylonite marble consistently with the observed grain fabric symmetry.     

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

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

An experimental approach to strain pattern and folding in unconfined and/or partitioned transpressional deformation

Three different series of experiments were carried out with pitch (bitumen) and/or composite pitch-plasticine models to observe the spatial and temporal changes of strain pattern and/or fold styles in a tectonic zone undergoing sinistral-sense unconfined transpression. In the first series, rectangular pitch models with circular strain markers, when subjected to vertically and laterally unconstrained transpression, showed that the circular markers deformed into ellipses with long axes (maximum instantaneous stretching axis: ISA_max) oriented <45° with the plane of simple shear at the onset of deformation. This initial angle decreased with increasing angle of convergence (α = 15°, 30°, 45° and 60°) between the boundary walls. In the second series, a longitudinally pre-cut pitch block simulated a pre-existing fault zone in the crust. Lubrication along the cut was varied to simulate varying ease of fault slip. During oblique convergence, transpressional strain was partitioned between the pre-existing cut (dominantly the simple shear component) and the pitch blocks (mostly the pure shear component). Partitioning was higher for higher lubrication and lower angle of convergence in these experiments, which envisaged interesting observations on the efficiency of natural ‘partitioned transpression’ systems. In series 3, folds in a thin plasticine layer placed over the pitch block initiated parallel to the long axis of the finite strain ellipse in transpression and rotated anticlockwise with increasing strain.