Removal of finite deformation using strain trajectories

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

三维参照变形及应变相研究评述

三维参照变形和应变相是最近构造地质学领域中取得的重要进展,三维参照变形是理想化的三维变形分类,每一参照变形是共轴级分（拉伸、压扁或纯剪）和与其垂直的简单切组分同时作用的产物,三种可能的面理取向和三种可能的线理取向的不同组合构成六咱应变相,三维参照变形和应变相研究证明糜陵面理未必平行剪切带,可与剪切带斜交,甚至垂直,线理未必与剪切方向一致,可与剪切方向斜交,甚至垂直,出现横向面理时,剪切指向标志位于该面理内,出现横向线理时,剪切指向出现在与线理垂直的ａｃ面理内,三维变形分析不公可解决三维分析难以解释的横向面理和线理,而且可确定共轴组分的类型及其与单剪组分的结合方式。     

Fabric evolution in a numerical direct shear test

This paper presents a three-dimensional discrete element study on the mechanical behaviour and fabric changes of elongated particle assemblages in a direct shear test. Assemblages with elongated particles are prepared under different preparation methods: random placing and deposition. Particles are generated inside a direct shear box with arbitrary orientations in the random placing method. In the deposition approach, two directions of deposition are used: perpendicular and parallel to the shear direction. Assemblages formed by different methods give significantly different initial particle orientations. Different dilatancy responses and shear strengths are resulted upon shearing. Change in fabric along the shear plane is noticeable and the magnitude of such rotation depends highly on the initial particle orientation.     

Application of Image Analysis to Observe Microstructure in Sandstone and Granite

Abstract: Advanced techniques are examined to observe microstructure of rocks using image analysis combined with methods such as the fluorescent approach and the application of optical characteristics of minerals. Analyzed are discrimination of grains in rocks, distribution patterns of grain orientation in sandstone, changes of grain shape as weathering advances and distribution patterns of microcracks in granite. In Shirahama sandstone, relatively large and flat grains are orientated parallel to the bedding on the plane perpendicular to the bedding, while grains on the plane parallel to the bedding show random patterns. In weathered granite, it is clarified that the grain surface becomes complex as weathering advances and differences among three major mineral species are identified. In Inada granite, intracrystalline cracks predominate over intercrystalline cracks and grain boundary cracks both in total length and number. Furthermore, three types of microcracks show different orientations; the intercrystalline cracks show a dominant orientation which coincides with the orientation of the rift plane, the easiest plane to split, while the intracrystalline cracks and grain boundary cracks show no preferred orientation.     

Some garnet microstructures: an illustration of the potential of orientation maps and misorientation analysis in microstructural studies

The microstructures of two contrasting garnet grains are mapped using automated electron backscatter diffraction. In both cases there is a very strong crystallographic preferred orientation, with measurements clustered round a single dominant orientation. Each garnet grain is divided into domains with similar orientations, limited by boundaries with misorientations of 2° or more. In both samples most of misorientation angles measured across orientation domain boundaries are significantly lower than those measured between random pairs of orientation domains. One sample is a deformed garnet that shows considerable distortion within the domains. Lines of orientation measurements within domains and across domain boundaries show small circle dispersions around rational crystallographic axes. The domain boundaries are likely to be subgrain boundaries formed by dislocation creep and recovery. The second sample is a porphyroblast in which the domains have no internal distortion and the orientation domain boundaries have random misorientation axes. These boundaries probably formed by coalescence of originally separate garnets. We suggest that misorientations across these boundaries were reduced by physical relative rotations driven by boundary energy. The data illustrate the potential of orientation maps and misorientation analysis in microstructural studies of any crystalline material.     

The effects of Dissolution–Precipitation Creep on quartz fabrics within the Purgatory Conglomerate,Rhode Island

Quartz Crystallographic Preferred Orientation (CPO) patterns are most commonly a result of deformation by dislocation creep. We investigated whether Dissolution–Precipitation Creep (DPC), a process that occurs at lower differential stresses and temperatures, may result in CPO in quartz. The Purgatory Conglomerate is part of the SE Narragansett basin where strain intensity increases from west to east and is associated with top-to-the-west transport and folding during the Alleghanian orogeny. Within the Purgatory Conglomerate, DPC led to quartz dissolution along cobble surfaces perpendicular to the shortening direction, and quartz precipitation in overgrowths at the ends of the cobbles (strain shadows), parallel to the maximum extension direction. Quartz c-axis orientations as revealed by Electron Backscatter Diffraction (EBSD) methods were random in all analyzed domains within the cobbles and strain shadows irrespective of the intensity of strain or metamorphic grade of the sample. Quartz dissolution probably occurred exclusively along the cobbles' margins, leaving the remaining grains unaffected by DPC. The fact that quartz precipitated in random orientations may indicate that the strain shadows were regions of little or no differential stress.     

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.     

贝壳珍珠层的X射线衍射研究

珍珠层中文石晶体的择优取向是珍珠层具有优异力学性能的重要原因之一，通过对三角帆蚌贝壳珠层的X射线衍射研究，表明珍珠层除存在公认的（002）面网平行珍珠层面的强烈定向文石晶体外，还有强烈的（012）面网平行珍珠层面的亲新定向文石晶体，且在同一贝壳的不同位置，该两种取向的择优取向度变化较大，充分说明了珍珠层结构的不均匀性。     

The relationship between regional stress field,fracture orientation and depth of weathering and implications for groundwater prospecting in crystalline rocks

In a uniform granite gneiss study area in central Zimbabwe, lineaments oriented parallel to the maximum regional compressive stress orientation exhibit the thickest regolith development, while lineaments oriented perpendicular to the maximum compressive stress show the shallowest development of weathered regolith. The principal fracture set orientations were mapped using aerial imagery. The regional stress field, estimated from global stress maps, was used to determine the stresses acting on each principal lineament orientation. Multi-electrode resistivity profiling was carried out across fractures with different orientations to determine their subsurface regolith conditions. The results indicate that the 360 and 060° lineaments, which are sub-parallel to the principal compressive stress orientation (σ₁) exhibit maximum development of the regolith, while 130° lineaments perpendicular to σ₁ do not exhibit significant regolith development. Since regolith thickness has been positively correlated with groundwater resources, it is suggested that fractures with orientations sub-parallel to the principal compressive stress direction constitute favourable groundwater targets. Knowledge of the regional stress field and fracture set orientations can be used as an effective low cost tool for locating potentially higher yielding boreholes in crystalline rock terrains.     

Quartz [0001]-axes preferred orientation, Bluff, New Zealand: Origin elucidated by grain-size measurements

Permian volcanic sediments at Bluff have been strained and thermally metamorphosed by Permian intrusives to metasediments of hornblende—hornfels facies. Quartz, which crystallised as a secondary mineral during metamorphism, has an unusual preferred orientation with c-axes either forming paired maxima in the plane containing the lineation (=maximum principal strain axis = direction of extension) and the perpendicular to schistosity (=minimum principal strain axis = shortening direction) or a broad maximum parallel to the lineation; the paired maxima are approximately 30° either side of the lineation. Some quartz grains are markedly elongate parallel to the lineation, and according to hypotheses of preferred orientation involving crystal plasticity, there should be some correlation between the shape of such grains and their c-axis orientations. Grain-size and shape analysis of Bluff quartz demonstrate that no such correlation exists; the analyses show that the preferred orientation results from oriented nucleation in the residual stress field immediately following the bulk straining of the rocks, with the distribution of c-axes as predicted by Kamb's hypothesis (1959). The time relationships of rock deformation, thermal metamorphism, and nucleation and growth of quartz are discussed.     

Sheath fold morphology in simple shear

Sheath folds are highly non-cylindrical structures often associated with shear zones. We investigate the formation of sheath folds around a weak inclusion acting as a slip surface in simple shear by means of an analytical model. We present results for different slip surface orientations and shapes. Cross-sections perpendicular to the shear direction through the sheath fold display closed contours, so called eye-structures. The aspect ratio of the outermost closed contour is strongly dependent on the initial slip surface configuration. The center of the eye-structure is subject to change in height with respect to the upper edge of the outermost closed contour for different cross-sections perpendicular to the shear direction. This results in a large variability in layer thickness across the sheath fold length, questioning the usefulness of eye-structures as shear sense indicators. The location of the center of the eye structure is largely invariant to the initial configurations of the slip surface as well as to strain. The values of the aspect ratios of the closed contours within the eye-pattern are dependent on the strain and the cross-section location. The ratio (R′) of the aspect ratios of the outermost closed contour (R_yz) and the innermost closed contour (R_y′z′) shows values above and below 1. R′ shows dependence on the slip surface shape and orientation but not on the number of involved contours. Using R′ measurements to deduce the bulk strain type may be erroneous.     

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.     

Grain misorientations in partially molten olivine aggregates: an electron backscatter diffraction study

In polycrystalline aggregates of olivine with mean grain sizes above 35 μm plus a low basaltic melt fraction, both wetted and melt-free grain boundaries are observed after equilibration times at high pressures and temperatures of between 15 and 25 days. In order to assess a possible dependence of the wetting behaviour on the relative orientation of neighbouring grains, a SEM based technique, electron backscatter diffraction (EBSD), is used to determine grain orientations. From the grain orientations relative orientations of neighbouring grains are calculated, which are expressed as misorientation axis/angle pairs. The distribution of misorientation angles and axes of melt-free grain boundaries differ significantly from a purely random distribution, whereas those of wetted grain boundaries are statistically indistinguishable from the random distribution. The relative orientation of two neighbouring grains therefore influences the character of their common grain boundary. However, no clustering towards special (coincident site lattice) misorientation axes is observed, with the inference that the energy differences between special and general misorientations are too small to lead to the development of preferred misorientations during grain growth. Received: 8 December 1997 / Revised, accepted: 6 April 1998     

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.     

Strain distribution in superposed buckling folds and the problem of reorientation of early lineations

Two series of experiments were carried out with soft model-materials in order to assess the relative importance of initial homogeneous strain, external rotation and late-stage strain in reorienting early lineations during superposed buckle-folding. In the first series cylindrical buckling folds were produced in embedded planar sheets containing a “lineation”. In the second series noncylindrical folds were produced by compression of a set of cylindrical folds. The experiments indicate that the ratio of buckle shortening to layer-parallel strain is much smaller when the principal extension is parallel to the fold-axis than in the case when the principal extension is perpendicular to the fold-axis. In very competent rocks, the reorientation of old lineations is mainly by external rotation and by the associated concentric longitudinal strain. In moderately competent rocks, the orientation of early lineations always changes by initial homogeneous strain before buckling becomes significant. Because of the unlike amounts of initial strain in layers of different competences, orientations of unrolled lineations may not be parallel in disharmonically folded layers of unlike competences. Under certain conditions the early lineation may become virtually parallel to the later fold-axis. The experiments indicate that the effects of late-stage strain in buckle-folding are largely restricted to the incompetent layers of a multilayer. Hence, if orientation data of early lineations in both competent and incompetent rocks are lumped together, the pattern of orientation may become quite complex. Even for a single competent layer, the pattern of early lineations can locally become complex because of the complex nature of concentric longitudinal strain (and strain resulting from stretching of middle surface of the layer), development of conical folds, development of shear strain along hinge zones of deformed early folds and also because of the development of different orders of folds in both the first and the second deformations.     

Using fault displacement and slip tendency to estimate stress states

We suggest that faults in high slip tendency orientations tend to develop larger displacements than other faults. Consequently, faults that accumulate larger displacements are more likely to be reliable indicators of the longer term stress field and should be weighted accordingly in paleostress estimation. Application of a stress inversion technique that uses slip tendency analyses and fault displacements to interpret populations of coherent normal faults within the Balcones Fault System of south-central Texas provides stress estimates that are consistent with established regional stress analyses. Although the method does not require measurement of slip directions, these data, where available, and sensitivity analyses of the angular mismatch between measured slip directions and those predicted by inverted stress states provide high confidence in the stress estimates generated using slip tendency analyses. Close inspection of the fault orientation and displacement data further indicates that subpopulations of faults with orientations different from the regional pattern have formed in response to stress perturbations generated by displacement gradients on an adjacent seismic scale fault.     

Measurement and correlation of microstructures: the case of foliation intersection axes

Recent studies have used the relative rotation axis of sigmoidal and spiral‐shaped inclusion trails, known as Foliation Inflexion/Intersection Axis (FIA), to investigate geological processes such as fold mechanisms and porphyroblast growth. The geological usefulness of this method depends upon the accurate measurement of FIA orientations and correct correlation of temporally related FIAs. This paper uses new data from the Canton Schist to assess the variation in FIA orientations within and between samples, and evaluates criteria for correlating FIAs. For the first time, an entire data set of FIA measurements is published, and data are presented in a way that reflects the variation in FIA orientations within individual samples and provides an indication of the reliability of the data. Analysis of 61 FIA trends determined from the Canton Schist indicate a minimum intrasample range in FIA orientations of 30°. Three competing models are presented for correlation of these FIAs, and each of the models employ different correlation criteria. Correlation of FIAs in Model 1 is based on relative timing and textural criteria, while Model 2 uses relative timing, orientation and patterns of changing FIA orientations, and Model 3 uses relative timing and FIA orientation as correlation criteria. Importantly, the three models differ in the spread of FIA orientations within individual sets, and the number of sets distinguished in the data. Relative timing is the most reliable criterion for correlation, followed by textural criteria and patterns of changing FIA orientations from core to rim of porphyroblasts. It is proposed that within a set of temporally related FIAs, the typical spread of orientations involves clustering of data in a 60° range, but outliers occur at other orientations including near‐normal to the peak distribution. Consequently, in populations of FIA data that contain a wide range of orientations, correlation on the basis of orientation is unreliable in the absence of additional criteria. The results of this study suggest that FIAs are best used as semiquantitative indicators of bulk trends rather than an exact measurement for the purpose of quantitative analyses.     

A note on sand dyke orientations

The techniques of the statistics of directional data are applied to the orientations of sand dykes and bedding in strata unaffected by significant penetrative tectonic strain or compaction. From the records of the orientations of both bedding and sand dykes in 578 individual cases it is shown that the mean bedding-dyke dihedral angle for each group of dykes recorded in the literature ranged from 87.7 to 89.6°. The hypothesis is accepted that these dykes tended to form perpendicular to bedding. This has beneficial implications for their use as strain markers in compaction studies and in tectonic strain analysis.     

基于有限元法的层状岩体破裂规律探讨

层状岩体的非均质性及各向异性导致其破裂方式及规律与均质岩体有显著不同。对层状岩体分别进行不同方式的单轴、双轴、三轴试验, 分析应力-应变曲线特征; 再利用ANSYS有限元软件进行数值模拟, 观察应力、应变在岩体上的分布, 通过曲线和图件的对比分析, 并结合岩石破裂理论, 总结不同应力状态下层状岩体的破裂方式、顺序及规律; 最后以富台地区为例, 对分析结果进行验证。研究结果表明, 不同受力方式对层状岩体破裂的影响体现在施加的载荷及约束与层面的方位。当应力方向与岩层面平行时, 强度大的石灰岩岩体发生集中应力, 首先破裂; 而应力与岩层面垂直时, 强度小的泥岩岩体首先破裂。岩石试验、数值模拟结果以及实例均成功验证了这个规律。