Normalized center-to-center strain analysis of packed aggregates

The resolution of conventional techniques of center-to-center strain analysis is limited by the degree of original anticlustering of centers on the analyzed plane. However, the three-dimensional anticlustering of packed objects does not result in equivalent anticlustering on two-dimensional planes through these aggregates. Size variations due to imperfect sorting further decrease the anticlustering of natural aggregates. For the Fry all-object-object separations method, these problems are manifested in vague point-density distributions and ambiguously defined strain ellipses.Normalization of center-to-center distances allows more precise determination of small initial and tectonic anisotropies in packed aggregates. On planes through packed aggregates, object spacing is a function of object size, shape and the distance between object margins. Dividing the center-to-center distance between two objects by the sum of their average radii eliminates variations due to object size and sorting. Analyses of synthetic aggregates of packed spheres and statically recrystallized iron show that normalized Fry diagrams form better-defined vacancy fields and sharper rims of maximum point density regardless of the original sorting and anticlustering in the aggregate. Normalized strain analyses of deformed aggregates also show greatly increased resolution, with variable initial and tectonic ellipticity resulting in a wider ring of high point-density.     

Rapid extraction of central vacancy by image-analysis of Fry plots

The Fry method, based on the relative movement of different material points, typically grain centers, with reference to each other graphically yields a point distribution that displays the finite strain ellipse as a central vacancy. The diffused nature of the central vacancy induces subjectivity in strain estimation, particularly, if the point population when undeformed lacked an isotropic anticlustered distribution. Most existing methods use analytical and/or iterative approaches for improving the sharpness of the central vacancy and positioning the best-fit strain ellipse in a Fry plot. We provide an image-analysis method that is independent of any iteration or analytical solution. It is also an efficient technique for extraction of the central vacancy without any subjectivity. The method is more direct, simple and easy-to-use than most existing techniques.The image-analysis method uses Gaussian blur filter for distinction between the areas of largest and smallest pixel intensities in a Fry plot image. It then applies the optimal threshold value and an inversion filter for extraction of the sharp central vacancy. The method also searches for the best-fit strain ellipse through the extracted central vacancy and displays axial ratio and orientation of the ellipse in a separate window. The validity of the method is tested using several computer-simulated and natural examples.     

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.     

Strain analysis from objects with a random distribution: A generalized center-to-center method

Existing methods of strain analysis such as the center-to-center method and the Fry method estimate strain from the spatial relationship between point objects in the deformed state. They assume a truncated Poisson distribution of point objects in the pre-deformed state. Significant deviations occur in nature and diffuse the central vacancy in a Fry plot, limiting the its effectiveness as a strain gauge. Therefore, a generalized center-to-center method is proposed to deal with point objects with the more general Poisson distribution, where the method outcomes do not depend on an analysis of a graphical central vacancy. This new method relies upon the probability mass function for the Poisson distribution, and adopts the maximum likelihood function method to solve for strain. The feasibility of the method is demonstrated by applying it to artificial data sets generated for known strains. Further analysis of these sets by use of the bootstrap method shows that the accuracy of the strain estimate has a strong tendency to increase either with point number or with the inclusion of more pre-deformation nearest neighbors. A poorly sorted, well packed, deformed conglomerate is analyzed, yielding strain estimate similar to the vector mean of the major axis directions of pebbles and the harmonic mean of their axial ratios from a shape-based strain determination method. These outcomes support the applicability of the new method to the analysis of deformed rocks with appropriate strain markers.     

Influence of object concentration on finite strain and effective viscosity contrast: insights from naturally deformed packstones

Deformed conglomerates and ooidal/oncoidal packstones are commonly used to evaluate finite strain in deformed sedimentary successions. In order to obtain a correct estimate of finite strain, it is necessary to consider not only the different behaviour of matrix and objects, but also object concentration. The analysis of two-component rocks characterised by high values of packing commonly results in a substantial underestimate of bulk strain and of viscosity contrast between objects and matrix. In this study, the effects of the volumetric fraction of competent inclusions on both object and bulk measured finite strain, as well as on apparent viscosity contrast, have been investigated in naturally deformed packstones characterised by variable object concentration on the scale of the hand specimen (and hence for homogenous viscosity contrast). Object finite strain has been obtained by Rf/ analysis, whereas the Fry method provides a measure of whole-rock strain that is also a function of inclusion concentration. Therefore, the finite strain measured by the Fry method is better termed effective bulk strain. In order to investigate the role of object concentration, this parameter has been plotted against object and effective bulk strain, and also against viscosity contrast. These diagrams show that: (i) for high values of packing, measured object and effective bulk strain show values that are significantly lower with respect to the calculated maximum value (that would result in the ideal case of no particle interaction and represents therefore the real bulk strain of the samples); (ii) the viscosity contrast shows lower values with respect to the calculated maximum one (that is equal for the three principal sections of the finite strain ellipsoid), and as packing reaches the maximum value, the viscosity contrast approaches a unit value. Empirical equations have also been found that link object concentration with both object and effective bulk finite strain.     

A PACKING GENERATION SCHEME FOR THE GRANULAR ASSEMBLIES WITH PLANAR ELLIPTICAL PARTICLES

In this paper, a new generator algorithm and a computer program PG2D is introduced for 2D numerical simulation of packing configuration in a granular material composed of elliptical particles of different a/b aspect ratios. Each elliptical particle is approximated by four connected arcs. The centre co-ordinates and radius of each arc and co-ordinates of connecting points can be determined from the formulae derived by entering the major axis length, 2a, and the eccentricity. The domain to be filled with particles can be a polygon of any shape. Given the size of the packing domain, geometrical information and numbers of particles to be generated, the packing location of each particle and the co-ordinates of contact points along with contact normal rose diagram can be generated as outputs. Simulation results show that this new algorithm can provide quite a reasonable packing model in accordance with the initial input required for the analysis of the mechanics of granular material. This generation scheme has the potential to cover packing generation and behaviour analysis of 3D sphere or ellipsoidal shaped granular materials. © 1997 by John Wiley & Sons, Ltd.     

Measuring fracture orientation at exposed rock faces by using a non-reflector total station

Measurements of fracture orientation are usually taken by using a compass-inclinometer device on exposed rock faces. The drawbacks when using this method is that it is time-consuming if many fractures are measured and that measurement might be impossible if the rock face cannot be safely reached physically. To improve field mapping of rock fractures, a method for applying a non-reflector total station to measuring fracture orientation is presented in this paper. A non-reflector total station is a geodetic device that captures three-dimensional co-ordinates of target points without using a reflector. Therefore, physical touching the rock surfaces is no longer required. To determine a fracture orientation, co-ordinates of a set of points on the exposed fracture surface are captured at a distance from the rock face. The best-fit plane of the exposed fracture surface is defined by the co-ordinates of the target points, and the orientation (e.g. dip angle and dip direction) of the fracture surface is determined as that of its best-fit plane. This paper presents the technical procedure and a portable system designed for the field mapping of fracture orientation. Results of a case study performed at an exposed rock face are also included.     

在CorelDRAW平台上进行Fry法有限应变测量的新技术

有限应变测量是韧性变形定量估算的主要手段。Fry法有限应变测量以变形颗粒,如变形砾石、鲕粒、近等轴矿物等为应变标志体,地质实践中较容易获得,是岩石有限应变测量的最普遍手段。其传统测量过程极为繁琐,难以满足大范围应变测量的要求,数据精度也较低,限制了Fry法的应用。前人曾根据Fry法原理设计了计算机程序,但其程序相关编程语言如今已基本不能使用。本研究基于目前地质上较普遍使用的制图软件CorelDRAW平台的VBA编程,设计出一个基于CorelDRAW平台的Fry法应变测量宏程序FRY-1。在新图层上将所有变形颗粒的中心点标出后,运行FRY-1宏程序即可快速得到精确的Fry图解,并计算出相对精确的R值。通过实例与“最邻近心对心法”的测量结果进行的对比,证明该方法简便易操作,并可大幅提高有限应变测量的效率和有效减小误差。     

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.     

Elastic shear stiffness anisotropy and fabric anisotropy of natural clays

Natural clays usually show anisotropic stiffness due to their deposition process and anisotropic in situ stress state. The stiffness anisotropy depends on both of the stress anisotropy and fabric anisotropy, while the latter can be quantified by the stiffness anisotropy at isotropic stress states. This paper measures the K₀ value (i.e., stress anisotropy) and elastic shear stiffness anisotropy of natural Shanghai clay in a triaxial apparatus with horizontal and vertical bender elements. The results show that the K₀ value of Shanghai clay lies in the range of 0.40–0.66, and an empirical equation is proposed to estimate the K₀ value based on the plasticity index and initial void ratio. The fabric anisotropy of natural Shanghai clay lies in the range of 1.2–1.4 with a stronger fabric in the horizontal plane. Moreover, the experimental data of the stiffness anisotropy and fabric anisotropy of different clays in the literature are reviewed and analyzed. It reveals that the stiffness anisotropy generally increases, while the fabric anisotropy remains nearly the same during K₀ consolidation. For normally consolidated clay, the fabric anisotropy generally lies in the range of 1.1–1.7. For overconsolidated clays, the fabric anisotropy generally increases as the overconsolidation ratio increases. Empirical equations are proposed to approximately estimate the fabric anisotropy of clays based on its stress normalized elastic shear stiffness.

     

Fitting the void: Data boundaries,point distributions and strain analysis

An objective method is developed for identifying the shape of the void generated by analysis of the spatial arrangement of point distributions using the Fry method and in particular Delaunay triangulation nearest neighbour data. The method works by numerically minimising a weighted non-linear least-squares formulation which tends to selectively apply higher weights to points below the boundary and less weight to points above the boundary. Sampling errors are estimated using a bootstrap procedure. A simulation study indicates that the method works best for closely packed object arrangements and tends to underestimate the strain axial ratio at high imposed strains. Overall the method is found to be objective and consistent.     

A packing generation scheme for the granular assemblies with 3D ellipsoidal particles

This paper introduces a new generator algorithm and computer program for 3-D numerical simulation of packing configuration in a granular assemblies composed of ellipsoidal particles of different a/b aspect ratios. Each ellipsoidal particle is approximated by the revolution of an ellipse, formed by four connected arcs, about the major axis passing through its centroid. The centroid co-ordinates, major axis direction and lengths of the major and minor axes are the essential data for the packing generation and associated contact detection. The domain to be filled with particles can be a polyhedron of any shape. The packing program was coded based on a newly proposed scheme which obeys the no interpenetration kinematics of solid bodies. New contact detection algorithms for any two ellipsoids in the packing space were developed. Though simple, these algorithms effectively determine the contact condition and contact point without solving the simultaneous equations of the two ellipsoidal surfaces. Each particle's packing location, contact-point co-ordinates, and three-dimensional graphs can be created using the packing domain given boundaries, along with numbers, and geometrical information of particles to be generated. Simulation results show that this new algorithm provides an effective packing model as a required initial input for analysing the mechanics of granular material. This generation scheme potentially can explore the complex 3-D behaviours of material composed of discrete particles. Copyright © 1999 John Wiley & Sons, Ltd.     

3D DEM simulation of principal stress rotation in different planes of cross-anisotropic granular materials

Three-dimensional Discrete Element Method simulations have been performed to study the deformation of cross-anisotropic granular materials under principal stress rotation (PSR), for rotation planes oriented at different angles θ with respect to the bedding plane. The simulations have been conducted with a novel technique for applying specified stresses at three-dimensional boundaries. The results are qualitatively in agreement with experimental results from literature. Cumulative volume contraction is always observed under continuous PSR and increases with increasing θ. The dilatancy rate decreases with increasing number of PSR cycles, tending to zero. The noncoaxiality angle between the strain increment and the stress in the PSR plane increases with increasing number of cycles, reaching the same asymptotic value for samples of various densities and for various θ. Periodic oscillations of the dilatancy rate and noncoaxiality angle within each PSR cycle are observed with an increasing oscillation magnitude with increasing θ, due to the larger fabric anisotropy within the PSR plane. When θ = 30 or 60°, significant noncoaxial strain accumulation occurs in the plane perpendicular to the PSR plane due to the oblique angle between the PSR plane and the bedding plane, echoing the major principal fabric direction's being neither parallel nor perpendicular to the PSR plane. The macroscopic behavior of the samples is related to the microscopic parameters including coordination number and fabric anisotropy. With increasing number of cycles, the difference between normalized stress/strain/fabric increment tensors tends to become constant, with only a small lag between each pair, irrespective of θ.     

岩石有限应变测量反向轮法的计算机CSD软件设计

用计算机实现反向轮法测岩石有限应变的过程, 可提高有限应变测量的准确性和效率.开发出了用Visual Basic 5.0编写的CSD软件, 利用岩石薄片图像或者矿物颗粒轮廓图测出岩石有限应变的大小及应变椭圆长轴方向, 同时测出矿物颗粒分布的优选方位.其操作相当简便, 首先统计出图像中0°~180°各方向上矿物颗粒的边界数目; 然后在方位-边界坐标系中投点, 并利用最小二乘法进行数据点的多项式曲线拟合; 最后求出曲线的极值点坐标, 并根据坐标绘制相应的应变椭圆.软件运行中, 统计出的边界数据和拟合曲线以及应变椭圆图形都是可视的, 并能进行相应的保存.      

Constitutive modeling of inherent anisotropy in a strain space multiple mechanism model for granular materials

Consideration of fabric anisotropy is crucial to gaining an improved understanding of the behavior of granular materials. This paper presents a constitutive model to describe the sand behavior associated with fabric anisotropy within a framework of a strain space multiple mechanism model. In the proposed model, a second-order fabric tensor is extended by incorporating a new function that represents the effect of inherent (or initial fabric) anisotropy, along with three additional parameters: two of them, a₁ and a₂ , control the degree of anisotropy, and the second mode of inherent anisotropy can be expressed by introducing the parameter a₂ as well as the first mode by the parameter a₁ . The third parameter, θ₀ , expresses the principal direction of inherent anisotropy (eg, the normal vector direction of bedding planes relative to horizontal axis). The formulation of the dilative component of dilatancy (ie, positive dilatancy) is also extended to consider the effect of inherent anisotropy based on the interlocking mechanism. Experimental data on the complex anisotropic responses of Fraser River sand and Toyoura sand under monotonic loading is used to validate this model. The proposed model is shown to successfully capture anisotropic responses, which become contractive or dilative depending on different principal-stress directions, with a single set of anisotropy parameters; thus, the model is considered to possess the capability to simulate the anisotropic behaviors of granular materials. In addition to different loadings on the same fabric, the effects of different fabric anisotropies upon the sand behavior under the same loadings are also investigated.     

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.     

惯量投影椭球在构造变形分析中的意义初探

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

Some limitations of the Rf/φ technique of strain analysis

Finite strain determined from R_f/φ analysis of deformed ellipse distributions cannot be separated into compactional and tectonic parts on the basis of population asymmetry about the orientational mean, as was originally suggested. Furthermore, where there is a ductility contrast between objects and matrix, R_f/φ analysis yields an invalid estimate of the strain suffered by the objects.     

Strain analysis and microstructural evolution characteristic of neoproterozoic rocks associations of Wadi El Falek,centre Eastern Desert,Egypt

The estimation of finite strain in rocks is fundamental to a meaningful understanding of deformational processes and products on all scales from microscopic fabric development to regional structural analyses. The R_f/φ and Fry methods on feldspar porphyroclasts and mafic grains from 5 granite, 1 metavolcanic, 3 metasedimentary and 1 granodiorite samples were used in Wadi El Falek region. Finite-strain data shows that a high to moderate range of deformation of the granitic to metavolcano-sedimentary samples and axial ratios in the XZ section range from 1.60 to 4.10 for the R_f/φ method and from 2.80 to 4.90 for the Fry method. Furthermore, the short axes are subvertical associated with a subhorizontal foliation. We conclude that finite strain in the deformed granite rocks is of the same order of magnitude as that from metavolcano-sedimentary rocks. Furthermore, contacts formed during intrusion of plutons with some faults in the Wadi El Falek area under brittle to semi-ductile deformation conditions. In this case, finite strain accumulated during superimposed deformation on the already assembled nappe structure. It indicates that the nappe contacts formed during the accumulation of finite strain.     

The influence of matrix rheology and vorticity on fabric development of populations of rigid objects during plane strain deformation

The influence of vorticity and rheology of matrix material on the development of shape-preferred orientation (SPO) of populations of rigid objects was experimentally studied. Experiments in plane strain monoclinic flow were performed to model the fabric development of two populations of rectangular rigid objects with object aspect ratios (R_ob) 2 and 3. The density of the rigid object populations was 14% of the total area. Objects were dispersed in a Newtonian and a non-Newtonian, power law matrix material with a power law exponent n of 1.2. The kinematic vorticity number (W_n) of the plane strain monoclinic flow was 1, 0.8 and 0.6 with finite simple shear strain of 4.6, 3.0 and 0.9, respectively. In experiments with R_ob=3, the SPO is strongly influenced by W_n and the material properties of the matrix. Deformation of a power law matrix material and low W_n resulted in a stronger SPO than deformation of a linear viscous matrix and high W_n. Strain localization coupled with particle interaction plays a significant role in the development of a shape-preferred orientation. High strain simple shear zones separate trains of rigid objects that are surrounded by low strain zones with W_n lower than the bulk W_n. In fabrics involving populations of objects with R_ob=2, rheology of the matrix materials does not systematically influence the intensity of the SPO.