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1.
Derek Flinn 《Geological Journal》1979,14(2):87-98
The mathematics of irrotational deformation are simplified by presentation in matrix form. Matrix equations are easily programmed and are easily interpreted in geometrical terms. Graphical operations commonly carried out on orientation nets such as rotation of data can be translated into simple matrix equations for use with a computer. If the shape and orientation of the deformation ellipsoid for a pure shear are known, a matrix can be constructed for use as a deformation matrix. This can be used to deform other deformation ellipsoids to obtain a resultant ellipsoid. It can also be used to deform geological structures such as lineations and planes. The matrix equations for these operations are very simple, but their numerical solution often requires a computer. 相似文献
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Juan Ignacio Soto 《Mathematical Geology》1997,29(1):93-130
A deformation that is obtained by any simultaneous combination of two steady-state progressive deformations: simple shearing
and a coaxial progressive deformation, involving or not a volume change, can be expressed by a single transformation, or deformation
matrix. In the general situation of simple shearing in a direction non-orthogonal with the principal strains of the coaxial
progressive deformation, this deformation matrix is a function of the strain components and the orientation of shearing. In
this example, two coordinate systems are defined: one for the coaxial progressive deformation (xi system), where the principal and intermediate strains are two horizontal coordinate axes, and another for the simple shear
(x
i
t’
system), with any orientation in space. For steady-state progressive deformations, from the direction cosines matrix that
defines the orientation of shear strains in the xi coordinate system, an asymmetric finite-deformation matrix is derived. From this deformation matrix, the orientation and
ellipticity of the strain ellipse, or the strain ellipsoid for three-dimensional deformations, can be determined. This deformation
matrix also can be described as a combination of a rigid-body rotation and a stretching represented by a general coaxial progressive
deformation. The kinematic vorticity number (W
k is derived for the general deformation matrix to characterize the non-coaxiality of the three-dimensional deformation. An
application of the deformation matrix concept is given as an example, analyzing the changes in orientation and stretching
that variously-oriented passive linear markers undergo after a general two-dimensional deformation. The influence of the kinematic
vorticity number, the simple and pure shear strains, and the obliquity between the two deformation components, on the linear
marker distribution after deformation is discussed. 相似文献
4.
《Journal of Structural Geology》1988,10(3):273-281
Determination of the two-dimensional finite strain in turbidite bedding surfaces is important in both structural and sedimentological investigations of deformed sediments. The Meguma Group of Nova Scotia is one such sequence, in which the directional distribution of paleocurrent markers has been modified by strain. Previous strain determinations have used sand volcanoes as strain markers, which do not deform homogeneously with their matrix and, hence, may underestimate the true strain ratio. Intraclasts and concretions also suffer inhomogeneous deformation and are of limited use as strain markers. Hexagonal trace fossil networks can provide more reliable strain estimates but are rare. Widths of burrows exposed on bedding surfaces provide a new, straightforward means of measuring strain. Each burrow is represented by a pair of parallel lines. The separation of the lines is proportional to the burrow width, and their direction represents burrow orientation. The lines are tangential to the strain ellipse, which can be discerned despite initial scatter in burrow widths. 相似文献
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A method is presented for the determination of a triaxial ellipsoid (such as a strain ellipsoid)from three nonparallel plane sections of the ellipsoid. The sections need be neither orthogonal nor central sections of the ellipsoid. Measurement errors are used to adjust the observed plane ellipses so that they are exact sections of the nearest true ellipsoid, whose dimensions and orientation are then found by solution of a system of six linear equations. A solution of the inverse problem is also presented: given a triaxial ellipsoid with known orientation, to determine the shape and orientation of the ellipse on a plane section. The problem is solved by expanding the equation of an ellipsoid with rotated coordinates, then setting one dimension to zero. Also, a method is presented for the projection of a triaxial ellipsoid onto a plane surface. This is solved by taking the derivative of the ellipsoid equation in the direction of the normal to the plane surface. 相似文献
7.
Transpression 总被引：2，自引：0，他引：2
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. 相似文献
8.
构造出一种用于参数曲线自由变形的分段多项式函数,称之为缩放因子,将该因子作用于待变形曲线,从而使曲线发生形变。通过交互改变控制参数,可这到预期的变形效果。实验表明,这种方法数学背景简单,易于控制,操作方便,重复使用可获得丰富的变形效果,适用于几何造型、计算机动画等领域。 相似文献
9.
Deformation partitioning and porphyroblast rotation in meta-morphic rocks: a radical reinterpretation 总被引：1，自引：0，他引：1
T. H. BELL 《Journal of Metamorphic Geology》1985,3(2):109-118
Abstract Most porphyroblasts never rotate during ductile deformation, provided they do not internally deform during subsequent events, with the exception of relatively uncommon but spectacular examples of spiralling garnets. Instead, the surrounding foliation rotates and reactivates due to partitioning of the deformation around the porphyroblast. Consequently, porphyroblasts commonly preserve the orientation of early foliations and stretching lineations within strain shadows or inclusion trails, even where these structures have been rotated or obliterated in the matrix due to subsequent deformation. These relationships can be readily used to help develop an understanding of the processes of foliation development and they demonstrate the prominent role of reactivation of old foliations during subsequent deformation. They can also be used to determine the deformation history, as porphyroblasts only rotate when the deformation cannot partition and involves progressive shearing with no combined bulk shortening component. 相似文献
10.
The eigenvalues and eigenvectors of a direction cosine dispersion matrix may be used to characterise sets of orientation measurements in either two or three dimensions. General equations are presented that relate ratios of the eigenvalues of such a measured set to the finite strain ratios of the set assuming that the original orientation distribution, before a deformation, was uniform. The measured linear features in the rock are assumed to have behaved passively during a homogeneous deformation. The application of these relationships to strain estimation is discussed and tables for the three-dimensional case are given. 相似文献
11.
Mostafa E. Mostafa 《Tectonophysics》1984,110(3-4):317-331
A method is presented for the determination of the shape and orientation of the ellipse on a central plane section across a triaxial ellipsoid (such as a strain ellipsoid) of known orientation and magnitude. Conditional extremal values of longitudinal strain (λ) are obtained in the plane section and these are the principal axes of the strain ellipse. In solving this variational problem, the arbitrary cosntants (two undetermined Lagrange multipliers) are found to have physical meaning in themselves as proved in a complementary geometric solution. The two roots of the first constant directly determine the magnitude of the principal axes of the strain ellipse in terms of the normal to the cross section. The two arbitrary constants together define shear strains and orientation of these axes.
Strain computation uses simple, single-line equations in either frames of strained or unstrained states. 相似文献
12.
The deformation matrix and the deformation ellipsoid 总被引：1，自引：0，他引：1
Derek Flinn 《Journal of Structural Geology》1979,1(4):299-307
Homogeneous strain can be computed most easily by the methods of matrix algebra. Lines, planes and ellipsoids represented in matrix form can be homogeneously deformed by simple matrix multiplication by linear transformation matrices, the elements of which are the coefficients of the transformation equations. Deformation matrices or linear transformation matrices which cause geological-type homogeneous strain are divided into four classes based on the presence or absence of symmetry and/or orthogonality. The nature of the homogeneous strain caused by each class of deformation matrix is examined. Orthogonal-symmetrical and orthogonal matrices cause rotation. Symmetrical matrices cause irrotational strain with co-axial strain as a special case. Matrices which are neither orthogonal nor symmetrical cause many different types of rotational strain, some of which are examined. 相似文献
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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。 相似文献
15.
The purpose of this paper is to collect, clarify, augment and modify the authors' previous work on the subject of finite strain compatibility. The derivations of the fundamental equations are reviewed so that the geometric meaning of each step can be explained. Besides providing a basis for the geological interpretations of the equations, these derivations also lead to a useful new form of the strain compatibility equations.We begin by showing that compatibility is a geometric property of continuous and smooth coordinate grids, and we derive and explain the coordinate grid compatibility equations. We then use the fact that every finite deformation may be described by two coordinate grids to derive finite strain compatibility equations in principal coordinates and Cartesian coordinates. The resulting strain compatibility equations are not easily solved for general strain fields in any coordinate system. Nonetheless, we show that many common geological strain patterns have simple geometries for which the compatibility equations can be interpreted. For example, if a deformation has constant strain in one direction, as most shear zones do, then compatibility provides an iterative method for determining the strain throughout the deformed region if the strain is initially known at any one point. Some of the other strain geometries to which we apply compatibility in this paper include simple shear, inhomogeneous pure shear, parallel and similar folding. 相似文献
16.
A method of analysing finite strain from an initial random distribution of lines or planes is presented. The shape of the strain ellipsoid is mapped to the eigenvalue ratios of an orientation matrix. The method is used to interpret the andalusite fabrics from the aureole of the Ardara granite, County Donegal, Ireland. The high flattening strains determined in the aureole are accompanied by oblate strains in the granite margins and support the concept of deformation produced by expansion of the pluton. 相似文献
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岩石磁化率椭球体的三个轴与应变椭球体的三个轴方向相平行,并具有一定的共构关系.变形岩石的磁组构参数P、T、F、L以及磁化率椭球体主张量方向等可以用来定量地表征岩石变形的性状及期次.本文通过实测和计算27个样品的磁组构参数,研究了博白-合浦断裂带的变形性状与期次,结果表明:博白-合浦断裂带大致经历了三期构造变形作用,不同时期具不同性质的构造变形.变形性状分别表现为韧性、韧-脆性及脆性变形,应变行为分别表现为平面、非平面和线性应变. 相似文献
18.
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。 相似文献
19.
Soil–structure interaction problems are commonly encountered in engineering practice, and the resulting linear systems of equations are difficult to solve due to the significant material stiffness contrast. In this study, a novel partitioned block preconditioner in conjunction with the Krylov subspace iterative method symmetric quasiminimal residual is proposed to solve such linear equations. The performance of these investigated preconditioners is evaluated and compared on both the CPU architecture and the hybrid CPU–graphics processing units (GPU) computing environment. On the hybrid CPU–GPU computing platform, the capability of GPU in parallel implementation and high-intensity floating point operations is exploited to accelerate the iterative solutions, and particular attention is paid to the matrix–vector multiplications involved in the iterative process. Based on a pile-group foundation example and a tunneling example, numerical results show that the partitioned block preconditioners investigated are very efficient for the soil–structure interaction problems. However, their comparative performances may apparently depend on the computer architecture. When the CPU computer architecture is used, the novel partitioned block symmetric successive over-relaxation preconditioner appears to be the most efficient, but when the hybrid CPU–GPU computer architecture is adopted, it is shown that the inexact block diagonal preconditioners embedded with simple diagonal approximation to the soil block outperform the others. 相似文献
20.
The simulation of fabric development during plastic deformation and its application to quartzite: the influence of deformation history 总被引：1，自引：0，他引：1
The effect of deformation history on the development of crystallographic preferred orientation in quartzities has been simulated using a computer program based on the Taylor-Bishop-Hill analysis. Model quartzities with different combinations of glide systems have been subjected to various coaxial and non-coaxial deformation histories. It is possible to obtain information from the fabrics that develop during simple histories; for example, the location of the axis of extension is generally associated with a pole free area on a c-axis plot, and progressive axial shortening, plane strain and axial shortening produce characteristic fabrics. In progressive simple shear the fabric skeleton becomes asymmetric relative to the sense of shear and a-axes preferentially align in the flow plane parallel to the flow direction. However, this example illustrates that the fabric orientation and characteristics are controlled by the kinematic framework and bear only an indirect relationship to the finite strain accumulated to that point in the history.The imprint of the closing stages of deformation limits to some degree the use of crystallographic fabrics as a tool for structural geologists, but in favourable circumstances data can be obtained concerning characteristics of the deformation history, on the scale of the hand-specimen, for the last part of this history. 相似文献