Strain analysis using deformed line distributions

This paper describes a new technique of strain analysis using passively deformed linear markers which were initially uniformly or symmetrically distributed in a plane. The axial ratio of the strain ellipse is determined from the quartiles or other percentiles of the deformed distribution. The technique permits statistical testing of assumptions and rapid reconstruction of the undeformed state.     

Compatibility of two-dimensional strains and rotations along strain trajectories

A study of the compatibility of strains and rotations is necessary for a proper understanding of deformation patterns. Principles of compatibility also underlie all methods for removing the effects of deformation and finding the pretectonic shape of deformed regions. The object of this paper is to derive compatibility equations in a direct way and in a simple form which can be interpreted geometrically. This is done using sets of orthogonal curvilinear coordinates parallel to strain trajectories. As a result, all shear components vanish and only principal strains appear in the equations. Such advantages are obtained at the expense of the extra complexity of curvilinear coordinates compared with Cartesian ones. Orthogonal curvilinear coordinates are described first, with particular reference to curvature. Compatibility of strains and rotations is then discussed by comparing curvatures of trajectories in the undeformed and deformed states. Results are compared with previous theoretical work and numerical methods.     

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.     

Density distribution techniques and strained length methods for determination of finite strains

For a homogeneously deformed rock composed initially of an isotropic distribution of object shapes, finite strain may be determined from the correlation between the orientations of either two-dimensional or one-dimensional sample cuts and the frequencies with which they intersect marker objects. Mimran previously published an incorrect method for planar samples under the heading ‘density distribution technique’. Methods are described by which the three-dimensional strain may be directly determined from six general samples, either linear or planar. Construction of two-dimensional ellipses as an intermediate step is unnecessary and enforces practical difficulties.These methods may be simplified by use of samples parallel to known principal axes or planes of the finite strain. In this case the same large errors may arise from slight misorientation of samples as with other methods of strain measurement. A new quick method is proposed, combining linear and planar measurements of frequencies of intersected objects, which is thought to be the first method to circumvent a large part of the error from this error source. For example, if true X:Z ratio is 9 : 1, and orientations in the XZ plane are misjudged by 8°, normal methods give 38% error where the new method gives, with care, an error of 1.9%. For methods of strain measurement such as are described here the concept of strain ellipsoid is unnecessarily limiting, and should be abandoned.     

用线状被动标志物向分布估测岩石总体应变的最优化方法

在一些假设下提出了利用线状被动标志物的方向数据估测岩石总体应变的最优化方法。该方法等同地考虑到每个测量数据对应变估计的贡献,从而能够较好地反映出岩石的总体应变。它具有较好的稳健和广泛的适用对象。实验表明,观察数据大约在80个或更少时估测的应变椭圆轴比的相对误差小于20％,超过160个观察数据一般小于5％－10％。     

A comparison of intracrystalline deformation in naturally and experimentally deformed quartzites

A relatively undeformed quartzite sample from the Weverton formation was experimentally deformed in plane strain at a temperature of 700° C, a confining pressure of 15 kb and a constant strain rate of 10⁻⁶/sec, in a modified Griggs apparatus. A comparison of the known experimental strain for the sample with that measured from deformed rutile needles within the quartz grains shows fairly close agreement between the two values. This confirms the validity of using the needles as intracrystalline strain markers. A comparison has been made of the microstructures and preferred orientations in the experimentally deformed sample and a naturally deformed sample of the same quartzite which has undergone the same strain. The experimentally deformed sample exhibits more inhomogeneous intragranular deformation and a “double funnel” pattern of c axes, while the naturally deformed sample exhibits more homogeneous intragranular deformation and a broad great circle girdle of c axes normal to the foliation and lineation.     

Deformation path concepts in strain analysis: Applications for study of ductile shear zones

The internal fabric of a deformed rock represents the state of finite strain. In some special cases the fabrics also record the strain history of the deformed body. These special cases can profitably be utilized to compare the predictions of dynamic models and strain paths in natural deformations. In this contribution, the concept of deformation path in the study of ductile shear zones has been demonstrated.     

利用惯量椭圆进行岩石有限应变分析

提出了利用惯量椭圆对任意形状矿物颗粒进行描述的方法,并利用惯量椭圆理论,计算了岩石薄片中任意形状矿物颗粒的惯量椭圆,通过颗粒面积对椭圆参数进行标准化,得到每一矿物颗粒的等效应变椭圆。等效应变椭圆能够有效地反映对应矿物颗粒的优选方位以及变形特征,进而利用椭圆的矩阵参数形式对等效应变椭圆进行统计分析,获得岩石的有限应变椭圆;同时给出了相应的数值计算方法,编制了软件Straindesk,并得到了成功的应用。该方法克服了先前应变测量中的局限性,方便实现计算机的自动分析,具有较强的有效性和广泛的适用性。     

Flow pattern within a Permian submarine slump recorded by oblique folds and deformed fossils, Ulladulla, south-eastern Australia

The flow pattern within a slump in Permian marine rocks of the southern Sydney Basin, Australia, is recorded by folds and deformed fossils. Abundant brachiopod and bryzoan fossils in the slumped rocks are relatively undeformed, but fossil crinoid stems have been deformed by relative rotation of individual ossicles. Measurement of the strain indicates that the deformation of the crinoids is consistent with flexural flow folding within the slump. Previous models assume that curved slump fold axes remain parallel to the enveloping bedding surface of a slump sheet. Detailed measurements of the orientation of slump folds in this study found fold axes to be oblique to bedding, which is interpreted as a result of folds plunging downward towards the flanks of the slump or slump lobes. In the present model, fold axes are not generally parallel to the strike of the fold axial surface, and this can explain differences between the orientations of slump fold axes and axial surfaces when these are used as directional indicators of slump movement.     

非线性几何场论在开采沉陷预测中的应用

以拖带坐标描述法和S－R分解定理的非线性几何场论为基础，分析了初始位形主断面上倾斜、曲率和水平变形计算公式的不合理性，建立了实时位形上的计算公式；利用平均整旋角概念和裂纹产生与扩张的几何准则，建立了确定实时位形上，水下采煤导水裂隙带高度的方法。与经典断裂判据相比，该方法计算简单，应用方便。     

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.     

假想变位法模拟构造应力场的理论与实践

把变形地质体作为弹塑性体,用增量理论计算变形地质体内应力和应变的分布是假想变位法的理论基础。用假想变位法模拟构造形成时变形地质体的物理边界条件,代替历来人为假设物理边界条件的方法,尽量减少人为因素,提高了模拟实验结果的可信性。 把这一方法应用到日本佐渡金矿控矿断裂形成机制的研究中,对控矿断裂成因的地质推断得到了理论验证。     

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.     

Quantification of microcrack anisotropy in quartzite — a comparison between experimentally undeformed and deformed samples

In this paper, microcrack patterns in a quartzite are quantified using fractal geometry based methods. Since the quartzite does not show a mesoscopic foliation, the fabric was recognized using anisotropy of magnetic susceptibility (AMS) analysis. Microcracks were investigated in thin sections prepared along the three principal planes of the AMS ellipsoid. Point load tests were performed on cores drilled parallel as well as perpendicular to the magnetic foliation. After experimental deformation, thin sections were prepared in two orientations — (a) parallel to the plane of failure (i.e., parallel to the direction of loading), (b) perpendicular to the plane of failure (i.e., perpendicular to the direction of loading), and microcrack patterns in these sections were investigated. The box-counting method of fractal analysis was first applied to microcracks traced from SEM images from each thin section of the experimentally undeformed as well as deformed samples to establish the fractal nature of the microcrack pattern. It was found that in thin sections perpendicular to the direction of loading, the box (fractal) dimension tends to marginally increase. This is inferred as a manifestation of the increase in complexity of the pattern. The software AMOCADO, which is based on the modified Cantor Dust method of fractal analysis, was applied to microcrack pattern from each thin section in order to quantify the pattern anisotropy. It is noted that the anisotropy significantly reduces in sections perpendicular to the loading direction. SEM data are presented to demonstrate that this reduction in anisotropy is on account of generation and/or growth of new cracks in random orientations. It is envisaged that the approach adopted in this investigation maybe useful in rock mechanics and mineral-resource applications in future.     

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.     

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.     

Finite strain and progressive deformation in models of diapiric structures

The distribution of strain within and around gravitationally produced diapiric structures was studied through the use of experimental models which were deformed in a large-capacity centrifuge. A new method of model construction was developed which is equivalent to building the model of initially square 1-mm elements. After deformation the elements assume shapes which approximate parallelograms and their finite strains can easily be calculated. If several initially identical models are deformed to different extents, the finite strain states of an element in each of the models define points on the deformation path of that element. The deformation path can be used to make estimates of the nature of the internal fabric which would be expected in the equivalent element of the natural structure.This method was applied to the study of the finite strain in diapiric ridges. The models demonstrate that the highest strain is always in the region above the diapir. Within the diapir initial vertical stretching is followed by vertical flattening. Large portions of the structure can be seen to suffer what would in natural examples be called polyphase deformation, even though all of the deformation was due to a single buoyant overturn of unstable density stratification. The strain patterns within the models support the contention that in salt diapirs the buoyant salt has a lower viscosity than the overlying sediments, but that in mantled gneiss domes the reverse is true.     

Directional properties of polygons and their application to finite strain estimation

An eigenmethod for characterising the directional properties of convex polygons is described. Given a polygon, regular in the undeformed state, its image following a homogeneous strain transformation is an irregular polygon whose eigensolution (an ellipse) has the axis ratio and orientation of the strain ellipse. The eigenmethod can therefore be used for finite strain estimation using polygonal strain markers, and is compared with alternative methods recently proposed by Roder (1977) and Sanderson (1977).