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.     

Determination of the strain ellipsoid from measurements on any three sections

An ellipsoid is defined by, and may be re-constructed from, any three sections through it. In the field, calculation of the strain ellipsoid from general sections (two-dimensional strain ellipses determined from measured strain markers) is complicated by the fact that, due to experimental error and/or strain inhomogeneity, the three ellipses may not come from the same ellipsoid. The ellipses must first be adjusted to make them compatible. A method is suggested by which an adjustment ellipse is determined analytically for each of the three sections. Application of these adjustment ellipses makes the three sections compatible, and the strain ellipsoid may be determined. The principal axes of the ellipsoid are derived from the ellipsoid matrix by eigenvector analysis. Examples are given of practical applications of this method.     

The pole of the Mohr diagram

The pole of a Mohr diagram, for the two-dimensional case, is a unique point on the Mohr circle which permits any point on the Mohr circle to be related to the direction in the physical plane associated with that point. A Mohr diagram can be constructed for any second rank tensor. To illustrate the simplicity of this geometrical construction two examples of the use of the pole are presented, one for the strain tensor and the other for the stress tensor.     

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.     

Analysis of triaxial ellipsoids: Their shapes, plane sections, and plane projections

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.     

Determination of fabric and strain ellipsoids from measured sectional ellipses—implementation and applications

Geologists examine fabrics to constrain models of formation or of deformation of rocks, and it is often convenient to summarise the results by a fabric ellipsoid. As fabric data are commonly collected on planar sections through the rock, estimating a fabric ellipsoid from sectional ellipses, often with arbitrary orientations, is an important task. An algebraic method to calculate such an ellipsoid, presented in an earlier paper, has been implemented with the program ellipsoid. It is used here on examples that illustrate questions and issues that arise when collecting, selecting and processing sectional fabric data, and when assessing the results. The quality of fit of the ellipsoid to the data is assessed in all cases. Examples include a case in which the average sizes of markers on individual sections can be used in the determination of the ellipsoid, and other cases in which such sizes are not useful; a case in which sectional ellipses are not obtained from closed markers; and a case in which data scatter and insufficient coverage of section orientations lead to a hyperboloid instead of an ellipsoid.     

Least squares best-fit circles (with applications to Mohr's diagram)

Procedures are outlined for the selection of a least squares best-fit circle to data points defined by rectangular Cartesian coordinates. Equations are derived to allow fitting of circles centered on the x-axis as well as off-axis Mohr circles. These procedures are applicable to the estimation of second-order tensors such as stress and strain by means of Mohr's diagram.     

Least squares best-fit circles (with applications to Mohr''s diagram)

Procedures are outlined for the selection of a least squares best-fit circle to data points defined by rectangular Cartesian coordinates. Equations are derived to allow fitting of circles centered on the x-axis as well as off-axis Mohr circles. These procedures are applicable to the estimation of second-order tensors such as stress and strain by means of Mohr's diagram.     

Application of strain analysis to estimate pressure solution processes in regional shear zones

This paper considers the basic principles of the strain analysis method based on the analysis of antitaxial regeneration fibrous fringes around linear rigid inclusions in a low-viscosity rock matrix. This method has been developed for pressure shadows composed of fibrous minerals, whose orientation is controlled by the major elongation direction rather than the orientation of rigid inclusions. This approach is applicable only for rocks exposed to uniform coaxial straining. The strain ellipse is calculated in two ways: for three variably oriented strain markers, it is calculated using Mohr’s circles, and for numerous strain markers by average body ellipse. The strain ellipsoid is calculated using the parameters of a few strain ellipses calculated with three and more non-parallel planes. This paper provides the data on the method testing in reference sites of Dora–Pil’ ore field in the Upper Indigirka district and Vangash area in the Yenisei Range. Regeneration fibrous fringes around fragments of fern fossils and linear rutile metacrystals were used as markers. The results of strain analysis obtained for the reference sites in the Upper Indigirka district made it possible to describe the signs of variable strain stages of developing strike-slip zones making up the Adycha–Taryn Fault Zone. Sublatitudinal ore-bearing strike-slip zones are characterized by a subvertical orientation of the elongation axes X of elongated strain ellipsoids, which are subperpendicular to quartz–carbonate veins and slope kink zones. NW-trending strike-slip zones are characterized by subhorizontal orientation of the Z shortening axes of flattened strain ellipsoids, which are subparallel to the normals of quartz–carbonate veins and veinlets. The results of strain analysis obtained for reference sites in the Vangash area made it possible to describe the thrust strain environment following the metamorphism stage and to reveal specific features in the formation of the strain textures of ore-bearing rocks based on their rheological properties.     

The sectional strain ellipse during progressive coaxial deformations

The two-dimensional strain history on a sheet which is inclined to the principal axes of the strain ellipsoid is considered. Even if the strain history in three dimensions is coaxial, the two-dimensional progressive deformation on the surface of the sheet is in general of a non-coaxial type. It is shown in this paper that the degree and sense of two-dimensional non-coaxiality is governed by the strain path followed during three-dimensional coaxial deformation. The general relationship is defined between the gradient of the strain path on the Flinn strain ellipsoid diagram and the nature of the two-dimensional strain increments. For most strain paths an asymmetrical arrangement of structures in the oblique sheet is to be expected. Hence, en échelon folds, transected folds and extension veins with curved fibres could be produced during three-dimensional coaxial deformation. Only if the strain path is of a rather special type will the deformation be coaxial in a two-dimensional as well as a three-dimensional sense.     

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.     

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.     

岩石磁组构在断裂变形性状与期次研究中的应用——以广西博白-合浦断裂为例

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.     

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.     

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.     

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.     

Rotation and strain of linear and planar structures in three-dimensional progressive deformation

Rotation and progressive strain have been studied for a sheet embedded in a matrix which undergoes rotational three-dimensional strain under constant volume conditions. The mathematics gives explicit information on the following features:

1. (1) The length and position (relative to a defined coordinate system) of the principal axes of the strain ellipsoid at any stage of the progressive deformation.

2. (2) The position and length of the principal axes in any plane intersecting the strain ellipsoid, also at any stage of the deformation.

3. (3) The position and length of passive markers which initially coincided with the principal axes in an intersecting plane. This is of consequence for the distinction between passively rotating structures and actively forming structures.

4. (4) The shear strain parallel to an intersecting plane or sheet, as indicated by the angular difference between the normal to an intersecting plane at any time and the marker at the same time which initially, however, was parallel to the normal. This layer-parallel shear causes boudins to rotate and the axial plane of buckles to tilt.

The relationships have been expressed quantitatively in the bulk of the paper and illustrated in diagrams. The analysis presented is basic for the study of the deformational behavior of competent sheets of rocks embedded in less competent ones.     

Particle paths, displacement and progressive strain applicable to rocks

If the particle paths are known for deforming continuous media such as rocks, the strain is determined at all stages of the deformation. The particle paths are studied for various types of simultaneous combinations of pure shear and simple shear. Any kind of progressive plane homogeneous strain can be expressed as a simultaneous superposition of pure shear and simple shear by selecting the proper ratio between the two strain rates and the proper angle between the slide direction of the simple-shear part and the principal axes of the pure-shear part. In the cases studied — except one — the angle between the slide direction of the simple-shear part and the principal strain rate of the pure-shear part is 45°. Several combinations of the simple-shear rate, γ, and the pure-shear rate, , are tested. These combinations give particle paths varying from sets of straight parallel lines to orthogonal hyperbolas. Distorted hyperbolas, ellipses and circles constitute the particle paths at intermediate ratios. From the particle-path equations — which are found by integration of the rate-of-deformation equations — the strain ellipse is readily determined at any stage of the deformation. One particularly intriguing result is the rotating and pulsating strain ellipse found in the cases when the particle paths are closed curves (ellipses). Application of the results to various fold-, thrust- and inclusion structures is suggested. In an appendix the treatment of rotational deformation as a superposition of irrttational strain and rigid rotation is considered for comparison.     

Changes in strain trajectories in three different types of plate tectonic boundary deduced from earthquake focal mechanisms

Principal strain orientations (minimum horizontal compression—ex and maximum horizontal compression—ey) were established at three different types of plate tectonic boundary: two transform faults, an oceanic ridge located on the Southeast Indian Ridge and a trench located close to the South Sandwich Archipelago. To establish the strain patterns in each zone, 104 earthquake focal mechanisms (centroid-moment tensor solutions for earthquakes with mb≥4; Harvard seismology data, CMT) were examined by fault population analysis. Despite the existence of only one tectonic process that controlled deformation in these zones (divergence, convergence or passive displacement), and only one main strain tensor, several coeval strain ellipsoids were found. These differed from the main strain tensor in the location of the principal strains. In general, permutations were observed between the principal strains, i.e., interchanges between the location of the principal strain axes maintaining the strain ellipsoids in the same 3D orientation. Only in some cases were changes in the ellipsoid orientation associated with major structures.