Nilo C. Bobillo-Ares

GEOMETRICAL ANALYSIS OF FOLDED SURFACES USING SIMPLE FUNCTIONS

Abstract Several functions have been chosen in order to approximate fold profile geometry. Some of them are valid mainly for alloclinal folds (interlimb angle>0), whereas others are mainly valid for isoclinal folds (interlimb angle=0). In all cases, a fold profile can be characterised by an aspect ratio (y0/x0) between the height and the width of a limb (fold amplitude), and a shape parameter characteristic of the considered function. The shape parameters have been mutually linked through the area beneath the fold profile. The geometrical analysis enables a graphical classification based on a shape–amplitude diagram in which the most common types of folded surfaces are represented: cuspate, chevron, sinusoidal, parabolic, elliptic and box folds. Any of the shape parameters can be used as x-axis of the diagram in order to approximate the geometries commonly exhibited by natural folds. In the diagram presented in this paper two shape parameters have been combined: the exponent n of a power function for alloclinal folds, and a parameter C/y0, defined from a function composed of an elliptic part and a line segment of length C for isoclinal folds. In order to show the suitability of the classification method, it has been applied to some examples of finite-element, experimental and natural folds.

On tangential longitudinal strain folding

Abstract Intrinsic two-dimensional analysis of the geometry of fold profiles originated by tangential longitudinal strain has allowed us to examine several features of this folding mechanism. The strain distribution in the folded layers indicates that the direction of the strain ellipse axes has a deviation with respect to the directions normal and tangential to the boundaries of the folded layer, which mainly depends on the curvature changes. The geometry of the folded layers involves layer thickening in the hinge zone (single hinge folds of class 1C, and double hinge folds composed of classes 1A and 1C); when migration of the neutral line is allowed, layer thinning occurs in the hinge zone (single hinge folds of class 1A, and double hinge folds composed of classes 1C and 1A). Nevertheless, in most cases, the folds are close to parallel folds. The bulk shortening due to tangential longitudinal strain mainly depends on the amplitude of the folds and scarcely depends on the shape of the folded surfaces. When the original thickness of a layer is large in relation to the radius of curvature, two gentle minor antiforms separated by a synform appear in the inner arc close to the hinge zone of the major antiform. This minor folding involves local thickening in the hinge zone, which is observed in some natural folds as a protuberance in the inner arc of the hinge zone.

FoldModeler: a tool for the geometrical and kinematical analysis of folds ☆

Abstract FoldModeler is a system constructed in the Mathematica ™ environment that enables strain analysis in the profile of layers folded by the simultaneous or successive superposition of several strain patterns (layer shortening, tangential longitudinal strain, flexural flow and flattening). The fundamentals of the system involve the deformation of an initial grid of quadrilaterals according to the folding mechanisms considered. The main inputs to the system define the number, shape and size of the quadrilaterals, the characteristics of the sequences of incremental strain patterns involved, and the successive variations in form of a reference line named the ‘guideline’. The main outputs of the program are the parameters defining the form of the folded guideline, the drawings of the folded layer with several markers showing the strain distribution, graphics showing the variations in the orientation of the principal directions and the aspect ratio of the finite strain ellipse as functions of the layer dip, and Ramsays classification of the folded layer. FoldModeler has two main geological applications: (a) to predict the geometrical properties of folds produced by the combination of several types of strain patterns and (b) to analyse the strain state in specific natural quasi-symmetrical folded layer profiles and the possible combinations of strain patterns that could give rise to such a fold. This can be done by searching, with a fit and error method, a theoretical fold with the same geometrical characteristics as a given natural fold. The second application requires the existence of cleavage in the natural folded layer, and the best results are obtained when some strain measures are available.

Analysis of folding by superposition of strain patterns

Abstract Two methods have been developed in this paper to model the strain state and the layer geometry of folds. These methods analyse the superposition of strain patterns due to layer shortening, tangential longitudinal strain, flexural-flow and fold flattening. The first method multiplies the deformation gradients of these strain patterns to model the successive superposition of mechanisms. The second method is more general and is based on the transformation of points from the initial configuration to deformed points according to the geometrical properties of the folding mechanisms involved. This method simulates the simultaneous and successive superposition of strain patterns. Both methods generate graphic outputs that describe the strain variation through the folded layer. Another application of these methods is to attempt to find theoretical folds that fit natural or experimental folds and to perform a geometric and kinematical analysis of these folds. Knowledge of the shape of the folded layer and the cleavage pattern is the most common basic information available in natural folds that can be used to perform the analysis. Additional strain data from the folded rocks are valuable for improving knowledge of the kinematical mechanisms involved in the folding.

FOLD PROFILER: A MATLAB ® -based program for fold shape classification

FOLD PROFILER is a MATLAB code for classifying the shapes of profiles of folded surfaces. The classification is based on the comparison of the natural fold profile with curves representing mathematical functions. The user is offered a choice of four methods, each based on a different type of function: cubic Bezier curves, conic sections, power functions and superellipses. The comparison is carried out by the visual matching of the fold profile displayed on-screen from an imported digital image and computed theoretical curves which are superimposed on the image of the fold. To improve the fit with the real fold shape, the parameters of the theoretical curves are changed by simple mouse actions. The parameters of the mathematical function that best fits the real folds are used to classify the fold shape. FOLD PROFILER allows the rapid implementation of four existing methods for fold shape analysis. The attractiveness of this analytical tool lies in the way it gives an instant visual appreciation of the effect of changing the parameters that are used to classify fold geometry.

Phasors, Always in the Real World

Complex phasors are reinterpreted as real operators in a real vector space of functions. This structure is linked with dilative rotations of the Euclidean plane. Finally we conclude with some methodological ideas related to the teaching of the complex numbers.

StrainModeler: A Mathematica™-based program for 3D analysis of finite and progressive strain

Abstract StrainModeler is a program constructed in the Mathematica ™ environment that performs 3D progressive strain calculations for lines and planes undergoing any sequence of homogeneous deformations. The main inputs to the system define the initial line or plane to be deformed and the deformation sequence to be applied, including combinations of simple shear, pure shear and volume change. For the deformation of lines, the output of the program is the change of attitude of the initial line, which can be represented by graphics or plotted in an equal-area projection. For the deformation of planes, the program has several outputs: (i) change of attitude of the initial plane; (ii) magnitudes and ratio of the semi-axes of the strain ellipse on the deformed plane; (iii) orientation of the major and minor axes of the strain ellipse on the deformed plane; (iv) orientations of the axial planes of the folds formed on the deformed plane, and (v) area change on the deformed plane. The variation of any of these parameters can be shown against a linear parameter only linked to the number of steps involved in the deformation, as a kind of “time” line, or it can be shown against the variation of a parameter of the strain ellipsoid (e. g.: major axis/minor axis ratio). A sequence of directions can be also visualized as a curve in an equal-area plot. Three applications of the program are presented. In the first, the deformation by simple shear of a plane with any orientation is analyzed. In the second, we explore the formation of recumbent folds in layers with different initial orientations for simple shear and pure shear deformations. In the third, we use StrainModeler to analyze the deformation of a set of folds located in a ductile shear zone in the Variscan Belt of NW Spain.