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_Journal of Structural Geology 32 (2010) 1170-1184_ _Contents lists available at ScienceDirect_ _Journal of Structural Geology_ _journal homepage: www.elsevier.com/locate/jsg_ _Kinematic analysis of asymmetric folds in competent layers using mathematical modelling_ _J. Aller,*, N.C. Bobillo-Ares, F. Bastida, R.J. Lisle, C.O. Méndez_ _Departamento de Geología, Universidad de Oviedo, Jesús Arias de Velasco s/n, 33005 Oviedo, Spain_ _Departamento de Matemáticas, Universidad de Oviedo, 33007 Oviedo, Spain_ _School of Earth and Ocean Sciences, Cardiff University, Cardiff CF10 3YE, UK_ _article info_ _Article history: Received 18 December 2009; Received in revised form 28 June 2010; Accepted 27 July 2010; Available online 5 August 2010_ _Keywords: Folding Strain Mathematical modelling Recumbent folds_ _abstract_ _Mathematical 2D modelling of asymmetric folds is carried out by applying a combination of different kinematic folding mechanisms: tangential longitudinal strain, flexural flow and homogeneous deformation. The main source of fold asymmetry is discovered to be due to the superimposition of a general homogeneous deformation on buckle folds that typically produces a migration of the hinge point. Forward modelling is performed mathematically using the software ‘FoldModeler’, by the superimposition of simple shear or a combination of simple shear and irrotational strain on initial buckle folds. The resulting folds are Ramsay class 1C folds, comparable to those formed by symmetric flattening, but with different limb lengths and layer thickness asymmetry. Inverse modelling is made by fitting the natural fold to a computer-simulated fold. A problem of this modelling is the search for the most appropriate homogeneous deformation to be superimposed on the initial fold. A comparative analysis of irrotational and rotational deformations is made in order to find the deformation which best simulates the shapes and attitudes of natural folds._ _Modeling of recumbent folds suggests that optimal conditions for their development are: a) buckling in a simple shear regime with a sub-horizontal shear direction and layering gently dipping towards this direction; b) kinematic amplification due to superimposition of a combination of simple shear and irrotational strain with a sub-vertical maximum shortening direction for the latter component. The modeling shows that the amount of homogeneous strain necessary for the development of recumbent folds is much less when an irrotational strain component is superimposed at this stage than when the superimposed strain is only simple shear. In nature, the amount of the irrotational strain component probably increases during the development of the fold as a consequence of the increasing influence of gravity due to tectonic superposition of rocks._ _© 2010 Elsevier Ltd. All rights reserved._ _1. Introduction_ _Kinematic studies of folding are concerned with different types of strain patterns that appear in folded layers and the geometrical evolution of the folded layers from the initial, undeformed stage to the final fold. Kinematic folding mechanisms can be considered as theoretical tools used in this analysis; they define rules that determine the displacements to be produced and the final strain pattern inside the folded layers. A reasonable starting point for establishing basic folding mechanisms is the experimental production of folds, comparable to natural ones. For example,_ _q The program code ("FoldModeler") can be found at http://www.geol.uniovi.es/investigacion/OFAG/Foldteam.html_ _* Corresponding author. Fax: +34 985103103. E-mail address: aller@geol.uniovi.es (J. Aller)._ _0191-8141 $ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.jsg.2010.07.008_ _classical folding experiments by Kuenen and de Sitter (1938) using sheets of paper and rubber revealed the operation of the mechanisms that we now refer to as flexural and tangential longitudinal strain mechanisms, respectively. From this type of experiments it is possible to define kinematic mechanisms as theoretical idealizations that can be mathematically analyzed. In this way, applying the conditions required by every mechanism it is possible, from an initial configuration of a layer, to model theoretically the geometry of a folded layer and its strain pattern. The folding necessarily involves heterogeneous deformation; however, the superimposition of a homogeneous strain during folding can significantly modify the geometry of folds and their strain pattern, and this type of modification can be included as a folding mechanism in its own right._ _An interesting aspect of folding kinematics is concerned with the development of asymmetric folds. According to the definition given by most authors, a fold is asymmetric when it lacks bilateral symmetry about the axial plane (Turner and Weiss, 1963, p. 122; De Sitter, 1964, p. 272; Whitten, 1966, p. 601; Ramsay, 1967, p. 351). Most of the folds found in rocks are asymmetric and in many cases the degree of asymmetry is very high. Good examples are the major recumbent folds common in the hinterland of orogenic belts, the folds developed in ductile shear zones, and parasitic folds developed on the limbs of major folds. These examples illustrate the importance of gaining understanding of the folding mechanisms operating during the formation of asymmetric folds in order to further our knowledge of orogenic deformation. Most of the available studies on natural asymmetric folds place special emphasis on the shape and asymmetry of the folded surfaces or layers, but the final strain patterns or characteristics of the progressive deformation in these folds are poorly known._ _The aim of this paper is to study some general aspects of the kinematics of asymmetric folds in competent layers. Theoretical folds are modeled using tangential longitudinal strain, flexural flow and homogeneous strain. The asymmetry can be introduced via all three mechanisms. Superimposition on pre-existing folds of a general homogeneous strain, with final principal directions freely chosen will be the main source of asymmetry in our models. This modeling yields information about the final strain distributions and the progressive deformation in asymmetric folds (forward problem), and the potential for ascertaining the kinematic mechanisms that operated in specific natural asymmetric folds (inverse problem) by comparison of the geometry of these folds with theoretically modeled folds. A special attention is devoted to large recumbent folds, because they are key pieces for understanding the structure of orogens._ _The analysis is two-dimensional, considering strain in the profile plane of the folded layer. Application of the theoretically modeled folds to the analysis of the mechanisms that operate in natural folds is limited by the availability of strain measurements in the rocks. Fortunately, detailed information on the natural fold geometry and the cleavage pattern provides very useful data that can be related to the strain state and the kinematic mechanisms involved in the development of asymmetric folds._ _fold limbs (Ramsay, 1967, p. 351), a simple measure of the asymmetry of a fold is the ratio between these lengths. Tripathi and Gairola (1999) define the degree of asymmetry of a folded surface as the sum of two parameters, one depending on the difference in amplitude and other depending on the difference in shape. A problem with this method is that this parameter does not express the extent to which the asymmetry is due to differences in shape or differences in amplitude. In order to represent graphically the fold asymmetry in a 2D coordinate system, it is necessary to describe this geometrical feature using only two parameters. A complete description of the asymmetry of folded surfaces is not really possible with only two parameters; in fact, Twiss (1988) proposed a classification which requires six parameters for profiles of general asymmetric folded surfaces. Nevertheless, the most relevant features of this asymmetry can be characterized by the following parameters (Bastida et al., 2005; Lisle et al., 2006):_ _Shape asymmetry: Sa = AF_AB_ _(1)_ _Amplitude asymmetry: Aa = y0F_y0B_ _(2)_ _where AF and AB are the respective normalized areas (Bastida et al., 1999) of the forelimb and the backlimb (defined as the steeper and gentler limb respectively), and y0F and y0B are the y0 parameters of the forelimb and the backlimb, respectively (Fig. 1a). The plot of these parameters in a graph of Sa against Aa for all the folds of a specific set allows visualization of the variation in asymmetry of these folds._ _In order to analyze the asymmetry of the folded layer, comparison of the curves representative of the two limbs in the classifications of Ramsay (1967, pp. 359-372), Hudleston (1973) or Treagus (1982) can give a qualitative idea of the shape asymmetry. A simple parameter to quantify this aspect of asymmetry is the thickness asymmetry (Ta) (Fig. 1b), defined as the ratio between the orthogonal thickness of the forelimb and that of the backlimb for the maximum dip in Ramsay’s classification._ _3. Modelling asymmetric folds: preliminary considerations_ _2. On the description of asymmetric folds_ _A qualitative description of fold asymmetry was made by Ramsay (1967, pp. 351-352) using the letters M (symmetric), S or Z (asymmetric), in such a way that the shape of the letter describes the shape of the fold. This method is useful for mapping major structures from parasitic folds. From a quantitative point of view, Loudon (1964) and Whitten (1966) proposed the use of the third statistical moment of the orientation distribution of the normals to the folded surface profile to express the asymmetry of folds. Since the asymmetry depends on the relative length of the_ Ключевые слова: price, simple shear, orthogonal thickness, central highlands, symmetric attening, major axis, rotational deformation, hinge zone, shear, donath, foldmodeler, shortening, folded layer, aller journal, rotational, axial surface, natural fold, earth, martnez cataln, modelling, strain, graph, parallel, kinematic mechanism, ?nal, theory, general character, maximum shortening, limb, case, pure shear, aspect ratio, parallel fold, superimposition simple, symmetric fold, equation, deformed guideline, homogeneous deformation, fold prole, program, aller, hinge, curve, nal result, sanderson, recumbent fold, formed, chevron fold, conic, analysis, general, recumbent, hudleston, layer dip, de?ned, longitudinal strain, bastida, bobillo ares, cleavage, pattern, geology, mechanism requires, longitudinal, good, major semi-axis, orogenic belt, paor, strain ellipse, simultaneous, folding, scale factor, relative length, srivastava, axial plane, principal direction, journal, asymmetry, dark circle, elsevier, axial, area, tangential, shear direction, kinematic amplication, strain distribution, eocene sandstone, journal structural, change, form, van, thrust sheet, rock, irrotational strain, buckle fold, fold developed, simple, superimposed strain, pure, principal, kinematic analysis, combination, asymmetry induced, deformation, strain pattern, tikoff, chicago, parameter, carreras, irrotational, ares, ratio, structural geology, reviews, tectonics, task, homogeneous, folds, thicker forelimb, inverse modelling, folded surface, active folding, outer arc, competent layer, geometry, springer berlin, fold prior, developed, direction, buckle, geological, maximum stretch, progressive deformation, thrust, wickham manz, dip, tangential longitudinal, progressive shearing, maximum, zone, foldmodeler software, shear irrotational, asymmetric fold, fold, asymmetric, sitter, folding journal, folded, rea curve, comparison, main source, qea curve, hinge point, model, initial fold, sub-simple shear, point, ramsay, sheared fold, area change, parabolic guideline, superimposition, component, appendix, ramberg, bucher, bobillo-ares, coordinate change, treagus, correspond, development, exural, london, dietrich, pre-existing fold, surface, journal structural geology, tay nappe, homogeneous strain, orthonormal basis, computers, combination simple, arc, matrix, fold formed, guideline, nappes, sheath fold, layer, ?exural, kinematic, considered, increasing inuence, mechanism, tectonophysics, geological society, initial conguration, black curve, application, superimposed, oblique attening, ha, geometrical analysis, natural, initial, exural fold, square root, ramsays classication, basis, fossen, structural, shear strain, coordinate, nb, toimil, geological structures, review, bobillo, ?ow, numerical fold, casey, rotational component, ellipse, rigid rotation