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_Journal of Structural Geology 32 (2010) 2009e2021_ Contents lists available at ScienceDirect Journal of Structural Geology journal homepage: www.elsevier.com locate jsg Monoclinic and triclinic 3D ?anking structures around elliptical cracks Ulrike Exner a,*, Marcin Dabrowski b a University of Vienna, Department of Geodynamics and Sedimentology, Althanstrasse 14, 1090 Vienna, Austria b Physics of Geological Processes, University of Oslo, Pb 1048 Blindern, 0316 Oslo, Norway Article info Article history: Received 28 August 2009 Received in revised form 26 July 2010 Accepted 13 August 2010 Available online 20 August 2010 Keywords: Fault-related folds Monoclinic ?ow Analytical model Eshelby’s solution Sheath folds Abstract We use the Eshelby solution modified for a viscous fluid to model the evolution of three-dimensional ?anking structures in monoclinic shear zones. Shearing of an elliptical crack strongly elongated perpendicular to the flow direction produces a cylindrical ?anking structure which is reproducible with 2D plane strain models. In contrast, a circular or even narrow, slit-shaped crack exhibits a reduced magnitude of the velocity jump across the crack and results in smaller offset and a narrower zone of de?ection than predicted with 2D-models. Even more significant deviations are observed if the crack axes are oriented at an oblique angle to the principal flow directions, where the velocity jump is oblique to the resolved shear direction and is modi?ed during progressive deformation. The resulting triclinic geometry represents a rare example of triclinic structures developing in monoclinic ?ow and may be used to estimate the ?ow kinematics of the shear zone. © 2010 Elsevier Ltd. All rights reserved. 1. Introduction A ?anking structure is a set of de?ections of marker surfaces adjacent to a slip surface, commonly a crack or vein (Passchier, 2001). Analytical (Kocher and Mancktelow, 2005; Grasemann et al., 2005; Mulchrone, 2007), numerical (Grasemann et al., 2003; Wiesmayr and Grasemann, 2005; Kocher and Mancktelow, 2006) and analogue studies (Exner et al., 2004) demonstrated the wide variability in ?anking structure geometry. This variability depends on (1) the initial orientation of the crack relative to the set of marker surfaces; and (2) the kinematics of the bulk flow. These studies established the possibilities and limits of the application of ?anking structures to determine shear sense and estimate the kinematic vorticity number and ?nite strain from geometrical characteristics. Similar to other shear sense indicators, field examples of ?anking structures are usually sought within two-dimensional sections oriented orthogonal to the layering and parallel to the tectonic transport direction (highlighted e.g. by a mineral lineation). For this kind of exposure, two-dimensional models of ?anking structures are appropriate, assuming that the crack is strongly elongated perpendicular to the shear direction and deforms under plane strain flow conditions. However, some outcrops present * Corresponding author. Fax: +43 1 4277 9534. E-mail addresses: ulrike.exner@univie.ac.at (U. Exner), marcind@fys.uio.no (M. Dabrowski). 0191-8141 $ e see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.jsg.2010.08.002 a three-dimensional, monoclinic structure, where a foliation plane is de?ected around a crack of limited length perpendicular to the shear direction. Notably, in such a geometric con?guration the mineral lineation is not laterally displaced along the crack, but can be followed continuously across the structure along a straight line (indicated in Fig. 1a). Alternatively, cracks may also be oriented at an oblique angle to the mineral lineation. In these cases, the lineation additionally is displaced laterally along the crack and a de?ection of the lineation is observed within the foliation plane (Fig. 1bed). Such effects around three-dimensional cracks cannot be studied with two-dimensional models, especially if movement occurs out of the observation plane. Thus, we apply an analytical solution of elliptical inclusions in viscous flow to investigate the effect of the crack’s (1) aspect ratio, (2) orientation and (3) the background flow kinematics on geometry of the resulting ?anking structure in three dimensions. 2. Model formulation We model a ?anking structure as the structure formed in the flow about an elliptical crack in a homogeneous isotropic viscous medium. Mechanical anisotropy or layers of different viscosity may be present in the natural structure, but they are neglected here. The contact between the crack surfaces is frictionless and a jump in the tangential component of velocity across the crack occurs at zero shear stress. In the model implementation, to achieve this, the crack is ?lled with an in?nitesimally thin layer of incompressible inviscid fluid. The velocity jump results in the relative displacement of pairs of particles that are initially adjacent on the two faces of the crack. Crack opening is prevented by the presence of the incompressible material ?lling it; a normal velocity jump is not allowed. Crack propagation does not take place, but the crack stretches and rotates during deformation of the viscous medium. Such a structure has been termed a stretching fault by Means (1989). To obtain the pattern of surfaces deformed about this inhomogeneity during a ?nite deformation, we track the deformation of sets of initial marker lines and planes. Only far-?eld plane ?ow in pure shear and in simple shear is considered here. In irrotational pure shear, the far-?eld velocity components are v?xN? ? D?xNx ?x; v?yN? ? 0; v?zN? ? ?D?xNx ?z (1) For this case, the marker planes are oriented perpendicular to the direction of maximum shortening (xy-planes) and the marker lines are parallel to the direction of maximum stretching (x-direction). In simple shear, the far-?eld velocity components are v?xN? ? 2D?xNz ?z; v?yN? ? 0; v?zN? ? 0 (2) The marker planes are parallel to xy-planes (shear plane) and the marker lines are parallel to the x-direction (shear direction). In both cases, simple and pure shear, marker planes and marker lines are placed at orientations that remain unchanged under the far-?eld flow and as such they approximate the late-stage foliation and lineation, respectively. 3. Analysis 3.1. Reference frames The far-?eld ?ows described in Eqs. (1) and (2) are referred to a! teh?ze.xoTehrdieernectrfaeatrciekonnsceeomfstyi-hsaetxeuemsni! xtayb,za!bwseaitnvhdetcththoeerusnn!oeitrx0m,b!aeasyl0evaevncedtcot!eorzr0!sco!fedaxe,rt! eoertymataiinnndge crack reference system x0y0z0. The lengths of the semi-axes are a and b. The center of the crack coincides with the common origin of these two reference systems (Fig. 2). The position of the crack center is ?xed during deformation due to the symmetry. Using the three Euler angles j1, 4, j2 allows us to describe an arbitrary crack orientation with respect to the fixed reference frame (e.g. Goldstein et al., 2002). Imagining that the two systems are initially coincident, the xyz system is brought into coincidence with the crack axes by a sequence of rotations: first around the axis !z ; then around the axis !y by an angle j1; and finally rotation around the axis !z 0 by an angle j2. The rotation axes and their order follow the zeyez convention used for Euler angles. Positive angles are measured clockwise. For expository convenience, we speak of the xy-plane as horizontal. Then, j1 measures the strike, or the angle between the U. Exner, M. Dabrowski Journal of Structural Geology 32 (2010) 2009e2021 y xy x ' y ' b z x y elliptical crack x marker plane marker lineation Fig. 2. Setup and model parameters: an elliptical crack with the semi-axes !a and !b (parallel to x0 and y0) is oriented at the angles j1, 4 and j2 to the external reference frame (xyz). A passive marker plane and the lineation (in x-direction) are oriented parallel to the flow plane (xy) and shear direction (x) of the homogeneous monoclinic background flow. The line !c marks the intersection between the crack surface (x0y0) and the xy-plane. y-direction and the line of nodes !c, where the latter is the intersection of the horizontal marker plane and the plane of the crack. The dip of the crack is given by the dihedral angle 4. The angle j2 measures the rotation between the semi-axis b and the line of nodes !c. We introduce the rotation matrix R whose columns are the direction cosines of the crack base vectors with respect to the fixed coordinates. In terms of the Euler angles, in?nitesimal length c determine a ?nite jump of the velocity across the crack (Kassir and Sih, 1975; Mura, 1987). The velocity ?eld in the matrix is obtained by reducing the elastic solution presented by Eshelby (1959) for the case of an incompressible matrix containing a flat ellipsoidal crack filled with an inviscid fluid. Details about the implementation of the solution and generation of 2D- and 3D-graphs are provided in Appendix B. The crack itself is found to undergo a homogeneous deformation that results in stretching and rotation but preserves an elliptical shape. This allows us to reuse the solution to evaluate the velocity ?eld after readjusting the shape parameters and reevaluating the components of the far-?eld velocity gradient in the rotating crack reference frame. We determine the evolution of a ?anking structure._ Ключевые слова: foliation, two-dimensional model, velocity, vector, passchier, proceedings, incl, simple, parallel, inclusion rate, offset, stretching, shear, lim lim, freeman, left hand, limiting case, reference, simple shear, normalized component, royal society, elliptical, mulchrone, rate, aspect ratio, treating displacement, homogeneous, component, strain, grasemann, exner dabrowski, nite deformation, aspect, yield, surface, crack rim, oriented oblique, velocity vector, angle, displacement eld, resolved shear, exner, experiment, current orientation, tensor, mesh, pure shear, lineation, mylonitic foliation, foliation plane, uniform strain, position vector, offset vector, length, crack center, ?anking structure, ?nite, matrix, structure, normalized offset, velocity jump, mandal, limiting, resolved, structural, triclinic symmetry, rotation, kinematics, goldstein, sense, de?ection, euler angle, natural structure, central velocity, crack, solution, rotation matrix, shear plane, stretch matrix, ratio, jump, ellipsoidal inclusion, triclinic ?anking, structure formed, iacopini, maximum, geology, central, mineral lineation, ellipsoidal, eshelby solution, dabrowski journal, structural geology, outcrop, deformation, shearing direction, viscous matrix, oriented perpendicular, appendix, center, sheath fold, dabrowski, monoclinic, mathematical, oriented, velocity ?eld, elastic eld, constant, fold, oblique, inclusion, viscous, crack axis, point lying, result, eshelby tensor, monoclinic symmetry, direction, shear strain, function, xed axis, accumulated offset, eq, symmetry, crack surface, triclinic, zone, euler, rotated, potential natural, inclination, ab, initial orientation, viscous inclusion, dimensional, vorticity tensor, velocity yield, ?anking, journal structural, elliptical crack, position, elongated perpendicular, axis, pure, lim, quinquis alsop, ax, analysis, linear marker, dn, london series, oblique angle, shewchuk, axis length, initial angle, perpendicular, mancktelow, crack aspect, marker, model, natural, case, isoclinal fold, rst term, central marker, eld, mura, fletcher, orientation, initial, shear zone, poisson ratio, background, ?ow, crack plane, anking structure, elsevier, normalized, shear direction, linearly dependent, sheath, eshelby, journal structural geology, plane, fault, journal, velocity eld, initial position, marker plane, condition, gradient, geometry, ?eld