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_Journal of Structural Geology 32 (2010) 15–27_ _Contents lists available at ScienceDirect_ _Journal of Structural Geology_ _journal homepage: www.elsevier.com locate jsg _Kinematics and dynamics of fault reactivation: The Cosserat approach_ _Jure Zalohar*, Marko Vrabec_ _University of Ljubljana, Faculty of Natural Sciences and Engineering, Department of Geology, Askerceva 12, SI-1000 Ljubljana, Slovenia_ _article info_ _Article history: Received 26 December 2008 Received in revised form 10 June 2009 Accepted 11 June 2009 Available online 21 June 2009_ _Keywords: Cosserat continuum Fault reactivation Fault-slip analysis Kinematic analysis Paleostress analysis_ _abstract_ _In the theory of Cosserat continuum, faulting-related deformation of rocks is described using translational and rotational degrees of freedom, producing definitions for a symmetric macrostrain tensor and a skew-symmetric relative microrotation tensor. The macrostrain tensor describes large-scale deformation of the region, while the relative microrotation tensor describes the difference between large-scale regional rotation and local systematic microrotations of blocks between faults. Faults are activated when resolved shear stress in the direction of movement exceeds frictional resistance for sliding, according to Amontons’s Law of Friction. The direction of slip along the faults depends on the Cosserat strain tensor, which is defined as the sum of the macrostrain tensor and the relative microrotation tensor. We develop a constitutive relation for faulting-related strain (cataclastic flow) based on the J-2 plasticity model for the Cosserat continuum, from which we derive the generally asymmetric stress tensor. We also develop the Cosserat stress–strain inverse method for fault-slip data analysis. We show that the geometry of fault systems is controlled by both the Cosserat strain tensor and the stress tensor, and present a field example of a fault system that conforms to the predictions of the Cosserat theory._ _© 2009 Elsevier Ltd. All rights reserved._ _1. Introduction_ _Paleostress and kinematic analyses of fault-slip data are generally performed within the frame of classical continuum theory, where deformation of a body is described by three degrees of freedom (components of the translation vector) and stresses and strains are assumed to be symmetrical (e.g., Jaeger and Cook, 1969; Angelier, 1994). Most techniques for fault-slip data analysis also suppose that: (1) the stress strain field at the time of faulting was homogeneous; (2) faults are independent and do not interact; and (3) blocks bounded by faults do not rotate (e.g., Angelier, 1994; Nemcok and Lisle, 1995; Nemcok et al., 1999; Zalohar and Vrabec, 2007). These assumptions are obviously oversimplified and are only acceptable in certain geological situations. In the last two decades considerable progress in understanding the effect of block (micro)rotations between faults has been made using Cosserat continuum theory (Twiss et al., 1991, 1993; Twiss and Unruh, 1998, 2007). Twiss et al. (1991, 1993), and Twiss and Unruh (1998, 2007) were the first to recognize the influence of block (micro)rotations on fault-slip patterns. In the Cosserat continuum theory, the direction of slip along faults depends on the Cosserat strain tensor, not on the stress tensor. From this it follows that patterns of slip along faults are related in a systematic way to global deformation. The (paleo)stress can therefore be reconstructed from fault-slip data provided the rheological behavior of rocks and the constitutive relation between stress and strain are known._ _Twiss et al. (1991, 1993) and Twiss and Unruh (1998, 2007) also studied the influence of relative microrotations on the geometry of fault systems. They showed that, in addition to symmetric fault systems with conjugate or orthorhombic geometry predicted by classical continuum theory, Cosserat theory predicts monoclinic and triclinic fault systems._ _This article aims to present a fault reactivation model for the Cosserat continuum. We show that strain is not the only parameter affecting the geometry of slip-capable fault systems; another controlling parameter is stress, which is not necessarily symmetric. We also develop an improved Cosserat stress–strain inverse method for fault-slip data analysis based on the Cosserat (or micropolar) strain inverse method of Twiss et al. (1991, 1993), and Twiss and Unruh (1998, 2007). Our method is implemented in the T-TECTO 2.0 computer program (available free of charge from: www2.arnes.si/~jzaloh-tecto_homepage.htm). The method was thoroughly tested on numerous artificial and natural datasets. We present an analysis of one selected natural fault system, which indicates that in some cases at least, faulted rocks can be successfully described within the frame of the Cosserat theory._ _16_ _J. Zalohar, M. Vrabec Journal of Structural Geology 32 (2010) 15–27_ _2. Kinematics of the Cosserat continuum_ _a_ _the kinematics (micro)rotational odfegthreeeCoofsfsreereadt ocmon!tfinCoususemrat is characterized by , which is independent of the translatory motion described by the displacement field !u (Fig. 1, Table 1). The field of continuum macro-rotations does not coincide with microrotations at each material particle (Iordache and Willam, 1998), and therefore we must consider two scales of deformation: the instantaneous macrodisplacement gradient and the instantaneous microrotation (e.g., Twiss and Unruh, 2007). The former is defined by the relative motions of the centroids of the blocks bounded by the faults, whereas the instantaneous microrotation describes rigid rotation of blocks independent of the large-scale maraocurnodrotthateiiornce!nftmroacidros. Therefore, in the Cosserat continuum, corresponding strain measures are the Cosserat strain tensor e and the torsion-curvature tensor k (Forest, 2000; Forest and Sievert, 2003):_ _e ? u ? WC_ _u5 V_ _3 !f Cosserat_ _k ? u5 V_ _3 !f Cosserat_ _V : 3_ _Here, 3 represents the third-order permutation tensor 1_2?i ? j??j ? k??k ? i?, and WC tensor is the Cosserat microrotation tensor, which describes the microrotation of the blocks. We have also introduced the deformation gradient tensor (or instantaneous vuj _vxi u5 V this tensor u ? uij de?nes the macrostrain, while the skew-symmetric part u(A) de?nes the instantaneous macrorotation. Here we use the sign convention from the rock mechanics and paleostress analysis literature, where strains and stresses are assumed positive for contraction compression and negative for extension tension._ _The torsion-curvature tensor k takes into account differential changes of microrotations in the neighborhood of a point. From the definition of Cosserat deformation measures (Eq. (1)), it follows that the torsion-curvature tensor and the gradient of Cosserat deformation are related by the equation (Toupin, 1962, 1964; Forest and Sievert, 2003):_ _Table 1 Explanation of the most important quantities used in the text._ _Symbol ! ! !fff Cmreoalscsreorat ! x83u! x83! x83! l1 ; l2 ; l3_ _l1, l2, l3_ _!n !m W L_ _U(s, m, R) f(s, m, R) p_q R_ _J2d_ _a1, a2, b1, b2 pr, a andb f1_ _f2 F0 s, D_ _sn and s e ep and ee_ _e(S) and e(A)_ _k s sd m md 3 WC u u(S)_ _u(A) and Wmacro A N T 1_ _Explanation_ _Cosserat microrotation vector Regional macrorotation vector Relative microrotation vector Translation of material point Kinematic axes or eigenvectors of the macrostrain tensor u(S) Principal strains or eigenvalues of the macrostrain tensor u(S) Normal vector to the fault plane Slip direction along the fault Relative microrotation parameter Distance between the centroids of the neighboring blocks pseudo-potential of dissipation Yield function Rate-of-plastic multiplier Material internal variable accounting for material hardening Thermodynamic force associated with the material internal variables Second invariant of stress and or couple-stress tensors extended to the Cosserat continuum Material parameters Parameters in the constitutive equation for the cataclastic flow Maximum possible angle of friction for sliding on pre-existing fault Angle of residual friction Object function in the inverse method Parameters related to inhomogeneity of the strain stress field Normal and shear stress along the fault Cosserat strain tensor Plastic and elastic parts of the Cosserat strain tensor Symmetric and skew-symmetric parts of the Cosserat strain tensor Torsion-curvature tensor Stress tensor Deviatoric part of the stress tensor Couple-stress tensor Deviatoric part of the couple-stress tensor Third-order permutation tensor Cosserat microrotation tensor Deformation gradient tensor Symmetric part of the deformation gradient tensor (macrostrain tensor) Skew-symmetric part of the deformation gradient tensor (macrorotation tensor) Relative microrotation tensor Second-order projection tensor Third-order projection tensor Fourth-order identity tensor_ _First used in Eq. (1) (6) (6) (1) (7)_ _(7)_ _(7) (7) (10) (13)_ _(19) (20) and (21) (20) and (24) (20)_ _(19)_ _(22)_ _(22) (28)_ _(39)_ _(39) (43) (42)_ _(16) (1) (18)_ _(5) and (26)_ _(1) (15) (22) (15) (22) (1) (5) (1) (3) and (5)_ _(5)_ _(5) (12) (12) (12)_ Ключевые слова: symmetric, besdo, mohr point, material, skew-symmetric, relative, geophysical, equation, standard deviation, international, asymmetric stress, gregoric?, willam iordache, press, cosserat theory, cosserat strain, reches, unruh, vrabec journal, shear, rational mechanics, inversion, numerical, paleostress analysis, faulted rock, deformation phase, applied, relative microrotation, choose, rel, wa, twiss unruh, fault slip, tensor stress, cosserat continuum, inverse, journal structural, skew-symmetric component, stress tensor, mechanica, ?eld, university, willam forest, individual fault, inversion result, fault-slip data, angular mist, couple stress, elsevier, geometry remarkably, fault, classical case, yin, angle, inversion parameter, shear stress, avce fault, twiss, willam willam, parameter represents, ranalli, strainstress eld, model, principal, borst, science, paleostress, slip direction, block, depends, compatible, classical continuum, forest, granular material, slip-capable fault, strain tensor, natural fault, faulting-related deformation, measure, compatible fault, constitutive parameter, deformation, plane, york, text, continuum, predicted, tangential component, tectonophysics, normal stress, p, gradient, dip angle, toupin, theoretical fault, geometry, highly asymmetric, microrotation, amontonss law, displacement eld, slipcapable fault, normal vector, relative magnitude, fault depends, angelier, deformation mechanism, calculated, internal friction, amontonswin, cosserat medium, paleostress determination, principal strain, observed, macrostrain, slip, journal, journal structural geology, cataclastic, ax, de?ned, parameter, kinematic, macrostrain tensor, constitutive, friction, e, fault reactivation, actual direction, dartevelle, solution, data, compatibility function, vrabec, nemcok, iordache, block bounded, structural geology, fault plane, reactivation, function, asymmetric, result, classical, strain, graphically illustrated, stress, method, deformation style, active fault, symmetric stress, mohr, compatibility, z?alohar vrabec, slip fault, undetermined constant, instantaneous microrotation, applied mechanics, structural, materials, geological, strain stress, rock, acta mechanica, frictional resistance, cosserat, mechanics, direction, willam, eq, mohr diagram, cosserat model, pre-existing fault, amontonswin program, geology, case, torsion-curvature tensor, stress deviator, couple-stress tensor, yield function, represents, positive, microrotations, object function, component, couple, constitutive equation, condition, compatibility measure, relative microrotations, transposed deviatoric, acta, boundary condition, mohr circle, inversion procedure, fault-slip, plasticity model, residual friction, modeling, theory, observed fault, neighboring block, engineering, direction slip, normal, increasingly asymmetrical, vector, optimal solution, tensor, sn, constitutive relation, analysis, orientation, diagram, z?alohar