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_Journal of Structural Geology 32 (2010) 93–106_ _Implications of complex eigenvalues in homogeneous flow: A three-dimensional kinematic analysis_ _David Iacopini a,*, Rodolfo Carosi b, Paraskevas Xypolias c a Geology and Petroleum Geology Department, King’s College, University of Aberdeen, Meston Building, AB24 3UE, UK b Dipartimento di Scienze della Terra, Università degli Studi di Pisa, 56100, Italy c Department of Geology, University of Patras, 26500 Patras, Greece_ _Article history: Received 14 September 2008; Received in revised form 3 October 2009; Accepted 5 October 2009; Available online 14 October 2009_ _Keywords: Flow kinematics Ghostvector Pulsating pattern Triclinic flow Non-isochoric deformation_ _Abstract_ We present an investigation of the kinematic properties of three-dimensional homogeneous flow defined by complex eigenvalues. We demonstrate, using simple algebraic analysis, that the clear threshold between pulsating and non-pulsating fields, fixed for \( W_n > 1 \) and valid for planar flow, is not easily defined in a three-dimensional flow system. In three-dimensional flows, one of the three eigenvalues is always real and gives rise to an exponential flow coexisting with a pulsating pattern defined by the other two complex conjugate eigenvalues. Due to this mathematical property, the existence of a stable or pulsating pattern depends strongly on the relative dominance of the real eigenvector with respect to the complex ones. As a consequence, the pattern of behavior is not simply imposed by the kinematic vorticity numbers but is also determined by both the amount of strain accumulation and the extrusion component. It is also shown that complex flow can occur locally within shear zones and can sustain some predictable hyperbolic strain paths. These results are applied to the kinematic analysis of some non-dilational and dilational monoclinic and triclinic flows. Some geological implications of this investigation, and the limit of applying these algebraic and kinematic results to real rocks fabric analysis, are briefly discussed._ _Crown Copyright © 2009. Published by Elsevier Ltd. All rights reserved._ 1. Introduction The geological structures can be classified and approximated analytically using basic critical flow patterns (Ramberg, 1974; Ottino, 1989; Passchier, 1997). In homogeneous and steady-state flows, the kinematics of structures can be described in terms of either velocity or displacement gradient. The application of these concepts to deformable rocks enables us to study the progressive deformation and related deformation path patterns in two-dimensional (2D) or simple three-dimensional (3D) systems. In detail, steady-state flow patterns are critically dependent on flow parameters like the relative magnitude of vorticity number (\( W_n \)), the dilatancy parameter (\( A_n \)) and the strain rate (Passchier & Trouw, 2005). Ramberg (1974) and McKenzie (1979) examined the possible vorticity values and strain rate ratios which can create several pulsating and non-pulsating strain paths and showed that for two-dimensional deformation, the threshold limit between the oscillatory and non-oscillatory fields is defined by the eigenvalues of the strain rate matrix. If the eigenvalues are all real numbers, the eigen-flows give rise to exponential deformation paths and the eigenvectors behave as attractors or as repulsors (Ruelle, 1981; Passchier, 1997). If the eigenvalues are purely imaginary, the eigenvectors do not behave neither as attractors nor as repulsors (Ramberg, 1974; Weijermars, 1993). The analytical and experimental works of Weijermars (1991, 1993, 1998) and Weijermars & Poliakov (1993) provide a complete description of such pulsating strain in two-dimensional flow systems. Some examples of three-dimensional pulsating path and strain history were first described analytically by Weijermars (1997). Introducing the concept of fabric attractor, Passchier (1997) described geometrically a flow path spectrum that is connected with all non-isochoric homogeneous monoclinic flows controlled by complex eigenvalues. Three-dimensional triclinic geometries in shear zones have been suggested and analytically studied (Jiang & Williams, 1998; Lin et al., 1998), but the possible pattern defined by complex eigenvalues was not discussed so far. This study re-examine previous results that describe the planar flow as a dynamical system and introduce a complete algebraic analysis of general three-dimensional flows focusing in domains where complex eigenvalues can occur. The kinematic meaning of the complex eigenvalues and their relevance in describing geological structures is also discussed._ _0191-8141 $ – see front matter Crown Copyright © 2009. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jsg.2009.10.003_ _D. Iacopini et al. Journal of Structural Geology 32 (2010) 93–106_ _2. Analytical description_ _2.1 Flow description_ According to Ramberg (1974) and McKenzie (1979), a three-dimensional flow can be described, with respect to a geographic reference system, by the velocity tensor (or flow matrix) \( L_{ij} \). The \( L_{ij} \) could be decomposed into a symmetric stretching tensor or strain rate matrix \( D_{ij} \) and an antisymmetric vorticity tensor \( W_0{ij} \) as follows: \[ L_{ij} = 1/2 (L_{ij} - L_{ji}) + 1/2 (L_{ij} + L_{ji}), \] \[ D_{ij} = W_0{ij}, \quad (1) \] In a steady-state and homogeneous flow, the vorticity tensor \( W_0{ij} \) describes the angular velocity of an orthogonal pair of material lines in the deformation medium with respect to a geographical reference system. Eigenvectors of the \( D_{ij} \) tensor represent the maximum, medium, and minimum Instantaneous Stretching Axes (ISA) of the flow pattern. The flow is considered to be ‘‘homogeneous’’ if \( L_{ij} \) is space-independent and ‘‘steady’’ if it is time independent. As pointed out by Jiang (1994), heterogeneous flows are likely non-steady. In this paper, we limit our analysis to steady and homogeneous flows. In a steady flow, ISA and the eigenvectors of \( L_{ij} \) represent directions that do not change orientation during progressive deformation. The component of vorticity varies according to the framework chosen as reference system. Generally, if we choose an ‘‘external’’ (or geographical) rotating coordinate system parallel to a certain marker (e.g., the boundary walls of a shear zone), an additional rotation component can be introduced. Thus, as extensively demonstrated by Astarita (1979) and Means et al. (1980), with respect to an inertial geographical reference system, \( W_0{ij} \) should be further split into two components: \[ W_0{ij} = W_{ij} + e_{ijk}U_k, \quad (2) \] and Eq. (1) can be rewritten as: \[ L_{ij} = D_{ij} + W_{ij} + e_{ijk}U_k, \quad (3) \] where \( e_{ijk}U_k \) represents the spin component, i.e., the rotation rate of the ISA with respect to the external reference system, while \( W_{ij} \) describes the ‘‘internal’’ rotation rate of material lines parallel to the ISA at any instant with respect to the ISA. If the reference system fixes to the ISA in the flowing body, the spinning component proportional to \( U \) vanishes. _In order to formulate the flow kinematics in more quantitative terms, some definitions are needed:_ _(a) The Instantaneous Stretching Axes (ISA) are defined as the orientation of the eigenvectors of the strain rate matrix \( D_{ij} \) (defined in Eq. 1). With respect to the ISA, the strain rate matrix \( D_{ij} \) is diagonal with a and b (and c in three dimensions) as diagonal elements._ _(b) The kinematic vorticity number \( W_0n \), introduced by Truesdell (1954), is defined as the ratio between the vortex velocity \( v \) (from the antisymmetric part of \( W_{jk} \)) and the trace (\( 2\text{Tr}D_{ij} \)) of the stretching tensor \( D_{ij} \): \[ W_0n = q / (2\text{Tr}D_{ij}), \quad (3a) \] McKenzie (1979) and Means et al. (1980) first used the intuitive concept of the kinematic vorticity number \( W_n \) to show that hyperbolic flow paths belong to the range 0 < \( W_n \) < 1 (Fig. 1a–d), while pulsating (or oscillating) flow paths belong to the complementary field where 1 < \( W_n \) < N (Fig. 1e–h). Since then, a complete description of patterns of planar homogeneous flows (hyperbolic and elliptical) as a function of flow parameters has been given by various authors (e.g., De Paor, 1983; Passchier, 1988; Weijermars, 1991, 1993; Weijermars & Poliakov, 1993)._ _In case of three-dimensional kinematics, some additional parameters should also be defined to describe in a complete way the flow path:_ _(a) If \( a \), \( b \), and \( c \) are the eigenvalues of the \( D_{ij} \) matrix with \( a > b > c \), and if the vorticity vector is parallel to the eigenvector corresponding to \( c \), a flow parameter called stretching rate \( s = (a - b)^2/2 \) can be defined (note that this strain parameter can be used in two-dimensional flow matrix as well, and is always positive)._ _(b) Based on the above hypothesis, the sectional dilatancy number \( A_n \) can also be defined as: \[ A_n = a - b / 2s, \quad (3b) \] This quantity represents the amount of dilatancy during the deformation history. Following the arguments of Jiang (1994), Passchier (1997), and Iacopini et al. (2007), \( A_n \) should be calculated along a section orthogonal to the vorticity vector. If the flow is non-dilatant, then \( A_n = 0 \)._ _(c) The elongation (or extrusion) parameter, which are defined as: \[ T_n = 3 / i / 2s, \quad i = 1; 3, \quad (3c) \] and represents the elongation rate along one of the three ISA axes._ _2.2 Two-dimensional flow_ _The planar flow pattern can be described in two dimensions (see Table 1 for the definition of all mathematical symbols) by the equation:_ \[ x_{18}x_{19} + x_{18}x_{19} - x_y = L \cdot (x y), h, 3_1 g_{21}, g_{12} 3_2 x y, \quad (4a) \] _where \( 3_ \) and \( g_ \) represent the pure and simple shear rates of deformation, respectively. In this case, the kinematic vorticity number becomes:_ \[ W_0n = r / (x_{16} + x_{17})^2 - g_{12}^2 / 4(3_{21}^2 + 3_{22}^2) \quad (4b), \] _which, for a non-spinning planar flow in the ISA system, simplifies to:_ \[ W_n = r / (x_{16} - g_{12} x_{17})^2. \quad (4c) \] _For a steady flow, the solutions \( x(t), y(t) \) of Eq. (4a) can be written in terms of the eigenvalues \( l_1 \) and \( l_2 \) of the velocity gradient tensor \( L_{ij} \), and are:_ \[ x = A e^{l_1 t}, y = B e^{l_2 t}. \] Ключевые слова: rst, shear, tn wn, spiral sink, strain, williams, tensor, solution curve, complex eigenow, asymptotic behaviour, previous, ottino, real, dilatant, respect, ?eld, mathematica, pulsating strain, geological structure, real complex, velocity, fact, trlij, implies, wn, case, truesdell, holdsworth, astarita, reference, property, real eigenvalue, extrusion, lij, behaviour, strain rate, type, spiral, negative, analysis, eigenvector, ?ows, general, attractor, analytical, de?ned, dilatancy, pattern, geology, function, jiang, case represents, mckenzie, paor, complex eigenvalue, extruding ows, ows, eigenvectors, ghostvectors, rapidly merge, journal, eqs, weijermars, fabric, number, journal structural, university, iacopini, rock, history, work, ghost eigenvectors, eat?cos?bt?, relative dominance, simple, accumulation, soto, deformation, tikoff, parameter, extruding, isochoric, triclinic ows, complete description, structural geology, table, vorticity vector, tectonics, homogeneous, stretching rate, path, solution, geometry, displacement path, initial condition, vorticity number, wn tn, direction, graphic showing, represent, shear zone, progressive deformation, ruelle, complex solution, source, pulsating pattern, zone, clear, asymptotic, positive, expected, relative, time, mathematical, eigenvalue, guilbeau, unstable spiral, external reference, rst eigenvalue, discussion, time increase, boundary wall, passchier, spiral saddle, iacopini journal, component, ramberg, stable, previous case, positive indicating, strain history, particle path, rate, paper, dij, kline, planar, swn, rock fabric, journal structural geology, represents, spiral source, triclinic, controlled, matrix, saddle, pulsating, kinematic, existence, behave, condition, tectonophysics, real eigenvector, tn, vorticity, result, situation, limit, eq, structural, strain accumulation, monoclinic, purely imaginary, stretching, complex, dilatancy parameter, ?ow, isa reference, isa