Mineral lineation produced by 3-D rotation of rigid inclusions in confined viscous simple shear

    Mineral lineation produced by 3-D rotation of rigid inclusions in confined viscous simple shear Fernando O. Marques PII: DOI: Reference:

S0040-1951(16)00050-0 doi: 10.1016/j.tecto.2016.01.013 TECTO 126907

To appear in:

Tectonophysics

Received date: Revised date: Accepted date:

14 February 2015 7 September 2015 8 January 2016

Please cite this article as: Marques, Fernando O., Mineral lineation produced by 3-D rotation of rigid inclusions in confined viscous simple shear, Tectonophysics (2016), doi: 10.1016/j.tecto.2016.01.013

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ACCEPTED MANUSCRIPT Mineral lineation produced by 3-D rotation of rigid

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Fernando O. Marques

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inclusions in confined viscous simple shear

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Universidade de Lisboa, Lisboa, Portugal

*Corresponding author:

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Tel.: 351217500000; Fax: 351217500064; E-mail address: [email protected]

Abstract

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The solid-state flow of rocks commonly produces a parallel arrangement of elongate minerals with their longest axes coincident with the direction of flow – a mineral lineation.

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However, this does not conform to Jeffery’s theory of the rotation of rigid ellipsoidal inclusions (REIs) in viscous simple shear, because rigid inclusions rotate continuously with applied shear.

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In 2-dimensional (2-D) flow, the REI’s greatest axis (e1) is already in the shear direction, therefore the problem is to find mechanisms that can prevent the rotation of the REI about one axis, the vorticity axis. In 3-D flow, the problem is to find a mechanism that can make e1 rotate towards the shear direction, and so generate a mineral lineation by rigid rotation about two axes. 3-D analogue and numerical modelling was used to test the effects of confinement on REI rotation, and, for narrow channels (shear zone thickness over inclusion’s least axis, Wr < 2), the results show that: (1) the rotational behaviour deviates greatly from Jeffery’s model. (2) Inclusions with aspect ratio Ar (greatest over least principle axis, e1/e3) > 1 can rotate backwards from an initial orientation

e1 parallel to the shear plane, in great contrast to

Jeffery’s model. (3) Back rotation is limited because inclusions reach a stable equilibrium orientation. (4) Most importantly and in contrast to Jeffery’s model and to the 2-D simulations,

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ACCEPTED MANUSCRIPT in 3-D the confined REI gradually rotated about an axis orthogonal to the shear plane towards an orientation with e1 parallel to the shear direction, thus producing a lineation parallel to the

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shear direction. The modelling results lead to the conclusion that confined simple shear can be

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responsible for the mineral alignment (lineation) observed in ductile shear zones.

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Keywords: mineral lineation; analogue and numerical modelling; 3-D inclusion rotation;

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confined flow; shape preferred orientation

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1. Introduction

Geological materials typically behave either as a (brittle) solid or as a viscous fluid (actual liquid like magma). In many situations, both behaviours can be found together in the

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same rock, as in crystallizing magma or in ductile shear zones, where the matrix behaves like a

fluid.

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viscous fluid (solid state creep) and the crystals behave like rigid solids in suspension in the

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Mineral lineations can form by a number of mechanical and chemical processes. Typically, crystals may undergo rigid rotation to a stable orientation, or grow under a favourable stress field, or trigger pressure shadows where new minerals grow to form an elongated composite aggregate, or deform plastically (stretch) to an elongated shape, or react with the neighbouring matrix to form a mantled porphyroclast system with long recrystallization tails. The present work concerns mineral lineation produced exclusively by mechanical rotation of elongate mineral grains under confined simple shear. Cloos (1946), in his classical work about lineations, discussed the relationship between the orientation of elongate mineral grains and the fluid flow in magma, and concluded that prismatic crystals are commonly parallel to the direction of flow, both in granitic batholiths and basaltic lava flows. He also recognized that the solid-state flow of rocks commonly produces a

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ACCEPTED MANUSCRIPT parallel arrangement of elongate minerals, with their longest axes parallel to the direction of flow. However, these observations contradict Jeffery (1922), who provided approximate

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analytical solutions that describe how rigid ellipsoids rotate in 3-dimensional (3-D) viscous

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flow.

Jeffery (1922) mathematically analysed composite systems of rigid ellipsoids in a

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viscous matrix, and produced an analytical solution that is applicable to: (1) an infinite embedding medium, thus considering that the inclusions are small when compared to the

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system (i.e. no confinement); (2) inclusions with no-slip contact with the matrix; (3) isolated,

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single inclusions; and (4) isotropic matrix. However, in many geological settings the flowing matrix (e.g. ductile shear zone) can be narrow relative to inclusion dimension, which has been called confinement (e.g. large crystals or boudins in narrow shear zones, Fig. 1). Confinement

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is defined by the ratio Wr between shear zone thickness and inclusion’s least axis. Therefore,

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the main objective of the present study was to investigate the effects of confinement on the rotational behaviour of rigid ellipsoidal inclusions (REIs) embedded in a viscous matrix. In

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order to accomplish this objective, 3-D analogue and numerical modelling were used. Bhattacharyya (1966) analysed the structures and textures of the deformed and metamorphosed pelitic, psammitic and basic rocks of a ductile shear zone, and found that the most prominent structural features consist of an axial plane schistosity, and a strong mineral lineation parallel to the longest axis of deformed (ellipsoidal) pebbles. The longest axis of the ellipsoidal pebbles represents the direction of major rock flowage, and therefore Bhattacharyya (1966) concluded that the mineral lineation is parallel to the direction of major flowage in rocks. Recognizing that the classical work of Jeffery (1922) predicts that REIs rotate continuously with a synthetic sense of motion under applied simple shear flow, Bhattacharyya (1966) mathematically analysed the problem in a different perspective, and concluded that the observed parallelism between lineation and flow direction can be the effect of REI interaction.

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Figure 1. Strong mafic granulite boudin in soft matrix mostly composed of calcite (currently dissolved by fresh water – dissolved carbonates). The ductile shear zone in the carbonate is bounded by strong mafic granulite (granulite wall-rock). The small ratio between the shear zone thickness and the boudin’s least axis (< 1.5) confined the flow of the matrix, and determined the boudin’s rotational behaviour. Photo taken in the uppermost allochthon of the Bragança Nappe Complex, NE Portugal (e.g. Dallmeyer et al., 1991; Marques et al., 1992, 1996).

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Jeffery’s (1922) mathematical analysis predicts that (i) REIs rotate continuously with a synthetic sense of motion under applied simple shear flow, and (ii) REIs with aspect ratio equal to one rotate at a constant rate, equal to half the strain rate. However, flow confinement can be an important additional factor in many natural phenomena. Several theoretical and experimental studies have shown that confinement can affect both matrix flow and rotation behaviour of REIs (e.g. Brenner, 1962; Cox and Brenner, 1967; Ganatos et al., 1980; SugiharaSeki, 1993; Marques and Cobbold, 1995; Biermeier et al., 2001; Marques and Coelho, 2001; Taborda et al., 2004; Marques et al., 2005a, b, 2014). Despite the great insight gained, there is a major drawback in the 2-D modelling of REI rotation in a viscous matrix: the REI has its greatest axis e1 already in the shear direction, therefore the lineation already exists, and only rotations about the Y-axis (Fig. 2) can be investigated. Therefore, in 2-D flow the problem is to find mechanisms that can prevent REI’s

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ACCEPTED MANUSCRIPT rotation and so produce a stable shape preferred orientation (SPO, e.g. Willis, 1977; Fernandez et al., 1983; Fernandez, 1987; Masuda et al., 1995; Pennacchioni et al., 2001). Several

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mechanisms have been proposed: (1) transpression or transtension (e.g. Reed and Tryggvason,

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1974; Ghosh and Ramberg, 1976; Passchier, 1987, 1997; Marques et al., 2003); (2) tiling by rigid inclusion interaction in dense inclusion suspensions (e.g. Ildefonse et al., 1992a,b; Tikoff

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and Teyssier, 1994; Mandal et al., 2001, 2004, 2005b; Piazolo et al., 2002; Samanta et al., 2003; Mulchrone, 2007); (3) slipping inclusion/matrix interface (e.g. Ildefonse and

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Mancktelow, 1993; Mancktelow et al., 2002; ten Grotenhuis et al., 2002; Ceriani et al., 2003;

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Bose and Marques, 2004; Marques and Bose, 2004; Marques et al., 2005a, 2007, 2014; Schmid and Podladchikov, 2004; Mulchrone, 2007; Mancktelow, 2013); (4) confined simple shear flow (e.g. Brenner, 1962; Cox and Brenner, 1967; Ganatos et al., 1980; Sugihara-Seki, 1993;

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Marques and Cobbold, 1995; Biermeier et al., 2001; Marques and Coelho, 2001; Taborda et al.,

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2004; Marques et al., 2005a, b, 2014); and (5) rheology of the matrix (e.g. Mandal et al., 2000, 2005a; Pennacchioni et al., 2000; Dabrowski and Schmid, 2011).

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In great contrast to the 2-D rotational behaviour of REIs, in 3-D the REI can rotate about axes other than the Y-axis (Fig. 2), in particular the one perpendicular to the shear plane (Z-axis in Fig. 2), which is related to rotation of e1 towards the shear direction. Therefore, inclusion rotation about the Z-axis is the main problem to solve, and that is the reason why it was mostly investigated here, in order to try and understand the likely mechanisms leading to a mineral lineation parallel to the shear direction. Given the main problem and the objective, we narrowed down the large number of simulations needed for a full parametric study, by focusing on: (1) finding the value of Wr (confinement ratio, Fig. 2); (2) then on finding whether confinement can make e1 rotate towards the shear direction, by putting e1 in the initial stage at the most critical orientations; and (3) finally analyse the effects of REI’s aspect ratio (Ar, Fig. 2) on Wr.

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Figure 2. Schematic drawing with reference frame (X, Y and Z axes), geometrical relationships, studied angles of rotation ( and ), axes of the ellipsoid (e1, e2, e3, in colour), and applied shear sense. Wr represents confinement, through the ratio W/e3. Ar is the aspect ratio of the ellipsoid.

Previous 3-D studies investigated the problem of rigid inclusion rotation, but they address infinite shear zones (e.g. Freeman, 1985; Passchier, 1987; Ježek, 1994; Ježek et al., 1994; Ježek et al., 1996). In the present work, the focus is the formation of a lineation in 3-D confined simple shear flow, i.e. not a full parametric study of the 3-D rotational behaviour of REIs in confined flow, which is much more complex and is still under way. Despite the complexity, the conclusion that 3-D confined flow can produce a mineral lineation stands, because confined simple shear can make e1 rotate about the Z-axis towards the shear direction, as modelled here.

2. Modelling procedure 2.1. Analogue Modelling and boundary conditions

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ACCEPTED MANUSCRIPT Common analogue materials used as the viscous matrix in this kind of study are typically silicone putties, such as transparent polydimethylsiloxane (PDMS, manufactured by

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Dow Corning under the trade name SGM 36; for PDMS properties see Weijermars, 1986). The

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density is ca. 965 kg m-3, and the viscosity around 5x104 Pa s at room temperature. Hard synthetic materials were used as REIs, with a density similar to that of PDMS to avoid

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buoyancy problems.

In contrast to the 2-D experiments, the Cartesian reference frame comprises 3 axes (X,

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Y and Z, Fig. 2), and the inclusions used in the present study were not cylindrical or prismatic.

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The REIs had two types of axial ratios (Fig. 2): (1) e1 > e2 = e3 (prolate ellipsoid of revolution), and (2) e1 > e2 > e3 (triaxial ellipsoid). In 2-D models there is one single aspect ratio Ar = greatest/least axis, but in the case of a triaxial ellipsoid there are three Ar (Fig. 2). The dihedral

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angle between the instantaneous orientation of the inclusion’s greatest axis (e1) and the shear

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plane (X-Y) is  (Fig. 2). The dihedral angle between the instantaneous orientation of e1 and the X-Z plane is  (Fig. 2). In most experiments the REI started with their e1 at different initial 

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(i), but with initial i = 0º. The experiments were carried out in a simple shear rig similar to the one used by Marques and Coelho (2001) and Marques et al. (2008), at room temperature (ca. 21 ºC), and at a constant far-field strain rate of ca. 1.25E-4 s-1. According to Weijermars (1986), this value falls in the interval appropriate for experiments with a linear viscous matrix. To ensure homogeneous simple shear, PDMS sticks strongly to the walls that drive simple shear flow (parallel to the shear plane), and is thoroughly lubricated in the contacts with the remaining four walls. The model dimensions were 500 mm along X, 120 mm along Y, and variable Z in the confined simulations. Given the previous knowledge on the 2-D rotational behaviour in confined flow, the analogue experiments were run with the inclusion embedded in a channel with a thickness already close to the Wr and/or Ar for which the REI rotates backwards. This procedure avoided

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ACCEPTED MANUSCRIPT carrying out an excessive number of analogue experiments. Given the main problem addressed in the present work (can confined simple shear flow produce a mineral lineation?), we focused

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on running experiments close to the critical end-members, for instance in terms of Wr, initial

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REI orientation, and Ar. The analogue results guided the numerical modelling and made it much more efficient, especially in terms of the number of simulations, and therefore time.

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Nevertheless, a good number of analogue experiments and numerical simulations had to be carried out in order to find the Wr for a given Ar, for which the REI starts rotating backwards

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and stabilizes. Many numerical simulations were also run to test the effects of initial orientation

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and ellipsoid shape, but only the more critical and relevant results for the objective of this paper are shown. A full parametric investigation is currently being carried out, which is not the

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objective of the present work.

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2.2. Numerical modelling, mathematical formulation and boundary conditions Incompressible viscous fluid rheology is widely accepted in the literature as a simple

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but effective approximation to the behaviour of rocks undergoing ductile deformation. In the case of steady-state flow of a viscous, incompressible Newtonian fluid at very low Reynolds number, the dynamic Navier-Stokes equations of flow reduce to the Stokes approximation. The mathematical model used in the present work is based on the two-dimensional steady-state incompressible Navier-Stokes equations for steady-state incompressible viscous flows:

(1) (2) where u is the velocity vector, p the pressure,  the density,  the dynamic viscosity and F the external body force ( and  are constant, and F will be assumed negligible in this model). Then, defining the scaled variables x = x L, u = u U , p = p P and t = t T , in terms of the

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ACCEPTED MANUSCRIPT characteristic length L, velocity U, pressure P and time T = L/U, Eqs. (1) and (2) become:

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(3)

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(4)

where Re = UL/ and Eu = P/U2 are, respectively, the Reynolds and Euler numbers. For

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flows at low characteristic velocity U and high viscosity , we have Re  1 and Eu  1, and all

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inertial terms in Eq. (3) become negligible. We thus obtain the Stokes approximation of the

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momentum equation for quasi-static (creeping) flows, which in dimensional form reads:

(5)

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The Stokes equations were solved on the 3-D parallelepipedic domain illustrated in Fig. 2, which was filled with an incompressible viscous linear fluid. A rigid ellipsoidal body,

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defined by the three principal axes e1 > e2 ≥ e3 (main aspect ratio Ar = e1/e3), was positioned at the centre of the domain (x, y, z = 0). In the no-slip mode, a no-slip condition at the inclusion–

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matrix interface was defined. Therefore, the fluid velocity was assumed equal to the inclusion surface velocity, with a magnitude equal to

, where r() = |r| is the radial distance from

the centre to the surface of the inclusion. The flow equations were solved using the finite element program FEMLAB (2002) developed by Comsol, which runs with MATLAB software. The flow equations, with the boundary conditions specified, were solved in the primitive variables u(u,v) and p over a finite element mesh, using the algorithm for incompressible Stokes flows implemented in FEMLAB. Further details on the computational method and numerical procedures are provided by Gresho and Sani (2000). The boundary conditions needed to complete the mathematical formulation and define a simple shear flow were (i) velocity set to values ± Vtop at Y = ±W/2 (Fig. 2), and (ii) velocity set

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ACCEPTED MANUSCRIPT to vary linearly between top and bottom velocities (with zero-mean) at the left and right end boundaries (straight-out condition). These settings made the vorticity (V) of the fluid to be –1

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where undisturbed by the inclusion. The above equations were solved in the 3-D

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parallelepipedic domain illustrated in Fig. 2, which was filled with an incompressible linear viscous fluid with matrix viscosity () set to 1E+4 Pa s. The high-viscosity fluid-modelled

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inclusions (1E+12 Pa s) were positioned at the centre of the domain (x, y, z = 0). The boundaries between inclusion and matrix were set to neutral (i.e. perfectly bonded). The

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thickness of the computational domain (W, representing the shear zone thickness) was varied

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according to the desired confinement (Wr = w/e3) by varying W, and L was set to about 100 times the greatest inclusion axis (e1). Therefore, the rotation of the REI is the result of the

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velocities applied at the top and bottom boundaries.

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3. Modelling Results 3.1. Analogue experiments

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Experimental analogue modelling was carried out first, and the critical experimental results relevant for the formation of a lineation are presented in Fig. 3. They show that decreasing Wr significantly perturbs the inclusion rotation behaviour relative to the theoretical predictions for unconfined flow. The inclusion rotates slower with decreasing Wr, but rotation is still synthetic with applied simple shear until Wr = 1.3. For Wr < 1.3 the inclusion rotates antithetically about the Y-axis until a stable orientation (Fig. 3), which depends on Wr and Ar. Simultaneously, e1 also rotates about the Z-axis toward the shear direction (X-axis), therefore decreasing the initial . As shown in Fig. 3, the REI’s e1 rotated ca. 9º (from  ≈ 21 or 17º, to  ≈ 12 or 9º, respectively), either clock or anti-clockwise, from an initial orientation oblique to the shear direction.

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Figure 3. Images of initial and final stages after  ≈ 7. The rigid inclusion is a prolate ellipsoid of revolution, whose e1 ≈ 4.3 and Ar ≈ 2.9. Note the simultaneous rotation of the REI’s e1 toward the shear direction (X-axis) on the XY-plane (upper panels), and the antithetic rotation on the XZ-plane (bottom panel).

3.2. Numerical modelling After having tested experimentally the effects of confinement on REI rotation, we first tested it numerically for a prolate ellipsoid of revolution. The reliability of the numerical procedure was verified using results obtained for circular (Ar = 1) and ellipsoidal (Ar = 1.5 and

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ACCEPTED MANUSCRIPT 3.0) inclusions using large Wr values. Both results agree remarkably well with theoretical predictions: for a circular inclusion, a constant angular velocity equal to half the strain rate was

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, with a pattern in close agreement with the analytical solution.

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obtained, while the elliptical inclusion rotated at a varying angular speed, which is a function of

Figure 4. A – Aspect Ratio Ar = 3.0 and Wr = 1.28. Note that the rigid ellipsoid rotates with the flow in the matrix (arrow at REI’s tip pointing downwards). B – Ar = 3.0 and Wr = 1.27. Note that the prolate spheroid rotates against the flow in the matrix (arrow at REI’s tip pointing upwards). Top to the right sense of shear Testing Wr at constant Ar In order to find the Wr for which the REI’s rotation switches from synthetic to

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ACCEPTED MANUSCRIPT antithetic, we tested Wr values until the switch occurred. For a prolate spheroid with Ar = 3.0, similar to the ellipsoid used in the analogue experiments, we found Wr ≈ 1.275 as the

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confinement necessary to make the REI to rotate antithetically (Fig. 4).

Figure 5. A. At Wr = 1.3 and Ar = 3.0, a prolate spheroid rotates forward. B. At Wr = 1.3 and Ar = 3.0, a triaxial ellipsoid rotates backwards. Testing REI shape at constant Wr and Ar In order to test the effects of ellipsoid shape, we compared the rotational behaviours of

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ACCEPTED MANUSCRIPT prolate spheroidal and triaxial ellipsoids, initially with e1 parallel to X and on the XY-plane. The results (Fig. 5) show that, for identical Wr = 1.3 and Ar = 3.0, the triaxial ellipsoid rotates

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antithetically, and the prolate spheroid rotates synthetically.

Testing  at constant Ar and Wr

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We also tested the effects of changing , and the results show that, whatever the 

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value, the REI rotates simultaneously backward and towards the X-axis (Fig. 6).

Figure 6. Velocity field in the XY plane showing the rotation of the axes and the overall rotation of the triaxial ellipsoid, with its e1 rotating towards X, for initial  = 45º (A), 60º (B) and 75º (C).

4. Discussion From 2- to 3-D modelling there is a major jump, because one angle of rotation in 2-D ( 14

ACCEPTED MANUSCRIPT angle measured on the XZ-plane perpendicular to the vorticity Y-axis) becomes two angles in 3-D (two rotation axes,  and ), and, especially, the ellipse becomes an ellipsoid, which means

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that: (1) Ar is not unique as in 2-D, because there are 3 different axes in tri-axial ellipsoids; (2)

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the tri-axial ellipsoid can rotate about its own three axes; and (3) there are many different ellipsoid shapes that may affect the formation of an SPO. Given this major jump from 2- to 3-

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D, it is not so intuitive that confinement can generate a mineral lineation by rigid rotation of prolate REIs. The modelling results reported here show that there are two fundamental steps in

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inclusion rotation: the REI rotates backwards from an initial orientation of its e1 parallel to the

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shear plane, as a guarantee of confinement, and simultaneously towards the shear direction. It is this latter rotation that ultimately guarantees the development of a lineation parallel to the shear

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direction.

Jeffery (1922) showed that rigid ellipsoids rotate indefinitely and synthetically in simple

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shear. Therefore, the main problem with the interpretation of SPOs was, until recently, that SPOs generally show inclusion stabilization at shallow positive angles (dip opposite to shear

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sense), which could not be explained by theories that assume an isolated REI that is perfectly bonded to the matrix in infinite simple shear flow. This mismatch between field data and models has been analysed in a number of studies that show the possibility of inclusion stabilization at shallow angles to the shear plane. Most studies have addressed this problem using a 2-D approach, and concluded that REI rotation can be prevented by a number of mechanisms as presented in the Introduction section. However, it is not guaranteed that such mechanisms are viable in 3-D flow. Transpression and transtension seem to work in 3-D as viable mechanisms (e.g. Passchier, 1987, 1997), but a slipping inclusion/matrix interface or flow confinement are not guaranteed to work as mechanism capable of producing a lineation parallel to the shear direction. Preliminary 3-D analogue and numerical modelling using a slipping inclusion/matrix interface shows a very complex REI behaviour, and is therefore still

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ACCEPTED MANUSCRIPT in a working stage. In contrast, a viable mechanism, as reported here, is confinement. Given that confinement is a sine qua non condition in the present investigation, the REI

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initial orientation is critical. For instance, if the REI is a prolate ellipsoid with a relatively large

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Ar, and its e1 is initially at high angle to the shear plane, then the REI will rotate synthetically because Wr is not enough to prevent continuous forward rotation. Therefore, in order to

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guarantee backward rotation to a stable  angle, the initial REI orientation should be such as to guarantee the confinement effect.

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Where can we find confinement in nature? It can be found in layered rocks like

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sedimentary, metamorphic or magmatic rocks. Layers of contrasting mineral composition usually show different rheological behaviour (Fig. 7), indicating that flow can be markedly

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heterogeneous due to viscosity anisotropy (e.g. Mandal et al., 2005; Fletcher, 2009; Dabrowski and Schmid, 2011; Griera et al., 2013). The different rheological behaviour between adjacent

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layers can concentrate shear deformation in narrow layer-parallel ductile zones of softer (lower viscosity) rock bounded by stronger (higher viscosity) rock, and lead to confined flow. The

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photograph of a natural ductile shear zone in Fig. 1 illustrates confinement. Again, different rheological behaviour between layers of contrasting mineralogy leads to shear concentration: the softer mylonites inside the ductile shear zone are mostly composed of calcite (± silicates), bounded by stronger mafic granulite rich in garnet, pyroxene and plagioclase. A REI of mafic granulite tilted opposite to the shear sense, which is top to right, is present within the shear zone, and has its e1 parallel to the shear direction, deduced from the mineral stretching lineation. Our interpretation is that this is a stable equilibrium orientation resulting from confined flow. Information on the geological setting of the rocks illustrated in Fig. 1 can be found in Marques et al. (1996).

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Figure 7. Photograph of a natural example of a layered mylonite from the Bragança Massif, NE Portugal, which is a mafic granulite deformed and retrogressed to amphibolite facies. Note

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that, for the same PT conditions, layers behave quite differently. For instance, while the darker

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amphibole-rich layers deformed by folding (white arrow), the lighter plagioclase-rich layer deformed by fracturing. A similar situation is observed with layers still rich in high-grade metamorphic minerals like pyroxene and garnet. This strain partitioning between adjacent layers can concentrate shear deformation in narrow, softer bands inside the wider shear zone, and lead to confined flow. See Marques et al. (1992, 1996) for the geological setting. Sense of shear is top to the right.

5. Conclusion Modelling results allows drawing the following conclusions: 1. Confinement greatly affects the ideal rotational behaviour of REIs expected from Jeffery’s model. 2. The prolate spheroid of revolution and the triaxial ellipsoid rotate differently, which means

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ACCEPTED MANUSCRIPT that shape matters. 3. Confinement can make the REI rotate backwards from an orientation with the greatest

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principal axis parallel to the shear plane and stabilize dipping opposite to the shear sense.

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The stable orientation depends on inclusion shape and aspect ratio.

4. In contrast to the 2-D models, the 3-D models show that confinement can make the REI’s

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greatest axis rotate, about an axis perpendicular to the shear plane, toward the shear direction.

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5. We conclude that rigid rotation in confined simple shear can be responsible for the mineral

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alignment (lineation) observed in ductile shear zones. 6. The analogue and numerical models can explain many features observed in natural ductile

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shear zones, and can be used to extract relevant physical information.

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Acknowledgements

The author acknowledges a sabbatical fellowship awarded by FCT, Portugal

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(SFRH/BSAB/1405/2014).

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Mineral lineation is common in shear zones but does not conform to Jeffery’s theory 3-D models show the effects of confinement on rigid ellipsoid rotation Ellipsoids rotate antithetically until a stable orientation Confinement can make the prolate ellipsoid rotate toward the shear direction Confinement can produce the mineral alignment observed in ductile shear zones

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