Strain localisation in mechanically Layered Rocks , insights from numerical modelling

L. Le Pourhiet, B. Huet, P. Agard, L. Labrousse, L. Jolivet, and K. Yao UPMC Univ Paris 06, UMR 7193, ISTEP, 75005, Paris, France CNRS, UMR 7193, ISTEP, 75005, Paris, France Department for Geodynamics and Sedimentology, University of Vienna, Althanstrasse 14 1090 Vienna ISTO, UMR CNRS 6113, Université d’Orléans, Campus CNRS, 1A rue de La Férollerie, 45071 Orléans Cedex, France Mines ParisTech, Centre de Géosciences, 35 rue Saint Honoré, 77305 Fontainebleau, France


Introduction
Mechanical layering is present at all scale in rocks as a results of sedimentation, diagenesis, inherited strain (nappe stacking, folding, foliation, schistosity), intrusion of magmatic bodies.Shear localisation in these layered media results in strain partitioning between high strain domains shear bands or kink bands and low strain domains where the initial layering (Sn) is deflected but preserved.The scale of these structures ranges from grain size (hundreds of µm) up to hundreds of km (Hippertt, 1999).These structures constitute a classical tool to reconstruct the kinematics of the deformation (Simpson and Schmid, 1983;Hanmer and Passchier, 1991) larger scale kinematics and draw conceptual geodynamic models at the plate boundary scale; e.g.Cloos (1946); Ramsay (1980); Choukroune et al. (1987); Cosgrove (2007).Kinematic models based on flow partitioning related to the layering of the media (Platt, 1984;Platt and Vissers, 1980;Lister and Williams, 1983;Michibayashi and Murakami, 2007) lead to non-unique solutions because strain consists of more independent components than kinematic constraints allow to solve for.As a result, several kinematic models may be proposed to explain the same finite deformation (Carreras, 2001).The only solution to recover the full dynamics of the system with a unique solution is to introduce rheological relationships in order to compute the complete stress/strain evolution.
As it was shown by Cobbold et al. (1971), models with alternating stiff and weak isotropic viscous layers displays an effective anisotropic behaviour at larger scale allowing the formation of kink bands or shear bands depending on the orientation of the initial layering orientation.The orientations of these bands are successfully reproduced by mathematical models based on linear stability analysis established for anisotropic media (Biot, 1964;Cosgrove, 1976), which are yet applicable only for very small strain and peculiar orientation of the initial layering versus the maximum compressive stress σ 1 .Attempts have been made to numerically model large strain in anisotropic media (Muhlhaus et al., 2002(Muhlhaus et al., , 2004)).These studies were focusing on folding and mantle convection.The main results were that anisotropy enhances the shear instabilities.Small scale perturbations of the anisotropy compared to the layer thickness favoured the growth of mullions over folds, which develop for larger scale perturbations (Muhlhaus et al., 2002).However, models using anisotropic constitutive laws rather than a layered media have the disadvantage not to provide any physical length-scale.
Following previous studies (Mancktelow, 2006;Frehner et al., 2011), we make use of numerical codes that the momentum equation for stress and strain to perform dynamic numerical models of previously stratified rocks.The paper concentrates on the impact of a pre-existing layering on the occurrence, the dynamics and the preferred orientations of new localised band of deformation and new flattening planes in medium (10 m) Figures

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Full to large scale (1 km) shear zones deforming in a regime which is closed to the brittle ductile transition.As Mancktelow (2006) showed plastic yielding was an important parameter for localisation, we introduce a maximum yield strength using morh coulomb rheology (Vermeer and De Borst, 1984;Le Pourhiet, accepted).Exhumed examples of such shear zones are described in all geodynamic settings (compression e.g.Alps (Handy et al., 2005), extension eg Betics (Platt and Vissers, 1980) Aegean (Jolivet et al., 2010), strike-slip e.g.red river fault (Leloup et al., 1995), cap de creus spain (Fusseis et al., 2006)).Based on the example of the western Gulf of Corinth (Greece), where an active basal detachment zone is suspected to accommodate the extension at around 10-15 km depth, the models use a reference shear rate of 1 cm yr −1 distributed over 1 km thick zone (Avallone et al., 2004) and a confining pressure of 400 MPa (intended to simulate a shear zone buried at 15 km depth).In these conditions, the background strain rate approaches 3 10 −13 s −1 and brittle ductile interaction are expected to occurs for effective viscosity as low as 10 20 Pa s −1 .
After describing the model set up, we describes how varying the orientation of the layering versus the bulk shear direction may enhance or retard strain localisation and how the layering controls the orientation of the new localised shear zones when they occurs.The modelling results are analysed in term of finite strain, evolution of pressure and deviatoric stresses in the strong and weak layer of the system with shear localisation.In a second part, we fix the orientation of the layering versus shear direction for a case that strongly favour shear localisation in order to study in details the impact of length scale, effective viscosity contrast, plasticity, and strain rate on the degree of localisation.Introduction

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Full 2 Influence of the initial orientation on the deformation pattern

Model set up
The mechanical anisotropy is introduced following the approach of Cobbold et al. (1971) rather than modelling true anisotropic media in order to impose a length scale that overrides the one of the numerical mesh but still allows the formation of several localised band in the models.Numerical experiments allow modelling background strains of the order of 100 %.We focus the quantitative analysis on large strain simple shear numerical experiments using fully visco-elasto-plastic rheology.We present the large panel of micro structures that can form for different degrees of obliquity between the main anisotropic direction and the shear direction.The description of these structures is accompanied by the measure of the large scale softening or hardening associated to their formation.This complements the pre-existing linear stability analysis approach since the numerical simulation are run for larger strain and for orientations which are not initially aligned with the instantaneous stretching and shortening axes.Therefore most of the simulations are done at a given 4 × 1 km scale with viscosities that are believed to be close to those at the top of the ductile crust, i.e. 10 19 and 10 21 Pa s −1 for the weak and strong layers respectively.has been chosen in accordance with The initial dip δ 0 of the layers Sn (Fig. 1a) is counted positive when they dips towards the right of the model.The models have a large aspect ratio to avoid the impact of boundary conditions.The grey and white layers on Fig. 1a represent the alternance of low and high viscosity layers.Each layer is 30 m thick and corresponds to 3 elements.Boundary conditions correspond to simple shear simulating an infinitely long body by fixing the velocity as parallel to the direction of shearing on the lateral boundary.The vertical gradient of horizontal velocity is constant on the lateral boundaries to simulate large scale simple shear at the scale of the model.Throughout the paper, the intensity of bulk shear strain γ bulk is given in % and corresponds to the tangent of the angle γ on Fig. 1a.Horizontal velocity varies from 0.5 cm yr −1 to the left at the top of the model to Introduction

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Full -0.5 cm yr −1 to the right at the bottom (Fig. 1b).To avoid the impact of lateral boundary conditions, the results shown correspond to portions located at the center and top of the simulations (see the grey and white boxes in Fig. 1b).
We have deliberately simplified the problem by removing gravity and temperature dependance of viscosity from the model to keep the problem general and independent from the orientation of the shear direction versus gravity and thermal gradients.All the layers have the same visco-elasto-plastic rheology with only viscosity varied and the code approximates the solution of the momentum equilibrium using a mesh of 400 × 100 elements.The conservation equation relates the change of volume to the pressure via the elastic compressibility of the media.Internal friction angle (φ) is set to • , cohesion (C 0 ) to 20 MPa, the elastic bulk (K ) and shear (G) moduli are respectively set to 50 and 30 GPa.
At γ bulk = 100 %, a large range of microstructures can form depending on the original orientation δ 0 of the layering.They include symmetric boudins, asymmetric boundins, true S-C structures and symmetric folds or kink folds (Fig. 3).Type I and II correspond to positive values of δ o and include the models in which the strong layers are initialy stretched.They have been separated based on the predominance of shear bands in the type II models.Types III and IV correspond to models in which the initial foliation shortens initially (δ o < 0).They have been separated based on whether the deformed initial layering (Sn) is being tilted towards positive values of δ during the simulation (Type IV) or not (Type III).

Results at one glance
At γ bulk = 100%, a large range of microstructures can form depending on the original orientation δ 0 of the layering.They include symmetric boudins, asymmetric boundins, true S-C structures and symmetric folds or kink folds (Fig. 3).Type marked by the formation of boudins at small obliquity (Type I, Fig. 4a) and shear bands at large obliquity (Type II, Fig. 5a).The models with layers that shorten during shearing (δ 0 < 0), all accommodate the shortening by folding, but, in the models presented here, the finite wavelength of the folds vary with the initial dip of the layers.For initial dips ranging from δ 0 = −10 • to δ 0 = −30 • the foliation is folded at small scale compared to the size of models.The asymmetry of the fold limbs increases as the obliquity of the initial layering increases (Type III, Fig. 6a).At δ 0 = −30 • , the asymmetry and the anisotropy are such that kink folds develop (Type IIb).On the opposite, the models with δ 0 < −30 • have their foliation partially or completely (δ 0 = −45 • ) transposed toward positive values of δ 0 .In these cases, C structures begin to develop in the transposed part of the model (Type IV, Fig. 7a) at large γ bulk .Only the models with low negative obliquity (δ 0 = −10 • or −20 • , i.e.Type IIa) display hardening in the early deformation stages.The models with Sn dipping in the shear direction, i.e. to the right, tend to show no or little softening.
There are large pressure gradients between the weak and strong layers of all of the models.In the types I and II, the pressure is always higher than lithostatic in the weak layers.In the type III case, it is the opposite.The type IV case shows a reversal in the early deformation stages and thereafter mimics the type II case, once some parts of the model are transposed.Only type IV shows a pressure gradient switch during the Introduction

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Full early deformation stages.During this transient event, the deviatoric stresses are about 100 MPa although the pressure gradient is zero.For the initial and final conditions of type IV as well as for all the other cases, the average pressure difference between the weak and the strong phase is approximately equal to the deviatoric stresses in the strong phase, i.e. 100 to 200 MPa.
3 Detailed mechanical insights

Type I
The finite strain reflects the change in strain partitioning with increasing obliquity of δ o .This partitioning is outlined both by the large amount of finite shear strain in the weak layers (Fig. 4a) and by the oblique orientation of the MSD versus the current Sn (Fig. 4b).It is particularly visible at δ 0 = 10 • , where the white lines of the stretching direction are systematically parallel to the layering in the strong layers (pure stretching in the Sn) and oblique to the layering in the weak layers (simple shear flow parallel to the Sn).
Close to the applied shear, the strong layers tend to be cut at high angles by small left dipping shear bands which form a regular geometric pattern of strain in all the models.At small scale, all the Type I cases correspond to boudinage of the strong layers.However, as the obliquity increases, the asymmetry of the boudins increases and the MSD tends to become parallel to the bulk direction of shear in the high strain zones (Fig. 4b, up) rather than being aligned to the Sn (Fig. 4b, down).At high obliquity, the high strain zones tend to coalesce to form incipient top to the left shear bands that dip to the left (hence opposite to the dip of the initial foliation).At low obliquity, we interpret the alignement of the MSD with the Sn as the formation of symmetric boundins within simple shear boundary conditions.The increasing obliquity also correlates with an increasing depth of penetration of the deformation of the strong layers.Fig. 4a shows that the strong layers tend towards the Introduction

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Full green colour even in the center of the model at δ 0 = 30 • , while at δ 0 = 10 • only the top 10 % to 20 % of the model undergoes noticeable deformation in the strong layers.The depth of penetration of the deformation is also correlated with the rise of the average shear stress in the strong layers as obliquity increases (Fig. 4c).All the models that fall into the Type I class show very little softening with bulk strain.This class is also characterised by a constant pressure gap of 100 MPa with a gradient directed from the weak towards the strong layers.
The left lateral displacement is distributed at all depths from the early stage of the bulk deformation (curves on Figs.4a and 4d for γ bulk = 30 %).The rate of displacement with depth is constant with increasing bulk strain.However, the initial rate of displacement may either be constant with depth (case δ 0 = 10 • ) resulting in a linear finite displacement profile at γ bulk = 100 % or more concentrated at the top of the column (case δ 0 = 30 • ) resulting in a non linear finite displacement profile.Therefore, there is no true localisation of strain with time, i.e. no abandoned shear zones.This steady state behaviour is consistent with the absence of strain weakening demonstrated by the constant deviatoric stress measured in the strong layers.

Type II
Type II (Fig. 5) differs from Type I by the clear predominance of shear bands over simple shear in the plane of the Sn.not significantly change with increasing bulk strain although the lateral displacement is more concentrated at the top of the profile.In the δ 0 = 50 • case, the shear bands also appear early in the strain history.In that case, the rate of left lateral displacement decreases with increasing bulk strain in the lowest part of the model indicating that strain localisation is occuring from 60 % bulk strain on.In the δ 0 = 45 • case, the localisation of the shear bands seems to occur between 20 % and 40 % of bulk strain when oscillations begin to appear on the left lateral displacement profile.As the amplitude of those oscillations increases, the rate of displacement at depth decreases denoting again that strain localisation occurs at the scale of the model.As for Type I, shear bands form early in the strain history (Fig. 5d).However, the activity of the shear bands located at depth decreases with time for the models with initial layering obliquity of 45 • and more.
The progressive strain localisation highlighted in the displacement profiles is compatible with the slight strain weakening with increasing bulk strain observed in the evolution of the deviatoric stress (Fig. 5c).This weakening is maximum for δ 0 = 45 • and corresponds to the drop of average pressure in the weak layers with time.On average, the pressure is equal to the lithostatic pressure but there is a difference of 150 MPa to 100 MPa between the weak and strong layers.The pressure gradient is directed from the weak to the strong layers, as in Type I.The pressure difference between the strong and weak layers is very close to the average deviatoric stress invariant in the strong layers.

Type III
The Type III models are characterised by small scale folding and a higher pressure in the strong layers than in the weak layers (Figs. 3,6).Thereby this class defines models which are exactly opposite to the Type I pattern.Type III is separated in two sub-types because small obliquity models (Type IIIA) are characterised by strain hardening, while δ 0 = −30 • is characterised by slight strain weakening (Type IIIB, Fig. 6c).After 100 % bulk shear, the amplitude of the folds is higher at high obliquity than at small obliquity (Fig. 3), indicating that the growth rate of the folds rises with increasing obliquity.Introduction

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Full Similarly, the strong layers are still in the deep blue (low strain) at the center of the model for δ 0 = −10 • while for δ 0 = −30 • the entire medium deforms with kinks folds.As the initial obliquity of the layers increases, the penetration depth of folding instability increases (Fig. 6a) and the deviatoric stresses in the strong layers increases (Fig. 6c).In all the cases, the MSD is roughtly normal to the strong layers (Fig. 6b).Therefore, the shortening of the strong layers is composed of both a folding component and a pure thickening component.For the Type IIIA the MSD is oblique to the weak layers indicating that those layers deform in simple shear.However, for the Type IIIB the MSD is parallel to the layers that dip toward the right within the kink bands and oblique to the weak layers that slightly dip towards the left between the kink bands.The pure stretching domains therefore correspond to upright limbs of the kink folds (Fig. 6b).
The depth-displacement profiles indicate that the deformation is distributed at all depths from the early phase of the deformation.Those profiles have a tendency to linearise with increasing bulk strain (especialy δ 0 = −20 • ) which would imply that the shear rate increases downward with increasing bulk deformation to become perfectly distributed though the whole medium.This indicates that a natural shear zone forming in such a configuration would widen with increasing strain.In these models, the pressure is very close to lithostatic in the weak layers and 100 to 150 MPa higher in the strong layers.As noticed earlier for the Types I and II, the strain weakening/hardening is directly related to the drop/rise of the pressure gradients.In the δ 0 = −30 • case, the pressure drops at the onset of the high angle, right dipping, kink bands i.e. at γ bulk = 50 %.In the hardening cases, the pressure tends to increase with increasing strain in the strong layers (Fig. 6c).In the Type IIIb case, the softening does not correspond to strain localisation at bulk scale but to the rotation of the soft layers towards directions that are more compatible with the bulk strain direction.Introduction

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Full  7b).At δ 0 = −45 • , the strong layers do not deform internally.This lack of micro-structures at δ 0 = −45 • forbids the initial structural softening (Schmalholz et al., 2005) that affects the strong layers in the two other cases.As a result, the strong layer are effectively stronger and rotate passively in a simple shear regime as outlined by the obliquity of the MSD to the strong layers.In the weak layers, the shearing is so intense that the MSD is parallel to the current Sn (Fig. 7b).
The pressure gradient between the strong and weak layers switches from directed towards the weak layers, to directed towards the strong layers as the transposition of the layers occurs on Fig. 7c.This switch is synchronous with strain weakening and as the pressure begins to rise in the weak phase, a slight strain hardening occurs in the models.The strain weakening is linear for the δ 0 = −40 • case and very abrupt for the two other cases.The shear bands form late in the history of the deformation.

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Full They appear as the layers get transposed and are marked by an increase of the lateral displacement at the surface of the depth deformation profiles.It results that, in these cases, the shear localisation propagates downward as the layers get transposed.In the δ 0 = −40 • the depth deformation profile is separated into two areas of constant displacement gradient, i.e. two linear profiles crossing at the base of the transposed domain.In the δ 0 = −50 • case the gradient of displacement decreases more smoothly with depth as marked by the curvy shape of the depth displacement profile.Note, however, that the shear bands formed in the later deformation stages are not as continuous as the shear band that formed in the type II case.They localise at the tip of the mullions and folds and then coalesce progressively.
4 Scalability of the results for δ 0 = 40 In the models presented here, the viscosity contrast was set to 100.It is beyond the scope of the paper to perform a full parametric study of the scaling of the results for all the orientations.Nevertheless, we have concentrated our effort on the δ 0 = 40 • model, which best displays the formation of shear bands associated to extensional crenulation cleavages (ECC).Since we use a linear viscous rheology in the experiments, we can define an average maximum viscous shear stress as τmax = µ γ. (1) In our reference set up, the background strain rate is γ = 3 10 −13 s −1 and the viscosities of the weak and strong layers are respectively 10 19 and 10 21 Pa s −1 reaching average maximum viscous shear stress of 3 MPa and 300 MPa.Because we also use a pressure sensitive plastic yield criteria, it is worth comparing these values with the 300 MPa confining pressure applied to the models.In the reference case, it is clear that the strong layers are close to plastic yielding and the weak layer are very far from reaching the yield criteria.

SED Introduction Conclusions References
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Full Firstly, to test whether the results were independent of the length scale, we have run an experiment in which the dimensions were divided by a factor 100 yielding a total model dimension of 10 by 40 m and initial layer thickness of 30 cm (Fig. 8a).We have kept the viscosities and background strain rate similar as in the reference experiment (Fig. 8f).The results obtained are similar enough to conclude that our models are valid at least from the kilometers scale to the outcrop scales.
The series of experiments framed in the red box on Fig. 8(d, e, f, g) aims at characterising the impact of the strong layer viscosity on the outcome of the models.For larger viscosities, the orientation and the spacing of the shear bands do not change.There is one more shear band in the experiments d and e than in the reference model, and the displacement profiles shows that at the scale of the experiment, the strain is much more localised for viscosity contrast greater or equal to 1000.In the experiment in which the strong layer viscosity is decreased by a factor 10 (Fig. 8g), the displacement profile indicates that strain localisation is weak at the scale of the experiment and the finite strain does not feature proper shear bands.The viscosity of the strong layer is therefore a decisive factor for shear banding to occur at small scale and strain localisation to occur at large scale.The large similarity of the experiments d and e indicates that the localisation process saturates at high viscosity.
The series of experiments framed in the blue box on Fig. 8(b, f, i) aims at verifying the effect of the background strain rate on strain localisation.At lower strain rate, shear banding does not occur nor strain localisation.At higher strain rate, the shear bands are distributed through the entire medium, and their spacing decreases roughly down to the thickness of the Sn.Eventhough the strain is localised at small scale, it is distributed at the scale of the model.For these higher strain rates (Fig. 8i),the shear stress in the weak layers is on the order of 30 MPa and is therefore no longer negligible compared to the plastic yield strength of the strong layers.As a results, the effective viscosity contrast between the layers drops and much less strain occurs in the weak layers probably causing the lack of localisation at large scale.Introduction

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Full The experiment of Fig. 8h was produced by increasing the viscosity of both the weak and strong layers by a factor of 100, and decreasing the background strain rate by a factor of 100 compared to the experiment i.The results obtained are very similar and lead to the conclusion that our results are applicable to a wide range of viscosities and strain rates as long as the viscous stress in the strong layer exceeds the plastic yield strength.
Amongst all the models presented in Fig. 8, only the models b and g do not produce shear bands.Both cases correspond to lowest bound of the maximum viscous shear stress of the strong layer, i.e. 30 MPa, hence one order of magnitude lower than the confining pressure.On the other hand, when the maximum viscous shear stress of the strong layer largely exceeds the yielding criteria, the results are not modified (e.g.experiments d and e).We therefore postulate that although plastic strain is not predominant in the reference model, it is necessary to include a constant yield strength in the rheology of the rock in order to form the localised ECC.In order to confirm this hypothesis, we have run a simulation which is identical to the reference experiment except that plastic yielding was disabled.The results (Fig. 8c) lack the formation of shear bands as did the b and g experiments.
In all the models, one observes the same alignement of large strain structures with a dip of approximately 10-15 • towards the left, which indicates that the dip of the C structure is controled by the anisotropy at the beginning of the deformation rather than by the scale, the strain rate or the viscosity contrast.At constant strain rate, our models are lengthscale independent.At large strain rate, shear bands are favored but the strain is more distributed in the model.As the strain becomes more penetrative the flow pattern becomes more linear at large scale.In all the cases, true plastic deformation is a necessary condition for the localisation of shear bands.
These observations suggest that the knowledge of the yield strength of the strong layers is a prerequisite to infer the viscous property of the rocks from field observations.In the model presented here, we have used Mohr Coulomb yield criteria.This choice can be discussed since it is pressure dependent while localised shear bands form Introduction

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Full through the entire lithosphere in nature.This problem can be ruled out if we assess that the low permeability of the lower crust and upper mantle favours undrained conditions and that non-hydrostatic fluid pressure can limit the depth dependence of frictional yield criterion (Sibson, 1994).We also acknowledge that non-newtonian stress dependent rheology could allow the deformation to localise.Dislocation creep would enhanced the growth rate of the small scale shear instabilities but in order to localise shear bands at great depth, a mechanism with a fixed yield strength like Peierls creep (Goetze and Evans, 1979) would probably be better a candidate.

Direction of anisotropy and type of structures formed
The paper considers a foliated media where the strength of the weaker layers is two orders of magnitude less than the strength of the stronger layers.Assuming that stresses are continuous across the interface between the strong and the weak layers, one finds that the shear stress and the normal stress acting on the interface of the layers, i.e. the local Sn plane, must be the same in both layers.The shear strength of the weak layers being negligible, it may be considered as a good approximation to say that the shear stress along the Sn is close to zero.This implies by definition that, at the smaller scale, the Sn is one of the principal stress axis within the strong layers.Therefore, at small scale the hypothesis of Biot ( 1964) is always satisfied.A second implication of the continuity of stress is that the deviatoric stress in the strong layer must be equal to the gap of pressure between the strong and weak layer.Implication that is confirmed by the monitoring of the stress and pressure presented in Figs.4c, 5c, 6c and 7c.It results that the sign of the pressure difference defines if the orientation of the Sn (reported with a square on Fig. 9) supports σ 1 or σ 3 .
Accounting for this local orientation of the principal stress axis, we may draw the maximum shear stress axes within the strong layers (a and b on Fig. 9) and define Introduction

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Full the kinematics of the corresponding shear zones according to the local principal stress orientation.After reporting the a and b axis on the representation of shear strain at γ bulk = 10 %, and comparing with the preferred shear orientations at γ = 100 % on Fig. 3, we show that for the types I, II and IV, the orientation of the newly formed shear bands is acquired at the early beginning of the deformation.The fact that the shear localisation directions are acquired at very small strain indeed explains a posteriori why the small strain formulation of Biot (1964) allows retrieving the orientation of large strain feature like kink bands and ECC (Cosgrove, 1976).At the macro scale, the C structures correspond only to one of the two conjugate local maximum shear orientations, i.e. the one compatible with the larger scale flow direction.In the Type I, the chosen orientation is relatively similar to the extensional crenulation cleavage first described by Platt and Vissers (1980) or to the C' orientation in primarily isotropic medium (Berth é et al., 1979).In the Type II and IV case, the chosen orientation correspond to one of a neoformed shear band and forms S-C structures together with the Sn.Finally, in the Type III case the orientation b corresponds to a true crenulation cleavage.In conclusion, all types of micro-structures can form within the same boundary conditions.While the boundary conditions control the overall kinematics, the direction of the initial anisotropy controls the type of micro-structures that form and the effective strength of the sheared media.Considering the relation between meso-scale and microscale structure, it is obvious that as structure rotates at the mesoscale, the rocks may go through cycle of hardening (folding) and weakening (thining) at the microscale and give rise to anastomosed shearzone as was already pointed out by Ghosh and Sengupta (1987).

Petrological implications of the pressure gradients
The weak and strong layers of the models sustain very different pressure values which deviate by one fourth to one half of the confining pressure applied at the boundary.We have found that the amplitude depends on the effective strength of the strong layers and that the sense of the gradient depends on its orientation versus the shear direction.The Introduction

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Full pressure gap we computed are half of the amplitude found for very elongated ellipse clasts and the relationship with the orientation shows similar trends as Schmid (2005) found around rigid inclusions.These so called tectonic over/under pressures are rarely reported in the rocks (with exception of (Vrijmoed et al., 2009)) but predicted by all the models in use in geodynamics (see Mancktelow (2008) for review).Our models predict that for γ bulk > 120 % the pressure is very close to lithostatic in the weak deforming part of the model, regardless of initial orientation.Accounting that dynamic recrystallisations and therefore synkinematic paragenesis are more likely to occur in the weakest part of the rock, our model could provide an explanation for the scarcity of proven examples of tectonic over/under pressure in rocks.Incidently, the pressure difference we predict is on the order of the strength of the stronger layers of the rocks, i.e. close to 100 MPa, which compares with the error bar of thermobarometric methods.
On the other hand, large pressure differences at small scales induce large pressure gradients, which, according to our model, are maintained as long as deformation continues.Pressure gradients contribute to chemical potential differences and mineral solubilities without increasing the permeability of the crust.For example, this mechanism induces the dissolution of a mineral like quartz in high-pressure domains and its precipitation in low-pressure domains (Fletcher, 1977).We postulate that, in anisotropic media, veins could form by diffusive small-scale mass transfer enhanced by large and long-lived local pressure gradients rather than by large-scale advective mass transfer which requires seemingly unrealistic quantities of external water (Ferry and Dipple, 1991).Following this hypothesis, our model would thus predict that elements migrate toward the strong layers in Types I and II regime and that quartz veins may form in the weaker layers in type III and in the early deformation stages for Type IV orientations.
More work is needed in order to quantify the scale at which this process may become effective.

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Reconstructing the strain history
This theoretical study has implications on the interpretation of deformation in anisotropic rocks observed in the field.We have shown that for a given simple shear kinematics, all kind of micro-stuctures can form as a function of the initial orientation of the pre-existing foliation or stratification relative to shear strain field.In all cases, the kinematics indicated by the shear-bands or the asymmetry of folds has been shown to be consistent with the sense of shear at the boundary, whatever the initial direction of the layers.We note however that due to the anisotropy of the media, the neoformed shear bands are rarely parallel to the direction of the shear (except for δ 0 = ±45 • ).As a result, although simple shear was applied at the boundary, the shear bands dipping towards the sense of shear display a normal kinematics and those with an opposite dipping sense display a reverse kinematics.Similarly, we have shown that the observation of a lineation parallel to the Sn between boudins does not proove that the larger scale deformation is in pure shear mode (see the δ 0 = 20 • case).
The models presented here show that the sense of shear on the neoformed shear bands is always compatible with the large scale sense of shear.As the orientation of the shear bands is acquired very early in the deformation history, the geometry of the shear bands provides information on the inital orientation of the layering.We provide below such examples of natural S-C structures where the intial orientation can be constrained.Our models also show that localised deformation that crosscuts the initial foliation, i.e. shear bands, only occurs for a restricted range of initial orientation of the layering (types II and IV).On the one hand, the models have shown that the occurrence C structures requires that the strong layers yield plastically, implying that where µ strong , τ y and γ stand respectively for the viscosity of the strong layers, the plastic yield strength and the background strain rate.On the other hand, large scale strain localisation requires that the weak layer does not yield plastically, and hence where µ weak is the visosity of the weak layer.Therefore, providing that both the plastic yield strength and the large scale kinematics are known, the microstructure and the observation of strain localisation at larger scale in turn provide bounds on the viscosity of the strong layers.Conversely, one could use the laboratory data on creep parameters and plastic yield strengths to estimate the large scale strain rate.

Conclusions
We have run numerical experiments to simulate the deformation of mechanicaly stratified medium in simple shear in order to provide quantitative insights on the microstructures forming in that context as a function of the initial orientation of the layers relative to shear strain field.The results obtained are representative of the diversity of natural microstructures and highlight that the formation of shear bands, extensional crenulation cleavage, symmetric boudinage, micro-folds and kink bands is independent of the geological context but rather depends on the initial orientation of the anisotropy.This

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Full implies that one must be careful in interpreting the shape of the boudins or folds as indicators for simple shear or pure shear at large scales.The simulations have been separated into four groups depending on the absolute value of the initial dip, δ 0 , and its sign.The absolute value of δ 0 controls strain partitioning between the weak and strong layers and its sign controls the effective shortening or stretching of the layers as they rotate within the shear zone.We herein demonstrate that the initial orientation of the anisotropy controls the dip of the shear bands, the degree of localisation and the microstructures that can form in a large scale shear zone.At local scale, the maximum shear stress orientation in the strong layers that is the closest to the large scale flow is always selected to localise the strain.Therefore, at the regional or model scale, the shear sense on the microstructure is fixed by the plate kinematics or the boundary conditions.However, depending on the orientation of the initial anisotropy, the selected maximum shear direction corresponds alternatively to extensional crenulation cleavage (type I), C structures (type II and IV) and true crenulation (type III).
All the experiments show that anisotropy results in non-lithostatic pressure values which are proportional to the deviatoric stresses in the strong layer.Since this strength is limited by the plastic yield strength of the layer, for a confining pressure of 4 kbar, we never found deviations of more than 1.5 kbar at the onset of the deformation.These deviations are in the error bars of thermobarometric estimations based on thermodynamic methods.However, given the scale independance of the results, we speculate that these large pressure jumps can result in large pressure gradients at small scale and therefore influence the chemical potential of elements in the crust and their rate of migration.
We have also shown that our results are scale independant and that the orientation of the shear bands is acquired in the early deformation stages and remains independant of the strong layer viscosity or the strain rate.We demonstrate here that viscous rheology alone is not sufficient to model the diversity of microstructures observed in the ductile crust and that introducing some finite yield strength (e.g.brittle processes) Introduction

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Full in the initial configuration (see Fig. 2).Similarly, in the final configuration the passive markers define four vectors The initial and final vectors are related to one an other through the deformation matrices defined in Ramsay and Huber (1983).At large strain, the final mesh is not always a parallelogram (Fig. 2b).Therefore the four matrices S i differ from each other.For visualisation purposes, an equivalent matrix S must be defined for each quadrilateral element.We compute it using a least square inversion procedure for the four equations defined in eq.B3.
In the second step we compute the parameters of the finite strain from the equivalent strain matrix (Ramsay and Huber, 1983).The orientation of the maximum stretching axis of the finite strain tensor (θ ) is infered from tan (2θ ) = 2 (ac + bd ) This direction is represented with white line on the details snapshot of the models.The intensity of stretching in that direction is ) and the intensity of maximum shortening (normal to the direction of θ ) is The ellipticity is a measure of the intensity of the finite strain which varies between 1 (no strain) and +∞ (infinite strain).For visualisation purposes, we prefer to use ) which renormalises the finite strain on a hyperbolic scale that varies between 0 (no strain, blue on the figures) and 1 (infinite strain, red on the figures).Full . Based on a representative set of measurements, it allows one to upscale the field observations to Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | I and II correspond to positive values of δ o and include the models in which the strong layers are initialy stretched.They have been separated based on the predominance of shear bands in Discussion Paper | Discussion Paper | Discussion Paper | the type II models.Types III and IV correspond to models in which the initial foliation shortens initially (δ o < 0).They have been separated based on whether the deformed initial layering (Sn) is being tilted towards positive values of δ during the simulation (Type IV) or not (Type III).When |δ 0 | < 30 • (Type I and III), the shear strain primarily occurs in the weak layers of the pre-existing foliation.However, secondary structures begin to appear as soon as an initial obliquity of |δ 0 | = 20 • is reached, whilst an obliquity of |δ 0 | = 40 • is needed for true S-C structures to develop.When shear bands form, they do not form parallel to the direction of shear except for the special δ 0 = 45 • case.The dip of the C structures rotates with the initial dip of the foliation and forms an angle close to 45 • to the Sn.The models with layers stretched during the shear (δ 0 > 0) are Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Fig. 3 illustrates that the dip of the shear bands depends on the Sn orientation, with left dipping shear bands for δ 0 = 40 • , horizontal shear bands for δ 0 = 45 • and slightly right dipping shear bands for δ 0 = 50 • .Within the shear bands, the MSD tends to flatten with the intensity of shear strain and to become parallel with the large scale kinematics applied on the boundary of the models (Fig. 5b).At γ bulk = 100 %, the penetration of the strain with depth is relatively similar in all the models but the number of shear bands and their spacing change with the dip of the shear bands.Looking in more detail at the depth-displacement profiles with increasing bulk strain, one sees that in the δ 0 = 40 • case the rate of displacement with depth does Discussion Paper | Discussion Paper | Discussion Paper | Screen / Esc Printer-friendly Version Interactive Discussion Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | characterised by large wavelength folding which rapidly deflects the initial layering (Fig.3).In the three cases presented, at γ bulk = 100 %, the shear banding is only incipient for the δ 0 = −40 • and δ 0 = −50 • .For the case at δ 0 = −45 • , shear banding occurs along a horizontal direction only after γ bulk = 130 % (not shown).The incipient shear bands (dashed lines on Fig. 7a) dip to the right in the δ 0 = −40 • model and slightly to the left in the δ 0 = −50 • case.The dipping direction of the shear bands obtained is opposite to those obtained for the same obliquity in the Type II but the absolute value of their dip are roughly similar.δ 0 = −45 • marks a transition in the small scale mode of deformation of the strong layers that is best seen in the grey scale part of Fig. 3 (red circle).In the δ 0 = −50 • case the strong layers form mullions with periodic thickening of the black (weak) layers, whereas for δ 0 = −40 • the small scale deformation corresponds to asymmetric folding with a very short reverse limb.This change of small scale deformation is also outlined by the MSD in the weak layers which is almost normal to the layers in the δ 0 = −50 • case and oblique to the layers in the δ 0 = −40 • case in the deepest part of the detailed cross section (Fig.
Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Fig. 10a and b displays examples from the Betics in Spain which scaling, orientations and localisation are described in detail by Agard et al. (2011).According to our models, the lack of rotation of the S between the C indicates that the original foliation was dipping in the opposite direction to the sense of shear.10a strictly resembles type II while the slight variation of dip and the length scale of the shear band on picture b would fall in between type I and II suggesting that the initial schistosity was dipping with slightly smaller angle.10c and d are examples from the Tende shear zone in Corsica, France.In both cases, the foliation is folded between the shear zone indicating that the initial foliation would have been dipping toward the sense Discussion Paper | Discussion Paper | Discussion Paper | of shear initially.While Fig. 10d resemble a type III, the Fig. 10c is a clear example of type IV S-C structures.
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