Tectonophysics 484 (2010) 36–47

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Factors that control the angle of shear bands in geodynamic numerical modelsof brittle deformation

Boris J.P. Kaus ⁎Geophysical Fluid Dynamics, Department of Earth Sciences, ETH Zurich, SwitzerlandUniversity of Southern California, Los Angeles, USA

⁎ Geophysical Fluid Dynamics, Department of EaSonneggstrasse 5, 8092 Zurich, Switzerland. Tel.: +41 41065.

E-mail address: kaus@erdw.ethz.ch.

0040-1951/$ – see front matter © 2009 Elsevier B.V. Adoi:10.1016/j.tecto.2009.08.042

a b s t r a c t

a r t i c l e i n f o

Article history:Received 18 February 2009Received in revised form 24 June 2009Accepted 31 August 2009Available online 12 September 2009

Keywords:Lithospheric deformationRheologyPlastic deformationViscoelastoplasticityViscoplasticityNumerical modellingLong-term tectonics

Numerical models of brittle deformation on geological timescales typically use a pressure-dependent (Mohr–Coulomb or Drucker–Prager) plastic flow law to simulate plastic failure. Despite its widespread usage ingeodynamic models of lithospheric deformation, however, certain aspects of such plasticity models remainpoorly understood. One of the most prominent questions in this respect is: what are the factors that controlthe angle of the resulting shear bands? Recent theoretical work suggest that both Roscoe (45°), Coulombangles (45+/−ϕ/2, where ϕ is the angle of internal friction) and Arthur angles (45°+/−(ϕ+Ψ/4) where Ψis the dilation angle), as well as all intermediate angles are possible. Published numerical models, however,show a large range of shear band angles with some codes favoring Arthur angles, whereas others yieldCoulomb angles.In order to understand what causes the differences between the various numerical models, here I performsystematic numerical simulations of shear localization around an inclusion of given length scale. Bothnumerical (element type), geometrical and rheological (viscoplastic versus viscoelastoplastic) effects arestudied. Results indicate that the main factor, controlling shear band angle, is the non-dimensional ratiobetween the length scale of the heterogeneity d and the size of the numerical mesh Δx. Coulomb angles areobserved only in cases where the inclusion is resolved well (d/Δx>5–10), and in which it is locatedsufficiently far from the boundary of the box. In most other cases, either Arthur or Roscoe orientations areobserved. If heterogeneities are one element in size, Coulomb angles are thus unlikely to develop irrespectiveof the employed numerical resolution.Whereas differences in element types and rheology do have consequences for the maximum obtainablestrain rates inside the shear bands, they only have a minor effect on shear band angles. Shear bands, initiatedfrom random noise or from interactions of shear bands with model boundaries or other shear bands, result instress heterogeneities with dimensionless length scales d/Δx~1–2. Such shear bands are thus expected toform Roscoe or Arthur orientations, consistent with the findings in previous numerical models.

rth Sciences, ETH Zentrum,4 633 7539; fax: +41 44 633

ll rights reserved.

© 2009 Elsevier B.V. All rights reserved.

1. Introduction

Numerical modelling of the deformation of brittle crust on geo-logical timescales is an important but challenging topic that hasreceived an increasing amount of attention over the last two decades(Braun et al., 2008; Buck et al., 2005; Buiter et al., 2006; Fullsack,1995; Gerbault et al., 1999; Hobbs and Ord, 1989; e.g., 2008;Lyakhovsky et al., 1993; Moresi et al., 2007; Poliakov et al., 1996;Poliakov et al., 1993; e.g., Poliakov et al., 1994; Popov and Sobolev,2008). Experimentally, it has beendemonstrated that brittle rock failurecan be described by a pressure-dependent yield stress (‘Byerlees law’),

irrespective of rock type (e.g., Byerlee, 1978; Scholz, 1990; Ranalli,1995). In numerical models, this is frequently approximated by a(phenomological) Drucker–Prager or Mohr–Coulomb yield criteria inwhich the maximum differential stress of rocks is a function of itsnormal stress, the internal angle of friction ϕ and the cohesion C. Inaddition to this, modelling of plasticity requires the specification of a(phenomological) plastic flow rule. Typically, a flow rule is used whichhas the dilation angle Ψ as material parameter. Rocks have Ψ≪ϕ oreven Ψ~0° (so called non-associated plasticity) which results inspontaneous localization of deformation into narrow bands (Rudnickiand Rice, 1975; Vermeer and de Borst, 1984). Besides a plastic yieldcriteria (and corresponding plastic flow potential), one also has tospecify the rheology of non-yielded materials. Here, differences existbetween various modelling groups, where some employ viscousrheologies, whereas others treat non-yielded materials as viscoelastic.

Despite the widespread implementation of plasticity in geody-namic numerical models, some seemingly basic questions about

37B.J.P. Kaus / Tectonophysics 484 (2010) 36–47

plasticity remain unclear. One of the most prominent is: which shearband angle (θ) should we expect in a brittle Mohr Coulomb material?

From a theoretical point of view, this question was studied byVardoulakis (1980) and Vermeer (1990) for elastoplastic rheologies.Vermeer (1990) demonstrated with a bifurcation analysis and witha post-failure analysis that a range of angles are mechanically stable:

θ = 45-Fψ2

ð1aÞ

(Roscoe angle — Roscoe, 1970)

θ = 45-Fϕ2

ð1bÞ

(Coulomb angle — Coulomb, 1773)

θ = 45-Fϕ + ψ

4ð1cÞ

(Arthur or intermediate angle — Arthur et al., 1977)

The Coulomb angle results in the fastest rate of pressure reductioninside the shear band. The Arthur (or intermediate) angle is relatedwith (elastic) unloading of material outside the shear band and hencecauses the fastest rate of macroscopic softening upon shear bandformation. The Roscoe angle, on the other hand, is admissible but doesnot result in softening. Angles outside the Roscoe–Coulomb range arenot expected to occur, since formation of such angles would result in amacroscopic strain hardening of the material. Yet, all angles betweenthe Roscoe and Coulomb angle are feasible.

A more recent analysis by Lemiale et al. (2008) focused on a bifur-cation analysis of a viscoplastic solid (without post-failure deforma-tion) and found that both the Roscoe angle and the Coulomb angle arepossible solutions.

Numerically,Moresi et al. (2007) found theRoscoe angle to be stable,using a viscoplastic finite element approach with linear elements andmoving integration points. Lemiale et al. (2008), however, found theCoulomb angle to be stable using the same numerical code. Thedifferences between the two studies were attributed to differences innumerical resolution (which was considerably higher in the Coulombangle case), although the authors noted that it could not be excludedthat parameters such as type of integration schememight affect results.

Popov and Sobolev (2008) employed an arbitrary Lagrangian–Eulerian FEM code with a viscoelastoplastic rheology and found thatshear bands, initiated from random noise, initially develop at a rangeof orientations (including the Coulomb angle), but that the Arthurangle dominates after several percents of strain.

Hansen (2003) employed a meshless finite element formulationwith an elastoplastic rheology and obtained shear band angles close tothe Coulomb orientation. Mancktelow (2006) performed viscoelas-toplastic simulations using a finite element formulation and obtainedArthur shear band orientations.

Gerya and Yuen (2007) employ a viscoelastoplastic finite differencemethodology and obtain Arthur shear band orientations in extensionalexperiments both in incompressible cases and in cases with a nonzerodilation angle.

Buiter et al. (2006) performed numerical experiments in orderto reproduce laboratory experiments of brittle failure under bothextension and compression. Although a qualitative agreement bet-ween numerical and analogue models was found, differences existbetween numerical models with respect to location, spacing and dipangle of shear zones, which ranged from Roscoe to Coulomb angles.

Interestingly, the wide spread in shear band orientations can also beobserved in laboratory experiments in which sand is employed tosimulate brittlematerials. A literature summary given inVermeer (1990)indicates that Roscoe angles form in coarse grained sands whereas

Coulomb angles can be observed in fine grained sands, independent ofthe boundary conditions (Wolf et al., 2003). Note, however that thereason for the different shear band angles in experiments is likely to becaused by differences in material properties, whereas in numericalmodels they occur due to the non-uniqueness of the plasticity models.

This brief (and necessarily incomplete) summary thus shows thatthere is currently no consensus on what the stable shear band orien-tation is, and how this orientation is affected by the numerical methodand the employed rheologies. The goal of this work is therefore tosystematically test the effects of numerical parameters, initial con-ditions and rheology on the angle of shear bands obtained in a simplesetup of compression and extension of a brittle material. These in-sights might than help to interpret and understand the results inmorecomplex numerical models.

2. Mathematical and numerical formulation

2.1. Mathematical formulation

An incompressible Boussinesq approximation is employed whichis given by

εii = 0; ð2Þ

− ∂P∂xi

+∂τij∂xj

= ρgi ð3Þ

where εij = 12

∂vi∂xj

+ ∂vj∂xi

� �indicates strain rate, vi velocity, P = − σii

3

pressure, σij stress, τij=σij+P deviatoric stress, xi spatial coordinates(in the 2D case employed here: x1=x, x2=z) ρ density and gi thegravitational acceleration (the sign convention taken here assumesthat compressive stresses are negative).

The rheology of rocks is assumed to be Maxwell viscoelastoplastic,which (for an incompressible rheology) is given by

εij =12η

τij +12G

DτijDt

+ λ∂Q∂σij

; ð4Þ

where η is the effective viscosity, G the elastic shear module, t time, λ theplasticmultiplier andQ the plasticflowpotential (e.g.,Moresi et al., 2007).The Jaumannobjectivederivativeof thedeviatoric stress tensor is givenby

DτijDt

=∂τij∂t + vk

∂τij∂xk

−Wikτkj + τikWkj; ð5Þ

whereWij = 12

∂vi∂xj

−∂vj∂xi

� �is the vorticity.

Rocks fail plastically if differential stresses exceed the yield stress.A commonly used, phenomenological, yield stress formulation underlow temperatures is Mohr Coulomb plasticity, for which (in 2D) theyield function F and plastic flow potential can be written as (e.g.,Vermeer and de Borst, 1984)

F = τ⁎−σ⁎ sinðϕÞ−c cosðϕÞ;

Q = τ⁎−σ⁎ sinðψÞ; ð6Þ

where τ⁎ =ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiσxx−σzz

2

� �2 + σ 2xz

q;σ⁎ = −0:5ðσxx + σzzÞ, c is cohesion,

ϕ friction angle, ψ the dilation angle (zero in incompressible cases).ϕ is typically 30° for dry and less for wet rocks, such as serpentine(e.g., Ranalli, 1995). Plasticity is activated only if stresses are above theyield stress:

λ≥ 0; F≤ 0; λ F = 0: ð7Þ

38 B.J.P. Kaus / Tectonophysics 484 (2010) 36–47

It should be emphasized at this point that it is crucial to employthe dynamic pressure for the evaluation of the yield function (Eq. (6)).If this is not done, resulting the shear bands will form at 45°.

2.2. Numerical formulation

In order to solve the equations numerically, the time derivativein Eq. (4) is discretized in an implicit manner, which results in

τij = 2μveðεij− εplij Þ + χτoldij ; ð8Þ

with

μve =1

1η + 1

GΔt

;

χ =1

1 + GΔtη

;

τoldij = τold

ij + voldk∂τoldij

∂xk−Wold

ik τoldkj + τoldik W oldkj ;

where τijold are deviatoric stresses from the last time step, and Δt isthe employed time step (Kaus, 2005; Kaus & Podladchikov, 2006). Ifstresses are below the yield stress (F<0), plastic strain rates εijpl are zeroand rheology is viscoelastic. Note that the viscoelastic viscosities μνe aredependent on the employed numerical time step. A purely viscousrheology (τij=2ηεij) is recovered if G→∞ (since then χ=0, μνe=η).

In a typical time step one first assumes that plastic strain rates arezero and the rheology is fully viscoelastic. If initial stresses are abovethe yield stress (F(τij)>0, plastic strain rates should be computed insuch a manner that stresses are brought back to the yield surface(F=0). The rheological expression (Eq. (8)) is than given by

τy = 2μveðε2nd− εpl2ndÞ + χ τold2nd; ð9Þ

which can also be written as

τy = 2μ vepε2nd + χτold2nd; ð10Þ

where ε2nd =ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiεxx−εzz

2

� �2+ ε2xz

r; τold2nd =

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiτoldxx − τoldzz

2

� �2+ ðτold

xz Þ2r

; τy =

σ⁎ sinðϕÞ + c cosðϕÞ is the yield stress and μvep the viscoelastoplasticeffective viscosity, given by

μ vep =τy−χτold

2nd

2ε2nd: ð11Þ

The full viscoelastoplastic rheology employed here can thus bewritten as

τij = 2μ vep εij + χτoldij ; ð12Þ

with

μ ivep =

τy−χτold2nd

2ε2nd; if F ðτ initial

ij Þ > 0

μve; if F ðτ initialij Þ≤0

8>><>>:

9>>=>>; ð13Þ

The present formulation can deal both with viscoelastoplastic aswell as with viscoplastic rheologies (if G→∞). Typically, iterations(indicated by the superscript i in Eq. (13)) are required to computeplastic viscosity since changing the effective viscosity locally affectsthe stress distribution in surrounding points. In the present version

of the code, this is done by Picard iterations. More efficient strategiesexist, in particular Newton–Rhapson iterations with a consistent tan-gent stiffness matrix (de Souza Neto et al., 2009). This is not yet im-plemented in the current version of the numerical code. A recentcomparison of Newton–Rhapson versus Picard iterations suggeststhat the width of the shear bands is smaller and have larger strainrates in the case of Newton–Rhapson iterations (Popov and Sobolev,2008). Shear band angles, however, seem to be unaffected by the typeof iterations and the main conclusions made here are thus likely toremain valid.

Strain weakening is applied to cohesion, according to

c = c0 + ðc∞−c0Þmin 1;γpl

γ0

� �; ð14Þ

where γpl = ∫εpl2nddt indicates plastic strain and γ0=0.1 is a criticalvalue.

2.3. Numerical method

Spelled out in two dimensions, the strong form of the governingforce balance equations is

∂vx∂x +

∂vz∂z = − p

κ; ð15aÞ

−∂P∂x +

∂∂x 2μ i

vep∂vx∂x

� �� �+

∂∂z μ i

vep∂vz∂x +

∂vx∂z

� �� �

= − ∂τoldxx

∂x +∂τoldxz

∂z

!;

ð15bÞ

−∂P∂z +

∂∂z 2μ i

vep∂νz

∂z

� �� �+

∂∂x μ i

vep∂νz

∂x +∂νx

∂z

� �� �

= ρg− ∂τoldxz

∂x +∂τoldzz

∂z

!;

ð15cÞ

where κ is a penalty parameter used to enforce incompressibility.Eq. (15a–c) are solved numerically with a recently developed finiteelement code MILAMIN_VEP, which solves the unknowns using avelocity pressure formulation. MILAMIN_VEP is based on the efficientMATLAB-based Stokes solver MILAMIN, which employs an iterativepenalty method to enforce incompressibility (Cuvelier et al., 1986)and which is described in detail in Dabrowski et al. (2008). The mainmodifications to the MILAMIN solver are τ ij

old-related terms inEq. (15b,c) which have to be added to the right hand side, as well asan iterative scheme that is required to determine μνep

i in case ofplastic failure. The error of plasticity iterations ep are computed atevery iteration step i with

eip =∥vi−vi−1∥∞

∥vi∥∞

where vi indicates nodal velocities at iteration step i. Iterations areterminated once ep<10−4 or once more than 25 iterations have beenperformed. Typically, the maximum number of iterations is reachedduring the first few time steps, but after that, iterations convergemorerapidly (presumably due to the effects of geometric and strain soft-ening). Tests with smaller cutoff values and more iterations revealedthat the shear zones obtain a slightly larger strain rates. This, however,does not change the conclusions with respect to shear band angles. Inall simulations, a constant time step of 30,000 years was employed.

Compared to the original MILAMIN code, I have also addedadditional element types and shape functions, thermo-mechanical

Fig. 1. Model setup. See Table 1 for material parameters.

Table 1Parameters employed in the models.

Parameter Symbol Dimensional value Dimensionless value

Width W 40 km 4Height H 10 km 1Weak inclusion size d 400 – 3200 [800] 0.04–0.32 [0.08]Background strain rate εbg (+/−)10−15s−1 (+/−)1Density times grav.acceleration

ρg 27,000 kg m−2s-2 2700

Viscosity inclusion μi 1020Pas 1Viscosity of viscoplasticlayer

μ 1025Pas 105

Initial cohesion c 40×106Pa 400Cohesion after weakening c∞ 5×106Pa 20Friction angle inclusion ϕi 0° 0°Friction anglevisco(elasto)plastic layer

ϕ 0–45° [30°] 0–45° [30°]

Minimum yield stress(cutoff value)

σy,min 20×106Pa 200

Maximum yield stress(cutoff value)

σy,max 1000×106Pa 104

Elastic shear module G 5×1010Pa or ∞ 5×105 or ∞

Characteristic values for length, viscosity and time are L⁎=10 km, μ=1020Pas, andt=1015s−1, respectively. Cohesion is weakened in a linear manner between plasticstrain=0 and 0.1.

39B.J.P. Kaus / Tectonophysics 484 (2010) 36–47

coupling, particle-based advection of material properties, remeshing,phase transitions, sedimentation. Both quadrilateral and triangularelements are implemented. The quadrilateral elements use bothbilinear (4-node) shape functions for velocity and a constant shapefunction for pressure (Q1P0), or (9-node) biquadratic shape functionsfor velocity and linear discontinuous shape functions for pressure(Q2P−1). Triangular elements are identical to the MILAMIN imple-mentation, which uses quadratic shape functions for velocity andlinear, discontinuous shape functions for pressure (P2P−1).

Lower andupper cutoff viscosities of 1019 and1025Pas are employed.The penalty factor κ (Eq. (15a)) was taken to be 1000 times the maxi-mum viscosity in the domain, which results in an incompressiblesolution in a few iteration steps. Viscoelastoplastic simulationshave zeroinitial differential stresses. In all simulations, the code is employed in apurely Lagrangian manner and simulations are performed until theelements become too deformed (as indicated by negative jacobians forthe elements), or until an overall strain of 2% is reached.

In the case of viscoplastic simulations, it is important to use aninitial stress field that is in agreementwith the physics of the problem,since otherwise the plasticity algorithm fails to return to the (correctpart of the) yield surface. One way to do this, is to compute the initialhomogeneous state of the problem (without heterogeneity), and usethis state of stress as an initial guess (Lemiale et al., 2008). A dis-advantage of this method, however, is that it can be cumbersome toderive the correct initial state of stress for more complicated boundaryconditions or initial geometries. For this reason, I use a differentapproach, in which the boundary velocity vb is incrementally increasedduring the first time step, according to

vb =knk

� �3v0b

where v0b is the desired boundary velocity, k the iteration step and nkthe total number of iterations. It was found that shear band orien-tations are sensitive to the value of nk during the first time step (nk istaken to be one in all subsequent steps). Values employed in mostsimulations are nk=25 for extension and nk=1 for compression.These values are selected, as they maximize the potential for Coulombangles, while simultaneously ensuring physically realistic stress statesat the end of the first time step. Employing significantly smaller nkvalues during extension results in unphysical initial stress states,whereas the use of larger nk values during compression increasesshear band orientation towards the Arthur angle (see Fig. 9 andrelated discussion).

No boundary velocity iterations are required for viscoelastoplasticsimulations, as the initial stress evolution in these simulationsincreases with time as a consequence of viscoelasticity. The timestep in these simulations, however, should be chosen sufficientlysmall to prevent failure of the whole model domain during the firsttime step. With the values employed here, it takes 5–10 time stepsuntil the system fully yields. Larger values of elastic shear modulus Gthus require a proportionally smaller time step.

3. Results

Shear bands do not initiate in numerical models with a perfectlyhomogeneous setup. To localize deformation, one therefore has tointroduce some form or heterogeneity. This can consist of randomvariations in material properties, a heterogeneity of specified size, orrandom noise that develops due to round-off errors in the numericalapproach (as is typically for explicit geodynamic numerical models).Here, it is demonstrated that the length scale of suchheterogeneities hasa large impact on the resulting shear band angles. For this reason, themodel setup considered here consists of a single viscous heterogeneityof length scale d, embedded in a viscoplastic or a viscoelastoplastic crust(Fig. 1). The box is subjected to either compression or extension with

constant background strain rate. Material parameters are given inTable 1 and are chosen in such a manner that shear localization occurs,starting at the heterogeneity.

In all simulations, the angle of the resulting shear bands wasdetermined in an automated manner by computing the location ofmaximum strain rate invariant at a distance of d/4 and d/4+2000 mfrom the sides of the inclusion. The average value from the left andright shear band is reported. Experiments in which the samplinglocation and width was varied revealed that the angle can bereproduced within +/−3°. Unless stated otherwise, Q1P0 elementsare employed.

3.1. Effect of friction angle

The effect of the employed numerical resolution on the shear bandangle is shown on Fig. 2 under both extensional and compres-sional boundary conditions. At small numerical resolutions, resultingshear band angles are close to the Arthur angle (52.5° respectively37.5°), whereas at larger numerical resolutions they are closer to theCoulomb angle (60° respectively 30°). Maximum obtained strain ratesinside the shear zone are larger for larger numerical resolutions.Convergence of the results with respect to shear band angle isobtained at a resolution of ~400 by 100 elements. The aspect ratio ofthe elements does not appear to play a major role, suggesting thatmesh alignment effects (frequently occurring in strain-softeningproblems; see e.g. De Borst, 1991) are of minor importance here.

A more systematic analysis on the effect of friction angle on theresulting shear band angle (Fig. 3), confirms that shear band anglesare closer to 45° at smaller numerical resolutions. Small numericalresolutions and friction angles smaller than ~20° result in Roscoe

Fig. 2. Effect of numerical resolution on shear band angle θ for both compressional (left) and extensional (right) cases for a viscoplastic rheology with ϕ=30°. The inclusion lengthsize d=800 meter, and Q1P0 elements are employed. Thin colored lines indicate the Roscoe, Arthur and Coulomb orientation. Thick white lines indicate the measured shear bandorientation. At small numerical resolutions (for which the heterogeneity is numerically less well resolved) the orientations are closer to the Arthur angle, whereas at largerresolutions they are closer to the Coulomb angle. Changing the aspect ratio of the elements has a minor effect on shear band angles (bottom row).

40 B.J.P. Kaus / Tectonophysics 484 (2010) 36–47

angles, whereas Arthur shear band angles are observed for largervalues of the internal friction angle. At larger numerical resolutions,the shear band angles are between Arthur and Coulomb angles.

Fig. 3. Effect of friction angle on resulting shear band angle for two different numerical resoland symbols indicate measured shear band angles from numerical simulations. The inclusio

These results thus confirm the findings of Lemiale et al. (2008) thatlarge numerical resolutions are required for Coulomb angles todevelop.

utions and viscoplastic rheology. Lines indicate the Roscoe, Arthur and Coulomb angles,n length scale d=800 m. All other parameters as in Table 1.

Fig. 4. a) Effect of normalized inclusion size on shear band angle for a resolution of 400×100 nodes, Q1P0 elements and viscoplastic rheology with ϕ=30°. Simulations are performedwith a resolution of 400×100 nodes and with increasing inclusion length scale. b) Effect of increasing the numerical resolution while keeping the inclusion length scale constant.Numbers in braces indicate the employed numerical resolutions. Coulomb angles are selected if the initial heterogeneity is numerically well resolved (d/Δx>~5–10).

41B.J.P. Kaus / Tectonophysics 484 (2010) 36–47

3.2. Effects of resolution and length scales

Although numerical resolution clearly matters, it remains unclearwhether this is because the heterogeneity is better resolvednumerically or whether another effect plays a role. Therefore threeadditional sets of experiments have been performed. In the firstexperiment, the numerical resolution was kept constant but theinclusion size d was increased such that the non-dimensionalparameter d/Δx increases (where Δx is the spacing between twohorizontal grid points). Results show a weak dependence of shearband angle on d/Δx with small values (i.e. small numerical resolu-tions) favoring Arthur angles and larger values favoring Coulombangles (Fig. 4a).

A potential problem with this experiment is that with increasinginclusion size, the results might be influenced by the proximity to theboundaries of the model box. For this reason, a second experiment

Fig. 5. Effect of model box size on shear band orientation. In all simulations, the inclusion leincreasing box size, such that the inclusion is resolved with the same number of elements inillustrate the steepening of the shear band angle with increasing box size. b) results of systemindicate a rotation of the shear band angle towards the Arthur angle for a model height<1

was performed in which the inclusion size was kept constant(d=800 m), but the numerical resolution was increased (such thatd/Δx increases). Results show a clear dependence of shear bandangle on d/Δx with Coulomb angles for values of d/Δx>10–15, and atrend towards Arthur or Roscoe angles at smaller values of d/Δx(Fig. 4b). These results thus suggest that shear band angles are verysensitive to the manner in which the heterogeneity, from which theshear band initiates, is resolved numerically.

In order to study effects of the proximity of the heterogeneity tothe upper boundary, a third series of experiments was performed,in which d/Δx and d were kept constant but in which the size of themodel box was increased. Results indicate that the proximity to theupper boundary indeed influences the shear band angle (Fig. 5). Theclearest results are obtained for compressional boundary condi-tions, which show that the model box height should be at least10 times larger than the inclusion size for Coulomb angles to

ngth scale is maintained constant (d=800 m). Numerical resolution is increased withall simulations (d/Δx=8). Material is viscoplastic and ϕ=30°. a) strain rate plots thatatic simulations as a function of box size under both compression and extension. Results0 km.

42 B.J.P. Kaus / Tectonophysics 484 (2010) 36–47

develop. Simulations with the same setup but without twice thegravitational acceleration (not shown here) gave a similar trend,which suggests that the effect observed here is not due to anincrease of confining pressure, but can be solely attributed to theproximity to the boundaries.

The results obtained here thus suggest that Coulomb shear bandangles can be obtained if (1) the heterogeneity from which the shearband initiates is resolved numerically with at least 5–20 elements, (2)the heterogeneity is sufficiently far from the boundaries of the modelbox (at least 10 times the heterogeneity size).

3.3. Effect of element type

So far results have been obtained with a single element type(Q1P0). Since this is a non-admissible element (see e.g., Cuvelier et al.,1986; Hughes, 1987), it is interesting to see how results change ifhigher-order elements are employed. A comparison of linear andquadratic shape functions for quadrilateral elements (Fig. 6), revealsthat higher order shape functions result in shear band angles that areslightly closer to the Coulomb angle (34° instead of 36°). Triangularelements can be either fitted to the viscous inclusion (‘fitted P2P−1’),or can be employed with a more or less regular distributed mesh(‘non-fitted P2P−1’). Both cases result in shear band angles that arecomparable to results obtained with Q2P−1 elements. It can thus beconcluded that the type of element (triangular versus quadrilateral)

Fig. 6. Effect of element type on shear band angle for a viscoplastic rheology with ϕ=25° andthe Coulomb angle is 32.5° and the Arthur angle 39°. Higher order elements thus result in slelement shape (triangular of quadrilateral) has little effect on the shear band angle.

has little effect on the shear band orientation, but that the type ofshape function (linear versus quadratic) has a minor effect.

3.4. Viscoplastic versus viscoelastoplastic rheologies

As outlined in the introduction, there are significant differencesbetween various geodynamic modelling groups when it comes to thetreatment of non-plastic rheologies. Whereas some use a viscousrheology, others use a (Maxwell) viscoelastic rheology. Rockdeformation experiments clearly indicate that rocks behave elasticallyat small strains (e.g., Ranalli, 1995) which argues in favor of vis-coelastic models. Yet, if one is interested in modelling brittle (plastic)behavior only, it can be argued that non-plastic behavior is irrelevant.Furthermore, implementing elasticity in viscoplastic numerical codesrequires additional work, since stress tensors have to be advectedalong with the numerical grid. Moreover, a previous study found littledifferences between viscoplastic and viscoelastoplastic rheologies(Buiter et al., 2006), although the setup employed in this comparisonwas relatively complicated.

Yet, it remains unclear to which extend elasticity influences resultsin a simpler setup. For this reason, a number of additional experi-ments have been performed in which the two rheologies are com-pared. Results for an extensional and compressional setup with afriction angle of 25°, reveals that there are little differences in theangle of the resulting shear bands (Fig. 7). There are however

a compressional setup. A resolution of ~40,000 nodes is used in all cases. For this setup,ightly steeper shear band angles as well as larger strain rates inside the shear band. The

Fig. 7. Comparison of viscoplastic versus viscoelastoplastic simulations under extension and compression (after 2% strain, with ϕ=30° and a numerical resolution of 800×200nodes). In both cases, viscoelastoplastic shear band orientations are slightly closer to the Coulomb angle. Under compression, elasticity results in larger strain rates inside the shearband, and in smaller matrix strain rates. Under compression, the situation is reversed and shear bands are more pronounced in the case with viscoplastic rheology.

43B.J.P. Kaus / Tectonophysics 484 (2010) 36–47

differences in the sharpness of the shear zones (which morepronounced in viscoplastic cases), as well as in the distribution ofstrain rates. Under compression, maximum obtained strain rates arean order of magnitude larger in viscoelastoplastic cases, whereas theopposite effect is observed under extension.

A more systematic study on the dependence of shear band angleon fiction angle for viscoelastoplastic rheologies (Fig. 8a) gives verysimilar results as for viscoplastic cases (Fig. 3b). Similar trends are alsoobserved for simulations in which the inclusion length scale is keptconstant (d=800 m) and the numerical resolution is increased(Fig 8b). Simulations, in which the inclusion is poorly resolvednumerically, favor Arthur (or Roscoe) angles, whereas simulationswith larger numerical resolutions result in Coulomb angles. The maindifferences between viscoelastoplastic and viscoplastic cases are

Fig. 8. Simulations which address the effect of elasticity on shear band orientation. a) Frictioresolution of 400×100 nodes. b) Shear band orientation versus normalized numerical red=800 m and with increasing numerical resolution. As in viscoplastic cases, low resolutiolarge resolutions.

observed in compressional setups, where the tendency for Arthurangles is stronger in the presence of elasticity. This is consistent withthe findings of Vermeer (1990), who showed that the Arthur angle isrelated to unloading of stresses outside the shear band. If elasticity ispresent, this unloading depends on the available elastic stresses(which are larger for compressional setups). Viscous deformation isirrecoverable and hence unloading should be less pronounced.

3.5. Effect of initial iterations in viscoplastic simulations

A (computational) advantage of viscoelastoplastic over viscoplas-tic simulations is that stresses build up gradually in during the courseof a numerical simulation. Plastic failure in the vicinity of the viscousheterogeneity thus only occurs after a finite amount of deformation,

n angle versus shear band orientation for a viscoelastoplastic rheology for a numericalsolution. Simulations have been performed with a constant inclusion length scale ofns result in Roscoe or Arthur shear band angles, whereas Coulomb angles develop at

Fig. 9. Effect of number of initial iterations nk on shear band orientation for viscoplasticrheologies and Q1P0 elements. Extensional cases with nk<10 resulted in numericalartifacts and are not shown. The manner of initializing the stress-state thus has animpact on the results, with compressional cases favoring small numbers of iterations,whereas extensional cases require 10 or more.

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which is approximately what one would expect in physical experi-ments. In viscoplastic simulations, however, the complete model failsin a plastic manner during the first time step, which might causedifficulties for the plasticity algorithm. For this reason, I employ analgorithm in which the applied boundary velocity is slowly increasedduring the first time step. It is interesting to study the dependence ofthe resulting shear band angle on the number of iterations that areperformed. Results (Fig. 9) indicate that at least 7 iterations arerequired for extensional cases and that results converge after ~15iterations. Compressional cases, on the other hand, can be performedwith a single iteration, and results in slightly steeper angles if morethan ~10 iterations are employed.

3.6. Consequences for strain rate inside shear bands

The effect of numerical resolution on themaximumstrain rate insideshear bands was studied for both viscoelastoplastic and viscoplastic

Fig. 10. Effect of numerical resolution on maximum strain rate at a distance d abov

rheologies under extension and compression (for d=800 m andϕ=30°). In these simulations, the maximum strain rate was computedat a height of 1200 m. Results show that under compression, vis-coelastoplastic cases always result in larger strain rates inside theshear band (Fig 10). Interestingly, both viscous and viscoelastoplasticsimulations result in a strain rate which saturates with increasingresolution.

Extensional cases, however, show such saturation behavior onlyfor viscoplastic rheologies (which have larger strain rates thanviscoelastoplastic simulations at low resolutions). These results thussuggest that the viscoplastic simulations performed here do not havea strong mesh dependency. More work is required, however, to fullyunderstand these effects.

3.7. Multiple heterogeneities

All simulations performed in here focused on a relatively simplesetup in which a single heterogeneity was responsible for strainlocalization. It is instructive to see how such simulations can be com-pared to more realistic cases in which several heterogeneities arepresent. Therefore an experiment was performed in which 4 hetero-geneities with length scale d=800 m were randomly distributed.Results for a viscoelastoplastic rheology as a function of strain(Fig. 11) show that shear bands with Arthur to Coulomb angles areinitiated at the corners of the heterogeneities (somewhat dependingonwhere the inclusion is locatedwith respect to the upper boundary).Interactions between two shear bands that initiated at two hetero-geneities, however, might even induce shear bands at an angle thatis significantly shallower than the Coulomb angle. Moreover, shearbands are inmany cases curved. An interesting observation is also thatshear bands, which are reflected at the lower boundary, increase theirshear band angle. In the light of the simulations presented here, thiscan be understood in the following manner: Once a shear bandreaches a boundary, it serves as a location to initiate a new shear band.The heterogeneity in the stress field, however, is only a few elementsin size and thus significantly smaller than d, with as a result shearband angles that are closer to the Arthur or Roscoe angle (e.g. Fig. 4).

4. Discussion

The simulations performed here thus highlight the importanceof resolving the initial viscous heterogeneity sufficiently well nu-merically. Further insight in this effect can be obtained by plottingthe second invariant of the deviatoric stress field as a function of

e the inclusion. Vertical resolution is proportional to the horizontal resolution.

Fig. 12. Second invariant of deviatoric stress tensor around viscous heterogeneities computed with different numerical resolutions, Q1P0 elements and a viscous rheology. Small d/Δxvalues result in smaller absolute stress values.

Fig. 11. Shear localization as a function of bulk strain for a model with 4 randomly distributed viscous inclusions with d=800 m. Rheology is viscoelastoplastic, ϕ=30° and aresolution of 1200×300 nodes is employed with Q1P0 elements. Magenta, black and yellow lines indicate Roscoe, Arthur and Coulomb angles respectively. Shear bands initiated frominclusions have an Arthur to Coulomb orientation. Once they reflect from side boundaries, however, their orientation rotates towards the Roscoe angle (see for example white arrowsat 2.0% strain).

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numerical resolution for a given heterogeneity size for a purely viscousrheology (Fig. 12). For small values of d/Δx, the two high stress ‘lobes’ atthe corners of the inclusion are not well resolved. As a result, maximumstresses are smaller than in cases with larger d/Δx. In addition, stressdifferences (maximumminusminimum stresses) are smaller. Thus, theinitial (pre-plastic) stress field is more homogeneous at smallernumerical resolutions, which appears to favor Roscoe or Arthur angles.

The findings of this work can be used to reconcile previous nume-rical experiments on shear localization, which appear to have givencontradictory results. Most numerical simulations reported shearband angles from simulations in which localization was induced byrandom noise (in either the viscosity field or in the plastic param-eters). Doing such, induces stress heterogeneities that are on the orderof one or several elements in size. The simulations performed here,suggest that in this case, resulting shear bands should have the Arthurorientation unless friction angles are 20° or smaller, where Roscoeangles dominate. Both Lemiale et al. (2008) and Hansen (2003)obtained Coulomb orientations in their simulations, but in both casesa heterogeneity was present that was numerically resolved suffi-ciently (d/Δx~5 in Lemiale's model, whereas d/Δx~5 in Hansen'smodels).

Several parameters, however, remain to be studied. Firstly, I haveemployed a very specific algorithm to enforce incompressibility (aniterated penalty method). The code used by Lemiale et al. (2008) usesa different algorithm to compute incompressibility and it is unclear towhich extend this affects model results. Secondly, some of the resultsobtained here seem to indicate that, at least in viscoplastic cases, themaximum obtainable strain rate inside shear zones saturates to agiven value with increasing numerical resolution. This indicates thatviscoplastic shear localization is not mesh sensitive, as opposed tonon-regularized elastoplastic localization (e.g., Vermeer and de Borst,1984; de Souza Neto et al., 2009).

In the computational mechanics literature there is a rich traditionof numerically solving of strain-softening problems. Here, it wasrealized several decades ago that non-uniqueness of the solutionsresults in mesh-sensitivity of the results particularly for rate-independent elastoplastic materials (for which no internal lengthscales exists). Several solutions have been proposed to overcome thissensitivity. One way, called the Cosserat continua, is to add additionalrotational degrees of freedom in the system (e.g., Muhlhaus andVardoulakis, 1987; De Borst, 1991; Deborst and Sluys, 1991). Anotherway is to add viscosity (which effectively makes the system visco-elastoplastic) which works in both quasi-static and quasi-dynamiccases (see, e.g. Needleman, 1988; Wang et al., 1996; Wang et al.,1997). This is many ways similar to what is done in geodynamicalmodels, although it should be noted that inertial forces are typicallynegligible in models of tectonic deformation and that thereforeonly the quasi-static results are directly applicable to such setups(unless one wants to include the earthquake cycle). Needleman(1988) performed a one-dimensional analysis of shear localization,and demonstrated that solutions are regularized once a rate-depen-dent material is added to the setup. He however also demonstratedthat the length scale of localization is related to the length scale ofthe initial heterogeneity, which is unfortunate if applied to geodyna-mic cases as we can never be sure what the initial heterogeneitylength scale was. The results obtained here (Fig. 10) also demon-strate that convergence with respect to strain rate occurs inviscoplastic simulations. Yet, this topic deserves more attention inthe future, in particular for viscoelastoplastic rheologies in a geo-dynamic context.

5. Conclusions

Numerical simulations have been performed to address the effectsof numerics, geometry and rheology on the orientation of shear bandsin geodynamic models. It was demonstrated that it is possible to

obtain the Coulomb orientation. This, however, only occurs if theheterogeneity, which induces this shear band, is resolved with asufficiently large numerical resolution. Effects of rheology, type ofelement and iteration strategy were found to play a second ordereffect. If the heterogeneity is not resolved with sufficiently largenumerical resolution (at least 5–10 elements), or if it is situated closeto a boundary of the numerical box, resulting shear band orientationsare closer to the Arthur or Roscoe angle, independent of the employedresolution.

Acknowledgements

I thank Hans Muhlhaus, Taras Gerya , and Dave May for helpfuldiscussions and feedback on this work. The constructive reviews ofLuc Lavier, Dave May and an anonymous reviewer helped to clarifythe text. MILAMIN is available from www.milamin.org.

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