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Earth and Planetary Science Letters 287 (2009) 241–254 Contents lists available at ScienceDirect Earth and Planetary Science Letters j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / e p s l Subduction erosion modes: Comparing finite element numerical models with the geological record Duncan Fraser Keppie a,⁎, Claire A. Currie b,1, Clare Warren b,2 a b Department of Earth and Planetary Sciences, McGill University, Montreal, Quebec, Canada Geodynamics Research Group, Department of Oceanography, Dalhousie University, Halifax, Nova Scotia, Canada a r t i c l e i n f o Article history: Received 7 January 2009 Received in revised form 31 July 2009 Accepted 6 August 2009 Available online 6 September 2009 Editor: T. Spohn Keywords: subduction subduction factory subduction erosion subduction angle Andes forearc removal a b s t r a c t During subduction erosion, the upper plate is tectonically eroded by the subducting plate and carried into the mantle. The geological record suggests that subduction erosion is a fundamental process at subduction margins; however the underlying causes are not well constrained. Finite-element numerical models of ocean–continent subduction are used to investigate the roles of crustal frictional strength, subduction angle, and convergence rate in subduction erosion processes. Subduction erosion occurs in models in which the plate boundary zone is moderately strong, due to either high frictional strength or shallow angle of subduction. The models exhibit two distinct modes of subduction erosion: (1) steady, with slow trench migration rates (<4 km Ma− 1), subsidence in the remaining forearc, and a decrease in the angle of subduction, in which the edge of the continental plate is eroded in small blocks, and (2) unsteady, with fast trench migration rates (⋙15 km Ma− 1), subsidence at the end of the process, and an increase in the angle of subduction, in which a large block of continental forearc is removed. The unsteady mode is compatible with the sudden migration of the volcanic arc into the continental interior, a concurrent hiatus in arc volcanic production, and geochemical signatures showing crustal (forearc) contamination in the magma source region when arc volcanism renews in its new location. Both modes are inhibited by the presence of a thick sediment layer within the subduction zone, and neither mode requires the presence of topographic asperities on the lower plate. In natural subduction zones, subduction erosion may initially occur through steady erosion at the edge of the continental plate. As material is removed, the subduction angle may gradually decrease, increasing the strength of the plate boundary zone. The increased strength may lead to failure in the continental interior, unsteady removal of a large forearc block and relocation of the subduction zone into the upper plate where the cycle may repeat. The proposed cycle may explain observed patterns in the Andean margin, including steady and unsteady erosion recorded in the geological record and along-strike variation in present-day subduction angles. © 2009 Elsevier B.V. All rights reserved. 1. Introduction Truncation of geological units in the upper plate, crustal thinning, and relative trench migration towards the upper plate in ocean– continent subduction zones suggest the removal of material from the upper plate by tectonic processes. When geological candidates for the missing material cannot be identified elsewhere, subduction erosion processes must be considered (von Huene and Scholl, 1991; Clift and Vannucchi, 2004). It has been suggested that more than half of ⁎ Corresponding author. E-mail addresses: [email protected] (D.F. Keppie), [email protected] (C.A. Currie), [email protected] (C. Warren). 1 Present address: Department of Physics, University of Alberta, Edmonton, Alberta, Canada. 2 Present address: Department of Earth and Environmental Sciences, The Open University, Milton Keynes, MK7 6AA, UK. 0012-821X/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.epsl.2009.08.009 modern subduction zones undergo subduction erosion and net loss of upper plate material (Clift and Vannucchi, 2004). The results of a recent global survey suggest that subduction erosion is limited to subduction zones in which trench sediments are less than 1 km thick and where convergence is faster than 60 km Ma− 1 (Clift and Vannucchi, 2004; Kukowski and Oncken, 2006). Beyond this, there appears to be no obvious first-order correlation between the rate of subduction erosion and key subduction parameters (Clift and Vannucchi, 2004). Despite the general consensus that tectonic subduction erosion is a key process in the subduction factory, the controlling mechanisms are hotly debated. Models of tectonic subduction erosion require the mechanical entrainment of frontal and/or basal upper-plate basement by the subducting lower (oceanic) plate (von Huene and Lallemand, 1990). Debate focuses on the causes and styles of relative mechanical weakening of the upper plate that necessarily precede the mechanical entrainment. Weakening may arise primarily due to elevated stresses 242 D.F. Keppie et al. / Earth and Planetary Science Letters 287 (2009) 241–254 acting across the plate interface (e.g. Uyeda and Kanamori, 1979; Adam and Reuther, 2000; Clift and Vannucchi, 2004; Kukowski and Oncken, 2006) or due to material softening in the upper plate above the plate interface (e.g. Le Pichon et al., 1993; von Huene et al., 2004; Sage et al., 2006). Due to the wealth of data in the literature and the broad spectrum of reported observations, the subduction of the Farallon plate and its descendants beneath the Americas has created an ideal region in which to formulate and test models of subduction erosion. At one end of the spectrum, upper-plate material appears to be eroded from the upper plate in a gradual fashion (i.e., small pieces of forearc) at relatively slow rates (Clift and Vannucchi, 2004). For instance, trench migration rates between 0 and 3 km Ma− 1 towards the continent have been estimated in the central Andes due to subduction erosion (Kukowski and Oncken, 2006). Elsewhere, higher rates of ca. 10 km Ma− 1 have been associated with topographic asperities on the lower plate (e.g. Clift et al., 2003). Possible mechanisms for this mode of subduction erosion include ductile ablation of the upper plate due to viscous drag (e.g. Tao and O'Connell, 1992; Pope, 1998), brittle and ductile abrasion of the upper plate by topographic asperities on the lower plate (e.g. Clift et al., 2003; Vannucchi et al., 2003; Clift and Vannucchi, 2004; Hampel et al., 2004; Bangs et al., 2006), and brittle hydro-fracturing arising along the base of the upper plate due to fluid overpressures associated with dehydration of subducted sediments and oceanic crust (von Huene et al., 2004; Sage et al., 2006). In other localities, the geological record suggests more rapid removal of larger blocks of forearc material, possibly due to subduction erosion (Moran-Zenteno et al., 1996; Calvert et al., 2003; MoranZenteno et al., 2007; Malavieille and Trullenque, 2009). Minimum arc migration rates of >12 km Ma− 1 have been associated with two inferred episodes of forearc removal in southern Chile during the Miocene (Kay et al., 2005), and minimum trench migration rates of ca. 18–75 km Ma− 1 are implied for the late Oligocene/early Miocene forearc removal from southern Mexico (Moran-Zenteno et al., 1996, 2007). Along the Andean margin, geological evidence suggests that both gradual and rapid subduction erosion may have operated in the past or still be operating today (e.g. Kay et al., 2005; Clift and Hartley, 2007). More than one mechanism may therefore control subduction erosion and in any one place the mode may also change over time. In this study, finite element numerical models are used to investigate possible mechanisms that may lead to the mechanical failure, entrainment and eventual subduction of upper-plate material. The models focus on parameters that have previously been suggested to be important in subduction erosion processes: subduction angle, convergence rate, sediment cover on the oceanic plate, and the brittle strength of the upper plate. The aim is to address the following questions: 1. What is the relative importance of various parameters to the failure and subsequent entrainment of upper-plate material? 2. What are the key mechanisms of failure and entrainment associated with these parameters? 3. What are the kinematic expressions of these processes? Does upper plate material fail and entrain in relatively small or large blocks? What are the rates of trench migration and/or material removal? Does the remaining forearc undergo subsidence in response to subduction erosion processes? 2. Numerical modelling methodology 2.1. Basic principles The numerical modelling approach closely follows previous work (Currie et al., 2007; Warren et al., 2008b). Ocean–continent subduction is modelled using a 2-dimensional thermal–mechanical finite element code (SOPALE) that assumes vertical cross-section plane strain (Fullsack, 1995). The Arbitrary Lagrangian Eulerian (ALE) methodology (Fullsack, 1995) calculates large-deformation flows with free upper surfaces on an Eulerian finite-element grid that stretches vertically to conform to the material domain. A Lagrangian grid and passive tracking particles, advected with the model velocity field, are used to update the mechanical and thermal property distributions on the Eulerian grid. Conservation equations for mass assuming incompressibility (Eq. (1)), momentum assuming creeping (Stokes) flow (Eq. (2)), and (heat) energy (Eq. (3)) are solved in the model domain. ∂νi = 0 i = 1; 2 ∂xi ∂σij′ ∂xi ρcp − ∂P + ρg = 0 i; j = 1; 2 ∂xj ∂T ∂T ∂ ∂T + A + ν2 αgTρ =k + νi ∂t ∂xi ∂xi ∂xi ð1Þ ð2Þ ð3Þ where xi,j are spatial coordinates, νi are the components of velocity, σ ij′ is the deviatoric stress tensor, P is dynamic (mean stress) pressure, ρ is density, g is the gravitational acceleration, cp is the specific heat, T is the absolute temperature, t is time, k is thermal conductivity, A is radioactive heat production per unit volume, and α is the volumetric thermal expansion coefficient. The last term in the heat balance equation is the temperature correction for adiabatic heating when material moves vertically at velocity ν2. The mechanical computation calculates the velocity field, strain rate, deformation, and stress, subject to specified velocity boundary conditions and internal buoyancy forces (associated with compositional and thermal density variations). The thermal computation solves the heat balance equation on the Eulerian grid subject to the mechanical velocity field in the model domain and the thermal boundary conditions. 2.2. Model setup and mechanical and thermal boundary conditions The 2000 km × 660 km model domain is represented by a 200 × 115 Eulerian grid. Eulerian cell dimensions are 10 km × 2 km in the upper 120 km and 10 km × 10 km below (Fig. 1). Oceanic lithosphere is separated from continental lithosphere by an inclined weak seed used to initiate subduction. Mechanical boundary conditions are stress-free (upper), no-slip (sides) and free-slip (base). A uniform velocity boundary is applied to the oceanic lithosphere (VP, Fig. 1); the continental lithosphere boundary is held stationary (VR, Fig. 1). Material fluxed into the model domain by the specified convergence is balanced by a small uniform leakage flux out of the sides of the model domain beneath the lithosphere (VF, Fig. 1, after Currie et al., 2007). Material is not permitted to flow through the base of the model. The current models do not include surface processes. The initial steady-state two-dimensional temperature field is calculated for a fixed surface temperature (273 K), insulated side boundaries, and a fixed basal temperature (1833 K) (Fig. 1). Initial oceanic and continental lithosphere geotherms are illustrated in Fig. 1 and corresponding thermal parameters are listed in Table 1. In the sublithospheric mantle, convective heat transport is modelled conductively (52 W m− 1 K− 1) by scaling the conductivity by the Nusselt number for marginally supercritical upper mantle convection (Pysklywec and Beaumont, 2004) using an adiabatic temperature gradient of 0.4 K km− 1. The current models do not include shear heating (Warren et al., 2008b) or viscous strain weakening. D.F. Keppie et al. / Earth and Planetary Science Letters 287 (2009) 241–254 243 Fig. 1. Finite element numerical model setup showing mechanical and thermal boundary conditions. Ocean lithosphere reflects the thermal structure of a 70 Ma oceanic plate (Parsons and Sclater, 1977; Ranalli, 1987; Stein and Stein, 1992). Continental lithosphere reflects the thermal structure of average 120 km thick non-cratonic continental lithosphere (Vitorello and Pollack, 1980) including the combined effects of lithospheric thermal conductivity (W m− 1 K− 1) and crustal radioactive heat production (µW m− 3). 2.3. Model materials invariant of deviatoric strain. Frictional strength is reduced from ′ ϕeff = 15° to ϕeff = 2°–6° through the strain range 0:5 < ðI2 Þ < 1:5 for all model materials except the weak seed and ocean sediment (Table 1). Viscous flow deformation is specified by thermally-activated power law creep (Eq. (6)). 1 2 Both frictional-plastic and viscous flow rheologies are specified for each model material. Frictional-plastic failure deformation (loosely termed brittle) is specified by a pressure-dependent Drucker–Prager yield criterion (Eq. (4)). ′ 12 ð4Þ J2 = P sinϕeff + C; with v P sin ϕeff = ðP−Pf Þ sin ϕ ð5Þ where J 2′ is the second invariant of deviatoric stress, ϕeff is an effective internal angle of friction, C is the cohesion, P is the dynamic (mean stress) pressure, Pf is the pore fluid pressure, and ϕ is the internal angle of friction. Frictional-plastic strain softening is implemented through a linear decrease in ϕeff with ðI2′ Þ , where I 2′ is the second 1 2 ⁎ ðð1−nÞ = 2nÞ ηeff = f ðB ÞI˙2′ Q + PV ⁎ nRT ð6Þ e where ηveff is an effective viscosity, f is a linear scaling factor, B⁎ is the pre-exponential factor converted from uniaxial experiments to plane strain, İ2′ is the second invariant of deviatoric strain rate, n is the stress exponent, Q is the activation energy, V ⁎ is the activation volume and R is the universal gas constant (Table 1). Table 1 Finite element numerical model parameters. A. Parameters varied in the model experiments Parameter Description Units Value ZS ϕS θP VP Sediment thickness Strain-softened ϕeff Subduction angle Convergence rate km deg deg km Ma− 1 0, 4 2, 3, 4, 5, 6 10, 15, 20, 25, 30 25, 50, 75, 100, 125, 150 B. Mechanical and thermal material properties Parameter Units Ocean sediment Ocean crust Continent crust (upper) Continent crust (lower) Mantle lithosphere Sub-lithospheric mantle Thickness km Zs 8 24 12 To 660 km 2800 473 2900 – 8−1 0 Wet quartzite 1 4 2.92 × 106 223 0 750 2.25 1.0 × 10− 6 3.0 × 10− 5 1.15 2900 273 3300 – 15 − ϕS 0 Dry diabase 0.1 4.7 1.91 × 105 485 0 750 2.25 1.0 × 10− 6 3.0 × 10− 5 0 2800 473 2850 2900 15 − ϕS 2 Wet quartzite 5 4 2.92 × 106 223 0 750 2.25 1.0 × 10− 6 3.0 × 10− 5 1.15 2950 773 3100 – 15 − ϕS 0 Dry diabase 0.1 4.7 1.91 × 105 485 0 750 2.25 1.0 × 10− 6 3.0 × 10− 5 0.55 82 (ocean) 84 (cont.) 3250 1609 – – 15 − ϕS 0 Wet olivine 10 3 1.92 × 104 430 1.0 × 10− 5 1250 2.25 0.6 × 10− 6 3.0 × 10− 5 0 Ref density Ref density T HP density UHP density ϕeff − ϕS Cohesion Flow lawa f n B⁎ Q V⁎ Heat capacity Thermal conductivity Thermal diffusivity Thermal expansion Heat production a −3 kg m K kg m− 3 kg m− 3 deg MPa 1 Pa sN kJ mol− 1 m3 mol− 1 J kg− 1 K− 1 W m− 1 K− 1 m2 s− 1 K− 1 μW m− 3 Flow law references in text. 3250 1609 – – 15 − ϕS 0 Wet olivine 1 3 1.92 × 104 430 1.0 × 10− 5 1250 52 1.4 × 10− 5 3.0 × 10− 5 0 244 D.F. Keppie et al. / Earth and Planetary Science Letters 287 (2009) 241–254 The overall nonlinear solution for each timestep is solved using ηveff (Eq. (6), viscous flow) when the local flow stress is less than the Drucker–Prager yield stress and ηPeff (Eq. (7), frictional-plastic flow) when the Drucker–Prager yield stress is exceeded. Setting 1 P ηeff = J2′ 2 ð7Þ 1 2 2ðI˙2 Þ ′ 1 in regions that are on frictional-plastic yield (stress equals J2′ 2) and deforming at ðI˙2′ Þ allows the velocity field to be determined from the finite element solution for viscous creeping flows (Fullsack, 1995; Willett, 1999). Rheological parameters for viscous deformation in Eq. (6) (B⁎, n, Q , V ⁎) are based on experimental results for wet quartz (Gleason and Tullis, 1995), dry Maryland diabase (Mackwell et al., 1998), and wet olivine (Karato and Wu, 1993). Viscous flow laws for rocks that are stronger or weaker than the base set are constructed by linearly scaling the values of ηveff by f (Eq. (6)) (Beaumont et al., 2006). The use of a small set of reference materials permits some variation in flow properties but simplifies the interpretation of model results. This approach acknowledges inherent uncertainties in extrapolating laboratory results to natural settings (Patterson, 2001) but does not imply that corresponding Earth materials have the same composition as materials whose flow laws were used. Representative densities are specified for model materials at reference temperatures close to their average initial temperatures; densities are adjusted for thermal expansion during model evolution (Table 1). Sediment parameters follow Currie et al. (2007). Density changes for model materials are specified for eclogite (HP) and ultrahigh pressure (UHP) facies metamorphism (Table 1, following Warren et al. (2008b)). Phase-change densities are estimated by considering upper continental crust to contain 10% mafic and 90% felsic by volume (Walsh and Hacker, 2004); density changes correspond to 100% metamorphic transformation of the mafic component and 20% of the felsic component based on observations from natural UHP terranes (Krabbendam et al., 2000). Density changes are reversible. For density changes accompanying phase changes, the incompressibility equation (Eq. (1)) is modified for mass conservation. The associated volume change is calculated numerically by applying an δρ additional isotropic stress (pressure), δP = where βv is the 1 2 δρ βv ⁎ρ viscous bulk modulus and is the fractional density change, to finite ρ elements only at the moment of phase-related density change. This mitigates the long-term effect of over/under estimation of buoyancy forces due to fractional error in material volumes (Warren et al., 2008a). 2.4. Model evolution and parameter variation Prior to assessing model evolution for subduction processes, two stages prepare the models (Currie et al., 2007; Warren et al., 2008a,b). A first stage allows oceanic and continental domains to adjust isostatically. A second stage applies 500 km convergence at 25 km Ma− 1 to initiate subduction. Model times referred to below include the second preparation stage. The third stage investigates variations in shear traction on the plate boundary subduction shear zone by varying the effective angle of internal friction, the initial angle of the weak seed used to initiate subduction, the presence of sediment on the oceanic crust and the convergence velocity. In the models, shear traction in the brittle regime is primarily controlled by the effective angle of internal friction (ϕeff, Eq. (4)). Frictional-plastic strain-softening in nature is poorly constrained, but geological processes, including changes in mineralogical composition, foliation development and/or the impact of fluid-assisted processes, may contribute to a reduction in frictional strength in strongly deformed regions (Bos and Spiers, 2002; Huismans and Beaumont, 2003, 2007; Selzer et al., 2007; ). We implement frictional strainsoftening by varying the strain-softened effective internal angle of friction, ϕS, over a range of values (2° to 6°, Table 1) consistent with similar studies (Huismans and Beaumont, 2003; Ellis et al., 2004; Huismans and Beaumont, 2007). Variations in the angle of subduction may play a role in determining the onset of subduction erosion processes (Pilger, 1981; Pope, 1998; Gutscher, 2002; Kay et al., 2005; Clift and Hartley, 2007). If subduction erosion does not occur, the evolution of slab dip at crustal depths does not vary much through model time. Varying the angle of the weak seed used to initiate subduction (from 10° to 30°, Table 1) is therefore a useful proxy for subduction angle at shallow (i.e., crustal) depths until subduction erosion processes modify the subduction zone. Subduction erosion appears in nature to be limited to those margins where trench sediments are less than 1 km thick (e.g. Clift and Vannucchi, 2004; Kukowski and Oncken, 2006). We have run a set of numerical models both with and without a 4 km (2-element thick) layer of weak oceanic sediment to test the potential effect of trench sediment on subduction erosion processes. This thickness represents the upper limit of sediment supplied to natural subduction zones (e.g. Mishra et al., 2004; Currie et al., 2007). Erosive margins appear to be limited to those at which convergence is faster than 60 km Ma− 1 (Clift and Vannucchi, 2004; Kukowski and Oncken, 2006). We test a possible dependence on convergence rate by varying orthogonal convergence rates over a range of values consistent with Cenozoic subduction of the Farallon plate and its descendents beneath the Americas (25 km Ma− 1 to 150 km Ma− 1 Table 1) (Müller et al., 2008). 3. Results 3.1. Tectonic modes Four tectonic modes are observed in the numerical models (Fig. 2). Models where the plate boundary subduction shear zone is weak (i.e., low maximum shear traction) show no subduction erosion. In these models, the subduction zone remains stable throughout model evolution, i.e., no forearc is removed, the trench doesn't migrate and no change in subduction angle is evident. In models where the plate boundary subduction shear zone is strong (i.e., high maximum shear traction), plate convergence is accommodated through repeated failure of the oceanic plate in a manner inconsistent with modern plate tectonics. Numerical models where the strength of the plate boundary subduction shear zone is intermediate show two distinct modes of subduction erosion. (1) The term edge-weakening subduction erosion (EWSE) is introduced in this paper to describe the steady and gradual removal of upper plate (continental) material from the edge of the upper plate; both lower (Fig. 3A) and upper (Fig. 3B) crustal material is removed. Initially, the ductile lower continental lithosphere of the upper plate is carved away by the subducting plate (Fig. 3A), possibly due to viscous drag (Tao and O'Connell, 1992). This results in a shallowing of the subduction angle at depth. Subsequently, the brittle upper continental lithosphere of the upper plate at the continent edge adjacent to the plate boundary may deform and strain-soften and will ultimately fail when the frictional-plastic yield stress is achieved. It is then entrained with the lower plate and subducted (Fig. 3B). Subducted continental material may transiently underplate the continental lithosphere before being subducted into the deep mantle (Fig. 3C). Trench migration rates are generally <4 km Ma− 1 (Fig. 2A and C). Rates >4 km Ma− 1 are observed only for very shallow and relatively fast subduction zones (Fig. 2C). Over time, the subduction angle generally shallows. D.F. Keppie et al. / Earth and Planetary Science Letters 287 (2009) 241–254 245 Fig. 2. Numerical model results showing tectonic styles for variations in (A-series) strain-softened internal angle of friction and subduction seed angle (no sediment), (B-series) strain-softened internal angle of friction and subduction seed angle (4 km sediment), and (C-series) orthogonal convergence rate and subduction seed angle (no sediment). Example numerical models from A-series discussed in the text are MA1(ϕS = 4°, θP = 30°) for edge-weakening subduction erosion and MA2(ϕS = 3°, θP = 15°) for internal-weakening subduction erosion. (2) The term internal-weakening subduction erosion (IWSE) is introduced in this paper to describe when the upper (continental) plate weakens within the orogenic zone (Fig. 4A) and eventually fails (Fig. 4B). During failure, the plate boundary relocates from the ocean–continent interface to the shear zone formed in the continental interior. Rapid, unsteady removal of continental forearc happens as the forearc between the old and new subduction zones is entrained with the lower plate and subducted (Fig. 4C). The trench (or ocean–continent interface) migrates towards the upper plate at the orthogonal convergence rate. Some of the subducted forearc underplates the upper plate at depth (Fig. 4D). For models converging slowly with shallow angles of subduction, some of the underplated forearc subsequently exhumes up the subduction channel. The subduction angle of the new subduction zone is generally steeper (ca. 30° to 40°) than the abandoned subduction zone (ca. 10° to 30°). The internal-weakening subduction erosion mechanism observed in our numerical models occurs when moderate strength on the active subduction shear zone promotes bending (and thickening) in the orogenic zone of the upper plate (Fig. 5A). Bending (and thickening) deformation triggers brittle strain-softening which progressively weakens this region (Fig. 5A). Given sufficient weakening, failure initiates on several, variously oriented, shear zones (Fig. 5B) which then connect and form a new subduction zone across the weakened continental interior (Fig. 5C). Continental forearc trapped between the old and new subduction zones is entrained with the lower plate and subducted (Fig. 5D). 3.2. Effect of parameter variation on tectonic modes Fig. 2A shows the distribution of tectonic modes for variations in the strain-softened frictional angle and subduction angle for a convergence rate of 25 km Ma− 1 and no oceanic plate sediment. As the strain-softened internal angle of friction (ϕS) is increased, the strength or maximum shear traction on the plate boundary subduction shear zone increases. The tectonic mode shown by the models changes from stable subduction through edge-weakening subduction erosion to internal-weakening subduction erosion. At large ϕS, the plate boundary subduction shear zone is so strong that convergence is accommodated through failure of the oceanic plate. This behaviour is also dependent on subduction angle (θP). For larger θP, boundaries between the modes are shifted to larger ϕS. For cases in which internal-weakening subduction erosion occurs, the distance between the old and new subduction shear zones changes with the subduction angle such that the width of the removed forearc block decreases from ~ 300 km-wide for θP = 10° to ~50 km-wide for θP = 30°. The above experiments were repeated for a model with a 4 km layer of sediment on top of the oceanic plate (Fig. 2B). The addition of rheologically weak sediment inhibits subduction erosion, and field boundaries for internal-weakening subduction erosion shift towards increased friction and shallowed subduction. Edge-weakening 246 D.F. Keppie et al. / Earth and Planetary Science Letters 287 (2009) 241–254 Fig. 3. An example of edge-weakening subduction erosion. Key observations include: (1) the initial entrainment and subduction of the lower continental crust followed by the entrainment and subduction of the upper continental crust, and (2) the decrease in subduction angle at crustal depths over time. See discussion in text. subduction erosion is not observed in any of the models with sediment. Our numerical models examined only one possible sediment model; future work will address variations in sediment thickness and sediment rheology. Fig. 2C shows the results for variations in orthogonal convergence rate and subduction angle for a strain-softened friction angle of 2° and no sediment. At lower subduction angles, internal-weakening subduction erosion is the stable tectonic mode. At higher subduction angles, no subduction erosion is seen at low convergence rates, and edge-weakening subduction erosion is seen at higher convergence rates. 3.3. Trench migration and forearc subsidence for subduction erosion modes The migration of the trench towards the continent and the subsidence of the forearc may be tracked during model evolution, to allow comparison with data from natural subduction zones. For tracking purposes, we define the position of the model trench as the left-most surficial Eulerian cell containing continental material. Fig. 6 shows plots of the position of this Eulerian cell over time for models showing edge-weakening subduction erosion (Fig. 6A) and internal-weakening subduction erosion (Fig. 6B). The horizontal (migration) and vertical (subsidence) components of trench/forearc motion are shown in the left and right plots, respectively. For edge-weakening subduction erosion, the trench migrates steadily towards the continent at rates typically <4 km Ma− 1, and subsides during the subduction erosion process (Fig. 6A). The trench migration rate due to edge-weakening subduction erosion scales with the orthogonal convergence rate for a set of models with the same initial subduction angle (Fig. 2C). The scaling relationship is greatest for the set of models with the smallest initial subduction angle (10°) where trench migration rates are observed to be roughly 10% of the orthogonal convergence rate (Fig. 2C). For internal-weakening subduction erosion, subduction erosion occurs during a 2 to 10 Ma event. During this event, the trench migrates towards the upper plate at the orthogonal convergence rate and subsides at the end of the subduction erosion process (Fig. 6B). Fig. 6B (left plot) also shows trench migration back towards the oceanic plate during exhumation/unroofing of previously subducted material. 3.4. Fate of tectonically eroded continental crust for subduction erosion modes Subducted crustal material is tracked in the models as it is transported into the mantle. During edge-weakening subduction erosion, most of the crustal material is transported into the deeper mantle (Fig. 3). In contrast, following internal-weakening subduction erosion, a significant proportion of tectonically eroded material underplates D.F. Keppie et al. / Earth and Planetary Science Letters 287 (2009) 241–254 247 Fig. 4. An example of internal-weakening subduction erosion. Key observations include: (1) the formation of a new subduction shear zone across the deformed and softened continental interior, (2) the entrainment and subduction of a wide forearc block in a single event, and (3) the net effect of subduction angle steepening at crustal depths. See discussion in text. Legend as in Fig. 3. the upper-plate lithosphere (Fig. 4). In some of the models, most notably those which have shallow subduction and converge at slow rates, subducted crustal material may subsequently be exhumed up the subduction channel (Fig. 6B). 4. Discussion 4.1. The relative strength of subduction shear zones The numerical models show that plate convergence is accommodated by lithosphere-scale shear zones which operate at the weakest section of lithosphere at any time, i.e., subduction at the ocean– continent interface (plate boundary zone), failure in the continental crust (leading to subduction erosion) or oceanic plate failure. These variations may be explained by theoretical calculations of the strength of potential shear zones across each section of model lithosphere. Fig. 7A shows Brace–Goetze strength envelopes for the continental and oceanic plates and the plate boundary zone assuming a strain rate of 1 Ma− 1 (3.17 10− 14 s− 1); details of the calculations are in Appendix A. These theoretical profiles are in good agreement with strength profiles extracted from the numerical models (Fig. 7B). Integration of each strength profile over the surface area of the shear zone provides a measure of the total strength of the lithosphere. The behaviour of the subduction system is governed by the relative strengths of the plate boundary zone, oceanic and continental lithospheres. In the models, a weak plate boundary arises with a low effective internal angle of friction of 2° (Fig. 7A). Plate convergence is accommodated through localized deformation along the narrow shear zone. This is the stable subduction mode. The strength of the plate boundary zone will increase with either a larger strain-softened effective angle of internal friction (Fig. 7C) or a decrease in subduction zone angle, which increases the length of the shear zone and increases the effective frictional strength of the shear zone materials (Fig. 7D). At low subduction angles and high internal friction angles, the oceanic plate is the weakest lithosphere section in the system and geologically-unreasonable oceanic plate failure occurs (Fig. 7A). Between these two modes, the plate boundary has moderate strength, and subduction erosion is observed. Edge-weakening subduction erosion occurs for a plate boundary zone that is slightly stronger than the stable subduction case (e.g., higher internal angle of friction). Enhanced shear traction from the plate margin induces viscous drag and brittle failure and entrainment of the adjacent 248 D.F. Keppie et al. / Earth and Planetary Science Letters 287 (2009) 241–254 Fig. 5. Evolution of cumulative strain in the internal-weakening subduction erosion mechanism. Bending in the continental interior causes sufficient strain to trigger strain-softening. The strain-softened region in the continental interior ultimately fails as a new subduction shear zone. See discussion in text. Velocity arrows as in Fig. 3. continental crust. The rate at which continental material is removed through edge-weakening depends on the magnitude of shear traction, which is related to the subduction angle and the orthogonal convergence rate (Fig. 7A). An even stronger plate boundary zone exhibits internal-weakening subduction erosion, as increased shear traction promotes bending and thickening in the interior region of the continental plate in a broad region from 50 to 500 km inboard of the plate margin depending on the geometry of the subduction zone (e.g., Fig. 5A). This region undergoes frictional-plastic strain-softening, which reduces its strength (Fig. 7C). Once this section of lithosphere is weaker than the plate boundary zone, it begins to fail (Figs. 7B and 5C). The locus of subduction shifts landward a distance of 50 to 300 km depending on model geometry when continued failure in the orogenic zone forms a new subduction zone (Fig. 5C). Continental forearc trapped between the old and new subduction zones is entrained with the lower plate and subducted (Fig. 5D). Strain softening is one mechanism by which the continental interior may be weakened. Natural weakening mechanisms could also include thermal weakening due to arc magmatism and activation of pre-existing weak planes. Subduction of rheologically weak sediments will modify the behaviour of the plate boundary (Fig. 2B). At steep subduction angles, sediments act to localize deformation within the plate boundary zone and inhibit subduction erosion. This behaviour is consistent with observations that significant subduction erosion does not occur where trench sediments are greater than 1 km thick (Clift and Vannucchi, 2004). At shallower subduction angles, shear traction along the interface increases, and internal-weakening subduction erosion may occur for conditions that allow stress to be transmitted into the plate interior (e.g., higher internal angle of friction). At large values of internal friction, the continent becomes too strong to deform, leading to oceanic plate failure. 4.2. Comparing the numerical modelling results with natural subduction zones 4.2.1. Trench migration rates In nature, trench migration is inferred from changes in volcanic arc position (e.g. Moran-Zenteno et al., 1996; Kay et al., 2005) and from estimates of forearc removal from forearc subsidence records and other proxies (e.g. Clift and Vannucchi, 2004; Clift and Hartley, 2007). Rates of trench migration towards the upper plate estimated for natural examples fall into two categories: 1. slow, ca. 0–3 km Ma− 1 (ranging up to 10 km Ma− 1 when topographic asperities on the lower plate appear to enhance forearc removal), for which several subduction erosion mechanisms have been proposed including ductile ablation (e.g. Tao and O'Connell, D.F. Keppie et al. / Earth and Planetary Science Letters 287 (2009) 241–254 249 Fig. 6. Representative behaviour for the change in trench/forearc position and elevation through time in numerical models of subduction erosion. (A) An example of edge-weakening subduction erosion (EWSE, Model MA1, Fig. 3). (B) An example of internal-weakening subduction erosion (IWSE, Model MA2, Fig. 4). 1992), topographic abrasion (e.g. Clift and Vannucchi, 2004) and hydro-fracturing due to fluid over-pressures (e.g. von Huene et al., 2004), and 2. fast, ca. 10–100+ km Ma− 1, for which subduction erosion processes have been invoked (Moran-Zenteno et al., 1996; Kay et al., 2005; Moran-Zenteno et al., 2007) but mechanisms were not discussed. We suggest that the edge-weakening and internal-weakening subduction erosion mechanisms may provide explanations for the slow and fast mode of natural subduction erosion respectively. Rates of trench migration due to edge-weakening subduction erosion in the numerical models are generally slow, ca. 0–4 km Ma− 1, for most of the tested parameters. Critically, this erosion mechanism does not require the presence of topographic asperities on the lower plate. Model trench migration rates in excess of 10 km Ma− 1 are observed for crustal subduction angles <15° and convergence rates >100 km Ma− 1 (Fig. 2C). There is currently not enough data in the literature for natural examples to test whether this prediction is also the case in nature. The current numerical models are not designed to address the details of magmatic processes at the volcanic arc, but it is expected that edge-weakening subduction erosion should be accompanied by landward migration of the volcanic arc at approximately the trench migration rate. In the models, internal-weakening subduction erosion occurs during a short (2 to 10 Ma) event during which a large (50 to 300 km wide) forearc block is subducted. During this period, model trench migration rates are generally equivalent to the imposed model convergence rate (i.e., 25–150 km Ma− 1). 4.2.2. Forearc subsidence records Our numerical models show that for margins experiencing edgeweakening subduction erosion, subsidence takes place during the process (Fig. 6A). This result matches the correlation between forearc subsidence and inferred slow subduction erosion inferred along the Middle American and Andean subduction zones (e.g. Vannucchi et al., 2003; Clift and Vannucchi, 2004; Clift and Hartley, 2007). In models that experience an internal-weakening subduction erosion event, subsidence occurs suddenly at the end of the process (Fig. 6B). This result is similar to the record in southern Mexico where 250 m of forearc subsidence has been suggested to have taken place between 23 Ma and 19.5 Ma (Clift and Vannucchi, 2004). The subsidence in southern Mexico took place just after the inferred removal of a ca. 250 km wide forearc block (Moran-Zenteno et al., 1996; Keppie and Morán-Zenteno, 2005). Constraints on the size and timing of forearc removal in southern Mexico come from the concurrent hiatus and migration of the volcanic arc from the coast to the Mexican interior at the end of the Oligocene (Keppie and Morán-Zenteno, 2005; MoranZenteno et al., 2007; Gomez-Tuena et al., 2008). 4.2.3. Where does tectonically eroded crust end up? The fate of subducted continental crust has important implications for crustal recycling processes at convergent margins. The models suggest that subducted material may be transported into the deep mantle, may underplate the continental lithosphere or may be subsequently exhumed up the subduction channel. Crustal material returned to the deep mantle will affect the Earth's chemical fractionation through time (e.g. Clift and Vannucchi, 2004; Hawkesworth and Kemp, 2006). Underplated material may be returned to the surface through arc magmatism and will affect the chemistry of the magma source region (e.g. Kay et al., 2005; Currie et al., 2007). Although most exhumed UHP material is thought to be sourced from the subducting plate (Warren et al., 2008a), the results of this and other studies suggest that exhumed material may also be sourced from subducted upper plate continental crust (Gerya et al., 250 D.F. Keppie et al. / Earth and Planetary Science Letters 287 (2009) 241–254 Fig. 7. A: Calculated profiles for non-softened ocean, non-softened continent and a softened subduction zone dipping at 15°. These profiles represent reference estimates for the corresponding sections of lithosphere in the numerical models initiated with a 15°-dipping subduction zone just prior to varying parameters in order to study their effect on ′ 1 subduction zone evolution(i.e., at the end of setup stage). The calculations assume a constant strain rate of ð˙I2 Þ2 = 3:17 × 10−14 s−1 (see Appendix A for details). B: Strength profiles for different sections of lithosphere obtained in Model MA2 (θ = 15°, ϕeff = 3°, Fig. 2A). Profiles shown are for non-softened ocean, non-softened continent, softened subduction zone, and softened continent obtained from vertical, vertical, 15° inclined, and the weakest non-linear trajectory through the softened orogenic zone starting at the surface of the model at 300 km, 1700 km, 600 km and 800 km model distance for 22 Ma, 22 Ma, 22 Ma, and 30 Ma model time, respectively. C: Changes in subduction zone strength and continental lithosphere strength as a function of the effective angle of internal friction given a constant angle of shearing with respect to the horizontal (37.5° for the continental lithosphere and 15° for the subduction zone, See Appendix A for details). D: Changes in subduction zone strength as a function of upper slab subduction angle given the effective angle of internal friction used in the setup stage (2°). 2002; Gerya and Stockhert, 2006). Further study will assess the relative importance of subduction erosion mode to other factors, such as material density, in determining the fate of subducted material. Where the subduction and possible underplating of large forearc blocks have been inferred in nature, arc magmatism is observed to experience a hiatus of several million years before resuming in a position farther into the continental interior (Kay et al., 2005; MoranZenteno et al., 2007). Geochemical signatures from the earliest magmas produced after the hiatus show spikes in trace element signatures (Kay et al., 2005) as well as adakitic signatures (GomezTuena et al., 2008). One explanation of the geochemical signatures could be large amounts of crustal material supplied to the magma source region after subducted forearc blocks underplate the upper plate (Kay et al., 2005). 4.3. Subduction erosion cyclicity The two subduction erosion modes represent end-members in a kinematic continuum (Fig. 8A). Sizes of continental forearc blocks removed from the upper plate range from less than the model resolution (i.e., 10 × 2 km2) up to roughly 300 × 15 km2 in cross-section. Rates of trench migration range from <1 km Ma− 1 up to the model convergence rate (25 to 150 km Ma− 1). Removal of small forearc blocks and slow trench migration is characteristic of edge-weakening subduction erosion, whereas removal of a large forearc block and rapid trench migration is characteristic of internal-weakening subduction erosion. In the Andean forearc, several studies suggest that periods of steady (slower) subduction erosion have been interrupted by periods of non-steady (faster) subduction erosion (Kay et al., 2005; Clift and Hartley, 2007). Variations in subduction angle have also been observed along the Andes. Subduction geometry has been linked to plate velocity (e.g. Lallemand et al., 2005), distance from a symmetry plane inferred for the Central Andes (Gephart, 1994), resistance to bending in the middle of large oceanic plates, (Schellart et al., 2007; Schellart, 2008) and topographic asperities on the subducting slab (Gutscher, 2002). These explanations, however, do not always hold for the whole Andean subduction system (Fig. 9). We propose a conceptual model linking the two numerical model modes and use this conceptual model to propose an explanation for the observed cyclicity in subduction erosion and variation in subduction D.F. Keppie et al. / Earth and Planetary Science Letters 287 (2009) 241–254 251 Fig. 8. A: Edge-weakening and internal-weakening subduction erosion are end-members in a possible kinematic continuum of subduction erosion modes. B: A conceptual model of subduction erosion cyclicity predicts alternation between edge-weakening and internal-weakening subduction erosion modes. See discussion in text. angle in the Andes (Fig. 8B). This model is based on the observation that edge-weakening subduction erosion is the preferred mode for steeper subduction angles but acts to shallow the subduction system, whereas internal-weakening subduction erosion is favoured for shallow subduction angles but acts to steepen the subduction system. The cyclical model starts with a subduction system showing little to no subduction erosion. Over time, external boundary forces such as plate velocity (Lallemand et al., 2005) or resistance to plate bending (Schellart et al., 2007; Schellart, 2008) may cause the subduction angle to shallow. Edge-weakening subduction erosion is initiated, resulting in tectonic erosion of the side and basal edges of the upper plate and a decrease in subduction angle. The combination of shallowing and cooling by the subducting plate acts to increase the strength of the plate boundary shear zone. Once the strength of the active subduction shear zone exceeds the strength of the continental interior, internalweakening subduction erosion is triggered. A large block of forearc material is subducted and the plate margin relocates inboard to a new subduction zone which initiates at moderate angles of 35° to 45°. The external forcing of the shallow subduction angles governs ongoing evolution, re-engaging edge-weakening subduction erosion and the sequence repeats. The numerical models themselves show partial cyclicity, from edge-weakening to internal-weakening back to edgeweakening subduction erosion, in the absence of an external control renewing shallow subduction angles. The proposed cycle may explain, in part, the present-day alongstrike variation of near-surface subduction angle observed in the Andean margin (~15° to ~ 40°, Fig. 9). At a given time, different segments of the margin could be at different stages in the proposed cycle. Segments of the system may progress through the cycle at different rates depending where factors such as supply of sediment to the trench and pulses of arc magmatism vary along strike (e.g. Ernst, 2004; Kukowski and Oncken, 2006). Cascadia (Calvert et al., 2003), a forearc block subducted beneath the Luzon arc (e.g. Chemenda et al., 1997; Malavieille and Trullenque, 2009), and the forearc removed from southern Mexico during the late Oligocene/early Miocene (Moran-Zenteno et al., 1996, 2007). Further evidence comes from the rapid step-wise shift of arc magmatism into the continental interior as inferred in the Andes and in southern Mexico (Kay et al., 2005; Moran-Zenteno et al., 2007). Proposed explanations for the observations include the removal of the forearc by strike-slip tectonics in the southern Mexico example (e.g. Pindell et al., 1988; Silva-Romo, 2008) and slab shallowing causing arc migration in the Andean example (e.g. Clift and Hartley, 2007). Subduction erosion is an alternative possibility that should be explored. In the numerical models, internal-weakening subduction erosion occurs above relatively strong shallowly-dipping subduction zones (Fig. 2). Early stages of the process include thrust-sense shearing in the weakened continental interior on several disconnected and variously oriented shear zones (Fig. 4). The global CMT earthquake catalogue (Global-CMT-Catalog, 2008), filtered for thrust-sense earthquake focal mechanisms (i.e., rake > 45°) along the Andean margin, suggests that thrust-sense earthquakes in the continental interior are apparently restricted to the segments of relatively shallow subduction (i.e., from 0°–14° S and from 27°–32° S, Fig. 9). It is important to note that earthquake focal mechanisms do not discriminate between subduction-synthetic and subduction-antithetic thrusts. These earthquakes may be explained by thin-skinned foldand-thrust belts (e.g. Kley et al., 1999) or subduction-antithetic thrusts where the South American craton is underthrusting the Andes. However, they may also indicate that internal-weakening subduction erosion is incipient above the Andean shallow subduction zone segments, where the shallow subduction angle results in a strong plate boundary where stress can be transmitted inboard to the weaker orogenic zone. 4.4. Does internal-weakening subduction erosion occur in nature? 5. Conclusions The evidence for the entrainment and subduction of large forearc blocks at ocean–continent subduction zones is only sparsely preserved in the geological record, and thus has not yet been fully recognized in subduction margin evolution. Candidate examples include a forearc block interpreted in seismic images beneath Finite-element numerical models of ocean–continent subduction have been used to investigate the roles of crustal frictional strength, subduction angle, and convergence rate in subduction erosion processes. The models exhibit two distinct modes of subduction erosion, 252 D.F. Keppie et al. / Earth and Planetary Science Letters 287 (2009) 241–254 Fig. 9. The Nazca Plate and the South American Andes and the Cocos Plate and the Mexican Sierra Madres are shown in (A) plan view, and (B) for several vertical sections constructed orthogonal to the subduction zones. The Global GMT earthquake catalogue filtered for thrust-sense earthquakes (i.e., rake > 45°) (Global-CMT-Catalog, 2008), slab-top contours (in km, Syracuse and Abers, 2006)), and topography (Farr et al., 2007) are plotted. Regions of shallow thrust-sense earthquakes in the continental interior occur above shallow subduction angle segments. neither of which require topographic asperities on the subducting plate. 1. Steady, with slow trench migration rates (<0–4 km Ma− 1) and a decrease in subduction angle, in which most tectonically eroded material moves in small blocks that are principally transported into the deep mantle. During this process, the forearc undergoes progressive subsidence and the volcanic arc may migrate inland at the trench migration rate. 2. Unsteady, with fast trench migration rates (⋙15 km Ma− 1) and an increase in subduction angle to moderate values (ca. 35°–45°), in which a large (50 to 300 km wide) block of continental forearc is removed in a single tectonic event. The remaining forearc may record evidence of thrust deformation at the beginning of this process and sudden subsidence timed with the end of this process. After the unsteady removal of a large forearc block, the volcanic arc may relocate inland into the upper plate a distance approximately equal to the width of the removed forearc block. The volcanic arc may appear to jump to its inland position since migration may coincide with a hiatus in the production of arc volcanics. The first arc volcanics produced in the new location may also show geochemical signatures consistent with forearc material in the arc magma source region. The occurrence of subduction erosion is governed by the relative strength of the plate boundary zone and the interior of the upper (continental) plate. If the plate boundary is weak due to low frictional strength or the presence of rheologically weak sediments, subduction erosion does not occur. A moderate increase in plate boundary strength (e.g., increased frictional strength, decreased subduction angle or increased convergence rate) will increase shear traction on the continental plate, initiating subduction erosion. The steady mode, termed edge-weakening subduction erosion, is characterized by the entrainment of small blocks of continental material adjacent to the plate boundary. The non-steady mode, termed internal-weakening subduction erosion, occurs at stronger plate boundary zones where plate convergence induces failure within the interior continental orogenic zone. Subduction erosion behaviour in the numerical models is compatible with geological evidence from the Mexican and Andean subduction zones. D.F. Keppie et al. / Earth and Planetary Science Letters 287 (2009) 241–254 A cyclical model has been proposed in which the two subduction erosion modes alternate. If regional tectonics favour shallowlydipping subduction zones, then repeated internal-weakening subduction erosion events, which temporarily increase the subduction angle, may occur in response to edge-weakening subduction erosion, which acts to continually shallow and strengthen the active subduction zone. This cycle may provide an explanation for the observed alternation between steady and non-steady subduction erosion as well as alongstrike variation in subduction angle at natural subduction zones such as along the Andean margin. Acknowledgements DFK acknowledges support from a Canada Graduate Scholarship from the Natural Sciences and Engineering Research Council of Canada. Chris Beaumont of the Geodynamics Research Group at Dalhousie University is thanked for his generosity, guidance and support, without which this work would not have been possible. Douglas Guptill (Geodynamics Research Group, Dalhousie University) is acknowledged for technical support. Two anonymous reviewers are acknowledged for their constructive feedback. Appendix A. Strength profile calculations Strength profiles plotting differential stress with depth (e.g. Ranalli, 1987) are produced to illustrate the strength of different sections of lithosphere in the numerical models for the case of failure on thrustsense shear zones (Fig. 7). In Fig. 7B, values for the differential stress are extracted directly from the numerical models. In Fig. 7A, B, and D, we calculate values for the differential stress using a theoretical calculation and some simplifying assumptions. Details of this approach are as follows. For principal stress axes, we assume σ1 is horizontal and σ2 is vertical and equal to the lithostatic pressure. This represents simple stress conditions compatible with thrust-sense shearing (Anderson, 1951; Sibson, 1974). For strain rate, we assume a constant value of 1 Ma− 1 (i.e., 3.17 ⁎ 10− 14 s− 1). This represents a geologically reasonable strain rate for the purposes of illustration (Ranalli, 1987), although it is understood that strain rate is variable in the numerical models and nature. For differential stresses, at each depth, we calculate the differential stress that would be sustained on a thrust-sense shear zone inclined at a constant angle with respect to the horizontal. For each material, differential stress for frictional-plastic (Eq. (4)) and viscous (Eq. (6)) rheological equations of state are calculated and the minimum value of the two is used (Ranalli, 1987). Brittle anisotropy occurs in the numerical models because the subduction zone is initiated at a given angle with weak material and model materials strain-soften in dynamically produced shear zones. The finite element method handles brittle anisotropy directly. In the 1 theoretical calculations, we use Eq. (4) ⁎sinð2θ + ϕeff Þ to account for brittle anisotropy, where θ is the angle of the shear plane with respect to σ1 and ϕeff is the effective angle of internal friction. For isotropic materials, brittle failure occurs on the optimal failure plane which satisfies the condition 2θ + ϕeff = 90°. The required differential stress will be larger for deformation along planes at other orientations unless the material is anisotropic and the effective frictional strength is sufficiently low for weak planes with non-optimal orientations (Jaeger, 1960). For potential shear zones crossing undeformed oceanic and continental lithosphere, we assume the same initial rheological and thermal profiles as in our numerical experiments (Fig. 1) and ϕeff = 15°. If these sections of lithosphere were to fail and shear, the Drucker–Prager yield criterion (Eq. (4)) for material with ϕeff = 15° predicts failure planes at an angle of θ = 37.5° from the horizontal maximum stress, σ1. A value of θ = 37.5° is used in the calculations to illustrate the strength of shear zones crossing initially undeformed oceanic and continental litho- 253 sphere, and for simplicity, it is assumed that the shear zone maintains this angle, even if the material undergoes strain softening. 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