Initiation of localized shear zones in viscoelastoplastic rocks

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 111,, doi: /2005jb003652, 2006 Initiation of localized shear zones in viscoelastoplastic rocks Boris J. P. Kaus 1,2 and Yuri Y. Podladchikov 3 Received 28 January 2005; revised 29 July 2005; accepted 16 January 2006; published 25 April [1] Shear localization is a process of primary importance for the onset of subduction and the evolution of plate tectonics on Earth. In this paper we focus on a model in which shear localization is initiated through shear heating. The rheology employed is linear Maxwell viscoelastic with von Mises plasticity and an exponential dependence of viscosity on temperature. Dimensional analysis reveals that four nondimensional (0-D) parameters control the initiation of shear zones. The onset of shear localization is systematically studied with 0-D, 1-D, and 2-D numerical models, both under constant stress and under constant velocity boundary conditions. Mechanical phase diagrams demonstrate that six deformation modes exist under constant velocity boundary conditions. A constant stress boundary condition, on the other hand, exhibits only two deformation modes (localization or no localization). Scaling laws for the growth rate of temperature are computed for all deformation modes. Numerical and analytical solutions demonstrate that diffusion of heat may inhibit localization. Initial heterogeneities are required to initiate localization. The derived scaling laws are applied to Earth-like parameters. For a given heterogeneity size, stable (nonseismic) localization only occurs for a certain range of effective viscosities. Localization is inhibited if viscosity is smaller then a minimum threshold, which is a function of the heterogeneity size. The simplified rheological model is compared with a more realistic and more complex model of olivine that takes diffusion, power law, and Peierls creep into account. Good agreement exists between the models. The simplified model proposed in this study thus reproduces the main physics of ductile faulting. Two-dimensional late stage simulations of lithospheric-scale shear localization are presented that confirm the findings of the initial stage analysis. Citation: Kaus, B. J. P., and Y. Y. Podladchikov (2006), Initiation of localized shear zones in viscoelastoplastic rocks, J. Geophys. Res., 111,, doi: /2005jb Introduction [2] Strain localization is one of the long-lasting research topics in theoretical geodynamics. Whereas localization in the upper, brittle, part of the lithosphere is relatively well understood [e.g., Scholz, 2002; Brace and Kohlstedt, 1980], the formation of shear zones in deeper parts of the lithosphere is more puzzling [e.g., Regenauer-Lieb and Yuen, 2003]. Yet, numerical models of mantle convection indicate that localized deformation in these parts of the lithosphere is of crucial importance for the generation of plate tectonics. So far two main groups of models have been employed: (1) models in which a damage rheology results in localization [e.g., Bercovici, 1993, 1996; Tackley, 1998, 2000; Bercovici, 2003; Auth et al., 2003; Ogawa, 2003] and (2) models in which a localization is caused by shear heating in combination with elasticity, diffusion creep, 1 Earth Sciences Department, ETH Zürich, Zürich, Switzerland. 2 Now at Department of Earth Sciences, University of Southern California, Los Angeles, California, USA. 3 Physics of Geological Processes, University of Oslo, Oslo, Norway. Copyright 2006 by the American Geophysical Union /06/2005JB003652$09.00 dislocation creep and low-temperature plasticity [Regenauer-Lieb et al., 1999; Regenauer-Lieb and Yuen, 2000a, 2000b; Regenauer-Lieb et al., 2001; Regenauer-Lieb and Yuen, 2004]. Whereas the first group of models should still be regarded as being somewhat ad hoc, the second group of models sofar suffered from the fact that too many parameters were involved to make incorporation into typical convection codes feasible. [3] In this work we focuss on the second group of models. The main purpose is (1) to parameterize the relatively complex, laboratory-based, rheology as employed by Regenauer-Lieb and Yuen [1998] by a simple viscoelastoplastic rheology, and (2) to systematically study the parameterized rheology and determine parameters that control the initiation of ductile shear zones. Despite its simplicity, the simplified rheology still catches the main physics of shear-heating-driven shear localization. Kaus [2005] gives rules (in the form of MATLAB scripts) to transform realistic rheologies (as, e.g., employed by Regenauer- Lieb et al. [2001]) into the parameterized rheology employed in this work. By employing these rules and scaling laws derived in this work, we can qualitatively explain the observation of Regenauer-Lieb et al. [2001] 1of18

2 Table 1. Variables and Parameters Used in This Work Variable Units Meaning Variables v ms 1 velocity x,z m space coordinate s Pa stress P Pa pressure t Pa deviatoric stress _e s 1 strain rate _e vis s 1 viscous strain rate _e el s 1 elastic strain rate _e pl s 1 plastic strain rate t s time _l Pa 1 s 1 plastic multiplier jtj = (0.5t ij t ij ) 0.5 Pa effective stress F Pa von Mises yield function T K temperature T 0 K initial temperature T di = T T 0 K temperature difference s 0 Pa initial stress _e BG = (0.5 _e ij _e ij ) 0.5 s 1 background strain rate R m heterogeneity radius B vis = 2m 0 _ebg - Deformation number s0 B pl = s0 - Plasticity number sy Pe = pffiffiffiffiffiffiffiffiffi R - Peclet number km 0 =G Br = s2 0 g rcpg - Efficiency of shear heating U ms 1 applied 1-D velocity q 0-D Ks 1 0-D growth rate q 1-D Ks 1 1-D growth rate q 2-D Ks 1 2-D growth rate q diff Ks 1 diffusion growth rate T bg K background temperature Br * = s2 0 H 2 g km 0 - classical Brinkman number De = 2m 0 _e G - Deborah number Parameters m Pa s effective viscosity G Pa elastic shear modulus s y Pa yield stress m 0 Pa s viscosity at T = T 0 g K 1 e-fold length of viscosity k m 2 s 1 thermal diffusivity r kg m 3 density c p Jkg 1 K 1 heat capacity f s - shear parameter L m thickness of 1-D slab 0 T i K initial thermal perturbation k Wm 1 K 1 thermal conductivity that subduction at passive continental margins occurs more readily for wet rheologies than for dry rheologies. 2. Mathematical Model and Nondimensionalization 2.1. Governing Equations [4] The rheology employed in this work is viscoelastoplastic with an exponential temperature-dependent (linear) viscosity. The parameters that enter this rheology (temperature dependence of viscosity, initial viscosity, elasticity and yield stress), can be computed from a more realistic rheology that employs diffusion creep, dislocation creep and Peierls (low temperature) plasticity [Kaus, 2005]. Comparisons between the real and the parameterized rheology give satisfying results for a large number of nondimensional (0-D) (constant strain rate) experiments (see also section 4.1). [5] The rheology is i ¼ 0 here v i is velocity and x i the spatial coordinates (see Table 1 for symbols). Assuming that the effects of inertia and gravity can be ignored, force equilibrium j ¼ 0 where s ij = Pd ij + t ij is stress, P = 1/3s ii is pressure, d ij the Kronecker delta and t ij are deviatoric stresses. The strain rate is defined as _e ij ¼ i j ð3þ i The rheology is viscoelastoplastic: _e ij ¼ _e vis ij þ _e el ij þ _e pl ij ¼ 1 2m t ij þ 1 Dt ij þ lt 2G Dt _ ij where Dt ij /Dt = _t ij W ij t ij + t ij W ij denotes the objective Jaumann derivative of the stress tensor, W ij ¼ j i is the material spin tensor, m is the shear viscosity, G is the elastic shear modulus and _ l is a to be determined variable to ensure that the yield criterion is not violated. Laboratory experiments [e.g., Goetze and Evans, 1979] indicate that the stress of rocks in the upper mantle is limited, either by the Peierls creep mechanism or by a different semibrittle yielding mechanism. Although these two mechanisms are significantly different from a physical point of view, they are both characterized by a relatively weak dependence of differential stress on strain rate [e.g., Regenauer-Lieb and Yuen, 2004]. Numerically, we approximate this effect by employing a pressure-independent von Mises yield criteria [Simo and Hughes, 2000], which can be written as F ¼jtj s y where jtj = (0.5 t ij t ij ) 0.5 is the effective stress and s y is the yield stress. Viscosity is assumed to be temperaturedependent according to the Frank-Kamenetzky approximation [Frank-Kamenetzky, 1969]: ð1þ ð2þ ð4þ di gt m ¼ m 0 eð Þ ð5þ where T di = T T 0 is the temperature difference compared to the initial temperature, T is temperature, m 0 is initial viscosity, and T 0 is initial temperature. Conservation of energy is given di þ v i ¼ 2 þ c j t ij _e vis ij rc p þ _e pl ij ð6þ 2of18

3 where k is thermal diffusivity, r is density, and c p is heat capacity, and the last term denotes shear heating due to dissipative, nonrecoverable processes with c being its efficiency. Experimental work suggests this an efficiency of 0.4 at 5% strain to 0.9 at large strains [Chrysochoos and Belmahjoub, 1992]. The factor 2 difference of this parameter is small compared to variations in strain rate and viscosities and therefore in this work we have assumed c = Nondimensionalization [6] We will consider both cases in which the far-field stress s 0 = (0.5s ij s ij ) 0.5 and cases in which the far-field strain rate _e BG = (0.5_e ij _e ij ) 0.5 is maintained at a constant level. Nondimensionalization is performed by choosing s 0, m 0, k and g as characteristic values. This gives the following characteristic values for stress, time, temperature, and length, respectively: s* ¼ s 0 ; t* ¼ m 0 G ; T* ¼ 1 g ; L* ¼ rffiffiffiffiffiffiffi km 0 G The rheological equation (equation (4)) in nondimensional form (with tilde denoting nondimensional variables) can be written as _e ij ¼ s T 0e ~ di ~t ij þ s 0G 1 D~t ij þ s 0 l~tij _ 2m 0 m 0 2G D~t 2m 0 which may be simplified to 2m 0 _e ij s 0 ¼ e ~ T di ~t ij þ D~t ij þ l~t D~t _ ij Since the background strain rate is maintained at a constant value, the terms on the left-hand side can be collected into a nondimensional parameter B vis =2m 0 _e BG /s 0 which indicates the magnitude of the steady state viscous stress versus the initial stress. The energy equation in nondimensional form is given ~T ~T di þ ~v i 2 j þ Br~t ij ~_e vis ij þ ~ _e pl ij ð7þ ð8þ ð9þ ð10þ where Br = s 0 2 g/rc p G. If, additionally, a heterogeneity length scale R is introduced, four nondimensional numbers are present: B pl ¼ s 0 ; Br ¼ s2 0 g s y rc p G ; Pe ¼ r R ffiffiffiffiffiffiffi km 0 G ; B vis ¼ 2m 0 _e BG s 0 ð11þ Physically, B pl denotes the importance of the initial stress versus the yield stress, Br denotes the efficiency of shear heating, Pe denotes the ratio of an heterogeneity length scale over the diffusion length scale, and B vis denotes the ratio of steady state viscous stress over initial stress. [7] Upper bounds for the nondimensional parameters given above can be estimated by assuming that G Pa, k =10 6 m 2 s 1, g = , rc p = [see, e.g., Turcotte and Schubert, 1982]. Viscosity is assumed to vary from m 0 = Pa s. Moreover, the theoretical yield strength of rocks (given by the point at which bonds between atoms start to break) is around 1 10 of the shear modulus, giving a value of s y Pa [e.g., Scholz, 2002]. Typical yield strengths of rocks vary between 500 and 1000 MPa [Regenauer-Lieb and Yuen, 2003]. Physical meaningful results can only be obtained if the initial stress, s 0, is less than or equal to the yield stress. The stress state of the lithosphere may be estimated from stress released from earthquakes (typically around 10 MPa [Lachenbruch and Sass, 1991]) and from rheologyindependent force balance considerations about the average lithospheric stress required to support mountain belts (around 100 MPa [Jeffrey, 1959]). Maximum stress values occur in a preloaded lithosphere and will have values close to the yield stress (1000 MPa). From these considerations, the initial stress is assumed to vary from s 0 = MPa. Strain rate estimates for geodynamic processes give _e 0 = s 1. An upper bound for the size of the heterogeneity is obviously the thickness of the lithosphere O(100) km, so R = m. Using these values, the following parameter range can be estimated to be realistic for the nondimensional numbers given in equation (11): 10 4 B pl 0: B r Pe B vis ð12þ The characteristic length scale L * varies between 30 m and 10 5 km. 3. Numerical Methods [8] The initiation of shear localization is studied with 0-D, 1-D, and 2-D models. In this section the numerical techniques employed for the different models are described, whereas results obtained with these methods are given in section 4. All values are in nondimensional units; we have dropped ~ for readability. Transformation of the nondimensional into dimensional units can be done by multiplying them with the characteristic values (equation (7)) The 0-D Model [9] In order to study the process of 1-D and 2-D shear localization, one should first study the effects purely due to the employed rheology. A 0-D model, which assumes that the process is entirely adiabatic (i.e., no heat leaves the system), serves this purpose by ignoring any spatial derivatives. In the case of a constant stress boundary condition (dt/dt = 0 and l _ = 0), the model is describe by a single ordinary differential equation di 2 et ¼ f s Brt 2 ð13þ where f s is a parameter that describes the type of the applied shear; f s = 1 for simple shear and f s = 2 for pure shear. Moreover, t = 1 due to the chosen nondimensionalization. 3of18

4 An analytical solution of equation (13) (assuming that T di (0) = 0) is 0 T di ðþ¼ln 1 1 A ð14þ f s Br t 2 f sbr The solution of equation (14) always tends to blow up after a blowup time t 2/f s Br (the point where the solution becomes imaginary). A constant stress boundary condition will thus lead to thermal runaway, a wellestablished fact [e.g., Turcotte and Schubert, 1982; Kameyama et al., 1999]. [10] In the case of a constant strain rate boundary condition, two coupled ODEs di ¼ f s Brt _e BG _e dt dt ¼ 2_e BG e T di t 2 lt _ ð15þ here _e BG = B vis /2 and _e el = 1 2 (dt/dt). If dt/dt is discretized as (t t old )/dt, an expression for l _ is given by 8 < 0; t s y _l ¼ : 1 etdi 2dt 2 þ _ebg s y þ 1 2dts y t old ; t > s y ð16þ where s y = B 1 pl. A general analytical solution for equations (15) does not exist. Therefore we have chosen to solve the system of equations numerically (see Kaus [2005] for MATLAB source codes) The 1-D Model [11] The 1-D model (Figure 1a) assumes that a slab of thickness L is subjected to simple shear, with either a constant shear stress or a constant velocity (equivalent to a constant background strain rate). Localization is initiated by increasing the initial temperature to T 0 i at z =[( R/2)(R/2)]. The commonly made assumption of small strains and small rotations [Cherukuri and Shawki, 1995] is employed, which ignores rotational parts of the Jaumann derivative (equation (4)). In this case, equations (2) can be simplified ¼ 0 and equation (10) can be simplified to ð17þ Figure 1. (a) Setup of the one-dimensional model. (b) Setup of the two-dimensional model. Under a constant stress boundary condition, elastic strain rates are zero, since dt xz /dt = 0. In this case, the velocity at the boundary U will change (typically increase) with time. Time discretization of equation (20) gives L t xz t old xz dt which can be rewritten as ¼ U t xz Z L=2 L=2 e T di þ l _ dz T 2 þ Brt xz _e xz _e el xz ð18þ t xz ¼ Udt L þ hdt þ L L þ hdt told xz ð22þ Note that equation (17) implies constant t xz over the model domain. Rheology is given by dt xz dt ¼ 2_e xz e T di t xz 2 _ lt xz ð19þ Shear stress can be obtained by integrating equation (19) with respect to z: L dt xz dt ¼ U t xz Z L=2 L=2 e T di þ l _ dz ð20þ with h ¼ Z L=2 L=2 e T di þ l _ dz ð23þ In the case of constant strain rate boundary conditions, the problem is determined by equations (18), (22), and (23). In the case of constant stress boundary conditions, the lefthand side term in equation (19) disappears, and the problem is reduced to one with quasi-viscous rheology governed by equations (18) and (19) (with dt xz /dt = 0). 4of18

5 [12] Boundary conditions for temperature are adiabatic both at the upper and the lower boundaries T di = 0). The initial condition of temperature is uniform (T di = 0), except in the center of the domain between R z R, where the temperature is raised to T 0 i. The stress at t =0is given by t(t =0)=s 0 /s 0 = 1. The governing equation for temperature is discretized with a standard second order implicit finite difference method on a nonuniform grid with a numerical resolution of at least 2001 grid points. The governing equation for momentum is advanced in time by numerical integration of the temperature field [Fleitout and Froidevaux, 1980; Kameyama et al., 1999]. The time step is variable and similar to the one employed in the ODE solver [Kaus, 2005]. Resolution tests have been performed to ensure that the temporal and spatial resolution are sufficient The 2-D Model [13] In order to test whether the one-dimensional simulations are applicable in two-dimensional settings, we have performed over 500 two-dimensional simulations. For this we have used two numerical codes, recently developed in the frame of the thesis of Kaus [2005]. The first code, GANGO, is a finite difference/spectral method which solves the governing equations for viscoelastoplasticity in an Eulerian/Lagrangian framework, using a staggered finite difference discretization in the vertical direction and a spectral approach in the horizontal direction. Lateral variations in viscosity are treated iteratively. Stress advection (and rotation), which is required for the treatment of viscoelasticity, is done by using a semi-lagrangian advection scheme. The energy equation is solved on a grid that has a resolution two times higher then the mechanical part of the code. Time movement is done by an implicit approach, which additionally checks that the temperature increase per time step is not larger then a specified value. The code has been extensively benchmarked versus a range of different setups (thermal advection, diffusion, shear heating, stress around a cylindrical inclusion, viscoelastic buckling, stress rotation). Further details of the numerical code are given by Kaus et al. [2004], and a summary of benchmark results can be obtained from B. J. P. Kaus Web page. Relevant for the current work are comparisons of stress distribution around a 2-D elliptical viscous inclusion in a viscous matrix, for which 2-D analytical solutions were described by Schmid and Podladchikov [2003]. Moreover, comparisons have been made with the 0-D ODE solutions [Kaus, 2005]. [14] Most of the 2-D simulations in this work have been performed with GANGO. However, selected simulations have been compared with results obtained with a recently developed finite element code (SloMo). Excellent agreement exists between the two methods. [15] Two different 2-D model setups are employed. The first setup consists of a circular inclusion of radius R/2 0 where the initial temperature is increased to a value T i (see Figure 1b). Two different boundary conditions are employed: (1) A pure shear background strain rate condition, during which the model is extended at a constant strain rate _e bg. (2) A constant stress boundary condition, which is numerically treated by iteratively changing _e bg until the second invariant of the stress tensor at the lateral boundaries is within 0.1% of the required boundary stress. The thermal boundary conditions are periodic in the lateral directions and zero flux on the lower and upper boundary. Mechanical boundary conditions are periodic in the horizontal direction. The size of the domain is times larger then the radius of the inclusion, to ensure that the boundaries do not influence the results. Our 2-D simulations have been performed under extension, to prevent possible effects due to the occurrence of a buckling instability [e.g., Schmalholz and Podladchikov, 1999]. The initial heterogeneity is chosen to be circular, since this represents the worst-case scenario for shear localization: differently shaped inclusions generate larger stress perturbations [e.g., Schmid and Podladchikov, 2003], and hence simplify initiation of localization. The second 2-D setup is almost identical to the first setup, with the difference that a layer of air is added on top, and a layer of asthenosphere at the bottom of the model. Again a circular heterogeneity is introduced in order to initiate localization. The lower boundary condition is free slip and the viscosity of the asthenosphere is Pa s. 4. Initiation Stage 4.1. The 0-D Model [16] Stress and temperature evolutions computed for the 0-D (adiabatic) case with a dry olivine rheology are presented in Figure 2 for various initial lithosphere temperatures. Results are compared with the parameterized rheology employed in this work (parameters m 0, g and s Y of the simplified rheology have been computed from the dry olivine rheology using rules spelled out by Kaus [2005]). High initial temperatures result in relatively small stresses and thus relatively small amounts of shear heating (Figure 2a). Lower initial temperatures result in an increase of stress up to the yield stress after which the stress remains constant or slightly decreases due to the decrease of the temperature-dependent effective viscosity (Figure 2b). Temperature increase in the adiabatic case is maximum if elastic strain rates are zero (see equation (15a)). Elastic strain rates are proportional to the change in stress versus time (dt/dt, e.g., equation (4)). This thus explains why during the viscoelastic stress buildup stage, relatively little shear heating occurs (most of the deformation goes into reversible elastic stress increase, dt/dt _e BG ). Once the yield stress has been reached, however, dt/dt = 0 and all further deformation is done by dissipative plastic and viscous mechanisms. A sudden increase in the amount of shear heating thus occurs, once the yield stress is reached. This is reflected by a kink in the temperature-time evolution (Figure 2b). [17] A comparison of the 0-D model results for the parameterized rheology (having three parameters) with the dry olivine rheology (having 12 parameters) shows that the results are in fairly good agreement concerning the overall temperature and stress evolutions (for additional comparisons, see Kaus [2005]). The remaining part of this paper therefore concentrates on the parameterized rheology. [18] In order to quantitatively compare various 0-D models, we define a growth rate q 0-D given by q 0-D ¼ DT Dt ð24þ where Dt is the time interval required to raise the temperature by a given unit DT. Throughout most of this 5of18

6 Figure 2. General behavior of the 0-D model. Models are computed for dry olivine using G = Pa, _e = s 1, rc p = Jkg 1 m 3, s 0 = 1 MPa and pure shear deformation. The following parameters apply for the different models: (a) m 0 = Pa s, g = K 1, s y = 104 MPa, B vis = 39, Br = and B pl =110 3 ; and (b) m 0 = Pa s, g = 0.05 K 1, s y = 470 MPa, B vis = , Br = and B pl = MATLAB scripts given by Kaus [2005] have been used to create this plot. work, we will concentrate on the initiation stage of shear localization, and typically assume DT 0.1 (which corresponds to a few degrees in dimensional units). All numerical simulations have been integrated until an increase DT in temperature was detected. [19] In the nondimensional 0-D model, T di (t = 0) = 0, and the relevant nondimensional parameters are B vis, Br and B pl. A contour plot of log 10 (q 0-D ) versus Br and B vis for two different values of B pl (Figure 3) demonstrates that five different domains occur, each with its own dependence of q 0-D on the nondimensional parameters (see Table 2 for empirically determined scaling relationships for q 0-D ). The boundaries between the different domains are relatively sharp, and can be computed from the scaling laws for q 0-D. The domains represent different modes of deformation and have significantly different stress-time evolutions (see insets in Figure 3). We can distinguish the following: [20] 1. In the viscous mode, significant stress buildup occurs, with little overall increase in temperature (Figure 3, inset). Stress changes from the initial value to the viscous stress (t = B vis ). After this, stress slightly drops due to the rise in temperature. The expression for growth rate in this domain can be estimated from equation (15) by realizing that the stress is predominantly viscous (t = B vis, _e el = 0). Thus equation (15a) can be simplified di /@t / BrB 2 vis, which is similar to the empirically determined growth rate (see Table 2). Note that if B vis < 1, the viscous steady state stress is smaller than the initial stress and stress relaxation occurs. [21] 2. For the elastic mode, shear heating is very efficient (large Br). The initial temperature rise is larger then the increase in stress, with the result that stress remains nearly constant during the initiation stages. The expression for growth rate is thus nearly identical to the one in the case of a constant stress boundary condition (equation (13)), and given by q / Br (in excellent agreement with the empirical result, see Table 2). [22] 3. For the viscoelastic mode, stress increases almost linearly with time, according to t = B vis t (for t 1). Stress buildup is nearly completely elastic. Heating, however, is caused by viscous dissipation only. By substituting the above expression for stress evolution, one can rewrite equation (15a) di /@t = 0.5f s Br (B vis t) 2 exp(t di ). Under simple shear and for exp(t di ) 1, temperature evolution is given by T di = 1 6 BrB vis 2 t 3. The corresponding linear growth rate for T di = DT can than be computed to be q 0-D =6 1/3 (DT) 2/3 Br 1/3 B 2/3 vis, in good agreement with numerical results (Table 2). [23] 4. For the plastic 1 mode, stress increases linearly from the initial stress to the yield stress (s y = B 1 pl ). Yielding occurs after only a small amount of bulk heating, and the predominant deformation mode is plastic. Once the material yields, elastic strain rates are zero (since _e el = dt/dt) and all further deformation occurs by dissipative plastic and viscous processes. The amount of shear heating thus sharply rises once yielding occurs (see also Figure 2b). The growth rate expression (q 0-D / BrB vis B 1 pl ) can be estimated from equation (15) by setting t = B 1 pl. [24] 5. For the plastic 2 mode, the overall behavior is similar to the plastic 1 mode, with the difference that yielding now occurs at the end of the numerical simulation, when the overall temperature is almost DT. [25] The growth rates in most domains (with exception of the viscoelastic and the plastic 2 domain) are relatively insensitive to the total temperature increase DT used to determine q 0-D (Table 2), meaning that equation (24) is appropriate. [26] In the case of a constant stress boundary condition, only a single domain exists with growth rate q 0-D / f cs Br. This expression is identical to the elastic case with a constant strain rate boundary condition (Table 2). In Appendix A it is demonstrated that the elastic mode can also be obtained from the classical analysis of shear localization. [27] The boundaries between different modes of deformation are well defined and relatively sharp. Most boundaries are continuous and their location can be computed analytically by requesting that the growth rates of two adjacent domains are equal at the phase boundary (see 6of18

7 Table 2. Empirically Determined Scaling Laws for Growth Rate q 0-D in the 0-D Setup With Constant Strain Rate Boundary Conditions for Different Values of DT a Domain Equation q 0-D = DT = 0.01 DT = 0.05 DT = 0.1 Viscous 2 f BrB vis Elastic fbr Viscoelastic fbr B vis Plastic 1 1 f BrB vis B pl Plastic 2 fb vis B pl a Growth rates with a constant stress boundary condition are identical to the expressions in the elastic domain. f timescale, the growth rate of temperature inside is larger than the growth rate outside the perturbation. In the adiabatic limit this will thus always result in a weak zone and thus in localization of shear. Thermal diffusion, if sufficiently efficient, will stop localization. This effect is analyzed both analytically and numerically, and scaling laws are derived for the onset of localization in the presence of diffusion. [30] In the 1-D setup, a small perturbation either in temperature or in viscosity is introduced (Figure 1a). If the initial matrix viscosity is m 0, its temperature will increase as T bg (t) =q 0-D tg/m 0. If additionally the temperature evolution of the perturbation with initial viscosity m i 0 is given by T(t) =q 0-D tg/m i 0 (note the difference in timescale), one can write the temperature difference as Tt ðþ T bg ðþ¼ t m 0 G m 0 1 q 0-Dt i m 0 ð25þ Figure 3. Numerically computed growth rate q 0-D as a function of B vis, Br, and B pl for a constant strain rate boundary condition and a 0-D simple shear setup. Computations are performed for DT = 0.1. Insets show stress evolution versus time at various locations. (a) B pl = (no plasticity). (b) B pl =10 2. Scaling relationships for q 0-D in the various domains are summarized in Table 2. Expressions for the boundaries between different deformation modes (straight white lines) are given in Table 3. Table 3). A discontinuous boundary occurs between the viscous and the elastic domain, which is accompanied with a sharp increase in growth rate. [28] All growth rates determined in this section are positive. Any deformation will thus result in a temperature increase. In order to understand whether or not this will result in shear localization, a 1-D model is required The 1-D Model [29] In this section we study the process of shear localization in a 1-D setup, which consists of a matrix and a layer of slightly increased initial temperature (Figure 1a). We start by ignoring the effects of thermal diffusion (adiabatic assumption) and demonstrate that due to a difference in where m 0 i is the initial viscosity of the shear zone (m 0 i < m 0 ) and T bg (t) is the temperature evolution of the matrix far away from the shear zone (which can essentially be described with a 0-D model; see equation (24)). [31] If an initial thermal perturbation of magnitude T 0 i is present, equation (25) can be used to define a 1-D growth rate q 1-D as Tt ðþ T bg ðþ¼ t Ti 0 þ e T i 0 1 q 1-Dt ð26þ If the initial thermal perturbation is small, Taylor expansion around T i 0 = 0 gives Tt ðþ T bg ðþ¼t t 0 ð1 þ q 1-DtÞ ð27þ Note that if q 1-D > 0, the heating rate inside the perturbation is larger than the background heating rate. This results in a Table 3. Expressions for the Phase Boundaries in Figure 3 a Boundary Expression DT = 0.01 DT = 0.05 DT = 0.1 Viscous-elastic Br = f Viscous-viscoelastic B vis = fbr Elastic-viscoelastic B vis = fbr Viscous-plastic 1 1 B vis = fb pl Plastic 1-plastic 2 2 Br = fb pl Plastic 2-viscoelastic 3 B vis = f BrB pl a The expressions of all boundaries (with exception of the viscous-elastic boundary) have been computed from the growth rate expressions of Table 2; using the requirement that at the boundary between two domains, their corresponding growth rates should be equal. i f 7of18

8 Figure 4. (a) Normalized growth rate versus Pe for a constant stress boundary condition with Br = 1000, DT = 0.1, and T i 0 = Diffusion inhibits localization if Pe < (b) Numerically computed growth rate q 1-D versus Pe and Br with DT = 0.1. Only positive growth rates are visualized; diffusion dominates in the white area. Expressions of the black lines are given in the text. low viscosity zone and shear localization. If, on the other hand, q 1-D < 0 the initial temperature perturbation diffuses away and the whole domain will heat with the same rate. In this case, shear localization is inhibited. [32] The growth rate q 1-D has been computed numerically by integrating the governing equations (18) and (19) until T(t)>DT + T i 0, with DT = 0.1. In order to obtain insight on the influence of the nondimensional parameters on the initiation of shear localization, more then 100,000 numerical simulations have been performed. Convergence tests showed that the spatial and time resolution employed was sufficient. [33] In addition to the nondimensional parameters B pl, B vis and Br, two new parameters occur in the 1-D setup, namely, Pe = R/L * and R/L. Pe describes the ratio between the size of the initial heterogeneity R and the diffusion length scale (equation (7)). L/R is the ratio between the size of the domain and the size of the heterogeneity. If L > R is sufficiently large, this ratio does not influence the dynamics of the initial stages. Otherwise, the boundaries will influence the dynamics at the center of the deforming domain and the growth rate is reduced. The critical L/R is dependent on L * and was found to be around 2 to 3 for the setup considered here. This is rather small compared to typical shear zones, and therefore all simulations presented here are performed for larger L/R Constant Stress Boundary Condition [34] The only nondimensional parameters that may influence the problem in the case of a constant stress boundary condition are Br and Pe. An empirically derived growth rate expression for sufficiently large Pe is given by q 1D ¼ 0:55Br ð28þ [35] Diffusion starts to reduce the growth rate if Pe < 0.12 and inhibits localization (q 1-D <0)ifPe < 0.06 (Figure 4a). The boundary between diffusion dominated deformation and localization is given by (Figure 4b) Pe ¼ cbr 0:5 ð29þ where c = 2.3 under simple shear and c = 1.7 under pure shear boundary conditions Constant Velocity Boundary Condition [36] The 1-D growth rate in simulations with a constant velocity boundary condition is computed for various Pe, B pl, B vis and Br numbers (Figure 5). The 0-D and the 1-D growth rates are similar (Table 4). Differences, however, are related to the effect of diffusion, which may inhibit localization if it is faster than the rate of heat production. [37] Diffusion is proportional to Pe 2. The equations for the phase boundaries that involve diffusion demonstrate that this boundary can be predicted under the assumption that the diffusion growth rate is given by q diff 2Pe 2 (the requirement q diff = q 1-D than predicts the phase boundary) Localization and Diffusion [38] In the previous sections, the effect of diffusion on 1-D localization was studied through numerical simulations. In this section, we employ 1-D analytical solutions of diffusion in the presence of a finite zone of constant heat production to obtain insight in the localization process. The empirically derived scaling laws, described in the previous two sections, are derived analytically. [39] An analytical solution for the temperature evolution with time at the center of a 1-D shear zone of width R, with initial temperature T 0 i and with a constant heat production Q = q 1-D T 0 i is given by [e.g., Carslaw and Jaeger, 1959; Cardwell et al., 1978] Tt ð n Þ T bg ðt n Þ T 0 i ¼ q 1-DPe 2 t n þ 1 1 þ 1 erf p 8 4 ffiffiffi t n þ q pffiffiffi 1-DPe 2 t p n 2 ffiffiffi exp 1 2 p! p 4 ffiffiffi q 1-DPe 2 t n 8 ð30þ where t n = t/pe 2 is the rescaled time and it is assumed that no heat is produced outside the center zone. Diffusion will be more efficient than heating if at a given time t n, the left-hand side of equation (30) becomes <1. The transition between diffusion-dominated and heating-dominated temperature evolution is thus 1 1 erf ffiffiffi p q 1-DPe 2 4 t ¼ n t n þ 1 8 erf p1 ffiffiffi þ ffiffiffi p t n ffiffi p exp 1ffiffiffi 2 ð31þ p 1 4 t n 2 p 4 t n 8 8of18

9 Figure 5. Growth rate versus Br and B vis for various B pl and Pe numbers in a 1-D constant velocity model (DT = 0.1, T 0 i = 0.01). Scaling laws for q 1-D are indicates in Table 4 for simulations with Pe >1. The white areas are areas with negative growth rates (diffusion-dominated). The bottom right plot is a case where diffusion also influences the quasi-elastic domain (see text for discussion). At small times, diffusion is not very efficient (Figure 6a). At later stages (larger t n ) diffusion becomes more efficient, and may eliminate the initial increase in temperature in the center of the shear zone (Figure 6a). Interestingly, an upper limit of q 1-D >3/Pe 2 exists above which diffusion is unable to inhibit localization. [40] The time t rise, required to increase the temperature of the perturbation from T i 0 to T i 0 + DTin the absence of diffusion, is dependent on the background growth rate of temperature as (see equation (27)) where T i 0 1. t rise ¼ DT q 0-D þ Ti 0q DT DT 1-D q 0-D q 1-D ð32þ Table 4. Scaling Laws for Growth Rate q 1-D in the 1-D Setup With Constant Strain Rate Boundary Conditions for Different Values of DT a Domain Equation q 1-D = DT = 0.01 DT = 0.05 DT = 0.1 Viscous 2 f BrB vis Elastic fbr Viscoelastic fbr 1 2 3B vis Plastic 1 1 f BrB vis B pl Plastic 2 fb vis B pl a Growth rates with a constant stress boundary condition are identical to the expressions in the elastic domain. Expressions have been derived for Pe 1. f 9of18

10 Figure 6. (a) Growth rate versus t n (equation (31)). Grey area is diffusion dominated. (b) Growth rate versus DT. The black line is computed analytically from equations (32) and (31). The 1-D numerical simulations have been performed to verify the analytical solution and are indicated by crosses (T(t rise )>T 0 i ) and circles (T(t rise )<T 0 i ). Maximum occurs at DT 0.34, q 1-D Pe 2 3. [41] Substituting equation (32) into equation (31) results in an implicit equation for q 1-D. The critical growth rate required to overcome the effect of diffusion is given by q 1-D ¼ f ðdtþ Pe 2 ð33þ where f(dt) is a constant that is solely dependent on DT (which can be obtained numerically from equation (31); see Figure 6b). [42] Equation (33) can be used to predict the boundary between the viscous deformation mode and the diffusion mode as shown in Figure 5. For DT = 0.1, f(0.1) The condition for inhibiting localization is thus given by BrB vis = Pe 2 or B vis =2Pe 1 Br 0.5, in excellent agreement with empirical results (Table 5 and Figure 5). Similarly, one can predict that diffusion inhibits localization if Pe < 0.06, in the case of a constant stress boundary condition and with the parameters of Figure The 2-D Model [43] The results presented in the last section were valid for the initiation stage in a 1-D simple shear setup, or in other words for deformation along an existing weak zone. The purpose of this section is to understand how such weak zones can be created in 2-D pure shear extensional setting (see Figure 1 for the setup). Shear localization is initiated by increasing the initial temperature to T i 0, or decreasing the viscosity to m i 0 in a circular heterogeneity with radius R. Two cases will be considered: (1) the background strain rate is maintained constant and (2) the far-field stress is maintained at a constant level. [44] Three examples of shear localization under constant strain rate boundary conditions are shown in Figure 7. In these examples, plasticity is not activated. As in zero dimension and one dimension, the following deformation modes can be distinguished: [45] 1. The viscous simulation reaches the viscous steady state stress (s 2nd = B vis ) soon after the onset of extension. A flower-like shear heating distribution in and around the inclusion is obtained at this stage, in agreement with analytical results [Schmid and Podladchikov, 2003; Schmid, 2002]. Shear heating is maximum in the lobes around the inclusion that are oriented at 45 of the principle axis. [46] 2. The viscoelastic simulation shows a distinctly different behavior. In this case, shear heating mainly occurs in the center of the heterogeneity. This can be understood by realizing that the Maxwell relaxation time (= m/g) inside the inclusion is smaller than in the matrix. Thus the material in the center behaves more viscous than the material in the matrix, and hence it is heating at a larger rate. [47] 3. The elastic simulation behaves like the viscoelastic simulation with the difference that stress, and therefore shear heating, stays at an almost constant level (in agreement with 0-D and 1-D models). [48] The viscoelastic mode ultimately switches into a viscous mode (heating around instead of inside the inclusion) but only when the viscous steady state stress has been reached (after 2 3 Maxwell times). As will be shown later, this does not occur for realistic parameters, since the differential stresses are too large. Once this switch occurs, however, linear low viscosity shear zones are initiated from the inclusion. [49] If plasticity is considered in the simulations, the pattern of shear heating changes. Three different plastic deformation modes have been observed (Figure 8): [50] 1. The plastic mode corresponds to the plastic 1 and plastic 2 modes in the 1-D setup. During the preyielding stages of the simulation, deformation is viscoelastic (with maximum amounts of heating inside the inclusion). At yielding, brittle shear zones initiate at an angle of 45 from the inclusion and cut the model domain (Figure 8). Continuing deformation may result in weak shear zones at those locations, akin to the 1-D case. Table 5. Expression for the Phase Boundaries in Figure 5 Boundary Expression f DT = 0.1 Viscous-elastic Br = f 0.43 Viscous-viscoelastic B vis = fbr Elastic-viscoelastic B vis = fbr 8.7 Viscous-plastic 1 1 B vis = fb pl 1.0 Plastic 1-plastic 2 2 Br = fb pl 0.20 Plastic 2-viscoelastic 3 B vis = f BrB pl 1.77 Diffusion-viscous B vis = fpe 1 Br Diffusion-plastic 1 B vis = fb pl Pe 2 Br of 18

11 Figure 7. Examples of the initiation stage in a 2-D pure shear setup (Figure 1) for three different, nonplastic, deformation modes. Colors indicate the amount of shear heating. (a) Evolution at the time when T max T 0 i 0.033, (b) T max T 0 i 0.067, (c) T max T 0 i 0.1. (d) Time-dependent parameters (second invariant of the stress tensor and temperature). B pl =10 20, Pe =10 3 in all simulations. Other parameters are viscous, Br =10 5, B vis = 10; viscoelastic, Br =10 3, B vis =10 3 ; and elastic, Br =10 1, B vis = of 18

12 Figure 8. (a-c) Snapshots of shear heating at three different times for the different plastic modes that occur in the 2-D model. Pe = 103 in all simulations. Other parameters are static-plastic, Br = 10 4, Bvis = 10, Bpl = :6; elastoplastic, Br = 10, Bvis = 1, Bpl = 1:0001; plastic, Br = 10, Bvis = 10, Bpl = 50. The static-plastic mode yields only in part of the domain. With ongoing deformation, viscosity decreases and the stress drops below the yield stress (switches to the viscous mode). The elastoplastic mode also yields in part of the domain only but otherwise behaves elastically. The plastic mode results in whole domain yielding. 12 of 18

13 Figure 9. Two-dimensional simulations of shear localization around circular inclusions. T i 0 = 0.01, DT = 0.1. Black lines are phase boundaries derived with 1-D simulations (Figure 5). [51] 2. The static-plastic mode occurs when the yield stress is slightly larger than viscous steady state stress of the matrix. The circular inclusion causes a stress amplification compared to the far-field stress [see, e.g., Schmid, 2002]. Yielding thus occurs in the vicinity of the inclusion. During ongoing deformation, however, shear heating reduces differential stress below the yield stress. In the static-plastic domain, yielding thus occurs for a finite amount of time only. [52] 3. The elastoplastic mode occurs when the yield stress is very close to the initial stress. Shear heating inside the inclusion is maximum. Because of stress amplification, four yielding zones exist in the vicinity of the inclusion, propagating outward. Ongoing deformation results in thermal runaway. [53] Simulations performed with a constant stress boundary condition are similar to the elastic mode under constant strain rate conditions. [54] In order to understand whether the different deformation modes described above form for the same range of parameters as the corresponding 0-D and 1-D modes, we have performed systematic 2-D simulations. The results (Figure 9) indicate that this is the case. Diffusion in two dimensions is slightly more efficient than in one dimension, but differences are small and the diffusion boundaries derived in the 1-D case can be applied to 2-D cases. [55] In two dimensions, shear heating is maximum in either the center of the inclusion (e.g., Figure 7a), or in one of the lobes around the inclusion (e.g., Figures 7b and 7c). To further compare the 2-D with the 1-D results, growth rates have been determined both inside and outside the inclusion (Table 6). The viscous cases produce more heat outside, and the viscoelastic and elastic cases more heat inside the inclusion (in agreement with Figure 7). [56] For small viscosity contrasts, the dependence of the growth rate on inclusion temperature or viscosity is similar to the 1-D case (equation (25)). For viscosity contrasts larger than , the growth rate in the viscous domain saturates, whereas the viscoelastic mode undergoes a switch from heating inside to heating outside the inclusion. This can be attributed to the fact that the amount of stress amplification around an inclusion saturates after a certain viscosity contrast [Schmid, 2002]. [57] The 2-D simulations thus point out the importance of plasticity in creating linear zones of enhanced shear heating. Plasticity has two effects: First, it limits the stress level to reasonable values. Secondly, the onset of plastic yielding is accompanied by the creation of linear zones in which shear heating is slightly larger than the background level. Although the initial temperature rise in these zones is generally small (<1K), it does help localization at later stages. 13 of 18

14 Table 6. Scaling Laws for Growth Rate in the 2-D Setup With Constant Strain Rate Boundary Conditions a Domain Growth Rate in Center Growth Rate Outside Viscous q 2-D 0 out q 2-D Elastic q 2-D = 1.1Br out q 2-D = 0.25B 2 vis Br 0 Viscoelastic q cen 2-D = 0.15Br 1 3 B 2 out vis3 q 2-D 0 Plastic q cen 1 out 2-D = 0.05BrB vis B pl q 2-D 0 a Growth rates have been determined both in the center of the inclusion and outside the inclusion (stars in Figure 7a) for Pe 1 and DT = 0.1. This localization process is essentially 1-D, and the scaling laws of section 4.2 are applicable. [62] Two-dimensional simulations, have been performed to verify the results of the linear stability analysis (Figure 11) in the more complicated setup of a lithosphere under extension. For a given strain rate of s 1, linear stability analysis requires a minimum viscosity of O(10 23 ) Pa s to initiate localization, in agreement with numerical simulations (e.g., Figures 11a and 11b). [63] A larger initial viscosity of the lithosphere results in a faster and more vigorous localization. This can be attributed to the increased Maxwell relaxation time of the lithosphere and thus an increased importance of elasticity. Simulations, performed for viscosities much larger then those shown in 5. Possible Implications for Localization on Planets [58] We have demonstrated that shear localization due to shear heating may occur by different deformation modes. Sofar we have concentrated on the effect of various nondimensional parameters on the localization process. In this section, diagrams are presented in dimensional units. The results demonstrate that shear localization is possible for Earth-like parameters (Figure 10). However, the onset of localization is critically dependent on the background strain rate, the effective viscosity and the size of the initial heterogeneity. Small background strain rates and/or small effective viscosities require large preexisting heterogeneities. [59] In order to verify whether the scaling laws, derived in this work for initial stages, also describe the behavior during late stages, we have performed additional 1-D numerical simulations integrated until DT 10 (corresponding to K). The results indicate that the diffusion-viscous and the viscous-viscoelastic boundaries are indeed correctly predicted from the initial stage analysis. The plastic-diffusion boundary, however, does not seem to occur during later stages. An analysis of the numerical simulations showed that significant stress drop occurs during late stages of simulations within the plastic-viscous mode. Thus this mode becomes very similar to the viscous mode, which explains why the transition to localization is described by the diffusion-viscous boundary. [60] The onset of localization is independent on the initial stress and the elastic shear modulus of the lithosphere but is a strong function of heterogeneity size (Figure 10): _e BG ¼ 1:4 R rffiffiffiffiffiffiffiffiffi krc p m 0 g ð34þ The initial stress and the elastic shear modulus, however, influence Br (efficiency of shear heating), which in turn influences the growth rate and is thus proportional to the amount of strain required to raise the temperature by a certain value. [61] A new effect occurs during late stages, namely, the transition from aseismic creep to seismic creep, the latter being characterized by a rapid increase in temperature and strain rate in the center of the shear zone. For the parameters of Figure 10, neither the viscoelastic nor the elastic modes are present. The elastic mode is absent since the initial stress of s 0 = 10 MPa yields Br , which is too small for the elastic mode. The viscoelastic mode is absent since B pl is sufficiently large (B pl 0.01). Figure 10. Deformation modes as a function of strain rate and effective viscosity for an inclusion size of (a) R =10km and (b) R = 1 km (with g = 0.1 K 1, s 0 = 10 MPa, rc p = Jm 2 Ks 1 kg 1, G =10 11 Pa). Boundaries between modes are derived from the 1-D initial stage scaling laws for a simple shear, constant velocity boundary condition (dotted lines are boundaries with a twice larger or smaller growth rate, and thus take, e.g., the effect of pure shear into account) and have been verified by numerical simulations. One-dimensional late stage numerical simulations (integrated until DT 10 and with R/L = 0.01, T 0 i = 0.01) are indicated by symbols. The condition for thermal runaway is max(_e) 10 8 _e BG. 14 of 18

15 Figure 11. Two-dimensional results for a lithosphere with an initially circular weak inclusion of 5 km radius that is extended with a pure shear constant strain rate of s 1. The initial viscosity is (a) m0 = , (b) m0 = , (c) m0 = (d) Model with the same parameters as model in Figure 11b but with plasticity deactivated. Other parameters are G = 1011, g = 0.1, rcp = 3 106, k = 10 6, s0 = 50 MPa, sy = 1000 MPa. The free surface is approximated by a layer of m = 1019 Pa s in the top 50 km of the model (with a no-stress boundary condition on top of the layer). A moderate resolution of is employed. Black lines represent a passive marker grid. Gravity is not present. Figure 11c, are numerically difficult due to the very high strain rates that develop together with a fast drop in stress and large increase in temperature. This phenomenon is physically most likely related to the development of earthquakes and was also observed by other authors [e.g., Regenauer-Lieb et al., 1999; Regenauer-Lieb and Yuen, 2000a, 2004]. [64] The role of plasticity in the localization process is twofold. First, it limits differential stresses, which may otherwise be unphysically large. Secondly, it generates linear zones of increased shear heating (Figure 8), which facilitates localization during later stages (compare Figures 11b and 11d). It is however theoretically possible to initiate localization without invoking plasticity, both by viscous and viscoelastic (Figure 11d) mechanisms. However, shear zones in the viscous regime require large heterogeneities, whereas viscoelastic deformation tends to develop unrealistically large differential stresses. That the formation of shear zones around weak inclusions is difficult in the viscous domain was also found by Mancktelow [2002] for a strain weakening rather then a thermal weakening rheology. [65] It is interesting to compare the linear stability results obtained here with previous numerical simulations. Regenauer-Lieb et al. [2001] presented a model in which subduction was initiated by adding 10 km of sediments on top of an oceanic plate. They obtained lithospheric-scale shear zones for wet olivine rheologies only. Whereas their model setup and rheology differs in detail from the one employed in this study, we can still obtain some insight in the process by employing the scaling relationships derived here. The initial growth rate of the shear localization instability under a constant stress, simple shear, boundary condition is q1-d = 0.55Br. Diffusion may inhibit shear localization only if q1-d < 3/Pe2. By using the definitions of the nondimensional parameters (equation (11)), it can be shown that diffusion will influence shear localization if m0 > 15 of 18