Are intermediate depth earthquakes caused by plastic faulting?

Earth and Planetary Science Letters 382 (2013) 32–37

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Are intermediate depth earthquakes caused by plastic faulting? Carl E. Renshaw a,∗ , Erland M. Schulson b a b

Department of Earth Sciences, Dartmouth College, Hanover, NH, USA Thayer School of Engineering, Dartmouth College, Hanover, NH, USA

a r t i c l e

i n f o

Article history: Received 24 January 2013 Received in revised form 5 September 2013 Accepted 6 September 2013 Available online 25 September 2013 Editor: P. Shearer Keywords: earthquakes semi-brittle faulting adiabatic instabilities

a b s t r a c t The increase in confining pressure and temperature with depth precludes frictional sliding beyond a few tens of kilometers, yet earthquakes occur to depths greater than 600 km. Although rock failure is often conceptualized as either brittle under low pressure and temperature or ductile at higher pressure and temperature, an intermediate brittle-like failure mode, sometimes called “transitional” or “semi-brittle faulting”, is well known experimentally. In contrast to brittle and ductile failure, models for semi-brittle terminal failure strength are lacking. We show that semi-brittle failure is consistent with the dictates of adiabatic instabilities that result in failure via plastic faulting. Although adiabatic instabilities have been suggested as a possible cause of deep (> 500 km) earthquakes, extrapolation of laboratory semi-brittle faulting data using the theory of plastic faulting with a rheology and geotherm typical of subducting oceanic crust indicates that adiabatic instabilities are more likely at intermediate depths to about 300 km. © 2013 Elsevier B.V. All rights reserved.

1. Introduction Frictional sliding is well known to limit brittle failure (Byerlee, 1978; Renshaw and Schulson, 2001; Schulson et al., 1999). Yet even under confining pressure sufficient to suppress frictional sliding, brittle-like failure is still observed, both in the laboratory (e.g. Caristan, 1982; Carter and Tsenn, 1987; Dobson et al., 2004; Hirth and Tullis, 1994; Kirby, 1987; Meade and Jeanloz, 1991; Riggs and Green, 2005; Shelton et al., 1981; Tullis and Yund, 1992; Violay et al., 2012), and in the field (e.g. Frohlich, 1989; Hobbs et al., 1986; Lund et al., 2004; Persh and Houston, 2004; White, 1996). Failure is “brittle-like” in the sense that failure is accompanied by both the localization of strain along a macroscopic faulting plane and a sudden drop in the load-bearing ability of the material. The geophysical implications of high-confinement brittle-like failure are not limited to deep earthquakes. Localized shear failure under high-confinement may also provide limits on the steady state flow stress of highly confined rock and ice mixtures, constraining the thermomechanical evolution of certain moons and planets (Durham et al., 1992; Mangold et al., 2002), and limit the stability of engineered structures in ice-infested waters where loads induced by interactions with ice are governed by both brittle and semi-brittle compressive failure processes (Kim et al., 2012). Three mechanisms have been proposed as potentially responsible for high-confinement brittle-like failure: (i) dehydration embrittlement (Meade and Jeanloz, 1991); (ii) faulting induced by

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a phase transformation between one mineral form and another, denser form (Green and Burnley, 1989; Kirby, 1987); and (iii) adiabatic shear instability (Orowan, 1960), also referred to as plastic faulting (Renshaw and Schulson, 2004; Schulson, 2002). Both dehydration embrittlement and transformational faulting have extensive experimental support in crystalline materials (Burnley and Green, 1989; Dobson et al., 2004; Green and Houston, 1995; Jung and Green, 2004; Jung et al., 2004; Kirby, 1987; Kirby et al., 1996; Liu et al., 2004; Meade and Jeanloz, 1991; Mosenfelder et al., 2001; Riggs and Green, 2005). In contrast, while adiabatic shear instabilities are well known in metals (Basinski, 1957), polymers (Winter, 1975) and metallic glasses (Perez-Prado et al., 2001), they remain little explored, at least experimentally, in more nominally brittle crystalline materials (Frohlich, 1989), limiting our ability to assess the viability of plastic faulting in geologic materials in nature. Although rock failure is often conceptualized as either brittle under low pressure and temperature or ductile at higher pressure and temperature, an intermediate brittle-like failure mode, sometimes called “transitional” or “semi-brittle faulting”, is well known experimentally. Here we show that under confinements sufficient to suppress frictional sliding, laboratory semi-brittle failure data are consistent with the dictates of adiabatic instabilities that result in shear failure via plastic faulting. Using the theory of plastic faulting as a guide, we extrapolate the laboratory semi-brittle high-confinement faulting data to the field using a rheology and geotherm typical of subducting oceanic crust, and we demonstrate that plastic faulting is possible in subducting crust to depths of several hundred km.

C.E. Renshaw, E.M. Schulson / Earth and Planetary Science Letters 382 (2013) 32–37

2. Theory 2.1. P-faulting failure stress In plastic faulting, or P-faulting, deformation is localized due to the instability that develops when thermal softening exceeds strain and strain rate hardening (Renshaw and Schulson, 2004). By plastic deformation we mean von Mises or volume-conserving plasticity. The requirements for P-faulting are twofold; a critical inelastic strain is required to generate sufficient heat, and a critical strain rate is required to ensure approximately adiabatic conditions. These requirements can be quantified and combined through a balance of the effects of strain and strain rate hardening and the effect of thermal softening (Frost and Ashby, 1982), yielding the critical strain rate ε˙¯ c required for P-faulting

ε˙¯ c =

√ −2 2 aκ mw f  w d2 (∂ σ¯ /∂ T )

(1)

where a is a geometrical factor of order unity, κ is the thermal conductivity, w f is the width of the fault, w d is the characteristic length of heat diffusion, and ∂ σ¯ /∂ T quantifies the thermal softening of the material. Consistent with our observations in ice (Golding et al., 2010), microstructural analyses of a P-fault (our words) in quartzite by Hirth and Tullis (1994) revealed that the majority of the shortening that occurred in the sample was accommodated by displacement across the fault. If all the displacement at the sample boundaries is accommodated by displacement across a fault oriented at 45◦ to the direction of shortening, the local strain rate within the fault is related to the strain √ rate measured at the edges of the sample by a factor of / 2 w f , where  is the characteristic length of the sample/body. The ratio / w f thus reflects that concentration of the strain rate measured at the global or macroscopic scale to the rate of strain across the width of the fault. In the laboratory where a single fault generally develops within sub-meter sized test specimens,  is taken as the length of the specimen along the direction of shortening. In the field  may correspond to the spacing between faults. The parameter m is the work-hardening exponent in the expression σ¯ ∝ ε¯ m , where σ¯ and ε¯ are the effective stress and effective strain, respectively (Golding et al., 2010). The work-hardening exponent reflects the degree to which plastic deformation increases the stress required to maintain plastic deformation at a constant strain rate; i.e. m is a measure of the work/strain hardening. A value of m = 1.0 represents a linear work-hardening elastic material while m = 0 represents a perfectly plastic solid that exhibits zero strain hardening. Typical values of the work-hardening exponent in metals are between 0.1–0.5 (Callister, 2005). Values for geologic materials are less well established. Golding et al. (2012) determined m = 0.4 for granular and columnar S2 ice over the temperature range −40 to −10 ◦ C, while Boidy and Pellet (Boidy and Pellet, 2002) estimate that m = 0.2 for shale. Using these values as a guide, we assume here that m = 0.3. Our results are not highly sensitive to this assumption; halving or doubling this assumed value changes the predicted failure stress only by a few percent or less. The effective stress, defined using Von Mises’ criterion for a plastically isotropic material, that is required for P-faulting can be estimated by combining the critical strain rate ε˙¯ c with a rate dependent plastic flow law. At relatively low stresses the relationship between effective stress σ¯ and effective strain rate ε˙¯ in rock is typically described using an Arrhenius power-law equation of the form ε˙¯ = A p σ¯ n e − Q /(RT ) , where A p is a materials property, Q is an apparent activation energy, R is the gas constant, and n is the stress exponent. At higher stresses the creep-ratelimiting process is thought to transition from climb-controlled to

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glide-controlled dislocation motion (Frost and Ashby, 1982). The power-law relationship between effective stress and effective strain rate may break down in this high-stress creep regime and deformation may instead be characterized by an exponential flow law of the form ε˙¯ = A e e β σ¯ e − Q /(RT ) , where A e and β are materials properties (Evans and Goetze, 1979; Tsenn and Carter, 1987). The transition from a low-stress power law to a high-stress exponential flow law can by described by a single hyperbolic creep law of the form ε˙¯ = A h [sinh(α σ¯ )]n e − Q /(RT ) , where A h is a materials property and the parameter α is the reciprocal of the effective stress at which low and high stress creep contribute equally to the global creep rate (Garofalo, 1963; Wong and Jonas, 1968). For values of α σ¯ less than about 0.8, this expression reduces to the Arrhenius power-law equation with A h = A p /αn . For α σ¯ greater than about 1.2 this expression reduces to an exponential flow law (Barnes et al., 1971). The corresponding P-faulting failure stress when ductile deformation is governed by the hyperbolic creep law is

σ¯ =

1

α

sinh−1



√   −2 2 aκ mw f 1/n Q /nRT e A h  w d2 (∂ σ¯ /∂ T )

(2)

where the thermal softening term is estimated by differentiating the hyperbolic creep law, yielding:

∂ σ¯ =− ∂T

Q (ε˙¯ / A h )1/n e ( Q /(nRT ))



αnRT 2 (ε˙¯ / A h )2/n e(2Q /(nRT )) + 1

(3)

2.2. Application to Earth’s crust Variants of the adiabatic instability mechanism have been proposed to operate in nature in geologic materials (Griggs and Handin, 1960; Hobbs and Ord, 1988; Ogawa, 1987). Until recently, these proposals have been largely theoretical, leading some investigators to question whether P-faulting (our words) is possible under conditions typically present in the Earth (Green and Marone, 2002). The concern, derived from existing adiabatic instability models that assume a pre-existing local perturbation of material properties or temperature, is that if the system is not pre-organized into an appropriately-oriented shear zone, thermal softening from shear heating will lead to stress relaxation rather than runaway heating, and any thermal anomaly will dissipate rather than leading to localized failure. The P-faulting model differs from previous adiabatic instability models in two ways: (1) it does not posit pre-existing local perturbations of either material properties or temperature; and (2) model predictions are validated, for the first time, against laboratory observations of adiabatic instabilities in a geologic material. Also, unlike most previous models, the P-faulting model provides a closedform analytical prediction of failure stress based on independently measurable parameters. Braeck et al. (2009) also developed a closed-form analytical prediction for failure stress due to adiabatic instability, but the increase in failure stress with increasing temperature predicted by this model is inconsistent with predictions of the P-faulting model and, more importantly, with laboratory observations of decreasing failure stress with increasing temperature (Golding et al., 2012). While the P-faulting model does not predict the temperature field at failure, such as is possible using numerical solutions (Braeck et al., 2009; Kameyama et al., 1999; Kelemen and Hirth, 2007), the experimentally-measured heating within P-faults is broadly consistent with predictions of the Pfaulting model (Golding et al., 2010). However, in contrast to the periodic shear-heating model of Kelemen and Hirth (2007), the P-faulting model does not explicitly address the question of repeating adiabatic instabilities. As developed above, the P-faulting model would not apply to repeating instabilities should faulting change the values of the creep parameters of the material within

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C.E. Renshaw, E.M. Schulson / Earth and Planetary Science Letters 382 (2013) 32–37

Table 1 Deformation parameters. Granite

Quartzite

Anorthosite

Diabase

Gabbro

Creep parameters A p (MPa−n s−1 ) n Q (kJ mol−1 ) α (MPa−1 )

1.3 × 10−9 3 100 0.0068

4.0 × 10−10 3 99 0.0030

5.0 × 10−3 2.8 245 0.010

8 4.7 485 0.013

2.0 × 1010 4 644 0.01

Sample properties G (MPa) κ (W m−1 K−1 ) P (kg m−3 ) C p (J Kg−1 K−1 ) W f = W d (m)  (m)

23 800 0.33 2650 1000 1 × 10−6 0.016

38 700 0.33 2650 1000 5 × 10−6 0.005

62 500 0.36 2900 1000 1 × 10−5 0.0126

40 900 0.21 2800 1000 1 × 10−5 0.02

30 000 0.3 3000 1000 1 × 10−5 0.01

Creep law parameters from Renshaw and Schulson (2007) (granite and quartzite), Carter and Tsenn (1987) (anorthosite), Mackwell et al. (1998) (diabase), Zhou et al. (2012) (gabbro).

the fault zone. Such a change might be expected because of the extensive dynamic recrystallization observed along P-faults which, at least in some metals, appears to serve as a localizing “soft spot” (Osovski et al., 2013). However, at least in ice, where repeating adiabatic instabilities have been observed experimentally, pre-existing recrystallization does not appear to affect P-faulting failure stresses (Golding et al., 2012). If semi-brittle failure is controlled by P-faulting, then plastic faulting theory can be used to guide the extrapolation of experimental semi-brittle faulting observations to the field. To examine the plausibility of this approach, we determined, as a function of depth in the crust, the effective stress required for (i) frictional sliding using Byerlee’s law (Byerlee, 1978), (ii) P-faulting, and (iii) globally ductile flow using the Arrhenius power-law. We define the dominant failure mode as the one that requires the lowest stress to operate. The distribution of the dominant failure mode with depth depends on both the crustal rheology and geotherm. Here we consider failure in oceanic crust with a rheology assumed to be similar to that of gabbro (Table 1). We generally keep the rheology constant to allow for direct comparison between models, but briefly also consider slabs having the eclogite rheology given by Zhang and Green (2007). We consider two geotherms, one representative of a non-subducting 60 Myr slab and the other representative of a subducting slab. The geotherm of a non-subducting oceanic slab is determined as the superposition of the cooling of a semiinfinite half-space and an adiabatic temperature gradient (Turcotte and Schubert, 2002) of (dT /dz)adiabatic . The resulting temperature profile as a function of depth z is (Turcotte and Schubert, 2002)



T ( z) = T o + ( T m − T o ) erf

z

√

2 λt





+z

dT dz



(4) adiabatic

where T o = 273 K is the surface temperature, T m = 1500 K is the reference melting temperature of oceanic crust, λ = 1 mm2 s−1 is the thermal diffusivity of oceanic crust, and t = 60 Myr is the age of the slab. Following Turcotte and Schubert (2002), we assume (dT /dz)adiabatic = 0.3 K km−1 . In a subducting slab the temperature along the upper surface of the slab is determined from the shear heating along the upper boundary of the descending slab, giving (Turcotte and Schubert, 2002)

T ( z) = T o +

2τ

κ



uλz

π sin(θ)

1/2 (5)

where, again following Turcotte and Schubert (2002), τ = 180 MPa is the shear stress acting along the upper boundary of the descending slab, κ = 4 W m−1 K−1 is the thermal conductivity, u = 1 cm/yr is the slab velocity, and θ = 45◦ is the dip of the subducting slab.

3. Results 3.1. Comparison to laboratory observations Golding et al. (2012, 2010) (hereafter cited as Golding et al.) report results from systematic experiments on laboratory-grown polycrystalline freshwater ice loaded triaxially under confinement sufficient to suppress frictional sliding. Under lower strain rates deformation of the ice occurred homogeneously (i.e., not localized along a macroscopic fault plane), there was not a sudden drop of load-bearing ability, and the relationship between effective stress and applied strain rate closely followed an Arrhenius power-law with n ≈ 3. In contrast, at higher applied strain rates, but still under confinement sufficient to suppress frictional sliding, a distinct fault, oriented close to 45◦ from the direction of the greatest principal stress, developed at terminal failure and was accompanied by a nearly instantaneous drop in loadbearing ability. Of all the mechanisms commonly cited to explain high confinement brittle-like failure (Green and Burnley, 1989; Meade and Jeanloz, 1991), only P-faulting was consistent with the features observed in these high strain rate, high confinement experiments. Thus Golding et al. identified these faults as P-faults. No similar explicit identification of P-faulting exists in rock. However, based on characteristics observed in ice, we note that P-faulting is consistent with characteristics of the “high pressure” faults in dry Man-nari granite observed by Shimada and Cho (1990) and Shimada (1992) (hereafter collectively cited as Shimada et al.). In contrast to faults that form under low confinement, Pfaults in ice and the “high pressure” faults in granite: (i) only formed under confinements sufficient to suppress frictional sliding, (ii) were not attended by significant gouge nor concentration of microcracks near the main fault, (iii) were very narrow, and (iv) were oriented close to the plane of maximum shear stress. Furthermore the terminal failure stresses observed by Shimada et al. as a function of the temperature adjusted strain rate parameter (Fig. 1) are consistent with the failure stresses predicted using the P-faulting model, providing quantitative support for our hypothesis that these “high pressure” faults were P-faults. Microstructural analyses of other types of crystalline rock that failed via a semi-brittle mode when confinement was sufficient to suppress frictional sliding reveal features similar to the plastic faults in ice and in Shimada et al.’s “high pressure” faults in granite, including limited gouge or microcracking near the main fault and a narrow fault oriented, in most cases, close to the plane of maximum shear stress (Caristan, 1982; Hirth and Tullis, 1994; Shelton et al., 1981; Tullis and Yund, 1992). Also, as in P-faulted ice (Golding et al., 2010, 2012), semi-brittle faulting in rock often onsets after significant inelastic strain (Shelton et al., 1981). That semi-brittle faults may be plastic faults is further supported by the close fit between the experimentally observed failure stresses and

C.E. Renshaw, E.M. Schulson / Earth and Planetary Science Letters 382 (2013) 32–37

Fig. 1. Open symbols indicate the terminal failure stresses of Man-nari granite observed by Shimada et al. and of Heavitree quartzite for those samples in Hirth and Tullis (1994) where failure was due to semi-brittle faulting and where confinement was sufficient to suppress frictional sliding. For comparison, the shaded squares indicate the ductile (i.e., non-localized) failure stresses of dry Westerly granite observed by Tullis and Yund (1977) and the semi-brittle flow failure stresses of Heavitree quartzite observed by Hirth and Tullis (1994). We acknowledge the different types of materials used in this plot, but argue that the mineralogical differences between the rock types is only a secondary control on the deformation mechanism relative to the orders of magnitude variation in temperature-corrected strain rate. As in the case of ice (Golding et al., 2012), the ductile failure data are well described (narrower line) by an Arrhenius power-law with n ≈ 3 (parametric values are given in Table 1). Given the power law creep parameters (n, A p and Q ) and following Golding et al. (2012) in assuming w d ≈ w f , an estimate is made of the one adjustable parameter (α ) in the hyperbolic creep law (Eq. (1)) using a least-squares best fit (wider line) to the data of Shimada et al.

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Fig. 3. Effective stress within the (a) non-subducting and (b) subducting oceanic crust required for failure via frictional sliding (narrower line), P-faulting (wider line), and ductile flow (broken line). Shading indicates dominant failure mode, defined as mode with lowest effective failure stress. In determining these depth distributions of dominant failure mode we have assumed a typical global strain rate (Carter and Tsenn, 1987) of 10−14 s−1 and a characteristic length and fault width similar to those of laboratory samples. However, the results are not highly sensitive to these assumptions (Section 3.2).

with limited microcracking and extremely high dislocation densities, semi-brittle faults that formed under lower confinement exhibited abundant closely spaced microcracks in grains adjacent to the fault and thus were unlikely to be P-faults. In our experience, P-faulting only occurs when confinement is sufficient to suppress frictional sliding. 3.2. P-faulting in the Earth’s crust

Fig. 2. Solid symbols indicate the experimentally-observed semi-brittle terminal failure stresses and the open symbols the failure stresses predicted using the best fit P-faulting model. Parametric values are summarized in Table 1. Data are from indicated sources. Only those data for which the confinement ratio R was sufficient to suppress frictional sliding (assuming a coefficient of friction μ = 0.75) are included.

those predicted by the P-faulting model obtained using the same procedure described above (Figs. 1 and 2). In other words, the similarities between semi-brittle faults and P-faults suggest a common origin. We caution, however, that not all faults classified as “semibrittle” are P-faults. Hirth and Tullis (1994) report that narrow, “semi-brittle” faults formed in some quartzite samples even when confinement was low enough to allow frictional sliding. However, in contrast to semi-brittle faults that formed under confinement sufficient to suppress frictional sliding and which were associated

In both the non-subducting and subducting oceanic crust, in the low-confinement, near-surface region (to depths of several tens of km) frictional sliding is the strength-limiting mechanism. In the non-subducting slab, for all global strain rates greater than 1014 s−1 , there is a narrow window, starting at a depth of 42 km, over which P-faulting dominates (Fig. 3a). The width of the window increases from 2 km at a strain rate of 10−14 s−1 to 6 km at a strain rate of 10−12 s−1 . At greater depths failure occurs in a ductile manner. Possible values for the characteristic length are bounded by the maximum slab thickness, which is of order 102 km. At a strain rate of 10−14 s−1 increasing the characteristic length  seven orders of magnitude, from 10 mm to 102 km, slightly elevates the depth at which the P-faulting window begins (to 39 km) and widens the depth of the window to 6 km. Increasing or decreasing the fault width by an order of magnitude changes the transitional depths by less than one km. In short, the results for the non-subducting slab are not highly sensitive to the strain rate, to characteristic length , or to fault width. For all global strain rates greater than 10−14 s−1 , and for all plausible values of fault width and characteristic length, there is a narrow depth-window of a few km starting at a depth of about 40 km within which P-faulting is the strength-limiting mechanism. Above this depth-window failure is dominated by frictional sliding, while below failure is ductile.

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C.E. Renshaw, E.M. Schulson / Earth and Planetary Science Letters 382 (2013) 32–37

Fig. 4. The depth window over which P-faulting is the dominant failure mode (Fig. 3) corresponds well with the exponential falloff in frequency of moderate to large earthquakes (Frohlich, 1989) after its peak at 86 km (the depth at which failure transitions to P-faulting in the subducting slab model) to ∼300 km, similar to the depth at which, using the same geothermal model, transformational faulting is possible (Green and Burnley, 1989).

In the subducting slab P-faulting is similarly the transitional failure mode between frictional sliding in the near-surface region and ductile flow at depth, but as we show in Fig. 3b, the operative window for P-faulting begins deeper (starting at 86 km) and extends deeper. The transition to ductile flow deepens as strain rate increases, from 283 km at a strain rate of 10−17 s−1 to 431 km at a strain rate of 10−12 s−1 . At a strain rate of 10−14 s−1 , increasing the characteristic length  to 102 km slightly broadens the depth-window over which P-faulting dominates, raising the depth at which failure transitions to P-faulting to 80 km and lowering the depth at which failure transitions to ductile flow to 375 km. Increasing or decreasing the fault width changes the transitional depths by less than a few km. In summary, as for the non-subducting slab, our results for the subducting slab are only modestly sensitive to the characteristic length  and to fault width. Increasing strain rate increases the size of the depth-window over which P-faulting dominates, from about 200 km at a strain rate of 10−17 s−1 to more than 300 km at strain rates greater than 10−13 s−1 . Using the eclogite rheology given by Zhang and Green (2007) gives qualitatively similar results, although the size of the depth-window for P-faulting is generally about half that for the gabbro rheology. The modest sensitivity of both non-subducting and subducting slab models to the characteristic length  and to fault width reflects the dominant control of the temperature gradient on model results. For example, at a strain rate of 10−14 s−1 , decreasing the shear stress along the subducting slab by 50%, resulting is a corresponding 50% decrease in the average temperature gradient, increases the maximum depth for P-faulting from ∼ 270 km to more than 600 km. Increasing the shear stress along the subducting slap (and thus temperature gradient) by 50% decreases the size of the depth window over which P-faulting is expected from 270 km to 90 km. 4. Discussion and conclusions The transitions in dominant failure modes in our model of the subducting oceanic crust correlate well with breaks in depthdependent trends in the frequency of moderate to large earthquakes (Fig. 4). These results do not imply that P-faulting is the only possible cause of intermediate depth earthquakes (Meade and Jeanloz, 1991), but do indicate that semi-brittle failure via Pfaulting is a viable mechanism at these depths. Observations of possible repeating intermediate-depth earthquakes (Prieto et al.,

2012) are also consistent with experimental observations of repeating P-faults (Golding et al., 2012). If intermediate-depth earthquakes are due to P-faulting, then the decrease in earthquake frequency with depth may reflect the higher strain rates needed for the deepest P-faults as well as the possibly reduced maximum depth of P-faulting associated with a transition to an eclogite rheology. However, our observation that the eclogite rheology of Zhang and Green (2007) reduces the size of the depth-window for P-faulting should be taken cautiously given the significant uncertainty in eclogite rheology. While our results may seem to argue against P-faulting as a cause of deep (>500 km) earthquakes, caution is warranted because the transition depths shown in Figs. 3 and 4 are sensitive to both the assumed rheology and the geotherm. Conditions resulting in high rates of deformation and relatively low temperatures at depth (Karato et al., 2001) might permit plastic faulting to occur at depths >300 km, perhaps explaining the variation in deep earthquake seismic source properties with the temperature of the subducting slab (Tibi et al., 2003). The P-faulting failure stress in geologic materials is similar to 1/α , or up to a few hundred MPa (Fig. 2, Table 1). This stress is similar to the conditions under which pseudotachylyte formed that is interpreted to be associated with plastic instabilities in mylonites (White, 1996). Comparison of the stress drop associated with P-faulting and inferred from seismic data is complicated by inconsistent laboratory observations. Hirth and Tullis (1994) observed stable sliding (i.e., little to no stress drop) during the semibrittle failure of quartz aggregates while both Golding et al. and Shimada et al. observed significant stress drops with the formation of P-faults. Loading curves in Golding et al. indicate stress drops of about one-third of the failure stress during P-faulting. For failure stresses of a few hundred MPa this corresponds to stress drops on the order of 101 MPa, which is similar to those inferred from seismic data for intermediate depth earthquakes (Green and Marone, 2002; Prieto et al., 2012). Finally, consider the physical significance of the transition stress in the hyperbolic creep law, 1/α . In crystalline olivine and granite deformed under high temperature and pressure the exponential creep law has been inferred to apply when creep is controlled by the resistance to dislocation glide (Evans and Goetze, 1979; Renshaw and Schulson, 2007). In this case (Ashby and Verrall, 1978; Frost and Ashby, 1982), the resistance to glide can be quantified by the Peierls stress σ p = 1/β . When α σ¯ is greater than about 1.2 the hyperbolic creep law (Eq. (1)) reduces to the exponential flow law with β = nα (Barnes et al., 1971). Typical values of the Peierls stress in crystalline rock are of order 10−1 G, where G is the shear modulus (Renshaw and Schulson, 2004). In contrast, values of 1/α that best fit the experimental P-faulting data are of order 10−3 G, similar to the normalized stresses at which the Arrhenius creep law breaks down in metals, ice, and non-silicate rocks (Tsenn and Carter, 1987; Renshaw et al., in review). Thus the transition stress in the hyperbolic creep law for P-faulting appears to be several orders of magnitude lower than the Peierls stress. Even though the physical significance of the transition stress remains unclear, the similarity in the experimentally determined values of α G ≈ 103 across four different crystalline rock types and for freshwater ice, where again (Golding et al., 2012) α G ≈ 103 , may provide, in the absence of experimental P-faulting data, a convenient approximation of the transitional stress for other crystalline materials. Acknowledgements This work was supported by NFS grants EAR-0911071 and EAR-071019.

C.E. Renshaw, E.M. Schulson / Earth and Planetary Science Letters 382 (2013) 32–37

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