Corresponding author: C. He, State Key Laboratory of Earthquake Dynamics, Institute of Geology, China Earthquake Administration, Jia #1, Huayanli, Chaoyang District, Beijing 100029, China. (email@example.com)
 We investigated properties of frictional sliding of plagioclase, pyroxene, and their mixture gouges with a small amount of hornblende, biotite, and quartz as accessory minerals, under hydrothermal conditions with an effective normal stress of 200 MPa, pore pressure of 30 MPa, and temperatures from 100°C to 600°C. Axial loading rate was stepped between 0.001 and 0.0001 mm/s to acquire the rate dependence. Both plagioclase and pyroxene gouges showed velocity-weakening behavior in the whole temperature range except the velocity-strengthening behavior of pyroxene at 200°C. For temperatures above 400°C, both plagioclase and pyroxene gouges showed oscillatory slips, as a result of small dc values of 3–4 µm which make the critical stiffness rise remarkably and approach the system stiffness. Above 300°C, the direct effect of plagioclase shows an increasing trend with temperature, indicating control of the deformation process by thermally activated mechanisms. As the difference of a and b values here are only 20% at most, this trend also applies to the evolution effect. Our analytical derivation based on the theory of pressure solution shows a log-linear contact area growth with time that corresponds to an evolution effect, and estimations based on this encompass the plagioclase data, though the identification of actual mechanisms is not easy. Finally, it is found that a little quartz (3–5%) added to the plagioclase (60–62%)-pyroxene (35%) mixture has a strong stabilizing effect, leading to a transition from velocity weakening to velocity strengthening. These results may help constrain the depth range of seismic slips on deep faults in the lower crust of gabbroic composition.
 Laboratory-derived rate and state friction laws have been important in modeling the processes of earthquake nucleation on planar faults [Dieterich, 1992; Lapusta and Rice, 2003; Rubin and Ampuero, 2005; Ampuero and Rubin, 2008], periodic slow slips that occur at the interface of subduction zones [Liu and Rice, 2005, 2007, 2009], and in giving insightful portraits of earthquake cycles through numerical simulations[e.g., Tse and Rice, 1986; Lapusta et al., 2000; Lapusta and Rice, 2003].
 In the rate and state friction laws, the frictional strength τ of the sliding surfaces or gouge is described as a function of sliding rate V and a state variable θ [Dieterich, 1979, 1981; Ruina, 1983], as follows:
where σeff is the effective normal stress, and V* is a reference velocity at which the friction coefficient has the steady state value μ*. The term a ln(V/V*) that describes the direct rate effect on the frictional strength, with a constitutive parameter a, is now interpreted to reflect an Arrhenius-type thermal activated process related to creep at the asperity contacts [Nakatani, 2001; Rice et al., 2001], which includes dislocation glide and subcritical cracking. The state-dependent part of frictional strength is proportional to the logarithm of the state variable θ and the constitutive parameter b. θ has dimensions of time and can be interpreted as the average age of the population of contacts between two sliding surfaces, or their average lifetime when the fault surfaces or gouge are in motion. There are two commonly used evolution functions for the state variable θ, namely, the slowness law (or “aging law”) and the slip law [Ruina, 1983; Linker and Dieterich, 1992], as follows:
 In the slowness law (2a), θ evolves with time, even when the frictional surface is under truly stationary contact (V = 0). For the slip law (2b), it is evident that the state variable changes with slip alone because (dθ/dt = 0) during stationary contact (V = 0). Because of observations of time-dependent healing phenomena between sliding surfaces [e.g., Dieterich and Kilgore, 1994, 1996], the slowness law is now more commonly used than the slip law.
 Despite the difference between the above two evolution laws and other differences especially in evolution patterns of the state variable θ during stick-slip motions [He et al., 2003], the two evolution laws have many features in common. At a steady state (dθ/dt = 0) under sliding rate V, the friction coefficient reduces to
for both evolution laws. Moreover, linear stability analysis shows that spring-slider systems with the two evolution laws have a common critical stiffness kcr = − (a − b)σ/dc that defines the stability boundary under small perturbations [Ruina, 1983]. Thus, for both evolution laws, a spring-slider system under small perturbation is always stable when a-b > 0 (velocity strengthening), but only stable when k > kcr for a-b < 0 (velocity weakening). Spontaneous seismic slip nucleation is found to be related only to the velocity-weakening behavior [Dieterich, 1992; Lapusta and Rice, 2003; Rubin and Ampuero, 2005], i.e., the steady state rate dependence a-b with the minus sign.
 Because of the importance of the sign of a-b and its actual variation with depth on active faults, experiments have been pursued on the depth variation of a-b since the mid-1980s. These were experiments with temperature conditions that cover the seismogenic zone on gouges of quartz [Chester and Higgs, 1992], granite [Lockner et al., 1986; Blanpied et al., 1995, 1998], gabbro [He et al., 2006, 2007], and talc [Moore and Lockner, 2008], along with experiments on serpentinites [Moore et al., 1997] and natural gouges in lower temperature ranges [Mizoguchi et al., 2008; Verberne et al., 2010; Lockner et al., 2011; Zhang and He, 2013] or at temperatures up to 430°C [Tembe et al., 2009]. Though the depth at which a-b changes from velocity weakening to velocity strengthening is an important transition point in this context, the mechanisms responsible for the transition are not always clear. For the case of quartz and granite, the transition at about 350°C is found to be related to a certain fluid-assisted creep as manifested by the strong velocity-strengthening behavior beyond the transition point [Chester and Higgs, 1992; Blanpied et al., 1995], whereas such a transition for gabbro [He et al., 2007] proved to be not related to any types of prominent creep which would show strong velocity-strengthening behavior with the a-b value greater than 0.01. With the two types of slip behavior beyond the respective transition temperatures, questions arise as to what kind of mechanism controls the transition from velocity weakening to velocity strengthening for the gabbro case and, further, whether one of the major mineral phases can be a key constituent through which the predominant mechanism functions. Answer to these questions is useful in general in understanding the mechanical behavior of some typical crustal rock types where quartz is in negligibly small amount, such as gabbro, diorite, and amphibolite, among others, and thus helps improve earthquake modeling. In this study we have tried to run experiments that give clues to these questions. The experiments were performed first on the major constituents of gabbro, i.e., plagioclase (Pl) and pyroxenes (Px) under hydrothermal conditions; preliminary result of which has revealed velocity-weakening behavior without transition to velocity strengthening [Luo and He, 2009]. Experiments then were performed on Pl-Px mixtures with or without accessory minerals of a small amount added in order to examine the effect of mixing and effect of the minor phases. These experiments successfully isolated the mechanical behaviors of the major mineral phases in gabbro and clearly demonstrated the effect of minor mineral phases in small amounts mixed in gabbro, and these results lead to the understanding of detailed contributions of different mineral phases to mechanical behavior of gabbro and provide clues for responsible mechanisms behind the mechanical behavior. Based on analysis on the experimental results, we also performed theoretical analyses for comparison to understand the mechanisms that control the frictional sliding.
2 Experimental Procedure
2.1 Sample Preparation
 The samples of plagioclase and pyroxene were prepared through crushing a piece of gabbro sample described in a previous study [Zhou et al., 2012] and separating the mineral phases from the mixture, and then the separate samples were ground and sieved using a 200-mesh sieve, deriving a particle size of <75 µm. The output Pl and Px gouges have median particle sizes of 62.5 and 14.4 µm, respectively. The minor phases that need to be checked are hornblende, quartz, and biotite, each of which has volume proportions less than 2%, respectively, in the gabbro sample tested previously [He et al., 2006]. The hornblende sample was separated from an amphibolite, and the quartz sample was from a piece of fresh quartzite collected in Deshengkou of Changping, northern suburb of Beijing. The biotite sample was collected from Yingli of Lingshou, Hebei. All these samples were ground and sieved in a same procedure as applied to the Pl and Px samples, with the final median particle sizes as listed in Table 1.
Table 1. Median Particle Sizes of the Starting Gouge Materials Obtained From Measurements Based on Laser Diffraction
Median particle size (µm)
2.2 Experimental Method
 In the previous study on frictional sliding of gabbro under hydrothermal conditions [He et al., 2007], transition from velocity weakening to velocity strengthening was found under 200 MPa effective normal stress and 30 MPa pore pressure with elevated temperatures. To take these results as the basis for comparison, our new frictional sliding experiments were conducted in the same effective normal stress and pore pressure on a 1 mm thick gouge layer sandwiched between gabbro forcing blocks, using the same testing system employed in previous studies [He et al., 2006, 2007]. In the initial stage of shortening, confining pressure of 152 MPa was applied, allowing increasing normal stress up to 230 MPa from which point servo control was switched from constant confining pressure control to constant normal stress control. The initial confining pressure was chosen to be the confining pressure corresponding to 230 MPa normal stress around the “yielding” point of the shear stress versus displacement curve. With a pore pressure of 30 MPa, a normal stress of 230 MPa corresponds to an effective normal stress of 200 MPa. Some experiments with a constant confining pressure of 130 MPa were also performed for cross-check, which corresponds to an effective normal stress of around 200 MPa during sliding as both confining pressure and axial stress make up a portion of normal stress. Our experiments have been run with nominal temperatures of 100°C–600°C measured at the top of the sample, and the temperatures quoted hereafter are temperatures at the middle of the sample. The pore fluid has a direct access to the gouge through a highly permeable brass filter, and the pressure of the inlet is controlled by an independent servo mechanism of quick response (Figure 1); see He et al. [2006, 2007] for other details of the experimental procedure and the testing system.
3.1 Results on Plagioclase and Pyroxene Gouges
3.1.1 Basic Features: Oscillatory and Stable Sliding Behaviors
 Typical friction coefficient (μ = τ/σeff) versus shear displacement data for plagioclase and pyroxene gouges are plotted in Figures 2-5. The curves change from initial elastic loading gradually to steady slip strengthening from displacement of ~0.25 to ~1.25 mm. The slopes of the steady slip strengthening range from 0.01 to 0.06/mm, with an average of 0.04/mm. Loading velocity was stepped back and forth between 1.22 and 0.122 µm/s, and responses in the steady slip-strengthening stage were examined to investigate steady state rate dependence.
 In the high temperature range from 398°C to 607°C, all sliding experiments on plagioclase and pyroxene gouges showed sustaining oscillatory slips as seen in the friction coefficient versus shear displacement curves (Figures 2-4, Table 2), which are basically a manifestation of velocity-weakening behavior as predicted by the rate and state friction law [Ruina, 1983; Gu et al., 1984; Gu and Wong, 1994] in the vicinity of the critical point of stability when the critical stiffness associated with the friction parameters approaches the system stiffness. Plagioclase around 400°C was basically stable (plw04 in Figure 2, plw12 in Figure A6), which exhibited decaying oscillatory slips at a loading rate of 1.22 µm/s and stable, monotonic sliding behavior at 0.122 µm/s. Pyroxene around 400°C was similar to plagioclase, but the oscillatory slips at 1.22 µm/s seem to be more complicated (pyw02 in Figure 3). During the oscillations, stress changes faster in the descending phase than in the rising phase, but the stress drops typically in 5–7 s at a loading rate of 1.22 µm/s and in several times longer period at loading rate of 0.122 µm/s for the plagioclase cases. Even in the fastest cases of plagioclase, as seen in the initial responses following an upward rate step, the duration of stress drop was above 4 s. For the oscillatory slips of pyroxene gouge, the duration of stress drop was relatively shorter, but it was above 2 s at a loading rate of 1.22 µm/s and was 5–10 s at loading rate of 0.122 µm/s. Thus, for both plagioclase and pyroxene cases, the stress drops during the oscillations were not inertial instabilities but basically quasi-static processes. Moreover, because of the relatively faster stress decrease during the oscillation of pyroxene, control overshoots occurred during the experiments (Figure 3).
Table 2. List of Experiments on Plagioclase and Pyroxene Gouges
Steady state value of friction coefficient at 1.5 mm axial permanent displacement at 1.22 µm/s (or average value for cases of oscillatory slips).
Parameters obtained by fitting typical data to the slowness law.
Parameters obtained by fitting to the initial part of the response to a downward rate step. SS = stable sliding, Osc = sustaining oscillation, Trans = transitional slips between stable sliding and sustaining oscillation.
Extrapolated value along trend beyond the jacket failure point.
Steady state rate dependence (a-b) obtained using an empirical equation (5) which relates the average shear stress, b-a, and velocity ratio inferred with numerical simulations with the slowness law [Verberne et al., 2010];
dc values roughly estimated by equating actual stiffness to the critical stiffness kcr = (b-a)σeff/dc using the a-b data derived from the empirical equation (5). (a-b) s.d. is the standard deviation of (a-b) for measurements of more than two points.
 The minus sign of steady state rate dependence during these oscillatory slips can be seen from the variation of average friction coefficient ((peak + trough)/2) with respect to changes in the loading rate (Figure 1, 2), as predicted by the rate and state constitutive law for sustaining quasi-static oscillation [Verberne et al., 2010]. The same oscillation also occurred to plagioclase gouge at 201°C, with a hardly discernible sign of steady state rate dependence.
 The occurrence of oscillatory slips of plagioclase above 400°C and at 200°C was confirmed by supplementary experiments performed under 100 MPa effective confining pressure (with a comparable effective normal stress of around 200 MPa beyond the yield point) to exclude the possibility of artifact due to servo manipulation on the normal stress that may cause the occurrence of oscillation (Figure 4). It is clear from these experiments that the velocity-weakening behavior is true and it also occurs at much slower rates tested here (down to 0.0488 µm/s).
 The “sustaining” oscillations at 500°C and 600°C are actually more complicated than the rough impression. Careful check on the first oscillation series of plw14 at 503°C (Figures 4 and A7) reveals a gradual approach toward a periodical oscillation, which, in the context of rate and state friction laws, implies evolution of constitutive parameters with displacement. In such an evolution, the critical stiffness seems to increase significantly and approach the system stiffness from below.
 Stable sliding was observed for plagioclase at 100°C and 300°C (Table 2, Figure 5). Similarly, stable sliding occurred to pyroxene gouge at temperatures up to 300°C (Table 2, Figure 5).
 While the stable sliding cases are the easy part for data analysis, parameter acquisition from the oscillation cases is hard with simple methods. One of the possible approaches is to approximate the critical stiffness by the system stiffness, i.e., k ≈ (b − a)σeff /dc, which may give a rough estimate of an upper bound of b-a for the sustaining oscillatory slip cases. However, the validity of this estimate depends on the proper estimate of characteristic slip distance dc. The quantity of dc may be measured directly from the stable sliding cases at the lower temperatures, but it is not guaranteed that dc is roughly a constant in the whole temperature range and in the whole sliding process at each temperature. Moreover, even for the stable sliding cases, only a-b values can be read directly from the data; other parameters like a and dc values cannot be read accurately from the data. Under this situation, numerical procedures were employed to analyze the data with both constant normal stress cases and variable normal stress cases under constant confining pressure to extract the unknown constitutive parameters.
 For the variable normal stress cases, a constitutive relation incorporating varying normal stress is needed for analyses. To this end, the formulation below by Linker and Dieterich  was used for the evolution laws:
The general relation remains the same as the constant normal stress case except that the interpretation of the effective normal stress is now a variable rather than a constant. As inertial instability may be triggered by 10-fold rate stepping for systems obeying the slip law with stiffness in the vicinity of the critical value [Gu et al., 1984], slowness law was used in our analyses to reproduce the quasi-static oscillations in our experiments.
 In the numerical analyses, downward steps were chosen to fit whenever possible for two reasons: (a) The slowness evolution law produces asymmetric stress changes responding to upward and downward rate steps, and the response to 10-fold downward steps is more similar to that of the slip law, which fits better the transient sliding behavior after a step change of slip rate [Bayart et al., 2006; Ampuero and Rubin, 2008]. (b) The normal stress control in our system follows the stress changes well at the slow rates used here.
 It is noted that a well-defined initial condition is needed for each numerical analysis; thus, for quasi-static oscillations, the rate steps started from a steady state sliding were chosen for data fitting. As the series for plagioclase with constant confining pressure control satisfies this condition and has good control quality during the whole process, typical rate steps in each experiment of this series were chosen for data fitting. Details for the numerical method and results are given in Appendix A. Bearing in mind that the constitutive relations do not reproduce the full range of frictional sliding behavior and that the actual sliding process includes evolution of parameters, the results below obtained by data fitting are considered here as rough estimates.
3.1.2 Steady State Rate Dependence
 For the stable sliding cases, the steady state velocity dependence (a-b) can be quantified using a finite difference form of equation (3), i.e., by calculating the ratio of the difference in steady state friction coefficient Δμss at high and low loading rates to the difference in natural logarithm of the shear displacement rate (Δln V). To ensure that the sliding is well within the steady slip-strengthening stage, data of the final three steps were used for calculation. All these stable sliding cases showed velocity-weakening behavior (Table 2, Figure 6) except the experiment on pyroxene (pyw06) at 201°C, which showed weak velocity strengthening with an average (a-b) value of 0.0014.
 For the cases with decaying oscillation at 1.22 µm/s, the a-b value can be calculated with the same method as in the stable sliding cases by taking the final mean stress as an approximation of the steady state value.
 For the sustaining oscillatory slips under constant normal stress control at temperatures over 500°C, a steady state was not accessible before the triggering of oscillations. To get a rough estimate of a-b in these cases especially for pyroxene gouge where data fitting to the friction model was impossible, an empirical equation obtained from numerical simulations of a spring-slider system at the critical point (k = kcr) [Verberne et al., 2010] was used. With the slowness law, the equation relates b-a, mean stress τm of the oscillation, effective normal stress σeff, and velocity ratio V0/V* of 0.1–11 as follows:
where v0 = V0/V* and τ* is the mean shear stress at the reference velocity.
 From the direct readings, data fittings and estimations using the empirical equation (5), respectively, for different cases, a-b values were obtained (Figure 6, Table 2). All the data points show velocity-weakening behavior except one case of pyroxene at ~200°C. While temperature dependence is not significant, it seems that the a-b values roughly have a lower limit of −0.005 for plagioclase. The pyroxene case seems to be different in low and high temperature ranges. For temperatures up to 300°C, the data points lie in the vicinity of the velocity neutral line, but for temperatures above 400°C, (a-b) has values ranging from −0.002 to −0.006, with a minimum at ~600°C.
3.1.3 Direct Rate Dependence
 Due to interaction between the frictional sliding and the elastic surroundings, direct reading of the a value from the mechanical data is difficult, especially when the system stiffness is close to the critical point as in the cases of oscillatory slips. To give better estimates of the direct effect, all the data here are obtained from data fitting, both for stable sliding cases and oscillatory slips (Table 2, Figure 7).
 The limited data of pyroxene up to 300°C show a step change from a low level below ~0.008 to a much higher level of ~0.017 when temperature is elevated from 200°C to 300°C (Figure 7). A similar jump can be seen for the plagioclase case, but the a value increases to a relatively lower level of ~0.015.
 For temperatures above 300°C, the plagioclase data show an increasing trend with temperature, with a slope of ~1.5 × 10−5 K−1.
3.2 Mixture Gouges
3.2.1 Steady State Rate Dependence
 The velocity-weakening behavior of plagioclase and pyroxene gouges at temperatures over 400°C means absence of transition from velocity weakening to velocity strengthening, in contrast to the occurrence of such a transition reported for gabbro gouge tested previously in the same hydrothermal conditions [He et al., 2007]. That is, none of the essential mineral constituents of gabbro reproduces the transition in rate dependence that occurred in gabbro for temperatures ≥400°C. Because of this, the focus of this study turned to effect of accessory minerals in small amount as identified in the gabbro sample, i.e., hornblende, biotite, and quartz.
 The effect of different accessory minerals was first examined by mixing 5 wt % of these minerals separately to a base Pl-Px mixture with 33–35 wt % pyroxene and 60–62 wt % plagioclase (Table 3), and experiments were performed on these mixtures at 503°C. The base mixture (65 wt % Pl + 35 wt % Px) was tested first for reference with temperatures from 400°C to 600°C, and the results showed velocity-weakening behavior similar to Pl and Px gouges (Table 3, Figure 8). The mixture with 5 wt % hornblende (mix01) also showed similar oscillatory slips, indicating no significant change to the velocity-weakening behavior when 5 wt % hornblende is mixed (Figure 8). The addition of 5 wt % biotite into the base mixture showed a significant effect on the sliding behavior, which experienced a change from oscillatory slips to stable sliding (mix08), but the sign of velocity dependence remains unchanged with an average (a-b) = −0.0024 (Figures 8 and 9).
 The accessory mineral that has the most prominent effect on sliding behavior turned out to be quartz (mix02), which made the sliding behavior change from velocity weakening (sustaining oscillatory slips) to velocity strengthening with sliding mode of stable sliding (Figure 8). Further tests were carried out on the same mixture gouge for temperatures ≥400°C, deriving (a-b) values ranging from 0.001 to 0.0025.
 Influence of the amount of quartz was tested on a mixture with quartz content reduced to 3 wt % (mix10-12, 3% Qz + 60% Pl + 37% Px) for temperatures of 402°C, 505°C, and 605°C. The results showed stable, velocity-strengthening behavior, with (a-b) values ranging from 0.0016 to 0.0020, quite similar to the case with 5% quartz mixed (Figure 9, Table 3).While the overall sliding behavior is stable and velocity strengthening when quartz is mixed in small amount, it should be noted that the mechanical data tend to be a little rough with small bumps (Figures 8 and A5).
 The effect of the quartz proportion is further discussed below through examining available data on the direct effect.
3.2.2 Direct Effect
 All the direct effect data here were derived from numerical data fitting based on the slowness law, which has been limited to cases with well-defined initial conditions before the rate step change. Thus, data for mixtures with quartz and biotite are available here (Table 3, Figure 10). As shown in Figure 10, direct effect of the mixture with 5% quartz increases linearly with temperature in the tested temperature range, with a slope of ~8.4 × 10−5/K. When the quartz proportion is reduced to 3%, the direct effect data exhibit saturation above 500°C, showing values similar to those of plagioclase.
 As biotite of 5% acted as a stabilizer to the frictional sliding of the Pl-Px mixture, it is of interest to examine the direct effect of the mixture. As seen in Figure 10, the a value associated with this mixture is significantly lower than that of plagioclase (~0.011 versus ~0.017). Moreover, the b value inferred for this case is also quite lower than the plagioclase data.
 As the steady state rate dependences (a-b) for the 5% quartz and 3% quartz mixtures have comparable values, the b values drop a similar amount as a values do when quartz proportion is reduced from 5% to 3%. These changes in the rate dependencies indicate that both the direct effect and the evolution effect are strongly affected by the presence of quartz.
4 Microstructural Features of Deformed Samples
 Microstructures of deformed gouge were observed with scanning electron microscopy (SEM) on thin sections cut perpendicular to the gouge layer and along the sample axis. Using the terminology of Logan et al. , the gouge after deformation generally shows Riedel shears (R1), boundary shears (B), and Y shears (Figures 11a, 12a, and 13a) along with pervasive shear deformation (Figure 12b). R1 and associated B shears are the most predominant in the localized shear bands. Moreover, grain size is remarkably reduced, both in the shear bands (Figures 11b and 11c) and in the matrix after pervasive shear deformation (Figure 12b), as compared to the undeformed sample (Figure 12c). The majority of the grains in the matrix or in shear bands are in the submicron scale. The fine-grained matrix is mixed occasionally with larger clasts with an upper bound of ~76 µm in their short axes.
 As dc values and the sliding behavior change remarkably in the temperature range up to 300°C, it is of interest to examine the difference in microstructures. The microstructure of plw07 deformed at 101°C shows curved but through-going R1 shears (with some steps) coordinated with B shears as major shear zones (Figure 11a), and short, discontinuous R1 and Y shears were also developed (close-up image of the rectangle area shown in the inset).
 Microstructure of plw02s (deformed at 201°C) shows through-going, low-angle (angle lower than the 101°C case) and straight R1 shears coordinated well with Y shears and boundary shears. Relatively straight shear bands in submicron particles can be seen ubiquitously, associated with the major R1 or Y shears or developed independently (Figure 11b).
 Microstructure of plw03 (deformed at 298°C) has a prominent low-angle, through-going R1 shear subparallel to prominent boundary shears, along with a lot of secondary R1 shears, mostly with low angles to the boundary, discontinuous and in curved nature (Figure 11c). The inset of Figure 11c shows shear bands with submicron particles.
 The SEM image of pyroxene deformed at 201°C (pyw06) shows a major through-going Riedel shear with small steps and prominent boundary shears (Figure 13a). The microstructure of the sample deformed at 405°C (pyw02) shows a single smoothly curved R1 shear and secondary high-angle Riedel shears, also curved (Figure12b), with some fine-grained shear bands (inset of Figure 13b). Multiple through-going R1 shears can be seen in the sample deformed at 503°C (pyw01), piecewise straight but has some steps on the whole (Figure 13c). Fine-grained bands can be seen in Y or P directions (inset of Figure 13c).
 While intergranular pressure solution may be a possible process under the hydrothermal conditions, it is of interest to see the relevant evidences from the microstructure in deformed samples. However, as continuous contact time at the contacts between grains is short during granular flow in shear deformation, observable indentations due to pressure solution are not expected ubiquitously. Actually, indentation was only observed in very limited localities where large clasts are in contact and they seem to have experienced longer contact time than in other localites, as seen in pyroxene gouge deformed at 503°C (pyw01, Figure 14). The limited occurrence of such microfabrics shows that intergranular pressure solution is a possible process, but it is not enough to be an evidence for a major controlling process.
 As quartz in small amount showed strong effect in controlling the velocity dependence, it is of interest here to see the shapes and distribution of quartz grains in the gouge after deformation. Through careful search by SEM and energy-dispersive X-ray spectroscopy (EDS) analysis, quartz grains are only found in submicron matrix of relatively dark gray areas in backscattered electron images (indicated with white arrows in Figures 15a and 15b). Clasts of quartz with size above micrometer was not found, indicating that quartz grains are more easily crushed than plagioclase and pyroxene grains during the shear deformation.
5.1 Rate Dependencies as Results of Thermally Activated Processes
 As mentioned above, the direct effect in the rate dependence can be interpreted as a thermally activated creep under high stress at the contacts [Nakatani, 2001; Rice et al., 2001] in an exponential form:
where T is the absolute temperature, k is the Boltzman constant, Q is the activation energy, τc is the average shear stress at the contacts, Ω is the activation volume, and V0 is the product of the attempt frequency and the jump distance. As demonstrated by Nakatani  and Rice et al. , this relation predicts an estimate of the direct effect by a = kT/Ωσc with σc as the average normal stress at the contacts.
 The creep governed by (6) has been considered to include multiple mechanisms such as dislocation glide and subcritical cracking [Nakatani, 2001]. The estimate of the a value inferred by Nakatani  from his Na-feldspar data is compared with our results on plagioclase in Figure 16. While his data show a slope of ~3.41 × 10−5 K−1, data of this study show a smaller temperature dependence of the order 1.5 × 10−5 K−1 in the temperature range above 300°C. The difference may have resulted from a different deformation process under the hydrothermal conditions associated with our data, though identification of the related mechanism is not easy. Moreover, a much higher slope of the a-T relation can be seen in granite data of Blanpied et al.  (Figure 16), which may be associated with the enhanced plasticity of quartz under hydrothermal conditions [Blanpied et al., 1995].
 Actually, the exponential plastic flow mentioned above is also related to the evolution effect of friction (b value in the constitutive relations) if creep in the normal direction is considered, as demonstrated by Bréchet and Estrin . By a constraint of no volume change during contact creep, they successfully derived an expression for b value by considering the contact area growth with time, as follows:
where the S parameter is the strain rate sensitivity of the flow stress σtr (with subscripts tr and tr0 indicating true stress and initial true stress, respectively), and it is equivalent to kT/Ω in (6).
 As τtr/σtr0 is on the order of 0.6–0.8 for hard rocks at room temperature, the b value depends mostly on S/σtr0.
 It should be noted that the direct effect can be any deformation mechanisms which reproduce the direct rate dependence, whereas the evolution effect requires mechanisms that cause contact area growth with time. Microindentation experiments at temperatures of 400°C–600°C show that plagioclase basically exhibits brittle behavior, manifested by load-dependent hardness and lateral cracking [Huang et al., 1985]. This may be related to the highly reduced particle size as seen in our SEM observations, which implies fracture processes of the grains during shearing deformation, and such a process is hard to be related to contact area growth.
 In addition to exponential plastic flow, pressure solution is another well-known process that causes contact area growth under hydrothermal conditions [e.g., Yasuhara et al., 2003]. While the identification of the dominant mechanism is not easy, it is of interest to compare our mechanical data with a prediction based on the pressure solution process.
 To this end, evolution effect associated with pressure solution is considered here. By considering identical spherical grains and dissolution process at the contacts and bridging the strain rate and contact area growth through geometrical relations, we derived an approximation of evolution effect of friction due to dissolution at the contacts and contact area growth, as follows (see Appendix B for details):
where R is the molar gas constant, Vm is the molar volume of the solid, and subscript IPS indicates association with intergranular pressure solution. This expression is similar to (7) in form, but molar volume rather than activation volume is used here according to pressure solution theory. With σtr0 = 2.05 and 3.68 GPa, respectively, as lower and upper bounds of the strength of plagioclase estimated from elastic constants of plagioclase An56 [Bass, 1995] (see Appendix B for details), the estimations by (8) are converted to the a value for comparison using the b/a ratio of 1.06 typical in our experiments (Figure 16). Evidently, both estimations by pressure solution (red lines) are smaller than the Na-feldspar data of Nakatani , but the range limited by the upper and lower bounds includes our plagioclase data under hydrothermal conditions.
 From the derivations of Bréchet and Estrin , this study, and demonstration by experiments [Dieterich and Kilgore, 1994], the evolution effect should be related to a certain creep of the contacts (in single or multiple mechanisms), which may be plastic flow and/or pressure solution in the well-known mechanisms.
 On the other hand, the strikingly reduced particle size implies that fracture of the granular grains is also a major deformation mechanism during the process of frictional sliding, which should have a contribution to the direct effect. Accordingly, it seems that multiple thermally activated deformation mechanisms may operate concurrently in the whole process of deformation.
 As identification of concrete deformation mechanisms responsible for contact creep is not easy with the current data, further work with carefully designed experiments is needed to this end.
5.2 Stabilizing Effect of a Small Amount of Quartz
 From the experiments on the mixtures, both quartz and biotite have a stabilizing effect to the frictional sliding of the Pl-Px mixture. In the biotite case (mix08 at 503°C), the 5% biotite stabilized the frictional sliding by increasing the dc value to a much higher value (from 4 to 27 µm), but the a-b value remains negative and similar to the range of plagioclase data. Because a larger dc value corresponds to a smaller critical stiffness kcr = (b-a)σeff/dc, this kind of effect stabilizes the system only conditionally by reducing the critical stiffness to a level away from the critical point (from k ≈ kcr to k > kcr), but it does not exclude unstable slip nucleation on actual faults [Dieterich, 1992]. Thus, such an effect is not crucial when nucleation of seismic slip is concerned.
 On the other hand, the addition of a small amount of quartz into the Pl-Px mixture led to a more “intrinsic” change in the steady state rate dependence from velocity weakening to velocity strengthening (Figure 9).
 As seen in Figure 10, reducing the quartz proportion from 5% to 3% corresponds to a drop in direct effect (a value) from a relatively high value to the level similar to that of plagioclase (Figure 10), whereas (a-b) remains positive with values similar to the case of 5% quartz. This fact implies that the presence of a small amount of quartz weakens the original evolution effect to a lower level that is enough to change the sign of steady state rate dependence (a-b). In other words, the presence of quartz reduces the logarithmic growth rate of contact area, whatever the original mechanism is. As quartz is much more soluble in water than feldspar and pyroxene; then if the original mechanism related to evolution effect is pressure solution, saturation of silica may occur due to the presence of quartz that may shut down the pressure solution process. On the other hand, intervention of the fine quartz grains between the otherwise “clean” contacts may also change the process at the contacts by “poisoning” the original mechanism, though the details for this mechanism is much less specific. No matter what the mechanism is, it is the presence of water that allows fluid-assisted chemical processes to occur at the grain contacts, as seen similarly in previous studies [e.g., Moore and Lockner, 2013].
5.3 dc Value: The Major Parameter That Determined the Sliding Behavior
 As the critical stiffness kcr is proportional to (b-a) and inverse of dc, significant change of these parameters may cause a change in the sliding behavior. While the direct readings and inferred values of (b-a) of plagioclase were found to be below 0.005 in the whole temperature range, the inferred dc values change strikingly in different types of sliding behavior. dc inferred from the initial part of sustaining oscillatory slip cases showed small values ranging from 3 to 4 µm, regardless of the temperatures at which the oscillations occurred. Under the background of small (b-a) (<0.005), the occurrence of oscillatory slips in our experiments seems to be controlled by the small dc values at ~200°C and above 500°C.
 The stable sliding cases of plagioclase analyzed below 400°C show dc values ranging from 5 to 71 µm, with a maximum at ~300°C and a minimum at ~100°C. The initial part of decaying oscillation of plw12 at ~400°C showed a dc value of 10 µm, which is between the typical values for stable sliding and sustaining oscillations.
 It is of interest to note that the dc value in the stable sliding of plw12 (405°C) at the third step (47 µm) is much larger than that in the subsequent decaying oscillation (10 µm), indicating considerably quick evolution of dc with displacement to lower levels.
 Nothing seems to be special with the dc value (23–35 µm) for the stable sliding cases of pyroxene below 300°C, indicating a different behavior compared to the plagioclase cases in this temperature range. However, for temperatures above 500°C, roughly estimated dc values from the oscillatory slips using the critical condition k = kcr are quite similar to the plagioclase case (2–4 µm, Table 2).
5.4 Implications to Frictional Sliding of Gabbro and Seismic Slips in the Lower Crust
 Before the following discussion, it should be noted that the velocity-weakening behavior of plagioclase and pyroxene at temperatures from 400°C to 600°C is a typical brittle behavior, which is essentially due to the employed loading rate that is much higher than typical tectonic loading (0.0488–1.22 µm/s in this study versus ~0.001 µm/s typical for San Andreas Fault). At a given tectonic loading rate, brittle-plastic transition may occur at a temperature between 400°C and 600°C depending on the contrast between frictional strength and creep strength. While frictional strength is basically controlled by effective normal stress that is a function of pore fluid pressure and depth, temperature-dependent creep strength is typically controlled by crystalline plasticity or plastic deformation associated with chemical processes such as pressure solution. The frictional sliding behavior as seen in our experiments may transition to plastic deformation as loading rate decreases to a certain critical level that depends on the temperature condition.
 As seen in a previous study [He et al., 2007, Table 2], the frictional sliding of gabbro under similar hydrothermal conditions in this work showed weak velocity dependence for temperatures above 400°C, with│a-b│ <0.006. Comparison of the gabbro data with the results in this work suggests that the small amount of quartz (approximately 2%) played an important role in influencing the mechanical behavior of gabbro, which may otherwise exhibit velocity-weakening behavior in the whole temperature range if quartz is not included. The gabbro gouge exhibits transition from velocity weakening to velocity strengthening for temperatures over 400°C, with the transition temperature at approximately 510°C, which may correspond to the bottom of the seismogenic layer or unstable slow slips in the lower crust in this case. In this regard, modeling by Liu and Rice  shows that the gabbro friction data are the best choice in the limited data sets to explain the GPS measurements on surface deformations due to episodic slow slip transients on the subduction interface in northern Cascadia, indicating a possible control by frictional sliding of gabbro with a small amount of quartz.
 On the other hand, it should be noted here that for mafic rocks without occurrence of quartz or with its occurrence in trace amount, the frictional sliding will exhibit velocity-weakening behavior for temperatures over 400°C and at least up to 600°C, according to our experiments. Under this situation, the bottom of the seismogenic layer in the lower crust may be constrained by brittle-plastic transition, which is considered here to be controlled mainly by pore pressure in the fault zone and creep strength below the frictional portion. Considering grain size reduction due to strong shear deformation, the creep strength of the lower crust may be constrained by diffusion creep of a mafic aggregate composed of 75% anorthite and 25% diopside [Dimanov and Dresen, 2005], which has much lower stress than prediction by the flow law of dislocation creep synthesized using data of anorthite [Rybacki and Dresen, 2000] and clinopyroxene [Chen et al., 2006] according to the formulation by Tullis et al.  (Figure 17). With pore fluid pressure ranging from hydrostatic pressure to a level of 2.5 times of that, the brittle-plastic transition depth may vary from 27 to 30 km from the simple illustration in Figure 17. Though the estimation is rough, the brittle depth range overlaps with the depth range of tremor locations on the San Andreas Fault (SAF) in central California given by Shelly and Hardebeck . Moreover, the P wave velocity data [Lin et al., 2010] of this area indicate that the bottom depth is still in the lower crust, with temperature of about 550°C according to the thermal model by Lachenbruch and Sass  (model B). Thus, the mechanism for the tremors in the lower crust is perhaps related to unstable frictional behavior of gabbroic rocks revealed in this study.
 To understand the mechanism that controls the frictional sliding of gabbro and mafic rocks in general in continental lower crust under hydrothermal conditions, first we tested the major mineral phases of a previously studied gabbro sample [He et al., 2007] and then tested the mixture gouges of the major phases with separate accessory minerals in small amounts. From the experiments and our analyses it is concluded that
 The sliding behavior of both plagioclase and pyroxene gouges is velocity weakening in the whole temperature range except that of pyroxene at 200°C which showed weak velocity strengthening. Direct readings from the stable sliding cases and inferences from the oscillatory slips show (a-b) values for plagioclase ranging from −6.5 × 10−4 to −4.0 × 10−3. Similar estimation of the velocity-weakening behavior for pyroxene gouge shows (a-b) values ranging from 1.7 × 10−4 to −5.9 × 10−3. While the (a-b) value of plagioclase shows no significant temperature dependence, the pyroxene case shows stronger velocity-weakening behavior for temperatures above 400°C than in the lower temperature range. The similar sliding behaviors of both plagioclase and pyroxene above 400°C fail to reproduce the rate dependence of a gabbro sample documented in a previous study [He et al., 2007].
 For temperatures over 400°C, both plagioclase and pyroxene gouges showed oscillatory slips. Numerical fittings of plagioclase data to the rate and state constitutive relation associated with the slowness law indicate that the oscillatory slips are not due to a relatively stronger velocity-weakening behavior but resulted from much smaller dc values of 3–4 µm in these cases compared with the stable sliding ones. dc values tend to get smaller especially for temperatures over 400°C and also tend to evolve to smaller values with displacement.
 Direct effect (a value) inferred from numerical fittings is found to increase linearly with temperature for temperatures over 300°C, indicating that the deformation process is related to thermally activated mechanisms in this temperature range.
 With a small amount of accessory minerals added separately to the plagioclase (60–62 wt %)-pyroxene mixture, it is found that quartz is the only one which has a prominent stabilizing effect on the velocity-weakening behavior of the mixture. Quartz of 3–5 wt % added to the mixture lead to an acute transition from velocity weakening to velocity strengthening.
 Reducing the added amount of quartz from 5% to 3% did not make a significant difference in the steady state rate dependence of the mixture, but the 3% quartz had a stronger influence to the evolution effect than to the direct effect, implying that the evolution effect was weakened by the presence of quartz.
 In the microstructure of deformed samples, quartz can only be seen as very fine fragments with a grain size of <0.4 µm, much smaller than the widespread large clasts of plagioclase and pyroxene with size of 2–3 µm, probably due to easier crushing.
 Effect of pressure solution at the contacts is examined theoretically through analytical derivation based on standard exponential equation of pressure solution. The analytical result shows that the b value due to pressure solution at the contacts is proportional to a first-order estimate of coefficient of friction and absolute temperature, and inversely proportional to yield stress or true fracture strength (whichever is smaller) of the material multiplied by molar volume of the material. Rough estimates for temperatures over of 400°C by this result encompass the data of plagioclase, though such a comparison and the limited microstructural observations are not enough to identify the actual mechanism.
 Combining with the previous gabbro data, these new results may help constrain depth range of seismic slips in the mafic lower crust, for quartz-bearing or quartz-free situations. The bottom of the seismogenic lower crust may be constrained by a temperature of 510°C for cases with occurrence of quartz in small amount, or deeper for quartz-free cases by the brittle-to-plastic transition which is controlled by a combination of actual creep strength and actual pore fluid pressure. The bottom depth of 30 km at SAF as indicated by tremor locations [Shelly and Hardebeck, 2010] corresponds to temperature of approximately 550°C around which brittle-to-plastic transition may occur with high pore pressure.
Appendix A: Data Fitting to the Slowness Law
A1. Numerical Procedures
 Fitting of the slowness law to the experiments is based on forward calculation of a lot of combinations of constitutive parameters and their comparison to actual data. Forward calculations of the sliding process of a spring-slider system with constant normal stress (Figure A1) and variable normal stress with constant confining pressure (Figure A2) are considered here. The equation of motion of the system is expressed by
where δ is the slip distance of the block, δ0 is the load point displacement, τ is the shear stress acting on the system, m is the mass of the block, and k is the spring stiffness.
 For the variable normal stress case with constant confining pressure, effective normal stress and shear stress are coupled by
where σ3eff is the effective confining pressure, and ϕ is the fault angle as in Figure A2. The governing equations are the general equation (1) combined with (A1) and the evolution laws of state variable, i.e., (2a) and (4a), respectively, for the constant normal stress cases and the variable normal stress cases, and (A2) is also necessary for the latter. Numerical integration of the full set of differential equations represented by equations ((1)), ((2a)) or ((4a)), and ((A1)) (and also (A2) for the variable normal stress case) was performed employing a procedure identical to that of He and Ma  and He et al. .
 For data fitting, slip strengthening of the experimental data was detrended and a segment of the μ-δ0 curve representing the response to a rate step change was chosen to be compared with the numerical results. A reduced data set from smoothed data (10-point adjacent averaging) was used here. The output data of the numerical simulation with much smaller sampling interval were aligned to the experimental data by linear interpolation with two closest points on the left and right sides of an actual displacement point.
 For stable sliding behavior, a-b is known; thus, only b/a and dc are to be determined for the constant normal stress cases. One more parameter, α, is to be determined for the variable normal stress cases.
 With dci as a tentative estimate from the mechanical data, the best fit is searched through exhaustive exploration of the solution space for dci −0.015 mm < dc < dci +0.015 and simultaneously for 1.01 < b/a < 1.5 for the stable sliding and velocity-weakening cases with constant normal stress, with precisions of 0.001 mm and 0.01 for dc and b/a, respectively. For variable normal stress cases, the α value is usually searched from 0.01 to a value smaller than the initial steady state coefficient of friction, and the lower bound is allowed to be extended to smaller values if the α value at the minimum hits 0.01. Similar explorations were performed for velocity-strengthening behavior with 0.01 < b/a < 1.0.
 For the oscillatory slip cases, b-a value is also unknown and to be determined. As the variable normal stress cases (with constant confining pressure) for plagioclase above 400°C have well-defined initial conditions, a specially designed scheme was applied for this series. To explore the b-a value, the ratio of k/kcr is explored from 1.01 to 5 with k measured from the mechanical data, with
where μss(V0) = μ* − (b − a)ln(V0/V*) is the steady state coefficient of friction at the new loading rate V0 [He et al., 1998]. For given k/kcr values and other given parameters, b-a is solved numerically from k = (k/kcr)kcr(V0) for the current model calculation. The precision of k/kcr here is 0.01.
 Efficiency of calculation during exploration was generally guaranteed by initial rough exploration on the unknown parameters and subsequent focus on relatively narrow bands of the parameters, and the range of parameters was reduced one by one in sequence from b/a, dc, k/kcr, and α.
 For each case of calculation, the root mean squared error is calculated and logged with the model parameters as a residual. The parameter set with the minimum residual is adopted as the best fit after the whole exploration.
A2. Results for Stable Sliding Cases
A2.1 Constant Normal Stress Cases
A2.1.1. Pyroxene Cases up to 300°C
 The pyroxene gouge showed stable sliding from 100°C to 300°C (pyw05, pyw06, and pyw04) under normal stress control with good mechanical data. These cases are the only access to the direct effect (a value) and characteristic slip distance dc in the whole temperature range.
 The sliding was basically stable in this series, though small bumps occurred around 300°C. The fittings are basically satisfactory (Figure A3). However, a sharp spike and subsequent trough in pyw05 (102°C) were not reproduced by the model; probably, one more state variable is needed to represent the data. A much higher direct effect (a value) can be seen around 300°C compared to the values at lower temperatures.
A2.1.2 Pl-Px Mixture with 5% Quartz Above 400°C
 This series was performed under constant normal stress control, and the overall sliding behavior is stable despite a lot of bumps that make the curve rough. Sharp spikes in the curves are due to the delay of normal stress control responding to bumps during sliding, and the spikes are not included in the smoothed data.
 The fittings are quite satisfactory in reproducing the major features of the sliding behavior (Figure A4).
A2.2 Variable Normal Stress Cases with Constant Confining Pressure
 This part includes data of Pl-Px-3%Qz and Pl-Px-5%Bt mixtures and plagioclase cases at 100°C and 300°C. The data quality is better in these experiments because the control of a constant confining pressure is much easier than the constant normal stress control.
A2.2.1 Pl-Px Mixture with 5% Biotite Around 500°C
 Small bumps can also be seen in this case, especially after upward step change of loading rate to 1.22 µm/s. The fitting is quite satisfactory (top of Figure A5).
A2.2.2. Pl-Px Mixture with 3% Quartz Above 400°C
 The fittings for the three cases are quite satisfactory (Figure A5).
A2.2.3. Plagioclase Cases
 For plw09 (102°C), though a “best fit” was obtained for the fifth step (downward), satisfactory result was not found for this case. The fourth step, an upward step, was fitted instead (Figure A6). While the peak was fitted well, the best fit model deviates from the data in the final portion of the descending part.
 For plw11 (302°C; third step), this fitting is satisfactory, which has a much larger a value than that around 100°C. The dc value of 71 µm is also much larger than that at ~100°C (5 µm).
 For plw12 (405°C; third step), this fitting is satisfactory, which shows a smaller dc value than the case at ~300°C (plw11).
A3. Results for Oscillatory Slip Cases
 This series includes the oscillatory slip cases of plagioclase with constant confining pressure control, where the oscillatory slips were preceded by stable sliding. In addition to the oscillatory slips above 400°C, the relatively weak oscillatory slips around 200°C were also analyzed (Figure A7).
 plw13 (607°C), third step: This case is closest of all to the critical point, i.e., k = kcr(V0). Hence, in this case, k = kcr(V0) is used as a constraint which is considered to correspond to an upper bound of b-a. The fitting is basically satisfactory. The dc value is found to be 3 µm in this case, much smaller than in the stable sliding cases above 300°C.
 plw14 (503°C), first step: As the first step triggered oscillatory slips, the sliding behavior is evolving quickly with displacement, making it difficult to fit the whole response. To get estimates of a and dc values, the initial part of this step was fitted. The dc value obtained is similar to that at ~600°C (plw13).
 plw12 (405°C), fourth step: This case is still subcritical in the whole process, but the sliding behavior is evolving (evolving constitutive parameters) with displacement. Hence, it was difficult to fit the whole response; thus, only the initial part was fitted, and the result turned out to be relatively rough. A significant difference to the preceded stable sliding case at the third step is a much smaller dc value (10 µm) compared to that of the stable sliding case (47 µm).
 plw10 (201°C), second step: This step is the only access to a steady state before the oscillatory slips at this temperature. Due to the small amplitudes of oscillation (i.e., a low signal to noise ratio), the fitting turned out to be the most difficult of all even with the initial part of the step response. However, the result shows a very small dc value (3 µm), similar to the sustaining oscillatory slips above 500°C.
Evolution Effect Associated With Pressure Solution Creep
 In nominally dry conditions, both the direct effect and the evolution effect for two sliding faces may be regarded as the manifestations of time-dependent behavior at the contacts. As separated by Dieterich and Kilgore  with approximations, the direct effect mainly reflects the viscous behavior of the shearing deformation of the contacts, whereas the evolution effect approximately reflects the growing contact area due to creep of the contacts in their normal direction.
 Under the hydrothermal conditions in our experiments, enhanced dissolution at contacts commonly known as intergranular pressure solution (IPS) may occur as seen in quartz granular aggregates [e.g., Elias and Hajash, 1992; Niemeijer et al., 2002], which generally leads to compaction (decreasing porosity) and increase of the contact area [Yasuhara et al., 2003].
 To examine the effect of IPS on the evolution effect, here we consider simply packed spherical grains. The IPS process includes three serial subprocesses, namely, dissolution within stressed grain contacts, the diffusion of matter along the grain contacts, and precipitation on the solid surfaces away from the contacts. Under steady state conditions, the slowest of these subprocesses controls the strain rate [e.g., Spiers et al., 2004]. Following Raj  and the modification by Revil , the reaction-controlled pressure solution creep can be described by the following equation for low deviatoric stress (uniaxial stress for example):
where Vm is the molar volume of the solid, is the strain rate, σeff is the true effective normal stress, k+ is the dissolution rate constant which is known to be temperature dependent in Arrhenius form, d is the grain size, T is the temperature, and R is the gas constant.
 While the strain rate controlled by precipitation has a form similar to (B1) [e.g., Niemeijer et al., 2002], the diffusion process along the contacts is described by a different viscous equation [Rutter, 1976]:
where C is the solubility of the solid surface in the pore water solution, h is the effective thickness of diffusion pathways at the grain contacts, D is the solute diffusivity along the diffusion pathways at the grain contacts, and ρg is the density of the solid. Estimates of strain rates by these two types of linear laws employing experimental parameters for quartz at 400°C indicate that strain rate of the diffusion process is several orders higher than the dissolution/precipitation processes with grain size of 3 µm; thus, IPS is basically controlled by the dissolution/precipitation processes. Niemeijer et al.  demonstrated that their IPS data of quartz for the temperature range of 400°C–600°C fitted well to the strain rate equation for dissolution process for porosities higher than 15%, indicating that the controlling mechanism of IPS is dissolution of quartz.
 Similar linear equations are derived and used to relate volumetric strain rate (i.e., compaction rate) with effective stress [Raj, 1982; Revil, 1999; Niemeijer et al., 2002; Yasuhara et al., 2003; Spiers et al., 2004]. The latter equations often include a function of porosity f (φ) which reflects the effect of geometric change during the compaction process [e.g., Revil, 1999; Spiers et al., 2004].
 While the linear equation (B1) is an approximation for small effective stress, the strain rate is better described by an exponential function of effective stress for large stress [Dewers and Hajash, 1995; Niemeijer et al., 2002]. According to these data, the dimensionless term Vmσeff/RT can be replaced with an exponential function [Niemeijer et al., 2002]:
where the use of σtr ≡ σeff instead of σeff is to emphasize true effective normal stress at the contacts.
 Following the basic derivation by Raj  and the general expression by Niemeijer et al. , the deviatoric creep equations can be written in a generalized exponential form:
where is the strain rate, with subscript x (s, p, d) indicating dissolution (s), precipitation (p), and boundary diffusion (d) mechanisms; Ax is a constant; Zx(T) is a temperature-dependent constant corresponding to each of the three mechanisms; Vm is the molar volume of solid; d is the diameter of the grain; and q is an exponent related to the specific mechanism, 3 for boundary diffusion control and 1 for the other two mechanisms. Note that this equation does not include any term of strain, different from the equations for compaction creep, which depends on volume strain (or porosity).
 While (B1) and (B2) are linear approximations, the replacement of the linear dimensionless stress term Vmσeff/RT by the exponential function restores its original form in the formulation of the fundamental equations [Niemeijer et al., 2002].
 For given temperature and geometry conditions, the creep rate equation can be written in a brief form as follows:
with Cx = AxZx(T).
 Based on the nonlinear equation ((B3)′) for steady state creep, here we derive the area increase with time due to pressure solution.
 For our experiments on frictional sliding, the true effective normal stress is high; thus, the −1 term in ((B3)′) can be omitted for a reasonable approximation. Thus,
 To obtain a geometrical relation between the normal strain and area change due to dissolution, consider a truncated sphere to represent the shape change after initial elastoplastic deformation at the contact (Figure B1). For the relatively short contact time during frictional sliding, the change in the contact area is small; thus, a linear approximation is possible. A cone-shaped body with its surface tangent to the sphere is used to approximate the spherical body for limited area change (Figure B1); thus,
where Acs is the contact area, with subscript cs denoting contact area growth by pressure solution; dz0 is the initial length of the truncated grain diameter; is the average diameter of the increased contact annulus due to dissolution; and tanα is the slope of the cone at . With initial diameter dc0 of the contact area and initial contact area Acs0, then , and , with ξ as a coefficient on the order of unity. Then we have
 Considering a small dissolution strain, from (B6), we have
where O(ε2) is a second-order term which can be omitted for approximation.
 With normal load w at the contact and σtr = w/Acs, and for a short time scale during frictional contact, d ≈ d0, then from (B4) and (B6), we have
 With the initial effective normal stress σtr0 = w/Acs0, and S2 = RT/Vm, then
 Integration over the two sides, we have
which is equivalent to a logarithmic form:
 Getting back to (B6′), we have the time-dependent area growth equation in a form similar to the contact creep shown by Bréchet and Estrin , as follows:
 With area growth function of (B8), the coefficient of friction can be written as
where Acs and Acs0 are contact area and its initial value, respectively, and tτ is a time constant. For t > > tτ (note that ) as in the case accessible in experiments, the b value by IPS (bIPS) is
 With this result, the evolution effect by IPS can be readily compared with the evolution effect of friction obtained in experiments.
 The true effective normal stress at the contacts is an important parameter for the b value estimation. As demonstrated by contact creep of transparent acrylic and soda-lime glass during stationary contact [Dieterich and Kilgore, 1994] and similar behavior of other materials including rocks (see review by Baumberger and Caroli ), the true normal stress applied at the contacts is comparable with the bulk yield stresses of the contacting materials under a nominally dry condition, which can be inferred from microhardness data when the dominant deformation mechanism of the indentation is plastic flow. In some cases, however, the deformation mechanism at the contacts may be cataclasis rather than plastic flow, and the true stress at the contacts is probably controlled by stress required for fracture. It should be noted here that there are a number of concurrent deformation mechanisms operating simultaneously during the whole deformation process. From the highly reduced particle size as seen the in microstructures of our deformed samples, it is evident that cracking and fracture is one of the major mechanisms at least; hence, friction should be related to fracture strength of the contacts. Similarly, friction should also be related to yield stress if plasticity occurs under the P-T conditions, although this is not sure for plagioclase. Theoretically, pressure solution is a process that can occur at any significant stress level (note that this is quite different from the theory of fracture and plasticity), and the driving force is the chemical potential that related to the stress in whatever level. In other words, it occurs both at high stress limited by fracture strength or yield stress if the stress is available from the specific process and at any actual low stress. For frictional sliding, the process needs to shear the contacts with high stress, which may lead to fracture of the grains and/or plastic flow at the contacts, depending on the deformation mechanism map of the material. Of course, at localities with much lower stress (such locations may appear and disappear dynamically during the granular flow), pressure solution occurs as well. What is important here is that only the pressure solution process associated with the high stress is related to constitutive relation of frictional sliding. Accordingly, though pressure solution processes can occur in many places with different stress levels, only the high-stress cases are related to the constitutive relation of friction. This is why the fracture or yield stresses can be used to calculate the evolution effect related to the pressure solution process.
 Using a relation by Johnson , Dorner and Stöckhert  inferred yield stresses for diopside from their microhardness data tested at 400°C–600°C, giving yield stresses from 3.44 to 3.79 GPa (Table B1), with a decreasing trend with increasing temperature.
Table B1. Initial True Stresses σtr0 at the Contacts and Derived bIPS Valuesa
Vm is molar volume(m3 mol−1), and R is molar gas constant; bIPS = 0.74S2/σtr0.
Yield stresses inferred from microhardness data.
Initial true stresses inferred from elastic coefficients cii (i = 4, 5, 6) for plagioclase an56 when strain is defined as tensor [Bass, 1995], which roughly reflect the relations between shear stresses and shear strains in single crystals. σtr0 is estimated by 0.2cii/2 (i = 4, 5, 6) following Rice et al. .
Molar volume for plagioclase is calculated with molecular formula of albite.
 Microhardness tests on feldspar single crystals by Huang et al.  show that the indentations are accommodated by cataclasis rather than plastic deformation for temperatures <600°C, indicating that the yield stress is higher than the resistance to fracture. In this case, the true contact stress of plagioclase grains is restrained by fracture strength. As suggested by the microhardness data, fracture strength of the crystalline grains at the contacts of small scale is very high and has a rough correlation with coefficients of elasticity [Rice et al., 2001]. Initial true stress here is estimated by 0.2cii/2 (i = 4, 5, 6) following Rice et al. , where cii (i = 4, 5, 6) are elastic coefficients for plagioclase an56 when strain is defined as tensor [Bass, 1995], which roughly reflect the relations between shear stresses and shear strains in single crystals. As temperature sensitivity of frictional strength was not observed in our experiments, temperature effect on the initial true stress is assumed to be negligible here. The mean fracture strength thus estimated for plagioclase (an56) is 3.05 ± 0.88 GPa (Table B1), but lower and upper bounds are used as a range of rough constraint.
 As the first part, τtr/σtr0 in (B10) reflects the coefficient of friction including the direct rate effect, and it is on the order of ~0.74 (Table B1) as first-order approximation. With these estimations and the calculation of S2 with corresponding temperatures and molar volumes, the bIPS value estimated for pyroxene gouge ranges from 1.44 × 10−2 to 2.05 × 10−2, whereas that for plagioclase gouge ranges from 1.13 × 10−2 to 2.62 × 10−2.
 For the absolute strain rate related to the IPS process, two points should be noted here. First, comparison of compaction creep data of quartz to model prediction using the exponential equation and parameters from Rimstidt and Barnes  made by Niemeijer et al.  and van Noort et al.  shows that actual compaction rate is 1 to 2 orders slower than the model prediction for temperatures of 300°C–600°C. van Noort et al.  also found that the activation energy inferred from their experiments (105 kJ mol−1) is higher than the value obtained from dissolution experiments. These discrepancies, especially that in the absolute compaction rate, may be due to the effects of additional mechanisms, such as subcritical microcracking, plasticity at the contacts, or the effects of grain boundary structure on the rate of pressure solution. Analysis by Van Noort and Spiers  indicates that microscale plasticity at grain boundary islands does slow down pressure solution and can help explain the discrepancies between observed and theoretical pressure solution rates. With the additional effects superimposed to the pressure solution process, equation (B3) may be used as an empirical flow law when the constants are inferred directly from experiments through data fitting, as seen in the work of Van Noort et al. . Second, the granular flow of the gouge in frictional sliding is a different situation where the strain accumulated at the contacts will be canceled out when the contacts are sheared and the old contacts are replaced by the new ones. The rearrangement of particle positions during such relative movement accompanies certain particle rotations. Thus, in this case, the overall strain rate is not necessarily related to the strain rate at the contacts, and it is meaningless to compare the two quantities.
 Bearing in mind the two points mentioned above, the absolute strain rate governed by ((B3)) is examined below for completeness of the topic discussion.
 Because systematic data on dissolution of feldspars and pyroxenes in pure water are not available in the literature [Chen and Brantley, 1997], an estimate of dissolution rate constant for feldspar of around 400°C is made based on dissolution data by Schloemer , who measured dissolution rates of quartz and orthoclase up to the solubility point at 375°C and fluid pressure of 22 MPa, from which a ratio of 303.9 between molar dissolution rates of quartz and orthoclase is obtained. This leads to an estimate of dissolution rate constant of 7.44 × 10−9 mol m−2 s−1 for orthoclase, which is taken here as a rough estimate of k+ value for plagioclase at 400°C. Using this parameter with d = 3 µm and taking equation (B3) as an empirical model, the coefficient As is on the order of 10−20 to 10−21 to reproduce a strain rate similar to that in our experiments, but the actual strain rate at the contact is not known here. Related to the small As coefficient, an estimate of tτ in ((B7)) using these parameters also shows a very small value on the order of 7 × 10−20 s. Such an estimate of tτ seems not to be sensible, but if this is qualitatively right, it means that the pressure solution process tends to be quick rather than slow.
 We owe Caiyun Lan and Zhen Lu for their assistance in performing some supplementary experiments. We also thank Wenming Yao for his technical support in maintaining and renovating the testing machine. This work has also benefited through discussions with a number of colleagues, among whom Chris Spiers, Teng-fong Wong, Toshi Shimamoto, Bunichiro Shibazaki, and Yajing Liu are especially acknowledged here. We thank Lei Zhang for his help in preparing most of the input data for numerical fitting. Thanks are also given to Andre Niemeijer and Nick Beeler for the first-round reviews and their constructive criticisms and to David Lockner and another anonymous reviewer for their recommendations and further comments. This work was funded by NSFC under grant 40574080 and was also supported in part by the State Key Laboratory of Earthquake Dynamics, Institute of Geology, China Earthquake Administration, under grant LED2012A01.