We shall now integrate the above mechanical, textural and chemical data to elucidate the physical processes controlling mechanical property of fault during frictional melting. A physical model of frictional melting is proposed at the end that will enable one to model the mechanical properties of faults during frictional melting.
4.1. First Slip Weakening and Subsequent Strengthening on Frictional Melting
 The present high-velocity friction experiments on Indian gabbro clearly brought about two stages of potentially unstable, slip-weakening behavior of a fault. Continued slip after the initial static friction causes slip weakening down to the frictional coefficient, μ, of 0.3 to 0.4 over several decimeters of displacement (first slip weakening; Figure 2b). This strength reduction and slip-weakening distance are much larger than those measured in conventional friction experiments with slow slip rates and limited displacements [e.g., Scholz, 1990; Ohnaka, 1992]. Indeed, the slip weakening distance during the first weakening is on the same order as that determined seismologically [Ide and Takeo, 1997; Mikumo et al., 2003; Fukuyama et al., 2003]. The first slip weakening is most likely due to the thermal weakening induced by flash heating at asperity contacts [Rice, 1999]. However, melt patches are already formed on a fault immediately after the first slip weakening (Figures 5a, 5f, and 6a) and temperature along the fault must have exceeded at least melting temperature of biotite (∼650°C). Melting must have gone further than flash heating at asperity tips to form melt patches already at this stage. Our preliminary data clearly shows the weakening distance during the first slip weakening decreases with an increase in the heat production rate, but we shall report on this after conducting more systematic studies.
 Tsutsumi and Shimamoto [1996, 1997a, 1997b] have shown experimentally that the onset of frictional melting abruptly increases friction or shear resistance along a fault. The same effect can be seen even more clearly on the frictional coefficient versus displacement curve (point c to d in Figure 2a). Textural observation and mechanical data have revealed that melt patches, with more or less similar geometry, first increases in number and causes rapid strengthening of fault (Figure 8). Melt patches then begin to coalesce and to grow in thickness, and a continuous molten layer forms at around the second peak friction.
 Intuitively, one may think that the onset of melting causes marked weakening of a fault, but this is opposite to what is observed. As an example, take a molten layer just after the second peak friction for which an average melt thickness is 33 μm, the average shear strain rate of molten layer is 2.6 × 104 s−1 for the slip rate of 0.85 m s−1, and the apparent viscosity of molten layer is 45 Pa s (Figure 10). This viscosity is rather low as typical of melt with basaltic composition, but the shear resistance of melt is not small owing to extremely large shear strain rate. The shear stress in this case is determined as 1.16 MPa by multiplying the apparent viscosity with the average shear strain rate, and this yields frictional coefficient of 0.83. Thus melt formation is more or less like putting highly viscous material into a fault zone whose dragging effect causes the strengthening. The marked fault strengthening also must have braking effect on seismic fault motion. Rare occurrence of natural pseudotachylytes might imply that the strength increase upon frictional melting acts as a brake to further fault motion, thereby suppressing pseudotachylyte formation.
 The logistic curve, equation (3), describes the overall melt growth very well (Figure 7a) and this indicates that the initial increase in average thickness of molten layer is exponential with respect to the displacement after the first peak friction. However, this does not mean that molten layer grows exponentially from a very, very thin molten layer. The initial thicknesses of melt patches are 8 to 10 μm (Figure 8b; see also Figure 6a) although the average initial thickness of melt patches is 2 μm or even less (Figure 9 and Table 2). Thus the initial increase in the average thickness of melt patches (open symbols in Figure 9a) is primarily due to the increase in the number of melt patches.
 Thickness of melt patches can dramatically affect the shear resistance of a fault. This can be illustrated by a simple two-dimensional example in Figure 12. Thickness and lateral width of a melt patch with viscosity η are 2H and L, respectively, in Figure 12a, whereas the same amount of melt is thinned to a half of this thickness and elongated twice as wide as this length in Figure 12b. If the slip rate is the same for both cases, the tangential shearing force required to shear the melt patch in Figure 12b is 4 times as large as that in Figure 12a. Initial melt patches occupy about 10% of fault area (Figure 8a). If the same amount of melt is spread over the fault surface, the thickness reduces to about one tenth, and the shear resistance will be about 100 times as large. Of course, such a large resistance is not observed. Shear resistance of infinitely thin molten layer will become infinitely large because shear strain rate (slip rate/melt thickness) becomes infinitely large. However, such an unlikely case is avoided by the formation of melt patches of finite thickness. An increase in the number of melt patches causes strengthening because a fault has to overcome shearing resistance of more viscous melt patches. We do not know at present what determines the initial thicknesses of melt patches, although roughness of fault surface and possibly grain size will play important roles.
Figure 12. Schematic diagrams showing (a) a thick melt patch and (b) a thin melt patch with the same area. Tangential shearing forces (shear stress multiplied by the area of molten layer) are given for both cases assuming unit length perpendicular to the figure.
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 From present data we do not know exactly how melt patches behaved under extreme shear. The thickness of melt patches is 10 to 20 μm (Figure 8b) and the amount of fault displacement during the melt patch formation causing fault strengthening is on the order of 10 m (Figure 2a), so the shear strain of early melt patches reaches 105 to 106 if they deform uniformly by simple shear. Thus intuitively, one may think that melt patches will be sheared to cover the entire fault surface immediately after their formation. In fact, very, very thin melt patches formed locally, but they do not appear to cover the entire fault surface. On the other hand, wetting angle of molten basalt to a few minerals is measured at 20–50° [Khitarov et al., 1979]. Their basalt has similar chemical compositions to those of our gabbro (compare Table 1 with Khitarov et al.'s Table 1), and hence it is unlikely that melt patches wet fault surfaces in our experiments. Thus another possibility is that melt patches rolled between host rocks more or less maintaining their shapes, just like a mercury drop rolls between two plates moving past each other. Melt patches for which we collected geometrical data in Figure 8b might have behaved like that, but we do not have clear image right now on how melt patches behaved under extreme shear. Systematic work is needed in the future on this.
4.2. Mechanisms for the Second Slip Weakening
 The second slip weakening occurs after host rocks are separated by a continuous molten layer, so that the shear resistance of a fault should be determined by the gross viscosity and shear strain rate of the molten layer. Both apparent viscosity of a molten layer and melt viscosity (open circles and solid diamonds, respectively, in Figure 10a) increase sharply at the initial stage of the weakening. Despite this increase in viscosities a fault loses its strength because the molten zone widens and the average shear strain rate sharply drops with increasing displacement (Figure 9b). In other words, the effect of about fivefold reduction in shear strain rate is greater than the effect of about threefold increase in apparent viscosity (compare Figures 9b and 10a). Thus the growth of molten layer is the primary cause of the second slip weakening during frictional melting.
 Another important aspect of frictional melting is melt loss from its generation zone into air in experiments (melt splashes from a fault) and into fractures in nature. Melt loss from a simulated fault is unclear soon after the second peak friction is exceeded, but melt splashes become visible under naked eyes as the steady state friction is approached (around point e in Figure 2a). The rate of increase in molten layer thickness declines with increasing displacement during the second slip weakening (Figure 9), so that the melt production should be more or less in balance with melt loss near the steady state friction.
 Jaeger [1969, pp. 140–143] solves a simple two-dimensional squeezing problem of a viscous material under a pressure between two plates and shows that the squeezing rate along parallel plates is proportional to the cube of the thickness-width ratio of the material. Although our specimen geometry is different from Jaeger's model, his analysis indicates that melt cannot move much along a fault when a molten layer is very thin, i.e., when marked slip weakening occurs and that melt migration and melt loss become progressively important as a molten layer thickens. Whereas reduction in shear stress and resulting drop in heat production rate should reduce the rate of melting, and a steady state will be attained when the melt production rate is balanced with the rate of melt loss. Melt escaping from the molten zone carries heat away from the molten zone (heat generation zone) and the viscous shear heating can no longer be used all for melt production after the melt loss becomes significant. A complete analysis of frictional melting process should incorporate not only the thickness change of the molten zone, but also the renewed energy budget associated with the melt loss.
4.3. Changing Melt Viscosity: Effect of Temperature or Melt Chemistry?
 Melt viscosity notably changes during frictional melting (Figure 10), and we have long thought that this is caused by temperature change in the molten layer, i.e., change in melt temperature due to overshooting of melting temperature as discussed in the next subsection. However, the problem is more complicated because melt chemistry changes during frictional melting (Figure 7) and this can affect melt viscosity as well.
 Change in melt chemistry in Figure 7 does appear to reflect selective melting. Percentage of K2O is very high over the whole displacement because minerals with low melting temperature such as biotite, hornblende melted selectively. This is consistent with the absence of biotite and hornblende in the fragments contained in the molten zone. Increasing contents of CaO, Fe2O3, MgO and TiO2 toward the second peak friction (shown as vertical stripe in Figure 7) suggest that clinopyroxene and ilmenite began to fuse toward the second peak friction. This is supported by partially molten clinopyroxene and ilmenite grains with rounded shape recognized after the second peak friction. Percentages of Al2O3 and Na2O first drop and then increase after the second peak friction, which implies that plagioclase (Ab38An62) starts melting early around the second peak friction.
 Shaw  proposed a simple, but useful empirical method to predict melt viscosity, including its temperature dependence, from melt chemistry. We calculated melt viscosity at various temperatures for host rock or protolith composition (Table 1) and for melt compositions of 14 specimens with different displacement (Table 3). Calculated viscosities at three temperatures are plotted against displacement in Figure 13a which indicates that viscosity can vary by more than 1 order of magnitude with displacement, reflecting changing chemical composition of melt. The estimated viscosity is high initially and tends to decrease toward the second peak friction, and this is mainly due to the change in SiO2 and Al2O3 contents. Chemical composition and predicted viscosity becomes uniform toward the steady state friction (Figures 7 and 13a).
Figure 13. Melt viscosity at different temperatures, as estimated from chemical compositions (Tables 3) using Shaw's  method, plotted (a) against the post-first peak displacement and (b) against the post-second peak displacement. Estimated viscosity for host rock (protolith) is also given at the left end of Figure 13a. Figure 13a also exhibits apparent viscosity, estimated neglecting the effects of bubbles and clasts, for melt patches (open circles; solid-solid frictional coefficient is assumed to be 0.4) and for molten layers (solid circles); see Figures 8c and 10a for original results. Maximum and minimum viscosities for molten layers, collected for bubbles and clasts, are plotted in solid and open diamonds, respectively, in Figure 13b; see Figure 10b for the original results.
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 To test whether viscosity changes of melt patches in Figure 8c and those of molten layer in Figure 10 are due to changing melt temperature or due to a change in melt chemistry, we have plotted apparent viscosity of melt patches, assuming solid-solid frictional coefficient is 0.4, in open circles and apparent viscosity of molten layers in solid circles in Figure 13a. Even taking into account the possible ranges of viscosity of melt patches (open circles and open diamonds in Figure 8c), the apparent viscosity notably increases as melt patches develop into molten layers and then it remains nearly constant toward the steady state friction. This initial increase in viscosity is opposite to the trend estimated from melt chemistry and cannot be explained by changing melt chemistry. We believe that the initial low viscosity is due to the high temperature of melt patches created by concentrated shear heating in thin melt patches.
 Figure 13b compares melt viscosity corrected for bubbles and clasts (Figure 10b) and viscosity estimated from melt chemistry for 6 specimens exceeding the second peak friction. The melt viscosity increases by about a factor of 2 soon after the second peak friction is exceeded (the first four datum points in Figure 13b). Those points fall roughly along equitemperature lines of around 1100°C, so that the viscosity change could be due to the changes in melt composition, rather than the temperature change in the molten layer. However, this change in viscosity is much smaller than the overall changes shown in Figure 13a. Thus the present data suggest that changes in melt viscosity depends primarily on changing temperature of melt patches and molten layers, but that melt chemistry affects local fluctuation of its viscosity.
 We have measured melt temperature of 1140°C at steady state regime using K-type thermocouples and a radiation thermometer, following the same procedures as Tsutsumi and Shimamoto [1997b]. This temperature value is on the same order as that estimated from measured viscosity and Shaw's  method (Figure 13).
4.4. Frictional Melting as a Stefan Problem
 On the basis of mechanical and observational data, we now propose a physical model of frictional melting. To state our conclusion first, frictional melting processes can be solved as a class of problems called “Stefan problem” [e.g., Alexiades and Solomon, 1993; Hill, 1987], handling moving boundaries due to melting, freezing and other phase changes [Fialko, 1999; Hirose, 2002]. After host rocks are separated by a continuous molten layer, (1) rock-on-rock friction can no longer be the heat source and viscous shearing of a molten layer becomes the primary heat source after the second peak friction, and (2) viscous shear resistance of the molten layer primarily determines the fault strength during the second slip weakening (Figure 14a). Unlike most melting problems treated in engineering, the heat source is distributed within the molten layer itself in the present case. Accumulation of clasts and bubbles in the molten layer is a complication factor.
Figure 14. (a) A schematic diagram showing physical processes involved with frictional melting, and (b) sketches for the melt escape from fault zones (left) for frictional melting experiments and (right)for frictional melting along a natural fault forming pseudotachylyte.
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 Melting absorbs heat due to latent heat and hence melting surfaces are big heat sink in the system (Figure 14a). If melting occurs very slowly, excess heat is taken away by latent heat and temperature in the system becomes homogeneous and stays at the melting temperature (equilibrium temperature). However, for rapid phenomena such as seismogenic fault motion, temperature distribution (viscosity distribution) and melting rate are determined dynamically due to the rates of heat generation, heat transfer and melting. The rate of melting or the growth rate of molten layer is determined primarily by the rate of heat transfer to the melting surfaces. If the rate of heat transfer to the melting surface is not rapid enough to take all excess heat away, overshooting of melting temperature within a molten layer will occur. Thus it is not surprising if the change in melt viscosity such as that in Figure 13 is caused by changing melt temperature during frictional melting.
 An analysis of heat transfer into melting surfaces and into host rocks from distributed heat sources yields temperature distribution in the system. Temperature distribution within a molten layer in turn determines viscosity distribution in the molten layer, and this viscosity distribution determines the mode of shearing deformation and hence the mode of distributed heat source. Melting does not require nucleation, so that overshooting of melting temperature rarely occurs at melting surfaces [e.g., Alexiades and Solomon, 1993]. It is thus often assumed that temperature stays at melting temperature at melting surfaces (moving boundaries) to solve the coupled problem (often called Stefan condition). Those coupled problem can be solved as a one-dimensional problem. However, when one attempt to compare theoretical prediction with experimental results, it should be kept in mind that frictional melting in laboratory specimens cannot be treated as a one-dimensional problem because heat transfer occurs in three dimension for the specimens (more rapid cooling than predicted by one-dimensional analyses). This effect causes an increase in the slip weakening parameter, dc.
 In addition, melt loss has to be incorporated in the analysis and it can also take heat away from melting surface (Figure 14b). Among the coupled processes, melt loss may be hardest to incorporate since melt injection into host rock itself is a complex problem accompanied by cooling and solidification [see Newall and Rast, 1970]. Melt loss in balance with melt production rate should primarily determine a stable thickness of molten layer and the level of steady state friction during frictional melting. Cracks in host rocks may not open under extremely large pressures, as expected at great depths where deep focus earthquakes occur, melt may not be lost from its generation surface causing continued loss of strength. In practice, the effects of melt loss can be examined as two extreme cases; one is “free melt loss” as in our experiments and the other is “no melt loss”. Real cases will fall in between the two.
 Significance of melt loss can be recognized in considering the data on the shear stress plotted against normal stress in Figure 4. At a glance, the relationship between the two appears to be a familiar linear relationship for friction, like Byerlee's friction law at high pressures [Byerlee, 1978]. However, the data in Figure 4 are not for solid-solid friction but for a molten layer separating host rocks. The range of normal stress is far too small to expect much effect on viscosity due to the pressure change. Then what is causing the pressure dependence of the shear resistance of the molten layer? It is likely that melt is squeezed more from a fault zone under a higher normal stress, resulting in a thinner molten layer. Then shear strain rate would become higher at a higher normal stress to increase the shear resistance. However, real processes would be more complex since this increase in the shear resistance may cause an increase in melt temperature and a drop in melt viscosity. Our present data are not sufficient to discuss the detailed underlying processes responsible for the normal stress dependence in Figure 4 beyond this rough sketch, and we plan to do a systematic work on this in the future. The effect of normal stress on melt patches formation is also significant. Our experimental results have clearly shown that the friction or shear resistance markedly increases with the formation of melt patches. Whether this strength barrier is indeed as large as that shown in Figure 2 even at great depths with much greater heat production rates must be reexamined based on detailed analyses of the normal stress dependence of frictional melting processes.
 Another important problem with respect to earthquake generation is the prediction of slip-weakening parameter, dc. This parameter is almost 10 m for the case in Figure 2a, but dc approaches seismically determined values (typically several decimeters to about 1 m [see Ide and Takeo, 1997; Mikumo et al., 2003; Fukuyama et al., 2003] with increasing slip rates (Figure 3). Higher heat production rate at a higher slip rate must have increased the melt production rate to shorten dc. Thus a big goal of the analyses of the above Stefan problem is to predict dc and the amount of strength reduction that are both critical to the stability analyses of fault motion [e.g., Dieterich, 1978]. Frictional melting is a highly nonlinear phenomenon for which slip history itself determines fault properties. We believe that physical processes during frictional melting are understood reasonably well, and we now throw the ball into the hands of theoreticians and modelers.