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few Angstroms. Solid surfaces in general form adsorbed films, such as molecules of water vapor and/or oxygen. The thickness of these films may be of the order of a few molecularly thick layers (4). The presence of adsorbed films between the mating solid surfaces interrupts the surface’s atom-to-atom interaction at the contacting portions. This is the reason why frictional strength is in general much lower than fracture strength.

However, adhesion or cohesion occurs at the contact areas where the adsorbed films have been broken up during the normal load application or during sliding. In this case, the shearing strength of these adhesive (or cohesive) junctions is the prime cause of frictional resistance (5). For rocks containing typical, hard silicate minerals such as quartz, feldspar, and pyroxene, the penetration of hard asperities into the films on the opposing surface easily occurs during the normal load application or during sliding, and the asperities at the contacting portions fail by brittle fracture, rather than by plastic shear (6). In this case, frictional strength is primarily due to brittle fracture of these asperities.

Let the sum of the solid-solid contact (film-broken) areas be denoted by A1, which is a fraction of the sum of the whole junction areas Ar, and the rest of the junction (solid-film-solid contact) areas by A2=ArA1. When two surfaces are pressed together by a normal load N, the total frictional force F is


where μ1 and μ2 are the frictional coefficients for the solid-solid contact and solid-film-solid contact portions, respectively (7). Average frictional coefficient μ between the surfaces is given by


The frictional coefficient μ1 represents the shearing strength of the solid material and μ2 represents the shearing strength of the adsorbed or intervening films. It has been found that μ1 is much greater than unity, but μ2 is less than unity; for instance, it has been estimated for Solenhofen limestone that μ1=4–18, and μ2=0.3 (7).

We have from Eq. 2 that μ=μ2 if A1=0, and that μ=μ1 if A2=0. When A1=0, μ has a minimum value, so that frictional strength is very low in this case. In contrast, at a higher normal load, larger deformation of the asperities that are in contact causes more asperities to get in contact, which results in larger Ar and A1. At a sufficiently high normal load at elevated temperature, the entire area Aa of fault surface may get into real, cohesive contact (A1=Ar=Aa), and in this case the shear frictional resistance comes equal to (but never exceeds) the shear strength of intact rock. In other words, the shear strength of intact rock can be regarded as the upper limit of frictional strength. This shows that frictional slip instability on a preexisting fault (of a case where A1=0) and shear fracture instability of intact rock are the two extreme cases of shear rupture, and therefore both instabilities should be treated unifyingly and quantitatively in terms of a single constitutive law for shear rupture. Indeed, there is considerable circumferential evidence that earthquake rupture instability that occurs in the Earth’s crust is a mixed process between what is called frictional slip and fresh fracture of initially intact rock.

The two extreme phases, that is, (i) stable, quasistatic rupture growth, and (ii) unstable, dynamic fast-speed rupture propagation, should also be treated unifyingly and quantitatively by the single constitutive law, because both phases are part of the rupture process. It has been demonstrated that unstable, dynamic rupture processes are neither time nor rate dependent (8) and that strong motion source parameters such as slip acceleration for dynamically propagating shear rupture are well explained by a slip-dependent constitutive law in quantitative terms (3), as will be discussed later. These show that the time or rate effect is of secondary significance to the constitutive law for shear rupture and that the constitutive law for shear rupture should primarily be slip dependent.

Shear rupture is essentially an inhomogeneous and nonlinear process where local concentration of deformation in a potential thin zone of imminent rupture results in the bond separation in the zone during slip, forming the macroscopic rupture surfaces and releasing the accumulated stress (and strain energy). In other words, shear rupture is the process where the shear strength eventually degrades to a residual stress level with ongoing slip displacement on the rupturing surfaces (Fig. 1A), and the zone behind the rupture front over which the shear strength degrades to the residual stress level is referred to as the breakdown zone (cf. Fig. 1B). The slip-weakening property in the breakdown zone is intrinsic to shear rupture of any type, any phase, and any size scale. That is to say, even if shear rupture occurs along the preexisting fault of weak zone with a finite thickness, which may be made up of gouge particles, or even if shear fracture is of intact material, essentially common is the slip-weakening property in the breakdown zone. This property is also common, despite vast differences in the size scale. Thus, if there is a constitutive law applicable for shear rupture of any type, any phase, and any size scale, the law should primarily be slip dependent.

FIG. 1. (A) A constitutive relation for shear rupture. (B) A physical model of the breakdown zone near the propagating tip of shear rupture in the brittle regime, derived from the constitutive relation shown in A. τi is the initial shear stress on the verge of slip, τp is the peak shear stress, τr is the residual friction stress, Da is the slip displacement at which the peak stress is attained, Dc is the critical slip displacement, and Xc is the breakdown zone size.

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