## Frictional Heating of Rocks

The observation that friction can raise rocks to incandescence is an ancient one: People have been starting fires with flint and steel for centuries. However, it took the science of the Industrial Age to properly relate mechanical work, heat and velocity. Amontons and Coulomb first formulated the law of solid friction (Palmer 1949; Scholz 1990). They discovered that the resistance to sliding of one block over another is proportional to the normal load and independent of the area of contact. In modern terms, the basic law of friction states that the shear stress x needed to move one block over another is proportional to the normal stress uN multiplied by the coefficient of friction p,:

For most rocks, p, is close to 0.85 at pressures below about 200 MPa. At higher pressure, a slightly more complicated expression applies; x = 50 MPa + 0.6ctn. This relation is often called "Byerlee's law" and holds over a wide range of pressures, temperatures and compositions (Byerlee 1978), so long as the rock is actually solid. At high temperatures this coefficient declines as the rock softens (Ohnaka 1995; Stesky et al. 1974).

When slip occurs at a steady rate, the work done is dissipated as heat. Following directly from Newton's deduction that work equals force times distance, the power W dissipated per unit area on a sliding fault is given by

where 5 is the differential rate of slip (= relative velocity) between the two blocks.

The total heat dissipated during sliding, as 8 increases from 0 to its final value 8/, is t/

For a constant shear stress x, this is just W = 8 x/. Fig. 4. Typical stress-strain curve for brittle rock. Intact rock (solid curve) shows a well-defined stress maximum, whereas fractured rock (dashed) does not. The peak stress for most rocks occurs at a strain of a few percent, and depends on ambient pressure and temperature, as well as rock composition (Jaeger and Cook 1969).

### Strain

Fig. 4. Typical stress-strain curve for brittle rock. Intact rock (solid curve) shows a well-defined stress maximum, whereas fractured rock (dashed) does not. The peak stress for most rocks occurs at a strain of a few percent, and depends on ambient pressure and temperature, as well as rock composition (Jaeger and Cook 1969).

There has been some confusion in the geological literature about the relative importance of displacement or displacement rate in determining the onset of frictional melting. Certainly, slip at too slow a rate allows time for the heat generated to diffuse away from the shear zone and prevents the attainment of high temperatures. A flint struck too slowly yields nothing but cold flint fragments. However, if the displacement is too small, not enough energy is dissipated to heat the rock. Both displacement and velocity must be within the right range for melting to occur (Jeffreys 1942; Sibson 1975, 1977).

McKenzie and Brune (1972) proposed a simple model of shear heating (a model that I will call the "physicist's model" because of its unrealistic idealizations) to examine the ability of earthquakes to produce melt. Although the original model featured a sophisticated mathematical analysis, such complexity is not necessary to illustrate the basic tradeoffs that lead to melting. The model's chief idealization is that sliding begins on an infinitesimally thin shear zone. As heat is dissipated it flows laterally away from the heated fault plane, obeying Fourier's law (Turcotte and Schubert 1982). It spreads out over a region of width y = , where t is the time after heating began and k is the thermal diffusivity, equal to about 10-6 m2/sec for most rocks. If we assume that slip occurs at a constant rate

S and that the energy dissipated fills the region of width y uniformly (this is not actually true—this is where the sophisticated mathematics comes in, but getting this exactly right is not essential to display the main structure of the solution), then the energy density (energy/unit volume) E in the slip zone is:

This energy appears as heat. The energy density necessary to achieve melting is given approximately as E = pCP (TS - T0), where p is the rock density, CP is the heat capacity at constant pressure, TS is the solidus temperature and T0 is the initial temperature of the ambient rock. Inserting this definition into equation (4) and noting that 8 = St, the amount of slip SM necessary for the onset of melting is:

Beware that this equation implicitly assumes that the frictional stress x is independent of temperature! If the frictional stress decreases as the temperature rises, the amount of slip will increase.

It is clear that the amount of slip needed for melting is inversely proportional to the slip velocity, so both factors play important roles. Inserting typical values of the parameters in equation (5) for rock, p = 3000 kg/m3, Cp = 1000 J/kg and (Ts - To) = 1000 K, and further suppose that x is equal to the lithostatic pressure of 3 km of rock, x = 108 Pa, we find that, at a slip rate S of 1 m/sec, melting begins after a total slip of only 1 mm. This occurs just 1 millisecond after slip begins. However, note that at this time the slip zone is only y = 30 micrometers wide—much smaller than the grain size in almost any rock. This is why I call the model an "unrealistic idealization".

Whatever the shortcomings of this simple model, it does indicate that thin films of melt should form readily during rapid slip of rocks at crustal depths, and it illustrates the basic tradeoffs that must be met for melting to occur at all. Most of the rest of the discussion in this paper is a refinement of the concepts embodied in this model.