Stresses and Strains in an Impact Event

The details of how stresses and strains vary as an impact crater opens and then collapses are highly complex functions of many variables, including details of the flow field, time after impact, rock structure and properties (Collins et al. 2004). Nevertheless, it is possible to give a general overview of the general magnitudes of stress and strain through this process. Impact crater formation is broadly divided into three major stages: Contact and compression, excavation, and collapse (Melosh 1989). Contact and compression, the stage in which the kinetic energy of the projectile is transformed into heat and kinetic energy of the target, is irrelevant to shear zone formation so long as the impact velocity exceeds a few km/s. Most rocks undergo at least partial bulk melting at shock pressures larger than about 50 GPa, a pressure that is exceeded in a basalt-on-basalt impact at a velocity of about 5.2 km/s. Since most impacts in the solar system occur at much higher velocities (the average asteroidal impact velocity on Earth is 17 km/s), I ignore this stage.

The excavation stage includes both the expansion and dissipation of the shock wave and the opening of the crater cavity. Although included in the same stage, these two processes occur at very different rates.

Stresses, strains, and strain rates are highest in the strong compressive shock wave that propagates away from the site of the initial impact. Unfortunately, it is not easy to say precisely how high they become. However, volume strains seldom exceed 2: In a very strong shock the heat added by compression increases the resistance to compression and thus limits the total volume strain achievable by shock (Zel'dovich and Raizer 1967). Strain rates are determined by the rate at which rock engulfed by the shock wave is compressed. This rate is related to the rise time, tr, of the shock wave or, equivalently, to the width of the shock front w divided by the wave speed U. The widths of shock fronts in gases are tiny: w is comparable to the mean free path of an air molecule, about 10-6 m (Zel'dovich and Raizer 1967). However, due to inelastic processes, in metals and single-crystal oxides shock widths are broader, about 10-3 m (Swegle and Grady 1985). Study study of the rise time of shock waves from underground nuclear tests suggests that w in rocks may reach hundreds of meters, but depends inversely on the particle velocity vp behind the shock front (Melosh 2003).

Direct gauge observations of the shock wave from the 62 kT PILEDRIVER nuclear detonation in the granitic Climax Stock at the Nevada Test Site showed that the rise time depends on the particle velocity vp according to the relation:

The particle velocity in these measurements ranged from 100 m/s down to 15 m/s. Because the particle velocity itself declines as approximately the -1.87 power of the range r from the explosion center (Melosh 1989), I estimate that in the impact of a projectile of radius a, the particle velocity declines as

where v, is the impact velocity. Combining equations (7) and (8), the strain rate in the shock is given by e = V^ = ^ w VPtr 0.5VP

3.74

where VP is the compressive wave speed in granite, about 5.1 km/s in this case. Inserting some typical values in this equation, the strain rate £ at a particle velocity of 100 m/s (the highest for which data exists) is about 13 s-1. If this can be extrapolated back to a particle velocity of 3.6 km/s, at which the shock pressure reaches the 50 GPa limit in granite, then £ exceeds 104 s-1. At the other end of the process, using equation (9) to estimate the strain rate at the edge of the transient crater at r ~ 20 a for a 20 km/s impact, <£" would be approximately 2 s-1. Thus, strain rates in the compressive shock wave itself range from more than 10,000 s-1 down to a few s-1.

The corresponding strains can be computed by multiplying the strain rate times the rise time. Strain magnitudes range from 0.7 at the highest velocity to 0.01 near the crater rim.

Stress may be estimated from the second Hugoniot equation, which states that the pressure jump across the shock (P-P0) = p0 vp VP, where p0 is the uncompressed density of the rock. These stresses range from 50 GPa, just below bulk melting of the rock, down to about 1 GPa at the crater rim.

Estimation of the strain rates in the excavation flow is more straightforward. It is well established that, during most of the excavation

Homogeneous Strain Heterogeneous Strain

Fig. 5. Homogeneous versus heterogeneous strain. Although the total strain is the same in both cases, the strain is localized across a number of narrow shear zones in the heterogeneous strain case. The characteristic spacing of fractures is given by length parameter L. Note that a minimum of two independent slip systems is necessary to accommodate arbitrary strain in the two dimensions shown here.

stage, the radius R grows as a power of the time, R(t)~ t2'5 (Melosh 1989). The entire duration of the growth process is given by the formation time of a gravity-dominated crater, Tf D /g, where D is the diameter of the transient crater and g is the acceleration of gravity.

The highest strain rate is near the rim of the growing crater. The overall strain is of the order of R/a and the maximum strain rate at any given time t is determined from the ratio of the derivative of the radius to the radius itself:

R 51

This strain rate applies within a distance R of the growing crater and falls as some power with greater distances (approximately given by the famous Z-model; Maxwell (1977)). Assuming that the power law growth model holds from the edge of the projectile, R = a, to the final transient crater diameter at R ~ 20 a, maximum and minimum strain rates can be derived:

min r rji

5 Tf

The formation time for a crater the size of Vredefort, in which the transient crater diameter D ~ 100 km (Turtle and Pierazzo 1998), is of the order of 100 sec and so the strain rate ranges from about 7 s-1 down to a low of 4 x 10-3 s-1. Strain rates in the excavation flow are thus generally lower than in the shock wave itself, but there is no large gap between the two regimes, except to note that the highest strain rates in both cases occur at the smallest radii. The innermost rocks, close to the impact site, always experience higher strain rates and larger strains than the more distant rocks.

Finally, during the collapse portion of the cratering flow, the overall time scale is the same as the formation time Tf. Because the transient crater starts out with a depth/diameter ratio of about 0.3 and collapses nearly flat (for large craters), the strain is of order 0.3 and so the strain rate is nearly the same as smin in equation (11). Although the magnitude of the strain rate during collapse is thus similar to that during excavation, the sign of the strain rate is reversed. Cratering motions are grossly outward during excavation, but inward during collapse (O'Keefe and Ahrens, 1999). As a result, strains that develop during excavation may be partly, or wholly, reversed during collapse. It is thus important to distinguish the maximum strain that occurs during the overall cratering process from the final strain after all motion ceases.

An estimate of the stresses that develop during the final stages of crater excavation and collapse is not straightforward. On the surface, I could simply accept the results of rock mechanics tests on real rocks (e.g., Fig. 4) and relate the stress directly to the strain. However, it has been known for some time that, if this is done, the observed collapse of large impact craters cannot be reproduced (Melosh and Ivanov 1999). It is clear that some kind of strength degradation must occur in the near vicinity of the crater (less than one crater radius) that drastically reduces the strength of the surrounding rock during collapse. The apparent strength must drop to only a few MPa (Melosh 1977) for most of the duration of crater collapse. I have proposed the idea that strong vibrations created by scattering of the shock wave may briefly reduce the strength of the rock by the process of "acoustic fluidization" (Melosh, 1979). Whether this is the detailed mechanism or not, it is clear that most of the strain during late excavation and collapse must take place at stress levels nearly 100 times lower than those expected from normal rock friction. For this reason, many of the results below must be taken with caution. However, in the final stages of flow when the vibrational energy has dissipated but while the rocks are still in motion, the normal friction coefficient may again assert itself and some high-stress shearing may take place, albeit at strains only equal to a fraction of those computed from the general motion of the rock debris.

Well outside the transient crater rim, no strength degradation occurs and large stresses may develop in this region in response to the mass deficiency in the crater interior. The rings that develop around large impact basins probably form in this regime. This is also the region where massive pseudotachylites have been reported (Spray and Thompson 1995).

Fig. 6. Unlike tension cracks, shear cracks cannot grow in their own plane and so have difficulty extending. Instead, stresses at their tips open tension cracks at a steep angle to the original shear fracture and parallel to the most compressive stress direction in this idealized picture of shear fault growth.

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