On the scaling of crater dimensions: 1. Explosive processes

Abstract

A theoretical examination of explosive crater scaling rules based upon dimensional analysis is presented. The distinction between the scaling of similar experiments as opposed to the more common practice of scaling nonsimilar experiments is emphasized. Three different sets of dimensionless •r groups commonly used in the literature (denoted here as the mass set, the energy set, and the gravity set) are compared, suggesting alternate ways to preserve similarity among experiments. Specific scaling rules using an assumed power law functional relationship among •r groups are derived for nonsimilar experiments. These scaling rules result from assumptions that cratering is independent of certain variables. The role of explosive energy density is examined and shown to be an essential variable which must be included in the analysis. None of the derived results lead exclusively to the requirement of cube root or quarter root scaling except under very special circumstances, which are shown to be common to all three formulations. In the usual series of experiments conducted in the same soil type with an identical type of explosive the scaling rules are shown to be bounded by cube root and quarter root rules. Other scaling exponent restrictions are presented.

Cratering History and Lunar Chronology

“…The first extensive use of dimensional analysis in the investigation of cratering phenomena was made by Chabai (1965) in studies of explosion craters; many subsequent investigators applied his work directly to impact events by equating explosive energy to the kinetic energy of the impactor. More than a decade later, however, the lack of a velocity (or momentum) dependence in Chabai's explosion relationships was noted by Holsapple and Schmidt, who developed an extensive suite of scaling relationships for application to impact cratering (Schmidt 1977(Schmidt , 1980Holsapple 1978, 1982;Holsapple 1980Holsapple , 1987Holsapple , 1993Holsapple and Schmidt 1980;1982;Housen et al 1983;Holsapple and Schmidt 1987;Schmidt and Housen 1987). Continuing work by other investigators has tested the scaling relationship predictions, suggested improvements, and incorporated more variables (e.g., Croft 1985;Schultz 1988;Cintala and Hörz 1990;Cintala et al 1999).…”

Exogenic Dynamics, Cratering, and Surface Ages

“…The instant shock front surface is much closer to a hemisphere with the center in the contact point than a truncated spherical surface. Holsapple and Schmidt (1980), Holsapple and Schmidt (1982), Holsapple and Schmidt (1987), and Schmidt and Housen (1987) had found a set of parameters to scale cratering dimensions, not explicitly considering target material compressibility yet. The coupling (parameter) of impact energy or momentum into the target depends on the specific target (rock) properties (e.g., solid-solid phase transitions, porosity, and effective strength).…”

Cratering saturation and equilibrium: A new model looks at an old problem

“…When this is the case, a number of power-law scaling relationships have been observed in experimental impacts, and derived mathematically as point-source solutions, that link impacts of different sizes, velocities and gravitational accelerations. The derivation of these crater scaling relationships is based upon the Buckingham p theorem of dimensional analysis (Buckingham, 1914) and have undergone extensive development over the years, as described in Holsapple and Schmidt (1980, 1982, Housen et al (1983), Schmidt and Housen (1987); and the review work, Holsapple (1993).…”