A shock model is developed that leads to an analytical expression for the Hugoniot of condensed media. In the analysis the final state is selected to coincide with the end of the shock transition so that the total energy change across the shock front is evaluated from changes in configurational energy only using an n-6 pair potential(shown to be valid for all n > 0) and a given lattice structure.Thermal energy changes are ignored because the dwell time of the molecules in the shock transition region is less than the thermal relaxation time. The total energy change is equated to the Hugoniot energy change in the Rankine-Hugoniot conservation relations. This together with the assumption of linear compression across the shock transition gives the desired expression for the Hugoniot.
In the "weak form" (WF) solution the Hugoniot depends on molecular (atomic) weight M and initial density ρo as well as thecollision diameter σ, depth of the potential well ε and repulsive exponent n of the pair potential. Extrapolation of this solution, under certain conditions, yields an expression for the sound velocityUo dependent on M, ε and n. In the "strong form" (SF) solutionthe Hugoniot depends only on Uo and n.
The shock data for 13 liquids and 23 metals are compiled and a selection process used to eliminate poor data and data affected by phase transitions. Using σ from the literature and ε from a melting point correlation, the WF solution Hugoniot is applied to the liquids and the "best" values of n determined using numerical fitting techniques. Excellent fits are obtained with values of n from6.2 to 11.7. A common value of 9.2 is found to fit the shock data for argon at four different initial states.Failure of the theory isnoted only for (di-)ethyl ether and water. The results are generally concluded to support the validity of the shock model. Values ofn for argon, mercury and nitrogen compare favorably with values reported in the literature. The WF solution does not yield accurate values ofUo.
The SF solution is expanded in Taylor series to eliminatesingularities and applied to the shock data for 10 fcc and 13 bccmetals and the "best" values of n determined. For the fcc metalsexcellent fits are found for values of n from 4.0 to 6.3. Based onthe "pseudo-atom" concept, it is concluded that metals have "softer"potentials than liquids. The results for the fcc metals are concludedto generally support the validity of the shock model. For the bccmetals excellent fits are found for n = 0.1 to 4.6 and it is concludedthat bcc metals are "softer" than fcc metals. Since σ is "notdefined" for all n < 3 it is speculated that the Hugoniot might notbe "well defined" in these cases. The theory is found to be notapplicable to Cs and Ba. The values of n found for Cu, Al and Pbagree well with values in the literature. The n for metals are foundto roughly correlate with the Grüneisen coefficient γ.
The major assumption of the theory, that the transition regionis "sufficiently thin," is analyzed and found to be reasonable. Themean number of molecular layers in the transition region is ~ 7 andthe mean residence time ~ 2 x 10-12sec. In addition the n-6 potentialis judged to be adequate for the present study.
The SF solution Hugoniot is shown to be compatible with theclassical linear U-µ relation (U = A + Bu) when x-1 « 1. Values ofthe slope B from experimental data are found to agree closely withthe derived relation B ≃ (n + 5)/6 for the corresponding metals.Recommendations for further studies with liquids, fcc and bcc metalsare made, including an evaluation of an explicit expression generally relatingthe pair potential to the shock data.
In several subsidiary studies a new method of computing temperatures along the Hugoniot is found, two statistical approaches tothe definition of "nearest neighbor distance" and its use as a measureof liquid structure are developed and the effect of phase transitionson the shock model is determined.
