Group analysis of shock wave phenomena in solids

The group theoretic, or Lie's method of continuous point transformations is applied to the study of shock wave propagation through solid media. Theoretically, this method allows for the systematic investigation of all solutions of the set of governing partial differential flow equations that leave the original set of equations invariant.This method is applied to determine the invariance properties of three different problems.First the cold compression case is analyzed for a general one-dimensional geometry (j = 0,1,2), then for each of the individual cases of rectangular (j = 0), cylindrical (j = 1), and spherical geometries (j = 2). This analysis explores mathematical forms of the specific internal energy which are functions of the density only.An important example of cold compression with a Mie potential is the Lennard-Jones 6-12 Potential. This next problem is explored for the general geometry condition. Each of the separate geometries is then examined individually.Finally, an analysis of the adiabatic shock wave problem involving an arbitrary adiabatic bulk modulus K$\sb{\rm s}$ is performed. This system is explored for the general geometry condition, and then for each of the specific geometries separately.Once the invariance properties are determined, the group generator is used to transform the governing equations from first order, linear, hyperbolic partial differential equations to non-linear, first order, ordinary differential equations.Classical self-similar motion for a spherical explosion is recovered as a subcase of the general motion. Numerical modeling of these reduced equations is used to validate the analysis.Knowledge of the invariance properties of the self-similar problem permits the study of a possible equation of state for aluminum, copper, and lead.U of I OnlyETDs are only available to UIUC Users without author permissio