Backward Integration of the Equations of Motion to Correct for Free Surface Perturbations

Window and free surface interfaces perturb the flow in compression wave experiments. The velocity of these interfaces is routinely measured in shock-compression experiments using interferometry (i.e., VISAR). Interface perturbations often must be accounted for before meaningful material property results can be obtained. For shockless experiments when stress is a single valued function of strain, the governing equations of motion are hyperbolic and can be numerically integrated forward or backward in either time or space with assured stability. Using the VISAR results as “initial conditions” the flow fields are integrated backward in space to the interior of the specimen where the VISAR interface has not perturbed the flow at earlier times and results can be interpreted as if the interface had not been present. This provides a rather exact correction for free surface perturbations. The method can also be applied to window interfaces by selecting the appropriate initial conditions. Applications include interpreting Z-accelerator ramp wave experiments. The method applies to multiple layers and multiple reverberations. For an elastic-plastic material model the flow is dissipative and the governing equations are parabolic. When the parabolic terms are small, the equations also can be successfully integrated backward in space. This is verified by using a traditional elasticplastic wave propagation code with a backward-derived stress history as the boundary condition for a forward calculation. Calculated free surface histories match the starting VISAR record verifying that the backward method produced an accurate solution to the governing equations. With our cooperation, workers at Los Alamos have successfully applied the Sandia-developed backward technique for the time-dependent quasielastic material model and are analyzing stress histories at a spall plane using the VISAR free surface velocity measurement from a “pullback” experiment.