Application of Bayesian Calibration to Improve Multiple Ballistic Impact Modeling

Abstract

Analytical impact models for steel penetration, such as the Alekseevskii-Tate and Lambert-Zukas models, are a combination of physics principles and empirically derived constants fit by trial data to represent a specific experimental condition. These models are very useful to predict material performance under single impact conditions of a non-deforming or hydrodynamic projectile given suitable experimental test data but were not developed to account for the effects associated with repeated impact loading. The uncertainty in multiple impact events comes from variability in the impact location, effected area after impact, inertia induced fracture, material response to heating, and many other factors. Because of the meaningful uncertainty in multiple impact modeling, it is useful to apply Bayesian updating to formally combine the predictive capacity of an impact model with limited available test data to improve the model’s accuracy for a specific application and better quantify the uncertainty in the estimates. In this report, existing experimental data for impacts of 0.223 caliber ammunition against AR500 steel panels with 2-inch ballistic rubber is used for Bayesian updating. The existing data from the U.S. Army Aberdeen Test Center was gathered by shooting a steel plate while cycling through sixteen independent locations until one location is perforated. The total number of shots delivered to the plate was recorded as the number of shots to failure. Because sixteen independent plate locations were fired on, however, there is useful data from both locations where failure was not reached and those that were perforated. After creating the prior distribution of plate failure for a range of total impacts test data from all 48 locations is incorporated using Bayes’ Theorem to create a posterior distribution which represents an updated model for plate failure. The posterior density of plate failure strength — measured in number of shots at the failure location — can then be used as one parameter in a model to determine the safe allowable total number of impacts on the target of interest. This future model must also consider parameters such as the distribution of shots across the plate and the area affected by each impact while making assumptions about the practical variability in impact velocity and obliquity. A model of this type will inform decision makers to develop safe inspection criteria and utilize a safe number of impacts in training for current and future ammunition.