The development of advanced small caliber weapon systems has resulted in rounds with more material penetration capabilities. The increased capabilities may mean that existing live-fire facilities will no longer be adequate for the training and certification of military and law enforcement personnel. Constraints on training in many live-fire shoot house facilities are already in place, with some allowing only single round impact during training. With little understanding of the probability of perforation, or failure, of existing containment systems, this study evaluates risk by studying the single round impact of small caliber ammunition against live-fire shoot house containment systems constructed from AR500 steel panels with two-inch ballistic rubber covering. An analytical and numerical study was conducted using an existing model for steel penetration developed by Alekseevskii-Tate and the EPIC finite element code. A modified form of the advancing cavity model for the ballistic resistance of the target material was used to account for the relatively unconfined material resulting from the studied impacts. These results are then compared to experimental tests conducted by Goodman for rounds of various small calibers impacting live-fire facility containment systems. Projectile and target characteristics were then modeled as continuous random variables, and Monte Carlo simulations were conducted using the validated analytical model to estimate the probability of a single round impact perforating the live-fire facility containment system. An importance sampling scheme was used to reduce the variance of the solution and provide a more accurate estimate of the probability of failure. The Alekseevskii-Tate model was found to provide accurate estimates of the depth of penetration when compared to experimental and numerical results at ordnance velocities and an estimate of the probability of failure is on the order of 1x10-5. This study provides useful tools for the analysis of existing live-fire facilities against future and existing ammunition, and for the design of new facilities. When coupled with Monte Carlo simulation techniques, a risk-based approach to certify live-fire facilities for use with any variety of small arms ammunition can be applied.

Periodic updates to small caliber weapon systems and projectiles used in military and law enforcement have resulted in consistently increasing material penetration capabilities. With each new generation, ballistics technology outpaces the lifecycle replacement of live-fire training facilities. For this reason, it is necessary to develop and maintain constitutive material models for use in analyzing the effects new threats will have on existing facilities and for designing new training facilities using numerical methods. This project utilizes material testing data to characterize cellular concretes used in the construction of live-fire training facilities with a 13-parameter Holmquist-Johnson-Cook (HJC) concrete constitutive model. Various statistical tools are used in this analysis to successfully describe the importance of each model parameter and quantify their uncertainty. First, Bayesian linear regression was used to calibrate the parameters in the strength and pressure components of the HJC material model given testing data of cellular concrete. These uncertain parameters were then used to construct computer simulations of penetration and perforation experiments that were previously conducted by Collard and Lanham. Then, Latin Hypercube Sampling of the parameter space was used to generate training data for a Gaussian Process surrogate model of the computer simulation. Using the surrogate model, a global variance-based sensitivity analysis of the material model was completed by computing main and total effect Sobol indices. Finally, a Bayesian calibration of the computer simulation based on the physical experiments was conducted to fully characterize the stochastic behavior of the material subjected to perforation impacts. These approaches can be used to inform decision makers about the potential risk associated with existing facilities and by designers of future live fire training facilities.

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.

In order to allow for the numerical modeling of impacts for the design of live fire facilities commonly used by military and law enforcement personnel against next generation and environmentally friendly ammunition currently in development, constitutive models for novel target materials must be developed. Many existing facilities are constructed from AR500 steel, coupled with a layer of cellular rubber to reduce impact velocities and contain projectile fragments. High strain rate models, such as the commonly used Johnson-Cook constitutive model, are widely available to characterize AR500 steel, but calibrated models do not currently exist to characterize the cellular rubber. This project seeks to address this shortfall and provide a suitable material model for designers of these facilities in order to ensure the safety of users and the public. Appropriate constitutive models that account for the large strain, high strain rates, and temperature effects experienced during ballistic events and the porosity of the material were researched and a plan developed for future materials testing. Three suitable models were selected for further analysis — A Non-Linear Elastic Model described by Johnson in his work with polyurethane coupled with a Mie-Gruneisen equation of state to account for the porosity of the material, an Osborn-Hull model developed for use with crushable solids, and the Holmquist-Johnson-Cook Model commonly used for cementitious materials.