The objective of this project was to develop physically based models to predict the penetration depth of common military munitions in various soil conditions. Ultimately, the models would be used to determine probable depths of munitions in the soil of formerly used defense sites in support of planning for remediation. The simulation results can be used to aid sensor detection and removal of these munitions.

To model munitions penetration, a meshfree framework based on the Reproducing Kernel Particle Method (RKPM) was developed for handling extremely large deformation. A particle-to-particle based contact algorithm was introduced to efficiently capture energy and momentum exchanges between munitions and soils and between soil fragments. A two-field (displacement and pressure) semi-Lagrangian formulation was developed and implemented in consideration of porous nature of soils. Stabilized nodal domain integration schemes that ensured the accuracy and stability of the numerical solution was developed for the two-field Galerkin formulation.

To accurately represent the behavior of the soil, a viscoplasticity model was developed with regularized softening to account for large deformation of the soils. The model accounted for the behaviors seen in penetrations problems, including nonlinear pressure sensitivity of shear strength, rate dependence, shear-enhanced dilation, and a compaction hardening due to pore collapse and grain crushing. The model was updated with a regularized softening with increasing porosity that can naturally transition to more fluid-like flow, including liquefaction that is sometimes observed during penetration. The viscoplasticity model was embedded in a saturated two-field meshfree code, with partially saturated framework to be completed in subsequent research. 

The two-field (displacement-pressure) formulation based on the Biot theory had been developed and implemented under the semi-Lagrangian RK framework, where displacement and pressure field were independently approximated by the semi-Lagrangian Reproducing Kernel (RK) shape functions. Some numerical schemes originally designed for the single-field formulation were modified and implemented for the two-field formulation, including the modified stabilized non-conforming nodal integration for the domain integration, stress update, and kernel contact algorithms. The central difference and forward Euler temporal integration schemes were applied to the displacement and pressure fields, respectively, in the two-field formulation, leading to an explicit time marching scheme.

The three-invariant viscoplasticity models were developed to effectively integrate tensile, shear, and compressive behavior. The evolution of volumetric plastic strain was explicitly connected to the void ratio, allowing the model to be integrated with a poromechanical framework. Novel hardening/softening laws were added to characterize strengthening and weakening in different loading regimes, and regularized the softening using viscoplasticity. The model also accounted for rate effects, differences in strength in triaxial extension and compression, compression hardening, and other effects. An efficient implementation using the spectral decomposition was employed to reduce the cost of the complicated model.

The model was verified against a Drucker-Prager model, and the meshfree and finite element implementations have been verified against each other to ensure proper implementation. The behaviors of the model were demonstrated in a numerical framework using reasonably simple example problems. The developed two-field meshfree code was employed to simulate penetration process into soil and predict the final penetration depth under different penetration angles. In the penetration simulations, the numerical results showed that the maximum penetration depth varies from 3m to 1m with penetration angles ranging from 90˚ to 45˚. The deformation of soil, e.g. soil splashing on the free surface, reflected the experiment observations after impact. 

The project benefits the Department of Defense by providing a robust numerical framework for modeling penetration into soils. Eventually, the results of the simulations will translate into a set of tables for probable depths of munitions based on soil conditions, projectile type, and firing conditions.

The modeling will also have broader impacts to the engineering and scientific communities. The framework will be able to model other penetration scenarios for soil, rock, and concrete, for applications as diverse as deep penetrators designed to target underground bunkers to meteor impacts on extraterrestrial bodies. The constitutive models will further the fundamental understanding of soil behavior as the researchers move toward physically based models for capturing observed soil responses. In addition, the numerical algorithms will enhance the set of tools available to solve many physical problems, especially those involved in large deformation, material separation, and coupled physics problems.