The objective of this project was to develop physics-based models and predict the penetration depth of common military munitions in various soil conditions. Ultimately, the models will 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. 

A robust meshfree code based on reproducing kernel (RK) approximation has been developed for penetration simulation. Several improvements have been developed and implemented to ensure solution accuracy and stability, including: 

• Developing the midpoint integration method, which offers better robustness for penetration simulation compared to existing methods. 
• Developing and implementing the absorbing boundary condition in the RK formulation to remove the reflected wave due to truncated boundary. 
• Developing the consistent semi-Lagrangian (SL) formulation to address the algorithmic issue due to the convective term. 
• Implementing the quasi-linear reproducing kernel formulation, which ensures the consistency of RK approximation under material separation. 
• Implementing a gradient-based stabilization technique to improve the stability when a nodal integration is used. 
• Implementing a pressure projection method for the u-p formulation, which resolves the instability due to the inf-sup condition. 

Soil constitutive models have been developed and implemented in the meshfree framework to simulate soil behaviors under different saturation conditions during the penetration process. Developments and improvements included: 

• Reformulating the cap surface to better reproduce experimental data. 
• Modifying the plastic potential function so that plastic compaction of the soil may occur even as the shear strength increases. 
• Implementing some algorithmic improvements to enhance the robustness and efficiency of the soil models. 
• Developing strategies to calibrate material parameters for soil models. 
• Performing the sensitivity analysis to identify critical parameters in the soil models. 

Soil properties tests and penetration tests under different saturation conditions have been conducted in the laboratory. The data have been used to verify and validate the developed soil models and meshfree framework. 

The laboratory-tested data and several penetration experiments from the literature have been used to validate the developed meshfree framework for predicting the penetration metrics under different saturation conditions.

Task 1 – Enhancement of Multi-Field Reproducing Kernel Computation Framework, Review of SL Reproducing Kernel Formulation. This section recapitulated the RK formulation, the main framework for the meshfree simulations in this project. It provided the necessary background to present the enhancements developed and implemented during the performance period. 

Task 2 – Soil Constitutive Model Improvement, Algorithm Improvement. The project team implemented improvements to the model to capture soil behavior more accurately. The project team reformulated and implemented the growth function for the cap surface so that it better reproduced experimental data. 

Task 3 – Soil Tests and Penetration Tests, Soil Tests for Calibration Soil Mechanical Properties. The US SILICA™ Flint #12 sand was selected as the granular material for the laboratory tests and validations. Some of the mechanical properties of the sand have been reported in the previous reporting years, including density determination, specific gravity determination by pycnometer, grain size analysis, etc. Moreover, Consolidated Drained Triaxial Tests were conducted for dry, unsaturated, and fully saturated sand samples to determine the shear strength parameters under different saturation levels. A downhill simplex scheme was employed for extracting the material parameters for simulations. 

Apparatus for Penetration Tests. An apparatus for performing penetration tests was built and placed in the University of Illinois (UIC) high-bay structural laboratory. This is the first apparatus for penetration tests at the UIC. The main goals of building the apparatus were to provide penetration testing data consistent with the soil testing data, validate the simulation results, and guide the modification of the current model. The system was powered by compressed air and could launch a one-inch diameter bullet with different shapes up to 100 m/s. The final impact speed was measured by a chronograph, and the projectile penetration depth was measured by a digital caliper. After the shot was completed, the projectile was cleaned to wash off the sand residual, and a sample of the target was taken from a typical impact zone to analyze its water ratio in the laboratory.

Penetration Tests for Dry, Unsaturated, Fully Saturated Sand. In each penetration test, the soil sample was prepared using the same conditions. The depth gauge was zeroed referenced to the top surface of the soil target. After the projectile was shot in the soil target, a thin steel rod on the digital caliper was used to carefully probe the impact area and find the lowest reading before encountering resistance, which indicates the distance from the soil surface to the top surface of the projectile. This probing reading was further verified after the projectile was dug out, and the verified distance reading was recorded as the depth of penetration in the final report. The penetration process was recorded with 240-fps videos to analyze the impact mechanics and validate the numerical simulations. Over 100 tests were completed for dry, unsaturated, and fully saturated soil targets, of which the corresponding nominal saturation ratios (SR) were 0, 0.5, and 1, respectively. Thirty-eight tests were carried out for dry sand with the impact speed ranging from 7 to 71 m/s. The depth of penetration (DOP) increased linearly when the final speed was less than 25.3 m/s. The DOP increase rate decreased and reached the maximum DOP at 53 m/s with an average DOP of 163.4 mm. The DOP dropped after the impact speed of 63.5 m/s, which may be attributed to the boundary effect from the bottom of the target. Overall, the DOP test data for dry sand showed good consistency, of which the standard deviation was less than 4 mm, and the maximum deviation is less than 8 mm. 

Fifty-three tests for the fully saturated sand were carried out with the impact speed in the range of 7 to 63 m/s. The DOP was much lower compared with that of the dry sand under a similar impact speed. The DOP increased linearly with the increase in impact speed up to 40 m/s and achieves the maximum value at around 80 mm. However, the DOP results fluctuated in comparison to the ones for dry sand. The standard deviation of the DOP with a similar speed was around 5 mm, and the highest standard deviation was about 10 mm. Due to the complexity of preparing partially saturated soil samples, only 12 tests were performed for sand with SR = 0.5, with the impact speed in the range of 30.8 to 66.4 m/s. The DOP results fluctuated among all SR conditions due to the uncertainty of preparing samples. Nevertheless, excluding a seeming outlier, the mean DOP results lay between those for dry and fully saturated sand. 

Overall, penetration test results showed that the DOP increased when the impact speed was low, reached the maximum value in the mid-speed range, and decreased slightly in the higher speed range. The slight decrease in DOP in the higher speed range in the tests could be attributed to the confining effect of the container and the wave reflection from the bottom. These DOP data and deformation patterns were used to validate simulation results. 

Penetration of an Ogive Projectile into Dry Sand. Experimental data in the literature was considered to validate the developed meshfree code for modeling penetration into the soil. An ogive projectile of length 41.66 mm and radius 2.083 mm made of aluminum was projected with various velocities into the coarse sand. Several impact velocities ranging from 24 m/s to 170 m/s were simulated. 

Penetration of a Spherical Ball into Sand. Experimental data in the literature was taken to validate the meshfree code for modeling penetration into dry soil. A spherical ball of radius 5.0 mm made of steel was projected with various velocities into the coarse sand target. Several impact velocities ranging from 40.66 m/s to 341.95 m/s were simulated. 

Ricochet of a Projectile on Sandy Soil. Experimental data in the literature was taken to validate the meshfree code for modeling penetration into dry soil with an impact angle. A spherical ball radius of 25.0 mm made of concrete was projected with an impacting velocity of 62.23 m/s into the coarse sand and an impact angle of 30º. 

After calibrating the material properties and parameters, the rest of the cases were simulated for validation, and rebound angles and velocities were calculated and compared with the referred experimental results. A higher impact angle and velocity resulted in a higher DOP. Overall, the numerical solution agreed with the experimental data. The maximum errors for exit velocity and exit angle were about 10% and 13%, respectively. 

Penetration of Spherical Projectiles into Saturated Soil. Experimental data in the literature was taken to validate the numerical model for penetration into the saturated soil. 

Penetration of Spherical Projectiles into Sand with Different Saturation Ratios. The laboratory test results were taken to validate the developed numerical modeling framework. A steel ball was shot into a circular bucket filled with sand with different SR. 

Ricochet and Penetration of 25 mm Projectile into Sand. Experimental data in the literature was taken to validate the developed numerical model for modeling high impact penetration into dry soil with an impacting angle. An ogive projectile of length 90.00 mm and radius of 12.5 mm made of aluminum was projected with an impacting velocity of 589 m/s and impact angles of 25º and 12.5º into the coarse sand. 

Simulations of Some Realistic Projectiles Penetrating into Sand Ground. This section presented simulations of several realistic projectiles penetrating into dry and compacted sand ground, similar to the coarse sand. The dimensions and properties of projectiles were taken from the literature, and the impact velocities and angles were estimated from the muzzle velocities of the launching platform. The model setting, initial conditions, and boundary conditions were set as close as possible to the realistic scenarios. Although the experimental data were not available for validation, the numerical results demonstrated that the meshfree method could simulate penetration processes of unexploded ordnance projectiles under realistic conditions. More importantly, the computational cost of these simulations provided insight into the costs of developing tabular DOP results for different sediments, projectiles, and impact trajectories.

The developed SL meshfree code effectively offered an accurate numerical solution for soil penetration. In addition to bulk penetration measurements, such as DOP, the numerical solution accurately predicted the soil damage pattern and crater shape and size. Nevertheless, the computational cost remained high. Model reduction techniques or machine learning techniques combined with extensive offline simulation data will be needed to expedite the simulation. 

The data from penetration experiments using dry, saturated, and partially saturated sands provided insight into how saturation affects the depth of penetration and other penetration measurements. These data, along with simulations of three experiments and from the literature, suggested that the meshfree modeling of penetration of smaller projectiles into sand was realistic. 

The mixed field formulation for modeling soil under different saturation conditions has been developed and validated. However, since the penetration process typically completes within a few hundredths of a second, the fluid in the pore does not have sufficient time to dissipate or transport in space. The continuity equation in such a case plays a minor role in predicting the overall penetration mechanics. Therefore, a single field formulation with a properly calibrated soil model may suffice for the DOP prediction for computational efficiency. 

While implicit implementation of constitutive models offers unconditionally stability, complicated formulas and conditioning issues in the tangent matrix can lead to issues in robustness. Semi-implicit methods improve robustness and, for applications that require small time steps, improve efficiency as well. Explicit methods can also be attempted, but the solutions tend to drift away from the yield surface or produce saw-tooth patterns that could reduce global stability for dynamic problems. 

In complex materials with many parameters, fitting can be a challenge. It helps to create a model with physically motivated parameters that can be fit sequentially from experiments. Sensitivity analysis can also greatly aid in determining which parameters are most important to a given set of simulations.