Thermoplastic polymers as intermediate layers in protective structures could offer hypervelocity impact (HVI) damage resistance without compromising on weight and cost. Moreover, the ability to accurately simulate the ultra-high strain rate response of polymers would enable protective structure development and optimization. In a previous study, HVI experiments were performed using a two-stage light gas gun to probe the HVI response of high-density polyethylene (HDPE). 6.35 mm thick, monolithic, square targets were subjected to a series of normal, 2.0–6.5 km/s, 10 mm diameter aluminum sphere HVIs. The debris cloud tip velocity, perforation radius, and target mass loss were found to increase with impact velocity. High-speed images displayed apparent HVI-induced bulk melting and large-scale plastic deformation in the HDPE targets. In the current study, the HVI experiments were simulated using established material models and equations of state (EOS) in the Elastic Plastic Impact Computation (EPIC) code. Appropriate material model and EPIC code simulation parameters were calibrated to improve the correlation of HVI damage metrics ( e.g., damage morphology, debris cloud geometry and velocity). The simulation results agree well with the experiments across most of the tested velocity regime and provide useful data to help infer the extreme target strain rates and temperatures induced by HVIs. Keywords: hypervelocity impact, high-density polyethylene, ultra-high strain rate, impact simulation 1. Introduction and Motivation The mechanical behavior of polymers is highly dependent on temperature, strain, and strain rate, as 1 well as manufacturing and processing history. These sensitivities, however, could potentially be leveraged 2 to develop the next generation of protective materials and structures for space applications ( e.g., microme- 3 teoroid/orbital debris protection) [1, 2]. Moreover, tuned thermoplastic polymeric materials as intermediate 4 layers in protective structures could offer hypervelocity impact (HVI) damage resistance while also provid- 5 ∗Corresponding author. J. Mike Walker ’66 Department of Mechanical Engineering, Texas A&M University, College Station, Texas, 77843, United States. Email address: telacyjr@tamu.edu (Thomas E. Lacy, Jr.) Preprint submitted to TBD August 25, 2024 ing potential weight and cost savings. The ability to accurately simulate the macroscopic ultra-high strain 6 rate response of polymers would accelerate protective structure development and optimization. Accurately 7 modeling the high- and ultra-high-strain rate response of materials, however, is generally challenging due to 8 HVI-induced irreversible phenomena, including projectile and target erosion, penetration and/or perforation, 9 inelasticity, fracture, melting, spallation, fragmentation, etc. [3]. Often, the development and validation of 10 material models and the assessment of simulation techniques requires direct comparison to full-scale HVI 11 experimental results. A significant portion of the HVI/penetration mechanics literature consists of empirical, 12 analytical, and numerical studies on metals. Investigations on non-metallatic materials have been generally 13 less frequent, focusing on composites [4–6], ceramics [7], and other materials [8]. 14 Interest in thermoplastics, specifically polyethylene (PE) and PE composites, has recently increased due 15 to their attractive energy absorption features [1, 9–13]. In addition, these polymers can be additively man- 16 ufactured and potentially tuned for anticipated impact threats. Even so, numerical predictions of the HVI 17 and ultra-high rate response of PE have been relatively unexplored, especially at higher velocities (and strain 18 rates). This gap in understanding is due, in part, to ( i) metals being the primary materials used in con- 19 ventional protective structures, ( ii) the difficulty associated with systematically generating ultra-high strain 20 rates, and ( iii) the complexity of polymer-microstructure-induced material behavior. The relatively low-rate 21 material behavior of PE has been well characterized; however, findings at lower rates cannot necessarily be 22 extrapolated to higher strain rates consistent with HVIs. Thus, the problem at hand is three fold: ( i) char- 23 acterizing the HVI response of polyethylene (PE) viaexperimentation, ( ii) using empirical data to inform 24 applicable material models and equation of state (EOS) for practical simulation and optimization efforts, 25 and ( iii) developing a fundamental understanding of how intrinsic polymeric chemistry, microstructure and 26 morphology influence the macroscopic damage formation and evolution. 27 Despite these challenges, some work has been done to model and simulate the low- and high-velocity 28 (<3.0 km/s) impact response of thermoplastics, specifically PE. For example, Shah et al. [14] developed 29 numerical models for high-density PE (HDPE) and low-density PE (LDPE) substrates each impacted by 30 a 50 µm-diameter polymer particle traveling at 0.2 km/s. This study was performed to develop a better 31 understanding of the cold spray deposition of polymer substrates, which is important for coating surfaces. 32 Pundhir et al. [15] used ANSYS Autodyn®and a Johnson-Cook (JC) strength model with orthotropic 33 softening failure criteria to simulate the impact response of a 200 x 200 mm, polymer composite plate (5- 34 9 mm-thick) comprised of HDPE and ultra-high molecular weight PE (UHMWPE) subjected to a steel ogive 35 projectile impact at 0.85 km/s. Simulation results showed that a higher percentage of UHMWPE led to 36 larger energy transfer to the target and greater Von Mises stress in the target at the velocities tested. Austin 37 [16] simulated 2.1 and 3.5 km/s impacts to UHMWPE using ANSYS Autodyn®for comparison with gas gun 38 2 experiments to improve model accuracy through considering thermal softening, not successfully accounted 39 for in previous UHMWPE models. The model led to simulation results that were more consistent with 40 experimental results while minimizing computational time. Ramezami et al. [17] focused on developing 41 an UHMWPE hydrocode model to improve ballistic efficiency and minimize the failure of equipment used 42 in armor and security applications. UHMWPE, specifically 5.5–16.2 mm thick, Dyneema®HB26, panels 43 were numerically impacted by 7.62 mm x 39 mm projectiles at approximately 0.68 km/s. The corresponding 44 results suggest that improvements could be made to modeling techniques for impacts with common battlefield 45 projectiles at these velocities. Several other studies have also conducted similar impact simulations on 46 UHMWPE composites, further exploring their ballistic performance and failure mechanisms, see e.g., [18– 47 22]. 48 The HVI ( ≥3.0 km/s) response of PE has also been somewhat explored. Bowman et al. [23] performed 49 molecular dynamics (MD) simulations of 6 nm thick, PE and polystyrene thin films to analyze the ballistic 50 response of nanoscale materials, which has not been widely investigated and is generally difficult to study 51 experimentally. A relatively large-scale ( e.g., millions of atoms) 30 nm spherical projectile impact simu- 52 lation was performed. A transition was found in the failure mode from petalling to fragmentation as the 53 impact velocity increased from 0.75 km/s to 3 km/s. Also, Bohannan and Fahrenthold [24] used membrane 54 particle-elements to model and simulate the response of 1 mm thick, aluminum-Kevlar targets subjected to 55 1.68–7.18 km/s HVIs by 2.8 mm and 3.2 mm diameter aluminum spheres. A new modeling method that 56 utilizes membrane particle-elements was developed to address previous inaccuracies associated with modeling 57 techniques for flexible fabrics. L¨ assig et al. [25] modeled and simulated 2.0–6.5 km/s, 6 mm diameter alu- 58 minum sphere HVIs to 200 x 200 mm, 0/90◦cross-ply Dyneema®HB26 laminate using ANSYS Autodyn®. 59 Two stage light gas gun (2SLGG) experiments were also performed at similar impact velocities to compare 60 with computational results, revealing a slight trend towards over-predicting debris cloud velocity for the 61 model with increasing impact velocity. Lastly, van der Werff et al. [26] and others [27] present a plasticity- 62 based model to improve modeling capabilities for HB26 UHMWPE, a layup of SK76 Dyneema®fibers. 63 3.0–6.0 km/s HVIs to 200 mm x 200 mm and 100 mm x 100 mm targets were simulated with two distinct 64 projectiles: a 6 mm diameter aluminum sphere and a 5.5 mm diameter steel sphere, respectively. Simulation 65 results using this nonlinear plasticity and shock compression model aligned well with experimental results. 66 These relevant results from the literature are qualitatively plotted in impact-velocity-impactor-scale space 67 in Fig. 1, emphasizing simulation conditions yet to be explored. Although the HVI response of UHWMPE 68 has been explored to some degree, the macroscopic HVI response of HDPE has not (to the knowledge of the 69 authors) been numerically simulated. 70 Rogers et al. [28] recently showed experimentally that the HVI response of UHMWPE and HDPE 71 3 Figure 1: A qualitative representation of relevant literature: numerical simulations of high-rate impacts to polyethylene and polyethylene composites. differ significantly both macroscopically and microscopically. Monolithic, 6.35 mm thick square plates were 72 subjected to 2.5–6.5 km/s HVIs by a 10 mm diameter 1050 aluminum spheres. UHMWPE was found to 73 fragment in a brittle-like manner (Fig. 2a–c) while HDPE exhibited large-scale melting (Fig. 2d–f). In 74 addition, the HVI-induced perforation radii were larger while the target mass loss was lower in HDPE when 75 compared to similar UHMWPE specimens. For each experiment, the debris cloud tipvelocity, target mass 76 loss, and perforation radius were found to increase with impact velocity. High-speed images show HVIs to 77 HDPE resulted in apparent bulk-melting of the material and large-scale plastic deformation. Additional 78 material characterization experiments ( e.g., DMA, TGA, DSC) were performed to measure key material 79 properties for use in material models. In the current work, the HVI experiments into HDPE presented by 80 Rogers et al. [28] were simulated (Fig. 1) using the Elastic Plastic Impact Computation (EPIC) code, 81 a hybrid finite element (FE) and meshless particle computational approach that is well suited for large 82 deformation impact simulations (more later) [29–31]. The preliminary findings of this effort were presented 83 in [32]. Appropriate material model and EPIC code simulation parameters were calibrated to improve the 84 correlation of HVI damage metrics ( e.g., damage morphology, debris cloud geometry and velocity). The 85 results show that simulation results agree well with the experiments across a majority of the tested velocity 86 regime. The simulation results were then used to infer local HVI-induced temperatures and strain-rates 87 4 within the HDPE plates. 88 Figure 2: Comparison of fully-developed ejecta and debris clouds at different targeted impact velocities within the tested range: (a-c) for UHMWPE and (d-f) for HDPE. As the targeted impact speed escalates from left to right, the average size of UHMWPE debris fragments reduces, while the extent of melting in HDPE intensifies. Figure reprinted from Ref. [28]. The experimental methodology is briefly presented, followed by a detailed discussion of the simulation 89 approach. Experimental and simulation results are then presented, compared, and discussed. Conclusions 90 are drawn concerning the modeling and simulation approach’s validity and limitations in this application, 91 and potential future improvements are suggested. 92 2. HVI Experimental and Simulation Methodologies 93 The simulation results presented and discussed in this paper were compared to HVI experimental results. 94 Here, the experimental and simulation methodologies are presented, and the material models and EOS 95 employed are introduced. 96 2.1. Subjecting HDPE Plates to HVI Using a 2SLGG 97 The Texas A&M University Hypervelocity Impact Laboratory1(HVIL) aeroballistic range 12.7 mm 98 diameter, smooth bore 2SLGG was used to launch the projectiles to the desired hypervelocities [33]. An 99 in-depth discussion of this range and its capabilities is presented in Ref. [33]. The monolithic, HDPE 100 targets were obtained from stock extruded sheets manufactured by King Plastic Corporation ([28, 34]; Table 101 1The HVIL website: https://telacyjr.engr.tamu.edu/facility/hypervelocity-impact-laboratory-hvil/ . 5 1). The thickness of all targets was 6.35 ±0.1 mm, and the in-plane dimensions of the targets were either 102 15.2 cm x 15.2 cm or 30.5 cm x 30.5 cm. The targets were secured between two apertured, A-36 steel plates 103 using fasteners before being fixed in the target tank. This target fixturing approach effectively induced 104 axisymmetric boundary conditions to facilitate computational simulations. The targets were impacted by 105 10 mm diameter, 1050 aluminum spheres possessing masses of 1.416 ±0.004 g. After the HVI experiments, 106 the ultra-high-speed camera images were analyzed using motion tracking software ( e.g., Tracker [35]) to 107 determine the debris cloud tipvelocity. This quantity was calculated across multiple frames and averaged 108 to obtain a nominal value. In addition to these velocity measurements, the ultra-high speed camera images 109 were analyzed qualitatively to determine the bulk material response of the HDPE targets during and after 110 impact. Post-impact analyses included measurements of the perforation radius, target mass loss, and overall 111 damage morphology, as well as scanning electron microscopy (SEM) image evaluation. These results served 112 as quantitative HVI response/damage metrics for the HDPE targets. A more comprehensive presentation 113 and discussion of the experimental methodology and results can be found in Ref. [28]. 114 2.2. Simulating HVIs into HDPE Plates Using EPIC 115 In this study, a hybrid computational code (EPIC), which uses finite elements and meshless particles to 116 represent large deformations, was used to simulate HVIs into the HDPE targets. Circular targets 6.35 mm- 117 thick and 203 mm in diameter were rendered and meshed in three-dimensional (3D) quarter-symmetry 118 using hexahedral elements. Each hexahedral element was then converted into twenty-four linear tetrahedral 119 elements (Fig. 3). The size and shape of the simulated target was consistent with the axisymmetric region 120 imposed on the experimental targets by the apertured target frame ( cf.Section 2.1). The target mesh was 121 biased such that the elements increased in size as a function of distance from the point of impact. This 122 biasing was done to increase simulation fidelity while keeping the problem size manageable. Moreover, the 123 1050 aluminum projectile elements were similar in edge length to those in the contact region ( ∼0.77 mm). 124 Element-to-particle conversion, a key feature of the EPIC code, was utilized, and the equivalent strain 125 required to convert an element to a particle in both materials was set to 30%, a common conversion strain 126 value for ductile materials [36]. 127 Table 1: Quasi-static material characterization data for HDPE resulting from TGA, DSC, and DMA. The extruded sheets of HDPE were manufactured by King Plastic Corporation. Data reprinted from Ref. [28]. Quantity Value Glass transition temperature, Tg(°C) -116.2 ±0.8 Melting temperature, Tm(°C) 131.14 ±0.58 Crystallinity, (%) 70.24 ±1.98 Density, (g/cc) 0.954 ±0.002 Initial decomposition temperature, IDT (°C) 423.2 6 Figure 3: Simulation snapshots showing projectile and target quarter-symmetric meshes before and after the conversion of each hexahedral element into twenty-four tetrahedral elements. A simple mesh refinement study was performed in 3D quarter symmetry. Specifically, the same impact 128 simulation, a 4 km/s impact, was performed with three different target and projectile mesh refinement levels 129 (roughly 18,000, 119,000, and 950,000 elements total) to observe any potential mesh-induced changes in 130 impact damage metrics ( e.g., perforation radius, debris cloud tip velocity, damage morphology). This mesh 131 refinement study showed that the most refined mesh (950,000 elements) and the second most refined mesh 132 (119,000 elements) returned perforation radii that differed by 0.5% (see Fig. A.1), and the corresponding 133 debris cloud tip velocities exhibited similar agreement. Even with this agreement, the more refined mesh 134 was selected for this study in the absence of significant additional computational burden. Specifically, the 135 highest impact velocity simulation (6.5 km/s), which corresponds to the shortest time step, ran to 80 µs in 136 40 min using the fine mesh geometry on 108 SGI Ice-X 2.6GHz cores. Simulation results using the chosen 137 quarter-symmetric mesh were also compared to similar simulation results using a 3D half-symmetric mesh of 138 the same fidelity; no significant differences in debris cloud tip velocity or perforation radius were observed. 139 Hence, the 3D quarter-symmetric mesh of 950,000 elements was chosen for all simulations presented and 140 discussed herein. 141 Both the aluminum projectile and the HDPE target were modeled using a JC constitutive model [37], 142 a JC fracture model [38], and a Mie-Gruneisen EOS [39]. The HDPE model uses the cubic form of the 143 Mie-Gruneisen EOS; this form can be shown as 144 P= (K1µ+K2µ2+K3µ3)(1−Γµ/2) + Γ Es(1 +µ), (2.1) where Pis the pressure, K1, K2,andK3are polynomial coefficients, Γ is the Gruneisen coefficient, Esis the 145 internal energy per unit volume, and µis given by the relationship µ=ρ/ρ0- 1 where ρis the current density 146 andρ0is the initial density. While the Mie-Gruneisen EOS is typically applied to metals, the polyethylene 147 7 fit from Ref. [40] (see Table 2) is comparable to experimental Hugoniot data for Marlex (a proprietary 148 commercial HDPE), as reported in [41] (see Fig. A.3). 149 The JC constitutive model gives the material dynamic flow stress ( σ) as 150 σ= C1+C2¯εN (1 +C3ln˙¯ε∗) [1−(T∗)m], (2.2) where C1is the yield stress, C2is the hardening coefficient, ¯ εis the equivalent plastic strain, Nis the 151 hardening exponent, C3is the strain rate coefficient, ˙¯ε∗is the dimensionless total equivalent strain rate 152 (normalized by a strain rate of 1 s−1),T∗= (T−T0)/(Tm−T0) is the homologous temperature, TandT0153 are the current and reference temperatures, Tmis the melting temperature, and mis the softening exponent. 154 Here, the target temperature is updated at the ( ith+1) timestep viaTi+1−Ti=βσ¯ε/(ρ0cv), where Ti=1=T0, 155 β= 1 is the Taylor-Quinney coefficient for HDPE ( i.e., fraction of plastic work converted to heat), and c v156 is the specific heat capacity of HDPE at a constant volume. The parameters in Eq. (2.2) were adjusted to 157 reflect the actual target and projectile materials used in experimentation. In the case of the 1050 aluminum 158 projectile, the density, yield stress, melting temperature, and shear modulus were adjusted to reflect the 159 values reported by the projectile supplier [42]. For the HDPE target, the density and melting temperature 160 values were adjusted in the model to reflect material characterization results of coupons taken from the 161 HVI-tested specimens [28]. Additionally, to better capture the constitutive response of HDPE, data from 162 a variety of strain rates and temperatures were sourced from Brown et al. [43] and were used to calibrate 163 the JC constitutive model constants (see Fig. A.2). In this previous study, quasi-static compression and 164 split Hopkinson pressure bar tests were performed for HDPE specimens across a range of temperatures to 165 record the true stress-true strain relationship for a significant span of strain rates (10−4–103s−1). The JC 166 constitutive model parameters were chosen to fit the almost perfectly plastic material response observed. 167 These values, as well as other important material parameters are reported in Table 2. 168 Additionally, a short parametric study was performed to understand the sensitivity effects of JC fracture 169 model coefficients on target perforation radius. A common form of the JC fracture model gives the equivalent 170 strain to fracture ( εf p) under constant conditions of ˙¯ε∗(cf.Eq. 2.2) as 171 εf p= D1+D2eD3σ∗ (1 +D4ln˙¯ε∗) (1 + D5T∗), (2.3) where D1–D5are fitting parameters, σ∗=σm/σeqis the pressure stress ratio (stress triaxiality [38]), σmis 172 the mean normal stress, and σeqis the equivalent stress. Damage in this equation is defined through the 173 sum of the accumulated equivalent strain divided by εf p. An element or converted particle is considered to 174 8 be fully damaged or fractured when Σ∆ εp/εf preaches a value of unity. 175 Initially, the spall strength of the HDPE model was adjusted to match data from the literature. Spall 176 can occur within the JC model when average tensile stress (negative pressure) reaches a critical value. The 177 equivalent strain to fracture ( εf p) varies linearly from σ∗= 1.5 to σ∗ spall at the minimum strain to failure 178 limit ( D1) where σ∗ spall=σspall/σeq. The spall strength ( σspall) for the HDPE model was chosen from the 179 data reported by Jordan et al. [44]. The data showed that for three flyer plate impact experiments with 180 velocities ranging from 0.43–0.80 km/s the spall strengths ranged from 50–70 MPa, implying that HDPE spall 181 strength is relatively strain rate independent compared to other polymers. To determine the appropriate spall 182 strength, three HVI simulations at an impact velocity of 6.5 km/s were conducted, adjusting the spall strength 183 of the HDPE target to values report in Ref. [44] (50, 60, and 70 MPa). The impact velocity of 6.5 km/s 184 was selected for the parametric study due to the significant changes in hole size and damage morphology 185 at higher velocities. The simulation results indicated that the spall strength was inversely proportional to 186 the perforation radius, however, the influence was small ( <3.3% difference between the 50 and 70 MPa 187 simulations). The simulation with a spall strength of 50 MPa exhibited more petalling on the rear surface 188 of the target compared to experimental observations in Ref. [28], while the simulation with a spall strength 189 of 70 MPa underpredicted the extent of petalling. Based on these results, a spall strength of 60 MPa was 190 selected as it provided the most accurate representation of the observed damage morphology. Additionally, 191 the minimum strain to failure limit, controlled by D1[Eq. (2.3)], was reduced from a value of 30% to a value 192 of 5%. This reduction had very little effect on the perforation radius. The negligible repercussions of this 193 Table 2: Mie-Gruneisen EOS and Johnson-Cook constitutive and fracture model parameters for the for HDPE target. Parameter Value Unit Yield stress, C1 31.27 [MPa] Hardening coefficient, C2 0.0 [MPa] Strain rate coefficient, C3 0.04 - Hardening exponent, N 0.75 - Softening exponent, m 0.85 - Shear modulus, G 275.8 [MPa] Specific heat capacity, cv 2.323 [J/g-◦C] Strain to failure coefficient, D1 0.05 - Exponential coefficient, D2 1.48 - Stress triaxiality coefficient, D3-1.5 - Strain rate coefficient, D4 0.25 - Temperature coefficient, D5 0.0 - Spall strength, σspall 60 [MPa] Bulk modulus, K1 7.9 [GPa] Quadratic M-G Coefficient, K2 2.296 [GPa] Cubic M-G Coefficient, K3 43.36 [GPa] Gruneisen Coefficient, Γ 0.86 - 9 change can be attributed to the relatively minimal share of spall-independent material plasticity that comes 194 from high triaxiality during HVI impacts. The effects of changes in strain rate coefficient, D4, were also 195 investigated: a total of eight HVI simulations were performed where v0was changed from 2.0 to 6.5 km/s for 196 each of the four D4values investigated. The parametric study showed slight changes in the perforation radii 197 for both the 2.0 and the 6.5 km/s impacts ( <2.9% and <1.7%, respectively). The most drastic variation is 198 likely attributed to the detachment of single particles (averaging 0.35 mm in diameter) around the damaged 199 rim of the perforation, increasing the radius slightly. Thus, the value of D4was chosen to be 0.25 for the 200 remainder of this study to optimally account for changes in impact-velocity-dependent damage response. 201 The predicted debris cloud tip velocities, perforation radii, and target mass loss values were calculated 202 using ParaView version 5.5.1 for comparison with experimental results [45]. The debris cloud tip velocity 203 was estimated by averaging the downrange velocities of the converted particle composing the debris cloud 204 tip at a given instant in time. This measurement technique differs slightly from the experimental method: 205 experimental debris cloud tip velocities were computed by averaging the downrange tip velocity over time. 206 This approach was necessary to account for minimal debris cloud deceleration induced by atmospheric 207 drag [ i.e., some sub-atmospheric pressure ( ∼100 Torr) existed in the test/target chamber]. The lack of 208 atmospheric drag in the simulations made the time-averaged velocity calculation step unnecessary. In fact, 209 a given converted particle’s downrange velocity remained constant unless acted upon by another particle. 210 Thus, the debris cloud tip velocity simulation estimation needed to only be performed after sufficient debris 211 cloud expansion to reduce potential particle-particle interactions (a time of 80 µs was found to be sufficient 212 for all impact velocities). The perforation radii were also determined viaParaView after the predicted 213 strain rate in the target elements reached a functionally low value ( e.g.,≤10−4s−1) for the highest impact 214 velocity. At this time (roughly 80 µs), a software threshold was placed on the target to hide all particles 215 with an accumulated damage/burn parameter of unity. With this filter in place, the impact perforation 216 radius was measured from the edges of the remaining converted particles that have not been detached from 217 the simulated target. The simulated target mass loss was evaluated in a similar, damage-based manner 218 that would appropriately reflect the experimental measurement methodology to achieve the most consistent 219 results. 220 3. A Comparison of EPIC Predictions to Experimental Measurements 221 For this study, impact simulations were executed at six different velocities across the experimentally tested 222 impact velocity regime, v0= 2.0,3.0,4.0,5.0,5.5,and 6 .5 km/s. The simulation results were postprocessed 223 to determine the debris cloud tip velocity, perforation radius, and target mass loss. Comparing these three 224 10 Figure 4: A comparison between experimental high-speed images and simulation snapshots for a 2.0 km/s, a 4.0 km/s, and a 6.5 km/s, 10 mm sphere HVI into 6.35 mm thick HDPE showing differences in debris cloud size and shape development at t= 35µs: (a)–(b) 2.0 km/s results, (c)–(d) 4.0 km/s results, and (e)–(f) 6.5 km/s results. parameters to experimental values serves as one simple method to assess the validity of the simulation. 225 Representative experimental high-speed images (Figs. 4a, 4c, and 4e) and time-corresponding simulation 226 snapshots (Figs. 4b, 4d, and 4f) show 10 mm diameter sphere HVIs into a 6.35 mm thick HDPE plates at 227 velocities of 2.0 km/s (Figs. 4a and 4b), a 4.0 km/s (Figs. 4c and 4d), and 6.5 km/s (Figs. 4e and 4f). Each 228 experimental and simulated snapshot was taken 35 µs after impact for this side-by-side comparison. The 229 simulation appears to capture the essential character (size, shape, etc.) of the debris cloud across the range of 230 velocities. Some notable differences, however, do exist. For example, the experimental ejecta cloud is larger 231 than the predicted cloud, particularly for the 6.5 km/s impact (Fig. 4e and 4f). The simulation also could 232 not precisely capture the fibrous (spider-web-like) features of the melted ejecta and debris clouds reported 233 by Rogers et al. [28]. This discrepancy is expected as melting and recrystallization phase transitions, as well 234 as solid-liquid interactions, are difficult to reproduce using FE-SPH computational methods. 235 11 Despite these differences, key quantitative metrics between the simulations and experiments agreed quite 236 well for a majority of the investigated velocity regime. The simulated debris cloud tip velocity (Fig. 5a) 237 matched the experimental measurements reasonably well across the entire velocity regime. Except at high 238 velocities ( ≥5.5 km/s), the predicted values tended to be slightly lower than the measured velocities. The 239 simulated perforation radii at impact velocities of 2.0 km/s, 3.0 km/s, 4.0 km/s, and 5.0 km/s are consistent 240 with experimental measurements (Fig. 5b). A noticeable discrepancy, however, exists between the simulated 241 and experimental values at impact velocities >5.0 km/s. The predicted target mass loss (Fig. 5c) agreed 242 reasonably with the experimental data with a slight tendency to over predict target mass loss in the velocity 243 range of 4.0 to 5.0 km/s. These observed discrepancies are likely due to the JC plasticity and fracture models 244 being calibrated using experimental data at strain rates <104s−1(i.e., whereas the average strain rates 245 induced during HVIs likely are on the order of v0/ht∼105–106s−1, where htdenotes target thickness). 246 Constitutive model insensitivity at ultra-high strain rates is well documented in the literature for a 247 variety of materials ( e.g., [3, 46]). In essence, the HVI response of projectile and target materials depends 248 more strongly on density, bulk modulus, projectile/target geometries, and pressure ( i.e., EOS) as v0(strain 249 rate) increases. Furthermore, a reduction in target performance at roughly 5.5–6.5 km/s (Fig. 5d) has 250 been reported to correspond to a transition from partial to full melt of projectile and target materials for 251 aluminum-on-aluminum impacts [28, 47, 48]. A similar transition could account for the observed decrease 252 in target performance for aluminum-on-HDPE impacts. As impact velocity increases, the heating from 253 adiabatic shock and rapid plastic deformation also rises, potentially causing complete melting of the target 254 material along the projectile path. The simulation results appear to deviate from experimental values as 255 the impact velocity exceeds this transition velocity. The ability to accurately predict target mass loss and 256 debris cloud tip velocity has implications on the prediction capabilities of overall debris cloud momentum and 257 kinetic energy, two key metrics in understanding the protective capabilities of a HVI protective structure. 258 Overall, these simulations are very effective in predicting the perforation radius, target mass loss, debris 259 cloud shape, and debris cloud tip velocity resulting from HVIs to HDPE plates over the impact velocity 260 range v0≈2.0–5.5 km/s at this target-thickness-to-projectile-diameter ratio ( ht/D≈0.635). 261 The relatively good agreement motivated the use of simulation results to probe the spatial distributions 262 of HVI-induced target strain rates and temperatures, which cannot yet be readily measured experimentally. 263 Simulation snapshots showing strain rate and temperature distributions at 1.6 µs after 2.0, 4.0, and 6.5 km/s 264 impacts are presented in Figs. 6a–6c and 6d–6f, respectively. Figures 6a, 6b, and 6c provide insight into the 265 strain rates achieved during an HVI event by displaying the strain rates in the simulations as they vary from 266 quasi-static ( <10−4) to ultra-high strain rate ( >106). A strain rate index showing characteristic regimes is 267 also provided for reference. These strain rates are consistent with observed and/or predicted rates in the 268 12 Figure 5: A comparison of simulation and experimental results: the (a) debris cloud tip velocity ( vDC), (b) perforation radius, (c) target mass loss, and (d) impact-velocity-normalized debris cloud tip velocity ( vDC/v0) measurements plotted as a function of impact velocity ( v0) for both experiment and simulation. The discrepancies between experiment and simulation roughly correspond to a transition evident in (d), also observed in Ref [28]. literature [49]. Furthermore, these results confirm that the strain rates generated in the simulation exceed 269 the strain rates used to calibrate the JC model’s strain rate dependency. There is a tendency for higher strain 270 rates to occur closer to the exit face for this particular target and projectile geometry. To provide even more 271 insight into the extreme conditions developed during HVIs, the temperatures recorded by the simulation are 272 displayed with a scale that ranges from room temperature (22.1 °C) to the melting temperature of HDPE 273 (Tm= 131 .1°C). The simulation predicts local temperatures sufficient to melt the polymer a very short 274 time after impact (1.6 µs). Extensive melting has also been observed in experimental high-speed camera 275 13 Figure 6: Elemental strain rate and temperature distributions of HDPE targets t= 1.6 µs after impact for (a,b) 2.0 km/s, (c,d) 4.0 km/s, and (e,f) 6.5 km/s. footage of HDPE impacts (Fig. 2d–f) [28]. Note that at t= 1.6µs, the majority of the elements at or 276 above melting temperature exist in the downrange half of the target’s thickness. This temperature and 277 strain rate distribution could partially explain why there is a slight increase in perforation radius between 278 the target’s impact face and exit face. The ability to accurately predicting through-thickness strain rates and 279 temperatures for a given impact scenario may enable the development of novel polymer protective structures, 280 where the material properties are optimally tuned for the local anticipated HVI-induced strain rates and/or 281 temperatures. 282 4. Conclusions 283 The hybrid FE-meshless particle code, EPIC, was used to simulate 10 mm diameter spherical 1050 284 aluminum projectile HVIs (2.0–6.5 km/s) into 6.35 mm thick HDPE plates. The simulations employed a 285 JC constitutive and fracture model and a Mie-Gr¨ uneisen EOS. The constitutive and fracture models were 286 calibrated using material data and relevant data from the literature. Three HVI damage metrics were used to 287 14 compare experimental and simulation results: debris cloud tip velocity, target perforation radius, and target 288 mass loss. The simulations captured the variation of these three quantities quite well over a large range of 289 the considered impact velocities (2.0–5.0 km/s). Simulation results deviated slightly from experimental data 290 above 5.5 km/s, likely due to inadequate materials data at the corresponding high strain rates (106s−1) and 291 the inability of the Mie-Gr¨ uneisen EOS to appropriately predict phase change as impact velocities increased. 292 A transition velocity has been previously reported in other HVI studies. The simulation results were used 293 to infer the HVI-induced spatial distribution of HDPE target strain rates and temperatures over a range of 294 velocities; to the authors’ knowledge such predictions have not been previously reported for polyethylene. 295 This crucial spatial data can inform ongoing and future experimental, analytical, and numerical studies. 296 The ability to reasonably estimate target mass loss and debris cloud geometry and tip velocity simul- 297 taneously is particularly useful. In designing novel protective structures, quantifying momentum/energy 298 transfer to underlying structures or assets for a given impact scenario is of paramount importance. Accurate 299 estimates of the momentum and kinetic energy of the ejecta/debris clouds, as well as further investigation 300 into impact geometry effects, could permit the application of this approach to the development of HDPE 301 strain-rate tuned and/or architected spacecraft protection systems. The current study serves as an important 302 step towards that goal. 303 Declaration of Competing Interest 304 The authors declare that they have no known competing financial interests or personal relationships that 305 could have appeared to influence the work reported in this paper. 306 Acknowledgments 307 This research is based upon work partially supported by the DoD Science, Mathematics, and Research for 308 Transformation (SMART) program. The authors also acknowledge support through the US Army Engineer 309 Research and Development Center (ERDC). The views and conclusions contained in this document are those 310 of the authors and should not be interpreted as representing the official policies, either expressed or implied, 311 of the Army Research Office or the U.S. Government. The U.S. Government is authorized to reproduce and 312 distribute reprints for Government purposes notwithstanding any copyright notation herein. 313 15