# Study of Hypervelocity Projectile Impact on Thick Metal Plates

Authors: Shawoon K. Roy; Mohamed Trabia; Brendan O’Toole; Robert Hixson; Steven Becker; Michael Pena; Richard Jennings; Deepak Somasoundaram; Melissa Matthes; Edward Daykin; Eric Machorro
Journal: Shock and Vibration
 (2016)

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## Abstract

Hypervelocity impacts generate extreme pressure and shock waves in impacted targets that undergo severe localized deformation within a few microseconds. These impact experiments pose unique challenges in terms of obtaining accurate measurements. Similarly, simulating these experiments is not straightforward. This study proposed an approach to experimentally measure the velocity of the back surface of an A36 steel plate impacted by a projectile. All experiments used a combination of a two-stage light-gas gun and the photonic Doppler velocimetry (PDV) technique. The experimental data were used to benchmark and verify computational studies. Two different finite-element methods were used to simulate the experiments: Lagrangian-based smooth particle hydrodynamics (SPH) and Eulerian-based hydrocode. Both codes used the Johnson-Cook material model and the Mie-Grüneisen equation of state. Experiments and simulations were compared based on the physical damage area and the back surface velocity. The results of this study showed that the proposed simulation approaches could be used to reduce the need for expensive experiments.


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## Body

## 1. Introduction

Hypervelocity impact events are ubiquitous in many areas, including micrometeoroid collision with spacecraft, projectile impacts, and when modeling effects of explosives on structures. Consequently, researchers have been studying various aspects of this problem for several decades. A common technique to study hypervelocity impact in laboratory settings is the two-stage light-gas gun [1, 2], which can accelerate a projectile to generate shock waves in a target similar to those created by detonating high explosives or meteorite collisions [3]. Swift [4] discussed the historical development of this type of gun.Under hypervelocity impact conditions, thin metallic plates tend to stretch and bend around the impact area, absorbing a significant part of the projectile’s kinetic energy before perforation occurs. On the other hand, thick plates experience several failure modes during impact, such as spalling, petalling, discing, and plugging [5]. These failure modes depend upon several factors, such as the impact velocity, the properties of the plate material, and the geometry of the projectile. Spalling, which is of a particular interest in this study, occurs when a triangular-shaped stress wave is reflected from the back of the target plate, thereby creating a tensile pressure that is greater than the material strength [6], which then results in an internal crack that progresses normal to the direction of the wave.Goldsmith et al. [7] studied and developed analytical models for the elastic-plastic plate deformation of aluminum plates impacted by a hard-steel cylindrical-nose projectile. Failure modes for rod penetration experiments under hypervelocity impact conditions have been studied only by a relatively small number of researchers [8]. Christman and Gehrig [9] studied the penetration mechanics and cratering processes in metallic and nonmetallic targets at impact velocities from 0.3 to 6.7 km/s. Sorensen et al. [10] studied penetration mechanics of semi-infinite rolled homogeneous armor (RHA) steel by monolithic projectiles and segmented penetrators at velocities ranging from 2.5 to 4.0 km/s. Subsequent studies on spall fracture and dynamic response of materials were reviewed by Oscarson and Graff [6]. Shockey et al. [11] reported that different projectile materials cause different types of physical damage to steel plates under hypervelocity impact. In addition, they reported microstructural changes and the pressure induced α⇄ε polymorphic transition of the steel plates during impact. The penetration behavior of long rod tungsten-alloy projectiles in high-hard armor steel plates was studied by Anderson et al. [12] at two impact velocities, 1.25 km/s and 1.70 km/s. Bond [13] suggested that spallation damage and phase-transition kinetics were important in damage studies of martensitic steel armor when subjected to hypervelocity impact.Merzhievskii and Titov [14] evaluated the perforation and deformation of thin steel plates at velocities from 3 to 9 km/s; they developed an analytical method to relate debris particles to the impact velocity. The ballistic limits of aluminum plates of various thicknesses were studied by Børvik et al. [15] using a compressed-air gas gun. An analytical model was developed based on those limits. Analytical models were developed to predict the cratering, depth of penetration, and perforation of semi-infinite target plates by long rod penetrators by Wen et al. [16–18].Recent work on impact dynamics and shock physics of materials has emphasized the use of velocimetry data in flyer plate experiments to characterize the equation of state, spall strength, polymorphic phase transition, and the Hugoniot elastic limit of materials. Velocimetry data can provide a clear representation of the response of materials under high pressure and high strain rates, which may lead to developing accurate predictive computational models for materials. The following is a brief overview of velocimetry diagnostic techniques. Barker and Hollenbach developed the homodyne interferometry technique: Velocity Interferometer System for Any Reflector (VISAR) system [19]. This technique became widely popular within the shock physics community. Numerous impact and shock studies have used VISAR as the primary diagnostic tool [20–28]. However, complexity, cost, and various issues that cannot be easily resolved, such as obtaining velocities from multiple points on a moving surface, prompted the development of an alternative diagnostic tool known as photonic Doppler velocimetry (PDV), a displacement interferometer that collects velocity data based on measuring displacement using optical fiber probes [29]. Advantages of PDV include its relative simplicity, robustness, cost, accuracy, and versatility [30]. Chau et al. [31] used PDV for shock Hugoniot measurements of a single-crystal copper plate during impact experiments.Accurate predictive modeling of impact events under hypervelocity conditions can provide a less expensive alternative to conducting actual experiments. However, these numerical simulations require the identification of a large number of parameters that are needed to describe the nonlinearities of the problem. In many cases, hypervelocity impact simulation models are based on many simplifications that affect the accuracy of these simulations. One feature of hypervelocity impacts is that solid materials behave somewhat like fluids after the initial elastic-plastic transition. Therefore, it is appropriate to model plate penetration using hydrodynamic techniques having a separately defined strength model. Recently, a considerable effort has gone into developing models that deal with perforation and penetration problems. For example, it was shown that an accurate computational model of impact depends on the selection of the proper physics models and input parameters [32]. Several constitutive models were developed over the years to simulate hypervelocity impact events. Johnson and Cook [33] developed a constitutive model for materials subjected to large strains, high strain rates, and high temperatures. Two other flow stress models were developed for plastic deformation of ductile materials: Zerilli-Armstrong [34] and Steinberg-Cochran-Guinan-Lund model [35]. A comparative study of all these models was performed by Banerjee [36]. Beside all of these constitutive relations, equation of state (EOS) models were developed to understand the dynamic behavior of materials under shock loading and were often used in this type of simulations. Børvik et al. simulated perforation phenomena in LS-DYNA [37], using a constitutive material model [38, 39] that combined viscosity and ductile damage. Additionally, plate perforation with various nose shapes for projectiles was simulated [40–42] using the viscoplastic relationship described in [39]. Eftis and Nemes simulated ductile spall fracture and postspall behavior of a circular plate, using the PRONTO finite element code where a viscoplastic constitutive relationship was implemented that included the microvoid volume fraction as a scalar material damage variable [43]. Recently, several computational packages based on hydrocode methods that simulate events are associated with extreme high-pressure dynamics [44]. This technique has been used for penetration modeling of different steel plates such as Eulerian-based CTH hydrocode [12, 45–47]. All these works [45–47] listed Mie-Grüneisen EOS in their simulation work.Another simulation approach is to use the Lagrangian-based method of smooth particle hydrodynamics (SPH) technique [48], a meshless Lagrangian numerical method that can address the problems associated with the large geometric distortions where typical grid-based mesh techniques have difficulties. The SPH technique has recently gained popularity in simulating impact-penetration problems [49–53]. Johnson-Cook model was used as the primary strength model in [49] and Mie-Grüneisen EOS was listed in [51].Although numerous studies describe the perforation and penetration mechanics of plates during hypervelocity impact, only a few discussed the plastic deformation of plates that do not experience complete penetration under such conditions [13].This study presents velocimetry data captured by a PDV system in experiments using a two-stage light-gas gun to understand the plastic-deformation behavior of A36 steel plates that have not had full penetration. Two computational approaches were developed to simulate the experiments. Results were compared with each other and to the experimental data.


## 2. Materials and Methods

### 2.1. Materials

All gas gun experiments in this study used Lexan projectiles, cylinders 5.6 mm in diameter and 8.6 mm in length. Depending on the fill pressure of the gas used, the projectile velocity varied from about 4.5 to 6 km/s. The dimensions of the ASTM A36 steel target plates were 152.4 × 152.4 × 12.7 mm. The thickness of the target plates was chosen to prevent their complete penetration due to projectile impacts at these velocities. Young’s modulus (E) for this Lexan and A36 steel used in all experiments was 2.54 GPa and 200 GPa, respectively.


### 2.2. Methods

A two-stage light-gas gun, Figure1, was used to perform the hypervelocity impact experiments. The main components of the gas gun are the powder breech, pump tube, central breech, launch tube, blast tank, and target chamber.Figure 1
Two-stage light-gas gun.The experiments were conducted according to the following steps. A cartridge, filled with gunpowder, was fired using a solenoid pin. The resulting explosion propelled a piston in the pump tube filled with pressurized helium gas. As the piston moved, it increased the pressure of the helium, which eventually breaks a petal valve. The gas then accelerated the projectile, which was placed immediately behind the petal valve in the launch tube, through the blast tank and the drift tube, until it would impact the target within the target chamber.The target plates were bolted using four 12.7 mm (1/2′′) SAE Grade 5 bolts onto a mounting plate that was attached to the walls of the target chamber, Figure2.Figure 2
Target plate bolted inside the gas gun chamber.The velocity of the projectile was measured by a laser intervalometer system having two stations separated by a fixed distance. Each station had a laser source that directed a beam through a port to a narrow band-pass filter to ensure that a 32-photodiode array was free of any contamination by external light. Measuring the time interval was initiated by the flight of the projectile across the first station and terminated when it passed through the second station. The time interval was recorded using a digitizer.Free surface velocity from the target was measured using a PDV system. The basic working principle of a single channel PDV system is described in Figure3 [29]; however, a brief overview is presented. A laser light is fed into an optical fiber probe. The reference source light (typically at a wavelength, λ, of 1550 nm) is reflected from the measured moving surface. The resulting Doppler shifted light is sent to a detector to produce fringes, each of which corresponds to the displacement of the surface by a wavelength difference of  λ/2 between reflected and reference lights. These displacement-time signals are recorded by a high-speed digitizer. The PDV was completely enclosed.Figure 3
Basic principle of a single channel PDV system (based on [29]).In all gas gun experiments, the PDV optical fiber probe was placed in a holder and aimed at the center of the back surface of the target plate, Figure4. The target surface was polished by using different-grit sand papers and ball roller beforehand to reflect light at certain intensity. The PDV system was triggered few microseconds after the projectile passed through the second station of the intervalometer. This delay was based on the projectile velocity and the distance to the target plate was used to initiate the PDV data collection. Once a digitizer recorded the data, a fast Fourier transform was performed to obtain the measured point velocity based on the recorded displacement data.Figure 4
Layout of the PDV data acquisition.Figure4 shows the layout of the experiment. After testing, physical measurements of the impact crater and the resulting back surface bulge were recorded. The target plates were sectioned to check for spall damage.


## 2.1. Materials

All gas gun experiments in this study used Lexan projectiles, cylinders 5.6 mm in diameter and 8.6 mm in length. Depending on the fill pressure of the gas used, the projectile velocity varied from about 4.5 to 6 km/s. The dimensions of the ASTM A36 steel target plates were 152.4 × 152.4 × 12.7 mm. The thickness of the target plates was chosen to prevent their complete penetration due to projectile impacts at these velocities. Young’s modulus (E) for this Lexan and A36 steel used in all experiments was 2.54 GPa and 200 GPa, respectively.


## 2.2. Methods

A two-stage light-gas gun, Figure1, was used to perform the hypervelocity impact experiments. The main components of the gas gun are the powder breech, pump tube, central breech, launch tube, blast tank, and target chamber.Figure 1
Two-stage light-gas gun.The experiments were conducted according to the following steps. A cartridge, filled with gunpowder, was fired using a solenoid pin. The resulting explosion propelled a piston in the pump tube filled with pressurized helium gas. As the piston moved, it increased the pressure of the helium, which eventually breaks a petal valve. The gas then accelerated the projectile, which was placed immediately behind the petal valve in the launch tube, through the blast tank and the drift tube, until it would impact the target within the target chamber.The target plates were bolted using four 12.7 mm (1/2′′) SAE Grade 5 bolts onto a mounting plate that was attached to the walls of the target chamber, Figure2.Figure 2
Target plate bolted inside the gas gun chamber.The velocity of the projectile was measured by a laser intervalometer system having two stations separated by a fixed distance. Each station had a laser source that directed a beam through a port to a narrow band-pass filter to ensure that a 32-photodiode array was free of any contamination by external light. Measuring the time interval was initiated by the flight of the projectile across the first station and terminated when it passed through the second station. The time interval was recorded using a digitizer.Free surface velocity from the target was measured using a PDV system. The basic working principle of a single channel PDV system is described in Figure3 [29]; however, a brief overview is presented. A laser light is fed into an optical fiber probe. The reference source light (typically at a wavelength, λ, of 1550 nm) is reflected from the measured moving surface. The resulting Doppler shifted light is sent to a detector to produce fringes, each of which corresponds to the displacement of the surface by a wavelength difference of  λ/2 between reflected and reference lights. These displacement-time signals are recorded by a high-speed digitizer. The PDV was completely enclosed.Figure 3
Basic principle of a single channel PDV system (based on [29]).In all gas gun experiments, the PDV optical fiber probe was placed in a holder and aimed at the center of the back surface of the target plate, Figure4. The target surface was polished by using different-grit sand papers and ball roller beforehand to reflect light at certain intensity. The PDV system was triggered few microseconds after the projectile passed through the second station of the intervalometer. This delay was based on the projectile velocity and the distance to the target plate was used to initiate the PDV data collection. Once a digitizer recorded the data, a fast Fourier transform was performed to obtain the measured point velocity based on the recorded displacement data.Figure 4
Layout of the PDV data acquisition.Figure4 shows the layout of the experiment. After testing, physical measurements of the impact crater and the resulting back surface bulge were recorded. The target plates were sectioned to check for spall damage.


## 3. Experimental Results

In all gas gun experiments, the Lexan projectiles disintegrated completely. A small crater was created as a result of the impact, with a bulge on the back side of the target plate. Crater details and bulge dimensions are listed in Table1. In all gas gun experiments, the impact location varied within 3 mm of the geometric center of the plate. A typical sectioned target plate showed that spall had occurred inside the material, Figure 5. Table 2 summarizes the spall cracking in the two experiments.Table 1
Physical dimensions of the crater and bulge after impact.
Test ID
Impact velocity (m/s)
Crater diameter(mm)
Penetration(mm)
Bulge (mm)
1000-016
5338
17.0
6.3
2.3
1000-017
5063
16.9
5.8
2.3Table 2
Details of spall crack in sectioned plates.
Test ID
Impact velocity (m/s)
Crack diameter (mm)
Crack width (mm)
Spall crack location with respect to free surface (mm)
1000-016
5338
18.5
1.8
2.3
1000-017
5063
18.5
1.7
2.4Figure 5
Typical target plate after an experiment.Figure6 shows typical velocimetry data from a single channel PDV. The duration of the projectile-target interaction typically lasted for 25 μs. The first 5 μs displayed the most characteristic features of the hypervelocity impact, namely, elastic precursor wave, HEL, plastic wave propagation, spall signature, and elastic loading and unloading. A brief discussion of these features follows.Figure 6
Typical free surface velocity data as obtained by a PDV system.When material is loaded in extreme pressure, the shockwave creates an elastic response up to certain limit. This limit is usually defined by HEL, which is usually known as the elastic precursor wave. After this limit, material flows plastically due to the strong shockwave propagation. In the case of uniaxial strain, the peak velocity is followed by multiple loading and unloading phase. The first drop from the peak velocity and subsequent loading-unloading zones are associated with the spall signature in metals. Typically, the first spall signature in metals is followed by a significant sharp drop in free surface velocity, which is defined as the elastic unloading stage.


## 4. Numerical Simulations

This study described two approaches to simulate the experiments described in the previous section: a smooth particle hydrodynamics (SPH) solver in LS-DYNA software and the combined hydro, radiation, and transport diffusion (CTH) hydrocode. The remainder of this section details model development.


### 4.1. Material Model

Both the LS-DYNA SPH and the CTH computer codes used the Johnson-Cook material model [10] for both the Lexan projectile and A36 steel plate. In the Johnson-Cook material model of plasticity, flow stress is expressed as(1)σy=A+Bεpn1+Cln⁡ε·∗⁡1-T∗m,where σy is the flow stress; A, B, C, n, and m were material constants; εp is the effective plastic strain; and ε·∗ is the effective total strain rate normalized by quasistatic strain rate. The homologous temperature, T∗, is defined as (2)T∗=T-TrTm-Tr,where Tr and Tm were room and the melting temperatures in Kelvin, respectively.All the parameters of the Johnson-Cook model for Lexan [54] and A36 steel [55] are listed in Table 3.Table 3
Johnson-Cook material properties for the projectile and the target.
Material
A (MPa)
B (MPa)
C
M
n
T
m (Kelvin)
ν
†
EPSO
Lexan [54]
75.8
68.9
0
1.85
1.004
533
0.344
1
A36 steel [55]
286.1
500.1
0.022
0.917
0.2282
1811
0.260
1
†
ν is Poisson’s ratio.


### 4.2. Equation of State (EOS)

Materials under shock wave loading needed a shock model that could account for the sudden pressure, temperature, internal energy, and density changes that occur in front of the shock waves. The equation of state (EOS) of a material is a general thermodynamic relation that is defined by the code user. Various forms of EOS were used to describe the volumetric compression or expansion behavior of different types of materials. One of the most commonly used EOS is the Mie-Grüneisen equation of state, which can be expressed [58] as (3)P=ρ0C02μ1+1-γ0/2μ-a/2μ21-S1-1μ-S2μ2/μ+1-S3μ3/μ+122+γ0+aμE,where P is the pressure; S1, S2, and S3 are the coefficients of slope of shock velocity-particle velocity curve, γ0 is the Grüneisen coefficient; a is the volume correction factor; ρ is the instantaneous density; C0 is the Hugoniot intercept of the metal; E is the internal energy; and μ=(ρ/ρ0-1), where ρ0 is the reference density. For materials under compression, a temperature-corrected form of the above equation is given in the following [59]: (4)P=ρ0C02μ1+1-γ0/2μ1-S1-1μ2+γ0E.Assuming a negligible change in density and internal energy, the above equations can be rewritten as(5)P=ρ0C022-γ0/21-S12.Both LS-DYNA and CTH included a Grüneisen EOS for Lexan and A36 steel; these were used in simulations during this study, using the input parameters listed in Table 4.Table 4
EOS parameters for the projectile and the target.
Material
ρ
0 (kg/m3)
C
0 (m/s)
S
1
γ
0
Lexan [56]
1190
1933
1.42††
0.61
A36 steel [57]
7890
4569
1.49
2.17
†
†
S
1 value for Lexan is suggested as 1.42 [55].


### 4.3. Estimation of the Hugoniot Elastic Limit and Spall Strength of A36 Steel

In most fundamental shock studies, velocimetry data are obtained from uniaxial strain experiments on the shock Hugoniot. This work considered the case of a projectile plate penetration experiment that can be described using axisymmetric assumptions. The Hugoniot elastic limit and spall strength were calculated based on the uniaxial case in the absence of axisymmetric data.The approximate Hugoniot elastic limit,σHEL, and spall strength, σspall, of A36 steel were calculated from this velocity profile by assuming that the impact was a one-dimensional localized phenomenon with the following relations [28]:(6)σHEL=12ΔUHρ0cl,σspall=12ΔUfsρ0cb,where ΔUH was the free surface velocity at elastic precursor wave; ΔUfs was the pullback velocity of free surface as shown in Figure 7; and cl and cb correspond to the longitudinal and bulk sound speed, respectively. These speeds depend on Young’s modulus, E, and the bulk modulus, K, of A36 steel. Equations are given as follows:(7)cl=Eρ,cb=Kρ,where cl and cb were calculated to be 5035 m/s and 4212 m/s, respectively, for values of 200 GPa [60] and 140 GPa [60] of E and K, respectively.Figure 7
HEL and spall strength calculation for typical test.Based on these equations, the Hugoniot elastic limit and the spall strength of A36 steel were approximated based on the experiments and the above analysis; results are shown in Table5.Table 5
Hugoniot elastic limit and spall strength.
Test ID
Impact velocity (m/s)
Δ
U
H (m/s)
σ
HEL (GPa)
Δ
U
fs (m/s)
σ
spall (GPa)
1000-016
5338
61.22
1.22
91.2
1.52
1000-017
5063
49.30
0.98
57
0.95Spalling of the material could be defined using either a pressure cut-off (Pmin) value in the Johnson-Cook model or a simple spall threshold parameter independent of the Johnson-Cook material-failure model. In this study, the first option was used to induce the spall in both LS-DYNA and CTH simulations. By taking the average of the two spall strength values of Table5, a pressure cut-off value of 1.23 GPa was determined.


### 4.4. Developing Simulation Models

The same two-dimensional (2D) axisymmetric geometry was used for both the SPH and the CTH simulations. The same material model, equations of the state, and boundary conditions were used in the CTH models. Because the shock did not reach the boundaries during the period of interest, it was decided that the target plates be modeled as cylindrical plates with a radius of 76.2 mm. This assumption allowed the use of a 2D axisymmetric SPH model.


#### 4.4.1. SPH in LS-DYNA

A body in an SPH model is represented by a set of particles that are placed in interpolation points within the body. The simulation primarily depends on an interpolation function called the smoothing length [49]. In addition, parameters such as particle density, bulk viscosity, and scale factors contribute to the quality of the solution.Artificial bulk viscosity was included to dampen the numerical ringing and oscillations of the shock front. The artificial bulk viscosity parameters Q1 and Q2 are assigned values of 1.5 and 1.0, respectively [61, 62]. No boundary conditions were defined for the model. Figure 8 shows the SPH model.Figure 8
A typical SPH model (zoomed in) with 0.05 mm particle spacing.Several SPH particle spacing arrangements were tested while ensuring that the mass of each projectile particle was approximately equal to the mass of each target plate particle. All the simulation models were tested in a 46-core CentOS 5.6 system having LS-DYNA R 7.0 MPP (massively parallel processing) version. A summary of the free surface velocity profiles for different particle spacing was presented in Figure9. The results showed that increasing the particle density resulted in a better representation of the elastic precursor wave. However, it was observed that increasing the particle density increases the peak velocity when compared with the experimental curve. Additionally, increasing the particle in the SPH model required significantly more computational time. Therefore, a particle spacing of 0.10 mm was selected for all subsequent SPH simulations.Figure 9
Typical SPH particle sensitivity study.


#### 4.4.2. CTH Model

A typical CTH model with a 0.10 mm × 0.10 mm zone is shown in Figure10. Zone-size studies recommended an optimal zone size of 0.10 mm × 0.10 mm for all CTH models (Figure 11) as this zone density reasonably described the elastic wave and the sharp rise of the plastic wave.Figure 10
A typical CTH model with 0.10 mm × 0.10 mm zone (zoomed in).Figure 11
A typical CTH zone-sensitivity study with a 1200 MPa spall strength.


### 4.5. Comparison of Simulation Results

Simulation results for LS-DYNA and CTH are shown in Figure12. The results were compared with the experiments in terms of crater dimensions, spall details, and free surface velocity profiles.Figure 12
Spall plane in Test 1000-017 simulations.As Table6 shows, both techniques accurately captured the dimensions of the crater. The shorter simulation time within CTH (10 μs) may explain the larger difference in the length of the back surface bulge dimension. LS-DYNA simulations showed higher magnitude of crater diameter and bulge for both experiments compared to CTH simulations. But LS-DYNA simulations underpredicted the crater depth in case of 5063 km/s impact experiment. The exact reason for this anomaly is yet to be understood; both codes captured the physical damage reasonably.Table 6
Physical comparison of the impact craters.
Impact velocity (m/s)
Test ID
Crater diameter (mm)
% difference
Crater depth (mm)
% difference
Back side bulge(mm)
% difference
5338
1000-016
17.0
N/A
6.3
N/A
2.3
N/A
LS-DYNA
17.1
+0.6
5.9
−7.1
2.1
−8.1
CTH
15.7
−7.6
5.7
−9.3
2.0
−19.5
5063
1000-017
16.9
N/A
5.8
N/A
2.4
N/A
LS-DYNA
15.9
−5.9
5.0
−13.8
2.2
−7.8
CTH
15.7
−7.1
5.7
−1.7
1.9
−12.9Additionally, spall damage was monitored in both simulations. These simulations were able to capture the spall behavior in the material (Figure12); the location of the spall plane is compared in Table 7. As simulation models were not run for longer times, spall cracks did not develop completely. Location of the spall plane was measured from the back surface of the target plate to the start of spall crack plane.Table 7
Comparison of spall crack in simulations.
Impact velocity(m/s)
Test ID
Spall crack location with respect to free surface (mm)
% difference
5338
1000-016
2.3
N/A
LS-DYNA (10μs)
2.8
−19.6
CTH (10μs)
2.6
13.0
5063
1000-017
2.4
N/A
LS-DYNA (10μs)
2.3
−3.3
CTH (10μs)
2.1
−12.5Comparison of the free surface velocity profiles for the first 5μs of both tests is presented in Figures 13 and 14. Free surface velocity profiles of the simulations and experiments were compared in terms of (a) elastic precursor wave and HEL, (b) plastic wave rise, (c) peak velocity, and (d) spall signatures, which were defined in Section 3. The values of these variables are listed in Table 8. The results show that both LS-DYNA and CTH simulations captured the elastic precursor. However, both of these simulations showed lower HEL. Both LS-DYNA and CTH simulation showed sharper rise in the plastic wave. The slope of the simulation curves was steeper than that of the experimental velocity profile. Both LS-DYNA and CTH simulations were able to capture the pullback velocity signal after the second peak velocity, which determines spall strength of the material. However, the magnitude of the pullback velocity signal was significantly different from what was observed in the experiment, which may be due to the fact that the spall strength values used in both CTH and LS-DYNA simulations were based on the assumed similarity with the one-dimensional flier plate experiments. In both cases, CTH overestimated second peak in velocity profiles when compared to the corresponding LS-DYNA simulations. This difference can be explained by the fact that Eulerian-based hydrocode often had issues modeling broader set of physical behaviors and Lagrangian or coupled Eulerian-Lagrangian method was preferred in those cases [63], though both of these codes had their limitations comparing to the experimental velocity profiles.Table 8
Comparison of free surface velocities in simulations.
Impact velocity (m/s)
Test ID
HEL (GPa)
% difference
1st peakvelocity (m/s)
% difference
2nd peak velocity (m/s)
% difference
5338
1000-016
1.22
N/A
243.0
N/A
195.0
N/A
LS-DYNA (5μs)
0.64
−47.5
232.0
−4.5
220.0
12.8
CTH (5μs)
0.57
−53.3
247.2
1.7
256.5
31.5
5063
1000-017
0.95
N/A
205.0
N/A
171.0
N/A
LS-DYNA (5μs)
0.64
−32.6
228.9
11.7
195.1
14.1
CTH (5μs)
0.57
–40.0
224.8
9.7
240.2
40.5Figure 13
Free surface velocity comparison of Test 1000-016 (impact velocity: 5.338 km/s).Figure 14
Free surface velocity comparison of Test 1000-017 (impact velocity: 5.063 km/s).


## 4.1. Material Model

Both the LS-DYNA SPH and the CTH computer codes used the Johnson-Cook material model [10] for both the Lexan projectile and A36 steel plate. In the Johnson-Cook material model of plasticity, flow stress is expressed as(1)σy=A+Bεpn1+Cln⁡ε·∗⁡1-T∗m,where σy is the flow stress; A, B, C, n, and m were material constants; εp is the effective plastic strain; and ε·∗ is the effective total strain rate normalized by quasistatic strain rate. The homologous temperature, T∗, is defined as (2)T∗=T-TrTm-Tr,where Tr and Tm were room and the melting temperatures in Kelvin, respectively.All the parameters of the Johnson-Cook model for Lexan [54] and A36 steel [55] are listed in Table 3.Table 3
Johnson-Cook material properties for the projectile and the target.
Material
A (MPa)
B (MPa)
C
M
n
T
m (Kelvin)
ν
†
EPSO
Lexan [54]
75.8
68.9
0
1.85
1.004
533
0.344
1
A36 steel [55]
286.1
500.1
0.022
0.917
0.2282
1811
0.260
1
†
ν is Poisson’s ratio.


## 4.2. Equation of State (EOS)

Materials under shock wave loading needed a shock model that could account for the sudden pressure, temperature, internal energy, and density changes that occur in front of the shock waves. The equation of state (EOS) of a material is a general thermodynamic relation that is defined by the code user. Various forms of EOS were used to describe the volumetric compression or expansion behavior of different types of materials. One of the most commonly used EOS is the Mie-Grüneisen equation of state, which can be expressed [58] as (3)P=ρ0C02μ1+1-γ0/2μ-a/2μ21-S1-1μ-S2μ2/μ+1-S3μ3/μ+122+γ0+aμE,where P is the pressure; S1, S2, and S3 are the coefficients of slope of shock velocity-particle velocity curve, γ0 is the Grüneisen coefficient; a is the volume correction factor; ρ is the instantaneous density; C0 is the Hugoniot intercept of the metal; E is the internal energy; and μ=(ρ/ρ0-1), where ρ0 is the reference density. For materials under compression, a temperature-corrected form of the above equation is given in the following [59]: (4)P=ρ0C02μ1+1-γ0/2μ1-S1-1μ2+γ0E.Assuming a negligible change in density and internal energy, the above equations can be rewritten as(5)P=ρ0C022-γ0/21-S12.Both LS-DYNA and CTH included a Grüneisen EOS for Lexan and A36 steel; these were used in simulations during this study, using the input parameters listed in Table 4.Table 4
EOS parameters for the projectile and the target.
Material
ρ
0 (kg/m3)
C
0 (m/s)
S
1
γ
0
Lexan [56]
1190
1933
1.42††
0.61
A36 steel [57]
7890
4569
1.49
2.17
†
†
S
1 value for Lexan is suggested as 1.42 [55].


## 4.3. Estimation of the Hugoniot Elastic Limit and Spall Strength of A36 Steel

In most fundamental shock studies, velocimetry data are obtained from uniaxial strain experiments on the shock Hugoniot. This work considered the case of a projectile plate penetration experiment that can be described using axisymmetric assumptions. The Hugoniot elastic limit and spall strength were calculated based on the uniaxial case in the absence of axisymmetric data.The approximate Hugoniot elastic limit,σHEL, and spall strength, σspall, of A36 steel were calculated from this velocity profile by assuming that the impact was a one-dimensional localized phenomenon with the following relations [28]:(6)σHEL=12ΔUHρ0cl,σspall=12ΔUfsρ0cb,where ΔUH was the free surface velocity at elastic precursor wave; ΔUfs was the pullback velocity of free surface as shown in Figure 7; and cl and cb correspond to the longitudinal and bulk sound speed, respectively. These speeds depend on Young’s modulus, E, and the bulk modulus, K, of A36 steel. Equations are given as follows:(7)cl=Eρ,cb=Kρ,where cl and cb were calculated to be 5035 m/s and 4212 m/s, respectively, for values of 200 GPa [60] and 140 GPa [60] of E and K, respectively.Figure 7
HEL and spall strength calculation for typical test.Based on these equations, the Hugoniot elastic limit and the spall strength of A36 steel were approximated based on the experiments and the above analysis; results are shown in Table5.Table 5
Hugoniot elastic limit and spall strength.
Test ID
Impact velocity (m/s)
Δ
U
H (m/s)
σ
HEL (GPa)
Δ
U
fs (m/s)
σ
spall (GPa)
1000-016
5338
61.22
1.22
91.2
1.52
1000-017
5063
49.30
0.98
57
0.95Spalling of the material could be defined using either a pressure cut-off (Pmin) value in the Johnson-Cook model or a simple spall threshold parameter independent of the Johnson-Cook material-failure model. In this study, the first option was used to induce the spall in both LS-DYNA and CTH simulations. By taking the average of the two spall strength values of Table5, a pressure cut-off value of 1.23 GPa was determined.


## 4.4. Developing Simulation Models

The same two-dimensional (2D) axisymmetric geometry was used for both the SPH and the CTH simulations. The same material model, equations of the state, and boundary conditions were used in the CTH models. Because the shock did not reach the boundaries during the period of interest, it was decided that the target plates be modeled as cylindrical plates with a radius of 76.2 mm. This assumption allowed the use of a 2D axisymmetric SPH model.


### 4.4.1. SPH in LS-DYNA

A body in an SPH model is represented by a set of particles that are placed in interpolation points within the body. The simulation primarily depends on an interpolation function called the smoothing length [49]. In addition, parameters such as particle density, bulk viscosity, and scale factors contribute to the quality of the solution.Artificial bulk viscosity was included to dampen the numerical ringing and oscillations of the shock front. The artificial bulk viscosity parameters Q1 and Q2 are assigned values of 1.5 and 1.0, respectively [61, 62]. No boundary conditions were defined for the model. Figure 8 shows the SPH model.Figure 8
A typical SPH model (zoomed in) with 0.05 mm particle spacing.Several SPH particle spacing arrangements were tested while ensuring that the mass of each projectile particle was approximately equal to the mass of each target plate particle. All the simulation models were tested in a 46-core CentOS 5.6 system having LS-DYNA R 7.0 MPP (massively parallel processing) version. A summary of the free surface velocity profiles for different particle spacing was presented in Figure9. The results showed that increasing the particle density resulted in a better representation of the elastic precursor wave. However, it was observed that increasing the particle density increases the peak velocity when compared with the experimental curve. Additionally, increasing the particle in the SPH model required significantly more computational time. Therefore, a particle spacing of 0.10 mm was selected for all subsequent SPH simulations.Figure 9
Typical SPH particle sensitivity study.


### 4.4.2. CTH Model

A typical CTH model with a 0.10 mm × 0.10 mm zone is shown in Figure10. Zone-size studies recommended an optimal zone size of 0.10 mm × 0.10 mm for all CTH models (Figure 11) as this zone density reasonably described the elastic wave and the sharp rise of the plastic wave.Figure 10
A typical CTH model with 0.10 mm × 0.10 mm zone (zoomed in).Figure 11
A typical CTH zone-sensitivity study with a 1200 MPa spall strength.


## 4.4.1. SPH in LS-DYNA

A body in an SPH model is represented by a set of particles that are placed in interpolation points within the body. The simulation primarily depends on an interpolation function called the smoothing length [49]. In addition, parameters such as particle density, bulk viscosity, and scale factors contribute to the quality of the solution.Artificial bulk viscosity was included to dampen the numerical ringing and oscillations of the shock front. The artificial bulk viscosity parameters Q1 and Q2 are assigned values of 1.5 and 1.0, respectively [61, 62]. No boundary conditions were defined for the model. Figure 8 shows the SPH model.Figure 8
A typical SPH model (zoomed in) with 0.05 mm particle spacing.Several SPH particle spacing arrangements were tested while ensuring that the mass of each projectile particle was approximately equal to the mass of each target plate particle. All the simulation models were tested in a 46-core CentOS 5.6 system having LS-DYNA R 7.0 MPP (massively parallel processing) version. A summary of the free surface velocity profiles for different particle spacing was presented in Figure9. The results showed that increasing the particle density resulted in a better representation of the elastic precursor wave. However, it was observed that increasing the particle density increases the peak velocity when compared with the experimental curve. Additionally, increasing the particle in the SPH model required significantly more computational time. Therefore, a particle spacing of 0.10 mm was selected for all subsequent SPH simulations.Figure 9
Typical SPH particle sensitivity study.


## 4.4.2. CTH Model

A typical CTH model with a 0.10 mm × 0.10 mm zone is shown in Figure10. Zone-size studies recommended an optimal zone size of 0.10 mm × 0.10 mm for all CTH models (Figure 11) as this zone density reasonably described the elastic wave and the sharp rise of the plastic wave.Figure 10
A typical CTH model with 0.10 mm × 0.10 mm zone (zoomed in).Figure 11
A typical CTH zone-sensitivity study with a 1200 MPa spall strength.


## 4.5. Comparison of Simulation Results

Simulation results for LS-DYNA and CTH are shown in Figure12. The results were compared with the experiments in terms of crater dimensions, spall details, and free surface velocity profiles.Figure 12
Spall plane in Test 1000-017 simulations.As Table6 shows, both techniques accurately captured the dimensions of the crater. The shorter simulation time within CTH (10 μs) may explain the larger difference in the length of the back surface bulge dimension. LS-DYNA simulations showed higher magnitude of crater diameter and bulge for both experiments compared to CTH simulations. But LS-DYNA simulations underpredicted the crater depth in case of 5063 km/s impact experiment. The exact reason for this anomaly is yet to be understood; both codes captured the physical damage reasonably.Table 6
Physical comparison of the impact craters.
Impact velocity (m/s)
Test ID
Crater diameter (mm)
% difference
Crater depth (mm)
% difference
Back side bulge(mm)
% difference
5338
1000-016
17.0
N/A
6.3
N/A
2.3
N/A
LS-DYNA
17.1
+0.6
5.9
−7.1
2.1
−8.1
CTH
15.7
−7.6
5.7
−9.3
2.0
−19.5
5063
1000-017
16.9
N/A
5.8
N/A
2.4
N/A
LS-DYNA
15.9
−5.9
5.0
−13.8
2.2
−7.8
CTH
15.7
−7.1
5.7
−1.7
1.9
−12.9Additionally, spall damage was monitored in both simulations. These simulations were able to capture the spall behavior in the material (Figure12); the location of the spall plane is compared in Table 7. As simulation models were not run for longer times, spall cracks did not develop completely. Location of the spall plane was measured from the back surface of the target plate to the start of spall crack plane.Table 7
Comparison of spall crack in simulations.
Impact velocity(m/s)
Test ID
Spall crack location with respect to free surface (mm)
% difference
5338
1000-016
2.3
N/A
LS-DYNA (10μs)
2.8
−19.6
CTH (10μs)
2.6
13.0
5063
1000-017
2.4
N/A
LS-DYNA (10μs)
2.3
−3.3
CTH (10μs)
2.1
−12.5Comparison of the free surface velocity profiles for the first 5μs of both tests is presented in Figures 13 and 14. Free surface velocity profiles of the simulations and experiments were compared in terms of (a) elastic precursor wave and HEL, (b) plastic wave rise, (c) peak velocity, and (d) spall signatures, which were defined in Section 3. The values of these variables are listed in Table 8. The results show that both LS-DYNA and CTH simulations captured the elastic precursor. However, both of these simulations showed lower HEL. Both LS-DYNA and CTH simulation showed sharper rise in the plastic wave. The slope of the simulation curves was steeper than that of the experimental velocity profile. Both LS-DYNA and CTH simulations were able to capture the pullback velocity signal after the second peak velocity, which determines spall strength of the material. However, the magnitude of the pullback velocity signal was significantly different from what was observed in the experiment, which may be due to the fact that the spall strength values used in both CTH and LS-DYNA simulations were based on the assumed similarity with the one-dimensional flier plate experiments. In both cases, CTH overestimated second peak in velocity profiles when compared to the corresponding LS-DYNA simulations. This difference can be explained by the fact that Eulerian-based hydrocode often had issues modeling broader set of physical behaviors and Lagrangian or coupled Eulerian-Lagrangian method was preferred in those cases [63], though both of these codes had their limitations comparing to the experimental velocity profiles.Table 8
Comparison of free surface velocities in simulations.
Impact velocity (m/s)
Test ID
HEL (GPa)
% difference
1st peakvelocity (m/s)
% difference
2nd peak velocity (m/s)
% difference
5338
1000-016
1.22
N/A
243.0
N/A
195.0
N/A
LS-DYNA (5μs)
0.64
−47.5
232.0
−4.5
220.0
12.8
CTH (5μs)
0.57
−53.3
247.2
1.7
256.5
31.5
5063
1000-017
0.95
N/A
205.0
N/A
171.0
N/A
LS-DYNA (5μs)
0.64
−32.6
228.9
11.7
195.1
14.1
CTH (5μs)
0.57
–40.0
224.8
9.7
240.2
40.5Figure 13
Free surface velocity comparison of Test 1000-016 (impact velocity: 5.338 km/s).Figure 14
Free surface velocity comparison of Test 1000-017 (impact velocity: 5.063 km/s).


## 5. Conclusion

Gas gun experiments were performed to measure the plastic deformation of A36 steel plates during hypervelocity impacts. The velocity of the back surface of plates was measured using a PDV system. Simulation models were developed in the LS-DYNA SPH solver and the CTH hydrocode. Both models used Johnson-Cook material model and the Mie-Grüneisen equation of state. A procedure for identifying the Hugoniot elastic limit and spall strength of A36 steel was presented. A study was conducted to determine SPH particle sensitivity, and CTH zone spacing studies were conducted to identify the best meshing strategy. The results showed that both simulation approaches were able to accurately match the physical measurements of impact cratering. Moreover, the simulations were able to predict the velocity profiles in the PDV experiments; however, some differences were observed. Additional experiments and fine-tuning of the simulation models were needed, including the use of more accurate material models and simulation parameters. Furthermore, studies were needed on the effect of the pressure inducedα↔ε phase transition that is known to occur in pure iron.



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*Source: 4313480-2016-01-10.xml*