I. Introduction

II. Materials and Failure Models

2.2. Failure Models

Failure models describing the onset and evolution of failure used in our numerical simulations were the Johnson-Cook (JC) [27] for the metals, Virtual User Material (VUMAT) [28] developed by Dassault Systems for the PMCs, and the Prony series and Virtual User Defined Field (VUSDFLD) for PU [27,29]. The script developed allows for the joint application of diverse material models when analyzing several different impact conditions.

2.2.1. Johnson-Cook Model

The material model and the calibrated Johnson-Cook (J-C) failure parameters were used in the verification script for the ALU2024. The material response is linear elastic up to its yield strength, after which it deforms plastically until its failure strain [27]. The yield stress (${\overline{\sigma }}_Y$) and the strain at failure (${\overline{\epsilon }}_f)$ [30] are expressed by equations (1) and (2).

${\overline{\sigma }}_Y\left(A, B, n, C, \dot{\overline{{\epsilon }_0},}{T}_m,T_0,m\right)=\left[A+B {\left({\overline{\epsilon }}_{pl}\right)}^n\right] \left[1+C ln{\displaystyle \frac{{\dot{\overline{\epsilon }}}_{pl}}{{\dot{\overline{\epsilon }}}_0}}\right]\left[1-{\left({\displaystyle \frac{T-T_0}{T_m-T_0}}\right)}^m\right]$ (1)

${\overline{\epsilon }}_f\left(d_1,d_2, d_3, d_4, d_5, \dot{\overline{{\epsilon }_0},} T_m,T_0\right)$ = $\left[d_1+d_2 \mathrm{e}\mathrm{x}\mathrm{p}\left(d_3\eta \right)\right] \left[1+d_4\ln \left({\displaystyle \frac{{\dot{\overline{\epsilon }}}_{pl}}{{\dot{\overline{\epsilon }}}_0}}\right)\right] \left(1+d_5{\displaystyle \frac{T-T_0}{T_m-T_0}} \right)$ (2)

In equations 1 and 2, η= -p⁄q represents the triaxiality, defined as the hydrostatic pressure divided by the deviatoric stress or (Von Mises stress), $\dot{\overline{{\epsilon }_0}}$ is the reference strain rate, T is the attained temperature of the material, T_m is the melting point, T₀ is the ambient temperature or the reference temperature, ${\overline{\epsilon }}_{pl}$ is the plastic strain, and ${\dot{\overline{\epsilon }}}_{pl}$ the plastic strain rate, C is the viscous effect constant, n is the strain hardening exponent, m is the thermal softening exponent, and $d_1,d_2, d_3, d_4, {\mathrm{a}\mathrm{n}\mathrm{d} d}_{5 }$are damage parameters.

2.2.2. Adiabatic Considerations

During impact, the high plastic strain rate causes the dissipation of energy as heat which raises the temperature (T) of each of the impacted elements in the model. The temperature rise is exclusive to each element and there is no heat conduction assumed between the elements. The inelastic heat fraction applied in the simulation was 0.9, that is, 90% of the energy dissipated by the plastic deformation was converted to heat.

Table 1. Properties of aluminum and steel plates used for numerical verification.

Table 1. Properties of aluminum and steel plates used for numerical verification.

Table 2. Johnson-Cook properties for 4340 steel and lead projectiles.

Table 2. Johnson-Cook properties for 4340 steel and lead projectiles.

2.2.2. Virtual User Material for Polymer Matrix Composites

The subroutine for composite progressive damage, Virtual User Material (VUMAT), was made available by Simulia [24] in their documentation. The material properties of CFRP linked to the VUMAT subroutine that was applied in this study were also from [24] and are summarized in Table III. The VUMAT subroutine was enhanced by utilizing Hashin’s damage model for composites [33] for the initiation and evolution of the through-thickness damage to the fiber and matrix for PMCs. In the subroutine, the Hashin damage criterion was specified for the fiber failure mode while the Puck criterion [24,34] was applied to the matrix damage modes. The relationship between fiber and matrix damage utilized in the VUMAT subroutine is given by [24]:

$\mathrm{F}\mathrm{i}\mathrm{b}\mathrm{e}\mathrm{r} \mathrm{d}\mathrm{a}\mathrm{m}\mathrm{a}\mathrm{g}\mathrm{e}\left(\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}\right): d={\left({\displaystyle \frac{{\sigma }_{11}}{X_{1tf}}}\right)}^2+{\left({\displaystyle \frac{{\tau }_{12}}{S_{12f}}}\right)}^2+{\left({\displaystyle \frac{{\tau }_{13}}{S_{13f}}}\right)}^2$(3)

$\mathrm{F}\mathrm{i}\mathrm{b}\mathrm{e}\mathrm{r} \mathrm{d}\mathrm{a}\mathrm{m}\mathrm{a}\mathrm{g}\mathrm{e}\left(\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}\right):d={\displaystyle \frac{\left|{\sigma }_{11}\right|}{X_{1cf}}}$(4)

Matrix tension and compression (Puck)

$d=\left[{\left({\displaystyle \frac{{\sigma }_{11}}{{2X}_{1tm}}}\right)}^2+{\displaystyle \frac{{\sigma }_{22}^2}{X_{2tm}.X_{2cm}}}+{\left({\displaystyle \frac{{\tau }_{12}}{S_{12m}}}\right)}^2\right]+{\sigma }_{22}\left({\displaystyle \frac{1}{X_{2tm}}}+{\displaystyle \frac{1}{X_{2cm}}}\right)$ (5)

• where ${\sigma }_{ij}{,and \tau }_{ij},$ are the effective stress tensors, $X_{1tf}, {X_{1cf}, S}_{12f}, {\mathrm{a}\mathrm{n}\mathrm{d} S}_{13f}$are the fiber tensile, compressive, and shear strengths. $X_{1tm}, \mathrm{a}\mathrm{n}\mathrm{d}{X}_{2tm},$ are the matrix tensile failure stress in the 1 and 2 directions (1 is the fiber direction), $X_{2cm}$is the matrix compressive failure stress in the 2-direction, and $S_{12m}$, is the matrix failure shear stress.

Table 3. CFRP and GFRP Unidirectional (UD) Laminae Properties.

Table 3. CFRP and GFRP Unidirectional (UD) Laminae Properties.

Table 4. Material Properties for Polyurea [29].

Table 4. Material Properties for Polyurea [29].

2.2.3. PU Material Model

The material properties of the particular PU (four parts of Versalink^®P1000 and one part of Isonate^®143L) investigated by [29] are listed in Table IV. PU exhibits a linear and viscous response in its stress/strain characteristics. The time-dependent behavior of PU is attributed to the material’s hard and soft segments, i.e., the motion of the chain at different time scales. PU has been modeled as a linear viscoelastic isotropic solid material by [29] using the material’s shear relaxation modulus G. The Prony series was used in [21] and in this work to model the relaxation of a polymer consisting of n decaying exponentials. Equations 6 and 7 describe the time dependent shear and bulk moduli of PU under high strain rates [35].

$G_t=\left[G_{\infty }+{\sum }_{k=1}^n{g}_k{G}_0.\mathrm{e}\mathrm{x}\mathrm{p}(-{\displaystyle \frac{t}{{\tau }_k}})\right]$ where $g_k$ = ${\displaystyle \frac{G_k}{G_0}}$ (6)

$K_t=\left[K_{\infty }+{\sum }_{k=1}^n{k}_k{K}_0.\mathrm{e}\mathrm{x}\mathrm{p}(-\frac{t}{{\tau }_k})\right]$ where $k_k$ = ${\displaystyle \frac{K_k}{K_0}}$ (7)

The terms g_k and k_k in equations (6) and (7) are the ratios of the shear and bulk moduli to the shear and bulk moduli of the kth element at the onset of deformation, and τ_k is the relaxation time of the kth element, that is the time it takes to ‘relax’ the stress to about 38% (1/e) of the initial applied stress. $G_0$, and $K_0$ are the shear and bulk moduli at time t = 0, $G_{\infty }, K_{\infty }$ are the shear and bulk moduli of the first element in the generalized Maxwell model. These are the viscous properties needed for input in a Prony series in the time domain in the Abaqus explicit simulation. In this work, 14 terms of the Prony series, provided by the experimental work of [29] were used in the simulation. Finally, a Virtual User Defined Field (VUSDFLD) [27] subroutine was used to assess the response of PU. VUSDFLD is a custom constitutive model used for modeling the high strain rate behavior of PU. The VUSDFLD subroutine computes the strain in the material at each time step using the Prony series and then compared it with a critical strain value at which the material fails. For our work, the critical strain at failure value for PU was set at 0.25, in the middle of the range of values for PU subjected to strain rates experienced in ballistic impacts at the prescribed velocity.

III. Independent Verification of FE Models

3.2. FE Model Verification on CFRP Panels

To confirm the validity of the PMC model, a comparison was made to an experiment conducted by [23]. In that work, a 5 mm diameter spherical steel projectile of mass 0.51 g was fired at three different velocities at a multi-ply composite panel and the residual velocities were measured. The composite panel was a [0,90,0] _ns 12 or 18 layer CFRP, in which n=2 or 3 respectively, and the panel thickness ranged from 1.8 mm to 2.7 mm. For this validation, the 12-layer (n=2) experiment was used. A PMC 200 x 200 mm panel was simulated with a target area of 50 x 50 mm to reduce computation costs. Taking advantage of symmetry, a quarter model was developed as seen in Figure 4a,b. The plies and thickness of the model followed those of the experiment [23]. The target was divided into sub-laminates, where each [0,90,0] sub-laminate was separated by a cohesive zone. The cohesive zones were modeled with cohesive elements (COH3D8) with maximum degradation set to 1 kJ. The finite element model is shown in Figure 4. The region of the plate outside the target area was modeled with shell elements (S4R) to reduce the computational cost. The target region was represented with first-order C3D8R continuum elements [24] (layer-by-layer) to capture the damage in detail. Edges of the plate were fixed in place. To make the model more versatile, a python script was used to develop the model where geometry conditions, mesh parameters, contact parameters, speed of the projectile, and boundary conditions were defined as variables.

The exact properties of the steel projectile used in the experiment in [23] were not stated. In our work cold- rolled steel was used to model the projectile as a deformable spherical solid following [24]. We specifically selected cold-rolled 4340 steel for the projectile. The projectile yield followed the Johnson-Cook parameters listed in Table II. These complete parameters may be used to simulate both the initiation and propagation of the damage in the projectile under elastoplastic conditions. However, this study limits the projectile to a perfectly elastic material in order to simplify the model.

A mesh sensitivity study was conducted for the target with mesh sizes of between 0.3 and 1.0 mm, and for the projectile, with mesh sizes of between 0.35mm and 0.55mm. A good correlation was found with the experimental result at mesh sizes of 0.85 mm in the planar directions and at one element per ply in the thickness direction (0.15 mm) for the target and 0.4 mm for the projectile. We utilized the tools developed by [24] but compared our simulation results directly with the experiment [23] due to some concerns about the geometry assumptions used by [24].

The bottom curve in Figure 5 is the fitted curve from [23] for the 12-layer panel. This fitted curve utilizes the function.

$V_R=\alpha \sqrt{{V_i}^2-{V_{bl}}^2}$ (8)

where V_i is the impact velocity, V_bl is the ballistic limit, and α is a function of the mass of the projectile and the masses of the fragments released from the plate upon impact (for more information see [23]). Certain points along that curve are specified for comparison with the model where those same velocities were simulated. A curve using the same parameters as the experiment (i.e., with α = 0.9) is fitted to those points and displayed in the top curve of Figure 5. It is to be noted that while the variation in the residual velocities decreases as impact velocity increases as shown in Figure 5, the difference in the residual kinetic energies of the projectile between the experiment and the simulation stays roughly constant. Using either measure, the simulation produces a conservative estimate of the energy dissipated by the composite plates and thus the actual results should be better than those predicted by the modeling.

Figure 5. Comparison of CFRP model with experimental results from [23].

Figure 5. Comparison of CFRP model with experimental results from [23].

IV. Impact SIMULATION of Low Carbon Steel Including the Effect of PU Coating

4.1. LCS Plate Response to Ballistic Impact at 400m/s

When a 10mm thick LCS plate, the standard material and thickness of modern LPT tanks, was impacted with a 5 mm diameter projectile made of lead at 400m/s, the projectile did not penetrate. Instead, a dent was created on the LCS plate as shown in Figure 6a. Mild steel is roughly 14 times stiffer than lead, making steel almost impenetrable to lead. When the projectile material was replaced with 4340 steel, a penetration was observed as shown in Figure 6b. Therefore, it is clear that penetration of 10 mm thick LCS could occur with high-velocity projectiles. Thus, there are two obvious solutions to this problem. One way is to increase the thickness of the steel. The other solution could be to add a ballistic protection coating. These two options are discussed in the sections below.

4.1.1. Option 1 – Thicker LCS Tank

With a thicker LCS tank, an arrest can be made. For a sufficiently thick tank, our simulation showed that a 5 mm diameter projectile made of 4340 steel impacting at 400 m/s would be arrested about 14 mm into the tank (please refer to the results in table VI and Figure 7). However, any increase in the thickness of the tank would come with significant weight penalties as discussed in [1].

4.1.2. Option 2 – Addition of PU Coating

An alternative solution to the penetration of LCS plates by 400 m/s projectiles (or higher) is to coat the plates with PU. Three thicknesses of coatings were investigated: 1, 2, and 3 mm. With a 1 or 2 mm coating, the plate was still fully penetrated. However, with a 3 mm coating, the projectile was arrested 6.2 mm into the mild steel tank portion (9.2 mm from the surface of the PU). Clearly, the coating effect on the LCS was significant. It is interesting to note that even if the critical strain on PU were set at the bottom of the 0.2-0.3 range, the required thickness of PU to prevent penetration in the simulation only increased to 4 mm. This demonstrates that PU could be effective even with conservative parameters.]

Figure 7. Quarter cross-section through LCS showing: (a) perforation through 10mm without coating, (b) an arrest through a 27.6 mm without coating, and (c) an arrest of the projectile with a 3mm coating of PU on 10 mm of LCS. All three are impacted with 4340 steel projectiles at 400 m/s.

Figure 7. Quarter cross-section through LCS showing: (a) perforation through 10mm without coating, (b) an arrest through a 27.6 mm without coating, and (c) an arrest of the projectile with a 3mm coating of PU on 10 mm of LCS. All three are impacted with 4340 steel projectiles at 400 m/s.

Table 6. Thickness and coating effect on the penetration of and arrest by LCS plates.

Table 6. Thickness and coating effect on the penetration of and arrest by LCS plates.

Thickness and coating	Structure Thickness (mm)	Arrest Depth I (mm)
Mild Steel	10	Penetration
3mm PU + 10 mm Mild Steel	13	9.2
Mild Steel	27.6	14.2

V. Impact Response of CFRP and GFRP Plates and the Effect of PU Coating

Figure 8. Penetrations of and arrests in CFRP plates using a 400 m/s projectile: (a) penetration in 10.8 mm plate, (b) arrest in a 27.6 mm thick plate, and (c) arrest in a 10.8 mm plate coated with 3 mm of PU.

Figure 8. Penetrations of and arrests in CFRP plates using a 400 m/s projectile: (a) penetration in 10.8 mm plate, (b) arrest in a 27.6 mm thick plate, and (c) arrest in a 10.8 mm plate coated with 3 mm of PU.

Table 7. Penetration results from 400 m/s test with 4340 steel projectile: a) CFRP and GFRP laminates, b) CFRP and GFRP hybrid composites, and c) CFRP, GFRP, and hybrid composites with PU coating.

Vi. Comparison of Ballistic Resistance Performance of Metals and Composites

VII. Conclusions

VIII. Acknowledgments

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