# Ballistic Penetration Analysis of Soft Laminated Composites Using Sublaminate Mesoscale Modeling

^{1}

^{2}

^{*}

## Abstract

**:**

^{®}cross-ply laminate in order to predict its behavior under ballistic impact. The sublaminate model is implemented within an explicit dynamic FE code to simulate the continuum response in each element. The sublaminate model assumes a through-thickness periodic stacking of repeated cross-ply configuration. In addition, a cohesive layer is introduced in the sublaminate model in order to simulate the delamination effect leading to the subsequent degradation and deletion of the elements. This new approach eliminates the widely used costly computational approach of using explicit cohesive elements installed at pre-specified potential delamination paths between the layers. Furthermore, in-plane damage modes (such as fiber tensile, and out-of-plane shearing) are also accounted for by employing failure criteria and strain-softening. The obtained quantitative results of ballistic impact simulations show good correlation when compared to a relatively wide range of experiments. Moreover, the simulations include evidence of capturing the main energy absorption mechanisms under high-velocity impact. The proposed modeling approach can be used as a useful armor design tool.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Material System

^{3}and its fiber volume content is high and up to the geometric limit, which is about 83% [22]. The targets in the experiments are made of Dyneema HB26 (HB stands for Hard Ballistics). Its areal density is between 0.257 and 0.271 kg/m

^{2}[23]. It is manufactured by the gel spinning process for producing its fibers, progress to ply fabrication towards thermo-pressing at high temperature and pressure to final laminate production according to the manufacturer’s (DSM) recommendation (the exact temperature and pressure are proprietary of Plasan Sasa Ltd.). Its fiber (SK76) tensile strength and modulus values are 3.6 and 120 GPa, respectively, with an elongation at failure of 3.8% [24].

#### 2.1.1. Mechanical Properties of Dyneema HB26

#### 2.1.2. Non-Linear Stress–Strain Relation in Tension and Shear

#### 2.2. Experimental Procedure and Results

^{2}, were firmly clamped on their edges. These composite targets were impacted on their center point by fragment simulating projectile (FSP) 2.85 gr (44 grain, MIL-P-46593) projectiles at varying velocities. A high-speed camera was used to record the impact events, one frame on each 27 µs, producing a side view of the projectile penetration process on the targets. This visual data were used to extract the basic test parameters for each impact test, such as the residual velocity (Vr), the bulge height (BH) and the bulge width (BW), which will be used for calibration with the FEM simulations. The shooting data and the results from the experiments are shown in Table 2.

#### 2.3. Method of Analyses (The Proposed Model)

#### 2.3.1. The Sublaminate Model

^{5}/s) and a very short loading period (µs) of the impact, these modes of failure are highly dominant, while the others are local and do not take place widely at these loading conditions. In other words, it seems that under ballistic impact, the cohesion between adjacent layers will fail before the failure of the bond between the fiber and matrix. In addition, the transverse impact on a fiber-reinforced material will cause extensive tension in the fibers, in an endeavor to stop the projectile from penetrating through the plate, while fiber kinking and matrix cracking are more likely to occur under long-term and local loads.

#### 2.3.2. The Model’s Structure

#### 2.3.3. The Sublaminate Mathematical Formulation

#### Perfect Bonding Conditions

#### Partial Inversion—The ABD Matrix

#### Effective Sublaminate Properties

_{,}and ${t}_{3}$ represent the thicknesses of layers 1–3. In this model: ${t}_{1}={t}_{2}\gg {t}_{3}$ (${t}_{3}$ is the cohesive layer) and $t=1$ (unit thickness).

#### 2.4. The Model’s Failure Criteria

#### 2.4.1. Distributed Delamination

#### 2.4.2. Tsai–Wu Failure Criterion for Anisotropic Materials

- ${X}_{c}\text{}=\text{}{X}_{t}/10$ • ${S}_{o}>{S}_{i}$
- $\Delta ={X}_{t}$ • ${\tau}_{o}={S}_{i}$
- ${Y}_{c}>>{Y}_{t}$ • $\beta \text{}=\text{}1,\text{}n\text{}=\text{}3$

#### 2.4.3. Implementation of the Tsai–Wu Failure Criteria on the Sublaminate Model

- Delamination—It has been identified as a leading energy absorption mechanism of the Dyneema composite, especially under ballistic impacts. After inspecting the data presented by the SDV, the delamination failure term (local failure—for element’s elimination) is:$$\left(\varphi \ge 1\right)\left(TW\left[3\right]\right)$$The first term in Equation (19) accounts for a total failure according to the Tsai–Wu polynomial. The second term accounts for the failure only under the condition of positive ultimate normal stresses, as these tend to separate the adjacent layers.
- Fiber tension—a major energy absorption mechanism is included in the tensile straining of fibers. In addition to the fact that composites possess high tensile energy absorbing capabilities, fibers that suffer high tensile straining directly under the impacting projectile tend to tear. The local tensile failure term (for element’s elimination) is:$$\left({\sigma}_{11}^{\left(k\right)}\ge {X}_{t}\right)({\sigma}_{22}^{\left(k\right)}\ge {X}_{t})$$
- First incident shear tearing—high OP pressure inflicted on the first impacted layers by the projectile, tear these layers and allow the projectile to penetrate through the plate, thus keeping the damage localized. This mode occurs in the first stage of the projectile’s impact when it possesses high kinetic energy causing early element deletion.The relevant term for this failure mode is defined by$$\left(\varphi \ge 1\right)\left\{\left(TW\left[10\right]\ge 1\right)or\left(TW\left[11\right]\ge 1\right)\right\}$$This mode of failure requires a combination of local Tsai–Wu ultimate state and ultimate OP shear stresses, $\left(TW\left[10\right]\ge 1\right)$ or $\left(TW\left[11\right]\ge 1\right)$.

#### 2.5. FE Structural Implementation

#### 2.5.1. Ballistic Impact Simulation

^{2}. Both the composite and projectile were modeled in full geometry to be able to simulate the stress waves propagating in the composite plate target starting from the impact point towards the clamped edges and back. This will exhibit how the composite will respond. The thicknesses of the Dyneema Plates are 8.7 mm. The model also includes a FSP 2.85 gr projectile with steel material properties, modeled as a rigid body (no deformation—an assumption verified by experimental visual data and previous studies, such as [2,8,29,30,31]).

^{3}.

#### 2.5.2. Success Parameter of the Model

## 3. Results and Discussion

#### 3.1. Parametric Study

_{t}and Y

_{c}for establishing the optimal parameters to be used in the model.

#### 3.1.1. Parametric Study: Changing Xt

_{t}. The chosen values are: X

_{t}= 1, 1.2, 1.4 and 1.6 GPa. All the other strength parameters are kept as given in Table 3. Five of the eight impact velocities: 606, 698, 708, 814 and 893 m/s, were calibrated, but for simplicity, only results for arrest case (606 m/s, see Figure 6), and the perforation case (708 m/s, see Figure 7) are described here. As indicated in Table 2, these results are shown at time 25 µs and 60 µs for the perforation and the arrest case, respectively.

_{t}. As can be seen from Figure 6a, after the impact takes place (0–55 µs), most of the plate’s displacement (~10 mm) occurs and it is close to the displacement obtained from the experiment (~11 mm). After that and until the arrest of the projectile, a slight increase in displacement is observed. Figure 7a shows a good correlation between the analysis and the experimental BH, and its value is around 11 mm.

_{t}. It is important to note that the missing experimental curve of the projectile’s velocity is attributed to the fact that the projectile is hidden from the high-speed camera while it penetrates the target. Therefore, the reported experimental residual velocity is only shown for the perforation cases. Figure 6b shows that the velocity reduces almost linearly soon after the impact takes place. It can be clearly seen that within 70 to 80 µs after the impact, the graph shows a plateau which indicates that the velocity is slightly constant at this range, moving toward the arrest of the projectile (similar to experiments). A rebound effect (negative velocity of the projectile) can be observed for the X

_{t}= 1.6 GPa simulated case towards the arrest of the projectile. The elastic rebound of the bulge can be attributed to the release of strain energy stored in the secondary yarns which convert into the kinetic energy of the projectile [15]. The highest SP value (SP = 84.2) in all the cases is found for the X

_{t}= 1 GPa. An advantage of the model is reflected by the repeatability of the curves on the graphs, meaning that small changes in the model’s parameters cause small changes in the model’s results.

#### 3.1.2. Parametric Study: Changing Yc

_{c}. The selected values are: Y

_{c}= 1.4, 1.6, 1.8, 2, 2.2, 2.4 and 2.6 GPa, together with X

_{t}= 1 GPa. All the other strength parameters given in Table 3 are kept constant. Here too, two impact velocities’ results are discussed: 606 m/s for the arrest case (see Figure 8) and 708 m/s for the perforation case of the projectile (see Figure 9). Similarly to the previous sub-section, Figure 8a and Figure 9a also show that most of the plate’s displacement (~10 mm) occurs after the impact takes place (0–55 µs) which is close to the measured displacement (~11 mm). Moreover, Figure 9a shows a good correlation between the analysis and the experimental BH as its value is around 11 mm. Figure 8b shows that the velocity exhibits sharp reduction soon after the impact takes place (0–20 µs). This indicates the good energy absorption ability of the Dyneema. Similarly to the experiment, the projectile did not manage to perforate the target in all cases, and stops after 80–90 µs. Figure 9b shows that residual velocity (Vr) is around 100 m/s (note that if Y

_{c}= 2.2 GPa is assumed, the resulting Vr is around 220 m/s) while in the experiment, Vr equals to 307 m/s. The linear behavior exhibited after 45 µs of the Y

_{c}= 2.2 GPa curve, may be explained by a detached node from the center back face of the model, which travels at a constant speed. After calculating the SP (see Section 3.4) in all the considered cases, it was found that SP = 89.5, which results from X

_{t}= 1 GPa and Y

_{c}= 2.2 GPa, and can be considered as the calibrated case.

_{t}= 1 GPa and Y

_{c}= 2.2 GPa coupling is by exhibiting the residual velocity vs. initial (impact) velocity graph, (see Figure 10). As can be seen, the closest slope to the experimental one is due to the coupling of X

_{t}= 1 GPa and Y

_{c}= 2.2 GPa, as found earlier. Note that the time steps of the high-speed photography and the simulation are not perfectly synchronized, which is caused by the difficulty of catching the exact moment of the impact on the camera between the projectile and plate. Additionally, there are different time steps between the simulation and experiments (5 and 27 µs, respectively).

#### 3.2. Predictions of the Calibrated Model

_{50}is around 650 m/s.

#### 3.3. Post-Impact Comparison

#### 3.4. Success Parameter Calculation

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Example of the repeating stacking sequence (bolded in light blue), which the sublaminate is using to describe an equivalent nonlinear homogeneous continuum.

**Figure 6.**Plate’s bulge height (

**a**) and projectile’s velocity (

**b**) vs. time at varying Xt, Vinitial = 606 m/s.

**Figure 7.**Plate’s bulge height (

**a**) and projectile’s velocity (

**b**) vs. time at varying Xt, Vinitial = 708 m/s.

**Figure 8.**Plate’s bulge height (

**a**) and projectile’s velocity (

**b**) vs. time at varying Yc, Vinitial = 606 m/s.

**Figure 9.**Plate’s bulge height (

**a**) and projectile’s velocity (

**b**) vs. time at varying Yc, Vinitial = 708 m/s.

**Table 1.**Dyneema HB26 orthotropic layer properties, adapted from [25] E11, E22, E33 are the Young’s modulus in each of the three axes; G12, G13, G23 are the Shear modulus in each plane; and ν12, ν13, ν23 are the Poisson ratio in each plane.

Property | Dyneema HB26 |
---|---|

E11 (GPa) | 81.39 |

E22 (GPa) | 1.84 |

E33 (GPa) | 1.85 |

G12 (MPa) | 723.5 |

G13 (MPa) | 720 |

G23 (MPa) | 680.9 |

ν12 | 0.28 |

ν13 | 0.28 |

ν23 | 0.36 |

Test no. | Thickness (mm) | Impact Velocity (m/s) | Time (µs) | Ballistic Performance (Threat: 44 Grain Projectile) | |||
---|---|---|---|---|---|---|---|

BH (mm) | BW (mm) | Perforation of Target | Vr (m/s) | ||||

1 | 8.8 | 413 | 60 | 7.2 | 26.4 | No | ----- |

2 | 8.6 | 436 | 60 | 7.2 | 27.9 | No | ----- |

3 | 8.8 | 530 | 60 | 9.4 | 32.7 | No | ----- |

4 | 8.7 | 606 | 60 | 11.5 | 34.1 | No | ----- |

5 | 8.7 | 698 | 25 | 9.8 | 23.5 | Yes | 271.3 |

6 | 8.6 | 708 | 25 | 8.3 | 23.1 | Yes | 307.2 |

7 | 8.6 | 814 | 25 | 9.9 | 20.9 | Yes | 521 |

8 | 8.7 | 893 | 25 | 9.8 | 22.8 | Yes | 624.7 |

X_{t}(GPa) | X_{c}(GPa) | Y_{t}(GPa) | Y_{c}(GPa) | S_{i}(GPa) | S_{o}(GPa) | Δ (GPa) | β | G_{12}(GPa) | τ_{0}(GPa) | n | |
---|---|---|---|---|---|---|---|---|---|---|---|

Dyneema HB26 | 1 | 0.1 | 0.05 | 2 | 0.03 | 0.38 | 1 | 1 | 0.73 | 0.03 | 3 |

TW[#] | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|

component | ${f}_{1}{\sigma}_{1}$ | ${f}_{2}{\sigma}_{2}$ | ${f}_{3}{\sigma}_{3}$ | ${f}_{11}{\sigma}_{1}{}^{2}$ | ${f}_{22}{\sigma}_{2}{}^{2}$ | ${f}_{33}{\sigma}_{3}{}^{2}$ |

TW[#] | 7 | 8 | 9 | 10 | 11 | 12 |

component | ${f}_{44}{\tau}_{4}{}^{2}$ | ${f}_{55}{\tau}_{5}{}^{2}$ | ${f}_{66}{\tau}_{6}{}^{2}$ | $2{f}_{12}{\sigma}_{1}{\sigma}_{2}$ | $2{f}_{13}{\sigma}_{1}{\sigma}_{3}$ | $2{f}_{23}{\sigma}_{2}{\sigma}_{3}$ |

Plate | Projectile | |
---|---|---|

Material | Dyneema HB26 | Steel, FSP 2.85 gr (44 grain) |

Dimensions (mm) | 400 × 400 × 8.7 | Ø7.62 × 8.84 (d × h) |

Element’s type | C3D8R, Linear hexahedral | C3D4, Linear tetrahedral |

Number of elements | 432,000 | 10,923 |

DOF | 1,368,351 | |

Boundary condition | The plate is clamped at its side edges and the projectile is given an initial velocity in the OP direction. A general contact interaction is defined and all element surfaces are allowed to come into contact with each other. |

Impact Velocity (m/s) | Perforated | Time (µs) | Residual Velocity (Vr) (m/s) | ||
---|---|---|---|---|---|

Model | Experimental | Error (%) | |||

413 | No | ----- | 0 | 0 | 0.0 |

436 | No | ----- | 0 | 0 | 0.0 |

530 | No | ----- | 0 | 0 | 0.0 |

606 | No | ----- | 0 | 0 | 0.0 |

698 | Yes | 50 | 217.6 | 271.3 | 19.8 |

708 | Yes | 65 | 229.9 | 307.2 | 25.2 |

814 | Yes | 30 | 359.8 | 521 | 30.9 |

893 | Yes | 25 | 535.7 | 624.7 | 14.3 |

**Table 7.**Calibrated case’s ballistic simulation results (BH, BW) compared to the experimental results.

Impact Velocity (m/s) | Time (µs) | Bulge Height (BH) (mm) | Bulge Width (BW) (mm) | SP | ||||
---|---|---|---|---|---|---|---|---|

Model | Experimental | Error (%) | Model | Experimental | Error (%) | (%) | ||

413 | 60 | 6.1 | 7.2 | 15.3 | 26.6 | 26.4 | 0.9 | 95.2 |

436 | 60 | 6.5 | 7.2 | 10.1 | 26.4 | 27.9 | 5.4 | 95.9 |

530 | 60 | 7.4 | 9.4 | 21.2 | 32.9 | 32.7 | 0.6 | 93.4 |

606 | 60 | 8.2 | 11.5 | 28.9 | 34.4 | 34.1 | 1.0 | 91.1 |

698 | 25 | 8.4 | 9.8 | 14.9 | 24.0 | 23.5 | 2.0 | 85.2 |

708 | 25 | 8.1 | 8.3 | 2.0 | 19.8 | 23.1 | 14.0 | 84.0 |

814 | 25 | 9.8 | 9.9 | 0.8 | 18.4 | 20.9 | 12.2 | 83.0 |

893 | 25 | 12.4 | 9.8 | 26.0 | 18.2 | 22.8 | 20.2 | 81.0 |

Avg. SP | 89.5 |

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**MDPI and ACS Style**

Chricker, R.; Mustacchi, S.; Massarwa, E.; Eliasi, R.; Aboudi, J.; Haj-Ali, R.
Ballistic Penetration Analysis of Soft Laminated Composites Using Sublaminate Mesoscale Modeling. *J. Compos. Sci.* **2021**, *5*, 21.
https://doi.org/10.3390/jcs5010021

**AMA Style**

Chricker R, Mustacchi S, Massarwa E, Eliasi R, Aboudi J, Haj-Ali R.
Ballistic Penetration Analysis of Soft Laminated Composites Using Sublaminate Mesoscale Modeling. *Journal of Composites Science*. 2021; 5(1):21.
https://doi.org/10.3390/jcs5010021

**Chicago/Turabian Style**

Chricker, Raz, Shaul Mustacchi, Eyass Massarwa, Rami Eliasi, Jacob Aboudi, and Rami Haj-Ali.
2021. "Ballistic Penetration Analysis of Soft Laminated Composites Using Sublaminate Mesoscale Modeling" *Journal of Composites Science* 5, no. 1: 21.
https://doi.org/10.3390/jcs5010021