**5. Results**

**5. Results**  After validating and comparing the results of both 1D and 3D element based models, a parametric study was carried out in order to analyze the influence of the projectile shape, the number of layers of the impacted panel and the impact angle. Even though both models are suitable to estimate the residual velocity, the 1D model was selected for the study because of its simplicity and After validating and comparing the results of both 1D and 3D element based models, a parametric study was carried out in order to analyze the influence of the projectile shape, the number of layers of the impacted panel and the impact angle. Even though both models are suitable to estimate the residual velocity, the 1D model was selected for the study because of its simplicity and low computational time. Results are discussed in the following sections.

### *5.1. Influence of the Projectile Geometry*

fabric generates a pyramid shape until failure.

*5.1. Influence of the Projectile Geometry*  In agreement with the literature related to low impact velocity in soft fabrics [47], the fabric area far from the impact zone, suffered a low deformation (a low displacement). Fixing the four sides of the fabric contributes to locate the stress and strain distributions around the impact area where the In agreement with the literature related to low impact velocity in soft fabrics [47], the fabric area far from the impact zone, suffered a low deformation (a low displacement). Fixing the four sides of the fabric contributes to locate the stress and strain distributions around the impact area where the fabric generates a pyramid shape until failure.

in [27].

Figure 7 shows the evolution of the ballistic curves in terms of variation of the residual velocity with impact velocity for different projectile geometries. It can be observed that the flat nose projectile presents the highest ballistic limit, *VBL* = 49 m/s (velocity at which all impact energy is absorbed by the fabric and no penetration takes place). The round and truncated conical nose projectiles presented the same ballistic limit valued at 44 m/s, and the conical projectile at 46 m/s. The velocity value is reduced by 11% and 8% respectively, compared with the flat nose. This result implies that any sharper projectile presents a lower ballistic limit compared to a flat projectile [4]. This can be explained because the sharp projectile favors the separation of the transversal yarns generating a hole through which the projectile can penetrate the fabric. At the same time, the contact area was reduced to a smaller surface, causing greater stress. Under this situation, the fibers are more easily ruptured. Figure 7 shows the evolution of the ballistic curves in terms of variation of the residual velocity with impact velocity for different projectile geometries. It can be observed that the flat nose projectile presents the highest ballistic limit, *VBL* = 49 m/s (velocity at which all impact energy is absorbed by the fabric and no penetration takes place). The round and truncated conical nose projectiles presented the same ballistic limit valued at 44 m/s, and the conical projectile at 46 m/s. The velocity value is reduced by 11% and 8% respectively, compared with the flat nose. This result implies that any sharper projectile presents a lower ballistic limit compared to a flat projectile [4]. This can be explained because the sharp projectile favors the separation of the transversal yarns generating a hole through which the projectile can penetrate the fabric. At the same time, the contact area was reduced to a smaller surface, causing greater stress. Under this situation, the fibers are more easily ruptured. with impact velocity for different projectile geometries. It can be observed that the flat nose projectile presents the highest ballistic limit, *VBL* = 49 m/s (velocity at which all impact energy is absorbed by the fabric and no penetration takes place). The round and truncated conical nose projectiles presented the same ballistic limit valued at 44 m/s, and the conical projectile at 46 m/s. The velocity value is reduced by 11% and 8% respectively, compared with the flat nose. This result implies that any sharper projectile presents a lower ballistic limit compared to a flat projectile [4]. This can be explained because the sharp projectile favors the separation of the transversal yarns generating a hole through which the projectile can penetrate the fabric. At the same time, the contact area was reduced to a smaller surface, causing greater stress. Under this situation, the fibers are more easily ruptured. 70

Flat

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Figure 7 shows the evolution of the ballistic curves in terms of variation of the residual velocity

**Figure 7.** Ballistic curves for different projectile noses. **Figure 7.** Ballistic curves for different projectile noses. Figure 8 represents the projectile velocity evolution from the instant of impact moment. The

Figure 8 represents the projectile velocity evolution from the instant of impact moment. The initial velocity of the projectile fixed at 60 m/s is above the calculated ballistic limit. A few observations were derived from the relationship depicted. The cone geometry offers the lowest rate of deceleration while the rest of the geometries have relatively higher or similar rates. The projectile velocity histories of the round nose and truncated conical nose were quite similar, because the second geometry can be considered as a rough approximation to the round case. The energy dissipated by friction between the projectile and woven fabric increases with the frontal contact area, which was higher for the conical frustum projectile. Finally, the cone nose projectile has a higher residual velocity due to the sharp nose of the impact. This geometry favors the penetration of the fabric, as can be seen Figure 8 represents the projectile velocity evolution from the instant of impact moment. The initial velocity of the projectile fixed at 60 m/s is above the calculated ballistic limit. A few observations were derived from the relationship depicted. The cone geometry offers the lowest rate of deceleration while the rest of the geometries have relatively higher or similar rates. The projectile velocity histories of the round nose and truncated conical nose were quite similar, because the second geometry can be considered as a rough approximation to the round case. The energy dissipated by friction between the projectile and woven fabric increases with the frontal contact area, which was higher for the conical frustum projectile. Finally, the cone nose projectile has a higher residual velocity due to the sharp nose of the impact. This geometry favors the penetration of the fabric, as can be seen in [27]. initial velocity of the projectile fixed at 60 m/s is above the calculated ballistic limit. A few observations were derived from the relationship depicted. The cone geometry offers the lowest rate of deceleration while the rest of the geometries have relatively higher or similar rates. The projectile velocity histories of the round nose and truncated conical nose were quite similar, because the second geometry can be considered as a rough approximation to the round case. The energy dissipated by friction between the projectile and woven fabric increases with the frontal contact area, which was higher for the conical frustum projectile. Finally, the cone nose projectile has a higher residual velocity due to the sharp nose of the impact. This geometry favors the penetration of the fabric, as can be seen in [27].

**Figure 8.** Typical velocity time history for different projectile geometries (*Vi* = 60 m/s). Time (ms) **Figure 8.** Typical velocity time history for different projectile geometries (*Vi* = 60 m/s). **Figure 8.** Typical velocity time history for different projectile geometries (*V<sup>i</sup>* = 60 m/s).

Based on the principle of energy conservation, the energy dissipated by the fabric (Figure 9) is equal to the loss of the projectile kinetic energy, given by Equation (1), where *mp*, represents the projectile mass, *V<sup>i</sup>* is the impact velocity and *V<sup>r</sup>* is the residual velocity. Based on the principle of energy conservation, the energy dissipated by the fabric (Figure 9) is equal to the loss of the projectile kinetic energy, given by Equation (1), where *mp*, represents the

projectile mass, *Vi* is the impact velocity and *Vr* is the residual velocity.

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$$
\Delta E = \frac{1}{2} m\_p \left( V\_i^2 - V\_r^2 \right). \tag{1}
$$

**Figure 9.** Variation of dissipated energy for different projectile geometries. **Figure 9.** Variation of dissipated energy for different projectile geometries.

It can be observed in Figure 9 that the front face of the projectile plays an important role in the yarn breakage. While a sharp front face favors the separation of the yarns (cone nose projectile), the flat nose projectile penetration is entirely governed by yarn breakage which involves a higher load in the fibers and reduces the residual velocity [27,36]. It can be observed in Figure 9 that the front face of the projectile plays an important role in the yarn breakage. While a sharp front face favors the separation of the yarns (cone nose projectile), the flat nose projectile penetration is entirely governed by yarn breakage which involves a higher load in the fibers and reduces the residual velocity [27,36].

#### *5.2. Influence of the Impact Angle 5.2. Influence of the Impact Angle*

The influence of the impact angle, defined as the angle between the projectile and the normal direction to the target (represented as α in Figure 2), was carried out with the round nose projectile. This projectile was chosen among the others because it is an intermediate geometry between flat and conic nose projectiles. It also has a close resemblance to common ammunition, like 9 mm "parabellun". The velocity magnitude, *V*, was split into directions *Y* and *Z* as follows: The influence of the impact angle, defined as the angle between the projectile and the normal direction to the target (represented as α in Figure 2), was carried out with the round nose projectile. This projectile was chosen among the others because it is an intermediate geometry between flat and conic nose projectiles. It also has a close resemblance to common ammunition, like 9 mm "parabellun". The velocity magnitude, *V*, was split into directions *Y* and *Z* as follows:

$$\begin{aligned} V\_Y &= V \cdot \cos(\alpha) \\ V\_Z &= V \cdot \sin(\alpha) \end{aligned} \tag{2}$$

impact velocity (sketched in Figure 10 and detailed in Figure 11): 1. Below the ballistic limit, 30 m/s < *V* < 45 m/s: The projectile reduces its velocity For a particular oblique angle, α = 45◦ , three cases could be distinguished as function of the impact velocity (sketched in Figure 10 and detailed in Figure 11):

For a particular oblique angle, *α* = 45°, three cases could be distinguished as function of the


Figure 11 illustrates the previous example. The left side plots correspond to impact velocities below the ballistic limit: the magnitude *V* (Figure 11a), *Y*-component *V<sup>Y</sup>* (Figure 11e) and Z-component *V<sup>Z</sup>* (Figure 11c). The right side plots correspond to impact velocities above the ballistic limit: the module *V* (Figure 11b), Y-component *V<sup>Y</sup>* (Figure 11f) and *Z*-component *V<sup>Z</sup>* (Figure 11d).

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**Figure 10.** Schematic representation of *VY* and *VZ* components as a function of the initial velocity *Vi*. (**a**) Impact velocity below the ballistic limit, (**b**) impact velocity close to the ballistic limit, and (**c**) impact velocity above the ballistic limit. **Figure 10.** Schematic representation of *V<sup>Y</sup>* and *V<sup>Z</sup>* components as a function of the initial velocity *V<sup>i</sup>* . (**a**) Impact velocity below the ballistic limit, (**b**) impact velocity close to the ballistic limit, and (**c**) impact velocity above the ballistic limit. **Figure 10.** Schematic representation of *VY* and *VZ* components as a function of the initial velocity *Vi*. (**a**) Impact velocity below the ballistic limit, (**b**) impact velocity close to the ballistic limit, and (**c**) impact velocity above the ballistic limit.

**Figure 11.** *Cont*.

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**Figure 11.** Velocity time histories for different initial velocities (impact angle = 45°): (**a**,**b**) magnitude of the velocity *V*, (**c**,**d**) velocity in the direction *Z* and (**e**,**f**) velocity in the direction *Y*. **Figure 11.** Velocity time histories for different initial velocities (impact angle = 45◦ ): (**a**,**b**) magnitude of the velocity *V*, (**c**,**d**) velocity in the direction *Z* and (**e**,**f**) velocity in the direction *Y*. component *VZ* (Figure 11c). The right side plots correspond to impact velocities above the ballistic limit: the module *V* (Figure 11b), Y-component *VY* (Figure 11f) and *Z*-component *VZ* (Figure 11d).

Figure 11 illustrates the previous example. The left side plots correspond to impact velocities below the ballistic limit: the magnitude *V* (Figure 11a), *Y*-component *VY* (Figure 11e) and Zcomponent *VZ* (Figure 11c). The right side plots correspond to impact velocities above the ballistic limit: the module *V* (Figure 11b), Y-component *VY* (Figure 11f) and *Z*-component *VZ* (Figure 11d). The evolution of the residual velocity as a function of the impact angle is represented in Figure The evolution of the residual velocity as a function of the impact angle is represented in Figure 12. Increasing the impact angle involves increasing the residual velocity in *Z* direction, due to the increase of *V<sup>Z</sup>* with the angle even in rebound cases (Figure 12a). Below the ballistic limit, rebound occurs for any angle (Figure 12b, *V<sup>i</sup>* = 30 m/s). Results show that the trend for *V<sup>Y</sup>* changes above the ballistic limit. It increases with the oblique angle, but at high angles the rebound may appear (*V<sup>i</sup>* = 60 m/s, α > 50◦ ). The evolution of the residual velocity as a function of the impact angle is represented in Figure 12. Increasing the impact angle involves increasing the residual velocity in *Z* direction, due to the increase of *VZ* with the angle even in rebound cases (Figure 12a). Below the ballistic limit, rebound occurs for any angle (Figure 12b, *Vi* = 30 m/s). Results show that the trend for *VY* changes above the ballistic limit. It increases with the oblique angle, but at high angles the rebound may appear (*Vi* = 60 m/s, α > 50°).

**Figure 12.** Residual velocity as a function of the impact angle (**a**) in *Z* direction (*VZ*) and (**b**) in *Y* direction (*VZ*). **Figure 12.** Residual velocity as a function of the impact angle (**a**) in *Z* direction (*VZ*) and (**b**) in *Y* direction (*VZ*).

The evolution of the absorbed energy during impact for different angles is represented in Figure

 (**a**) (**b**) **Figure 12.** Residual velocity as a function of the impact angle (**a**) in *Z* direction (*VZ*) and (**b**) in *Y* direction (*VZ*). The evolution of the absorbed energy during impact for different angles is represented in Figure 13. Around the ballistic limit, the fabric absorbed almost all of the impact energy to decelerate the projectile. It is observed that the ballistic limit increases with the angle. Keeping the impact velocity constant (*V*), the kinetic energy component due to velocity in the *Y* direction decreased with the angle (Figure 11). In practice, the impact component in this direction is what produces the rupture of the yarns. Consequently, the projectile would need a higher initial velocity to break the yarns. It was found that increasing the angle to 66% leads to an increment of the ballistic limit of 58%. The evolution of the absorbed energy during impact for different angles is represented in Figure 13. Around the ballistic limit, the fabric absorbed almost all of the impact energy to decelerate the projectile. It is observed that the ballistic limit increases with the angle. Keeping the impact velocity constant (*V*), the kinetic energy component due to velocity in the *Y* direction decreased with the angle (Figure 11). In practice, the impact component in this direction is what produces the rupture of the yarns. Consequently, the projectile would need a higher initial velocity to break the yarns. It was found that increasing the angle to 66% leads to an increment of the ballistic limit of 58%. *Materials* **2019**, *12*, x FOR PEER REVIEW 12 of 16

13. Around the ballistic limit, the fabric absorbed almost all of the impact energy to decelerate the

**Figure 13.** Variation of the dissipation energy with the impact velocity for different impact angles. **Figure 13.** Variation of the dissipation energy with the impact velocity for different impact angles.

For the hemispherical projectile, the relationship to estimate the dissipation energy and the

y = 0.0017x2 + 0.0128x + 6.9184 R² = 0.9966

Dissipation energy (J)

y = 0.0003x3 - 0.02x2 + 0.4611x + 42.9 R² = 0.9982

> Ballistic limit Dissipation energy Ballistic limit (fitted line) Dissipation energy (fitted line)

**Figure 14.** Variation of the ballistic limit and the dissipation energy with the impact angle.

0 20 40 60

Impact angle (º)

The combination of the impact angle and the number of layers provides a multiple-choice problem. An initial study was carried out with the numerical model implemented and based on the

A fitted equation calculated with the results obtained from the numerical model is presented in Equation (3), where *n* represents the number of layers and *θ* is the impact angle. The *R2* value was 0.92, highlighting the validity of the fitted model. The mechanistic model shows a good correlation between the ballistic limit obtained with the numerical model and the ballistic limit calculated from

= 22.250 + 0.458 · + 22.425 · + 0.335 · · <sup>ଶ</sup> = 0.927 (3)

*5.3. Influence of the Number of Layers* 

Ballistic limit (m/s)

Equation (3) as can be seen in Figure 15b.

round nose projectile.

Dissipation energy (J)

For the hemispherical projectile, the relationship to estimate the dissipation energy and the ballistic limit as a function of the impact angle were calculated (Figure 14). For both cases, the *R <sup>2</sup>* value was greater than 0.9, which supports the validity of the equations. For the hemispherical projectile, the relationship to estimate the dissipation energy and the ballistic limit as a function of the impact angle were calculated (Figure 14). For both cases, the *R2* value was greater than 0.9, which supports the validity of the equations.

0 20 40 60 80

Impact velocity (m/s)

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VBL (θ=0°)

VBL (θ=45°)

VBL (θ=60°)

VBL (θ=15°) VBL (θ=30°)

Impact energy

**Figure 14.** Variation of the ballistic limit and the dissipation energy with the impact angle. **Figure 14.** Variation of the ballistic limit and the dissipation energy with the impact angle.

#### *5.3. Influence of the Number of Layers 5.3. Influence of the Number of Layers*

**6. Conclusions** 

values and the experimental results.

lead to an increment of the ballistic limit.

with applicability in industrial environment.

The combination of the impact angle and the number of layers provides a multiple-choice problem. An initial study was carried out with the numerical model implemented and based on the round nose projectile. The combination of the impact angle and the number of layers provides a multiple-choice problem. An initial study was carried out with the numerical model implemented and based on the round nose projectile.

A fitted equation calculated with the results obtained from the numerical model is presented in Equation (3), where *n* represents the number of layers and *θ* is the impact angle. The *R2* value was 0.92, highlighting the validity of the fitted model. The mechanistic model shows a good correlation between the ballistic limit obtained with the numerical model and the ballistic limit calculated from Equation (3) as can be seen in Figure 15b. A fitted equation calculated with the results obtained from the numerical model is presented in Equation (3), where *n* represents the number of layers and θ is the impact angle. The *R <sup>2</sup>* value was 0.92, highlighting the validity of the fitted model. The mechanistic model shows a good correlation between the ballistic limit obtained with the numerical model and the ballistic limit calculated from Equation (3) as can be seen in Figure 15b.

$$V\_{\rm BL} = 22.250 + 0.458 \cdot \theta + 22.425 \cdot n + 0.335 \cdot \theta \cdot n \qquad \qquad \qquad \mathbb{R}^2 = 0.927 \tag{3}$$

**Figure 15.** (**a**) Variation of the ballistic limit as a function of the number of layers and the impact angle; (**b**) correlation of the ballistic limit between the estimated values using the 1D model versus values estimated from Equation (3). **Figure 15.** (**a**) Variation of the ballistic limit as a function of the number of layers and the impact angle; (**b**) correlation of the ballistic limit between the estimated values using the 1D model versus values estimated from Equation (3).

Figure 15a shows a response surface based on Equation (3). This Figure shows how the ballistic limit is enhanced by increasing the number of layers in the material. The ballistic limit value increases

Based on the work presented in this paper, the following main conclusions can be drawn: • A simplified model to study the impact in aramid fabrics at low velocities is been developed and parameterized. To validate the model, the results obtained were compared with the experimental tests reported in literature, obtaining a good agreement between the predicted

• The comparison of the 1D element based model with a 3D element based model demonstrated that the simplified models can reduce the computation time by 90%. This modelling methodology could be considered when designing personal protections with different woven structures and for various projectile geometries. The implementation of the numerical models in the industry, to help during the design process, requires simple and fast simulation tools. • The computational analysis was also able to delineate the influence of different factors such as projectile geometry, number of layers and impact angle. Sharper projectiles lead to a higher residual velocity and a lower energy absorption, because the specific geometric feature of the projectile causes a higher deformation of the fibers allowing an improved slip through the fabric and facilitating rupture of the fibers. An increase in the impact angle and the number of layers

• A mechanistic model developed for rapid estimation of the ballistic limit has been presented and validated with a very good confidence level. The expressions and surface diagrams obtained in this paper allowed to predict the critical velocity of impact once the number of layers and impact angle are known. This complementary analysis has elevated potential to be used in industry because of its simplicity. However, it is worth noting the necessity to carry out some previous work, both experimental and numerical, required to develop these types of mechanistic models

Figure 15a shows a response surface based on Equation (3). This Figure shows how the ballistic limit is enhanced by increasing the number of layers in the material. The ballistic limit value increases by 48%, 109% and 159% for two, three and four layers respectively. On the other hand, increasing the impact angle, raises the ballistic limit by nearly 27%, 63% and 95% for angles of 30◦ , 45◦ and 60◦ respectively. The evolution of the ballistic limit can be estimated rapidly using the fitted equation.
