Ballistic resistance of a novel re-entrant auxetic honeycomb under in-plane high-velocity impact

Abstract

Due to their high load-bearing capacity and excellent energy dissipation properties, metal honeycomb lightweight sandwich panels are commonly utilized as highly efficient weight-saving components in the automotive, aerospace, and military industries. Especially the auxetic honeycomb sandwich panels have higher yield strength, more robust shear modulus, fracture toughness, less fatigue expansion, and higher vibration and energy absorption. In this paper, the ballistic resistance and energy absorption mechanisms of a novel re-entrant auxetic honeycomb (RSH) sandwich panel are investigated. The ballistic limits and energy absorption of the re-entrant star-shaped honeycomb (RSH), star-shaped honeycomb (SSH), and re-entrant star-shaped honeycomb (RH) sandwich panels are compared and analyzed, as well as the deformation mechanism during projectile penetration. The results show that the RSH sandwich panel has the best in-plane ballistic performance among the three types of honeycomb sandwich panels. For the same relative density, the ballistic limit of the RSH sandwich panel is 17.4% and 7.1% higher than that of the SSH and RH sandwich panels respectively. In addition, the effects of different design parameters on the ballistic resistance of RSH sandwich panels are investigated by changing the panel thickness, the relative density of the core layer, the cell angle and the cell size. It can be concluded that increasing the thickness of the face sheet is more effective in improving the ballistic limit (perforation energy) of the RSH with a thinner core layer. However, increasing the relative density of the core layer is more effective in enhancing the ballistic limit (perforation energy) for thicker core layers. Cell size has a significant effect on the ballistic resistance of RSH sandwich panels compared to cell angle, especially at impact velocities close to the ballistic limit.

Keywords

sandwich structures auxetic honeycomb ballistic resistance energy absorption parametric study

Introduction

Ballistic penetration poses a significant threat to military armor, particularly vehicles, aircraft, and naval vessels. Thick metal plates are commonly utilized in the 20th century to prevent ballistic penetration of military armor. However, excessive armor can significantly reduce its maneuverability and cargo capacity. To enhance the ballistic performance and mobility of military vehicles, it would be beneficial to investigate the application of novel and inventive lightweight structures and materials.

Due to their high load capacity and superior energy absorption properties, honeycomb sandwich panels are extensively used as effective weight-reduction parts in the airplane, automobile, and military industries. The bullet impact response of sandwich panels has been extensively studied through experimental,^1–3 analytical,^4–6 and numerical methods.^7–9 Only the penetration resistance of conventional sandwich constructions has been researched in each of the mentioned studies. At present, the auxetic honeycomb cores have been extensively studied. Unique characteristics of the auxetic honeycomb enhance its physical properties and impact resistance.¹⁰ The re-entrant honeycombs (RHs) with conventional hexagonal honeycombs were compared and discovered that RHs can absorb more energy for the same range of impact strains.^11,12 It was widely acknowledged that there were two primary mechanisms for the negative Poisson’s ratio (NPR), namely re-entrant and rolling-up.¹³ The star-shaped honeycombs (SSHs) have been investigated for the equivalent mechanical behavior.^14,15 A novel re-entrant square honeycomb structure was conducted and the effect of geometrical characteristics on the in-plane dynamical crushed responses was further investigated.¹⁶ The research above evaluated auxetic structures’ the in-plane impact response to low-velocity impacts, while bomb explosions typically destroy targets by both blast wave impact and high-velocity fragmentation. When considering protective structures, the explosion resistance of auxetic honeycomb sandwich structures^10,17–19 and the low-velocity impact performance^20–24 were thoroughly researched. Overall, the results indicated that honeycomb sandwich panels with auxetic honeycomb were more resistant to blasts and low-velocity impacts owing to a partly NPR deformation. Nevertheless, the defense mechanism of auxetic honeycomb may be changed by high-speed impacts, such as ballistic impacts. The honeycomb was just slightly distorted before the projectile sheared off and penetrated, which indicated that the NPR effect was not significant. Therefore, evaluating the penetrable properties of auxetic sandwich constructions was valuable.

Unlike explosion and low-velocity collision, the projectile penetrating behavior of auxetic sandwich structures had received little attention from researchers. Affected by NPR effect, auxetic honeycomb structures showed stronger ballistic resistance than hexagonal and aluminum foam sandwich panels.^25,26 However, the ballistic characteristics of honeycomb structures of various diameters and geometries were evaluated.²⁷ The re-entrant structure provided the reverse cell wall with the poorest performance, according to the findings. Besides, the literature on materials with NPR was extensive. Star-shaped and re-entrant structures have outstanding NPR characteristics, and the combination of RH structures with other structures may result in the formation of novel structure. Hybrid design is a widely used approach for creating new honeycomb configurations. It involves combining existing cellular configurations to create new honeycomb structures that possess exceptional mechanical properties.²⁸

In summary, in this paper, a novel auxetic honeycomb structure (RSH) is proposed. The structure is created by combining star and reentrant permutations using a hybrid design. This study aimed to evaluate numerically the ballistic resistant performance of re-entrant star-shaped honeycomb (RSH) sandwich panels. In this research, a FE model was developed to examine the impact characteristics and make a comparison with the ballistic behavior of SSH and RH. The primary focus was on investigating the failure mode of honeycomb sandwich panels and analyzing the projectile velocity variations upon impact. The evaluation encompassed various ballistic performance indicators, such as residual velocity, ballistic limit, reach time, and energy absorption. Furthermore, a comprehensive investigation was conducted to analyze the impact of various factors, including impact speed, relative density, panel thicknesses, cell angle and cell size.

Computational models and validation

Geometric models

As depicted in Figure 1, sandwich panel comprised the honeycomb core ( $T_{c}$ = 60 mm), which was positioned between two panels of $T_{f}$ = 1.0 mm. The sandwich panel experienced an impact from a projectile of diameter $D_{p}$ = 15 mm and length $L_{p}$ = 25.5 mm. The projectile possessed a mass of $M_{p}$ = 1.44 g and was launched with an initial impact velocity, denoted as $V_{i}$ , ranging between 220 m/s to 420 m/s. To mitigate effect of boundary conditions, the width was widened (𝑤 = 300 mm), despite the fact that this factor is typically disregarded in high-speed impacts.⁹ The cell structure was also depicted in Figure 1. The geometry parameters of each cell element included horizontal wall length h, inclined wall length 𝑙, cell angle θ, and wall thickness 𝑡, perpendicular to the X-Y plane. In this research, re-entrant honeycomb (RH), star-shaped honeycomb (SSH) and re-entrant star-shaped honeycomb (RSH) have structural parameters h = 10 mm, 𝑙 = 5 mm, θ = 30°, and d = 160 mm. Other geometric dimensions of RSH, RH, and SSH are the same, including h, 𝑙 and θ. T varied with relative density.

Figure 1.

The honeycomb sandwich panels under in-plane projectile impact.

The prevalent honeycombs were produced through the expansion method using metallic foils that were uniformly adhered at regular intervals.²⁹ As a result, horizontal cell wall had twice the thickness (2𝑡) of inclined cell wall (𝑡) (Figure 2). Relative densities of RSH, RH and SSH can be calculated separately as

{\bar{ρ}}_{R H} = \frac{ρ_{R H}^{*}}{ρ_{s}} = \frac{1}{2} \frac{t}{l} \frac{(h / l + 2)}{(h / l - \sin θ)}

(1)

{\bar{ρ}}_{S S H} = \frac{ρ_{S S H}^{*}}{ρ_{s}} = \frac{t}{l} \frac{(1 + 4 h / l)}{{(1 + 2 \sqrt{2} \sin (π / 4 - θ) / h)}^{2}}

(2)

{\bar{ρ}}_{R S H} = \frac{ρ_{R S H}^{*}}{ρ_{s}} = \frac{1}{2} \frac{t}{l} \frac{(h / l - 2 t / l \cos θ + 8)}{(h / l - 2 \sin θ + 2 \cos θ) \cos θ}

(3)

Figure 2.

The representative geometric model of the sandwich panels (part) under projectile impact.

For comparison, h and 𝑙 were constant for each of the cytosolic configurations described in Figure 1. All three auxetic honeycomb densities, RH, SSH, and RSH, grew linearly or relatively linearly with increasing cell wall thickness but increased non-linearly with increasing angle θ. The RSH corresponded to the relative densities ${\bar{ρ}}_{R S H}$ are 0.05, 0.1, 0.15, and 0.2 for cell wall thicknesses of 0.1200, 0.2435, 0.3706, and 0.5022 mm, respectively.

Finite element models

FE method was utilized to systematically examine the mechanical properties of RH and RSH subjected to in-plane ballistic impact. The panel and substrate materials for the honeycomb core were aluminum alloy 2024 and the bullet material was 93W tungsten alloy. The Johnson-Cook model was selected for the material model to simulate the sandwich panel and the elastomeric material, whose yield stresses can be expressed as³⁰:

\bar{σ} = [A + B {({\bar{ε}}^{p l})}^{n}] [1 + c \ln ({\dot{\bar{ε}}}^{p l} / {\dot{ε}}_{0})] (1 - {\hat{θ}}^{m})

(4)

In equation (4),

\bar{σ}

is the plastic flow stress; A and B are respectively initial yield stress and strain hardening constantly;

{\bar{ε}}^{p l}

is equivalent plastic strain; n is strain rate hardening index; C is strain rate hardening term;

{\dot{ε}}_{0}

is nominal equivalent change at initial yield; and

{\hat{θ}}^{m}

is the dimensionless temperature. Table 1 gives parameters of 93W tungsten alloy and 2024 aluminum alloy, with D₁ to D₅ denoting the failure parameters.³¹

Table 1.

Parameters of materials.³⁴

Material	$ρ / g ∙ {c m}^{- 3}$	$G / G p a$	$A / G p a$	$B / G p a$	C	M	n	$D_{1}$	$D_{2}$	$D_{3}$	$D_{4}$	$D_{5}$
93W	17.7	160	0.731	1.67	0.03	0.82	0.91	2.0	1.7	−3.4	0.0	0.0
AL2024	2.698	77	0.175	0.380	0.034	1.0	0.43	0.13	0.13	−1.5	0.011	0.0

Considering reducing computational costs, quarter models were created for the sandwich panels and the shells. A mesh sensitivity analysis has demonstrated that an element size of 0.5 mm is enough for the damaged part closest to the path of the projectile, while an element size of 1.0 mm is enough for the other part. This makes sure that accurate results are produced. Offsets were established in order to prevent the initial penetration of the shell cells by creating a distance between the panel and the honeycomb core. Within the framework of the finite element model, certain cell walls were additionally taken into account as having double thicknesses as a result of the manufacturing procedure.²³ The utilization of contact modeling played a crucial role in the accurate prediction of the ballistic response exhibited by armored structures.³³ As a result of the significant contact between the honeycomb core and panel during bullet impact, this research established CONTACT_AUTOMATIC_SURFACE_TO_SURFACE. In addition, the contact algorithm COTACT_TIED_SURFACE_TO_SURFACE_OFFSET was chosen to simulate the gluing between honeycomb core and panel.³² Both static and dynamic friction coefficient were 0.1. In high-speed impacts, the effect of boundary conditions was usually negligible.³³ Therefore, it was assumed that symmetrical boundary conditions were applied to the impacted edge, while remaining two boundaries were specified as fixed boundary conditions. A partially representative FE model of a sandwich panel impacted by a projectile was shown in Figure 3.

Figure 3.

The representative FE model of the sandwich panels (part) under projectile impact.

Validation of model

In reference 34, spherical breakers were subjected to out-of-plane high-velocity impacts on the RH sandwich panel. The same structural characteristics, material model, and experimental circumstances as described in reference were utilized to validate the model’s predictions. It can be found in Figure 4 that when the fragments break the aluminum honeycomb sandwich structure, the front panel was depressions, and the front and back panels are deformed in a “conical” direction towards the exit hole. Because of the honeycomb core, the stresses exert on the front and back plates are distinct. The panel is always supported by the honeycomb core’s upper portion, which fit securely when the panel is damaged. The honeycomb core layer dissipate energy through bending deformation and fracture mechanisms, which effectively mitigate the total deformation of the panel. The localized region of the backplane experiences an increase in strain and a decrease in penetration resistance, resulting in a weakened penetration resistance of the backplane. Additionally, a noticeable occurrence of “debonding and delamination” is observed between backplane and the core layer. The simulation results show good consistency with the experimental results. The numerical outcomes exhibit a slight reduction in comparison to the experimental findings. This discrepancy can potentially be attributed to the presence of a binding agent situated between the sandwich layer and the front and rear panels during the experimental procedure. The analysis of the above results show that the utilization of the FE model enables the simulation of the penetration and can offer accurate data on the mechanical properties of RSH.

Figure 4.

Deformation of the target plate after penetration and numerical result. (a) Deformation mode, (b) numerical result.

Based on the structural parameters and experimental conditions of the corrugated sandwich panel³⁵ and the 2024 aluminium plate³⁶ impact test, the relevant numerical models were constructed using the Johnson-Cook material model and simulated their ballistic penetration tests to validate the numerical model further. From Figure 5(a), it is evident that when the bullet is in contact with the sandwich panel, the cells and the upper panel under the impact region experience buckling and plastic deformation. As the projectile’s displacement increases, the cells in the impact area are fully crushed, causing the lower panel to transition from slight deformation to significant fracture, ultimately resulting in complete perforation by the bullet. From Figure 5(b), it is observed that the experimental results closely match the numerical results. The contact force initially reaches a peak as the displacement increases, then rapidly decreases before climbing to a second peak, and finally decreases to zero as the corrugated sandwich panel is fully penetrated. As shown in Figure 5(c), cracks emerge in the aluminum plate when the entry velocity of the hemispherical bullet is 64 m/s. As the speed increased, the aluminum sheet exhibits plate deformation, petalling and plugging phenomenon. A comparison of the ballistic limits of the simulation and the test is presented in Figure 5(d), and it is evident that the experimental and numerical results closely align with each other. Therefore, the numerical study on the penetration resistance of RSH sandwich panels using the finite element method is adequately accurate in this study.

Figure 5.

Comparison between experimental results and simulation results. (a) Deformation mode and (b) load–displacement curves of the curvilinear corrugated-core sandwich, (c) Deformation mode and (d) Ballistic limit of the 2024 aluminium plate.

Ballistic resistance comparison

The simulations were performed to analyze the response of three distinct sandwich panel configurations under varying impact speeds of 220 m/s to 400 m/s. For each experiment, the residual velocity was recorded. This information was applied to assessing the ballistic resistance of various types of sandwich panels by calculating two critical quantitative indicators: ballistic limit $V_{b}$ and perforation energy $E_{p}$ . $V_{b}$ refers to the velocity at which a projectile becomes lodged in the rear panel or leaves using minimal speed. $E_{p}$ represents the dissipated energy by the panel during perforation.

Armor penetration process analysis

The trajectory of the projectile’s velocity during armor penetration of the RSH semi-armor-piercing projectile at impact velocity $V_{i}$ = 310 m/s is shown in Figure 6 as a representation of each impact scenario. The duration of armor penetration was found to be 320μs, denoting the interval from the initial projectile impact with the front panel to the point at which the bullet achieved full penetration of the sandwich panel. There are three phases to the piercing procedure. In phase I (0∼10 μs), the front surface induced an initial reduction in velocity, bringing the projectile down to a speed near 305 m/s before the collision with the honeycomb core. In phase II (10∼250 μs), the velocity exhibited a reduction as it traversed the honeycomb core. Upon reaching the rear panel, the velocity of the projectile was measured to be approximately 213 m/s. In phase III (250∼320 μs), the rear end face again induced a decrease in velocity, resulting in a residual velocity of nearly177 m/s. During the collision, the bullet experienced a reduction in its initial kinetic energy of 68%. The honeycomb core soaked up 84% of the total energy absorbed, while the front and back panels took up 9% and 7%, respectively. It was thus clear that the honeycomb core affected the anti-penetration ability of the sandwich panel structure.

Figure 6.

The evolution of projectile velocity at $V_{i}$ = 310 m/s.

Comparison of ballistic limit and energy absorption

Numerical simulation cases for the three models were further analyzed. The correlation between the residual velocity and impact velocity for the three sandwich panels was given in Figure 8. The Lambert - Jonas model may be used to determine the relationship between residual velocity $V_{r}$ and impact velocity $V_{i}$ .³⁷ The equation used to calculate the fitted curve shown in Figure 7 is as follows:

V_{r} = A ∙ {(V_{i}^{p} - V_{b}^{p})}^{1 / p}

(5)

In equation (5), 𝐴 and p are empirical parameters.

Figure 7.

Residual velocity versus impact velocity for three types.

Figure 8.

Absorbed energy versus impact velocity for the three types.

V_{b}

for the three types were presented in Table 2 for 𝐴 and p values and estimated by equation (5). The RSH sandwich panel exhibited highest ballistic limit, as determined to be 270 m/s based on equation (5). The

V_{b}

for SSH was 230 m/s when considering an in-ballistic impact, and the RHs was 252 m/s. The three trajectory curves converged as impact velocity increases in Figure 7. As an example, residual velocities of SSH, RH and RSH honeycomb sandwich panels at an impact velocity of 290 m/s were 183 m/s, 146 m/s, and 125 m/s, respectively. Upon increasing the impact velocity to 400 m/s, the residual speeds increased to 316 m/s, 312 m/s, and 308 m/s, respectively. The residual velocities of the honeycomb core become smaller as the impact velocity increases. Figure 8 presents additional information regarding energy absorption of three parts of each sandwich plate to various impact speeds. Across the scope of velocities investigated, it was observed that the energy absorbed by every part exhibited a near-linear relationship with the corresponding impact velocity. In contrast to the scenario observed in honeycomb sandwich structures exposed to out-of-plane ballistic impacts, where the panel assumes a prominent role in energy absorption,⁹ the findings of this investigation indicated that the honeycomb core primarily absorbed most of energy during in-plane impacts on honeycomb sandwich panels, as depicted in Figure 8.

Table 2.

Parameters in equation (5) and $V_{b}$ for three types of honeycomb sandwich panels.

Core configuration	A	p	Ballistic limit $V_{b}$ estimated by (5) (m/s)
SSH-type honeycomb	0.892	2.515	230
RH-type honeycomb	0.990	2.187	252
RSH-type honeycomb	0.959	2.273	270

In addition, for each variety of honeycomb sandwich panels, the honeycomb core absorbed more energy with increasing impact velocity than that of the panels. In most cases, the front panel absorbed more energy than the rear panel. The same situation has been found for honeycomb sandwich panels under out-of-plane ballistic impact.^9,38 Significantly, it is worth noting that the RSH exhibited a greater energy absorption capability over the others across the range of impact velocities investigated. However, the energy-absorbed capacities of the panels remained nearly identical for all studied types.

Ballistic impact process analysis

This section undertook an analysis of the projectile impact processes of three representative auxetic honeycomb sandwich panels to determine the factors contributing to their ballistic properties. The impact velocities considered were 240 m/s, 270 m/s, and 350 m/s.

Relatively low-velocity mode

Figure 9 given a snapshot of the typical deformation phase under $V_{i} = 240 m / s$ . As the impact speeds exceeded the ballistic limit for the SSH-type sandwich panel ( $V_{b} = 230 m / s$ ) and lower than those for the RH-type ( $V_{b} = 250 m / s$ ) and RSH-type ( $V_{b} = 270 m / s$ ) sandwich panels, the SSH-type panel was completely perforated (Figure 9(a)) through a whole perforation time $t_{p}$ of concerning 600 μs as well as a residual velocity $V_{r}$ = 33.8 m/s (Figure 10), while the RH- and RSH-type panels were just partly punctured (Figure 9(b) and (c)). During the initial stage of impact (e.g., 𝑡 = 30 𝜇s), only the cell walls of SSH, RH and RSH close to the perimeter of the warhead were the first to deform and collapse. Plastic hinges are observed to develop at both termini and the central region of all distorted cellular walls. The deformed procedures of SSH, RH, and RSH exhibited significant similarities. At the moment of penetration into the front surface and beginning interacting with its core, for example, 𝑡 = 220 μs, the honeycomb core displayed distinct dynamic characteristics due to the different cell element configurations and the local deformation evolution. For SSH-type honeycomb cores with star-shaped cellular elements, local necking occurred first in the cellular elements connected to the panels when impacted by the projectile. With the impact of the projectile, more distortion bands formed, and the cell walls began collapsing. The cells nearest the end of the bullet’s trajectory were predominantly subjected to shear loading Figure 9(a) and Figure 11(a)), and the number of cells affected by the projectile (i.e., the core area), was comparatively low. By contrast, the single-cell inclined walls on the ballistic track were predominantly subjected to tensile loads in the RH and RSH (Figure 9(b) and (c), Figure 11(b) and (c)). These tensile loads were transferred through the connected cell elements to remote core areas aside from the bullet route, resulting in the formation of deformation bands as shown by dashed lines in Figure 9(b) and (c). Increases in energy absorption and ballistic limitations are the result of the increased material used in this load transition. Additionally, Figure 11(b) and (c) showed that the sample exhibited dynamical nonuniformity and NPR properties when the auxetic honeycomb was compressed. The re-entrant cellular element in Figure 11(b), underwent lateral shrinkage, and the material flowed (towards) the vicinity of the impact. In contrast, Figure 11(c) showed that the re-entrant star-shaped cell element also underwent lateral shrinkage but was connected by two inclined walls between the cell elements, increasing resistance, and this topical deformation propagated layer by layer until reaching the fixed endpoint, leading to repeatable stress fluctuations. Eventually, the cell walls came into touch with one another, resulting in compressed densification just beneath the site of impact (indicated by the dashed circle Figure 9(c)), thus creating additional resistance to ballistic impact.

Figure 9.

Impact processes of (a) SSH-type, (b)RH-type, and (c)RSH-type by the projectile at $V_{i}$ = 240 m/s.

Figure 10.

Time history of the transient velocity of the projectile with $V_{b}$ = 240 m/s.

Figure 11.

Local instant node velocity (amplitude and direction are represented by length and direction of the arrow, resp) distribution of (a) SSH type, (b) RH type, and (c) RSH type under the projectile impact at $V_{i}$ = 240 m/s (projectile masked for better).

Relatively medium-velocity mode

While the initial velocity was raised to $V_{i}$ = 270 m/s, it was observed that both SSH-type and RH-type honeycomb sandwich panels penetrated completely (Figure 12(a) and (b)), with residual velocities of 138 m/s and 108 m/s, respectively (Figure 13). The cell shapes of both RH and RSH transformed into parallelograms during the projectile penetration. However, because the number of diagonal compression bars on the single cell sloping edge of the RSH differed significantly from that of the RH, with two struts per parallelogram sloping edge in the RSH and only one strut in the RH, more plastic hinges were formed during impact, reducing the stress concentration (Figure 11(c)), and the material concentration of the bullet tips allowed the RSH-type honeycomb sandwich panel to still withstand the projectile without perforation. Consequently, the speed of the bullet as it passed through the negatively expanded honeycomb sandwich panels fell more and faster than other two types of honeycomb sandwich structures for the same initial velocity (Figure 13).

Figure 12.

Impact processes of (a)SSH-type, (b)RH-type, and (c)RSH-type by the projectile at $V_{i}$ = 270 m/s.

Figure 13.

Time history of the transient velocity of the projectile with $V_{b}$ = 270 m/s.

Relatively high-velocity mode

Lastly, if the impact velocity was augmented to a value of $V_{i}$ = 350 m/s, it was observed that all three honeycomb sandwich panels underwent complete perforation (Figure 14). The honeycomb cores primarily failed due to local damage at high impact velocities. Due to the limited time available for lateral contraction of the cell wall at impact end, dynamic NPR properties were not notably present in the honeycomb sandwich panels. Hence, the cellular structure of core layer exhibited minimal impact on the ballistic properties. Residual velocities of SSH ( $V_{r}$ = 265 m/s), RH ( $V_{r}$ = 239 m/s) and RSH ( $V_{r}$ = 223 m/s) honeycomb sandwich panels exhibited no significant differences. Additionally, the RSH sandwich panels demonstrated relatively low residual velocities, which can be attributed to the extended duration of perforation as depicted in Figure 15.

Figure 14.

Perforation processes of (a) SSH-type, (b) RH-type, and (c) RSH-type by the projectile at $V_{i}$ = 350 m/s.

Figure 15.

Time history of the transient velocity of the projectile with $V_{b}$ = 350 m/s.

Parametric study of the RSH sandwich panels

Based on the simulation outcomes, the RSH sandwich panel exhibited the best ballistic resistance of the three types, particularly at comparatively low impact velocities. Consequently, a deeper examination was conducted with respect to its structural parameters, encompassing panel thickness, relative density, cell angle, and cell size. The objective was to offer further details about this novel ballistic design.

Effect of panel and core thickness

Three sets of simulations were conducted to evaluate the effect of panel and core thickness. In each group, auxetic honeycomb sandwich panels with the same core thickness ( $T_{c}$ = 40 mm, 60 mm, or 80 mm) and different face thicknesses ( $T_{f}$ = 0.5 mm, 0.75 mm, 1.0 mm, 1.25 mm, 1.5 mm) were impacted at four initial velocities of 275 m/s, 300 m/s, 325 m/s and 350 m/s, respectively. A cubic polynomial approximation of the residual velocity for each group of projectiles concerning the face thickness and impact velocity, as well as the response, was given in Figure 16. Consistent with expectations, the numerical findings demonstrated a consistent decrease in residual velocity as panel and core thickness increased. The influence of panel thickness was more pronounced for thinner honeycomb core (Figure 16(a)) than for thicker honeycomb core panels (Figure 16(c)). Furthermore, it can be seen from the plots that the cubic polynomial approximates the response well, indicating the high degree of non-linearity of the impact problem.

Figure 16.

Cubic polynomial approximations of the residual velocity of auxetic honeycomb sandwich panels with different face-sheet thicknesses.

Using equation (5) and the minimal perforation energy, ballistic limitations for each panel layout were derived from the simulated impact and residual velocities. Table 3 gave the panel specifications and results. Figure 17(a) showed the ballistic limits versus panel thickness. As expected, panels with thicker honeycomb cores resulted in higher ballistic limits. The ballistic limit exhibited a nearly linear relationship with the panel thickness across the three core thicknesses under consideration. For

T_{c}

= 40 mm, compared to the penetration limit of a 0.5 mm panel, the ballistic limits increased by 5.2%, 13.6%, 15.0% and 17.1% for 0.75 mm, 1.0 mm, 1.25 mm and 1.5 mm panels respectively. For

T_{c}

= 60 mm, the increases were 3.1%, 9.7%, 11.5% and 16.4% respectively while for

T_{c}

= 80, the increases were 1.7%, 4.6%, 7.9% and 11.8% respectively. It suggested that for honeycomb sandwich panel with a thin core, increasing the panel thickness was an efficient strategy to increase the ballistic limit. In addition, the minimal perforation energy exhibited a quadratic relationship with panel thickness for three core thicknesses in Figure 17(b).

Table 3.

Details and simulation findings for three groups with different panel thicknesses.

Group no	Panel thickness, $T_{f}$ (mm)	Core thickness, $T_{c}$ (mm)	Core relative density, $\bar{ρ}$	Ballistic limit, $V_{b}$ (m/s)	Minimum perforation energy, $E_{p}$ (J)
1	0.5	40	0.15	226.81	15.38
1	0.75	40	0.15	238.54	16.82
1	1	40	0.15	257.63	18.24
1	1.25	40	0.15	260.88	20.84
1	1.5	40	0.15	265.58	22.37
2	0.5	60	0.15	246.03	21.25
2	0.75	60	0.15	253.74	22.39
2	1	60	0.15	270.00	23.80
2	1.25	60	0.15	274.28	25.10
2	1.5	60	0.15	286.26	26.89
3	0.5	80	0.15	267.99	26.18
3	0.75	80	0.15	272.65	27.43
3	1	80	0.15	280.34	28.32
3	1.25	80	0.15	289.21	29.61
3	1.5	80	0.15	299.70	30.80

Figure 17.

Effects of face-sheet thickness on (a) ballistic limit and (b) minimum perforation energy.

Effect of relative density and core thickness

As a means to evaluate the influence of core thickness and relative density regarding ballistic performance, residual velocity was analyzed as a function of relative density ( $\bar{ρ}$ = 0.05, 0.1, 0.15 and 0.2). As depicted in Figure 18(a)–(c) , this analysis was conducted for various initial impact velocities and three different core thicknesses (40 mm, 60 mm and 80 mm). The manipulation of cell wall thickness was employed to induce variations in relative density. It clearly illustrated that there existed an inverse relationship between the residual velocity and the relative density for the three different core layer thicknesses examined. Specifically, it could be it can be discovered that as the relative density increased, the performance improved. Intriguingly, as the core layer thickness increased, the residual velocity showed more non-linearity with respect to core layer density and impact velocity. The specifications of the novel auxetic honeycomb sandwich panels used in this section, together with the calculated ballistic limits and minimal perforating energy, are listed in Table 4. Figure 19 visually illustrated the notable impact of relative density, thereby revealing an approximately linear correlation between these two variables. When comparing sandwich panels of different core thicknesses, it was clearly shown that ballistic limit of thicker structure increased more quickly as relative density increased. For example, when the core thickness was $T_{c}$ = 40 mm, ballistic limit increased from 231.26 m/s to 260.40 m/s as the relative density increased from 0.05 to 0.2, an increase of 14.33%. In contrast, the ballistic limit of the $T_{c}$ = 80 mm increased from 258.92 m/s to 324.32 m/s for same relative density, an increase of 25.26%. From another perspective, ballistic limits of all three sets were approaching as relative density decreased, implying that the change in core thickness had less effect on the ballistic limit with a low-density honeycomb core. Similarly, the quadratic connection the lowest shot hole energy and the relative density was clearly seen in Figure 19(b).

Figure 18.

Cubic polynomial approximations of the residual velocity of auxetic honeycomb sandwich panels with different core relative densities.

Table 4.

Details and simulation findings for three groups with different relative densities.

Group no.	Panel thickness, $T_{f}$ (mm)	Core thickness, $T_{c}$ (mm)	Core relative density, ρ	Ballistic limit, $V_{b}$ (m/s)	Minimum perforation energy, $E_{p}$ (J)
1	0.1	40	0.05	231.26	8.13
1	0.1	40	0.1	244.53	13.76
1	0.1	40	0.15	257.63	18.24
1	0.1	40	0.2	264.40	24.07
2	0.1	60	0.05	242.95	11.00
2	0.1	60	0.1	258.73	16.67
2	0.1	60	0.15	270	23.80
2	0.1	60	0.2	292.94	29.83
3	0.1	80	0.05	258.92	13.77
3	0.1	80	0.1	274.07	22.49
3	0.1	80	0.15	280.34	28.32
3	0.1	80	0.2	324.32	39.08

Figure 19.

Effects of core relative density on (a)ballistic limit and (b)minimum perforation energy.

Effect of cell angle and wall thickness

To facilitate comparison, the RSH sandwich panel studied had the same mass ( $T_{f}$ = 1.0 mm, $T_{c}$ = 60 mm, 𝜌 = 0.15) and cell wall length (h = 10 mm and 𝑙 = 5 mm), but differing cell angles (15°, 25°, 30°, 40°) and corresponding wall thicknesses (0.5052 mm, 0.4108 mm, 0.3706 mm, 0.2616 mm). Figure 20(a) showed the residual velocity of the projectile. It was clear that varying cell angles result in different residual velocities for the same impact velocity. Contrary to the monotonic influence of panel and relative density on the ballistic response (Figures 16–19), the variation of the cell angle led to a non-monotonic response. For all impact velocities considered, the smallest residual velocity was found at a single cell angle θ = 25°. The residual velocity was predicted using a cubic polynomial approximation equation for the range of cell angles and impact velocities examined, as shown in Figure 20(b). This approximate surface also showed the non-monotonic effect of cell angle on the residual velocity, reaching its minimum along the black dashed line near a cell angle of 25°. Using equation (5), the ballistic limits of the RSH sandwich panel based on various cell angles were obtained for sandwich plates in the range of about 270 m/s for the chosen angular variation and were plotted in Figure 20(c). Not surprisingly, the highest ballistic limit was $V_{b}$ = 274.95 m/s for a cell angle of 25° and the lowest ballistic limit was $V_{b} =$ 268.02 m/s for a cell angle of 40°. Table 5 summarizes ballistic limitations and minimal penetration energies of the RSH sandwich panel for five different re-entrant star-shaped single-cell angles and wall thicknesses. It was noteworthy that the minimum perforation energy for specimen 2 (θ = 25°) ( $E_{p}$ = 37.75 J) was almost twice that of specimen 4 (θ = 40°) ( $E_{p}$ = 20.15 J), which again illustrated that the single cell angle takes a major part to the ballistic performance of the RSHs (Figure 20(d)).

Figure 20.

Effects of unit cell angle and wall thickness on the ballistic responses: (a) residual velocity variations; (b) cubic polynomial approximation surface of residual velocity; (c) ballistic limit versus unit cell angle; (d) normalized minimum perforation energy versus unit cell angle.

Table 5.

Details and simulation findings for a group with different cell angles.

Group no.	Cell angle, θ(deg)	Cell wall thickness, 𝑡 (mm)	Ballistic limit, $V_{b}$ (m/s)	Minimum perforation energy, $E_{p}$ (J)
1	15	0.5052	272.23	28.22
2	25	0.4108	274.95	37.75
3	30	0.3706	270	23.80
4	40	0.2616	268.02	20.15

An introduction for the non-monotonic ballistic resistance of the RSHs based on single-cell shape changes is proposed. For example, when the cell angle was small (θ = 15°), the RSH sandwich panel was closer to the RH sandwich panel (Figure 21(a)), which was more prone to cell wall flexure and shear compared to RSH (Figure 9(a)and Figure 11(a)). The RH sandwich panel had the least ballistic resistance than the RSH sandwich panels. On the one hand, increasing the cell angle (absolute value) made the oblique wall of the cell more easily rotate around the joining margin of the cell wall, so that the NPR effect on the core can be fully exerted. The increase in cell angle also resulted in a decrease in the cell size while simultaneously increasing the number of cells in Z direction. The strong ballistic capacity of the RSH sandwich panel with a 25°cell angle was attributable to the findings of the investigations (Figure 21(b)). However, when the cell angle was too great, the cell walls were thinner and weaker, so they could be readily stretched beyond their breaking point (Figure 21(c)). Thus, the ballistic limit was less for a cell angle of θ = 40°than it was for θ = 15° (Figure 20(c)). With the impact velocity increased, the plastic collapse tended to localize (Figure 22), as the velocity was faster and stretching shear only occurs close to the projectile path, while the cell angle had less effect on its residual velocity variation (Figure 20(a)).

Figure 21.

Snapshots of impact processes of auxetic honeycomb sandwich panels with different reentrant unit cell angles by projectiles at t $V_{i}$ = 275 m/s: (a) θ = 15°, (b) θ = 25°, and (c) θ = 40°.

Figure 22.

Snapshots of impact processes of auxetic honeycomb sandwich panels with different reentrant unit cell angles by projectiles at $V_{i}$ = 350 m/s: (a) θ = 15°, (b) θ = 25°, and(c) θ = 40°.

Effect of cell size and wall thickness

The effect of single cell size of re-entrant star-shaped cell was considered. Maintaining same mass taken as in Section 4.3, all simulation examples involved honeycomb sandwich panel with

T_{f}

= 1.0 mm,

T_{c}

= 60 mm, 𝜌 = 0.15 and

θ = 30 °

. The ballistic resistance of the five RSHs with ℎ varying from 8.0 mm to 16.0 mm and inclined wall lengths 𝑙 being 1/2 ℎ, was evaluated numerically. The detailed parameters were shown in Table 6. Figure 23 showed graphically the residual velocity, ballistic limit, and minimal armor-penetrating energy for novel auxetic honeycomb sandwich panels with different cell sizes. It was worth noting that, as with cell angle, cell size also had a non-monotonic effect on the ballistic response of the panel, although this effect was relatively small. In the current environment, single cells with a horizontal wall length ℎ = 14.0 mm had the best ballistic resistance.

Table 6.

Details and simulation findings for a group with different cell wall lengths.

Group no.	Horizontal cell wall length, ℎ (mm)	Inclined cell wall length, 𝑙 (mm)	Cell wall thickness, 𝑡 (mm)	Ballistic limit, $V_{b}$ (m/s)	Minimum perforation energy, $E_{p}$ (J)
1	8	4	0.2887	260.83	17.36
2	10	5	0.3706	270	23.80
3	12	6	0.4331	294.34	28.59
4	14	7	0.5052	321.04	33.19
5	16	8	0.5825	291.79	27.17

Figure 23.

Effects of reentrant unit cell size and wall thickness on the ballistic responses: (a) residual velocity variations; (b) cubic polynomial approximation surface of residual velocity; (c) ballistic limit versus unit cell horizontal wall length; (d) normalized minimum perforation energy versus unit cell horizontal wall length.

Figure 24 gave a snapshot of the perforation process of the RSH sandwich panels for different cell sizes. For fixed relative density, a smaller single cell possessed a thinner wall. On one side, reducing the size of the single cell exposed more of the single cell to the projectile, thus reflecting the negative Poisson’s ratio properties to enhance its ballistic resistance (Figure 24(a)). However, as previously indicated, thinner cells were more susceptible to being strained to failure. Figure 24(b)) showed that for hemispherical head projectile with a diameter of $D_{p}$ = 15, cells with ℎ = 14 mm provided a favorable choice with cell size and wall thickness.

Figure 24.

Perforation processes of auxetic honeycomb sandwich panels with different reentrant unit cell sizes by projectiles at $V_{i} = 300 m / s$ :(a) ℎ = 8.0 mm, (b)ℎ = 14.0 mm, (c)ℎ = 16.0 mm.

In summary, this can suggest an optimal design for re-entrant star-shaped single cells with θ = 25° and ℎ = 14 mm to maximize the ballistic limit of RSH for a given mass under a defined bullet impact. It was imperative to acknowledge that the concept of “optimal design” discussed here was subject to certain limitations, as it applies to a relatively narrow range. In contrast, the notion of “global” design optimization encompassed a broader design domain and considered the intricate interactions among various design parameters.

Conclusion

In this research, a novel auxetic honeycomb structure is presented, and the impact behavior of RSH sandwich panels had been evaluated by FE simulation. The analysis focused on examining the penetrating process and energy dissipation mechanism to of the novel auxetic honeycomb sandwich panels.

The study showed that the honeycomb core absorbed over 80% of kinetic energy under in-plane impact situations. The cell configuration had a significant influence on its anti-penetration ability. In the range of impact velocities studied (220–420 m/s), RSH exhibited greatest ballistic performance, although its advantage was more pronounced at lower levels of impact velocity compared to higher levels. In particular, the ballistic limit of the RSH was 17.4% and 7.1% higher than that of SSH and RH, respectively.

In addition, considering excellent armor penetration resistance of the RSH sandwich panels, the influences of panel thickness, relative density, cell angle, and cell size on the ballistic resistance of sandwich panels were investigated. For a given core layer thickness, the residual velocity of projectile decreased with increasing panel thickness and relative density. Relationship between ballistic limit and panel thickness as well as relative density exhibited a nearly linear proportionality. In contrast, the minimal perforation energy demonstrated an almost quadratic relationship with these two variables. The ballistic limit of the RSH can be improved by increasing the panel thickness, notably if the core layer is kept thin. However, at greater core layer thicknesses, the ballistic limit can be more effectively improved by raising the relative density of the core layer. Meanwhile, RSH sandwich panels exhibited a non-monotonic ballistic response to changes in cell angle and cell size. In this study, the highest ballistic limit and the lowest perforation energy were observed at θ = 25° and ℎ = 14 mm, respectively. However, cell size had a relatively significant effect on the anti-penetration ability of RSH compared to cell angle, notably at impact speeds approaching the ballistic limit.

Footnotes

Authors’ contributions

Conceptualization, Qiang He; resources, Jiamei Zhu; software, Junlan Guo; writing—original draft, Junlan Guo; writing—review and editing, Lizheng Li; methodology, Dejun Yan.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by The National Natural Science Foundation of China (51705215), The Chinese Postdoctoral Science Foundation (2022M712932) and the Postgraduate Research & Practice Innovation Program of Jiangsu Province (NO. KYCX23_3858). The authors would like to express their thanks.

Consent to participate

All authors have read and agreed to the published version of the manuscript.

Consent for publication

All authors have read and agreed to the published version of the manuscript.

ORCID iD

Qiang He

Data Availability Statement

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

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