Spalling-like failure by cylindrical projectiles deteriorates the ballistic performance of multi-layer graphene plates

Accepted Manuscript Spalling-Like Failure by Cylindrical Projectiles Deteriorates the Ballistic Performance of Multi-Layer Graphene Plates

Zhaoxu Meng, Jialun Han, Xin Qin, Yao Zhang, Oluwaseyi Balogun, Sinan Keten PII:

S0008-6223(17)31068-0

DOI:

10.1016/j.carbon.2017.10.068

Reference:

CARBON 12495

To appear in:

Carbon

Received Date:

23 August 2017

Revised Date:

11 October 2017

Accepted Date:

20 October 2017

Please cite this article as: Zhaoxu Meng, Jialun Han, Xin Qin, Yao Zhang, Oluwaseyi Balogun, Sinan Keten, Spalling-Like Failure by Cylindrical Projectiles Deteriorates the Ballistic Performance of Multi-Layer Graphene Plates, Carbon (2017), doi: 10.1016/j.carbon.2017.10.068

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Spalling-Like Failure by Cylindrical Projectiles Deteriorates the Ballistic Performance of MultiLayer Graphene Plates Zhaoxu Meng1, Jialun Han1, Xin Qin2, Yao Zhang3, Oluwaseyi Balogun1, 3* and Sinan Keten1, 3* 1 Department

of Civil and Environmental Engineering, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208-3111, United States

2 Theoretical

and Applied Mechanics Program, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208-3111, United States

3 Department

of Mechanical Engineering, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208-3111, United States

* Corresponding authors: [email protected] (O. Balogun), [email protected] (S. Keten)

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Abstract Ballistic performances of ultra-thin graphene membranes have recently been investigated at the micro and nanoscale. Two open questions that remain unanswered are, how graphitic plates behave when they can no longer be treated as a thin membrane, and how the projectile shape influences the perforation resistance of plates of varying thicknesses. Through coarse-grained molecular dynamics simulations, we show that beyond a critical plate thickness, a cylindrical projectile penetrates the plate at a lower velocity than a spherical one. This counterintuitive phenomenon is explained by spalling-like failure, where the graphene layers at the bottom section undergo a wavesuperposition induced failure in the cylindrical case. Finite element simulations are carried out to show that in-plane tensile stress concentrates at the bottom section, resulting from the superposition of incident and reflected stress waves. A mechanics relationship is then proposed to describe the resisting pressure of the graphitic plate during ballistic impact. It indicates that the intensity of stress wave, which affects the spalling-like failure, depends on the projectile initial velocity, plate compressive modulus, and density. Our findings reveal the existence of a new failure mechanism for multi-layer graphene systems, and provide theoretical guidance for future dynamic mechanical property characterization of graphitic barriers.

Keywords Multi-layer graphene (MLG) plate, ballistic impact, coarse-grained molecular dynamics, projectile shape, stress wave, spalling

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1. Introduction Multi-layer graphene (MLG) barriers of nanoscale thickness have exhibited specific penetration energies that are 10 times higher than that of the steel sheets on an equal weight basis when quantified through microscale ballistic experiments [1, 2]. This is attributed to the cone waves induced by the deformation upon impact, which efficiently dissipate the kinetic energy of projectile to a much larger area, owing to the high modulus and low density of graphene [2-4]. In most studies to date, the ratio between thickness of the barrier and the size of projectile is usually smaller than 0.2, in which case the barrier can be treated as a very thin plate or membrane. Theoretical models developed for membranes have been highly successful in describing the cone wave speed, reactive force and ballistic limit velocity (V50) in agreement with experimental results [5, 6]. Previous simulation studies have examined how the width and thickness of the membrane can influence the ballistic performance, revealing that the cone wave can reflect from the clamped boundaries and result in easier perforation for a suspended thin graphene membrane with a few layers [3]. Scaling microscopic experiments up to larger scale, the barriers need greater thickness to be able to stop impact by much larger projectiles. Commonly deployed armor plates have comparable thickness to the caliber of the projectiles that they are designed to offer protection against. Considering this, the behavior of thicker MLG plates should also be studied for a complete assessment of their barrier properties. Macroscopically, it is already known that the penetration process and failure mechanisms become more complicated when the barriers become thicker [7, 8]. Also, the penetration process and failure mechanism have been shown to depend on the shapes and sizes of projectiles [9-13]. However, these investigations are mainly limited in the scope of macroscale fiber reinforced plastic plates or metallic plates [14-16]. It remains unclear in the micro- or nanoscale how the specimen dimension in the thickness direction and projectile shape influence the ballistic impact performance, and whether there are diverse mechanisms that emerge and affect the projectile impact process. The recent microscopic experiments also suggest that high-speed impact of microparticles is not simply a scaled-down version of macroscopic ballistic impact [1, 17]. An open question that remains in MLG ballistic impact is whether thicker MLG plates that cannot be treated as membranes would still exhibit superior ballistic performance. In

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addition, the effect of projectile geometry on perforation resistance of microscale MLG plates has not been studied yet. Simulation methods including all-atomistic (AA) MD simulations [18-20], coarse-grained (CG) MD simulations [3, 21], and finite element methods (FEM) [22-24], have been successfully applied to study the ballistic impact performance of different material systems ranging from the nanoscale to macroscale. However, for microscopic MLG system, AA MD simulations are usually limited to small specimen dimensions, and FEM’s usefulness highly hinges on the accuracy of the materials’ constitutive law [22]. On the other hand, CG MD models, in which several atoms are grouped into larger beads that interact through an effective force-field, show great promise to simulate mesoscale physical processes with great computational efficiency while retaining the atomistic detail of the system [25-30]. Our recently developed atomistically informed CG model of graphene has been shown to capture mechanical properties including in-plane anisotropy, fracture behavior and orientation dependent interlayer shear of MLG accurately, while accelerating the computational speed by at least two orders of magnitude [25]. The model has been used to capture the hysteresis during nanoindentation loading and unloading due to interlayer sliding and shear-lag law dependent mechanical properties of MLG paper [31-33]. Recently, the model has been used to study the ballistic impact response of thin MLG membranes [3], and it has successfully reproduced the petal-like failure mechanisms observed in microballistic experiments [2]. In this sense, CG MD simulations are accurate and efficient to investigate the MLG plates projectile impact behavior and provide new insights. They are also informative in the sense of revealing the underlying dynamic mechanism in the fast impact process and elucidating key factors for the ballistic impact performance. In this paper, we first investigate the ballistic impact behavior of MLG plates up to 25 layers using CG MD simulations. Multiple failure modes, including fragmentation, plugging and petal failure, are observed during the penetration process of the MLG plates based on two different projectile shapes, i.e., sphere and cylinder. The impact mechanisms are compared for these two projectile types as a function of plate thickness. A wave-induced dynamic failure at the bottom of MLG plates is observed for the cylindrical projectile case, which resembles macroscopic spalling failure, typically observed in dynamic failure of quasi-brittle materials such as concrete [34, 35]. A simple mechanics relationship for impact resistant pressure is then proposed, which is originally used to 4

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describe the impact mechanics relationship for macroscopic laminates [14]. The relationship is validated for the MLG plate system studied here, and it is helpful to understand the dependence of spalling-like failure and ballistic resistance on different physical parameters.

2. Materials and Methods We use our previously developed and verified CG model of graphene, which adopts a 4-to-1 mapping scheme and conserves the hexagonal lattice symmetry [25]. The force field parameters are calibrated according to MLG mechanical properties measured experimentally, as described in detail in our prior work [25]. The Morse bond potential used in the CG model gives rise to smooth bond breakage. We use bond cutoff length of 𝑟𝑐𝑢𝑡 = 3.5 Å. The bonds that are stretched beyond 𝑟𝑐𝑢𝑡 are broken and subsequently deleted in the simulations. Using this criterion, the graphene strength obtained from our CG model matches the nanoindentation predicted strength of 130 ± 10 GPa [31]. We generate square MLG plates composed of 3-25 layers, with the thickness ranging from 1 nm to 12 nm. The in-plane dimension size of the MLG plates is around 110 nm. Consistent Bernal stacking order is used for all the MLG systems [36]. Previous laser induced projectile impact tests on MLG membrane have used microscale silica spheres as projectiles [2], and no appreciable deformation is observed. Here we choose cylindrical projectile with flat nose as a direct comparison with the spherical projectile, to also investigate the effect of projectile shapes on the ballistic resistance of MLG plates. Both the spherical and cylindrical projectiles studied herein compose diamond cubic lattice beads with a lattice constant of 0.72 nm. The default projectile bead mass is 192 g/mole, which gives a density of 6.8 g/cm3. During the comparison of the two shapes of projectiles, the masses and radii of the projectiles are kept the same. In the simulations, the projectile is treated as a rigid body and the reactive forces act on the projectile as a single entity. The interactions between the projectile and graphene membrane are modeled by the same 12-6 Lennard Jones (LJ) potential as the ones employed between graphene beads in the original CG model: 𝜀𝐿𝐽 = 0.82 𝑘𝑐𝑎𝑙/𝑚𝑜𝑙 and 𝜎𝐿𝐽 = 3.46 Å. We have verified in our previous study that different LJ parameters governing the projectile interaction with the membrane have a negligible influence on the impact process, including the reactive force 5

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and projectile velocity evolution [3]. The projectile is first placed above of the center of MLG plate, and then an initial velocity orthogonal to the plate is assigned to the projectile. We employ the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) to carry out the MD simulations and Visual Molecular Dynamics (VMD) software to visualize the impact process [37, 38]. During the simulation, a timestep of 4 fs is used, which further provides us computational efficiency compared to AA MD simulations. First, the MLG plate is equilibrated with an NVT ensemble at 10 K for 200 ps. After the initial velocity is applied to the projectile, an NVE ensemble is used for the whole system to conserve the total energy, following previous protocols [3, 18-20]. A circular region in the center of the plate is modeled as suspended around the impact location, where the free span of the region is 100 nm in diameter. To resemble the experimental boundary condition that only the top and bottom surface is clamped, the beads outside of the circular region in the top and bottom layer are fixed, and the other beads are free without any constraints. We also carry out FEM analysis to illustrate the wave propagation and stress concentration mechanisms during the impact process. Commercial software ABAQUS is used to conduct the simulations. Axisymmetric model is used to represent the axisymmetric nature of the problem as well as to save computational costs. The properties of the MLG plate is transversely isotropic. The in-plane direction is linear elastic with Young’s modulus E = 0.9 TPa, and the thickness direction is also linear elastic with modulus E = 60 GPa, which are consistent with the linear elastic moduli measured in CG MD simulations. The failure stress 𝜎𝑓 for in-plane direction is 120 GPa, while the failure stress in thickness direction is infinite since we do not aim to capture the interlayer separation failure of the MLG system. The densities of the plate and projectile are 2.2 g/cm3 and 6.8 g/cm3, respectively, also consistent with MD systems. The interaction between projectile and plate is assumed to have exponential contact force varied with the gap. We set contact pressure to be zero between the impactor and the substrate when the gap is larger than 0.05 nm, and the repulsive force increase exponentially as they come closer, which becomes 50 GPa when they are in perfect contact. Our calculations have also shown that these parameters describing exponential contact force behavior have negligible influence on impact stress wave propagation and the resulting stress concentration. At the initial position, the impactor is set to be 0.1 nm above the

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substrate with initial velocity of 500 m/s. Both the impactor and the multilayer graphene are represented by 4-node bilinear quadrilateral axisymmetric (CAX4R) element in ABAQUS.

3. Results and Discussions We start our discussion by first considering the effects of the projectile shape. Snapshots of the penetration process from two CG MD simulations are shown for a spherical projectile in Fig. 1(a-c) vs. a cylindrical projectile in Fig. 1(d-f). Both systems comprise of a 10-layer graphene plate. The projectiles have the same total mass and same radius of 4 nm. By visualizing the penetration process of both systems with velocities slightly above V50, the lowest velocity that permits total penetration (at least 50% chance), we can see that there exist multiple failure modes for both systems. Since the initial stresses in top layers are very large due to very high velocity impact, localized failure is seen in the top layers. Specifically, local fragmentation (comminution) happens in the top layers impacted by the spherical projectile (Fig. 1 (b)), and local plugging develops for cylindrical projectile, where pieces of graphene flakes under the projectile are chopped off due to large shear stresses produced around the rim of the flat-nose (Fig. 1 (d)). During the penetration of the remaining layers, the velocity of projectile decreases dramatically. When the projectile reaches the bottom section of the plate, it no longer has adequate kinetic energy to induce localized failure. At this point, a cone-shaped deformation develops for the bottom layers and more global deformation with petal failure is found for both cases (Fig. 1 (c) and (f)). This transition in the deformation mode also results in delamination of the bottom layers from the upper layers during failure, which further increases the dissipated energy though creating new surfaces. As a result, the orientation and surface chemistry dependent interlayer shear interactions in graphitic layers might play a role in the ballistic performance of graphitic based barriers [26, 31, 39-41]. The different failure modes, i.e., fragmentation, plugging and petal, have been correspondingly observed in macroscopic ballistic penetration process [8]. In addition, we would like to note that the possible bond formation at the cracking site does not affect the failure mechanism or ballistic resistance of the plate since it occurs after the failure, although the surface bond formation has been shown to play an important role in other related structures [42, 43]. Next, we discuss the dependence of the impact process on the initial velocity of the projectile for both projectile types. We test a broader range of initial velocities (V0), and record their residual 7

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velocities (Vr) after impacting the 10-layer graphene plates. The V50 is recorded as the lowest initial velocity that results in positive residual velocity. The complete velocity diagram is presented in Fig. 2 (a), where there are three distinct regions, similar to the graphene membrane systems we have studied before [3]. The shadowed region is region II, and the left and right parts are region I and region III, respectively. The lighter shadow is for the spherical case and the darker one is for the cylindrical case. We note that the velocity range is relatively high comparing to the microballistic experiments [1, 2, 17]. However, we anticipate a general ballistic limit velocity increase with decreasing projectiles sizes, and previous MD simulations on graphene membrane also reported similar high velocity range [18, 19]. In region I, Vr is negative, indicating that the projectile bounces back instead of penetrating the plates with low velocities. However, unlike graphene membranes, where the bounce-back velocity increases with V0 and the membrane behaves more like a spring [3], for the plate system herein, the value of Vr does not necessarily increase with V0. By analyzing the impact process, we find that even though the projectile does not penetrate the whole plate, it may result in damage in the top section of the plate. The projectile kinetic energy is dissipated through bond breaking events, and this damage influences the rebound from the plate such that Vr is much smaller compared to V0. In region II, Vr suddenly changes from negative to positive, which indicates transition from partial penetration to total penetration. In region III, where V0 and Vr scale linearly, the plate experience local perforation at impact. The width of region II is larger for the spherical projectile than the cylindrical one. For spherical case, the failure is always successive from top to bottom layers. Even for relatively high V0, the cone wave still forms in the bottom section, and the MLG plate dissipates more kinetic energy of the projectile through extra deformation in the radial direction and new surface creation due to delamination, thus widening region II. The conclusion from this analysis is that MLG plates would exhibit excellent ballistic resistance, if V0 falls in region II. However, for a cylindrical projectile, the transition into region III is much sooner. This indicates that once the cylindrical projectile reaches a high enough velocity, it becomes much easier to penetrate the plates. Some other mechanisms must come into play to account for the reduction in the ballistic performance during cylindrical projectile impact. To further shed light on this issue, the same procedure is repeated for a different number of graphene layers to find out how V50 depends on plate thickness for both shapes and projectile radii, 8

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4 nm and 5nm. Generally, a higher velocity is required to penetrate thicker plates, and the V50 values decrease with increasing sizes of projectiles, for both spherical and cylindrical ones, as shown in Fig. 2 (b). The thinnest system we have tested is a 3-layer graphene plate, which is approximately 1 nm thick. The membrane theory is still applicable to the thinnest system since the thickness is less than 1/8 of the projectile diameter, where a previous study indicates that 0.2 is a critical ratio between plate thickness and projectile diameter, larger than which the barrier no longer behaves like a membrane [16]. For this thinnest system, the V50 required by a cylindrical projectile is about 50% higher than that by a spherical one. This can be straightforwardly explained by the fact that for a spherical projectile with a sharp and small contact area, a greater stress concentration is expected at the impact zone, which results in easier penetration. V50 values of increasing plate thickness for the spherical case follow a linear trend. On the other hand, the V50 values for the cylindrical case show a sigmoidal shape with increasing thickness, and the downturn starts at the 10-layer for the radius 4 nm projectile and 15-layer for the radius 5 nm projectile. Beyond the downturn at the critical plate thickness, the V50 for cylindrical projectile becomes lower than that for spherical projectile. This, again, indicates that a new mechanism emerges during the ballistic impact process of the cylindrical projectile for MLG plates beyond a critical plate thickness, which significantly reduces the barrier properties of the MLG plates. This new mechanism also seems to happen selectively in the MLG plates which no longer behave like membranes as the wave-dominated local response mode has replaced the global deformation mode [16].

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Figure 1. Snapshots of simulations comparing the impact of a spherical (a-c) vs. cylindrical (d-f) projectile onto a 10-layer graphene plate. For clarity, only zoomed-in square region with length 40 nm is shown here. The projectiles have the same mass and radius of 4 nm. The initial velocities are V0, sphere =3000 m/s, and V0, cylinder =3500 m/s. (a). Side view of the spherical system. (b). The local fragmentation of the top two layers. (c). The petal rupture failure of the bottom two layers. (d). Side view of the cylindrical system. (e). The local plugging failure of the top two layers. (f). The local petal rupture failure of the bottom two layers.

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Figure 2. (a). The relationship between residual velocity Vr and initial velocity V0 for both spherical and cylindrical projectile impacting on a 10-layer graphene plate. The shadowed region is region II (light: sphere and dark: cylinder). The left part is region I and the right part is region III. (b). Ballistic limit velocity V50 vs. plate thickness for spherical and cylindrical projectiles with radii of 4 nm and 5 nm.

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To search for the reason for the change in plate ballistic resistance with respect to cylindrical and spherical projectiles, we visualize the impact process for the 15-layer graphene system. We observe an early onset of failure at the bottom section of the plate impacted by cylindrical projectile even with V0 below V50. Fig. 3 shows the snapshots of the impact process with V0 = 3600 m/s, which is lower than the V50 of the system (3800 m/s). After contact between the flat nose of the projectile and the top section of the plate, the compressive wave propagates downward at a rate that is faster than the projectile velocity, as shown in Fig. 3 (a)-(b). A rough estimation of the speed of compressive wave is 5.2 km/s, given the elastic modulus of 60 GPa in the thickness direction and graphene density of 2.2 g/cm3. The actual speed could be even higher since there exists obvious strain hardening behavior for the MLG compressive deformation in our model. When the compressive wave reaches the bottom-most layer, part of the wave gets reflected. The reflected stress wave front interacts with the remaining portion of the incident pulse in such a manner that results in failure of several layers from the bottom, as shown in Fig. 3 (c)-(d). Specifically, the bottom-most layer keeps intact as it is the free boundary, while several layers above it exhibit cracks. Fig. 3 (e) shows a zoomed-in view of the cracks formed in the bottom three layers. This wave-induced dynamic fracture resembles the spalling that occurs in various materials. Spalling is a particularly well-known phenomenon for concrete, where the far-end of the structure fails by the impact of a flying object, blast, or thermal radiation such as those induced by lasers [35, 44]. This specific failure mechanism existing in the cylindrical projectile impact case facilitates the perforation of the whole plate. It explains the plate ballistic resistance change that cylindrical projectile needs lower V50 than a spherical one beyond a plate thickness. To further understand how the stress wave causes the in-plane cracks in the bottom section of the plate under cylindrical projectile impact, we carry out FEM simulations for the 15-layer MLG plate to visualize the stress wave propagation. The thickness of the plate in FEM is 5.25 nm, equaling to the MD case. However, the initial velocity of the projectile is set at 500 m/s, a lower value to ensure that failure is avoided in FEM, considering that the failure criteria set in the FEM are not calibrated to exactly capture the MLG failure properties or the discrete nature of the material. FEM simulations enable our observations on wave propagation and stress concentration. From Fig. 4 (b) and (c), we can clearly observe the propagation of the compressive wave as represented by the stress evolution of S22 (normal stress in the thickness direction), where the wave front shows a planar shape. In addition, the compressive wave intensity does not show significant 12

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attenuation during propagation. At the same time, there is in-plane normal stress S11 developed along with S22 due to the Poisson effect. After the reflection of the compressive wave at the bottom, we observe a sharp increase in S11 concentrated at the center of the bottom section of the plate due to stress wave superpositions, as shown in Fig. 4 (d). This concentration and sudden increase of S11 well explains the spalling failure observed in MD simulations. The wave superposition induced failure does not occur in spherical projectile cases. This is because the contact area between a spherical projectile and plate is comparatively small, and even with low V0, the top layers would fail by fragmentation after contact with the projectile. Therefore, the intensity of compressive wave is much lowered. In addition, the wave front of the compressive wave is in the shape of spherical surface, which leads to faster attenuation of the compressive wave intensity by 𝑟 ‒ 2 [45], where 𝑟 is the travelling distance, as also shown in Fig. 4 (f) and (g). The stress level at the wave front are compared between cylindrical and spherical case in Fig. 4 (c) and (g), and the stress in the wave front is much lower in the sphere case than that in the cylindrical case. Fig. 4 (h) further shows that for the spherical case, the in-plane stress S11 only concentrates in the local area immediately below the projectile. The values for S11 are very small in the bottom section of the plate. This observation explains the sequential failure mechanism starting from the top and proceeding to the bottom as seen in the CG MD simulations. However, we would like to note that the wave superposition induced failure could potentially be observed for microscale spherical projectiles, especially when the projectile is deformable, and the local curvature of the projectile nose becomes small so that the compressive waves become more concentrated with a relatively flat wave front.

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Figure 3. Spalling process of a cylindrical projectile impacting on 15-layer graphene plates, where zoomed-in center strip region with length 20 nm and width 4 nm is shown here. (a). Initiation of compressive wave in the thickness direction. (b). The compressive wave reaches the bottom-most layer. (c). Wave reflects from the bottom-most layer. (d). Cracks develop in the bottom section of the plates. (e). Zoomed-in view of the cracks in the bottom section.

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Figure 4. Axisymmetric FEM model comparison between cylindrical projectile (a-d) and spherical projectile (e-h) case. (a). Illustration of configuration and mesh of the cylindrical system. (b). Normal stress in the thickness direction (S22) at the initial contact instance. (c). S22 and wave front. (d). In-plane normal stress (S11) at the time when S22 wave front reflects from the bottom boundary. (e). Illustration of configuration and mesh of the spherical system. (f). S22 at the initial contact instance. (g). S22 and wave front. (h). S11 at the time when S22 wave front reflects from the bottom boundary.

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Figure 5. (a). Reactive force vs. time profiles for spherical projectile with different V0 obtained from CG MD simulations. (b). Force vs. time profiles for cylindrical projectile with different V0. (c). Peak force vs. V0 for both spherical and cylindrical cases.

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To identify the parameters that affect the spalling-like failure, we mechanistically investigate plate resisting pressure and reactive force during the impact event. The intensity of the compressive stress wave is the direct cause of the spalling failure observed above. The wave intensity is directly related to the initial impact stress or equivalently the reactive force exerted by the plate on the projectile [45]. Thus, we next analyze the reactive force evolution for both spherical and cylindrical cases and its dependence on different factors. First, the reactive forces on the projectiles are recorded for different V0. Fig. 5 shows the reactive force evolution of three different V0 for either a spherical projectile (a) or a cylindrical projectile (b) impacting on 10-layer graphene plate, consistent with the two cases in Fig. 1. A common feature among the reactive force curves is a sharp initial force pulse, where reactive force first increases to its maximum value and then drops. For the spherical projectile, initially the reactive force has a gentler increase compared to the cylindrical case, and the width of the force pulse is greater. On the other hand, for cylindrical projectile, the reactive force reaches its maximum at the instance of contact between projectile and plate. After the force pulse, the spherical case shows a more gradual decrease in the reactive force. For cylindrical case, the plate confined under the cylindrical projectile vibrates after the force pulse, and this gives rise to the subsequent force peaks for the cylindrical case, Fig. 5 (b). When V0 becomes much higher than V50, as in the 4000 m/s case for example, then the force pulse becomes the largest portion of force evolution. We obtain the initial peak force values for different V0 and plot them in Fig. 5(c). For relatively high V0, the initial peak force is the same with the highest point in the force curves given that there is always a sharp drop in the force after reaching the maximum, which is due to localized failure in the top section. For very low V0 cases, since there tends to exist a force plateau in the force pulse, only the initial peak values are taken as the peak force, as shown in Fig. 6 (a). Both spherical and cylindrical projectiles show a clear linear relationship. Also, the peak force values of cylindrical projectile are approximately double that of the spherical projectile at the same V0. This double relationship, arising from the effect of projectile shape, has been explained in the analysis of the mechanics of projectile perforation in metallic plates based on the different inertial forces acting normally on the projectile’s nose surface [7].

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Figure 6. (a). Reactive force vs. time profiles for cylindrical projectiles with different projectile densities (𝜌𝑝). (b). Reactive force vs. time profiles for the same cylindrical projectile with different plate densities (𝜌𝑡). (c). Peak force vs. 𝜌𝑡 results and theoretical fitting using 𝐹 = 𝑐1 + 𝑐2 𝜌𝑡. (d). Reactive force vs. time profiles for different interlayer interaction 𝜀𝐿𝐽. All the data are from CG MD simulations with projectile initial velocity 1000 m/s. Density unit is g/cm3, and 𝜀𝐿𝐽 unit is kcal/mol.

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The linear relationship between peak force and V0 observed is well explained by an empirical expression developed for the resistant pressure of the fiber reinforced plastic laminate plates during ballistic impact process [14-16]: 𝜎 = 𝜎𝑒 + 𝛽 𝜌𝑡𝜎𝑒𝑉0

(1)

where 𝜎𝑒 is the linear elastic limit of the laminates in through-thickness compression, and for laminates that have similar yield/strain hardening strain, 𝜎𝑒 is directly related to the elastic compressive modulus of the laminates, 𝜌𝑡 is the laminate density, and 𝛽 is an empirical constant, and for cylindrical projectile 𝛽 = 2 [16]. Eq. 1 indicates that the initial resistance pressure during ballistic impact depends on plate density 𝜌𝑡, while it is independent of projectile density 𝜌𝑝. To check whether this macroscopic relationship is valid for our micro or nanoscale system, we have changed the projectile density in the CG MD simulations by choosing different bead masses for the projectile while keeping the projectile radius (4 nm) and initial velocity (1000 m/s) constant, and the results in Fig. 6 (a) show that the initial peak force is indeed independent of 𝜌𝑝, although the general shapes of the force pulses change accordingly with 𝜌𝑝. Next, we alter 𝜌𝑡 by changing the bead mass of the CG graphene bead and alter 𝜎𝑒 by changing the interlayer LJ potential parameter 𝜀𝐿𝐽. In our CG model, 𝜌𝑡 linearly scales with graphene bead mass. Thus, we select different bead masses that give rise to a density range of 1.1 g/cm3 to 4.4 g/cm3 for the plate. Fig. 6 (b) shows that the peak force varies substantially for different 𝜌𝑡. In Fig. 6 (c), the corresponding peak force is well described by the relationship 𝐹 = 𝑐1 + 𝑐2 𝜌𝑡, where 𝑐1 = 𝜎𝑒𝐴, 𝑐2 = 𝛽 𝜎𝑒𝑉0𝐴, and 𝐴 is contact area, according to Eq. 1. Using leastsquare fitting method, it yields 𝜎𝑒 = 8.75 𝐺𝑃𝑎 and 𝛽 = 2.3. 𝜎𝑒 agrees well with the elastic limit stress of MLG system and the 𝛽 parameter is consistent with the macroscopic systems [16]. Although the quantitative relationship between 𝜎𝑒 and 𝜀𝐿𝐽 is complicated, 𝜀𝐿𝐽 positively alters the elastic compressive modulus of the MLG plates, which controls 𝜎𝑒. We do observe a positive relationship between the peak force and 𝜀𝐿𝐽 in Fig. 6 (d), which is also consistent with Eq. 1. We conclude that the empirical expression Eq. 1 developed for macroscopic ballistic impact is also valid for our nanoscale system when used as a simple scaling law. As a result, the intensity of 19

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stress wave linearly scales with projectile velocity, and it depends on the plate density and compressive modulus. More importantly, the relationships are independent of the system sizes across different size scales. Stronger compressive wave leads to higher reflected tensile waves, which would normally result in more obvious spalling-like failure. However, we note that that this wave-induced failure is more related to the failure strength of the material. For materials that have much larger compressive strength than the tensile strength, such as concrete, the wave-induced spalling failure is more easily to be observed [35]. For microscale MLG barriers, where defects such as grain boundaries exist, flat-nose projectile impact would induce more obvious spalling-like failure given that these defects decrease the in-plane tensile strength of graphene sheet greatly [46-49]. When the overall tensile strength is decreased, it also becomes easier to observe successive failure from top to bottom layers by the stress wave along compressive wave propagation, resembling the failure waves observed in glasses [50-52]. In addition, for macroscopic specimens, the stress wave could also lead to plastic deformation in compressive before the material is carried into tension. The change in the mechanical properties arising from plastic deformation would also affect the spalling behavior [34]. The mechanics relationship Eq. 1 also provides guidelines for designing better ballistic barriers since the initial resist pressure of the plate is also an index of the barriers’ projectile proof capability. Higher resist pressure often indicates a better projectile proof performance. As a result, a strategy for designing better ballistic barriers can be developed by increasing the quasi-static compressive performance 𝜎𝑒 by enhancing the cohesive energy density, which is the interlayer interaction in the MLG case studied herein, and increasing the barrier density 𝜌𝑡. Moreover, for nanocomposites based plate, the cohesive energy at the interfaces between different components might play an even bigger role, since the energy dissipation at these interfaces is a major source for energy dissipation that influences shock wave propagation. Introducing other energy dissipation mechanisms, such as reversible crosslinking bonds, interlayer friction and collision might also be beneficial for ballistic barriers. The use of MLG plates or other nanoscale reinforcements in complex systems to induce such dissipative mechanisms may be a fruitful direction of future research.

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4. Conclusions In conclusion, we conduct ballistic impact CG MD simulations of MLG plates composing of up to 25 layers. We show that when the barrier no longer behaves as a membrane, a combination of failure modes emerges during the impact process. When the projectile velocity exceeds V50, the top section of the plate usually exhibits localized failure with either fragmentation or plugging, and the bottom section shows more global deformation with petal failure. We compare different projectile shapes and show that projectile shape has a strong effect on the failure process and V50. The cylindrical projectile is capable of penetrating MLG plates with lower initial velocity than the spherical projectile beyond a critical plate thickness. This is because when the plate is thick enough, the wave superposition induced failure in the bottom section facilitates the perforation of the whole plate. This specific failure mode resembles the spalling failure usually observed in the concrete blunt-nose impact. To illustrate the evolution of the stress wave propagation, we carry out FEM simulations, which show clearly that under cylindrical projectile impact, the wave superposition induces stress concentration and increases the normal in-plane stress at the bottom section of the plate. An empirical relationship for the impact pressure developed for fiber reinforced laminates is applied to the MLG plates and shows good agreement as a scaling law. Both the relationship and our simulation results indicate that stress wave intensity depends on the initial velocity, the density and compressive modulus of the plates, which are also the key factors that govern the ballistic performance of plate barrier. The most notable contribution of this work is the observation of a new dynamic failure mechanism during the ballistic impact of MLG plates. The concepts and mechanistic principles presented and validated herein, however, are not limited to the graphene system, but are generally applicable to other layered system with related two-dimensional materials, and they should also provide guidelines for the design of nanocomposite based armor materials.

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Acknowledgements The authors acknowledge funding by the Army Research Office (award # W911NF1710430). The authors also acknowledge the support from the Departments of Civil and Environmental Engineering and Mechanical Engineering at Northwestern University, as well as the Northwestern University High Performance Computing Center for a supercomputing grant.

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