Kailu Xiao,Qiuyun Yin,Xianqian Wu,Chenguang Huang

a Institute of Mechanics,Chinese Academy of Sciences,Beijing,100190,China

b School of Engineering Science,University of Chinese Academy of Sciences,Beijing,100049,China

c Department of Applied Mechanics & Engineering,School of Engineering,Sun Yat-sen University,Guangzhou,510275,China

d School of Aerospace Engineering,Xi’an Jiaotong University,Xi’an,710049,China

e Hefei Institutes of Physical Science,Chinese Academy of Sciences,Heifei,230031,China

Keywords:Graphdiyne Ballistic limits Dynamic responses Wave propagation Energy dissipation

ABSTRACT The mechanical behavior of single-layer graphdiyne(SLGDY)subjected to high-velocity micro-ballistic impacts is analyzed by molecular dynamics(MD)simulations.The ballistic limits of SLGDY is obtained for the first time.The temperature deterioration effects of the impact resistance are also investigated.The results show that the ballistic limits can reach 75.4% of single-layer graphene(SLGR)at about 1/2 density,leading to approximately the same specific energy absorption(SEA)as SLGR.The ballistic limits of SLGDY and SLGR with single atomic thickness agree with the predictions of macroscopic penetration limits equations,implying the applicability of continuum penetration theories for two-dimensional(2D)materials.In addition,the dynamic responses involving stress wave propagation,conic deformation,and damage evolution are investigated to illuminate the mechanisms of the dynamic energy dissipation.The superior impact resistance of SLGDY and SLGR can be attributed to both the ultra-fast elastic and conic waves and the excellent deformation capabilities.This study provides a deep understanding of the impact behavior of SLGDY,indicating it is a promising protective material.

1.Introduction

High-performance materials are persistently required for engineering projects subjected to impulse loadings.In recent years,two-dimensional(2D)materials like graphene(GR)have attracted great attention due to their multitudinous excellent properties including superior stiffness,super high strength,and extremely low density[1–6].They have a variety of applications such as coating materials[7],solar cells[8],and nanoelectromechanical systems(NEMS)[9,10].In addition,they are regarded as promising bulletproof materials due to their superior impact resistance[11–14].Therefore,it is critical to understand the mechanical behavior of 2D materials under various loading rates.

The dynamic mechanical behavior of 2D materials has been studied intensively.Natsuki et al.[15]used general continuum mechanics theory to investigate the impact response of double-layer GR sheets(DLGSs)subjected to impact by nano-particles.Their results demonstrated that the impact response duration of DLGSs was exceedingly short,in the order of picoseconds,because of its super-fast in-plane elastic wave speed of about 21.07 km/s,much higher than that of general metals and bulletproof fibers[16–18].Lee et al.[14]studied the supersonic penetration behavior of GR with a thickness of 10–100 nm via laser-induced micro-particle impact experiments.They validated that under an impact velocity of 600 m/s,the specific penetration energy of the multilayer GR was approximately ten times that of steel due to the super-fast in-plane elastic wave speed.The study by Yoon et al.[19]revealed that besides the fast wave speed,the relatively large deformation of GR under impact contributes to its superior energy absorption capacity.Gao et al.[12]studied the mechanical behavior of DLGSs using fast nano-indentation complemented with density functional theory(DFT)analysis.Their results showed that the configuration of DLGSs beneath the diamond indenter transformed into diamond-like ultra-hard structures,providing the DLGSs with high resistance.Wang et al.[20]investigated the ballistic resistance of carbon nanotubes(CNTs).Their results indicated that the high stiffness and strength offered excellent impact resistance,and the failure of van der Waals interfaces between the CNTs provided new channels to dissipate the impact energy.Tian et al.[21]studied the dynamic response of monolayer hexagonal boron nitride(h-BN)using explicit finite element analysis(FEA).It was found that the h-BN hadgreat potential for armor application supported by excellent ballistic resistance.

Fig.1.Simulation models for(a)SLGDY and(b)SLGR,and total energy changes during equilibrium processes for(c)SLGDY and(d)SLGR.

As a carbon allotrope consisting of diacetylene and benzene,graphdiyne(GDY)has attracted great attention because the joining of acetylenic bonds changes the lattice structure while the hybridized type of GDY decreases the binding energy and modulates the electronic and optical properties,leading to lower atom density and superior performance in some aspects compared with GR and CNTs[22,23].GDY has been extensively studied,showing such potential applications as hydrogen purification[24],energy conversion,and storage[25,26].With a relatively low density(ρGDY)of 1.19 g/cm3[27],single-layer GDY(SLGDY)exhibits an elastic modulus(EGDY)of 513.8 GPa and intrinsic strength of 36 GPa using the self-consistent charge density-functional tight-binding calculation method[28].The in-plane elastic wave speed of GDY is(=)20.78 km/s,which is comparable to that of GR and much higher than that of most metals.In a recent study[29],the mechanical properties and fracture characteristics of GDY were obtained under quasi-static loading,showing that GDY provides an excellent balance between strength and ductility.The extremely high in-plane wave speed and large critical deformation might provide GDY with excellent impact resistance,making its future use as an advanced impact protective material possible.However,the key issues of ballistic limits,dynamic responses,and energy dissipation mechanisms of GDY under high-velocity impact conditions,which are essential to understand the protective behavior under high-velocity impact,have rarely been studied.

In this paper,the dynamic behavior and energy dissipation mechanisms of SLGDY under high-velocity micro-ballistic impact conditions are studied by molecular dynamics(MD)simulations.In addition,the impact behavior of single-layer graphene(SLGR)is investigated as a comparison to elucidate the relationship between the lattice structure and the ballistic impact resistance.The paper is organized as follows.In Section 2,the details of MD models of SLGDY and SLGR are provided.In Section 3,the ballistic limits of both SLGDY and SLGR are obtained,and the energy dissipation mechanisms are investigated based on dynamic responses including stress wave propagation,deformation behavior,and damage evolution,followed by discussions and conclusions.

2.Method

2.1.Potential

The adaptive inter-molecular reactive empirical bond order(AIREBO)potential given in Eq.(1)[30],which is a variant of the reactive empirical bond-order(REBO)potential and is regarded as one of the most effective potentials for carbon atoms,is used to investigate the carbon-carbon interaction of SLGDY and SLGR films.The AIREBO potential energy consists of multi-body interactions(REBO),pairwise interactions(LJ),and various dihedral angle preferences(Torsion items)respectively,

where VRijand VAijare repulsive and attractive pairwise potentials determined by the atom types of atoms i and j,respectively.Here bijdescribes the bond-order term modifying the strength of a bond due to changes in the local environment,which has a positive correlation with the bondorder.The interaction between the projectile and the target is described by LJ potential with σc=3.4?and εc=2.84 meV.The cutoff radius,rc,is set to be 2.0?[31].The AIREBO potential allows for covalent bond breaking and forming with appropriate changes in atomic hybridization[32]and has been widely used to study the properties of carbon-based nanomaterials such as CNT[33,34],GR[32,35],and GDY[29,36].

2.2.MD modeling

The large-scale atomic/molecular massively parallel simulator(LAMMPS)code[37],a widely adopted open-source software,isemployed to analyze the dynamic behavior of SLGDY and SLGR under high-velocity impact.The OVITO software is used to visualize the atomic configurations during the impact process[38].As shown in Fig.1(a)and(b),the SLGDY and SLGR films are constructed in the x-y planes,with the coordinate origin at the center of the film.The structure of GDY includes three types of bonds,single,aromatic(e.g.sp2),and triple(e.g.sp1).The original bond lengths for the single,aromatic,and triple bonds are set as 1.40?,1.44?,and 1.24?,respectively.The original bond length for GR is 1.42?according to the previous work[39–41].The elasticity,strength,and fracture characters of GDY and GR are captured to validate the computational models[29,42].As shown in Fig.1,the sphere projectile is constructed from diamond(sp3hybridized carbon atoms)and is considered as a rigid object with a diameter of 2 nm to achieve high computational efficiency.It is set to impact the centers of SLGDY or SLGR targets at normal incidence at various velocities.The initial distance between the projectile and the target is about 20?to avoid interaction before impact.

Fig.2.Relationships between Vr and Vi at 300 K for(a)SLGDY and(b)SLGR.(Different colored points represent the typical results).

Fixed boundary conditions are applied to the peripheries of the SLGDY and SLGR targets during impact.In our simulation model,as shown in Fig.1(a)and(b),the diameter of the projectile,d is 2 nm,and the minimum length of the MD simulation model,l is about 28 nm for SLGDY and 14.8 nm for SLGR.The ratio between l and d is about 14 for SLGDY and 7.4 for SLGR.Generally,when the ratio between l and d is larger than 5,the boundary effects could be negligible[43–45].The velocity-Verlet time-step method is adopted,with a 1 fs time increment to ensure the accuracy of the results.The system is fully relaxed using stress-controlled conjugate gradient energy minimization.The temperature effects on the dynamic behavior of SLGDY and SLGR are investigated using initial temperatures of 300 K and 400 K.During the thermal relaxation under canonical ensemble(i.e.NVT),the models are fully relaxed for 100 ps to ensure that they are in the equilibrium state with the lowest energy.To simulate the impact condition,the projectile is assigned an initial velocity after thermal relaxation and the whole impact process is conducted in the NVE ensemble,i.e.constant number,energy,and volume.

Based on the mechanical properties of SLGDY and SLGR(i.e.ESLGDY~513.8 GPa,σmSLGDY~36 GPa,ESLGR~981 GPa,and σmSLGR~117 GPa)[27,46],the projectile velocities are set at 20 to 90?/ps(i.e.2 km/s to 9 km/s)to capture the non-perforated and perforated morphologies.The total energies in the thermal equilibrium state are shown in Fig.1(c)and(d)for SLGDY and SLGR,respectively.The cell sizes for SLGDY in the X and Y directions are 9.53?and 8.253?,respectively,while for SLGR,they are 2.47?and 4.29?,respectively.The number of atoms used in the simulations is 19,152 for SLGDY,and 12,247 for SLGR.It should be noted that the ballistic limit of both materials needs to be computed and,considering the computational cost,only the single-layer results are obtained in this paper.

3.Results and discussion

3.1.Ballistic limits of SLGDY and SLGR

Based on a mass of simulations,the relationships between the initial velocity,Vi,and the residual velocity,Vr,at room temperature(300K)for SLGDY and SLGR can be divided into three regions[44],as shown in Fig.2.Region I represents the scenario that the projectiles cannot penetrate the targets and bounce back,as indicated by the nearly zero Vr.With increasing Vi,Vrchanges from zero to a positive value,indicating the onset of region II,in which an inelastic conic deformation with obvious perforation is formed during the impact and thus dissipates more projectile kinetic energy through a larger area.Further increasing the Vileads to the transition from region II to region III;Vrincreases almost linearly with the increase of Viand the projectiles perforate the targets locally.The relationship between Viand Vrin region II and region IIIshows a change from a non-linear to a linear tendency.The transition from region II to region III represents the failure mode from delocalization to localization.Based on Fig.2,the ballistic limit,Vb,of SLGDY is determined as 24.8?/ps,which is 24.6%lower than that of SLGR(32.9 ?/ps).

Fig.3.Relationships between Vr and Vi at 400K for(a)SLGDY and(b)SLGR.(Different colored points represent the typical results).

The dynamic behavior of SLGDY and SLGR at 400 K is also simulated,as shown in Fig.3,to investigate the effect of temperature.The Vbfor SLGDY and SLGR are determined as 24.2?/ps and 32?/ps,respectively.The Vbfor SLGDY and SLGR at 400 K decrease about 2.4% and 2.7%,respectively,compared to those at 300 K,due to the thermal softening effects,illustrating the deterioration of ballistic resistance at high temperatures.

The relationship between Viand Vrof a projectile can be fitted by the Lambert-Jonas formula[47],

where α is an empirical parameter that can be evaluated as α=mp/（mp+mplug）.Here mplugdenotes the mass of the shear plug ejected from the target,which can be neglected in the present study and therefore α=1.mprepresents the mass of the projectile and p is a parameter that can be fitted,as given in Table 1.

Table 1The mechanical properties and Vb of SLGDY and SLGR.

The Vbof SLGR impacted by projectiles with a radius of rpfrom 2 nm to 15 nm has been investigated by Meng et al.[45].A rough scaling law Vb=A+B/rpwas derived,where A and B denote the constants fitted by the relationship between Vband rp.Vbis estimated to be about 28.7?/ps for a projectile with a radius of 1 nm,which is close to Vb=32.9?/ps,as obtained in the present study.The acceptable difference should be ascribed to the size of the targets and the LJ potential parameters.

According to Wetzel et al.[11],Vb∝Λε1/3mΓ3/5for GR,and therefore

where εmis the maximum strain,Γ=ρ2DAp/Mpdenotes the mass ratio of the target to the projectile,ρ2Dis the areal density of the target,and Apand Mprepresent the area and mass,respectively,of the projectile.Λ=（U＊m.C1）1/3is a material parameter[48],where U＊m=Um/ρ2D(J/g)is the mass-normalized maximum tensile strain energy of the target and Um=42/3E[11–14],whileand E are the maximum tensile stress and elastic modulus,respectively,of the target.The values of the parameters are listed in Table 1.The Vbof SLGDY is predicted to be 70% that of SLGR,which approximately matches the present simulation results(24.8/32.9=75%)and it proves that our simulation results are independent of the in-plane size of the membranes.The Vbof SLGDY is approximately 8 times that of Kevlar 129 and 4 times that of CNT yarn[11],showing the excellent potential of SLGDY as an advanced bulletproof material due to its low mass density,high strength-to-weight ratio,high flexibility,and superior in-plane wave velocity.

Table 2Velocities of stress waves and crack propagation of SLGDY and SLGR.

To understand whether the penetration theory at the macroscale is applicable at the nanoscale,the ballistic limits of SLGDY and SLGR are estimated based on Gardner et al.[49]and Higginbotham et al.[50],

where h,ρ and σmdonate the thickness,density,and yield stress,respectively,of the target.dpand ρpare the diameter and density,respectively,of the projectile.σAl=0.069 GPa is the yield stress of aluminium.The results of Vbfor SLGDY and SLGR are estimated to be 19.6?/ps and 30.1?/ps,respectively,which shows a reasonable agreement between the present simulation and the predictions of Gardner et al.[49]and Higginbotham et al.[50],indicating the applicability of penetration limit equations at the macroscale for 2D materials.

3.2.Energy dissipation mechanisms of SLGDY and SLGR

The superior Vbof SLGDY indicates its excellent energy dissipation capacity.Here,the specific energy absorption(SEA),E＊p,which is normalized by the mass of the impact region,is used to evaluate the energy dissipation ability of SLGDY.The kinetic energy loss of the projectile,ΔEk,equals the energy to penetrate the membrane,Ep,while the energy loss of air drag can be ignored.Eprelies on the energy dissipation mechanisms of the target,including elastic deformation,fracture,and kinetic energy that transfers to the perforated debris.Based on Lee et al.[14],ΔEk=12m（Vi2-Vr2）and Ep=（ρAsh）Vi2/2+Ed.The frist term of the latter equation represents the kinetic energy that transfers to the targetwithin the deformation area As,and Edincludes other energy dissipation mechanisms.The SEA can be written as E＊p=Ep/ρAsh=V2i/2+E＊d,where E＊dis the delocalized penetration energy used to evaluate the outward propagation ability of the stress wave.The results are shown in Fig.4.The E＊pfor SLGDY and SLGR are as high as 31.3 MJ/kg and 31.8 MJ/kg,respectively,at Vb(i.e.24.8?/ps for SLGDY and 32.9?/ps for SLGR).The result of E＊pat Vi=50?/ps for SLGR agrees well with that of Haque et al.[51].As depicted in regions I and II,there is a transition region for the tendency of E＊p.With the increase of Vi,the E＊pfirstly decreases in region I(i.e.Vi<28?/ps for SLGDY and Vi<37?/ps for SLGR).Then it increases in region II(i.e.Vi≥28?/ps for SLGDY and Vi≥37?/ps for SLGR),implying the transformation of the energy dissipation mechanisms in the membrane.

Fig.4.SEA of(a)SLGDY and(b)SLGR.

Fig.5.Energy change of target and projectile.(a)and(b)Energy change of SLGDY at Vi=24.8?/ps and SLGR at Vi=32.9?/ps,respectively.(c)and(d)Energy change of projectile for SLGDY and SLGR,respectively.

It is noteworthy that E＊pof SLGDY and SLGR are approximately the same,indicating that the toughness of SLGDY originating from the unique joint of acetylenic bonds plays a key role in energy dissipation during impact,even though the strength of SLGDY is only one third that of SLGR.The E＊pof SLGDY and SLGR are about 30.8 MJ/kg and 32.0 MJ/kg,respectively,at 400 K,showing the deterioration of the ballistic resistance of SLGDY and SLGR at relatively high temperatures.

The energy evolution during impact is benefial for better understanding of energy dissipation behavior.According to the conservation of energy,the change of the impact kinetic energy of the projectile,ΔEp=EIp-ERp,converts to the kinetic energy,EKt,and the potential energy,EPt,of the target,where EIpand ERpare the initial and residual kinetic energies,respectively,of the projectile.The energy of the system is conserved,

The energy changes of SLGDY and SLGR at Vb(i.e.24.8?/ps for SLGDY and 32.9?/ps for SLGR)are analyzed in Fig.5(a)and(b).The entire impact process can be portioned in five typical regions by four characteristic times.Before t1both EPtand EKtincrease.From t1to t2,perforation occurs.During this process,the increase of EPtdue to cone wave propagation is faster than the reduction of EP

t resulting from crack propagation,leading to the continual increase of EtPand the decrease of EtK.From t2to t3,EtPand EtKkeep almost constant due to the appearance of multiple cracks,which balances the increase of EPtinduced by the conic wave propagation.From t3to t4,the conic wave reflects once it arrives at the fixed boundary,causing the re-bonding of some broken bonds and partly eliminating elastic deformation,leading to a slight decrease of EtPand a little increase of EtK.After t4,there is no interaction between the projectiles and the targets,and the targets approach equilibrium.It shows an obvious difference in the energy ratio between potential energy and kinetic energy for SLGDY and SLGR after t1,which should be ascribed to the different energy dissipation behavior involving conic deformation and crack propagation.Details of the process are given in Fig.6 and Fig.S1.

The histories of the impact energy of the projectiles are shown in Fig.5(c)and(d)for SLGDY and SLGR,respectively.If the impact energy is smaller than the critical penetration energy(i.e.281.49 eV at Vi=24.8 ?/ps for SLGDY,498.42 eV at Vi=32.9?/ps for SLGR),the kinetic energies of the projectiles decrease to almost zero after the projectiles contact the targets,implying the complete transformation of the impact energies to the targets.Once the impact energies exceed the critical penetration energy,the impact energies decrease quickly to the plateaus of the residual impact energies.

3.3.Dynamic response analysis for SLGDY and SLGR

3.3.1.Stress wave propagation under impact

For 2D materials,stress wave propagation is an effective pathway to dissipate the impact energy before perforation occurs.Once the projectile impacts the target,a longitudinal wave with a speed of C1,a transverse wave with a speed of C2,and a deformation-induced conic wave with a speed of C3are generated and propagate outward from the impact center,leading to partial dissipation of the initial projectile kinetic energy.For an elastic material under transverse impact,the speeds of the longitudinal,transverse,and conic waves can be calculated as follows[14],

Fig.6.Propagation of stress waves and dynamic failure behavior of SLGDY at Vi=28?/ps(a)A tensile wave propagates outwards and forms a cone zone.(b)and(c)A failure region is formed at the impact center.(d)~(f)The dynamic response of the membrane after penetration,including the snapback of broken bonds,propagation,and reflection of the stress wave.

where E and G are the linear elastic modulus and the shear modulus,respectively.ρ denotes density and μ is Poisson’s ratio of the target.As listed in Table 2,the wave speeds C1and C2are determined as 20.78 km/s and 12.4 km/s for SLGDY,and 21.07 km/s and 13.94 km/s for SLGR,which are much higher than that of steel[52](ρsteel=7800 kg/m3,Esteel=200 GPa,C1=5.06 km/s,C2=3.13 km/s)and Kevlar 129[11,53](ρKevlar=1440 kg/m3,EKevlar=96 GPa,C1=8.16 km/s,

C2=5.06 km/s).C3is dependent on Viand C1.From Eq.(6c),underimpact velocity Viranging from 20 to 45?/ps,the value of C3is about 17%–28%of C1for SLGDY and SLGR in the present study.The extremely fast in-plane wave speeds of SLGDY and SLGR imply excellent energy absorption capacities before failure.Fig.S2 depicts the shock wave propagation of SLGDY and SLGR for Vi

Fig.7.Out-of-plane displacements and diameters of conic deformation for SLGDY and SLGR at(a)and(b)Vi=Vb,and at(c)and(d)Vi=20?/ps,respectively.

Once Vi>Vb,perforation occurs.Taking Vi=28?/ps for SLGDY and Vi=35?/ps for SLGR as examples,the wave propagation and perforation processes are depicted in Fig.6 and Fig.S1.As shown in Fig.6(a),a tensile wave propagates outwards at C1,followed by conic wave propagation at C3.A conic zone is formed at the impact center.During the conic deformation,a highly deformed area of SLGDY near the impact center is observed,followed by the initiation of the breaking of bonds and eventually perforation(Fig.6(b)and(c)).After perforation,the snapback of the broken parts occurs due to the rapid release of the strain energy stored in the highly deformed impact region(Fig.6(d)).The momentum is continually dissipated by the stress wave propagating outwards(Fig.6(e))and the broken bonds mostly stay in broken morphology.The stress wave reflects when it reaches the fixed boundary(Fig.6(f))and the deformation degree in the out-of-plane direction is decreased compared to the on-going impact process(Fig.6(a)and(c)).The conic deformation behavior during impact can be seen in the front views of Fig.6.The conic wave speed of SLGDY is about 3.37 km/s,according to the simulation results at 1.0 ps and 1.6 ps,which is a little lower than the theoretical value(i.e.4.33 km/s according to Eq.(6c)),due to the fluctuation of the conical surface resulting from atomic vibrations.Then it slows down quickly to about 2.4 km/s at 2.2 ps after full perforation.The propagation of conic deformation will cease due to the broken and perforation induced release wave,while multiple waves reflect back and forth between the fixed boundaries and the cracked free edges near the impact center until the stored momentum in the target is fully dissipated.

Snapshots of SLGR under the impact velocity of 35?/ps are shown in Fig.S1.The stress wave propagation process is similar to that of SLGDY.During propagation of the conic wave,cracks initiate,followed by perforation and propagation of the cracks,as shown in Fig.S1(a)~(d).After full penetration,petals are formed around the impact center,as shown in Fig.S1(e);the number depends on the initial impact velocity.As shown in Fig.S1(f),some of the petals fold downwards and snap back several times due to the rapid relaxation of kinetic energies of the petals,leading to folded and creased patterns on the petals.The broken morphologies of SLGR are consistent with the study by Lee et al.[14]and much different from SLGDY,which is mainly dominated by the suspended chain network.

3.3.2.Deformation behavior of the targets

Dynamic deformation during impact is an efficient channel to dissipate the impact energy.Fig.7(a)and(b)show the evolution of out-ofplane displacements and the diameters of the conic deformation regions of SLGDY and SLGR,respectively,at Vb(i.e.Vi=24.8?/ps for SLGDY and Vi=32.9?/ps for SLGR).The maximum elastic displacements are 11.83?for SLGDY at 1.2 ps and 11.1?for SLGR at 0.9 ps as shown in Fig.7(a),indicating 6.2% higher deformation capability of SLGDY compared to that of SLGR.The bonds begin to break at the time depicted by the black arrows,followed by the occurrence of fracture.The increase in diameters with respect to time has been fitted by linear relations in Fig.7(b).The simulated average conic deformation velocities are determined as 3.71 km/s and 3.85 km/s for SLGDY and SLGR,respectively,at Vb,which are slower than the theoretical values(i.e.3.99 km/s for SLGDY and 4.85 km/s for SLGR)as given by Eq.(6c)due to the fluctuation of the conical surface resulting from the vibration of atoms.C3for SLGR is faster than that for SLGDY at Vband thus at a given instant,the diameter of the conic deformation for SLGR is larger than that for SLGDY.Therefore,during impact,a relatively smaller basal diameter but a deeper conic deformation region is formed for SLGDY compared to SLGR.

Fig.8.Fracture behavior of SLGDY and SLGR.(a)Vi=24.9?/ps for SLGDY and(b)Vi=33?/ps for SLGR.

The elastic out-of-plane deformation behavior of SLGDY and SLGR at Vi=20?/ps are given in Fig.7(c)and(d),respectively.At an impact velocity of 2 km/s,the conic radius increases,with average velocities of 3.10 km/s and 3.29 km/s for SLGDY and SLGR,respectively,which are slightly smaller than the theoretical values given by Eq.(6c).The maximum out-of-plane displacements attain 16.91?at 2.7 ps for SLGDY and 10.39?at 1.9 ps for SLGR.After that,the elastic deformation is gradually recovered.The maximum out-of-plane displacement of SLGDY is 38.7%larger than that of SLGR,indicating greater flexibility of SLGDY compared to SLGR.

3.3.3.Fracture behavior of the targets

At high impact velocities,cracks initiate and propagate in the targets due to the breaking of bonds,further contributing to the dissipation of the impact energy.The fracture morphologies of SLGDY at Vi=24.9?/ps and SLGR at Vi=33?/ps are shown in Fig.8.The color coding represents the potential energy of atoms.Several bonds of SLGDY near the impact center break at 0.7 ps,and the broken carbon atomic chains are rearranged after full perforation.There are two kinds of carbon chains at the peripheries of the penetration holes,as observed in Refs.[57,58].One is suspended free-hanging monatomic carbon chains with the end attached to the edge of the penetration holes,and the other is the carbon chain loops along the peripheries of the penetration holes.This phenomenon indicates that the line of carbon atoms is a stable structure that tends to stabilize the energy[59,60].The initial shape of the perforation hole is similar to that of the projectile,followed by further expansion along directions having relatively low binding energy.The complex sp2and sp hybridized carbon atoms in SLGDY make the fracture behavior of SLGDY somewhat like a rope network,with the final formation of an annular perforation hole,as shown in Fig.8(a).

The fracture behavior of SLGR at impact velocities 33?/ps and 34?/ps are shown in Fig.8(b)and Fig.S3,respectively.The impact-induced transverse wave propagates with C2as indicated by the black arrows.Unlike SLGDY,the typically damaged morphology shows asymmetrically shaped petals along with different radial angles,which is consistent with the results of Refs.[14,43,45,51].Fig.8(b)and S3 show that the average apex angles of petals of 89°±13°and 120°±27°for Vi=33?/ps and 34?/ps,respectively.Similar to SLGDY,two kinds of carbon chains of SLGR are observed at the peripheries of the penetration holes due to the rearrangement of broken carbon chains.It is to be noted that the rearranged chains after full perforation cannot enhance the impact resistance of a single-layer membrane.However,for multi-layer GDY and GR,the broken chains might form some new inter-laminar structures to improve the impact resistance of the materials,which will be investigated in the future.

According to Lee et al.[14],the maximum crack length,Lmax,can be used to estimate the maximum radius of the deformation cone(i.e.rmax=Lmax).The penetration time tp=Lmax/C3,and the average maximum tensile strain εmax≈（Vitp/Lmax）2/2.The results of Lmaxand εmaxfor SLGDY and SLGR are shown in Fig.9(a)and(b),respectively.The Lmaxof SLGDY increases sustainably with the increase of impact velocities for Vi≤60?/ps,then it attains a plateau for higher impact velocities due to the relatively stable fracture morphology,as shown by the atomicstructure of the inset in Fig.9(a).However,the tendency of Lmaxwith respect to Vifor SLGR is different.Initially,Lmaxincreases quickly to the peak value with increasing Vifrom 32.9 to 40.0?/ps.Then it keeps almost constant for Vi,ranging from 45.0 to 90.0?/ps.For Vi≤40?/ps,the triangular fracture morphology is observed for SLGR.It is intriguing to note that the fracture morphology of SLGR changed from triangle to quadrilateral with a further increase of Vito 45?/ps,resulting in the drastic decrease of Lmax,as shown in Fig.9(b).Then Lmaxof SLGR increases slowly with the increase of Vi,and the fracture morphology stays quadrilateral.For both SLGDY and SLGR,εmaxincreases with the increase of Vi.Taking Vi=35?/ps as an example,εmaxof SLGDY and SLGR are estimated to be 16% and 15.8%,respectively,showing that the large deformation abilities of SLGDY and SLGR provide essential contributions to the impact resistance.

Fig.9.Maximum crack length and tensile strain for(a)SLGDY and(b)SLGR.

4.Conclusions

The high-velocity micro-ballistic impact behavior of SLGDY and SLGR are investigated,and the related dynamic responses and energy dissipation mechanisms are obtained.The main conclusions are as follows.

1.The ballistic limits Vband specific penetration energy E＊pfor SLGDY and SLGR are obtained based on ballistic simulations.The results show that Vbof SLGDY can reach 75.4% of that of SLGR,and E＊pof SLGDY is approximately the same as that of SLGR,indicating the promising capability of SLGDY and SLGR as advanced bulletproof materials.Also,both Vband E＊pof SLGDY and SLGR show deterioration at relatively high temperatures.

2.The energy dissipation mechanisms of SLGDY and SLGR are carefully investigated.The superior impact resistance of SLGDY and SLGR is mainly attributed to the ultra-fast elastic and conic waves as well as excellent deformation capabilities during micro-particle impact.

3.The dynamic responses involving stress wave propagation,conic deformation,and damage evolution of SLGDY and SLGR are investigated.The transformation of the fracture morphology of SLGR from triangle to quadrilateral with the increase of impact velocities is observed and the critical impact velocity is determined.

4.The rearrangement of broken chains at the peripheries of the penetration holes is observed,which might improve the impact resistance of multi-layer GDY and GR by introducing some new inter-laminar structures.

5.The ballistic limits of SLGDY and SLGR reasonably agree with the predictions of macroscopic penetration limits equations,implying the applicability of continuum penetration theories for 2D materials.

Data availability statement

The data that support the findings of this study are available from the corresponding author upon request.

Declaration of competing interest

There is no conflict of business interests in this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China(Grant Nos.11672315,and 11772347),the Science Challenge Project(Grant No.TZ2018001),and the Strategic Priority Research Program of the Chinese Academy of Sciences(Grant Nos.XDB22040302,and XDB22040303).

Appendix A.Supplementary data

Supplementary data to this article can be found online at https://doi.org/10.1016/j.nanoms.2021.12.002.