Molecular dynamics study on temperature and strain rate dependences of mechanical properties of single crystal Al under uniaxial loading Cite as: AIP Advances 10, 075321 (2020); https://doi.org/10.1063/1.5086903 Submitted: 25 December 2018 Accepted: 06 July 2020 Published Online: 27 July 2020 Zhigao Li , Yongyi Gao , Shiping Zhan , Huihong Fang , and Zhongyi Zhang COLLECTIONS Paper published as part of the special topic on Chemical Physics, Energy, Fluids and Plasmas, Materials Science and Mathematical Physics AIP Advances 10, 075321 (2020); https://doi.org/10.1063/1.5086903 © 2020 Author(s) 10, 075321 AIP Advances ARTICLE scitation.org/journal/adv Molecular dynamics study on temperature and strain rate dependences of mechanical properties of single crystal Al under uniaxial loading Cite as: AIP Advances 10, 075321 (2020); doi: 10.1063/1.5086903 Submitted: 25 December 2018 • Accepted: July 2020 • Published Online: 27 July 2020 Zhigao Li,1,a) Yongyi Gao,1,2 Shiping Zhan,2,b) Huihong Fang,2 and Zhongyi Zhang2 AFFILIATIONS School of Mechanical Engineering, Hunan University of Science and Technology, Xiangtan 411201, China School of Physics and Electronic Science, Hunan University of Science and Technology, Xiangtan 411201, China a) b) Author to whom correspondence should be addressed: lzgsj1415@163.com spzhan86@163.com ABSTRACT Based on the embedded atomic method potential energy function, the uniaxial tensile and compressive deformation of nanocrystalline Al with different sizes in the crystal orientation ⟨100⟩ is studied by the atomistic molecular dynamics simulation approach at six different temperatures and three different strain rates The simulation results show that, under the same simulation condition, the stress–strain curves of nanocrystalline Al in the process of uniaxial tension and compression are asymmetric and there exists a significant difference in the late region of elastic deformation The reason for the asymmetry lies in the difference in the process that the work done by the surroundings converts into the strain energy of nanocrystalline Al in the deformation process At the same temperature and strain rate, the tensile elastic modulus and yield strength of nanocrystalline Al are greater than those of compression With the increase in temperature, the elastic modulus and yield strength of tension and compression gradually decrease and the two values tend to be the same with the increase in temperature The higher the strain rate, the greater the yield strength and the corresponding yield strain of the nanocrystalline Al Finally, the effects of size on the tensile and compressive properties of nanocrystalline Al are briefly discussed © 2020 Author(s) All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/) https://doi.org/10.1063/1.5086903., s I INTRODUCTION Since the nano-electromechanical systems (NEMS) became a hot topic, the structural mechanic behavior at the nano-scale had attracted many researchers.1,2 Because of the fundamental difference between the properties of structural mechanics at the macro-scale and the nano-scale, it is necessary to study the properties of structural mechanics at the nano-scale The molecular dynamics method, a tool to study the structural mechanics behavior at the nano-scale, has been widely used as an effective method in the studies of physics, chemistry, mechanics, materials science, etc Therefore, the study of structural mechanics behavior at the nano-scale by the atomistic molecular dynamics simulation approach has received considerable attention in the recent years AIP Advances 10, 075321 (2020); doi: 10.1063/1.5086903 © Author(s) 2020 Many researchers have carried out remarkable works in this field For example, Zhou et al used molecular dynamics simulations to investigate the deformation mechanisms of nanocrystalline copper.3–9 Li et al studied the deformation mechanisms in bcc Fe nanowires (NWs) by the atomistic MD simulation approach.10,11 Sainath et al conducted the study of the tensile deformation behavior of bcc iron nanowires at a temperature of 10 K and a strain rate of × 108 ps−1 by the atomistic molecular dynamics simulation approach.12 Setoodeh et al researched the mechanical properties of nickel nanowires at different temperatures by the atomistic molecular dynamics simulation approach.13 Koh et al investigated the effects of different strain rates on the tensile behavior of metallic nanowires.14 Chang et al employed molecular dynamics simulations to study the tensile behavior of single crystal titanium 10, 075321-1 AIP Advances for different strain rates (108 s−1 –1011 s−1 ).15,16 Fu et al analyzed the effect of size and temperature on the tensile mechanical properties of CdSe nanowires of the zinc blende by molecular dynamics simulations.17 Li et al used MD simulations to investigate the deformation behavior of nanocrystalline TiAl alloys under tensile loading conditions.18 Mojumder studied the effect of compressive loading on the plasticity of Al–Cu alloys.19 Yu et al employed the molecular dynamics method to study the mechanical properties of Ni3 Al nanowires (NWs) along different crystal orientations under tensile loading at an intermediate temperature.20 Komanduri et al carried out at a constant rate of loading (500 m s−1 ) on some single-crystal cubic metals, both fcc (Al, Cu, and Ni) and bcc (Fe, Cr, and W), to investigate the nature of deformation and fracture by the atomistic molecular dynamics simulation approach.21 In this paper, the relevant research achievements of the above scholars on the mechanical properties of other metals or Al alloys with the molecular dynamics method are presented Currently, the main research achievements on the tensile or compressive deformation of nano-metal Al using MD simulations are as follows For instance, Liu et al analyzed the deformation behavior of the Al nanorods and further explored the deformation difference among nanorods, nanowires, and bulk materials of Al.22 Yamakov et al predicted the mechanical properties of single crystal Al under tensile loading by the atomistic molecular dynamics simulation approach.23 Groh et al proposed a multiscale material modeling method to predict the deformation response of single crystal Al under compression.24 Chabba et al analyzed a pure Al model with MD to predict the bulk modulus for pure Al, finding that the bulk modulus of Al is in agreement with the experimental data.25 Rosandi et al studied the mechanical properties of Al nanowires and alumina coated Al at different strain rates under tension and compression.26 Yuan et al studied the tensile process of nano-single crystal Al for different strain rates and temperatures by the Morse potential The results showed that the stress–strain curves abruptly decrease after linearly increasing to the maximum because the first transition from elastic to plastic deformation and the slip took place.27 Xu et al studied the compressive process of Al nanopillars with different orientations using the MD, and the simulations showed that the initial dislocations always nucleate at free surfaces but the compressive orientation plays a decisive role in the subsequent microstructural evolution and stress– strain response of the pillars.28 Rezaei et al simulated the tensile and compressive behaviors of nano-single crystal Al for different temperatures by employing the MD method Their results showed that the employment of the embedded atomic method (EAM) potential is able to simulate mechanical properties including the modulus of elasticity and yield strength of single Al.29 Motamedi et al studied the mechanical properties of Al/carbon nanotube composites with the method of molecular dynamics simulations The results showed that the Young’s modulus of the composite decreases with the increase in temperature Meanwhile, the ultimate stress of the composite also decreases as the temperature increases.30 Xu et al employed large-scale molecular dynamics simulations to analyze the effect of size on the elastic mechanical properties of nanocrystalline pure metal Al.31,32 Tang et al carried out the molecular dynamics simulation of uniaxial tension and compression on single crystal Al with and without defects They found that, for perfect crystal Al, the ideal strengths and stress–strain curves obtained from the MD simulations agreed well with those obtained by first principles AIP Advances 10, 075321 (2020); doi: 10.1063/1.5086903 © Author(s) 2020 ARTICLE scitation.org/journal/adv calculations.33 Among the above references, Refs 28–30 and Refs 31–33 utilized the NVT ensemble and NPT ensemble, respectively, to carry out the simulations The results showed that the trend of the stress–strain curves obtained by the simulations was approximately the same, as well as the orders of magnitude for the stress The NPT ensemble was utilized in this work according to the research needs To sum up, many researchers have done a lot of simulations by the atomistic molecular dynamics approach, among which there are many valuable scientific research achievements in the field of the deformation of nanocrystalline Al However, these research studies did not use the work–energy transformation and the microstructure evolution method to study the tensile and compressive deformation processes of nanocrystalline Al Since the temperature and strain rate dependence is an important element in nanotechnology research and the microstructure of nanocrystalline materials is not easy to precisely control, the tensile and compressive properties of nanocrystalline Al at different temperatures and strain rates were studied by the atomistic molecular dynamics simulation approach in this work II MODELS AND METHODS The perfect nanocrystalline Al unit cell is the lattice in the form of the fcc structure The LAMMPS (Large-scale Atomic / Molecular Massively Parallel Simulator) molecular dynamics simulator which works based on parallel computing was used to conduct the atomistic MD simulations Three models were established, respectively, and the corresponding sizes are 20a × 10a × 10a, 40a × 20a × 20a, and 50a × 25a × 25a, where a is the lattice constant In this paper, the lattice parameter of nanocrystalline Al is 0.405 nm Each of the three models contained 8000, 64 000, and 125 000 atoms, respectively Figure shows only one of the models in which Fig 1(a) is the unit cell structure model of Al, whose X, Y, and Z axes are parallel to the crystal orientations [100], [010], and [001], respectively The periodic boundary condition was applied in the three orientations, and the loading direction of the strain rate was ⟨100⟩ crystal orientation The LAMMPS was used in the simulations of tension and compression The potential function in the model adopted the EAM potential function in Ref 34, and the Nosé–Hoover temperature control method was used for temperature control.35,36 The Parrinello–Rahman method was used for pressure regulation.37 The Verlet numerical integration algorithm was used to determine the atomic position and velocity.38 The time steps of the simulation were set to fs, and the NPT ensemble was utilized Since the metallic material will heat up during high strain rate loading, the NPT ensemble was used to maintain constant temperature and constant pressure.39 The uniaxial tensile and compressive loading was applied at three constant strain rates given in Table I along the ⟨100⟩ crystal orientation, while in the other two orientations, an NPT ensemble kept the stress around zero, which was more realistic in experiment Some values of loaded strain rates in Table I are large, and some are small compared with the previous studies by other researchers.10,15,16,29,33 Equilibrium is very important for molecular dynamics simulations, so for all the models in this study, before the simulations of tension and compression, the equilibrium process of 200 ps was carried out to make the nanocrystalline Al fully equilibrated, at which 10, 075321-2 AIP Advances ARTICLE scitation.org/journal/adv the atomic system can be expressed as follows: U = ∑ Fi (ρi ) + i ∑ ϕij (rij ) j≠i (1) The EAM potential divided the total potential energy of the system into two parts In the right of Eq (1), the first term ϕij represents the pair potential interaction, rij is the interval length between atom i and atom j, and the second term ρi represents the physical quantity of electron cloud density in the atom at position i, reflecting the multi-body interaction In this expression, 1/2 means that the pair potential is a two-atom common formula and ρi can be expressed as ρi = ∑ fj (rij ) (2) j≠i FIG Establishment of the simulation model (a) Face-centered cubic and (b) simulation model time the atomic configuration was in the initial state Then, the simulations of uniaxial tension and compression were carried out for each model along the ⟨100⟩ crystal orientation for the strain rates whose values are given in Table I, and the time step of the output thermodynamic data was set for the later processing of potential energy In addition, the microstructure evolution was analyzed with the visualization software OVITO, finding the similarities and differences in the process of tensile and compressive deformation.40,41 The stress values in this study were calculated based on the Virial theorem.42,43 This work utilized the Embedded Atomic Method (EAM) potential, and each atom in the material can be thought as embedded in other atoms.44,45 The EAM potential can be applied to the study of the mechanical properties of metal materials, such as material fracture, crystal surface defects, and alloy materials The total energy of In order to clearly describe the increase in the potential of the nanocrystalline Al in the elastic region during tensile and compressive processes, the work–energy transformation was utilized In addition, the difference in the stress–strain relationship between tensile and compressive processes was explained through the processing of the potential energy data of output and the usage of the visualization tool OVITO Suppose that the potential energy sequence of tension is {uT1 , uT2 , , uTi }(i = 1, 2, ) and the potential energy sequence of compression is {uC1 , uC2 , , uCi } (i = 1, 2, ), where T represents tension and C represents compression In the sequence, uTi and uCi represent the potential energy output values of the ith tensile and compressive deformation, respectively The differential of the sequences of potential energies {duTi } and {duCi } can be obtained from the following potential energy sequence: duTi = uTi+1 − uTi (i = 1, 2, ), (3) duCi = uCi+1 − uCi (i = 1, 2, ) (4) uTi , uTj and uCi , uCj represent the ith and jth potential energy output, and j > i According to Eqs (3) and (4), the cumulant between potential energy value points of the ith and the jth in the process of tension and compression is, respectively, sTij = uTj − uTi ( j > i, i, j = 1, 2, ), (5) sCij = uCj − uCi ( j > i, i, j = 1, 2, ) (6) TABLE I The strain rate and temperature of simulations Tension/compression Strain rate (ps) Temp1 (K) Temp2 (K) Temp3 (K) Temp4 (K) Temp5 (K) Temp6 (K) Tr a (ps) Ts b (ns) TSrate1/CSrate1 TSrate2/CSrate2 TSrate3/CSrate3 a b 1.0 × 10−4 1.0 × 10−3 1.0 × 10−2 10 10 10 100 100 100 200 200 200 300 300 300 400 400 400 500 500 500 200 200 200 0.3 0.03 Tr represents the relaxation time Ts represents the simulation time AIP Advances 10, 075321 (2020); doi: 10.1063/1.5086903 © Author(s) 2020 10, 075321-3 AIP Advances ARTICLE According to Eqs (5) and (6), it can be obtained that the cumulative difference of potential energy between any two points in the process of compressive and tensile deformation is C T sCT ij = sij − sij ( j > i, i, j = 1, 2, ) (7) III RESULTS AND DISCUSSION A Work–energy transformation explanation of the differences between tensile and compressive stress–strain relations Using the model shown in Fig 1, the tension and compression of nanocrystalline Al at the strain rate of TSrate3/CSrate3 = 1.0 × 10−2 /ps and the temperature of Temp4 = 300 K were studied The stress–strain curves obtained are shown in Fig By comparing the stress–strain curves of tension and compression in Fig 2, it can be seen that when the strain was within the range of 0–0.065, there was no significant difference between tensile stress and compressive stress When the strain was within the range of 0.065–0.1, the compressive stress was higher than tensile stress Accordingly, the abnormal strain region at the other five temperatures is shown in Table II The potential energy data of the simulation were imported into MATLAB, and the increased values of potential energy in the two deformation processes within the range of 0.065–0.1 were calculated by using the previous formulas (3)–(7) When the strain was 0.065 and 0.1, the corresponding potential energy output time steps were i = 65 and j = 100, and the following equations [Eqs (8)–(10)] were obtained: sTij = 142(eV) (i = 65, j = 100), (8) sCij = 166(eV) (i = 65, j = 100), (9) scitation.org/journal/adv C T sCT ij = sij − sij = 24(eV) (10) From formula (8), it can be seen that in the range of 0.065–0.1, the accumulated potential energy of the compressive process was more than that of the tensile process It can be seen from the relationship between inter-atomic force and atomic distance that when the atoms were stretched or compressed the same distance near the equilibrium point, the repulsion between atoms was close to the attraction With the increase of the degree of tension and compression, the repulsion between atoms increased faster than the attraction Therefore, as the strain value increased, the repulsion between atoms in the compression process was greater than the attraction in the tension process In this case, it took more work for compression to overcome the intermolecular forces than it did for stretching when changing the same inter-atomic distance According to the relationship between inter-atomic forces and potential energy, the compressive stress can be deduced higher than the tensile stress The relationship between potential energy and output time steps in tensile and compressive deformation is shown in Fig The reason why Fig 3(a) is different from Fig 3(b) is that the interatomic distance of nanocrystalline Al was different at 10 K and 400 K, and the interatomic dominant force at 10 K was repulsion Furthermore, at the same time step, compressive deformation required more work than tension, so the potential energy sequence value of compression was greater than that of tension, as shown in Fig 3(a) When the temperature increased, the interatomic distance of nanocrystalline Al gradually increased For example, at the temperature of 400 K, the potential energy of compressive deformation was divided into two regions It can be seen from Fig 3(b) that the potential energy sequence of compressive deformation was smaller than that of tensile deformation before the time step of × 104 This is because the FIG The simulated stress–strain curves under tensile and compressive uniaxial loading TSrate3 = 1.0 × 10−2 /ps, CSrate3 = × 10−2 /ps, and Temp4 = 300 K (8000 atoms) AIP Advances 10, 075321 (2020); doi: 10.1063/1.5086903 © Author(s) 2020 10, 075321-4 AIP Advances ARTICLE scitation.org/journal/adv TABLE II The abnormal strain region for all of the listed temperatures and strain rates The abnormal strain region Size (lattice) 20 × 10 × 10 40 × 20 × 20 50 × 25 × 25 −1 Strain rate (ps 1.0 × 10−4 1.0 × 10−3 1.0 × 10−2 1.0 × 10−4 1.0 × 10−3 1.0 × 10−2 1.0 × 10−4 1.0 × 10−3 1.0 × 10−2 ) 10 K 100 K 200 K 300 K 400 K 500 K 0.035−0.122 0.036−0.125 0.036−0.127 0.036−0.122 0.036−0.124 0.037−0.127 0.038−0.122 0.038−0.124 0.039−0.127 0.036−0.110 0.037−0.111 0.037−0.119 0.037−0.109 0.038−0.111 0.038−0.118 0.039−0.108 0.040−0.111 0.040−0.118 0.038−0.103 0.039−0.104 0.040−0.110 0.039−0.101 0.039−0.101 0.041−0.110 0.040−0.100 0.042−0.104 0.044−0.110 0.062−0.098 0.064−0.098 0.065−0.100 0.060−0.100 0.062−0.095 0.063−0.101 0.063−0.095 0.064−0.098 0.065−0.101 0.064−0.087 0.067−0.095 0.072−0.097 0.064−0.087 0.065−0.087 0.067−0.090 0.066−0.087 0.069−0.087 0.074−0.091 0.074−0.085 0.077−0.088 0.085−0.094 0.073−0.085 0.075−0.086 0.083−0.092 0.076−0.082 0.078−0.086 0.084−0.090 interatomic distance of nanocrystalline Al was large at 400 K, and at this point, the interatomic attraction was dominant Therefore, before the time step of × 104 (the first region), the value of the potential energy sequence of tensile deformation was greater than that of compressive deformation However, after the time step of × 104 (the second region), the interatomic distance of nanocrystalline Al in compressive deformation gradually decreases and the dominant force between atoms was converted from attraction to repulsion At this time, at the same time step, more work was needed to be done on nanocrystalline Al in compressive deformation than in tensile deformation Therefore, the relevant situation is shown in Fig 3(b) B The effect of strain rate on the stress–strain relationship of tensile stress Using the model shown in Fig 1, where the temperature is set as Temp4 = 300 K, the loading strain rates were chosen to be TSrate3/CSrate3, TSrate2/CSrate2, and TSrate1/Csrate1, respectively The tensile and compressive curves of nanocrystalline Al were discussed in detail The stress–strain curves for different strain rates were obtained, as shown in Fig It can be seen from Fig that the change in strain rates during the tensile and compressive process had a little effect on the elastic phase of the stress–strain relationship and had a greater effect on the plastic phase However, the yield point increased with the increase in strain rates After entering the yield zone, the tail of this stress–strain curve showed a zigzag shape, which was also found in some Refs 10, 22, 29, and 33 These values of elastic modulus were comparable with those reported in previous studies (44.8 Gpa and 70 GPa).22,46 The magnitude of nanocrystalline Al presented in this study was acceptable C The effect of temperature on the stress–strain relationship between the tensile and compressive process Using the model shown in Fig 1, at the strain rate of TSrate1/Csrate1 = × 10−4 /ps and all the temperatures given in Table I, the tension and compression of nanocrystalline Al were simulated and stress–strain relationship curves at different temperatures were obtained, as shown in Fig Figure 5(a) shows the AIP Advances 10, 075321 (2020); doi: 10.1063/1.5086903 © Author(s) 2020 stress–strain relationship curve during tension Figure 5(b) shows the stress–strain relationship curve under compression In addition, it can be seen from Fig that the tensile stress and compressive stress changed with the variation in temperature The stress of the elastic phase and yield strength decreased with the increase in temperature However, the stress with temperature varied more in the plastic phase than in the elastic phase The comparison between Figs 5(a) and 5(b) shows that for the same strain, the tensile stress was higher than the compressive stress After entering the yield zone, the tail of this stress–strain curve showed a zigzag shape, which was also found in Refs 10, 22, 29, and 33 D The effect of temperature on elastic modulus and yield strength Based on the established model with 64 000 atoms, using the strain rate and temperatures in Table I, the tensile and compressive processes were numerically carried out to obtain the tensile and compressive stress–strain curves for different temperatures and strain rates The tensile and compressive yield strength of nanocrystalline Al at different temperatures and strain rates were obtained by the stress–strain curves In this study, according to the general simulation rules, the maximum stress in the deformation process was taken as the yield strength47 and the corresponding strain was taken as the yield strain The elastic modulus was obtained by linear fitting of the stress–strain curve when the strain is