Molecular dynamics study of cylindrical nano-void growth in copper under uniaxial tension

Transcription

1 Molecular dynamics study of cylindrical nano-void growth in copper under uniaxial tension Kejie Zhao, Changqing Chen *, Yapeng Shen and Tianjian Lu The MOE Key Laboratory of Strength and Vibration, Xi an Jiaotong University, Xi an, P.R. China Abstract Molecular dynamics (MD) simulation with Embedded Atom Method (EAM) potentials is employed to investigate the cylindrical nano-void growth in face-centered cubic (FCC) single crystal copper. The problem is modeled by a periodic unit cell with centered nano sized hole, subjected to uniaxial tension. Effects of the initial void volume fraction, the size dependence of dislocation emission from void free surface, and the crystalline direction relative to the loading direction on the void growth are considered. Obtained numerical results show apparent size effects on the incipient yield strength, while the macroscopic Young s modulus is shown to be insensitive on the sample size for given void volume fraction. The observed size effects and defect pattern are expected to provide some helpful insights into the damage mechanism of ductile materials at micro scale. Keyword: MD, nano-void growth, single crystal copper, size effects * Corresponding author, cchen@mail.xjtu.edu.cn, Fax:

2 1. Introduction Nucleation, growth and coalescence of voids have been commonly accepted as the prime processes for the ductile failure of metals. All three successive stages are crucial for the strength of engineering materials. At macro-scales, ductile failure in terms of void growth has been extensively studied and various continuum models have been proposed to quantify the void growth in materials. 1-8 Although these continuum models are very helpful in understanding the mechanical behavior of void growth in metals, they are unable to capture the accompanying discrete events (e.g. dislocation emission) or to reveal the underlying mechanisms dictating the void growth in metals. Moreover, at the incipient stages of void growth, the void size is usually in the range of sub-micron or even nano-meter and it is still an open issue as to whether continuum models are appropriate for such small scales. Note that direct experiment study of void growth in metals at micro or nano scale is still a difficult task in view of currently available technologies. A number of micro and nano scale numerical simulations have been carried out aiming at a better understanding of physics picture of nano-void growth metals, including a comparison study of molecular dynamics and crystal plasticity predictions of nano-void growth in FCC single crystal Cu, 9 MD simulations of the effect of triaxiality on the void growth 10 and dislocation emission pattern in nano-voided Cu, 11 MD study of nano-void growth and coalescence in nickel, 12 quasi-continuum method study of nano-void growth in aluminum, 13,14 two dimensional discrete dislocation dynamics study 2

3 of dislocation emission from nano-voided copper, 15 among others. Also, laser shock experiment on the nano-scaled void growth in Cu 16 suggests dislocation emission instead of vacancy diffusion be the dominant mechanism for the void growth. Strong interaction between dislocation and void indicates a size/scale effect in the nano-scaled void growth in Cu, as it has been demonstrated in the mechanical behavior of small scale materials and structures by experiments. 17,18 However, the macroscopic consequence and especially the microscopic underlying mechanism for the effects of specimen size/scale on the nano-scaled void growth have yet to be fully clarified. The aim of this paper is to carry out MD simulations of cylindrical nano-void growth in Cu, in order to explore the size dependence of macroscopic incipient yield strength and stiffness, and the underlying mechanism for the nano-void growth under uniaxial tension. The outline of this paper is as follows: The simulation method and model are briefly introduced in section 2. Section 3 is focused on the simulation results and discussions. This paper is ended with conclusions. 2. Simulation method Note that in MD calculations the material behavior is completely determined by the interaction potential among the constituting atoms. Here, the EAM potential was used, in which the total energy E of an atomic system comprises two parts, i.e., the individual embedding energy i F of atom i in the atomic aggregate and the pair potential φ ij between atom i and its neighboring atom j, 3

4 n i i ij 1 ij ij E = F ( ρ ( r )) + φ ( r ) 2 (1) i j i ij, i j where ij r is the distance between the atoms i and j, n is the number of neighboring atoms of atom i, and ρ represents the electron density. Usually, the potential function can be determined by fitting the cohesive energy, equilibrium lattice constant, bulk modulus, cubic elastic constants, un-relaxed vacancy formation energy, bond length and the bond strength of the atomic molecule among others. The EAM potentials proposed by Foiles et al. 19 and Mishin 20 for copper were used in this study. The parameters for the potentials used in this work were fitted to the experimental data and made available for use in tabular form. Once the total energy E is determined, the force between any pair of atoms i and j can be calculated by f ij α ij E rα = (2) ij ij r r where subscript α indicates the directional component. The Virial formula for stress 21 is used to define the stress tensor for the atomic system as 1 p p ( ) i j α β ij ij σ αβ = + r f i α β V i m i j> 1 (3) where V is the occupied volume of atom i, and the first term on the right hand is the kinetic contribution of atom i with mass m i and momentum p i α and the second term is the microscopic virial potential stress. It should be emphasized that lower case supercripts i and j denote the atoms and subscripts α and β refer to the 4

5 directional component in the Cartesian coordinate system. The stress for the system of atoms is defined as the volume average of per-atom tensor. To study the nano-void growth in copper subjected to uniaxial impulse loading, periodic unit cell models as shown in Fig. 1 are employed, where a Cartesian coordinate system oxyz is defined and the uniaxial stressing loading direction is along the x direction. In Fig. 1, R is the radius of the through thickness cylindrical central hole, and L x, L y and L z are the dimensions of the unit cells in x, y and z directions, respectively. Accordingly, the void volume fraction is given by f v 2 π R = (4) L L x y In order to investigate the crystalline direction dependence of the obtained numerical results, two cases for different crystalline orientations were considered: [100] [010] [001] (i.e. case 1) and [110] [111] [112] (i.e. case 2). It should be noted that, in all the following simulations, periodic boundary conditions were employed in the x, y and z directions. To ensure periodicy of the unit cell models, values of L x, L y and L z can not be chosen arbitrarily. They must be greater than the potential cut-off radius r c (i.e nm) to eliminate the interference from the same atom in neighboring periodic cells. In addition, they must be multiple of the respective minimum periodic distance (denoted by a k with k = x, y and z ) which are crystalline orientation dependent. For the [100] [010] [001] crystalline orientation, the minimum periodic distance in the x, y and z direction is ak = a0 (k = x, y and z ) while 5

6 the corresponding minimum periodic distance for [110] [111] [112] are a x = 2a 0, a y = 3a 0 and a z = 6a 0, respectively, where a 0 = 0.361nm is the lattice constant for copper at room temperature. All simulations reported in this paper are performed using the Large-scale Atomic/Molecular Massively Parallel simulator (LAMMPS) developed by Plimpton and his co-workers. 22 A time step of 1fs is used to integrate the equations of motion for the atoms. Before tensile loading is applied in the x direction, the atomic systems are relaxed with the conjugate gradient method to reach a minimum energy state. The atomic systems are then loaded incrementally. During each increment, a small strain increment of 0.1% in the x direction is imposed to the outermost atoms (about 5 layers of copper atoms) from both ends of the unit cells in the x direction whilst other atoms were free to move in accordance with the Eq. (2). Right after each strain increment, the samples were simulated with NPT ensembles to ensure zero pressure in the y and z directions and a uniaxial stressing in the x direction. It should be noted that all the simulations were taken at 0K to avoid thermal activation during tension loading. The atomistic configurations during deformation were viewed using ATOMEYE developed by Li Results and Discussions 3.1 Strain rate effect It is noted that mechanical behaviors of most materials (including copper) are 6

7 loading rate sensitive. Even with the current available fastest computers, however, MD method is still too time-consuming to simulate experiments conducted at low strain rate (usually much lower than 10 6 /s -1 at laboratories) or to simulate deformation process lasting longer than 1μs. In fact, MD method is more suitable for high strain rate loading or extremely short time processes. Here, the effect of strain rate on the MD simulated macroscopic stress-strain responses of single crystalline copper with nano-void is investigated. The purpose is to quantify the strain rate effect. The crystalline orientation of the unit cell is given in the [100]-[010]-[001] system, with dimensions Lx Ly Lz = = 24a0 24a0 6a0, and the radius of cylindrical void is 3a 0. Five different strain rates, i.e., s -1, 10 9 s -1, s -1, 10 8 s -1, and s -1 are considered. Obtained strain-stress curves for different strain rates are shown in Fig. 2. Beyond the maximum strain rate s -1, it is found that as the strain rate increases from s -1 to 10 9 s -1 the peak strength increases from 6.26 GPa to 7.31 GPa while the initial slope of the stress-strain curve is strain rate insensitive. The failure strain (defined as the strain at the peak) also increases with the increasing of strain rate. Right after the peak loading, the stress decrease abruptly. It was also noted that higher strain rate loading always corresponds to greater fluctuation on the stress-strain curves after the peak, which is partly due to the inertia effect of atoms in high strain rate loadings. However, the inertia effect of atoms is not expected to play key role in governing the nano-void growth in copper at low to moderate strain rates and should be minimized. Numerical experiments show that a strain rate of ε = s -1 is a good trade-off between computational 7

8 efficiency and unexpected strain rate effect for MD simulations. In the following, results from simulations with strain rate ε = s -1 are reported. 3.2 Size effects In this part, effect of void size on the uniaxial stress-strain responses of nano-voided single crystalline copper is investigated. To this end, the unit cell model shown in Fig. 1 is employed, with the void volume fraction kept to be a constant by fixing the ratios R L and L / L while the void radius R systematically varied. Numerical results / x x y have shown that the dimension in the z direction has negligible effect on the stress-strain curves, provided that L z is greater than the potential cutoff radius r c (results are not shown for the sake of brevity). This is consistent with the fact that the nano-void considered here is cylindrically circular and periodic boundary conditions are applied in the z direction. Note that when non-periodic boundary conditions are used significant scale effect in the z direction may be expected. 9 Case-1: [100] [010] [001] oriented single crystal To study the size effects, differently sized models with constant L / L =1, x y R / L x =0.125 and Lz 6a0 = are constructed, with L x increases from 24 a 0 to 140 a 0. The models are rendered to be representative of single crystal copper with an infinite periodic array of voids. To highlight the induced defects during the void growth, centrosymmetry parameter defined by Kelchner et al 24 is employed. Note that the centrosymmetry 8

9 parameter for each atom is given by: 2 P = Ri + R i+ 6 (5) i where R i and i+ 6 R are the vectors corresponding to the six pairs of opposite nearest neighbors in the FCC lattice. Thus, P =0 represents the undisturbed state of the perfect lattice, and P will increases for any defects and for atoms close to free surfaces. For single crystal copper, 0.5 < P < 3 corresponds to the partial dislocation, 3 < P < 16 the stacking faults and P > 16 as the surface atoms. During loading process, it is found the centrosymmetry parameter is close to zero everywhere except for the surface atoms prior to the peak loading point in the stress-strain curves, indicating that the atomic system retains its near perfect lattice structure. Upon the peak point, partial dislocations start to nucleate from both the top and bottom sides of the void surface. This has also been found to be true for both crystalline systems with different void volume fraction: the peak loading points in the associated uniaxial stress-strain curves correspond to the initiation of partial dislocations (i.e., the incipient plasticity). Therefore, the peak stress can be defined as the incipient yield strength of nano-voided Cu under uniaxial tension. Fig. 3 shows the calculated incipient yield strength as a function of sample/void size with the void volume fraction fixed at 4.9% and 19.6%, respectively. The atomic interaction is modeled by the EAM potential function developed for Cu by Foils et al. 19 It is seen from Fig. 3 that the incipient yield strength of nano-voided single crystal 9

10 copper has an apparent size dependency: for given volume fraction, models with the smaller sample/void size behave stronger. In particular, the strength for smallest sample is about 30% and 28.2% larger than that of the biggest one when the void volume fraction is 4.9% and 19.6% respectively. For bigger samples than those given in Fig. 3, the size effect in the incipient yield strength diminishes. Since the initiation of dislocation emission is governed by the local stress, the stress concentration feature is studied, in order to explore the mechanism dictating the observed size dependence of the incipient yield strength. However, accurate determination of the local stress at atomic level is not straightforward, due to the difficulty in calculating the volume occupied by each atom. This is particularly true in regions with high stress gradient and within regions close to the free surface. Here, average atom volume is used to define local Virial stress. 21 The stress concentration factor K can then be defined as the ratio of the maximum local stress to the system average stress. Fig. 4 shows the size dependence of the maximum local stress and the associated stress concentration factor for [100]-[010]-[001] system. The results are shown at loading levels corresponding to the peak points in the stress-strain curves. As illustrated in Fig. 4 (a), for given void volume fraction, the maximum local stress required for dislocation emission is almost independent of the sample/void size. By contrast, the stress concentration factor increases with increasing sample/void size. Therefore, results given in Fig. 4 indicate that higher loading is required for dislocation emission in smaller voided samples. By using centroparameter method, the underlying deformation pattern 10

11 accompanying the observed size effect is also studied. A sequence of deformed atomic configurations for a sample with size of 32a0 32a0 6a0 and f = 4.9% is illustrated in Fig. 5(a)-(c) for the strain levels ε = 8.8%, 8.9% (failure strain) and 9.0%. In Fig. 5, atoms are colored according to their centrosymmetry parameter to give a rough picture of the defect pattern around the void. It is seen that soon after the incipient yielding point, dislocations propagate across the entire sample. It should be noted that a dramatic structural change occurs between Fig. 5(b) and Fig. 5(c) though their strain levels differ only by one loading increment (i.e., 0.1% in strain). In fact, deformation from Fig. 5(b) to Fig. 5(c) corresponds to the rapid and unstable process of partial dislocation growth. To investigate this unstable process further, we have shown in Fig. 6 four snapshots during the energy relaxation process between this strain loading levels. Only atoms with their centroparameter P in the range of 0.5 and 3 are visible, in order to show the defects in the form of partial dislocations. Atomic configurations of Fig. 6(a)-Fig. 6(d) correspond to relaxation instance of 0, 20, 50, and 90fs after the peak loading. It is clear from Fig. 6 that partial dislocations and dislocation loops accompany the unstable process between Fig. 5(b) to Fig. 5(c). This is consistent with the experiment and atomistic studies that dislocation loops emanating from nano-void in Cu. 11,16 Further after 90fs, developed dislocation network swaps through the entire system.. To check whether the aforementioned features of size effects and defect evolution is dependent upon the employed potential for the atoms, we re-calculate the uniaxial tensile 11 v

12 responses of different sample sizes using the EAM potential function developed by Mishin. 20 Again, similar size dependency of the incipient yield strength and defect pattern accompanied with nano void growth are observed. The results are not listed for brevity. It is thus concluded that the observed size dependency, dislocation nucleation and growth features are not sensitive to the employed atomic interaction potentials. Case-2: [110] [111] [112] oriented single crystal In addition to the [100]-[010]-[001] orientation, the [110] [111] [112] orientation is also of great interest for FCC materials by noting that <110> and {111} are the most closely packed direction and plane, respectively. To investigate the size effect in nano-void growth in this orientation, different sized samples are constructed. It is not possible to construct atomic structure with Lx = Ly while at the same time ensure the periodicity in both x and y directions. Nevertheless, models should have their L x approaching L y as closely as possible to eliminate the effect of different aspect ratio in the x and y direction. To this end, differently sized samples are constructed, with L x increases from 16a x to 96a x. Uniaxial tension is applied along the [110] direction and the axial direction of the cylindrical void is along the [112]. Also, two void volume fractions 4.9% and 19.6% are considered. The obtained size effect on the incipient yield strength is plotted in Fig. 7. It is seen that similar size effects to those shown for the [100]-[010]-[001] oriented system are also observed. Correspondingly, the size effect on the maximum local stress and stress concentration factor on the void free surface are 12

13 shown in Fig. 8(a) and Fig. 8(b) respectively. Together with Fig. 4, it can be concluded that the stress concentration factor could be an interpretation of the observed size effect of incipient yield strength. Similar to the preceding section in [100]-[010]-[001] system, the defect evolution process by partial dislocation emission between the loading levels 5.6% (failure strain) and 5.7% is studied. Atomic configurations at the intervals of 0, 30, 100 and 1000fs are plotted in Fig. 9, where only atoms with P in the range 0.5 < P < 3 are shown to illustrate the partial dislocation evolution pattern. Again, similar features to those reported for the [100]-[010]-[001] orientation are observed. Besides, with the simulated stress-strain response curve, the sample/void size effect on the macroscopic Young s modulus can be studied for both crystalline systems. The macroscopic Young s modulus derived from the stress-strain curves is defined as Ef dσ X = ε X 0 (6) d ε X Calculated results are shown in Fig. 10, it can be seen that at given void volume fraction, the macroscopic Young s modulus is insensitive to the sample/void size. 3.3 Effect of void volume fraction To study the effects of void volume fraction on the incipient yield strength and macroscopic Young s modulus of nano-voided single crystal Cu, samples with varying initial void volume fractions are constructed. For [100] [010] [001] and [110] [111] [112] oriented systems, sample sizes are fixed at 32a0 32a0 6a0 and 13

14 64a 52a 8a, respectively while their initial void radius is varied. EAM potential x y z developed by Foils et al. 19 is employed. Dependence of the macroscopic Young s modulus and incipient yield strength on the void volume fraction is shown in Fig. 11. Results are normalized by the corresponding Young s modulus and incipient yield strength of Cu without void. It can be seen from Fig. 11 that the Young s modulus and incipient yield strength of [110] [111] [112] oriented Cu are much more sensitive to the presence of void than those of [100] [010] [001] oriented Cu: introduction of a nano-scaled void with 10% volume fraction leads to about 22% drop in Young s modulus and 42% drop in yield strength for [110] [111] [112] oriented Cu while the corresponding decreases for [100] [010] [001] oriented Cu are only about 3.5% and 4%, respectively. 4. Conclusions Molecular dynamics simulations have been performed to study the initial void volume fraction influence on the mechanical properties of the cylindrical nano-voided Cu subjected to uniaxial tension. Obtained MD simulations show that the incipient yield strength has a strong size dependency in view of the void size. For given void volume fractions, the smaller void behaviors stronger. And it is concluded that the stress concentration on the void free surface dominates the observed size dependency of yield strength, providing the maximum local stress dictating the dislocation emission is constant for both crystalline systems. By contrast, it is found that the macroscopic 14

15 Young s modulus is insensitive with the sample size. Also, defect pattern accompanied with the nano void growth and dislocation nucleation and emission from the void free surface are observed. With increasing the void volume fraction, it is found that the Young s modulus and yield strength of Cu decrease, and the macroscopic Young s modulus and yield strength of [110] [111] [112] oriented Cu are much more sensitive to the presence of void than those of [100] [010] [001] case. Acknowledgement The authors are grateful for the financial support of this work by the National Natural Science Foundation of China ( and ) and by the National Basic Research Program of China (2006CB601202). 15

16 References 1 F. A. McClintock, J. Appl. Mech. 35, 363 (1968) 2 J. R. Rice and D. M. Tracey, J. Mech. Phys. Solids, 17, 201 (1969) 3 A. Needleman, J. Appl. Mech. 94, 964(1972) 4 A. L. Gurson, J. Eng. Mat. Technol, 99, 2 (1977) 5 V. Tvergaard, Int. J. Fract., 17, 389 (1981) 6 J. M. Duva and J. W. Hutchinson, Mech. Mater., 3, 41 (1984) 7 B. Liu, X. Qiu, Y. Huang, K. C. Hwang, M. Li and C. Liu, J. Mech. Phys. Solids, 51, 1171 (2003) 8 D. C. Ahn, P. Sofronis and R. Minich, J. Mech. Phys. Solids, 54, 735 (2006) 9 L. Farrissey, M. Ludwig, P. E. McHugh and S. Schmauder, Comput. Mater. Sci., 18, 102 (2000). 10 E. T. Seppala, J. Belak and R. E. Rudd,. Phy. Rev. B, 69, (2004). 11 L. P. Davila et al., Appl. Phys. Lett., (2005) 12 G. P. Potirniche, M. F. Horstemeyer, G. J. Wagner, P. M. Gullett, Int. J. Plasticity, 22, 257 (2006). 13 J. Marian, J. Knap, and M. Ortiz, Phys. Rev. Lett. 93, (2004). 14 J. Marian, J. Knap, and M. Ortiz, Acta Mater., 53, 2893 (2005). 15 M. Huang, Z. H. Li, and C. Wang, Acta Mater., 55, 1387 (2007) 16 V. A. Lubarda, M. S. Schneider, D. H. Kalantar, B.A. Remington and M. A. Meyers. 16

17 Acta Mater, 52, 1397 (2004) 17 N.A. Fleck, G. M. Muller, M.F. Ashby, and J.W. Hutchinson, Acta Mater. 42, 475.(1994) 18 W. D. Nix, and H. Gao, J. Mech. Phys. Solids, 446, 441 (1998) 19 S. M. Foils, M. I. Baskes and M. S. Daw, Phy. Rev. B, 33, 7983 (1986) 20 Y. Mishin, Phy. Rev. B, 63, (2000) 21 M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids (Oxford University Press, Oxford, 1987). 22 S. J. Plimpton. J. Comp. Phys, 117, 1 (1995) 23 J. Li. Modelling. Simul. Mater. Sci. Eng (2003) 24 C.L. Kelchner, S. J. Plimpton and J. C. Hamilton. Phy. Rev. B, 58, (1998) 17

18 List of table and figures R L y y x L x z Fig. 1 Unit cell model of cylindrical nano-voided Cu subjected to uniaxial tension in the x direction. 18

19 Stress(GPa) E10 1E9 2E8 1E8 2E Strain Fig. 2 Effect of strain rate on the MD simulated uniaxial stress-strain responses of [100]-[010]-[001] oriented single crystal Cu at 0 K. The loading is applied along the [100] direction, the model with dimensions being 24a0 24a0 6a0 and the radius of void being 3a 0 19

20 Incipient yield strength (GPa) f v =4.9% f v =19.6% L x /a 0 Fig. 3 Sample/void size effect on the incipient yield strength of nano-voided Cu under uniaxial tension in [100]-[010]-[001] crystalline system 20

21 Maximum local stress (GPa) f v =4.9% f v =19.6% Stress concentration factor f v =4.9% f v =19.6% L x /a 0 L x /a 0 (a) (b) Fig. 4 (a). Sample size dependence of the maximum local stress, and (b). Sample size dependence of the stress concentration factor of nano-voided Cu under uniaxial tension in [100]-[010]-[001] system. 21

22 (a) (b) [010] [100] (c) Fig. 5 Deformed atomistic configurations of a nano-voided sample with size 32a 32a 6a and void volume fraction f =4.9%. Crystal orientation is [100] [010] [001]. Atoms are color-coded according to the centrosymmetry parameter P in the range between 0 and 28. (a).strain levelε = 8.8%, (b). Strain level ε= 8.9% (failure strain), (c). Strain level ε = 9.0%. 22

23 [010] [100] (a) 0fs (b) 20fs (c) 50fs (d) 90fs Fig. 6 Plots of the partial dislocation growth during the relaxation process between the loading levels 8.9% and 9.0%. Only atoms with their centroparameter P in the range between 0.5 and 3 are visible. 23

24 Incipient yield strength (GPa) f v =4.9% f v =19.6% L x /a x Fig. 7 Sample/void size effect on the incipient yield strength of nano-voided single crystal Cu under uniaxial tension in [110] [111] [112] crystalline system 24

25 Maximum local stress (GPa) f v =4.9% f v =19.6% Stress concentration factor f v =4.9% f v =19.6% L x /a x L x /a x (a) (b) Fig. 8 (a). Sample size dependence of the maximum local stress, and (b). Sample size dependence of the stress concentration factor of nano-voided Cu under uniaxial tension in [110] [111] [112] system. 25

26 [111] [11 0 ] (a) 0fs (b) 30fs (c) 100fs (d) 1000fs Fig. 9 Plots of the partial dislocation at the intervals of (a) 0fs (b) 30fs (c) 100fs and (d) 1000fs of the relaxation process between the loading levels 5.6% (failure strain) and 5.7%. Only atoms with their centrosymmetry parameter P in the range between 0.5 and 3 are visible. 26

27 Macroscopic Young's modulus(gpa) f v =4.9% f v =19.6% Macroscopic Young's modulus (GPa) f v =4.9% f v =19.6% L x /a 0 L x /a x (a) (b) Fig. 10 Sample/void size effect on the macroscopic Young s modulus of nano-voided Cu: (a) [100]-[010]-[001] systems, and (b). [110] - [111] - [112] systems. 27

28 Normalized Young's modulus Case 1 Case Void volume fraction (f v ) Normalized incipient yield strength Case 1 Case Void volume fraction (f v ) (a) (b) Fig. 11 Dependence of the normalized (a) macroscopic Young s modulus and (b) incipient yield strength on the void volume fraction for Cu. 28