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, 710049 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, Email: cchen@mail.xjtu.edu.cn, Fax: 0086-29-83237910.
1
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 growth10 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
of dislocation emission from nano-voided copper,15 among others. Also, laser shock
experiment on the nano-scaled void growth in Cu16 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 F i of atom i in the atomic aggregate and the pair potential φ ij
between atom i and its neighboring atom j ,
3
n
E = ∑ F i (∑ ρ i (r ij )) +
j ≠i
i
1
φ ij (r ij )
∑
2 ij ,i ≠ j
where r ij is the distance between the atoms i and
(1)
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
fαij = −
∂E rαij
∂r ij r ij
(2)
where subscript α indicates the directional component. The Virial formula for stress21
is used to define the stress tensor for the atomic system as
σ αβ = −
pi p j
1
( ∑ α i β + ∑∑ rαij f βij )
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 mi and momentum pαi 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
4
refer to the
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 Lx , Ly and Lz are the dimensions of the unit cells in x , y and z directions,
respectively. Accordingly, the void volume fraction is given by
fv =
π R2
(4)
Lx Ly
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 Lx , Ly
and Lz can not be chosen arbitrarily. They must be greater than the potential cut-off
radius rc (i.e. 0.495nm) 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 ak 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
the corresponding minimum periodic distance for [110] − [111] − [112] are a x = 2a0 ,
ay =
3a0 and a z =
6a0 , respectively, where a0 = 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. 23
3. Results and Discussions
3.1 Strain rate effect
It is noted that mechanical behaviors of most materials (including copper) are
6
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 106/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 3a0 . Five
different strain rates, i.e., 1010s-1, 109s-1, 2 × 108s-1, 108s-1, and 2 × 107s-1 are considered.
Obtained strain-stress curves for different strain rates are shown in Fig. 2. Beyond the
maximum strain rate 1010s-1, it is found that as the strain rate increases from 2 × 107s-1 to
109s-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 =2 × 108s-1 is a good trade-off between computational
7
efficiency and unexpected strain rate effect for MD simulations. In the following, results
from simulations with strain rate ε =2 × 108s-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 / Lx and Lx / Ly while the void radius R systematically varied. Numerical results
have shown that the dimension in the z direction has negligible effect on the
stress-strain curves, provided that Lz is greater than the potential cutoff radius rc
(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 Lx / Ly =1,
R / Lx =0.125 and Lz = 6a0 are constructed, with Lx increases from 24 a0 to 140 a0 . 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
parameter for each atom is given by:
P = ∑ | R i + R i +6 |2
(5)
i
where R i and R i+6 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
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,
10
the
underlying
deformation
pattern
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 v = 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
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 Lx
approaching Ly 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 Lx
increases from 16ax to 96ax . 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
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
| εX
dεX
(6)
→ 0
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
64a x × 52a y × 8a z , respectively while their initial void radius is varied. EAM potential
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
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 (10472088 and 10425210) and by the
National Basic Research Program of China (2006CB601202).
15
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20
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22
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17
List of table and figures
R
Ly
y
x
Lx
z
Fig. 1
Unit cell model of cylindrical nano-voided Cu subjected to uniaxial tension in
the x direction.
18
10
1E10
1E9
2E8
1E8
2E7
Stress(GPa)
8
6
4
2
0
0
Fig. 2
0.02 0.04 0.06 0.08
Strain
0.1
0.12
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 3a0
19
Incipient yield strength (GPa)
7
6.5
fv=4.9%
fv=19.6%
6
5.5
5
4.5
4
3.5
3
20
40
60
80
100 120 140
Lx/a0
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
6.5
fv=4.9%
fv=19.6%
28
26
24
22
20
20
Fig. 4
40
60
80
100 120
Stress concentration factor
Maximum local stress (GPa)
30
5.5
5
4.5
4
3.5
20
140
fv=4.9%
fv=19.6%
6
40
60
80
Lx/a0
Lx/a0
(a)
(b)
100 120
140
(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
(b)
(a)
[010]
[100]
28.24
(c)
22.59
16.94
11.30
5.65
0.00
Fig. 5
Deformed atomistic configurations of a nano-voided sample with size
32a0 × 32a0 × 6a0 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 level ε = 8.8% , (b). Strain
level ε = 8.9% (failure strain), (c). Strain level ε = 9.0% .
22
[010]
[100]
(b) 20fs
(a) 0fs
(d) 90fs
(c) 50fs
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
Incipient yield strength (GPa)
7
6.5
fv=4.9%
fv=19.6%
6
5.5
5
4.5
4
3.5
3
40
80
120
160
200
Lx/ax
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
4.5
Stress concentration factor
Maximum local stress (GPa)
22
fv=4.9%
fv=19.6%
20
18
16
14
12
4
3.5
3
2.5
2
20
Fig. 8
fv=4.9%
fv=19.6%
40
60
80
100
20
40
60
Lx/ax
Lx/ax
(a)
(b)
80
100
(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
[111]
[11 0 ]
(b) 30fs
(a) 0fs
(c) 100fs
Fig. 9
(d) 1000fs
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
75
fv=4.9%
fv=19.6%
70
65
60
55
50
45
40
20
40
60
80
100 120 140
Macroscopic Young's modulus (GPa)
Macroscopic Young's modulus(GPa)
80
160
fv=4.9%
fv=19.6%
140
120
100
80
60
20
40
60
Lx/a0
Lx/ax
(a)
(b)
80
100
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.
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Normalized incipient yield strength
Normalized Young's modulus
1.2
Case 1
Case 2
1
0.8
0.6
0.4
0.2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Void volume fraction (fv)
1.2
Case 1
Case 2
1
0.8
0.6
0.4
0.2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Void volume fraction (fv)
(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