Journal of Modern Physics, 2012, 3, 1594-1601 Published Online October 2012 (
Evolution of Internal Crack in BCC Fe under
Compressive Loading
Dongbin Wei1, Zhengyi Jiang1, Jingtao Han2
1School of Mechanical, Materials and Mechatronic Engineering, University of Wollongong, Wollongong, Australia
2School of Materials Science and Engineering, University of Science and Technology Beijing, Beijing, China
Received August 5, 2012; revised September 3, 2012; accepted September 9, 2012
A molecular dynamics model has been developed to investigate the evolution of the internal crack of nano scale during
heating or compressive loading in BCC Fe. The initial configuration does not contain any pre-existing dislocations. In
the case of heating, temperature shows a significant effect on crack evolution and the critical temperature at which the
crack healing becomes possible is 673 K. In the case of compressive loading, the crack can be healed at 40 K at a load-
ing rate 0.025 × 1018 Pa·m1/2/s in 6 × 1012 s. The diffusion of Fe atoms into the crack area results in the healing proc-
ess. However, dislocations and voids appear during healing and their positions change continuously.
Keywords: Internal Crack; Evolution; BCC Fe; Simulation; Compressive Load
1. Introduction
Metals own the ability of self-healing. It is possible that
low strength materials might be made more reliable at a
low cost if the crack healing is applied at an intermediate
stage of their service life. The self-healing effect for
creep cavitation in heat resisting stainless steels has been
developed by alloy design, so micro-cracks can be con-
strained from propagating by automatically forming
compounds on the crack surfaces although they may not
be annihilated [1,2].
A lot of research has been conducted on crack healing
in inorganic and composite materials while less work has
been performed on metals. The damage evolution in
metals was analysed [3], and many kinds of heat treat-
ment methods can be used to decrease the number of
defects in metals after a long-term operation at high
temperatures [4]. By TEM observation, a crack in α-Fe
was found to be healed completely when the temperature
was increased to a critical value of 1073 K [5]. It was
suggested that the thermal or mechanical treatment pro-
vides the driving force to minimise the surface energy
resulting in the crack healing [6]. The healing of internal
pre-cracks obtained using plate impact technology was
studied in a vacuum at an elevated temperature [7,8], and
the inhomogeneous microstructure in crack healing area
was analysed [9]. It has been shown that the healing of
fine cracks (100 nm) begins immediately after heating
then proceeds according to the vacancy diffusion mecha-
nism in the surface layer about 10 μm thick [10].
The effect of hydrostatic pressure on the shrinkage of
cavities in copper introduced by creep at 773 K and situ-
ated on grain boundaries was studied [11]. The density of
the specimen showed a rapid increase when a hydrostatic
pressure was applied during annealing, while the change
was little when annealed under atmospheric pressure.
The similar result was also obtained in [12]. One of the
reasons why the ductility of experimental steels can be
improved over the whole working temperature is to anni-
hilate or heal micro-cracks that may occur during defor-
mation [13]. The internal crack can be healed completely
and rapidly under hot plastic deformation [14]. It is pos-
sible to intensify crack healing and/or refinement of the
microstructure by adjusting the parameters of plastic de-
formation or isostatic pressure [15].
Molecular dynamics (MD) has been adopted to dis-
cover the essence of crack healing in metals. MD simula-
tion was conducted to study crack healing in Al [16] and
Cu crystals [17], both of which are FCC. Dislocation
generation and motion occur during crack healing. The
critical temperature for crack healing depends upon the
orientation of the crack plane. BCC crystal is character-
ized by four close-packed directions, the <111> direc-
tions, and by the lack of a truly close-packed plane such
as the octahedral plane of the FCC lattice.
The crack healing behaviour in BCC-Fe crystal was
simulated, however, only heating has been considered
[18]. In this study, the molecular dynamics model has
been further developed to investigate the evolution of
nano scale crack in BCC Fe. Not only has the effect of
opyright © 2012 SciRes. JMP
D. B. WEI ET AL. 1595
temperature ranged 273 - 1173 K been considered, but also
the effect of compressive loading is taken into account.
2. Computational Method
The molecular dynamics method is built based on a set of
Newtonian dynamics equations for a number of particles
with the macro constraints and boundary condition. They
are solved by a numerical method to achieve the classical
trajectory and velocity. The macro physical properties
and motion routine can be obtained based on the statistic-
cal average of the results calculated for a long duration of
Assuming the total number of atoms is N, which fol-
lows Newton’s law.
tot1 2
 
rr r
where , Utot is the total potential energy of
N atoms, is the atom mass. The position i
r and
velocity i of every particle in a system can be obtained
by solving Equation (1). The position i and
velocity of every particle at time
1, 2,,i
vt t
depend on the position and velocity of that
particle respectively at time t, and the leapfrog algorithm
[19] is applied to calculate the positions and velocities of
The reliability of the molecular dynamics simulation
depends on the description of the inter-atomic potential.
The inter-atomic potential adopted here is the N-body
potential proposed by Finnis and Sinclair according to
the embedded atom method (EMA) [20], the total energy
of an assembly of atoms can be written as
 
where the second term on the right hand side of Equation
(2) is the conventional central pair-potential summation,
while the first term is the N-body term, in which
where both V and Φ are functions of the inter-atomic
distance rij. Constructed by [21], the following functional
forms have been adopted.
kk k
kk k
Vra rrHrr
  
where the parameters a1, a2, A3, A4, A5, A6, A1 and A2 are
36.559853, 62.416005, 13.155649, 2.721376,
8.761986, 100.0000, 72.868366 and 100.944815 re-
spectively. r1, R2, R3, r4, r5, R6, R1 and R2 are 1.180000,
1.150000, 1.080000, 0.990000, 0.930000, 0.866025,
1.300000 and 1.200000 respetively [22],
x is the
Heaviside unit-step function which gives the cut-off dis-
tance of each spline segment, r1 and R1 represent the cut-
off radii of V and Φ respectively.
3. Model Description
The x-axis is along the extension line of the central crack,
the y-axis along the normal of the crack plane, and the
z-axis along the 112
direction. The lattice constant a0
for BCC Fe is 2.8665 × 1010 m. The length of the calcu-
lation cell along the x-axis direction is 0
264 6a,
y-axis 0
48 2a, z-axis 0
66a. Every three layers
of atoms along the x-axis direction constitutes a cycle of
2a long, every two layers of atoms along y-axis di-
rection constitutes a cycle of 0
2a long, and every six
layers of atoms along z-axis direction constitutes a cycle
of 0
6a long.
The crack is expressed by removing some atoms in the
centre of the cell [23] and the angle between the
slip plane and the crack plane is set to be 60˚ randomly.
The crack is about 5.957 × 10–9 m long and over 1.500 a0
wide. The width is larger than the cut-off radii of Φ,
1.300 a0, which means that there exists no interplaying
force between the atoms on either side of the crack sur-
face. The total number of atoms is 12582.
Because of the limit of computation scale, the number
of atoms in the simulation is usually much lower than
that in a real system. Boundary conditions have signifi-
cant effect on the reliability of simulation results. In the
case of heating, fixed boundary condition [24,25] is
adopted along the x- and y-axis directions, which means
that the locations of the boundary atoms are constant
during the whole calculation process. In the case of load-
ing, displacement boundary condition is adopted along
the x- and y-axis directions, i.e. the boundary atoms have
displacement based on the biaxial plane strain linear
elastic displacement field in the solid containing crack
112 Relm
121 lmRe
where ux, uy are the displacement in x and y direction
respectively, E is Young’s modulus, γ is Poisson’s ratio.
 is a Westergaard function and
z, in which
is the applied stress, a is the half
Copyright © 2012 SciRes. JMP
length of the central crack and z = x + yi. In both cases,
periodical boundary condition [17,27] is used in z direc-
The initial velocity is the Maxwell-Boltzmann distri-
bution corresponding to a given temperature. The system
temperature is maintained constant by scaling the atom
velocities during the simulation.
The equilibrium of the atomic configuration is ob-
tained after a long time relaxing at a low temperature
near 0 K, as shown in Figure 1. No defects are set in this
initial configuration. dy representing the minimum verti-
cal distance between the atoms on top crack surface and
bottom crack surface is adopted for assessing healing
process [18]. It is 1.526 a0 in the initial configuration.
Crack healing becomes possible when dy is less than the
cut-off radii of V, 1.180 a0.
4. Results and Discussion
4.1. Heating
The simulated temperature in the model in this study has
to be below 1185 K for keeping the parameters of N-
body potentials for BCC Fe valid. Simulations were car-
ried out in the temperature range of 273 - 1173 K and the
increment is 50 K. The simulation spans 6.0 × 1011 s and
the time step is 1 × 1016 s. Δt
Relationships between dy and time at the temperatures
of 573, 623, 673, 773, 873 and 923 K are shown in Fig-
ure 2 for comparison. It can be seen that dy experiences a
dramatic decrease in the first 1.0 - 2.0 × 1012 s, then
increases and tends to fluctuate in a narrow range in all
cases. The dramatic decrease may result from the unsta-
ble calculation in this first 1.0 - 2.0 × 1012 s. The fluc-
tuation range moves down with an increase of tempera-
When the temperature is 573 K, the fluctuation may be
over the cut-off radii of Φ, R1 = 1.300 a0. When the tem-
perature is 623 K, the fluctuation range is between R1 =
1.300 a0 and the cut-off radii of V, r1 = 1.180 a0. When
the temperature is 673 K, the fluctuation range is around
r1 = 1.180 a0. When the temperature reaches 773 K, the
fluctuation ranges decrease rapidly to be around 0.500 a0.
When the temperature reaches 923 K, the atoms on top
and bottom surfaces move actively as shown in Figure 3,
and dy becomes 0 when the simulation time is 3 × 1011 s,
as shown in Figure 3(c). The reason why the fluctuation
ranges can decrease dramatically when the temperature
reaches about 773 K may be the atoms on top and bottom
surfaces become active enough at this temperature and
can be brought to within a close range abruptly. A thre-
shold value might exist but need to be delivered in the
future study.
The above results show that the critical temperature at
which the crack healing becomes possible is 673 K. A
=1.526 a
5.957 10
Figure 1. Initial configuration.
cut-off radii of
cut-off radii of V
Time, 10-11s
dy, x a0
Figure 2. Relationships between dy and time.
transition temperature may exist in the range of 673 - 773
K, above which the process of crack healing is acceler-
ated dramatically. The temperature at which a complete
healing may be achieved is 923 K. Figure 4 shows the
process of crack healing at 1173 K.
Only after 1 × 1013 s, dy decreases to 0.989 a0, which
is less than r1 = 1.180 a0, as shown in Figure 4(a); after
1.1 × 1012 s, the crack evolves into two sections, as
shown in Figure 4(b); after 1.8 × 1012 s, only two voids
are left at the position of two crack tips, as shown in
Figure 4(c); after 4.5 × 1012 s, the crack is healed com-
pletely, as shown in Figure 4(d); the state of crack heal-
ing remained stable after that.
4.2. Compressive Loading
The atomic configuration shown in Figure 1 is also
adopted as the initial configuration for simulating crack
evolution during compressive loading. The environment
temperature is 40 K. The time step Δts 1 × 1015 s and
the loading rate is 0.025 × 1018 Pa·m1/2/s.
After 5 × 1012 s, when the stress intensity factor KI =
1.25 × 105 Pa·m1/2, dy is 1.307 a0 and it is very close to
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D. B. WEI ET AL. 1597
Figure 3. Evolution of crack morphology at 923 K. (a) t = 6
× 1012 s; (b) t = 1 × 1011 s; (c) t = 3 × 1011 s.
the happening of crack healing, as shown in Figure 5(a).
After 6 × 1012 s, dy = 1.235 a0, which is less than R1 =
1.300 a0, which means that the crack tends to be healed
as shown in Figure 5(b). When the time is 3 × 1011 s,
the crack is healed completely, as shown in Figure 5(c).
Then it remains stable and the morphology when the time
is 6 × 1011 s is shown in Figure 5(d).
During crack healing process, atoms deviates from
their normal positions, which results in many defects
such as dislocations and voids similar to that in Al [16]
and Cu [17]. Some defects in the simulation cell at 4.5 ×
1012, 6.0 × 1012, 3.6 × 1011 and 6.0 × 1011 s after the
dy = 0.989 a0
Figure 4. Crack healing process at 1173 K. (a) t = 1 × 1013 s;
b) t = 1.1 × 1012 s; (c) t = 1.8 × 1012 s; (d) t = 4.5 × 1012 s. (
Copyright © 2012 SciRes. JMP
Copyright © 2012 SciRes. JMP
= 1.307 a
= 1.235 a
(a) (b)
(c) (d)
Figure 5. Crack healing process during compressive loading at 40 K. (a) t = 5 × 10-12 s, KI = 1.25 × 105 Pa·m1/2; (b) t = 6 × 1012
s, KI = 1.5 × 105 Pa·m1/2; (c) t = 3 × 1011 s, KI = 7.5 × 105 Pa·m1/2; (d) t = 6 × 1011 s, KI = 1.5 × 106 Pa·m1/2.
crack has been healed completely at 1173 K are marked
in Figures 6(a)-(d) respectively. The dashed rectangle
frames show original crack locations, and the circles
show the positions of some defects. The distribution of
the defects in the cell is inhomogenous. Defects can be
found to aggregate in the region where is with the origi-
nal crack, especially around the areas where are the crack
tips. The positions of these defects change continuously,
and the inhomogeneity of the distribution of the defects
decreases with an increase of time.
The crack can be considered a very large defect in the
simulation cell. The existence of nano scale crack surface
results in higher system energy than that without inner
crack, so there is a tendency to eliminate the crack to
reduce the system energy. However, the crack healing
process cannot happen when the temperature is low be-
cause an energy barrier exists between the state when the
crack exists and the state when the crack is healed. Im-
porting energy by temperature rise to activate atoms can
help overcome the energy barrier.
In the process of crack healing, the activated atoms
close to crack immigrate to the crack area and vacancies
appear in the positions from where atoms emigrated, then
the atoms adjacent to these vacancies need adjust their
positions. Because the crack size is much larger than the
size of any other defects appearing in the simulation cell,
it may cost much more time for the atoms immigrating
to the crack area to rearrange their positions than the
atoms that only need adjust their positions to accommo-
date adjacent vacancies. This results in that the distribu-
tion of defects is inhomogeneous in the early stage of
The healing of a crack means that a large defect
evolves into a lot of small defects and all the defects may
not be eliminated. However, the atoms adjust their posi-
tions continuously in order to reduce the system energy
even after the crack has been healed. This continuous
position adjustment results in more homogeneous distri-
bution of defects with an increase of time, and the quan-
tity of the defects in the crack area may be reduced.
Some defects in crack healing region when the time is
3 × 1011 and 6 × 1011 s during compressive loading are
marked in Figure 7. These defects also change their po-
sitions, but the change is not so dramatically. This may
be because the atoms are not very active due to the low
temperature. Most defects aggregate in the region where
D. B. WEI ET AL. 1599
(a) (b)
(c) (d)
Figure 6. Dislocations and voids in crack healing region at 1173 K. (a) t = 4.5 × 1012 s; (b) t = 6.0 × 1012 s; (c) t = 3.6 × 1011 s;
(d) t = 6.0 × 1011 s.
(a) (b)
Figure 7. Defects in crack healing region during compressive loading at 40 K. (a) t = 3 × 1011 s, KI = 7.5 × 105 Pa·m1/2; (b) t =
6 × 1011 s, KI = 1.5 × 106 Pa·m1/2.
Copyright © 2012 SciRes. JMP
the original crack was.
5. Conclusion
The critical temperature at which nano scale crack heal-
ing of BCC Fe crystal without pre-existing dislocations
becomes possible is 673 K. Temperature has a significant
effect on the evolution of nano scale crack. Healing pro-
ceeds rapidly at 1173 K. Compressive loading can result
in crack healing. Crack is healed completely at 40 K at
loading rate 0.025 × 1018 Pa·m1/2/s in 6 × 1012 s. Dislo-
cations and voids appear during healing process, and
their positions change continuously. In the case of heat-
ing, the distribution of the defects in the cell is inho-
mogenous in the early stage of healing. This inhomoge-
neity decreases and the quantity of the defects in the
crack area may be reduced with an increase of time. In
the case of compressive loading, most defects aggregate
in the region where the original crack is.
6. Acknowledgements
The authors thank the support of Australia Research
Council (ARC) Discovery Project.
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