Content uploaded by Javier Segurado
Author content
All content in this area was uploaded by Javier Segurado
Content may be subject to copyright.
IOP PUBLISHING MODELLING AND SIMULATION IN MATERIALS SCIENCE AND ENGINEERING
Modelling Simul. Mater. Sci. Eng. 15 (2007) S361–S375 doi:10.1088/0965-0393/15/4/S04
Computational issues in the simulation of
two-dimensional discrete dislocation mechanics
J Segurado
1
, J LLorca
1,3
and I Romero
2
1
Department of Materials Science, Polytechnic University of Madrid,ETSdeIngenieros de
Caminos, 28040 Madrid, Spain
2
Department of Structural Mechanics, Polytechnic University of Madrid,ETSdeIngenieros
Industriales, 28006 Madrid, Spain
E-mail: jllorca@mater.upm.es
Received 3 October 2006, in final form 25 January 2007
Published 1 May 2007
Online at
stacks.iop.org/MSMSE/15/S361
Abstract
The effect of the integration time step and the introduction of a cut-off velocity
for the dislocation motion was analysed in discrete dislocation dynamics (DD)
simulations of a single crystal microbeam. Two loading modes, bending and
uniaxial tension, were examined. It was found that a longer integration time
step led to a progressive increment of the oscillations in the numerical solution,
which would eventually diverge. This problem could be corrected in the
simulations carried out in bending by introducing a cut-off velocity for the
dislocation motion. This strategy (long integration times and a cut-off velocity
for the dislocation motion) did not recover, however, the solution computed
with very short time steps in uniaxial tension: the dislocation density was
overestimated and the dislocation patterns modified. The different response
to the same numerical algorithm was explained in terms of the nature of
the dislocations generated in each case: geometrically necessary in bending
and statistically stored in tension. The evolution of the dislocation density
in the former was controlled by the plastic curvature of the beam and was
independent of the details of the simulations. On the contrary, the steady-state
dislocation density in tension was determined by the balance between nucleation
of dislocations and those which are annihilated or which exit the beam. Changes
in the DD imposed by the cut-off velocity altered this equilibrium and the
solution. These results point to the need for detailed analyses of the accuracy
and stability of the dislocation dynamic simulations to ensure that the results
obtained are not fundamentally affected by the numerical strategies used to
solve this complex problem.
(Some figures in this article are in colour only in the electronic version)
3
Author to whom any correspondence should be addressed.
0965-0393/07/040361+15$30.00 © 2007 IOP Publishing Ltd Printed in the UK S361
S362 J Segurado et al
1. Introduction
There is compelling experimental evidence of a size effect on the resistance to plastic flow in
metals when the dimensions of the specimen or of the zone subjected to plastic deformation
are in the range of a few µm[1–4]. The analysis of this phenomenon is important from
the fundamental viewpoint, as the critical deformation and fracture processes in metals take
place at this length scale, and because of its implications on the development of micro-electro-
mechanical systems and in the micro-electronics industry [5]. The size effects in plasticity were
initially observed in conditions of constrained plastic flow (thin films, nanoindentation, torsion,
etc) in which strain gradients dominate the response and lead to the presence of geometrically
necessary dislocations [1–3]. More recent studies have reported, however, marked size effects
in the flow stress unconstrained compression of single crystals [4, 5], which were attributed to
a mechanism of dislocation starvation.
The origin of the size effect was traced in both the cases to the inhomogeneous nucleation
of individual dislocations upon loading and to the interaction of the dislocation structures
with the specimen and/or plastic zone boundaries. Discrete dislocation dynamics (DD), in
which the long range elastic interactions between individual dislocations and the influence of
the boundary conditions are taken into account within a continuum framework, emerges as
the ideal strategy to analyse this phenomenon. An important tool in studies of this kind is the
2D computational model developed by Needleman and Van der Giessen [6], which has been
applied to a number of problems including bending of small beams [7], nanoindentation [8],
uniaxial deformation of single crystals and polycrystals [9, 10], fracture of confined thin
films [11], etc.
Dislocations are 3D entities and it has been argued that 2D models cannot capture
many of the complex mechanisms of interaction between dislocations, and that the actual
dislocation structures which appear in 3D are different from those found in 2D simulations.
Although 3D dislocation dynamic simulations taking into account the boundary conditions
have been carried out [12, 13], the computational power required restricts the scope of these
simulations to very low strains in configurations which contain a limited number of dislocations.
On the contrary, 2D simulations easily treat problems including thousands of dislocations
up to strains of 10% and have provided realistic values for the size effect on the plastic
flow of single crystals in compression, once the physics of 3D short-range interactions
between dislocations are included in the 2D simulations through a set of constitutive
rules [14].
More often than not, the limiting factor in the computation of dislocation dynamics in two
dimensions is the time step rather than spatial resolution. The original 2D DD framework [6]
used an adaptative time-stepping algorithm so that the flight distance of any dislocation during
a time step remained within a user-defined maximum distance, which ensured that individual
events of dislocation nucleation, dipole annihilation and dislocation pinning by obstacles were
taken into account. This strategy led to an average time step below 0.01 ns, which restricted the
maximum achievable strain in the simulations to 1% even at very high strain rates (≈ 10
3
s
−1
).
This limitation was overcome afterwards by using a fixed time step of 0.5 ns and by introducing
a cut-off velocity of 20 m s
−1
for the dislocation speed to reduce the computational burden of
following the high velocity oscillatory motions within dislocation pile-ups [7]. As a result, the
number of time steps required to reach a given strain was reduced by two orders of magnitude
while the overall results were not influenced by these changes in a beam subjected to bending.
Since then, simulations based on this model have been carried out following this strategy
[8–10, 14], but the validity of these assumptions has not been published in detail and this is
the main aim of this investigation.
Simulation of two-dimensional dislocation mechanics S363
(a)
(b)
Figure 1. (a) Single crystal beam subjected to uniaxial tension. (b) Single crystal cantilever
subjected to bending. Slip planes oriented at an angle φ = 45
◦
with the axis x
1
are homogeneously
distributed in the shaded areas. The dimensions of the beam in the simulations presented in this
paper were characterized by L = 12 µm and L
= L/40.
2. Discrete dislocation dynamics model
The 2D DD simulations follow the model presented in [6–10] and only the most relevant
aspects will be recalled here. Basically, the crystals are taken as isotropic and elastic solids
characterized by the elastic modulus E and the Poisson’s ratio ν. Plane strain conditions are
assumed in the x
1
–x
2
plane (figure 1). The crystals have one slip system oriented at φ = 45
◦
with the positive x
1
axis and are initially free of dislocations. Dislocations are represented
by linear singularities perpendicular to the crystal plane with Burgers vector b. Dislocation
dipoles can be nucleated at discrete points randomly distributed on the slip planes, which
mimic the behaviour of Frank–Read sources in 2D, the Burgers vectors ±b being parallel to
the slip plane direction. Nucleation occurs when the magnitude of the resolved shear stress at
the source τ exceeds a critical value τ
nuc
during a period of time t
nuc
. The distance between
the two new dislocations, L
nuc
,isgivenby
L
nuc
=
E
4π(1 − ν)(1+ν)
b
τ
nuc
(1)
so that the resolved shear stress balances the attractive forces between them.
Once generated, dislocations slip in their respective glide planes and the speed V
i
of
dislocation i is given by
V
i
=
τ
i
b
B
, (2)
where τ
i
stands for the resolved shear stress on the glide plane and B is the drag coefficient.
Obstacles to dislocation motion are modelled as a random distribution of points in the glide
planes. Dislocations are pinned at these obstacles until the resolved shear stress exceeds
S364 J Segurado et al
(in absolute value) the obstacle’s strength τ
obs
. Moreover, two dislocations of different signs
gliding on the same slip plane are annihilated when they cross each other or if they are within
a distance L
anh
= 6b. Finally, if a dislocation exits the crystal, the dislocation is deleted from
the simulation and a displacement jump of b/2 is introduced along the slip plane.
The resolved shear stress on the glide plane at the line of dislocation i is computed as
τ
i
= n
i
·
ˆσ +
j=i
σ
j
·
m
i
, (3)
with n
i
the slip plane normal and m
i
the unit vector along the slip plane of dislocation i. σ
j
stands for the stress induced at the dislocation line for the dislocation j and it is computed
analytically from the expressions for the stress field induced by an edge dislocation on an
infinite, elastic and isotropic continuum. To this contribution it is necessary to add the field
ˆ
σ which includes the effect of the image forces induced by the crystal boundaries on the
dislocations. At a given stage of loading, the stress-rate and strain-rate fields in the crystal are
obtained by the superposition of the two fields, the first one given by the sum of those induced
by the individual dislocations in the current configuration and the second one that corrects for
the actual boundary conditions. This term is computed by solving a linear elastic boundary
value problem using the finite element method with the appropriate boundary conditions, as
detailed in [6–10].
The deformation process of the crystal is solved in an explicit incremental manner, using an
Euler forward time-integration algorithm for the equations of motion. Once the new positions
of all dislocations at time t have been computed, new dislocations are generated at the sources
according to (1) and dislocation pairs of opposite sign in the same slip plane are annihilated
when they are within L
anh
. The boundary conditions for the linear elastic boundary value
problem are computed from the new dislocation structure and the applied displacements for
time t +t. The resolved shear stresses on the dislocations are computed according to (3) from
the fields induced by the dislocations and the solution of the boundary value problem. Then,
the velocities of the dislocations are obtained from the corresponding resolved shear stress, as
givenby(2). The new positions are computed from these velocities and the dislocations that
meet an obstacle are pinned if the resolved shear stress is below τ
obs
.
It is important to note that the code is prepared to handle spurious numerical artefacts
which appear as a result of the explicit integration scheme. For instance, every time that a
dislocation implies that the dislocation has jumped over an obstacle, it is checked if the resolved
shear stress is higher than the obstacle strength to avoid dislocations bypassing obstacles. In
addition, there is a matrix which keeps track of the relative position of all dislocations gliding
in each slip plane. If there is a change in this position during any time step, this means that
dislocations of different sign have crossed each other and they are deleted.
2.1. Boundary conditions
Simulations were carried out on beams of length to depth ratio L/W = 3 subjected to two
different loading conditions: uniaxial tension and bending of a cantilever (figure 1). The
boundary conditions for the former are expressed as
u
1
= 0,T
2
= 0onx
1
= 0,
u
1
= U,T
2
= 0onx
1
= L,
T
1
= T
2
= 0onx
2
=±W/2, (4)
Simulation of two-dimensional dislocation mechanics S365
where T
i
= σ
ij
n
j
is the traction on the boundary with normal n
j
. The applied stress σ is
computed as
σ =
1
W
W/2
−W/2
T
1
(L, x
2
)dx
2
(5)
to obtain the uniaxial stress–strain curve (σ, U/3W). Loading was imposed by applying a
constant strain rate of ˙ε = U/L = 2000 s
−1
.
The corresponding boundary conditions for the cantilever beam subjected to bending are
given by
u
1
= u
2
= 0onx
1
= 0,
u
2
= U on x
1
= L and x
2
= W/2,
T
1
= T
2
= 0onx
2
=±W/2 and x
1
= L(x
2
= W/2). (6)
The applied force on the beam end was computed from the shear stresses acting along the
beam depth as
F =
W/2
−W/2
τ
12
(0,x
2
)dx
2
. (7)
Loading was imposed by applying a constant displacement rate
˙
U of 0.024 m s
−1
at the
cantilever end.
2.2. Material properties
The properties of the material in the simulations were E = 70 GPa, ν = 0.33 and b = 0.25 nm.
Only one family of slip planes (oriented at an angle of 45
◦
with the positive x
1
axis) was
considered, which is shown in the shaded region of figure 1. The distance between consecutive
slip planes in this region was 25 nm. The drag coefficient for dislocation glide, B = 10
−4
Pa s,
was representative of Al [15]. Dislocation sources and obstacles were randomly distributed
along the slip planes, the density of both being equal to 56 µm
−2
. The critical resolved shear
stress for dislocation nucleation was assigned randomly to the sources following a Gaussian
distribution with an average value of ¯τ
nuc
= 50 MPa and a standard deviation of 1 MPa. The
nucleation time for all the sources was 0.01 µs. The strength of the obstacles was 150 MPa.
These magnitudes are equal to those used in previous simulations [9,10].
2.3. Finite element analyses
The image stress ( ˆ
σ) and displacement (
ˆ
u) fields, which account for the effect of the boundary
conditions, were computed by the finite element method. These simulations were performed
using FEAP [16] with a rectangular grid of 50 by 50 bilinear quadrilateral elements for the
beam discretization. The image fields are not singular and this mesh was fine enough; this
point was demonstrated by simulations with finer and coarser meshes, which provided very
close results. The analyses to determine the image fields were linear and the stiffness matrix of
the beam was factorized once, at the beginning of the simulation. Computation of the stresses
and displacements in each time step only required the back-substitution with the new boundary
conditions
S366 J Segurado et al
(a) (b)
Figure 2. Evolution of the maximum bending moment, M, (normalized by the reference moment
M
ref
) versus maximum rotation angle θ in a microcantilever subjected to bending. (a) Influence
of the integration time step t.(b) Influence of the cut-off velocity on dislocation motion.
3. Results and discussion
3.1. Bending
A cantilever beam was chosen to study the behaviour of bending (figure 1(b)). Simulations
were carried out with time increments in the range 0.01–0.5 ns and the curves of maximum
bending moment M versus maximum rotation angle θ = U/L corresponding to t = 0.01,
0.05, 0.1 and 0.5 ns are plotted in figure 2(a). The bending moment was normalized by a
reference moment, M
ref
(as in [7]), which would result from a linear stress distribution of
magnitude ¯τ
nuc
x
2
/(W/2) along the x
1
= 0 section of the cantilever. Mathematically
M
ref
=
2
W
h/2
−h/2
¯τ
nuc
x
2
2
dx
2
=
1
6
¯τ
nuc
W
2
. (8)
The M–θ curves in figure 2(b) were practically superposed, and although those computed with
longer time increments were rougher, the differences were not significant, in particular if the
chaotic nature of dislocation dynamics is taken into account [17]. The dislocation structures
at the end of the analysis are plotted in figures 3(a) and (b) for the simulations carried out
with t = 0.01 and 0.5 ns. At first glance, both dislocations’ structures show similar features,
the dislocations being concentrated in particular planes in which plastic strain was localized,
and it should be noted that these results agree qualitatively with the experiments of Motz
et al [18] on single crystal Cu microbeams manufactured by the focused ion beam technique
and tested as cantilevers. Closer analysis of figures 3(a) and (b) showed some differences in the
dislocation structure induced by the differences in the time step. Dislocation nucleation started
near the lower axis of the beam and propagated along the slip planes and to the right as the
bending moment increased. Simulations carried out with t = 0.01 ns showed the formation
of dense dislocation pile-ups which slowed down the propagation of the dislocations along
the slip planes towards the upper-right regions of the beam. The dislocation pile-ups in the
simulations carried out with t = 0.5 ns were less dense and the dislocations reached the
upper-right regions of the beam. These differences were caused by the longer integration time
step: the displacement along the slip plane of a dislocation within a pile-up in one time step of
0.5 ns will be 0.1–0.2 µm, assuming that the resolved shear stress acting on a dislocation was
Simulation of two-dimensional dislocation mechanics S367
Figure 3. Dislocation structures in a single crystal microcantilever beam subjected to bending at
the end of the analysis (θ = 5%). (a) t = 0.01 ns. (b) t = 0.5 ns. (c) t = 0.5 ns and
V
cut−off
= 20 m s
−1
. Solid and open triangles stand for dislocations of different signs.
in the range 0.5τ
obs
to τ
obs
. As a result, the positions of the dislocations within the pile-ups
are very poorly resolved, leading to unrealistic non-equilibrium situations in which the stress
acting on the dislocation closer to the obstacle overcame spuriously the obstacle strength and
propagated along the slip plane. In addition, dislocations were able to jump over the neutral
axis of the beam (in which the applied stresses are low and the driving force for dislocation
motion decreased) as a result of the long displacements in each time step, and they reached
the upper-right region of the beam. This behaviour was unusual in the simulations carried out
at t = 0.01 ns (figure 3(a)).
These problems, associated with the long jumps of the dislocations when long integration
time steps were used, were cleverly solved by Cleveringa et al [7] by introducing a cut-off
velocity of 20 m s
−1
for the dislocation motion. The results obtained for the bending of the
microcantilever in the cases presented in figure 2(a) with the minimum (0.01 ns) and maximum
(0.5 ns) integration time steps are plotted again in figure 2(b) together with those computed
with the maximum time step of 0.5 ns and a cut-off velocity of 20 m s
−1
. These curves show
that the cut-off velocity did not modify significantly the M–θ curve, which was very close
to the one obtained with t = 0.01 ns and smoother than that computed with t = 0.5ns
without any limit for the dislocation velocity. Moreover, the dislocation structure at the end
of the analysis, shown in figure 3(c), was very similar to that obtained with t = 0.01 ns.
S368 J Segurado et al
Figure 4. Predictions of the evolution of the dislocation density as a function of the maximum
beam rotation θ during bending of a single crystal microcantilever in the simulations in figure 2(b).
Obviously, the cut-off velocity reduced the maximum dislocation displacement in each time
step by a factor of ≈ 20 and the position of the dislocations within the pile-ups were better
resolved. The density and the location of the dislocation pile-ups reproduced faithfully the
results obtained, at about fifty times the computational cost, with very short integration time
steps.
On the basis of these findings, Cleveringa et al [7] used longer integration time steps
coupled with a cut-off velocity in their dislocation dynamics simulations in bending and further
analyses reproduced their strategy in different loading conditions [8–11]. However, it should
be noted that bending is a very particular loading mode in which practically all the dislocations
are geometrically necessary and thus the dislocation density is controlled by the curvature of
the beam. This is shown in figure 4, in which the evolution of the dislocation density with
the maximum beam rotation angle is practically superposed in all the simulations plotted in
figure 2(b). Thus, it seems necessary to explore whether the strategy followed in [7] to increase
the integration time step without compromising the outcome of the simulations comes about
as an ‘intrinsic’ behaviour or whether it was conditioned by the deformation mode of the
cantilever beam which controlled the total dislocation density.
3.2. Uniaxial tension
The stress–strain curves for the crystals subjected to uniaxial tension up to 5% are plotted in
figure 5(a) as a function of the time step chosen for the simulations. The overall shape of the
curves obtained with t in the range 0.01–0.5 ns was very similar: an initial linear elastic region
followed by an ideally plastic response. The linear region ended when the first dislocation
was nucleated at a far-field applied stress slightly below 100 MPa, which is consistent with
the average nucleation strength of 50 MPa taking into account that the Schmidt factor for the
Simulation of two-dimensional dislocation mechanics S369
(a) (b)
Figure 5. Influence of the integration time step t on the simulation of the uniaxial deformation
of a single crystal microbeam. (a) Stress–strain curve. (b) Dislocation density.
slip system is (sin 2φ)/2 = 0.5. The dislocation density increased sharply after this event
(figure 5(b)) and reached a steady-state situation, as the number of dislocations nucleated was
equivalent to that which exited through the specimen lateral surfaces; the flow stress remained
constant under such conditions. The fluctuations in the stress–strain curve increased with the
integration time step and were particularly high when t = 0.5 ns, but the overall magnitude
of flow stress seemed to converge to the values obtained with shorter time steps at far-field
strains of 5%. The evolution of the dislocation density with the applied strain showed similar
features (figure 5(b)) and more oscillations were found for longer integration time steps. In
addition, the dislocation density grew faster and reached higher values at the initial stages
of deformation for shorter integration time steps and the steady-state dislocation density at
far-field strains of 5% seemed to increase slightly with t.
In order to smooth out the mechanical response and maintain long integration times steps,
the standard strategy has been to introduce a cut-off velocity in the dislocation motion [8–11].
The results obtained in terms of the stress–strain curve and the evolution of the dislocation
density with the applied strain are plotted in figures 6(a) and (b), respectively. The presence
of a cut-off velocity of 20 m s
−1
for the dislocations reduced dramatically the oscillations
in the stress–strain curve and the average value of the flow stress, which was closer to the
one computed with the shortest time step of 0.01 ns. However, the simulations with a cut-
off velocity of 20 m s
−1
severely overestimated (by a factor of 3.5 at 5% applied strain) the
dislocation density in comparison with the predictions of the models without cut-off velocity.
Thus, introducing a cut-off velocity for the dislocations reduced the oscillations in the
stress–strain curve but overestimated the dislocation density. Moreover, the dislocation
structures changed with the integration time step and were dramatically altered by the
introduction of a cut-off velocity of 20 m s
−1
. This is shown in figure 7 in which the dislocation
structures are plotted at ε = 5% for the simulations performed with t = 0.01 ns, t = 0.5ns
and t = 0.5 ns and a cut-off velocity of 20 m s
−1
. The simulations carried out with the shortest
time increment showed a limited number of active slip planes during the tensile deformation
of the single crystal, figure 7(a). The plastic deformation was mainly localized in these planes
in which dislocation pile-ups developed. The positions of the dislocations within the pile-ups
could not be computed accurately in simulations carried out with longer t (figure 7(b)) and
S370 J Segurado et al
(a) (b)
Figure 6. Influence of the dislocation cut-off velocity on the simulation of the uniaxial deformation
of a single crystal microbeam. (a) Stress–strain curve. (b) Dislocation density.
this led to the development of non-equilibrium situations which increased sharply the stresses
acting on the dislocation closest to the obstacle. As a result, the dislocation pile-ups were
unstable and the dislocations could not be stopped by the obstacles and propagated along the
slip planes, generating more dislocations in other planes and increasing the dislocation density.
Imposing a cut-off velocity of 20 m s
−1
on the simulations carried out with t = 0.5ns
changed completely the dislocation pattern (figure 7(c)). Long dislocation pile-ups were
formed in several slip planes and this led to noticeable differences in the deformation of the
beam between the simulations carried out with the shortest time step of 0.01 ns (figure 8(a))
and those performed at the longest time step of 0.50 ns with a cut-off velocity of 20 m s
−1
for the dislocations (figure 8(b)). In the absence of a cut-off velocity, plastic deformation
was concentrated in a couple of slip planes in which successive dislocations were nucleated,
propagated and left the crystal. The cut-off velocity generated long and dense pile-ups and
more dislocations were stored in the crystal. The stress field generated by the pile-ups nucleated
dislocations in nearby slip planes and the plastic deformation was spread over a wider region.
It has been shown in previous investigations that dislocation dynamics is chaotic [17]
and that the predictions obtained with dislocation dynamics are very sensitive to the details of
the simulations, such as the precise location of sources and obstacles, particularly when the
behaviour is dominated by statistical stored dislocations. In order to ensure that the results
presented in figures 6 and 7 are general, three simulations were carried out with and without
a cut-off velocity with different distributions of dislocations and sources. The predictions of
the dislocation density as a function of the applied strain are plotted in figure 9, and they show
that in all the cases the densities obtained with a cut-off velocity of 20 m s
−1
were about 3 to
4 times higher.
These differences are obviously a consequence of the cut-off velocity imposed on the
dislocations, which alters radically the dislocation dynamics. The effect of this modification
can be assessed from the results shown in figure 10(a), in which the average dislocation velocity
in each time step is plotted as a function of the applied strain in the simulations carried out with
t = 0.50 ns. The average speed was computed from all the dislocations in the model, taking
into account that the dislocations blocked by an obstacle are assigned null velocity, so that it
Simulation of two-dimensional dislocation mechanics S371
Figure 7. Dislocation structures in a single crystal microbeam subjected to uniaxial tension at
the end of the analysis (ε = 2%). (a) t = 0.01 ns. (b) t = 0.5 ns. (c) t = 0.5 ns and
V
cut−off
= 20 m s
−1
. Solid and open triangles stand for dislocations of different signs.
includes the dislocations within pile-ups and those sliding freely along the corresponding slip
planes. The spikes with average velocities above 50 m s
−1
are isolated events corresponding
to large instabilities and are not representative of the overall behaviour
4
. Nevertheless, the
average dislocation speed is comparable to the cut-off velocity of 20 m s
−1
most of the time
and approximately 50% of the mobile dislocations were slowed down throughout the analysis
as a result of this limit to the dislocation speed, figure 10(b). A portion of these slowed-down
dislocations is within pile-ups and the correction leads to more stable dislocation pile-ups by
smoothing out the high-frequency oscillatory motions. However, this correction also affected
many dislocations moving freely along slip planes (for instance, after they have broken free
from an obstacle), introducing an error in the dislocation dynamics simulations because the
forces acting on these dislocation are not used, in fact, to determine the dislocation speed.
Obviously, the higher the cut-off velocity, the lower the number of dislocations slowed
down and this should lead to better approximations. Increasing V
cut−off
to 100 m s
−1
led to
a stress–strain curve almost superposed on that computed with V
cut−off
= 20 m s
−1
, although
with more oscillations (figure 6). The dislocation density was significantly reduced, although
4
The curve contains 50 000 points and the spikes correspond to a few hundreds of time increments. Its effect is
magnified in the plot because of the thickness of the line.
S372 J Segurado et al
Figure 8. Deformed shape of the single crystal microbeam subjected to uniaxial tension. (a)
t = 0.01 ns. (b) t = 0.5 ns and V
cut−off
= 20 m s
−1
. The displacement magnification factor
was 2.
the steady-state value was almost twice that of obtained in the simulations with t = 0.01 ns.
These results are in agreement with the previous discussion and are supported by the changes
in the fraction of slowed-down dislocations with the cut-off velocity shown in figure 9(b).
In brief, the introduction of the cut-off velocity led to two effects on the dynamics of
dislocations. Firstly, the dislocation pile-ups are stabilized by filtering the high-frequency
oscillations which led to non-equilibrium situations when long time increments were used.
Secondly, it slowed down the overall motion of many dislocations moving freely along the slip
planes, and this modified the evolution of the dislocation density with the applied strain. The
curves plotted in figure 6(b) show that the dislocation density in uniaxial tension grew sharply
from the onset of plastic deformation and reached rapidly a steady state due to the balance
between the new dislocations generated at the sources and those which were annihilated or
exited the crystal through the lateral surfaces. This equilibrium is altered if the velocity
of the dislocations moving along the slip planes is reduced and as a result, the dislocation
density attained much higher values and—in our simulations—did not reach an equilibrium
value. These extra dislocations were accumulated in pile-ups (leading to the localization of
deformation in a few slip planes) and their stress fields contributed to the nucleation of more
dislocations in nearby slip planes.
The scenario depicted in the previous paragraph was not operative in the beams loaded
in bending, because the dislocations were geometrically necessary and the dislocation density
was proportional to the plastic curvature of the beam (figure 4). Only the first effect (stabilizing
the pile-ups) was operative and the insertion of a cut-off velocity for the dislocation motion
was a clever strategy to improve the computational efficiency of the dislocation dynamics
Simulation of two-dimensional dislocation mechanics S373
Figure 9. Influence of the initial distribution of sources and obstacles on the dislocation density
during uniaxial deformation of a single crystal microbeam. The results are presented for simulations
carried out with and without a cut-off velocity of 20 m s
−1
.
Figure 10. (a) Average dislocation velocity in each time increment in the simulations carried out at
t = 0.5 ns. (b) Fraction of dislocations slowed down in each time step in the simulations carried
out at t = 0.5 ns and V
cut−off
= 20 m s
−1
or 100 m s
−1
.
simulations without losing accuracy. In contrast, the dislocation density in uniaxial tension
is due to statistically stored dislocations and a reduction of the dislocation speed changes
the balance between dislocation nucleation and dislocations exiting the crystal, increasing
artificially the dislocation density.
S374 J Segurado et al
4. Concluding remarks
More often than not, temporal resolution is more critical than spatial resolution to achieve
meaningful results (either of fundamental or practical interest) in the numerical modelling of
the mechanical behaviour of materials. Two-dimensional discrete dislocation dynamics has
emerged in the last decade as an extremely useful tool to analyse size effects in plasticity when
the length scales involved are in the range of a few µm. The success of this simulation strategy
is partially due to its ability to treat rigorously the effect of complex boundary conditions and
to reach large strains (up to 10%) in problems containing hundreds of dislocations. This latter
result has benefited from ad hoc modifications in the Euler forward time-integration algorithm
for the equations of dislocation motion, which allowed longer integration time steps (up to
two orders of magnitude) without sacrificing accuracy, by introducing a cut-off velocity for
the dislocation motion.
It is well known that explicit time-integration algorithms are not always unconditionally
stable and this investigation was aimed at studying the influence of the integration time step
size and of the introduction of a cut-off velocity on the numerical results obtained for a beam
subjected to bending and uniaxial tension. Increasing the integration time step led in both
cases to similar results: a progressive increment of the oscillations in the numerical solution,
which would eventually lead to an erroneous solution because the details of the dislocation
nucleation and annihilation and of the interactions among dislocations were poorly resolved.
These problems were eliminated in the beam subjected to bending by introducing a cut-off
velocity for the dislocation speed and the solutions obtained with integration time steps of 50 ns
and a cut-off velocity of 20 m s
−1
were very close to those computed with integration time steps
fifty times smaller. However, the same strategy applied to uniaxial tension led to divergent
results, particularly in the dislocation density and in the deformation patterns. Basically,
the cut-off velocity for the dislocations reduced the rate of dislocations exiting the crystal
during deformation and increased by a factor of 3.5 the total number of statistically stored
dislocations. These dislocations were arranged in long pile-ups which led to the localization
of the plastic deformation in a narrow band of slip planes. This mechanism was not operative,
however, in bending because the dislocation density was controlled by the plastic curvature of
the beam. These results point out the need of detailed analyses of the accuracy and stability of
the dislocation dynamics simulations to ensure that the results obtained are not fundamentally
affected by the numerical strategies implemented to solve these complex problems.
Acknowledgments
The financial support from the Comunidad de Madrid through the program ESTRUMAT-
CM and the Spanish Ministry of Science and Education through grants DPI2006-14104 and
MAT2006-2602 is gratefully acknowledged.
References
[1] Fleck N A, Muller G M, Ashby M F and Hutchinson J W 1994 Acta Metall. Mater. 42 475
[2] Hutchinson J W 2000 Int. J. Solids Struct. 37 225
[3] Uchic M D, Dimiduk D M, Florando J N and Nix W D 2004 Science 305 986
[4] Greer J R, Oliver W C and Nix W D 2005 Acta Mater. 53 1821
[5] Schwarz K W and Chidambarrao D 2005 Mater. Sci. Eng. A 400–401 435
[6] Van der Giessen E and Needleman A 1995 Modelling Simul. Mater. Sci. Eng. 3 689
[7] CleveringaHHM,VanderGiessen E and Needleman A 1999 Int. J. Plast. 15 837
[8] Widjaja A, Van der Giessen E and Needleman A 2005 Mater. Sci. Eng. A 400–401 456
Simulation of two-dimensional dislocation mechanics S375
[9] Deshpande V S, Needleman A and Van der Giessen E 2005 J. Mech. Phys. Solids 53 2661
[10] Balint D S, Deshpande V S, Needleman A and Van der Giessen E 2006 Modelling Simul. Mater. Sci. Eng. 14 409
[11] Chng A C, O’Day M P, Curtin W A, TayAOAandLimKM2006 Acta Mater. 54 1017
[12] Weygand D, Friedman L H, Van der Giessen E and Needleman A 2002 Modelling Simul. Mater. Sci. Eng. 10 437
[13] Groh S, Devincre B, Kubin L P, Roos A, Feyel F and Chaboche J L 2005 Mater. Sci. Eng. A 400–401 279
[14] Benzerga A A and Shaver N F 2006 Scr. Mater. 54 1937
[15] Kubin L P, Canova G, Condat M, Devincre B, Pontikis V and Brechet Y 1992 Solid State Phenom. 23–24 455
[16] Taylor R L 2005 FEAP, A finite element analysis program, v 7.5. University of California
[17] Deshpande V S, Needleman A and Van der Giessen E 2001 Scr. Mater. 45 1047
[18] Motz C, Sch
¨
oberl T and Pippan R 2005 Acta Mater. 53 4269
