Content uploaded by Javier Segurado
Author content
All content in this area was uploaded by Javier Segurado on Jan 02, 2014
Content may be subject to copyright.
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. 16 (2008) 035008 (18pp) doi:10.1088/0965-0393/16/3/035008
Dislocation dynamics in non-convex domains using
finite elements with embedded discontinuities
Ignacio Romero1, Javier Segurado2,3and Javier LLorca2,3
1Departamento de Mec´
anica Estructural, Universidad Polit´
ecnica de Madrid,
E. T. S. de Ingenieros Industriales, 28006 - Madrid, Spain
2Departamento de Ciencia de Materiales, Universidad Polit´
ecnica de Madrid,
E. T. S. de Ingenieros de Caminos, 28040 - Madrid, Spain
3Instituto Madrile˜
no de Estudios Avanzados en Materiales (IMDEA-Materiales),
E. T. S. de Ingenieros de Caminos, 28040 - Madrid, Spain
Received 25 June 2007, in final form 29 February 2008
Published 26 March 2008
Online at stacks.iop.org/MSMSE/16/035008
Abstract
The standard strategy developed by Van der Giessen and Needleman
(1995 Modelling Simul. Mater. Sci. Eng. 3689) to simulate dislocation
dynamics in two-dimensional finite domains was modified to account for
the effect of dislocations leaving the crystal through a free surface in the
case of arbitrary non-convex domains. The new approach incorporates the
displacement jumps across the slip segments of the dislocations that have exited
the crystal within the finite element analysis carried out to compute the image
stresses on the dislocations due to the finite boundaries. This is done in a
simple computationally efficient way by embedding the discontinuities in the
finite element solution, a strategy often used in the numerical simulation of
crack propagation in solids. Two academic examples are presented to validate
and demonstrate the extended model and its implementation within a finite
element program is detailed in the appendix.
(Some figures in this article are in colour only in the electronic version)
1. Introduction
The understanding of microscale processes which control plastic deformation in engineering
materials has been based, to a considerable extent, on relatively simple models. The
expectations of nanotechnology—which have increased interest in the mechanical behavior
of components at micrometer-scale—together with developments in modeling tools have led
the way to more accurate and, to some extent, quantitative numerical simulations. In these
analyses, the microstructural details at the micrometer and nanometer scale are explicitly
taken into account to analyze the micromechanisms of deformation and fracture (see, for
instance, [2–5] and references therein). Dislocation dynamics has emerged as one of the
0965-0393/08/035008+18$30.00 © 2008 IOP Publishing Ltd Printed in the UK 1
Modelling Simul. Mater. Sci. Eng. 16 (2008) 035008 I Romero et al
most relevant strategies in this area and two different approaches have been developed. One
of them mainly focuses on the analysis of physical micromechanisms of plastic deformation
(dislocation interaction and multiplication, formation of dislocation structures, strain and forest
hardening, etc) [3,6], while the microstructure and the boundary conditions are kept simple
(e.g. an infinite single crystal subjected to homogeneous tension). The other is aimed at the
solution of boundary value problems at the micrometer scale and includes the analysis of
plastic deformation (and the associated size effects) during bending of small beams [7,8],
nanoindentation [9], uniaxial deformation of polycrystals [10], fracture of confined thin
films [11], etc.
The solution of boundary value problems within the framework of discrete dislocation
dynamics is a complex task which can only be accomplished, in general, through the
combination of analytical results and numerical simulations. An important tool in studies of
this kind is the model developed by Needleman and Van der Giessen [1], in which long-range
interactions between individual dislocations are accounted for using the analytical expressions
for the interaction between dislocations in an infinite and isotropic elastic solid. The effect of
finite boundaries is included through another term for the stresses acting on each dislocation,
which is computed by solving a complementary linear elastic boundary value problem using
the finite element method with appropriate boundary conditions. The dislocation dynamics
problem is solved in an explicit incremental way, using an Euler forward time-integration
algorithm for the equations of evolution. This paper presents an extension of this methodology
to deal with the effect of dislocations leaving the crystal through a free surface in the case of
arbitrary non-convex domains, in which the intersection of the slip plane with the domain is not
a continuous segment. The strategy used in the original framework to deal with dislocations
leaving the domain was to place these dislocations in a fictitious position along its slip plane but
far away from the domain [12] and to include the deformation field induced by these dislocations
(which is almost identical to the exact slip caused by the dislocation glide) to determine the
total displacements of the domain. However, this method is not always applicable in the case
of non-convex domains that contain defects with dimensions of the order of the Burgers vector
(such as sharp cracks, very thin slits or nm voids) because the deformation field induced in
the domain by a fictitious dislocation within the defect is very different from the actual one
due to dislocation leaving the domain. Examples of these situations can be found in micro-
electro-mechanical and micro-electronics devices [13] or in the analysis of relevant topics
in micromechanics such as cracked bodies, the growth of microvoids [14] or the mechanical
behavior of nanoporous metals [15,16]. The new approach is based on the use of finite elements
with embedded discontinuities following previous investigations [17–19].
The outline of the paper is as follows. After the introduction, the main features of
the Needleman and Van der Giessen model are briefly recalled with special emphasis on
the case of dislocations leaving the domain. The new approach for non-convex domains
is presented in section 3, while the numerical implementation using finite elements with
embedded discontinuities is detailed in section 4. Two examples of application are presented
in section 5to validate and demonstrate the extended model and the paper ends with the
conclusions section. An appendix is provided to facilitate the numerical implementation within
the framework of the finite element method.
2. Basic discrete dislocation dynamics framework
The 2D discrete dislocation dynamics simulations follow the model presented in [1],
subsequently employed, for instance, in [7,14,20], and which has been implemented by the
authors [8]. The model considers an elastic, isotropic crystal that might contain dislocations
2
Modelling Simul. Mater. Sci. Eng. 16 (2008) 035008 I Romero et al
Figure 1. Problem decomposition. The problem in the center is posed on an infinite domain with
the dislocations in it. The problem on the right is on the original domain , but the imposed
tractions and displacement boundary conditions are corrected. The three problems in the figure
will be denoted problems P, ˜
P and ˆ
P, respectively.
gliding on one or more families of slip planes, each one characterized by the unit normal and
tangent vectors, nand m, respectively. The crystal is assumed to remain in a state of plane
strain at all times, and its cross section occupies a domain ⊂R2. The boundary ∂ is
partitioned into disjoint sets ∂tand ∂u. Surface tractions Tare applied on the former, and
displacements Uare imposed on the latter. See figure 1for an illustration of these definitions.
The dislocations move on their slip planes, and their velocity can be accurately
approximated using a viscous law
v=τb
B(1)
in which τstands for the resolved shear stress on the glide plane, b=|b|is the modulus of the
Burgers vector and Bdenotes a viscous drag coefficient. The actual resolved shear stress at the
position of each dislocation is computed by solving a boundary value problem, which will be
referred to as problem P, to determine the stresses σin the elastic crystal due to the boundary
conditions and the dislocations. The resolved shear stress at the position of the ith dislocation,
located on a glide plane with unit normal nand unit tangent vector mcan be obtained by
projecting the stress σat the location of the dislocation according to
τi=m·
ˆσ+
j=i
˜σj
i
n.(2)
The tensor ˜
σj
iis the stress due to the jth dislocation in an infinite crystal. The analytical
expression for such stress has to be corrected with the term ˆ
σ, which includes the image forces
that appear as a consequence of crystal boundaries. An analytical expression for the correction
term does not exist and it has to be obtained numerically by solving a linear elastic analysis
with appropriate boundary conditions. The finite element method provides a convenient tool
for this task.
Figure 1illustrates this idea: instead of solving the original problem for displacement u
and stress σin the crystal (problem P), the two fields are decomposed as
u(x):=˜
u(x)+ˆ
u(x), σ(x):=˜σ(x)+ˆσ(x). (3)
Displacement ˜
uand stress ˜
σare obtained from an elastic solution in an infinite domain
containing all the dislocations, including those that have left the domain and are placed far
apart. This problem will be denoted problem ˜
P. The pair ˆ
u,ˆ
σis obtained by solving a
3
Modelling Simul. Mater. Sci. Eng. 16 (2008) 035008 I Romero et al
standard elastic boundary value problem on , without dislocations, and with the boundary
conditions of the original set-up corrected with the values obtained from the problem in the
infinite domain. This last problem will be referred to as problem ˆ
P. More details on this split
can be found in [1].
In the dislocation dynamics simulation, the quasi-static solution for the stresses in the
domain is coupled with the motion of dislocations. The resolved shear stress at each dislocation
can be calculated as explained above at every discrete instant of the time integration. Then,
a forward Euler method is employed to solve (1), using the computed stresses to move the
dislocations on their glide planes, taking into account the restrictions provided by a number
of constitutive rules which control the nucleation and annihilation of dislocations as well as
the interaction of dislocations with obstacles. Once the new positions of the dislocations are
obtained, the procedure continues recursively until the end of the simulation time.
The dislocations that reach the boundary can leave the domain, producing a relative slip
between the two regions in which the crystal is divided by the slip plane. This effect can be
approximately included by placing the dislocation which has exited the crystal in a fictitious
position along its slip plane but far away from the domain [12]. This dislocation is not taken
into account to compute the stresses in the domain but the deformation field (which is almost
identical to the exact slip caused by the dislocation glide) is included to determine the total
displacements of the domain. From a practical viewpoint, the fictitious dislocations are only
required for the solution of problem ˆ
P.
This strategy is a very accurate approximation in convex crystals but it cannot be
generalized in the case of non-convex domains as it might lead to inaccurate results in particular
cases. According to the idea previously described, the residual displacement associated with
the dislocations which have left the domain has to be introduced by placing a fixed, fictitious
dislocation on the extension of the slip plane, but far away from the boundary. This is always
possible if the prolongation of the slip plane from the exit point does not cut the domain but it
may not be so in non-convex domains if the prolongation of the slip plane cuts again the domain.
A relevant example can be found in figure 2(a) which considers a square domain with a sharp
crack. Dislocations can move in one slip plane (dashed line) which is limited by the crack
and the domain external surface. Let us assume that a dislocation dipole is generated within
the plane and both dislocations glide until they exit the domain. The residual displacement
field due to the dislocations which have left the domain is the horizontal displacement jump
of magnitude bat the slip plane while the vertical displacement is zero in the whole domain,
figure 2(c). In the original approach, this field has to be reproduced by placing two fictitious
dislocations far away from the domain. This can be done for the dislocation which has exited
the domain through the external surface, but not for the other one, which has to be placed within
the crack. The resulting displacement field due to both dislocations is shown in figure 2(b), and
it is obviously different from the exact solution. These differences are plotted in figure 2(d), in
which the exact (solid lines) and approximate (dashed lines) displacement fields are presented
for the solutions shown in figures 2(b) and (c), respectively.
3. Modeling dislocations leaving non-convex domains
Non-convex bodies, including those with voids, cracks and slits, often appear in micro-electro-
mechanical and micro-electronics devices or in micromechanics. It is desirable to extend the
dislocation dynamics framework described heretofore so that it can encompass any body,
convex or non-convex, with or without holes or cracks of any shape and size. The main idea
describing the proposed extension is described in this section and the mathematical formulation
and numerical implementation are provided in the following sections. The starting point is
4
Modelling Simul. Mater. Sci. Eng. 16 (2008) 035008 I Romero et al
-6
-4
-2
0
2
4
6
0 5 10 15 20 25
y/b
x/b
(d)
(a)
(b)
(c)
Figure 2. (a) Square domain with a sharp crack whose thickness is 2b. The dashed line represents
a slip plane with a dislocation dipole. (b) Total displacement field around the crack tip once the
dislocations have left the domain computed by placing one dislocation within the crack and the
other far away from the domain. (c) Exact solution for the total displacement field around the
crack tip when both dislocations have left the domain. (d) Approximate (----)andexact (——)
displacements fields in the cracked square domain around the slip plane which is located at y=0
for the solutions shown in (b) and (c), respectively.
a different procedure to account for the effect of dislocations leaving the crystal in convex
domains. Unlike the strategy detailed in the previous section, the new method is purely
numerical and works by modifying the statement of problem ˆ
P. It accounts for discontinuities
that exit the domain by accumulating the displacements due to plastic slip in the elements of
the finite element mesh.
For simplicity of exposition, let us consider only two dislocations of a loop moving on a
slip plane, as in the example of figure 2. If one of the dislocations reaches the boundary of
the crystal, the resulting displacement field is similar to the sum of the displacements due to
the remaining dislocation plus a discontinuous displacement field with a jump across the slip
plane of value b/2. This can be approximated by adding equal and opposite displacements
of magnitude b/4 above and below the slip plane, respectively. Once the second dislocation
leaves the domain, and an additional displacement jump is added, this approximate solution
becomes exact. We observe that placing the dislocations that have left the domain on the
prolongation of the slip line far away from the boundary essentially produces the same effect
as the proposed additional displacement jump, and the technique proposed in this section and
the original one yield identical results.
The new approach, however, has a drawback that needs to be corrected: in non-convex
bodies, slip planes do not always partition the domain into two regions, one above and another
below the slip plane (see, for instance, the domain in figure 2). Nevertheless, this new
approach can be extended to include the effect of dislocations reaching the boundary of any
two-dimensional domain. Let us consider first the analysis of a deformable crystal, free of
imposed displacements and tractions on its surface, with a pair of edge dislocations of opposite
sign and Burgers vector bmoving on a slip plane. The crystal may be non-convex and have
holes or cracks inside. When one dislocation reaches the boundary, the state in the body can
be approximated by superposing the effects of a stress-free slip of magnitude b/2 across the
plane and of the remaining dislocation. In physical terms, this situation can be reproduced by
5
Modelling Simul. Mater. Sci. Eng. 16 (2008) 035008 I Romero et al
Figure 3. Problem decomposition. The complete problem with discontinuities reaching the crystal
boundaries is approximated by the sum of three simpler problems. The four problems will be
referred to, from left to right, as Q, ˜
Q, ˇ
Q and ˆ
Q.
considering the stress field produced by the dislocation inside the body and by making a cut
in the body along the slip segment, moving apart the two sides at a relative distance of b/2
and welding them back together. The process of cutting and welding is purely kinematic and
no stresses would appear in the body if the two sides were free to move with respect to each
other. Nevertheless, the relative slip will cause stresses in the whole domain in many cases:
for example, when the domain includes holes or cracks and the slip segment extends from the
surface of the hole or crack to the external boundary.
The displacement and stress fields due to the dislocation inside the domain can be computed
using the methodology outlined in section 2. The slip across the plane remains to be included
somehow. To mathematically describe this state of deformation, let denote the line at the
intersection of the slip plane and the plane of the domain. If nis the oriented normal of and
xis a point of the domain lying on , the jump in any scalar, vector or tensor field φcan be
expressed in terms of the jump operator [[·]] defined as
[[ φ]] :=lim
→0+φ(x+n)−φ(x−n), (4)
where xis a point on the plane and nis its oriented unit normal.
If udenotes the displacement field of the body, the effect of a dislocation that reaches the
domain boundary is approximated by a jump expressed as
[[ u]] =b
2.(5)
When both dislocations reach opposite ends of , the effects of the slip add up as
[[ u]] =b,(6)
which is the exact solution. Of course, if both dislocations move to the same endpoint of
, their corresponding slips cancel and the jump in the displacement field disappears. The
procedure described is completely general and does not require convexity of the domain.
The most general case, denoted as problem Q, consists of a deformable elastic domain
with imposed displacements Uon the boundary ∂uand imposed tractions Ton its boundary
∂t, dislocations moving on their corresponding slip planes and dislocations that have reached
the domain boundaries and exited the crystal. The solution of this boundary value problem
can be obtained by superposition of three simpler problems (figure 3). The first one, similarly
to problem ˜
P, is posed over the whole plane R2and contains only the dislocations that remain
inside the crystal. This problem is referred as to ˜
Q. The second one, ˇ
Q, consists of the crystal
6
Modelling Simul. Mater. Sci. Eng. 16 (2008) 035008 I Romero et al
with displacement discontinuities across the slip planes in which one or more dislocations have
reached the boundary, and it is obtained by a linear combination of simpler solutions of the type
schematized in figure 3. The last problem, denoted problem ˆ
Q, is an elastic boundary value
problem on the crystal with the displacement boundary conditions of problem Q corrected
with the values obtained in problems ˜
Q and ˇ
Q.
The solution to problem ˇ
Q within the framework of the finite element method is described
in the next section. We note that in the case of a convex body such as the one in figure 3,
problem ˇ
Q has a stress-free solution but affects the global stress field through its displacement
ˇ
u, which appears in the displacement boundary conditions of problem ˆ
Q.
4. Elastic problems with displacement discontinuities
The 2D simulation of dislocation dynamics in arbitrarily shaped bodies in the presence
of dislocations that exit the domain can be achieved by incorporating in the analysis the
displacement jump across those slip segments on which these dislocations have glided. This
can be done systematically and efficiently by embedding the discontinuities in the finite element
solution, a strategy often employed in the numerical simulation of strain localization in solids.
The basic ideas are borrowed from the so-called strong discontinuity approach, a technique
pioneered by [21], and which has been extended and applied by many others [22–26]. A
related approach to model dislocations based on the extended finite element method should be
mentioned here [17].
Consider problem ˇ
Q, described in section 3. Its solution ˇ
uis a smooth function in
except on the segment , in which there is a discontinuity whose value is known a priori.
The displacement jump [[ˇ
u]] is not due to the action of any force or stress on the body and
thus the stresses are smooth across the discontinuity. The key idea for including the effect
of dislocations leaving the crystal is to incorporate the plastic slip by means of displacement
jumps in the finite element problem ˆ
Q. In practice, this amounts to solving problems ˆ
Q and ˇ
Q
simultaneously.
Let ucand σcdenote, respectively, the displacement and stress fields resulting from the
combination of problems ˆ
Q and ˇ
Q. These displacements and stresses account for the image
forces that correct the solution of problem ˜
Q and capture the effect of the discontinuities.
In order to formulate the elastic problem for ucin such a way that it can be later incorporated
in the finite element equations some new objects need to be defined. First, let be the set
:={x∈, distance(x,)<},with the condition ∩∂u=∅.(7)
This set is a band of width 2around the discontinuity . Although it is not necessary for
the derivation, it will be assumed that is much smaller than a characteristic dimension of the
body, for example, its diameter. See figure 4for an illustration of this idea.
Let ˇ
ube a known displacement function that verifies
ˇ
u(x)=0for x/∈e,[[ ˇ
u]] =b/2,[[ ∇ˇ
u]] =0.(8)
In addition, the function ˇ
umust be smooth in \. The strain due to ˇ
uis the tensor ˇ
ε
whose definition, away from ,is
ˇ
ε(x):=∇
Sˇ
u(x), (9)
where ∇Sdenotes the symmetric part of the gradient. At points xnot on the discontinuity, an
elastic and isotropic constitutive law defines the stress tensor by the relation
ˇ
σ(x):=Cˇε(x), (10)
in which Cstands for the fourth rank elastic stiffness tensor of the crystal.
7
Modelling Simul. Mater. Sci. Eng. 16 (2008) 035008 I Romero et al
Figure 4. Regions in which the effect of the displacement discontinuities is localized. Three
discontinuities are depicted.
The total displacement ucis the combination of the solutions to problems ˆ
Q and ˇ
Q and
can be written as
uc(x):=¯
u(x)+ˇ
u(x). (11)
¯
uis a smooth function, continuous everywhere in which must furthermore satisfy the
displacement boundary conditions of problem ˆ
Q. The total strain at x/∈can be expressed as
εc(x):=∇
Suc(x)=¯ε(x)+ˇε(x), with ¯ε(x):=∇
S¯
u(x), (12)
and the corresponding stress tensor is given by
σc(x):=Cεc(x)=¯σ(x)+ˇσ(x), with ¯σ(x):=C¯ε(x). (13)
It has been argued that the stress field in the whole crystal must be smooth. Since ¯
σis
smooth by construction, it follows from the previous equation that ˇ
σmust also be smooth.
The stresses at the discontinuity cannot be obtained from a constitutive equation; thus ˇ
σ(x)
when xbelongs to has to be obtained by finding the appropriate limit of the stress tensor
ˇ
σfrom either side of the discontinuity (which should be identical because, by construction,
[[ ∇ˇ
u]] =0see definition (8)).
By means of the split (11), the only unknown of the combination of problems ˆ
Q and
ˇ
Q is the smooth displacement field ¯
u. The following boundary value problem expresses the
equilibrium of the total stress σcand the boundary conditions for the total displacement uc
and tractions, but as a function of the two fields ˇ
uand ¯
u
−div(ˇ
σ+¯σ)=0in,
¯
σ=C¯εin ,
¯
ε=∇
S¯
uin ,
(ˇ
σ+¯σ)n=T−˜σnon ∂t,
¯
u=U−˜
u−ˇ
uon ∂u.
(14)
In these equations, ˇσis a known quantity, well defined at every point of . The advantage
of the proposed equations over the standard boundary value problem for the total displacement
ucis that the solution ¯
uof (14) is smooth and need not account for the jump across .
8
Modelling Simul. Mater. Sci. Eng. 16 (2008) 035008 I Romero et al
Figure 5. A finite element mesh of the crystal in figure 3. The set is identified.
Moreover, the weak form of (14) can be readily obtained. Let gbe a smooth function
defined on with g=U−˜
u−ˇ
uon ∂uand define the following function space:
V=w:→R2,w=0on ∂u.(15)
The weak solution of problem (14) is the function ¯
usuch that ¯
u−g∈Vwhich verifies
(ˇσ+¯σ)·∇ηd=∂t
(T−˜σn)·ηdS(16)
for all η∈V. In the previous equation, the dots stand for the inner products of the vectors
or tensor on which they operate. Expression (16) is just a statement of the principle of
virtual displacements for the field ¯
uwith an extra term ˇ
σthat accounts for the effects of
the discontinuity. The problem is linear; hence, the result of more than one discontinuity can
be obtained by superposition of additional known stress fields similar to ˇ
σ.
5. Finite elements with embedded displacement discontinuities
The variational formulation (16) can be discretized using the finite element method to find
an approximation for the unknown part of the displacement, the function ¯
u. The total
displacement in problem ˆ
Q can then be recovered by using (11). The only remaining difficulty
in discretizing (16) is to construct a discontinuous function ˇ
uwith the properties stated in (8).
This can be done easily and at an almost negligible cost.
Given a finite element mesh, the domain is partitioned into nel elements occupying a
subset e⊂and connected at nnode nodes of coordinates xa,a=1,...,nnode (figure 5). Let
be the set of all elements crossed by the discontinuity, i.e.
=
e∩=∅
e.(17)
The discontinuous function ˇ
ucan be defined in an element-by-element fashion and is
denoted ˇ
uh. First, let ˇ
uhbe identically zero in all the elements that do not belong to . The
discontinuity splits the remaining elements into two parts +
eand −
ewhich are, respectively,
above and below the discontinuity, with the sign provided by the normal n. The nodes of
element eare denoted Neand can also be divided into two groups: those located above the
discontinuity and those below it. They will be referred to as N+
eand N−
e, respectively (figure 6).
9
Modelling Simul. Mater. Sci. Eng. 16 (2008) 035008 I Romero et al
Figure 6. Discontinuity interpolation functions. The element ehas a discontinuity at line ,
located within the element.
Let H+
e(x)be the discontinuous function defined in ewhose value is one in +
eand zero in
−
e. The function ˇ
uin element eis defined to be
ˇ
uh(x)e
:=
H+
e(x)−
a∈N+
e
Na(x)
b/2,(18)
where Nastands for the standard finite element interpolation function associated with node a.
The support of ˇ
uis the band , continuous everywhere except at the discontinuity which
exhibits a jump in value b/2. Its gradient is also continuous across , thus satisfying all the
requirements put forward in (8). See figure 6for an illustration of this idea.
The description of the finite element formulation is completed with the discretization of
the trial space Vwhich is done in the standard fashion. Given the finite element mesh, the set
Vh⊂Vis defined as follows:
Vh:=˜
wh=
nnode
a=1
Na(x)wa,wh=0on ∂u.(19)
The finite element approximation to the displacement ucis denoted uhand,
mimicking (11), it can be split into a continuous and a discontinuous part:
uh(x):=ˇ
uh(x)+¯
uh(x). (20)
If ghis a known function that satisfies the displacement boundary conditions, the finite
element approximation ¯
uhis a function such that ¯
uh−ghbelongs to Vhwhich verifies
(ˇσ+¯σ)·∇ηhd=∂t
(T−˜σn)·ηhdS(21)
for all ηh∈Vh. The stress tensors are obtained as explained in section 4.
From a practical standpoint, including embedded discontinuities within the finite element
formulation amounts to adding an extra term ˇ
σin the evaluation of the internal forces of those
10
Modelling Simul. Mater. Sci. Eng. 16 (2008) 035008 I Romero et al
Figure 7. Schematic of the geometry and boundary conditions of a rectangular single crystal with
a hole subjected to uniaxial tension. There is only one slip plane with a dislocation source in the
middle.
elements which intersect the discontinuity. These terms do not depend on the unknown field ¯
uh
and thus they do not modify the value of the stiffness matrix. As a result, the stiffness matrix
should be computed (and factorized) only once regardless of the number of discontinuities
embedded in the domain. The total displacement uhhas a continuous contribution and a
discontinuous one, as indicated in (20).
It is interesting to note that the discontinuous part ˇ
uhhas a zero value at all the nodes of
the finite element mesh. Thus, at nodal coordinates xa,a=1,...,nnode ,
ˆ
uh(xa)=¯
uh(xa). (22)
This property is very convenient for postprocessing the solution, making the computation
of the total displacement field unnecessary. The total stress ˆ
σ=ˇσ+¯σis required, however, to
compute the resolved shear stress in the dislocations (2). As explained above, the extra term
due to the discontinuity has an almost negligible computational cost.
Since the nodal values of ˇ
uhare zero and this field only appears in the finite element
equations (21) through its strain, it turns out that the function itself is completely unnecessary.
In order to account for the effects of the slip, it suffices to compute the strain ˇ
εin the elements
belonging to . From (18), this extra strain can be computed as
ˇ
ε(x)e=−
b
2⊗
a∈N+
e
∇Na(x)
S
,(23)
where the notation (·)Sindicates the symmetric part of the tensor.
6. Examples of application
6.1. Tensile deformation of single crystal with a void
An academic application of this approach is depicted in figure 7: a rectangular single crystal
of 12 µm×4µm with a circular hole of 2 µm diameter at the center was subjected to uniaxial
tension. The beam was made up of a linear elastic and isotropic solid, characterized by its
shear modulus µ=26.32 GPa and Poisson’s ratio ν=0.33. Plane strain conditions are
assumed in the x1–x2plane and there was only one slip plane in the beam oriented at 45◦with
the positive x1axis. The Burgers vector was b=0.25nm and the slip plane was initially
free of dislocations. Dislocation dipoles were generated from one dislocation source in the
middle of the slip plane. The critical resolved shear stress for dipole nucleation was 50 MPa
11
Modelling Simul. Mater. Sci. Eng. 16 (2008) 035008 I Romero et al
0
0.01
0.02
0.03
0 0.005 0.01 0.015 0.02
hole
without hole
σ/µ
ε = u/L
I
II
III
I
II
Figure 8. The stress–strain curve of the rectangular single crystal with a hole of figure 7subjected
to uniaxial tension. The corresponding curve for the crystal without the hole is also plotted for
comparison.
and the nucleation time was 0.01 µs. Once generated, dislocations slip in the glide plane and
the speed of the dislocations is given by the resolved shear stress on the glide plane and the
drag coefficient B=10−4Pa s according to equation (1). These magnitudes are equal to those
used previously in [8]. There were no obstacles to dislocation motion in the slip plane.
The boundary conditions for uniaxial tension are expressed as (figure 7)
u1=0 and T2=0onx1=0,
u1=Uand T2=0onx1=L, (24)
T1=T2=0onx2=±W/2,
where Ti=σij njis the traction on the boundary with normal nj. Loading was imposed by
applying a constant strain rate of ˙=U/L =2000 s−1. The applied stress σis computed as
σ=1
WW/2
−W/2
T1(L, x2)dx2(25)
to obtain the uniaxial stress–strain curve (σ/µ,U/L), which is plotted in figure 8. The
corresponding curve for the crystal without the hole is also plotted for comparison. In this
latter situation, the slip plane extended through the whole section of the rectangular crystal,
and the dislocation source was located at the center of the slip plane.
The initial elastic modulus and yield strength of the crystal with a hole were lower than
those of the other one due to the hole. Both materials showed subsequent linear hardening
because the generation rate of new dislocation dipoles was limited by the nucleation time and
the applied strain rate has to be accommodated with elastic deformation in addition to the
plastic contribution provided by the dislocations. The simulations were carried out up to a
far-field tensile strain of 2%; the strain rate was then reversed and continued until complete
unloading of the crystal. The stresses acting on the slip plane were large enough to promote
dislocation nucleation and slip at the beginning of the unloading, and the initial slope of the
12
Modelling Simul. Mater. Sci. Eng. 16 (2008) 035008 I Romero et al
Figure 9. Residual stress distribution in the single crystal with a hole deformed in uniaxial
tension after complete unloading. The Von Mises stress is shown in the plot. The displacement
magnification factor is 1.
stress–strain curve upon unloading (region I in figure 8) was a compromise between the forward
deformation due to dislocation motion and the backward elastic strain. This was followed by
an elastic unloading (region II in figure 8) until zero stress in the crystal without the hole. The
region of the elastic unloading was, however, very short in the crystal with a hole and it was
followed by region III, in which the dislocation source was again activated and dislocations
slip in the opposite direction. As a result of the reversed plastic flow, the permanent strain of
the crystal with a hole after complete unloading was only one-fourth of the one of the crystal
without a hole. The differences in behavior upon unloading between the crystals with and
without a hole were due to the development of residual stresses in the crystal with a hole as a
result of the inhomogeneous strain field generated by plastic slip. The magnitude of residual
stresses after complete unloading of the crystal is shown in the contour plot of figure 9. The
plot also shows the deformation of the crystal and the localization of the plastic strain around
the slip plane. In the case of the crystal without a hole, the dislocations leaving the crystal
through the edges only induced a displacement jump at both sides of the slip plane but no
residual stresses.
6.2. Microvoid growth
Another example of application is the growth of a microvoid within a square single crystal
grain of dimensions 1 ×1µm2(figure 10). The microvoid radius Rwas 0.2185 µm, which
corresponds to an initial void volume fraction of 15%. The crystal has two slip systems
oriented at angles φ=±35.25◦with respect to the main loading axis x2. This orientation
corresponds to a planar model of a FCC crystal in which the x2direction is close to the [110]
direction and φstands for the angle between the x2axis and the [1 1 2] orientation [27]. The
elastic constants and Burgers vector of the crystal are the same used above, and the distance
between the slip planes was 81.7b. Dislocation sources were randomly distributed along the
slip planes with a density of 150 µm−2. The slip planes were initially free of dislocations. The
critical resolved shear stress for dislocation nucleation was assigned randomly to the sources
following a Gaussian distribution with an average value of 50 MPa, a standard deviation of
15 MPa and a nucleation time of 0.01 µs. There were no obstacles to dislocation motion within
the crystal surfaces but the boundaries were assumed to act as grain boundaries impenetrable
to dislocations, and so dislocations could only leave the crystal through the central void.
The kinetics of microvoid growth was analyzed in plane strain under three loading
conditions in which the hydrostatic stress component is very different: uniaxial tension,
uniaxial deformation and isotropic deformation in the x1x2plane. The boundary conditions
13
Modelling Simul. Mater. Sci. Eng. 16 (2008) 035008 I Romero et al
Figure 10. Schematic of the geometry, boundary conditions and slip systems of a square single
crystal with a central hole. See text for details.
for uniaxial tension are expressed as (figure 10)
u1=0 and T2=0onx1=0,
u2=0 and T1=0onx2=0,(26)
u2=Uand T1=0onx2=L,
T1=T2=0onx1=L,
while the last equation in (26) in the case of uniaxial deformation is substituted by
u1=0 and T2=0onx1=L(27)
or by
u1=Uand T2=0onx1=L(28)
for biaxial deformation. Loading was imposed by applying a constant strain rate of ˙=
U/L =2000 s−1up to a maximum strain of 1% under biaxial deformation and of 2% under
uniaxial strain and uniaxial tension.
The evolution of the void volume fraction with applied strain is plotted in figure 11(a) for
the three different loading cases considered. For the small deformations considered here and
after the elastic region, the growth rate was practically linear in all cases, but the higher the
triaxiality, the higher the void growth rate, as expected. Higher stresses facilitate the nucleation
of dislocation dipoles along both slip systems, and dislocation density increased with stress
triaxiality, figure 11(b). The dislocations of each dipole moved along opposite directions driven
by the resolved shear stress acting on each slip plane, and those which reached the crystal
boundaries were stopped, leading to the formation of long dislocation pile-ups (figure 12).
Many of those moving along the opposite direction reached the void surface and led to the
formation of a ledge. Voids grew by the accumulation of these ledges and the void perimeter
at the nanometer scale was tortuous, as shown in figure 13. Of course, the high number
14
Modelling Simul. Mater. Sci. Eng. 16 (2008) 035008 I Romero et al
15
15.2
15.4
15.6
15.8
16
0 0.5 1 1.5 2
Biaxial deformation
Uniaxial deformation
Uniaxial tension
Void volume fraction (%)
ε22 (%)
0
500
1000
1500
2000
2500
3000
3500
4000
0 0.5 1 1.5 2
Biaxial deformation
Uniaxial deformation
Uniaxial tension
ρ (µm-2)
ε22
(%)
(a) (b)
Figure 11. (a) Void volume fraction as a function of the applied strain for the single crystal with a
circular hole loaded in uniaxial tension, uniaxial deformation and biaxial deformation. (b) Evolution
of dislocation density within the single crystal under similar conditions.
Figure 12. Dislocation pattern in the single crystal subjected to a uniaxial deformation of 2%.
of dislocation dipoles generated under triaxial stress states increased the void growth rate.
As observed in previous simulations [7,8,10] and experiments [28,29] in micrometer-sized
single crystals, deformation is not homogeneous at this scale and tends to be concentrated in
particular slip systems. Similar results were found in the presence of a hole and the dislocation
activity was concentrated on a few slip planes, leading to the irregular shape of the void surface
(figure 13) and to the formation of intense slip bands, as shown in figure 14 for the crystal
subjected to uniaxial deformation.
Obviously, the results in this section show the potential of the new method for answering
many interesting topics in the micromechanics of void growth such as the influence of the
number and orientation of slip systems, density of dislocation sources and obstacles and void
and crystal size on the void growth rate for various loading conditions. Moreover, this type
of analyses can be used within a multiscale simulation strategy to compute realistic values for
15
Modelling Simul. Mater. Sci. Eng. 16 (2008) 035008 I Romero et al
-1.2
-0.8
-0.4
0
0.4
0.8
1.2
-1.2 -0.8 -0.4 0 0.4 0.8 1.2
Biaxial deformation
Uniaxial deformation
Uniaxial tension
y/R
x/R
ε22 = 1.0%
Figure 13. Influence of the stress triaxiality on the shape of the void.
Figure 14. Contour plot of the equivalent strain in the single crystal subjected to a uniaxial
deformation of 2%, showing the inhomogeneous pattern of deformation, which is mainly localized
along slip bands. The displacement magnification factor is 1.
the length scale which appears in the constitutive equations developed within the framework
of strain gradient crystal plasticity. These topics will be addressed in future investigations.
7. Summary and conclusions
The standard strategy for solving boundary value problems within the framework of discrete
dislocation dynamics in two dimensions was modified to account for the effect of dislocations
16
Modelling Simul. Mater. Sci. Eng. 16 (2008) 035008 I Romero et al
leaving the crystal through a free surface in the case of non-convex domains, in which the
intersection of the slip plane with the domain is not a continuous segment. This particular
configuration often arises in the simulation of the mechanical behavior of micro-electro-
mechanical and micro-electronics devices or in micromechanics. The new method incorporates
the displacement jumps across the slip segments of the dislocations that have exited the crystal
within the finite element analysis carried out to compute the image stresses on the dislocations
induced by the finite boundaries. This is done in a simple computationally efficient way by
embedding the discontinuities in the finite element solution, a strategy often employed in the
numerical simulation of crack propagation in solids. The mathematical foundations of the
new approach and its practical implementation within a finite element program are detailed
and two academic examples of application are presented. The first one shows the influence of
the residual stresses generated by an inhomogeneous slip during forward and reversed tensile
loading of a single crystal with a hole and the second one studies the growth of a void in a
micrometer-sized square crystal under different levels of triaxiality.
Acknowledgments
The financial support from the Comunidad de Madrid through the program ESTRUMAT-CM
and the Ministerio de Educaci´
on y Ciencia de Espa˜
na through grants DPI2006-14104 and
MAT2006-2602 is gratefully acknowledged.
Appendix. Numerical implementation of embedded discontinuities
This appendix provides the details of numerical implementation to embed discontinuities within
the context of the finite element method. The methodology requires one preprocessing step
and modification of the standard rule for evaluating the internal force in the elements.
Preprocess. The additional stress tensors ˇ
σ(see (21)) to be added to the elastic stress ¯σin
case a dislocation exits the crystal are stored in the preprocessing step.
Loop over all elements e:
Loop over all slip lines d:
Let bd=bmd, where mdis the oriented unit tangent vector on the line.
If the slip line dcrosses element ethen:
Compute (inactive) extra strain and stress, and store the stresses:
ˇ
εe,d =
a∈N+
e
−bd
2⊗∇Na(x)S
,ˇσe,d =Cˇεe,d.(A.1)
End if.
End loop over all slip lines.
End loop over all elements.
Computation of internal forces. The dislocations that leave the crystal modify the calculation
of the internal forces simply by shifting the value of the stresses at the integration points.
Loop over all elements e:
Compute stress due to smooth displacement ¯
σe=C∇S¯
u.
Loop over all slip lines d:
Set ˇ
σe=0.
17
Modelling Simul. Mater. Sci. Eng. 16 (2008) 035008 I Romero et al
Let χdbe the (signed) number of dislocations which were on plane dand have
left the crystal through the endpoint indicated by md.
If the slip line dcrosses element eand χd= 0 then
Add extra stress: ˇ
σe=ˇσe+χdˇσe,d.
End if.
End loop over all slip lines.
Combine stress contributions ˆ
σe=¯σe+ˇσe.
End loop over all elements.
The extra terms ˇ
σe,d do not depend on the unknown nodal variables ˆ
ua, hence the modifications
due to the dislocations which have left the domain do not contribute at all to the stiffness matrix.
References
[1] Van der Giessen E and Needleman A 1995 Modelling Simul. Mater. Sci. Eng. 3689
[2] Needleman A 2000 Acta Mater. 48 105
[3] Devincre B, Kubin LP, Lemearchand C and Madec R 2001 Mater. Sci. Eng. A309–310 211
[4] Gates T S, Odegard G M, FranklandSJVandClancy T C 2005 Comput. Sci. Technol. 65 2416
[5] Gonz´
alez C and LLorca J 2006 Acta Mater. 54 4171
[6] Bulatov V V and Cai W 2006 Computer Simulations of Dislocations (Oxford: Oxford University Press)
[7] CleveringaHHM,VanderGiessen E and Needleman A 1999 Int. J. Plast. 15 837
[8] Segurado J, LLorca J and Romero I 2007 Modelling Simul. Mater. Sci. Eng. 15 S361
[9] Widjaja A, Van der Giessen E and Needleman A 2005 Mater. Sci. Eng. A400–401 456
[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] Deshpande V S, Needleman A and Van der Giessen E 2003 J. Mech. Phys. Solids 51 2057
[13] Schwarz K W and Chidambarrao D 2005 Mater. Sci. Eng. A401–401 435
[14] Huang M, Li Z and Wang C 2007 Acta Mater. 55 1387
[15] Lee D, Wei X, Zhao M, Chen X, Jun J C, Hone J and Kysar J W 2007 Modelling Simul. Mater. Sci. Eng. 15 S181
[16] Biener J, Hodge A M, Hayes J R, Volkert C A, Hamza A V, Zepeda-Ruiz L A and Abraham F F 2006 Nano Lett.
62379
[17] Gracie R, Ventura G and Belytschko T 2007 Int. J. Numer. Methods Eng. 69 423
[18] Gracie R, Oswald J and Belytschko T 2008 J. Mech. Phys. Solids 56 200
[19] Lemarchand C, Devincre B and Kubin L P 2001 J. Mech. Phys. Solids 46 1969
[20] Deshpande V S, Needleman A and Van der Giessen E 2005 J. Mech. Phys. Solids 53 2661
[21] Simo J C, Oliver J and Armero F 1993 Comput. Mech. 12 277
[22] Armero F and Garikipati K 1996 Int. J. Solids Struct. 33 2863
[23] Borja R I 2000 Comput. Methods Appl. Mech. Eng. 190 1529
[24] Jirasek M 2000 Comput. Methods Appl. Mech. Eng. 188 307
[25] Oliver J, Huespe A E, Pulido M D G and Chaves E 2002 Eng. Fract. Mech. 69 113
[26] Sancho J M, Planas J, Fathy A, G´
alvez J and Cend´
on D 2006 Int. J. Numer. Anal. Meth. Geom. 31 173
[27] Benzerga A A, Br´
echet Y, Needleman A and Van der Giessen E 2004 Modelling Simul. Mater. Sci. Eng. 12 159
[28] Uchic M D, Dimiduk D M, Florando J N and Nix W D 2004 Science 305 986
[29] Motz C, Sch¨
oberl T and Pippan R 2005 Acta Mater. 53 4269
18
