An analysis of void growth using discrete dislocation dynamics J ...

**An**ales de Mecánica de la Fractura 26, Vol. 1 (2009)

**An** **analysis** **of** **void** **growth** **using** **discrete** **dislocation** **dynamics**

J. LLorca, J. Segurado,

Departmento de Ciencia de Materiales, Universidad Politecnica de Madrid & Instituto Madrileno de Estudios Avanzados en

Materiales (IMDEA-Materiales) ETS de Ingenieros de Caminos, 28040 Madrid, Spain

E-mail: jllorca@mater.upm.es

RESUMEN

El crecimiento de huecos en un metal controla la ductilidad del material. El crecimiento se produce por la deformación

plástica de la matriz metálica que rodea el hueco y, debido al tamaño micrométrico de éste, esta deformación plástica

presenta un efecto del tamaño que debe tenerse en cuenta en la simulación del fenómeno.

En este artículo, el crecimiento de un hueco en un monocristal aislado se ha estudiado mediante dinámica de dislocaciones

discretas en dos dimensiones. Las simulaciones están basadas en el modelo de Van der Giessen and Needleman

[1], extendido por los autores para el caso de dominios no convexos mediante la introducción de elementos finitos con

discontinuidades fuertes embebidas [2]. Monocristales de forma cuadrada y tamaños entre 0.5 µm y 2.5µm con un poro

cilíndrico en su centro ocupando un 10% del área fueron deformados de forma biaxial bajo condiciones de deformación

plana para estudiar el efecto del tamaño. Además, se aplicó tracción uniaxial en diferentes direcciones cristalográficas

para estudiar el efecto de la orientación del cristal. Se han encontrado dos efectos del tamaño, uno en el límite elástico

y endurecimiento, del tipo “más pequeño más resistente” y otro en la velocidad de crecimiento del hueco, del tipo’ “más

pequeño más lento”. Además se ha encontrado un importante efecto de la orientación del cristal respecto a la dirección de

la carga aplicada en caso de tracción uniaxial, tanto en el comportamiento mecánico como en la velocidad de crecimiento

del hueco.

ABSTRACT

Void **growth** in metals controls the overall ductility **of** the material. The **growth** is produced by the plastic deformation **of**

the matrix sourrounding the **void**, and because **of** the micron size **of** the **void**, this plastic defomation presents a size effect

that has to be taken into account in the simulations.

In this paper, the **growth** **of** a **void** in isolated single crystal was studied **using** two-dimensional **discrete** **dislocation**

**dynamics**. The simulations were based on the methodology developed by Van der Giessen and Needleman [1], which

was extended by the authors to non-convex domains through the use **of** finite elements with embedded discontinuities

[2]. Square single crystals (in the range 0.5 µm to 2.5 µm) with an initial **void** volume fraction **of** 10% were deformed

under plane strain conditions and applying biaxial deformation in order to account size effect. In adittion, uniaxial load

was applied in different crystalographic directions in order to check the effect **of** lattice orientation. Two size effects were

found, one on the initial flow stress and strain hardening rate **of** the **void**ed crystal (”smaller is stronger”) and another on

the **void** **growth** rate (”smaller is slower”). A big effect **of** lattice orientation with respect to the applied load in uniaxial

traction was also found in both mechanical behavior and **void** **growth** rate.

ÁREAS TEMÁTICAS PROPUESTAS:

Métodos numéricos.

PALABRAS CLAVE:

Plasticidad, dinámica de dislocaciones,elementos finitos.

1 INTRODUCTION

Ductile failure in metals normally takes place by the nucleation,

**growth** and coalescence **of** micron-sized **void**s.

Once the **void** has been nucleated by debonding or cracking

**of** a second-phase particle or inclusion, the stress concentration

at the stress-free **void** surface enhances plastic

flow around the **void**, which undergoes volumetric **growth**

and shape change. Overall ductility is mainly controlled

by the evolution **of** the **void** **growth** from the nucleation

stage up to the point at which fracture is triggered by the

sudden coalescence **of** neighboring **void**s into a crack.

The first model **of** **void** **growth** was due to Rice and

Tracey [3], who computed the shape changes during deformation

**of** a spherical **void** embedded in an infinite matrix

**of** a rigid-plastic non-hardening material. Many analytical

models and numerical simulations have been developed

afterwards to account for the effect **of** different

factors on **void** **growth**, including strain hardening capacity

and sensitivity **of** the ductile matrix, stress triaxiality,

**void** shape and spatial distribution, plastic anisotropy **of**

single crystals, etc. [4, 5, 6, 7, 8]. These analyses were

carried out in the framework **of** classical continuum plasticity,

and cannot predict any intrinsic effect **of** the **void**

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size on **growth** rate.

Nevertheless, there is compelling experimental evidence

**of** a size effect on the resistance to plastic flow when the

characteristic dimension **of** the problem (i.e. **void** diameter)

or **of** the zone subjected to plastic deformation is in

the range **of** a few µm [9, 10, 11]. This size effect can

be taken into account through non-local plasticity theories,

which are mainly based on strain gradient plasticity

(SGP) formulations, and these have been used to analyze

the influence **of** **void** diameter on **void** **growth** rate

[12, 13, 14, 15]. These models consistently predict a reduction

in **void** **growth** rate for smaller **void**s but quantitative

estimations depend on the magnitude **of** intrinsic

length scale included in the strain gradient plasticity formulation

which has to be obtained by fitting to experimental

results. Nevertheless, to the authors’ knowledge,

reliable experimental data on **growth** rates for microsized

**void**s are not available. As a result, the accuracy

**of** these quantitative predictions remains unknown.

**An**other solution for this limitation can be obtained from

”virtual tests”, in which the mechanical response **of** a

**void**ed crystal is simulated **using** **discrete** **dislocation** **dynamics**

(DDD). Hussein et al. [16] provided estimations

**of** the intrinsic length scale in the strain gradient plasticity

model for a periodic array **of** rectangular **void**s subjected

to biaxial tension and pure shear by fitting the predictions

for the flow stress with those obtained through DDD. Excellent

agreement between SGP and DDD was obtained

in terms **of** the plastic deformation patterns for different

slip system orientations and **void** volume fractions with

an intrinsic length scale **of** 0.325 µm but no predictions

**of** **void** **growth** were reported, and this is the focus **of** this

paper. Using the DDD framework developed by the authors

[2], a bi-dimensional study **of** the **void** **growth** is

made **using** a cylindrical **void** embedded in a square single

crystal for different **void** and crystal sizes. The effect

**of** lattice orientation respect to the applied load has also

been studied.

2 Numerical simulation strategy

The numerical simulation **of** the **growth** **of** a cylindrical

**void** within a single crystal is carried out within the

framework **of** DDD developed by Van der Giessen and

Needleman [1]. This strategy was extended by the authors

to non-convex domains through the use **of** finite elements

with embedded discontinuities [2] and it is briefly

recalled here. The model considers an elastic, isotropic

crystal **of** dimensions L×L with a **void** **of** radius R at the

center (Figure 1). Plane strain conditions are assumed in

the x 1 - x 2 plane (corresponding to crystalogrphic plane

(¯110) and the crystal contains several slip systems made

up **of** families **of** parallel slip planes with different orientations.

Dislocations are represented by linear singularities perpendicular

to the crystal plane with Burgers vector b

whose modulus is given by b. Dislocation dipoles can

be nucleated at **discrete** points randomly distributed on

the slip planes, which mimic the behavior **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 over a period **of** time t nuc . The distance

between the two new **dislocation**s, L nuc , is given

by

E b

L nuc =

4π(1 − ν 2 (1)

) τ nuc

so that the resolved shear stress balances the attractive

forces between them.

Figure 1: Schematic **of** the boundary value problem for

the **analysis** **of** **void** **growth** in a single crystal. See text

for details

Once generated, **dislocation**s 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 projection **of** the Peach-Koehler

force acting on the **dislocation** i in the sliding direction

and B is the drag coefficient. Moreover, two **dislocation**s

**of** different sign gliding on the same slip plane are annihilated

when they cross each other or if they fall 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 shear stress component **of** the Peach-Koehler force

acting on the ith **dislocation**, located on a glide plane with

unit normal n and unit tangent vector m, can be obtained

by projecting the stress at the location **of** the **dislocation**

according to

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⎛

τ i = m · ⎝ˆσ + ∑ j≠i

˜σ j j

⎞

⎠ · n . (3)

where ˜σ j j stands for the stress due to the jth **dislocation**

and it is computed analytically from the expressions for

the stress field induced by an edge **dislocation** on an infinite,

elastic and isotropic continuum. It is necessary to

add the field ˆσ which includes the effect **of** the image

forces induced by the crystal boundaries on the **dislocation**s,

to this contribution. At a given stage **of** loading, the

stress and strain 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 **dislocation**s 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 [1, 17].

The deformation process **of** the crystal is solved in an explicit

incremental manner, **using** an Euler forward timeintegration

algorithm for the equations **of** motion. Once

the new positions **of** all **dislocation**s at time t have been

computed, new **dislocation**s 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 + δt. The resolved shear stresses on the **dislocation**s are

computed according to (3) from the fields induced by the

**dislocation**s and the solution **of** the boundary value problem.

Then, the velocities **of** the **dislocation**s are obtained

from the corresponding resolved shear stress, as given by

(2). The new positions are computed from these velocities

and the **dislocation**s that meet an obstacle are pinned

if the resolved shear stress is below τ obs . More details

about the numerical implementation **of** the model can be

found in [17, 2]. It should be noticed that the current

version **of** the program has been parallelized to improve

performance in multiprocessor shared-memory computers.

3 RESULTS

The model described in the foregoing section was applied

to simulate the mechanical response **of** an isolated square

single crystals with different sizes, (side length **of** the single

crystal, L, ranging from 0.5 µm to 2.5 µm) with a

cylindrical **void** at the center, covering 10% **of** the crystal

area (Figure 1). The crystal was made up **of** a linear elastic,

isotropic solid, characterized by its shear modulus µ

= 26.32 GPa and Poisson’s ratio ν = 0.33. The modulus

**of** the burgers vector was 0.25 nm, the source densisty

150 µm −2 and activation mean stress **of** the sources was

50 MPa. The rest **of** the numerical parameters are taken

from previous studies [17, 18].

3.1 Study **of** the size effects

For this section, two slip systems are considered oriented

at angles φ = ±35.25 ◦ with respect to the main loading

axis x 2 (see Fig. 2). This orientation corresponds to

a planar model **of** a FCC crystal in which the x 2 corresponds

to the [001] direction and φ stands for the angle

between the x 2 axis and the [112] orientation [19].

In this study and for the sake **of** brevity, only biaxial deformation

is considered, being this the case where **void**s

grow faster with the applied deformation and size effects

are maximal [3]. The averaged σ 22 - ɛ 22 curves obtained

under biaxial deformation are plotted in Fig. 2 for each

crystal size.

σ 22

(MPa)

2000

1600

1200

800

400

2.5 µm

1.75 µm

1 µm

0.5 µm

0

0 0.005 0.01 0.015 0.02

ε 22

Figure 2: Effect **of** the crystal size on the mechanical

response **of** **void**ed crystals. Biaxial deformation (ɛ 11 =

ɛ 22 ).

The crystals were initially free **of** **dislocation**s and the initial

response was linear elastic and independent **of** size, as

the domains were geometrically homotetic. Non-linear

deformation was triggered by the nucleation **of** **dislocation**

dipoles at the sources, which moved along their corresponding

slip planes. The point where the crystals start

their inelastic deformation for each crystal was slightly

dependent on crystal size following a size-effect **of** the

type smaller is stronger. This first size effect comes about

because the number **of** sources available to generate **dislocation**s

decreases with size as the source density is constant,

and it is amplified by the inhomogeneous stress

state within the **void**ed crystal because the maximum resolved

shear stresses on the slip planes are localized in a

small region sourrounding the **void**s. The onset **of** plasticity

is controlled by the source with the lowest critical

resolved shear stress for nucleation located in the high

stress region, and this magnitude increases by statistical

reasons as the number **of** sources decreases in smaller

specimens.

Once dipoles have been nucleated at the sources, dislo-

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cation slip takes place until the **dislocation**s reach the

central **void** or the crystal boundary. The former exit

the domain and lead to a step on the surface. The latter

are stopped at the boundary and form **dislocation** pileups.

These pile-ups reduce the mobility **of** **dislocation**s

and generate a back stress which also reduces the activity

**of** the sources, and both factors were responsible for

the strain hardening **of** the **void**ed crystals after yielding.

The resulting **dislocation** structure at a far-field applied

strain **of** 2% for the largest 2.5 µm crystal is plotted in

Fig. 3. The figure shows that the **dislocation** activity is

concentrated in a **discrete** number **of** slip planes containing

sources that can generate **dislocation**s, leading to the

localization **of** the plastic deformation in these planes.

by the initial **void** area, ɛ v , was plotted as a function **of** the

applied strain for the crystals with different size subjected

to biaxial deformation (Fig. 4).

Figure 4: Evolution **of** the **void** area (divided by the initial

**void** area), ɛ v as a function **of** the applied strain, ɛ 22 for

crystals **of** different size. Biaxial deformation (ɛ 11 = ɛ 22 )

Figure 3: Dislocation patterns at a far-field strain **of** 2%

in the **void**ed crystals subjected to biaxial deformation, L

= 2.5 µm. Green and red crosses stand for **dislocation**s

**of** different sign while dots represent the position **of** the

sources.

Although the deformation micromechanisms are equivalent

in large and small crystals, the strain hardening rate

increased as the size decreased, leading to a marked size

effect (Fig 2). This behavior is related to the evolution **of**

the **dislocation** density ρ with the applied strain which

increased more rapidly in smaller crystals indicating a

higher number **of** **dislocation**s within the pile-ups that

lead to a increase in the crystal flow stress.

The global deformation **of** the different crystals have

an elastic and plastic contribution, and the higher strain

hardening **of** the smaller samples suggests a higher contribution

**of** the elastic deformation in theses crystals.

This fact has been proved by unloading the crystals and

observing than the recovered strain (elastic) was bigger

in the smaller samples.

The effect **of** the crystal size on the **void** **growth** rate was

studied computing the area **of** the **void** from the displacements

**of** the nodes at the **void** surface at different stages

**of** the simulation. The increase in the actual area divided

The **void** **growth** curves under biaxial deformation did

show a clear size effect **of** the type ”larger is faster”. The

origin **of** this size effect can be explained by taking into

account that the total volumetric strain (given by ɛ 11 +

ɛ 22 ) is the addition **of** the volumetric elastic strain and the

**growth** **of** the **void**, because the plastic deformation only

induces shear strains. As explained in the previous section,

the smaller the crystal the higher the elastic stresses

and thus the contribution **of** the volumetric elastic strains

to ɛ 11 + ɛ 22 is also smaller. As a result, the contribution **of**

**void** **growth** to the total volumetric strain increased with

size leading to the size effect on **void** **growth** depicted in

Fig 4.

3.2 Effect **of** lattice orientation

The square single crystal with L=2.5µm used in previous

section is loaded here along different directions in order

to check the effect **of** lattice orientation. Three slip systems

are considered, forming angles φ = ±35.25 ◦ and

90 ◦ with respect to the direction [001]. As in the previous

section, the x 1 − x 2 plane represents the crystalographic

plane (¯1 1 0), but in this section, the crystal orientation

is rotated and the loading direction, x 2 , is oriented parallel

to different directions included in the plane (¯110):

[001], [¯112], [¯111] and [¯110]. The simulations were carried

out under uniaxial tension (σ 11 =0), because it has

been observed and corroborated in the literature [7] that

under higher triaxiliaties, the effect **of** lattice orientation

on **void** **growth** and mechanical behavior is much smaller.

The stress-strain curves (σ 22 - ɛ 22 ) **of** the **void**ed crystals

deformed under uniaxial tension are plotted in Fig. 5.

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Each curve corresponds to one **of** the four lattice orientations

and is the average **of** three simulations obtained with

different distributions **of** the sources within the crystal.

150

0.03

0.024

[1 1 2]

[0 0 1]

[1 1 1]

[1 1 0]

120

[1 1 2]

[0 0 1]

[1 1 1]

[1 1 0]

σ 22

(MPa)

0.018

0.012

90

σ 22

(MPa)

60

0.006

30

0

0 0.005 0.01 0.015 0.02

ε 22

0

0 0.005 0.01 0.015 0.02

ε 22

Figure 5: Averaged σ 22 −ɛ 22 curves **of** the **void**ed crystal

under uniaxial traction for the different orientations.

The differences found in the hardening rate can be explained

in terms **of** a number **of** factors. Low hardening

was found in orientations in which the most active

slip planes (high Schmidt factor) were in regions where

the elastic stresses levels were high and had free surfaces

(that attract **dislocation**s) at both ends **of** the slip planes.

However, higher stresses have to be applied if the most

active slip planes have to pass above and below the **void**

(precisely the region where the applied stresses are minimum)

and ended at the upper or lower crystal surfaces,

where the image forces due to the imposed displacement

repel the **dislocation**s.

The lattice orientation have also an effect in the **void**

**growth** rate, and in order to investigate this effect, the

area **of** the **void** was computed at different instants **of** the

simulation and plotted on Fig. 6.

The physical mechanisms **of** plastic deformation in uniaxial

tension changed with the crystal orientation and this

led to the changes observed in **growth** rates. In general,

the hardest orientations ([¯112] and [001]) presented the

highest **growth** rates while the **void**s grew slower in the

s**of**test orientations ([¯111] and [¯110]). The differences in

**growth** rates between both groups were **of** the order **of**

2, although the overall **growth** rate under uniaxial tension

was small. This behavior (stronger crystals led to higher

**growth** rates) was opposite to the one found in previous

section where the smaller crystals presented the highest

flow stress and the lowest **void** **growth** rate. The reason

is that in the former case, the behavior was related to

the build-up **of** elastic stress as the crystal size decreased

because the total number **of** **dislocation**s available in the

smallest crystals was limited by crystal size and source

density.

Figure 6: Evolution **of** the **void** area (divided by the initial

**void** area), ɛ v , as a function **of** the applied strain, ɛ 22 for

crystals with different lattice orientations under uniaxial

tension (σ 11 = 0).

On the contrary, when the crystal size is maintained and

only the orientation changes, the level **of** elastic stresses

found was very similar. The plastic deformation was localized

in different slip systems for each orientation and

they contributed differently to the changes in the crystal

and **void** shape. While the **void** elongation in the x 2

direction was similar for all lattice orientations (it was

mainly imposed by the imposed deformation along this

axis), the deformation **of** the equator **of** the **void** depended

on the orientation **of** active slips systems and their interaction

with the lateral free surfaces. Overall, the [¯112]

and [001] crystal orientations which induced higher **dislocation**

densities (and thus stronger crystals) also led to

higher **void** **growth** rates.

4 CONCLUSIONS

The effect **of** the crystal size and lattice orientation on

the mechanical behavior **of** a single crystal with a central

cylindrical **void** (with initial size corresponding to 10%

**of** the crystal area) was studied **using** a two-dimensional

**discrete** **dislocation** **dynamics** framework. Four different

crystal sizes (in the range 2.5 µm to 0.5 µm) were analyzed

to account for the size effects, and four different

orientations with respect to the applied load were used to

investigate the effect **of** lattice orientation.

Biaxial deformation was applied to the cells **of** different

sizes and it was found that the initial yield stress and

strain hardening rate increased as the size decreased, although

for different reasons. The initial yield stress was

controlled by critical resolved shear stress for nucleation

**of** the sources sourrounding the **void** and this magnitude

increases by statistical reasons as the number **of** sources

decreases in smaller specimens. The size effect on strain

hardening was due to the combination **of** two different

mechanisms. Firstly, the **dislocation** density increased

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**An**ales de Mecánica de la Fractura 26, Vol. 1 (2009)

more rapidly as the crystal size decreased which resulted

in a higher strain hardening rate. Secondly, the total number

**of** **dislocation**s available in the smallest crystals was

limited by crystal size and source density, and the applied

strain has to be partially absorbed through elastic

deformation, leading to another contribution in the hardening

rate. The effect **of** **void** size on **void** **growth** rate was

also examined and a clear size effect **of** the type ”smaller

is slower” was found. This behavior was expected and

predicted by strain gradient plasticity models but it was

confirmed for the first time in these **dislocation** **dynamics**

simulations.

Respect to the effect **of** lattice orientation, crystals loaded

in uniaxial tension along the [001] and [¯112] directions

were stronger and presented higher **growth** rates than

those deformed in [¯110] and [¯111]. Low hardening and

**void** **growth** rates were found in orientations in which the

most active slip planes (high Schmidt factor) were in regions

where the applied stresses were high and had free

surfaces (that attract **dislocation**s) at both ends **of** the slip

planes. However, higher stresses has to be applied if the

most active slip planes have to pass above and below the

**void** (precisely the region where the applied stresses are

minimum) and ended at the upper or lower crystal surfaces,

where the image forces due to the imposed displacement

repel the **dislocation**s.

REFERENCES

[1] Van der Giessen E, Needleman A, Discrete **dislocation**

plasticity: A simple planar model, Modelling

and Simulation in Materials Science and Engineering

3, 689 (1995)

[2] I. Romero, J. Segurado, J. LLorca, Dislocation **dynamics**

in non-convex domains **using** finite elements

with embedded discontinuities, Modelling and Simulation

in Materials Science and Engineering 16

(2008) 035008.

[3] J. R. Rice, D. M. Tracey, On the ductile enlargement

**of** **void**s in triaxial stress fields, Journal **of** the Mechanics

and Physics **of** Solids 17 (1969) 201–217.

[4] A. L. Gurson, Continuum theory **of** ductile rupture

by continuum **void** nucleation and **growth**. part I:

yield criteria and flow rules for porous ductile media,

Journal **of** Engineering Materials and Technology

99 (1977) 2–15.

[5] M. J. Worswick, R. J. Pick, Void **growth** and constitutive

s**of**tening in a periodically **void**ed solid,

Journal **of** the Mechanics and Physics **of** Solids 38

(1990) 601–625.

[6] M. Gologanu, J. Leblond, G. Perrin, J. Devaux, Recent

extensions **of** gurson’s model for porous ductile

metals, in: Continuum Micromechanics. CISM

Courses and Lectures no. 377., Springer, 1997, pp.

61–130.

[7] G. P. Potirniche, J. L. Hearndon, M. F. Horstemeyer,

X. W. Ling, Lattice orientation effects on

**void** **growth** and coalescence in single crystals, International

Journal **of** Plasticity 22 (2006) 921–942.

[8] L. Briottet, H. Klocker, F. Montheillet, Damage in a

viscoplastic material Part I: cavity **growth**, International

Journal **of** Plasticity 12 (1996) 481–505.

[9] N. A. Fleck, G. M. Muller, M. F. Ashby, J. W.

Hutchinson, Strain gradient plasticity: Theory and

experiment, Acta Metallurgica et Materialia 42

(1994) 475–487.

[10] M. D. Uchic, D. M. Dimiduk, J. N. Florando,

W. D. Nix, Sample dimensions influence strength

and crystal plasticity, Science 305 (2004) 986–989.

[11] C. Motz. C, T. Schöberl, T, R. Pippan, Mechanical

properties **of** micro-sized copper bending beams

machined by the focused ion beam technique, Acta

Materialia 53 (2005) 4269–4279.

[12] J. Y. Shu, Scale-dependent deformation **of** porous

single crystals, International Journal **of** Plasticity 14

(1998) 1085–1107.

[13] B. Liu, Y. Huang, M. Li, K. C. Hwang, C. Liu, A

study **of** the **void** size effect based on the taylor **dislocation**

model, International Journal **of** Plasticity

21 (2005) 2107–2122.

[14] Z. Li, P. Steinmann, Rve-based studies on the coupled

effects **of** **void** size and **void** shape on yield behavior

and **void** **growth** at micron scales, International

Journal **of** Plasticity 22 (2006) 1195–1216.

[15] U. Borg, C. F. Niordson, J. W. Kysar, Size effects on

**void** **growth** in single crystals with distributes **void**s,

International Journal **of** Plasticity 24 (2008) 688–

701.

[16] M. Hussein, U. Borg, C. F. Niordson, V. S. Deshpande,

Plasticity size effects in **void**ed crystals,

Journal **of** the Mechanics and Physics **of** Solids 56

(2008) 114–131.

[17] J. Segurado, J. LLorca, I. Romero, Computational

issues in the simulation **of** two-dimensional **discrete**

**dislocation** **dynamics**, Modelling and Simulation in

Materials Science and Engineering 15 (2007) S361–

S375.

[18] V. S. Deshpande, A. Needleman, E. V. der

Giessen E, Plasticity size effects in tension and

compression **of** single crystals, Journal **of** the Mechanics

and Physics **of** Solids 53 (2005) 2661–2691.

[19] A. A. Benzerga, Y. Bréchet, A. Needleman,

E. V. der Giessen, Incorporating three-dimensional

mechanisms into two-dimensional **discrete** **dislocation**

**dynamics**, Modelling and Simulation in Materials

Science and Engineering 12 (2004) 159.

117