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

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

Anales 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 voided 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 voids.

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 voids 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 voids 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

voids are not available. As a result, the accuracy

of these quantitative predictions remains unknown.

Another solution for this limitation can be obtained from

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

voided 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 voids 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 dislocations, 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, 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 projection of the Peach-Koehler

force acting on the dislocation i in the sliding direction

and B is the drag coefficient. Moreover, two dislocations

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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Anales de Mecánica de la Fractura 26, Vol. 1 (2009)


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 dislocations,

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 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 [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 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 + δt. 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 given by

(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 . 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 voids

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 voided crystals. Biaxial deformation (ɛ 11 =

ɛ 22 ).

The crystals were initially free of dislocations 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 dislocations

decreases with size as the source density is constant,

and it is amplified by the inhomogeneous stress

state within the voided crystal because the maximum resolved

shear stresses on the slip planes are localized in a

small region sourrounding the voids. 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 dislocations 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 dislocations

and generate a back stress which also reduces the activity

of the sources, and both factors were responsible for

the strain hardening of the voided 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 dislocations, 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 voided crystals subjected to biaxial deformation, L

= 2.5 µm. Green and red crosses stand for dislocations

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 dislocations 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 voided crystals

deformed under uniaxial tension are plotted in Fig. 5.

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Anales de Mecánica de la Fractura 26, Vol. 1 (2009)

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 voided 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 dislocations) 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 dislocations.

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 voids grew slower in the

softest 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 dislocations 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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Anales 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 dislocations 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 dislocations) 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 dislocations.

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