Atomistic simulation study on key factors dominating dislocation nucleation from a crack tip in two FCC materials: Cu and Al

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Atomistic simulation study on key factors dominating dislocation nucleation

from a crack tip in two FCC materials: Cu and Al

Y. Cheng

a,c

_{, M.X. Shi}

b,c

_{, Y.W. Zhang}

c,⇑

a

Shengli Geophysical Research Institute of SINOPEC, Dongying, Shandong 257000, China

b

School of Mechanics and Engineering, Southwest Jiaotong University, Chengdu, Sichuan 610031, China

c

Institute of High Performance Computing, A*STAR, Singapore 138632, Singapore

a r t i c l e

i n f o

Article history:

Received 30 December 2011 Received in revised form 4 June 2012 Available online 24 July 2012 Keywords:

Atomistic simulation Fracture

Brittle to ductile transition Dislocation nucleation

a b s t r a c t

Dislocation nucleations from crack tips in FCC copper and aluminum are studied using atomistic simula-tions. It is shown that the critical load for dislocation nucleation predicted by Rice’s model (Rice, 1992) based on the Peierls concept of dislocation can either be under- or over-estimated in reference to the sim-ulation results. Such discrepancies have not been fully resolved by existing improved nucleation models, due to the complicated atomic environments at crack tips. Based on our simulation results, it is proposed that such discrepancies can be reconciled by the competition of two coupling processes at a crack tip: the tension-shear coupling, which facilitates the dislocation nucleation, and the nucleation-debonding cou-pling, which retards the dislocation nucleation. In addition, the two couplings are applied to explain the paradoxical observation: easy dislocation nucleation at a blunted crack tip. The present work provides a detailed picture to justify future improvements on Rice’s model for dislocation nucleation and to accu-rately predict intrinsic brittle to ductile transition for crystalline materials.

1. Introduction

Description of a fracture process generally requires a multi-scale approach, which is capable of simultaneously dealing with nucleation and propagation of dislocations, and the initiation and extension of a crack. Fracture failure in ductile materials has been studied extensively by modelings, simulations and experiments at various length scales, see some examples in latest publications (Szczepinski, 1990; Zhou et al., 1997; Li and Chandra, 2003; Khan and Khraisheh, 2004; Ohashi, 2005; Potirniche et al., 2006; Gourgiotis and Georgiadis, 2008; Tarleton and Roberts, 2009; Song et al., 2010; Michot, 2011; Terentyev et al., 2012). Conventionally, separation of two surfaces is used as manifestation for an intrinsi-cally brittle material, while dislocation nucleation from a crack tip is used as manifestation for an intrinsically ductile material. There-fore, the understanding of dislocation nucleation from a crack tip is fundamental in determining the brittle to ductile transition of crys-talline materials.

A variety of continuum models had been proposed to describe dislocation nucleation from a crack tip (Armstrong, 1966; Kelly et al., 1967; Rice and Thomson, 1974; Mason, 1979; Anderson and Rice, 1986; Argon, 1987; Cheung et al., 1991; Rice, 1992; Sun et al., 1993; Rice and Beltz, 1994; Sun and Beltz, 1994; Xu et al.,

1995; Schoeck, 1996, 2003). Meanwhile experimental investiga-tions on dislocation generation and propagation along slip systems leading to material fracture had been carried out as well (Ohr, 1985; Chiao and Clarke, 1989; Wang and Anderson, 1991; George and Michot, 1993; Serbena and Roberts, 1994). Comparisons with continuum models were aimed at in these experimental studies, though usually only qualitative remarks could be made, and occa-sionally improvements were suggested on the theory models. Among all above mentioned continuum models, the theory model proposed by Rice (1992)has attracted great attentions since its publication. In Rice’s model, a dislocation at a crack tip was as-sumed to be described by the Peierls model (Peierls, 1940), and the stress intensity at the crack tip provides the driving force for dislocation nucleation. A dislocation is considered to have been nucleated once the slip barrier, that is, the unstable stacking fault energy, is overcome. According to the model, the energy release rate necessary for dislocation nucleation from a crack tip under Mode-I loading is

Gdisl¼ 8

c

usf½1 þ ð1

m

Þ tan2/=½ð1 þ cos hÞ sin

2

h; ð1Þ

where

c

usfis the unstable stacking fault energy,

m

is the Poisson’s

ra-tio, / and h are the angles as indicated inFig. 1(a). The slip plane of an incipient dislocation and the crack plane intersect each other at a line of n, namely the crack tip, which is also the line sense of the incipient dislocation. / is the complementary angle between n and the Burgers vector b of the incipient dislocation. The intrinsic

http://dx.doi.org/10.1016/j.ijsolstr.2012.07.007

⇑ Corresponding author. Tel.: +65 64191478; fax: +65 64674350. E-mail address:[email protected](Y.W. Zhang).

Contents lists available atSciVerse ScienceDirect

International Journal of Solids and Structures

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / i j s o l s t r

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fracture behavior of a crystalline material can be predicted by com-paring the value of Gcleav with the above Gdislfor candidate slip

sys-tems, where Gcleavis the energy release rate for cleavage. According to the Grifﬁth theory,

Gcleav¼ 2

c

s; ð2Þ

where

c

_sis the surface energy and the factor of 2 before

c

_saccounts for the contributions from two separated surfaces. A material is classiﬁed to be intrinsically brittle if Gcleav is less than Gdislbecause

cleavage is favorable in comparison to dislocation nucleation. Otherwise, the material is classiﬁed to be intrinsically ductile.

Since fracture at atomistic level ultimately requires the rear-rangement of atoms involving bond breakings and remakings (Zhang et al., 1995), atomistic simulations serve as an ideal tool to validate Rice’s model by capturing the detailed information of fracture process, and thus generating quantitative descriptions. Detailed comparisons have shown that although the above nucle-ation models are able to differentiate the intrinsic fracture proper-ties of materials in a general way (Rice, 1992; Sun et al., 1993; Rice and Beltz, 1994), often the quantitative agreement with existing atomistic simulation results has not been reached (Sun and Beltz, 1994; Zhou et al., 1994; Cleri et al., 1997; Knap and Sieradzki, 1999; Zhu et al., 2004).

A number of attempts have been made to improve Rice’s model with consideration of the following factors: (1) the creation of ex-tra surfaces (also called ledge formation or new surface produc-tion) during dislocation nucleation (Cheung et al., 1991; Zhou et al., 1994; Gumbsch, 1995; Gumbsch and Beltz, 1995; Xu et al., 1995; Schoeck, 1996; Schoeck, 2003), (2) the tension-shear cou-pling effect through which slip is facilitated by tension perpendic-ular to the slip plane of an incipient or nucleated dislocation (Sun et al., 1993; Xu et al., 1995; Schoeck, 1996; Schoeck, 2003; daSilva

et al., 2003), (3) the specific unstable stacking fault energy consid-ering the crack tip nonlinearity (Zhang et al., 1995; Cleri et al., 1997), and (4) the specific crack tip configuration and its corre-sponding stress field (Knap and Sieradzki, 1999). Research activi-ties on the first two factors will be discussed more in details next since they have gained more attentions among researchers of the field.

Because a pre-existing edge dislocation of width l was consid-ered in the study of dislocation nucleation from a crack tip in

a

-Fe (Cheung et al., 1991), a corresponding energy term Ulð Þ ¼l

c

sbl

was added to the whole free energy change to account for the ef-fect of new surface production. Both stress softening and ledge for-mation factors were included in this work though the investigation was carried out indirectly by introducing two equivalent stress softening functions based on EAM (

a

-Fe) potential. Based on a lat-tice Green’s function method (Thomson et al., 1992; Zhou et al., 1993),Zhou et al. (1994)tried to investigate the ledge formation effect by evaluating the force balance with major components in-cluded (major force components were judged according to obser-vations in their atomistic simulations). Surprisingly, they concluded that the criterion for brittle-to-ductile transition was independent of the surface energy. This conclusion is truly against the traditional understanding of energy competition in the brittle-to-ductile transition. Their ledge crack mechanism is liable to ques-tioning since it has not been proved to be the main dislocation emission mechanism (George and Michot, 1993).

Sun et al. (1993)worked on tension-shear coupling effect theo-retically almost right after the publication of Rice’s continuum model (Rice, 1992). In their efforts, a potential Wgoverning the decohesion and relative gliding between two atomic planes was proposed based on the universal binding law of metals (Rose et al., 1981). Then the energy potential controlling separation and shearing along slip planes as dislocations are being nucleated can be obtained, from which one can get the unstable stacking fault energies for dislocation nucleation on slip planes under different loading cases.Xu et al. (1995) pointed out that, predictions of unstable stacking fault energies given inSun et al. (1993) were overestimated for some sampled metals with speciﬁc slip systems. The reason is simply that the well-known Frenkel’s sinusoidal model (Frenkel, 1926) for shear stress against shear displacement along slip planes, which was employed in the work as the starting point, leads to a higher unstable stacking fault energy when com-pared to the simulation results from density functional theory (Kaxiras and Duesbery, 1993). In order to improve this drawback, a generalized sinusoidal model in the form of Fourier series (Xu

Nomenclature

b Burgers vector

i dummy index denoting the shear resistance mode in sinusoidal form

l dislocation width

n normal vector to crack front s distance to new surface corner

u displacement along slip direction normalized by Burgers vector

v opening displacement normalized by L C22 elastic modulus along y direction G general term for energy release rate Gappl energy release rate applied Gcleav energy release rate of cleavage

Gdisl energy release rate of dislocation nucleation L characteristic length of the decohesion process Ly dimension in y direction of simulation model

S speciﬁc surface energy from ledge formation Ul the free energy from ledge formation

e

yy applied strain along y direction

c

s surface energy

c

usf unstable stacking fault energy

k decay depth of shear resistance to ledge formation

m

Poisson’s ratio

/ complementary angle between crack front and Burgers vector

s

r shear resistance to slip

s

ðiÞ

r shear resistance to slip for the sinusoidal mode i

s

s shear resistance to ledge formation

h complementary angle between crack and slip planes n crack front or line sense of an incipient dislocation

W decohesion energy b slip plan e crack plane x y z [110]

(a)

(b)

Fig. 1. (a) Schematic of the geometries of the crack plane, the slip plane of a nucleated dislocation and the emitted dislocation. (b) Schematic of the simulation box containing a crack.

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et al., 1995),

s

rð Þ ¼u P1i¼1

s

ðiÞ

r sin 2ið

p

uÞ was suggested to describe

shear resistance to slip, where

s

ðiÞ

r is the undetermined coefﬁcient

for each resistance mode i and u is the displacement along slip direction normalized by Burgers vector (note: the deﬁnition of u remains the same throughout this paper, and this rule of deﬁnition consistency for all symbols is valid throughout the document as well). Furthermore, they postulated the shear resistance to ledge formation in excess of the shear resistance to slip alone to be

s

s¼kbcsexp ksb

1 cos 2ð

p

uÞ

½ , where the nondimensional

param-eter k denotes the decay depth of

s

sand s is the distance to the new

surface corner. In the end they arrived at the energy for pure glid-ing on slip plane of an incipient dislocation as

W

ðu; sÞ ¼ k

c

s u 1 2

p

sin 2ð

p

uÞ exp ks b : ð3Þ

Based on the formulation for dislocation nucleation from crack tips with the tension opening and ledge formation effects taken into ac-count, they studied dislocation nucleations from crack tips under mixed modes of loadings (Xu et al., 1995). It will be shown in the section of our simulation results that both the proﬁle feature and the amplitude of this gliding potential in Eq.(3)are in good agree-ment with our simulation results.

Another improvement on the model (Sun et al., 1993) was given bydaSilva et al. (2003)to take into account the ledge formation. In this effort, in addition to the basic energy functionW, one extra en-ergy term was added and it was proposed asDW¼ 2

c

s

p

u

v

aev,

where

v

is the normalized tension opening. The unknown parame-ter a was fitted through comparisons on shear and normal stresses between the model and atomistic simulations. This extra term was not justified well enough in the formulation and its weakness was revealed in the poor yielding of shear stress profile, when the amplitude of shear stress was focused on in the fitting process (daSilva et al., 2003).

Schoeck considered ledge formation and tension-shear coupling effects on dislocation emission from crack tips by taking variations of the free energy of whole structure (Schoeck, 1996, 2003), in which the limitation of employing Peierls dislocation model to for-mulate the shear displacement discontinuities and shear stress across slip plane of a nucleated dislocation was circumvented. However, trial functions such as arctan type had to be used in order to take advantage of Ritz method to solve the problem numerically. One energy term, ð

c

_s G=2Þuð0Þ inSchoeck (1996), [where G is the energy release rate and the plus-minus sign refers to situa-tions respectively of the emission or immission of atoms relatively between the two slip planes of nucleated dislocations], and uð0ÞnS inSchoeck (2003), [where n is normal vector to the crack front and S is speciﬁc surface energy for the situation of emission of atoms relatively between the two slip planes of nucleated dislocations], was added to the global free energy formulation respectively, based on which the variational technique was then employed for further investigations. This simple inclusion of ledge formation en-ergy by incorporating only the crack tip gliding displacement u 0ð Þ is equivalent to an addition of Heaviside energy function of the rel-ative gliding displacement, i.e. H u x½ ð Þ u 0ð Þ. Thus a singular force component can not be avoided in the corresponding equilibrium force equation (Zhou et al., 1994; Gumbsch and Beltz, 1995).

As investigations from ab initio simulations are concerned, cred-its should be given to Kaxiras’ research group for the continuous efforts to study generalized stacking fault energies and the brittle-to-ductile transition issue for selected materials (Juan and Kaxiras, 1996; Juan et al., 1996; Sun and Kaxiras, 1997; Lu et al., 2000). They incorporated the ﬁrst-principles study results into continuum model and then the analysis indicated that ledge for-mation would lead to a rise of critical load by up to 20% for Silicon.

Each of above mentioned studies has considered the effects from the four factors, especially the two tension softening and ledge formation factors, individually or combined in some way on dislocation nucleation from crack tips. However, to the best knowledge of the authors, there is still lacking a comprehensive understanding of all these factors in a quantitative sense, espe-cially on the first two factors which have been considered in most cases. In this paper we use atomistic simulations to study the dis-location nucleation process from crack tips with an aim to identify the main influential factors and to assess their contributions to dis-location nucleation. The atomistic models employed in our simula-tions will be introduced along with the simulation procedures in next section. Then, the simulation results will be presented and major observations will be discussed in reference of above cited publications. It is found in our simulations that Rice’s model can either under-estimate or over-estimate the critical loads for dislo-cation nucleation from a crack tip, depending on the detailed atomic configuration at the crack tip and loading conditions. In or-der to reconcile the differences in the predictions from Rice’s mod-el and from atomistic simulations, two competing factors dominating the nucleation process are suggested as: the tension-shear coupling and the nucleation-debonding coupling. In the end a brief summary is given.

2. Simulation setup

The embedded-atom method (Daw and Baskes, 1983, 1984; Foiles et al., 1986) with interatomic potentials for copper (Mishin et al., 2001) and aluminum (Mishin et al., 1999) are used in our simulations. The potentials are based on ab initio calculations at large strains and are proved to be in good consistency in predicting material fracture behavior and free surface energy (Foiles et al., 1986; Mishin et al., 2001; Cheng et al., 2010). Therefore they are suitable for the studies of dislocation nucleations and fractures. The size of the simulation box in the xy plane is approximately 400 400Å2 (Fig. 1(b)). The box size along the z axis is 3 3:615=pffiffiffi2Å, where periodic boundary conditions are imposed. The crack front direction is along the z axis. For the simulation model, one can refer toCheng et al. (2010)for details. For Mode-I loading, uniaxial simple tensile strain

e

yyis applied by

homoge-neously straining the simulation cell at each time step. Energy relaxation is carried out subsequently by constraining the motions of the atoms belonging to boundary layers in all three directions, except for the atoms in a few layers of the left and right surfaces which could move only in y and z directions. (Note: due to the Pois-son effect, when the simulation cell shown inFig. 1(b) is stretched uniformly in the vertical direction, the atom motion constraint on left and right surface layers will result in a distribution of horizon-tal normal stress. Since this stress is in-plane of the crack surface and perpendicular to the crack tip line, its inﬂuence on dislocation nucleation in the Mode-I loading circumstance may be neglected because there is no energy contribution from this stress. What is more, inﬂuence from the possible non-uniformity in the stress dis-tribution can also be ignored when the crack tip is prescribed far enough from the two boundary surfaces.) The applied energy re-lease rate is calculated as the stored elastic energy

Gappl¼

1 2C22

e

2

yyLy; ð4Þ

where C22is the elastic modulus along y direction, and Lyis the box

size in y direction excluding the length of those clapped boundary layers.

The relaxation method GLOC (Bitzek et al., 2006) is used after every strain step to allow the system to reach the minimum energy state. AtomEye (Li, 2003) is used to visualize the atomic structures,

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where atoms are colored by their central symmetric parameters. To form an initial crack tip conﬁguration, the whole system is ﬁrst homogeneously strained in y direction to about two thirds of the critical loading for either cleavage or dislocation nucleation. Then a certain number of planes of atoms around position Ly=2 are

re-moved to make a slightly blunted crack tip, which can avoid crack tip closure. Atoms are then first relaxed to form an elliptical crack tip configuration, whose long axis is about two fifths of the box size in x direction. After energy relaxation, the whole system can be loaded further for the potential dislocation nucleation or crack cleavage.

Highly accurate results have to be ensured in our simulations in order to carry out a quantitative analysis. First, owing to our load-ing scheme, the applied energy release rate is constant, which is independent of crack length as long as the crack tip is far away from the left and right boundaries. Thus this setup is suitable to study crack propagation. Second, to avoid the influences from the left and right boundaries, the initial crack needs to be kept at a rea-sonable length. That is to say, it should be small considering the computation efficiency, and large enough meanwhile to avoid the interaction of the crack tip with the left boundary. Third, the influ-ences from the top and bottom clapped boundaries are vanishing as the system size is large enough, which was discussed in details inBitzek (2006). Last, as the critical strains are less than 3%, the nonlinearity of elasticity can be ignored from the continuum mechanics point of view in our calculation, together with the vol-ume expansion of the system. Thus predictions from Rice’s model (Rice, 1992) can be reviewed in reference of our simulation results. 3. Results and discussions

3.1. Detailed nucleation process

We ﬁrst show a simulation case in details for dislocation nucle-ation from a crack tip. The crack plane in Cu is ð111Þ as shown in

Fig. 2(a,b). Here it is necessary to point out that, in all the figures thereafter portraying crack tip atomic configurations for simula-tion cases the material focused on is Cu, except clarificasimula-tion is otherwise made for the specific case. The nucleation is on the slip plane ð111Þ and the dislocation line is along ½110. According to the previous simulations (Schiøtz and Carlsson, 2000; Zhu et al., 2004), an atomically sharp crack could not exist stably, which either healed or emitted dislocations depending on the magnitude of loading. In the present case, the crack tip is made slightly blunt by removing one layer of atoms on the crack plane. The nucleated dislocation has a pure edge characteristic and the Burgers vector is 1=6½112. This dislocation can be viewed as the leading partial dis-location of a full Burgers vector of 1=2½101, where the trailing par-tial dislocation is 1=6½211 and makes an angle of 30 with its dislocation line. Regarding the role of unstable stacking fault energy on nucleation of leading and trailing dislocations, a good discussion was given in Froseth et al. (2004), though in which

the nanocrystalline grain boundaries were the context for disloca-tion nucleadisloca-tion.

The critical configuration of the embryo of the dislocation being nucleated is shown inFig. 3(a). The embryo of the dislocation is de-picted by the Burgers vector density, which is the derivative of rel-ative displacement of atoms across slip plane of the nucleated dislocation. The constrained bvd (Burgers vector density) and dis-placement curves are referenced to Rice’s model (Rice, 1992). Though it is not shown inFig. 3, we have noticed that the maxi-mum slip displacement of the dislocation embryo (0.54Å) is still less than half of the magnitude of the Burgers vector of the lattice dislocation (0.74Å). The dislocation embryo grows nonlinearly with the loading increasing, but never reaches the magnitude of the Burgers vector density of the lattice dislocation. When the en-ergy barrier of the slip is overcome, there is a sudden increase in both the slip displacement and the Burgers vector density of the nucleated dislocation. Besides, the configuration of the dislocation embryo agrees well with the lattice dislocation independently ob-tained using the same potential, but disagrees with the constrained Burgers vector density (Rice, 1992), which is based on the assump-tion that atoms are homogeneously distributed along the slip direction. What is more,Fig. 3 also shows that the displacement profiles from our simulations and from Rice’s model have a large difference near crack tip but this discrepancy gradually decreases when departing from the tip along slip direction.

A quantitative result about the critical dislocation embryo is the tension opening as shown inFig. 3(b). The tension opening is the stretch of atomic bonds perpendicular to the slip plane of the nucleated dislocation compared to the perfect lattice bond. At the critical conﬁguration, it reaches maximum (approximately 0.12Å as shown inFig. 3(b)) at the immediate crack tip and de-creases to zero smoothly and monotonically along slip direction. This sharp decaying of decohesion proﬁle has also been revealed in similar studies (Gumbsch and Beltz, 1995; Xu et al., 1995). We will show that the stretch of bonds facilitates the nucleation greatly, resulting in a lower critical load for dislocation nucleation as discussed in the following sections.

The applied energy release rate for dislocation nucleation from this crack tip is higher than the prediction from Rice’s model. We attribute this to the coupling between dislocation nucleation and bond breaking (or debonding) at the crack tip. We will show that the debonding which occurs during dislocation nucleation hinders the dislocation nucleation, resulting in a higher critical load for dis-location nucleation.

3.2. Nucleation: coupled mechanisms

We have conducted a series of simulations with various crack orientations. The term variation of crack orientations means that two quantities are possibly changed among simulation cases: ﬁrst, the change of crack plane, and therefore the corresponding cleavage energy; second, the change of the angle between the slip

(a)

[ 1 2]

1 [1

1 2]

[1

1

1 ]

(1 1 0)

(b)

Fig. 2. Crack tip conﬁgurations immediately before (a) and after (b) dislocation nucleation from the crack tip. The crack plane is ð111Þ. Atoms marked by arrows get separated during dislocation nucleation.

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direction and the Mode-I loading direction, and therefore the ten-sion opening during dislocation nucleation. Three typical simula-tion cases are shown in sequence inFig. 4(a–f). The slip planes of nucleated dislocations and the corresponding Burgers vectors in pair for the three cases are: ð111Þ and 1=6½112; ð111Þ and 1=6½112; ð111Þ and 1=6½112. For each case the conﬁgurations right before and after dislocation nucleation are portrayed respectively in the left and right snapshots of the crack tip zone. For a crack with prescribed orientation (i.e. the normal of crack plane) under Mode-I loading, generally speaking, the dislocation nucleation pro-cess and characteristics are similar to one of the three cases shown

in Fig. 4. The details of all the simulation cases will not be pre-sented here one by one to save space. Instead, the major results are summarized inTable 1for all the simulation cases with tilt axis ﬁxed at ½110 as the crack tip line. In the table, for all the crystal ori-entations, h representing the slip plane of a nucleated dislocation or the cleavage plane, elastic modulus C22 and surface energy

c

s

are given in the ﬁrst three rows of data in sequence. Next, for all the dislocation nucleation or cleavage system of crack and slip planes, energy release rate for cleavage Gcleav (based on Eq.(2)) and the one for dislocation nucleation Gdisl(based on Eq.(1)),the

applied energy release rate Gappl (based on Eq. (4)), the ratios

Fig. 3. (a) The conﬁguration of the dislocation embryo inFig. 2(a) is indicated by both the Burgers vector density (bvd) and the relative shear displacement of atoms across the slip plane (black lines) of the incipient dislocation. The red curves are the predictions from Rice’s model. The blue curves are the lattice dislocation calculated using the same atomic potential. (b) The tension opening obtained at the crack tip immediately before nucleation. (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.)

[1

1 2

]

[0 0 ]

1 [ 1 0]

1 [1

1 2]

(1 1 0)

(b)

(a)

(c)

[ 1 2]

1 [1

1 4]

[2

2

1 ]

[ 1 ]

1

2 (1 1 0)

(d)

(e)

[1

1 2]

[2

2 1]

[1

1

4 ]

[ 1 2]

1 (1 1 0)

(f)

Fig. 4. Crack tip conﬁgurations immediately before and after dislocation nucleation from crack tips. The crack planes are ð110Þ (a,b), ð221Þ (c,d) and ð114Þ (e,f). Atoms marked by arrows get separated during dislocation nucleation.

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Gappl=Gdisland Gcleav=Gdisl, are all listed out then. In the bottom row

of data are the tension openings of all the cases. Among all the sim-ulation cases, one crack cleavage case in FCC Cu is observed shown in the third data column. For the last two column data in the table the FCC material is shifted to Al. It might be necessary to note that those data of stiffness C22and

c

s are obtained separately through

speciﬁcally designed simulation models. These quantitative results listed out in the table and the conﬁguration change right before and after dislocation nucleation from a crack tip provide a basis for us to analyze the major factors dominating the dislocation nucleation from cracks. Effects of these factors on dislocation nucleation will be discussed further through comparison with the predictions from Rice’s model.

The ratios between the applied and the predicted critical loads for dislocation nucleation are plotted inFig. 5for all the cases listed out in the table, as a function of tension-shear coupling, which is represented by the crack tip tension opening (labeled as ‘‘opening’’ in the ﬁgure), and nucleation-debonding coupling, which is de-scribed by Gcleav=Gdisl. It can be seen that the critical load can be

50% higher or lower than that predicted by Rice’s nucleation mod-el, depending on the combined effects of tension-shear coupling and nucleation-debonding coupling. It is shown that, with the ten-sion opening increasing, the applied critical load is lower than that predicted by Rice’s model. Due to the nucleation-debonding cou-pling, the relative shearing deformation throughout dislocation nucleation process leads to atomic debonding, i.e. forming ledge. As a result the debonding of atoms hinders dislocation nucleation as extra energy is required to overcome the bond breaking or

equivalently to produce a new surface. With the two couplings combined in dislocation nucleations, the critical applied loads for bond breaking accompanying dislocation nucleation are all lower than the Grifﬁth loads. Their ratios range from 0.50 to 0.87, as shown inTable 1.1_{Next, the effects on dislocation nucleation from} the two couplings and from crack tip blunting will be discussed fur-ther respectively.

3.2.1. Tension-shear coupling

A tension perpendicular to the slip plane of a nucleated disloca-tion usually exists at a crack tip, since the slip plane is generally in-clined to the tension loading direction. At the crack tip, the tension causes an extra opening or stretching of atomic bonds perpendicu-lar to the slip plane. In most cases, the opening facilitates the dis-location nucleation at the crack tip by effectively reducing the shear resistance and thus brings down the unstable stacking fault energy. This is known as the tension-shear coupling. The effect is also called tension softening. It is closely related to the true tri-axi-ality state of local stress right surrounding the crack tip.

The tension softening effect is further evaluated in the absence of a crack tip environment. The unstable stacking fault energies in both Cu and Al with various tension openings are focused on. It is

0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 0.1 0.15 0.2 0.25 0.3 0.35 0.6 0.8 1 1.2 1.4 1.6 Gcleav/Gdisl opening (Å ) a ppl /Gdisl G ppl

Fig. 5. The abscissa ‘‘opening’’ means tension opening. Gappl=Gdislas a function of both tension opening and Gcleav=Gdisl. It decreases with both an increasing tension opening

and a decreasing Gcleav=Gdisl.

Table 1

Elastic and energetic properties of crack tips with various orientations. For all directions of y; z is the fixed tilt axis ½110. The superscript b in the third data column of y axis denotes brittle behavior (cleavage) of the crack. The superscripts Al in last two data columns of y axes means that simulation material has shifted from Cu shown in the first eight data columns to Al in the last two study cases. For all the cases, in addition to Gcleavand Gappl, predictions based on Rice’s model are also included in the fifth row of data (indicated

by the label Gdisl). The ratios of Gappland Gcleavto Gdislare calculated and listed out then. In the last row of data the tension openings are also given for all the cases. The unstable

stacking fault energy is 0.158 J/m2

for Cu, and 0.168 J/m2

for Al. y ½110 ½221 ½001b ½332 ½111 ½112 ½113 ½114 ½110Al ½111Al h(degrees) 35.26 54.74 54.74 60.50 70.53 90.00 100.02 105.79 35.26 70.53 C22(GPa) 221 231 170 235 239 222 202 191 119 121 cs(J/m2) 1.475 1.411 1.345 1.360 1.239 1.434 1.472 1.466 1.006 0.870 Gcleav(J/m2) 2.950 2.822 2.690 2.720 2.478 2.868 2.944 2.932 2.012 1.740 Gdisl(J/m2) 2.087 1.202 3.606 1.118 1.067 1.264 1.578 1.876 2.220 1.134 Gappl(J/m2) 2.09 1.41 2.70 1.37 1.41 2.00 2.30 2.55 1.37 1.00 Gappl=Gdisl 1.001 1.173 0.749 1.225 1.320 1.582 1.458 1.360 0.617 0.882 Gcleav=Gdisl 1.414 2.348 0.746 2.433 2.322 2.269 1.866 1.570 0.906 1.534 opening (Å) 0.278 0.178 0.517 0.155 0.127 0.058 0.070 0.091 0.326 0.154 1

For the simulation with crack surface of ð112Þ, the position of dislocation nucleation is one layer behind the crack tip, where the nucleation does not need to break atomic bonds. This nucleation is suppressed by introducing a pre-existing stacking fault there so that dislocation nucleation at the immediate crack tip is ensured.

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achieved in two steps: first, atomic bonds crossing the slip plane are stretched, which is shown as the interface between two colored atom blocks in the insets ofFig. 6; then the length of the stretched atomic bonds is fixed while relative gliding between the two blocks is performed. The results of unstable stacking fault energies corre-sponding to monotonically increasing tension openings are shown inFig. 6(a). From the figure we can see that for both materials the slip is 4 times easier with an opening of approximately 0.3Å. This clearly demonstrates that tension opening plays a significant role in easing the dislocation nucleation. In our simulations for Al, the applied loads for dislocation nucleation are indeed much lower than the predicted values by Rice’s model. Due to the tension soft-ening effect, the predicted brittle behavior has been changed to ductile dislocation nucleation. In our simulations for Cu, however, the applied critical loads for dislocation nucleation are all higher than the predicted values by Rice’s model, and the ratios are in the range from 1.001 to 1.582. The ratio is closer to 1.0 with a high-er tension opening, as shown inFig. 5, indicating the facilitating ef-fect arising from the tension-shear coupling. But the ratio is always larger than 1.0, indicating that there must be other factors that cause difficulties for dislocation nucleation. We attribute this to nucleation-debonding coupling. Though tension opening is ele-gantly taken into account in the model (Xu et al., 1995), however, no unstable stacking fault energies can be found in the publication for different openings. Therefore, unfortunately, we can not judge specifically on their formulation of the tension-shear coupling ef-fect on dislocation nucleation.

Before we move onto the nucleation-debonding coupling to be discussed in next section, here we discuss a little further on the tension-shear coupling effect on material behavior. Though a vari-ety of studies on material behaviors have shown that tension can effectively reduce the shear resistance when dislocation nucle-ations are concerned from crack tips (Sun et al., 1993; Xu et al., 1995; daSilva et al., 2003) or from bi-crystalline interfaces (Spearot et al., 2007), it is always praisable to cautiously arrive at any con-clusions. A discussion is to be given next on advances of tension ef-fects on shear performance for Cu and Al based on a recent publication. The ﬁrst-principles study conducted byOgata et al. (2002) showed that, when pressurized in h110i directions, both Cu and Al got hardened in shearing behavior. However, when they were pressurized along the h111i directions, shearing behavior in Al would get hardened substantially while Cu showed slightly the opposite effect, i.e. softening. In other words, their observations imply that tension opening along one direction may give rise to shear hardening behavior along the perpendicular direction in Cu. This effect is contradictory to physical instinct and generally is very weak. Luckily, for our loading cases of Cu, no available

evidences show a shear hardening effect from tension opening. But, even if this tension-opening-give-rise-to-shear-hardening phenomenon occurred in the orientations of Cu in our simulation cases, tension opening would play a minor role in increasing the unstable stacking fault energy since the shear hardening from ten-sion was shown rather weak (Ogata et al., 2002). The ledge forma-tion normally plays an important role in increasing the unstable stacking fault energy by the new surface production resistance. This role would become more eminent if the abnormal phenome-non happened in some speciﬁc orientations. Anyway, it may need further ab initio investigations for Cu on this tension to shear either softening or hardening effects.

3.2.2. Nucleation-debonding coupling

In all the dislocation nucleation cases of Cu shown inFigs. 2 and 4and others listed out inTable 1, Rice’s model underestimates the critical load. To explain the underestimation, the idea of ledge for-mation is suggested. It is not a new concept and obviously it should not be restricted to Cu only. Based on the fact that a ledge of sur-face is created accompanying dislocation nucleation, extra energy is needed for this realization. Thus it would be physically reason-able to postulate that this extra energy works as a surface resis-tance role to restrict dislocation nucleation. However, since the ledge area at atomic scale is poorly deﬁned, hence it is difﬁcult to quantify the ledge formation effect by relating the surface en-ergy to the unstable stacking fault enen-ergy directly.

The coupling of dislocation nucleation with debonding has al-ready attracted attentions from quite a number of researchers in the ﬁeld recently (Zhou et al., 1994; Gumbsch and Beltz, 1995; Xu et al., 1995; Shastry and Anderson, 1997; Schoeck, 2003). In our simulations of Cu, the values of Gcleavare all much higher than Gdisl. In such scenarios, atom bonds will not break when Gdislis

ful-ﬁlled because Gcleavis not yet reached. Therefore, dislocation nucle-ation will be inhibited evidently as a result of the critical load being raised by bonds breaking. Based on above analysis, it may be more precise to use the nucleation-debonding coupling than the ledge formation in phrasing the situation for Cu, though usually the two terms are not so strictly distinguished and are rather used interchangeably.

It is shown inFig. 5that when Gcleavis closer to Gdisl, the

dislo-cation nucleation can be achieved easier. In such cases the applied critical load for dislocation nucleation Gapplis shown to be closer to

the theoretical load Gdisl. Based on this analysis, the debonding

inﬂuence can be evaluated using the ratio of Gcleavto Gdisl. Since this

coupling co-exists with the tension-shear coupling at a crack tip, it is truly difﬁcult to separate the inﬂuences from the two couplings. We have to show its effect together with the tension-shear

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Opening in [ 11]1 Al Cu us f (J/m ) 2

(a)

[ 1 ]1 2 [ 1 1]1 (Å) 0 0.5 1 1.5 2 0 1 2 3 4 5 Al Cu

(b)

Le dge formatio ne n e rg y( J/ m ) 2 [ 1 ]1 2 [ 1 1]1 1 2 3 (Å) Distance of affecting atoms

Fig. 6. (a) The unstable stacking fault energy as a function of extra tension opening relative to equilibrium state between the two colored blocks. The two isolated points are the results with atomic relaxation perpendicular to the slip plane. (b) The effect of the debonding between the yellow and the light grey atoms on the constrained slip. The distance is taken as zero when the light grey atoms are in an equilibrium stacking with the black atoms. (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.)

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coupling. But we can still make the nucleation-debonding coupling effect prominent as follows: with a similar tension opening, a smaller Gcleav=Gdisl in Al results in a much easier nucleation than

that in Cu, as shown inFig. 5. From the ﬁgure we can also see that, the applied load for dislocation nucleation can be much smaller than that theoretically predicted, if there is a large tension opening and Gcleavis close to Gdisl.

In absence of both tension-shear coupling and nucleation-deb-onding coupling, the applied load for dislocation nucleation will be predicted exactly by Rice’s nucleation model. Dislocation nucle-ation from a crack tip under Mode-II loading was shown as such an example (Zhou et al., 1994; Knap and Sieradzki, 1999). This is also evident in our simulations of constrained slips. As shown in

Fig. 6(b), there are three planes labeled by numbers 1; 2 and 3, respectively. The calculated unstable stacking fault energies on planes 1 and 2 using free boundary conditions in ½112 are the same as the one with periodic boundary conditions. In the case of plane 3, however, it is much higher where the nucleation-debonding coupling exists, because the slip needs bond breaking at the left surface.

To further quantify the influence of bond breaking on disloca-tion nucleadisloca-tion, addidisloca-tional simuladisloca-tions have been performed. One extra layer of atoms, that is, the grey atoms inFig. 6(b), are added to influence the constrained slip of the upper yellow atoms relative to the below black atoms. The distance is varying between the added atoms and the main body of black atoms, which is taken as zero at the equilibrium distance. With such added atoms, extra energy is needed to perform the slip between the yellow and the black atoms, in addition to the slip energy. The newly created sur-face area is the product of the length in the crack front direction with the slipped Burgers vector. This area is only an approximation since area is poorly defined at the atomic scale.Fig. 6(b) shows the extra energy per area of newly created surface as a function of the influencing distance. It is seen clearly that the debonding with the added atoms significantly increases the slip energy. The major influencing factor is the surface energy, which is one order of mag-nitude higher than the unstable stacking fault energy.

When we carefully examine the curve of ledge formation en-ergy along slip direction as portrayed inFig. 6(b), we can see that the energy function given by Eq.(3)(Xu et al., 1995) indeed cap-tures the major variation feature, i.e. gradual rising followed by a rapid decaying. When we come to estimate the amplitude of ledge

formation energy, we can set s 0 (since very close to crack tip) and u 1=2 (equivalently slip displacement b=2) in Eq.(3), then

W=

c

s k=2. This indicates that the ledge formation energy is on the

order of surface energy

c

s.Table 1shows that for Cu its surface

en-ergy

c

_sis approximately one order higher than the unstable stack-ing fault energy

c

usf. As a result the ledge formation energy can be

estimated as one order higher than

c

usf. This judgement agrees very

well with the observation of relative amplitudes of

c

_usf and ledge formation energy shown inFig. 6(a,b) respectively.

Furthermore, as effect of ledge formation on dislocation nucleation is concerned, Xu et al., 1995 showed that when 2

c

s=

c

usf¼ 10, the critical load for dislocation nucleation along a

constrained path was changed by approximately up to 50% when ledge formation was considered (in the sample k ¼ 1:0 was set for a representative study case on the factor). Though they focused on material

a

-Fe with a prescribed slip direction 45, this may serve as an example to justify our earlier remark when trying to understand the atomistic simulations on Cu that the ledge forma-tion can lead to about 50% increase of the critical load predicted based on Rice’s model.

3.3. Blunting of crack tips

Besides the effects from the two couplings discussed in above two sub-sections, the effect on dislocation nucleation from blunt-ing of a crack tip has also been studied speciﬁcally, though crack tip blunting can be viewed as the direct result of ledge formation accompanying the dislocation nucleations. Conventionally, it is generally believed that as the crack tip in materials becomes blunted, the critical load for fracture failure should increase. We will show that crack tip blunting can lower the critical load for the nucleation of dislocations.

The crack tip inFig. 2(a) is further blunted by removing another layer of atoms. Two conﬁgurations of the crack tip, as shown in

Fig. 7(a,b) and (c,d), are then obtained by different removals of atoms. For the former, there is no difference with the sharper crack tip (Fig. 2(a,b)) in both critical load and tension opening. For the latter, however, the critical tension opening is 0.061Å, which is about only one half of the former. In addition, the initial distance between the two atoms that are debonded in the latter case (marked by arrows inFig. 7(c,d)) is 3.00Å, which is smaller than that in the former case (marked by arrows inFig. 7Å. Therefore, a

(a)

[ 1 2]

1 [1

1 2]

[1

1

1 ]

(1 1 0)

(b)

(c)

[ 1 2]

1 [1

1 2]

[1

1

1 ]

(1 1 0)

(d)

Fig. 7. Crack tip conﬁgurations immediately before and after dislocation nucleation from two differently blunted crack tips (a,b and c,d), comparing toFig. 2(a,b). The crack plane is ð111Þ. Atoms marked by arrows get separated during dislocation nucleation.

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higher critical load would be expected for the latter case. However, this is not the real situation: the critical energy release rate for dis-location emission of the latter is only 85% of the former. To unveil the underlying reason, we further checked the atomic structures of the crack tips. ComparingFig. 7(c) withFig. 2(a), it is found that the atom indicated by the lower arrow is out of direct contact with its left neighbors after blunting, and therefore it is easier to move, thus facilitating the development of the dislocation front due to the removal of nucleation-debonding coupling. The same mecha-nism is also found in the case with the crack plane of ð221Þ. These atomistic simulations provide a detailed mechanism for contin-uum analysis of ductile versus brittle transition (Beltz et al., 1999). It is necessary to note that a different explanation for the critical load reduction has been made by measuring the stress ﬁeld of the blunted crack tip (Schiøtz and Carlsson, 2000).

Among simulations on different crack plane orientations, a brit-tle crack propagation is observed when the crack plane is set as ð001Þ, as shown inFig. 8(a,b). Its initial crack tip conﬁguration is made by removing one layer of atoms on the crack plane. Although simulations show it is brittle cleavage, the magnitude of the slip displacement of the dislocation embryo is relatively high right be-fore cleavage, which is almost half of the Burgers vector of a lattice dislocation (i.e. approximately 0:37Å). The tension opening is also large, 0.52Å. As the crack tip is further blunted by removing an-other layer of atoms as shown inFig. 8(c), it is not brittle any more, and a dislocation is observed being nucleated from the blunted crack tip as shown inFig. 8(d). A 60_{mixed dislocation is}

geomet-rically preferred to be nucleated, different from all other cases ana-lyzed above. Since the tension opening is only 0.40Å for this case, the major resistance to dislocation nucleation is from the nucle-ation-debonding coupling.

4. Conclusion

For an intrinsically ductile material, dislocation nucleation from a crack tip occurs earlier than the crack cleavage. On one hand, the breaking of atomic bond(s), which always accompanies dislocation nucleation, retards dislocation nucleation. This is due to the nucle-ation-debonding coupling effect. On the other hand, the opening of the slip plane of nucleated dislocations at a crack tip always facil-itates dislocation nucleation and bonds breaking. This is from the tension-shear coupling effect. The two couplings always co-exist

in the complicated crack tip environment and they have direct influences on dislocation nucleations from a crack tip in contradic-tory manners. The final critical load for dislocation nucleation is controlled by both (and even other factors), which explains the un-der- or over-estimation of Rice’s model when against our atomistic simulation predictions. Therefore, an improved theoretical model for dislocation nucleation process should take a comprehensive consideration of both coupling effects, far beyond the ideal Peierls model of dislocation nucleation. Our atomistic simulations provide detailed information necessary for clarifying the mechanism of individual factor dominating the dislocation nucleation at a crack tip. An improved theory model will be our next effort to work on in this field.

Acknowledgements

The work was supported by the A*STAR Visiting Investigator Program ‘‘Size Effects in Small Scale Materials’’ hosted at the Institute of High Performance Computing in Singapore under the academic supervising by Professor Huajian Gao. The author YC is grateful to discussions with Huajian Gao and Peter Gumbsch. References

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