a fatigue crack initiation model incorporating discrete...

Int J Fract (2007) 148:155–167DOI 10.1007/s10704-008-9190-x

ORIGINAL PAPER

A fatigue crack initiation model incorporating discrete dislocationplasticity and surface roughness

Steffen Brinckmann · Erik Van der Giessen

Received: 15 April 2007 / Accepted: 23 February 2008 / Published online: 26 March 2008© The Author(s) 2008

Abstract Although a thorough understanding offatigue crack initiation is lacking, experiments haveshown that the evolution of distinct dislocation distribu-tions and surface roughness are key ingredients. In thepresent study we introduce a computational frameworkthat ties together dislocation dynamics, the fields dueto crystallographic surface steps and cohesive surfacesto model near-atomic separation leading to fracture.Cyclic tension–compression simulations are carried outwhere a single plastically deforming grain at a freesurface is surrounded by elastic material. While ini-tially, the cycle-by-cycle maximum cohesive openingincreases slowly, the growth rate at some instantincreases rapidly, leading to fatigue crack initiation atthe free surface and subsequent growth into the crys-tal. This study also sheds light on random local micro-structural events which lead to premature fatigue crackinitiation.

Keywords Dislocation · Cohesive surface · Surfaceroughness · Fatigue initiation

S. Brinckmann · E. Van der GiessenZernike Institute for Advanced Materials, Universityof Groningen, Groningen, The Netherlands

S. Brinckmann (B)Department of Material Science, California Instituteof Technology, MC 308-81, Pasadena, CA 91125-8100,USAe-mail: [email protected]

1 Introduction

Even though fatigue can be considered a long-standingproblem, numerous experimental studies are being car-ried out even to date to explore fatigue crack initiation,predominantly in polycrystalline metals (e.g. Narasaiahand Ray 2008; Sackett et al. 2007; Polák 2007). Whilethey have identified certain key building blocks offatigue in fcc materials, their connection and interac-tion is not well understood. Mughrabi et al. (1979)and other researchers subsequently, have shown thatduring cyclic plastic deformation dislocations multiplyand form structures in the bulk of the material. Brownand Ogin (1984) actually attribute crack initiation tothe formation of a characteristic dislocation structurewith a logarithmic singularity at the free surface. Asthe dislocations escape the material, they leave behindcrystallographic surface steps (see, e.g., Vehoff 1994).Their accumulation leads to surface roughness whichplays a vital role in crack initiation; repetitive removalof the roughness has been found to lead to a consider-able increase in fatigue life (Hahn and Duquette 1978).On the other hand, Basinski and Basinski (1985) haveobserved fatigue in the absence of surface roughness.

Continuum models of fatigue crack initiation(e.g. Kratochvil 2001; Fine and Bhat 2007), by con-struction, cannot capture discrete microstructural events.Models that do include discrete crystalline events are,so far, mostly based on presumed dislocation structures(e.g. Antonopoulos et al. 1976; Essmann et al. 1981;Brown and Ogin 1984) or specific dislocation slip events

123

156 S. Brinckmann, E. Van der Giessen

(e.g. Neumann 1969). Based on the work by Essmannet al. (1981), Repetto and Ortiz (1997) predicted thegrowth of a surface protrusion by employing a contin-uum model and attributed crack initiation to this evolu-tion. One study where dislocation structures were notpresumed but were the outcome of discrete dislocationsimulations is that by Déprés et al. (2004) of a sin-gle ultra-fine grain with rigid grain-boundaries. In thisstudy also the surface roughness has been determined,but the possibility of crack initiation was not includednor the dependence of the surface steps on the disloca-tion behavior. To the best of the authors’ knowledge,no model is available at this moment that accounts forthe evolution of the discrete surface roughness and ofthe dislocation distribution.

The present article introduces a computationalframework that consists of three parts. The first, viz. thecohesive surface model of Xu and Needleman (1994),models the separation of atomic planes. The secondpart is an approximate elastic field due to the crys-tallographic surface roughness. Finally, a dislocationdynamics model represents plastic flow and incorpo-rates the long-range elastic fields due to the dislocationdistribution.

2 Model

2.1 Superposition

We model a sample of a plane strain specimen withan initially flat traction-free surface and the rest of itsboundary subjected to either a prescribed displacementu0 or traction t0. As illustrated in Fig. 1a, inside thisregion the model accounts for (i) plastic deformationby the motion of discrete dislocations, (ii) the pres-ence of cohesive surfaces and (iii) the development ofsurface roughness caused by dislocations exiting thematerial through the free surface. These three ingredi-ents are discussed in more detail subsequently and areintegrated by exploiting superposition.

The first ingredient of the approach, see Fig. 1b,is the two-dimensional dislocation dynamics model ofVan der Giessen and Needleman (1995), as outlined inthe following subsection. At each instant of time thispart provides the ( ˜) fields due to the instantaneousdistribution of dislocations.

The second part includes in an approximate waythe modification of the fields in the neighbourhood of

crystallographic surface steps. These correction fields,( ¯), are derived in Sect. 2.3 from the elastic solution ofa wedge.

Finally, the cohesive surface and the boundary con-ditions are incorporated through the third part, sketchedin Fig. 1d, whose solution is indicated by ( ˆ). A cohe-sive surface law is assumed with near-atomic proper-ties which is reversible, i.e. we neglect oxidation and allother processes which alter the evolving crack surfaceand prevent it from healing.

Since each of the three fields pertain to a linear elas-tic material, we can use superposition to express thetotal stress and displacement fields as

σ = σ + σ + σ , u = u + u + u. (1)

In Van der Giessen and Needleman (1995), the ( ˆ)fields correct the ( ˜) fields of the dislocation distri-bution for the boundary conditions; here, the ( ˆ) fieldsalso correct for the ( ¯) fields of the crystallographicsurface steps. However, the non-surface boundariesSext

are sufficiently far away that the ( ¯) fields can beneglected in correcting for the prescribed displacementsor tractions. Even though superposition can be appliedinside the elastic solid, the third subproblem is nonlin-ear because of the cohesive law. As a consequence, the( ˆ) solution is obtained in an incremental manner, asoutlined in Sect. 2.4.

As mentioned before, we assume small strains, asin Van der Giessen and Needleman (1995). Recently,Deshpande et al. (2003) have introduced a finite strainformulation for discrete dislocation plasticity, but thecombination with cohesive surfaces awaits implemen-tation.

2.2 Dislocation dynamics

Subproblem (a) in Fig. 1 pertains to a semi-infinite,two-dimensional strip containing a single grain near itsfree surface in which plasticity takes place; neighbor-ing grains in the polycrystalline material are assumedto be oriented such that no plastic flow takes place inthem, see Fig. 2. The rectangular grain has three slipsystems at 60◦ from each other (as a two-dimensionalrepresentation of an fcc crystal), one being favorablyoriented for slip (the so-called primary slip system) at45◦ from the remote tensile direction, which is parallelto the free surface.

Plastic flow inside the grain originates from themotion of discrete dislocations. Consistent with the

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A fatigue crack initiation model 157

Fig. 1 The proposedframework (a) is acompilation of a dislocationmodel (b), acrystallographic surfacestep model (c) and a modelto incorporate the cohesivesurface and the boundaryconditions (d)

Scoh

Sext

0u 0tu0 0tu0 0t

uu

~ ~

++

(a) (b) (d)(c)

~= u u

free surface45

grain boundary

obstacle

dislocation Frank−Read source

loading direction

Fig. 2 Half-infinite strip of material with a single grain insideof which dislocation dynamics is applied; the surrounding mate-rial is elastic. The grain has three slip systems, one of which isoriented for maximal shear at 45◦ from the free surface

plane strain condition, all dislocations are of edge char-acter with the Burgers vector in the plane of the modeland of length b. They are treated as singularities in a lin-ear elastic, isotropic continuum. Closed-form expres-sions are used for the long-range fields in the presenceof a traction-free surface (Freund 1994), so that thefree surface boundary condition is directly taken intoaccount. From these singular stress fields and the con-tributions of the other parts of the framework, Eq. (1),along with the free surface image stress on the disloca-tion (Hirth and Lothe 1968), the Peach-Koehler forceon each dislocation is calculated at each time step of theincremental calculation. This force governs the evolu-tion of the dislocation structure through a number ofconstitutive rules.

Dislocation motion is confined to be by glide, withthe velocity dependent on the Peach-Koehler forceaccording to a linear drag relationship with a drag coef-ficient B. Climb or cross-slip are not modelled. Thenucleation of new dislocations is incorporated throughtwo-dimensional Frank-Read sources. These are ran-domly positioned and generate a dipole when theresolved shear stress exceeds the source strength τnuc

for a sufficiently long time tnuc (see Van der Giessenand Needleman (1995) for details). Dislocation annihi-lation occurs when the distance between two disloca-tions of opposite sign is less than a critical distance of

6b. Furthermore, dislocations can escape from the grainat the free surface, leaving behind crystallographic sur-face steps. Finally, dislocations can get pinned at pointobstacles. These either represent small precipitates orforest dislocations. If the Peach-Koehler force on thepinned dislocation exceeds the strength of the obstacle,bτobs, the dislocation is released. Grain boundaries areassumed, for simplicity, to be impenetrable by disloca-tions.

The dislocation dynamics is simulated in an incre-mental fashion using straightforward Euler time inte-gration. The time step employed, �t = 0.1 ns, is smallenough to capture events such as dislocation nucleationand junction formation. The dislocation structure thatevolves is in no way presumed but is an outcome ofthe simulation, in which sources and obstacles are ran-domly distributed. With the constitutive rules outlinedabove, the uniaxial response of an isolated single grainis close to elastic-perfectly plastic.

2.3 Surface roughness

The surface roughness we include in the analysis is thatproduced by the accumulation of crystallographic sur-face steps that are left when dislocations leave the crys-tal. The interaction between surface steps is neglected,assuming that they are separated sufficiently well.Therefore, superposition of the fields of individual stepsis used, so that the stress field σ due to the surfaceroughness is given by

σ =∑

l

σ (l) (2)

where σ (l) is the stress due to an individual surfacestep l on a flat traction-free surface. The latter stress isdefined to be the deviation from the uniform stress fieldwhen the strip is subjected to uniaxial tension parallelto the surface, Fig. 3.

Within the framework of linear elasticity adoptedthroughout this work, the stress field caused by each

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σ

σ

bg

bg

h

Fig. 3 A single crystallographic slip step of height h on a freesurface caused by dislocations that have exited the crystal. Theσ (l) field in Eq. (2) is the local perturbation of the otherwiseuniform uniaxial stress σbg

step is self-similar. Therefore, we determine the fieldfor a representative surface step and scale all lengthsby the step height h. The field is approximated by anasymptotic solution similar to that of a wedge, and,as detailed in the Appendix, can be written in theform

σ (l) =∑

k

rλk−1Kk

∗f k (λk, θ), (3)

Kk = akσbgh1−λk . (4)

in polar coordinates (r, θ) from the tip of the wedge.∗f is a non-dimensional tensor of closed-form expres-sions for the θ -dependence of the stresses. λk are non-dimensional eigenvalues, given in the Appendix, and ak

are non-dimensional scalars to fit the analytical expres-sions for the stress field to numerical results. The back-ground stress is the sum of part (b) and (d) in Fig. 1 atthe root of the surface step, σbg = σ + σ . This non-local correction to the stress field is non-zero for allsurface step heights, even for atomic spacing height.However, for atomic distance sized features any con-tinuum approach looses its validity. This restriction isintrinsically taken into account in this framework as thestress intensity factor Kk decreases to zero as the stepheight approaches zero.

2.4 Cohesive surface

To model cleavage fracture as the initiation of a fatiguecrack, the cohesive surface model of Xu and Needle-man (1994) is adopted with properties that approachatomic separation (cf. Cleveringa et al. 2000). Thiscohesive law couples normal and tangential separationthrough the expression for the cohesive energy

� = �n − �n exp

(−�n

δn

)[(1 − r + �n

δn

)1 − q

r − 1

−(

q + r − q

r − 1

�n

δn

)exp

(−�2

t

δ2t

)](5)

where �n denotes the normal separation and �t thetangential separation. The corresponding normal andtangential components Tn and Tt of the traction vectorT are given by

Tn = − ∂�

∂�n

, Tt = − ∂�

∂�t

. (6)

The work of normal and of tangential separation aredenoted by �n = exp(1)σmaxδn and �t = √

exp(1)/2τmaxδt , respectively, while σmax and τmax are the normaland tangential strengths, respectively, and the charac-teristic lengths are δn and δt .

The material parameters q and r govern the couplingbetween the normal and tangential response as

q = �t

�n

, r = �∗n

δn

where �∗n is the value of �n after complete shear

separation with Tn → 0 (see Fig. 4a). Abdul-Baqi andVan der Giessen (2001) have investigated the influenceof q and r on the cohesive response and have found thatfor some combinations, the coupled response becomesphysically unlikely; r = q gives a physically mean-ingful response and is what we adopt here. The trac-tion–separation responses in tension and under shearare shown in Fig. 4.

2.5 Boundary value problem

As mentioned before, the nonlinear cohesive law (5)and (6) requires that the ( ˆ) solution of subproblem (d)in Fig. 1 is obtained in an incremental manner. We alsorecall that the ( ˆ) fields have to correct the ( ˜) and ( ¯)fields for the boundary conditions. For convenience,we introduce the ( ˘) fields which are defined as

( ˘) = ( ˜) + ( ¯) ,

so that ( ) = ( ˆ) + ( ˘).The virtual work for a body with a cohesive sur-

face, neglecting body forces and surface tension, canbe written as∫

V

σ δε dV −∫

Scoh

Tδ� dS =∫

Sext

Tδu dS. (7)

with T = σ n being the traction vector on the surfacewith unit outward normal n. Here, V is the volume of

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Fig. 4 Coupledtraction–separationresponses for (a) normaland (b) tangential directionnormalized by therespective strengths σmaxand τmax. The open arrowsindicate the effect of thecross-coupling on theresponse

T /

σn

max

=0t∆=t∆ δt

=t∆

=0n∆=n∆ δn

=n∆

T /

τt

max

∆ /δn n ∆ /δt t

∆ t

∆

(b)

n

(a)

−1

−0.5

0

0.5

0 1 2 3 4 5 6 7 8−1

−0.5

0

0.5

1

−4 −3 −2 −1 0 1 4

1

2 3

1/ 2r

the body analyzed. Scoh and Sext are the cohesive andexternal surface (Fig. 1), respectively. The boundaryconditions on Sext are prescribed in terms of prescribeddisplacements, δu = 0, or zero tractions T = 0. There-fore, the last term in (7) vanishes. As mentioned above,we neglect changes of geometry in order to avoid thecomplexities of a full finite strain formulation of dis-crete dislocation plasticity (cf. Deshpande et al. 2003).Yet the separation vector �, evidently, contains infor-mation about the deformed configuration.

Beside Eq. (7) at time t , we also consider virtualwork at t + �t . In view of the fact that the ( ˜) and ( ¯)parts of the decomposition (1) are self-equilibrating,virtual work at t + �t can be simplified to∫

V

σ (t+�t)δε dV −∫

Scoh

T (t+�t)δ� dS = 0. (8)

The term that requires due attention is the virtual workdone along the cohesive surface.

Cleveringa et al. (2000) were the first to use cohe-sive surfaces in conjunction with dislocation plasticityand proposed to adopt this Taylor expansion for thecohesive tractions about t + �t :

T(�(t+�t)

)= T

(�(t+�t) + �(t+�t)

)

= T(�(t+�t) + �(t) + �t

˙�

)

∼= T(�(t+�t) + �(t)

)

+K(�(t+�t) + �(t)

)�t

˙� (9)

where the superposed dot denotes differentiation withrespect to time t . Here, K is the instantaneous cohesivestiffness defined by

K = ∂T∂�

We substitute Eq. (9) into (8) along with a first orderexpansion for the stresses at t +�t . Subsequently, afterusing Eq. (7), we obtain

∫

V

˙σ (t)δε dV −∫

Scoh

K(�(t+�t) + �(t)

) ˙� δ� dS

= 1

�t

[ ∫

Scoh

[T

(�(t+�t) + �(t)

)

− T(�(t) + �(t)

)]δ� dS (10)

After introduction of the usual finite element interpo-lation functions, the first line of Eq. (10) leads to theleft-hand side of the linear system of finite elementequations Ka = f (with a the vector of ( ˆ) nodal dis-placement rates). The second line in (10) is the changein energy due to the cohesive surface opening, and givesrise to the right-hand side vector f.

3 Problem specification

We simulate the response of a 2µm×2µm grain at thefree surface, which is initially perfectly flat. A value ofB = 10−4 Pa s for the drag coefficient is adopted whilethe applied strain rate is 50,000/s (this is an unreal-istically high value, chosen merely from the point ofview of computing power). The grain is initially dislo-cation free, and dislocation sources and obstacles arerandomly distributed over the slip planes with densi-ties 100/µm2 and 140/µm2, respectively. The strengthof the obstacles is 150 MPa. Two realizations of dislo-cation sources are studied here; the obstacles are iden-tical in both realizations. The positions of the sourcesare identical but their strengths are slightly different,though from the same Gaussian distribution having anaverage value of 50 MPa and a standard deviation of10 MPa. As demonstrated by Deshpande et al. (2001a),dislocation dynamics is chaotic: small deviations leadto significantly different results in local behaviour. Wechose two slightly different distributions to restrict thisstudy not to a specific distribution and, therefore,specific result. The question is if small differences in

123


µx [ m]

µy

[ m

]

source strength[MPa]

20 30 40 50 60 70 80

0 0.5 1 1.5 2 0

0.5

1

2

1.5

Fig. 5 One of the two realizations of random dislocation sourcesin a 2×2µm grain. The second realization analyzed has the samedislocation positions, but a different selection from the sameGaussian distribution of source strengths

mµ0.5

mesh

grain

CZ 1

CZ 2

Fig. 6 Lower left-hand corner of a 2×2µm grain with twocohesive surfaces

the local properties lead to a significant change in theglobal response (Fig. 5).

In mere dislocation dynamics simulations, i.e. with-out cohesive surfaces and surface roughness, of a sim-ilar crystal we have observed that the highest in-planestress σ11 at the surface tends to occur between x = 0and x = 0.7µm. The long primary slip planes withthe highest dislocation activity, that intersect the freesurface roughly within 0 < x < 0.7µm, are the reasonfor the high stress at this location. One of the disloca-tions in the pair produced by a source tends to moveout of the crystal through this section, the sister disloca-tion remains inside. While dislocation motion relaxesstresses, on average, the long-range interaction of theinternally stored dislocations tends to produce a stressfield with a high stress at the end of the slip plane,as shown in Brinckmann and Van der Giessen (2004).Another reason for focusing on this area is that thesurface roughness is highest here because of the highdislocation activity on the long primary slip planes.

2 mµ

tT =0

y

x

tT =0

u=uapp11 mµ

grain

free surface

u=0

u=0

Fig. 7 Boundary conditions and total computational model

Based on these findings, we hypothesize that crackinitiation will take place in 0 < x < 0.7µm andemploy two cohesive elements near the center of thisarea, as shown in Fig. 6. The first cohesive surface ele-ment is normal to the free surface, because the normalstress in the element is maximum in this configuration.The second cohesive element is taken to follow the pri-mary slip system, i.e. 45◦ to the free surface.

The Poisson ratio ν = 0.33 and the shear modulusis taken to be µ = 26 GPa. For the cohesive surface weuse the same material parameters as have been used pre-viously to model fatigue crack propagation (Deshpandeet al. 2002): r = q = 0.5 and δn = δt = 2b = 0.5 nm.The cohesive strength is σmax = 600 MPa.

Plastic shearing along a slip system produces dislo-cation motion. However, it does not damage the mate-rial, i.e. it does not reduce the normal strength of thecohesive surface. Therefore, if a cohesive element isparallel to a slip system, e.g. the second cohesiveelement in Fig. 6, we do not allow for any tangentialopening, i.e. �t = 0, so that the normal cohesive prop-erties remain unchanged. For the cohesive element thatis normal to the free surface, the normal and tangentialmodes do interact in such a way that normal openingreduces the tangential strength, and conversely.

Finally, the tangential components of the traction onthe boundary are prescribed as zero everywhere. Sym-metry, uy = 0, is invoked along the the upper bound-ary of the computational model, Fig. 7. The remotelyapplied displacement uapp, parallel to the free surface,is taken to vary with time in a zig–zag fashion withumin = −umax. The maximum displacement is cho-sen such that it leads to a stress in the grain that is aquarter of the cohesive surface strength when disloca-tions are not included. Hence, crack initiation is onlypossible as a consequence of plastic deformation, as ischaracteristic for fatigue.

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∆ /δ

nn

norm

al s

epar

atio

n

cycle

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 120 140 160

realization 1

100

realization 2

Fig. 8 The normal opening of the first cohesive element �n atthe free surface (see Fig. 6) for two realizations of dislocationsources

4 Results and Discussion

In Fig. 8 the normal opening of the first cohesiveelement at the free surface is shown as a function oftime. For the first realization of dislocation sourcesthe opening increases slowly for roughly 120 cycles,after which it exhibits a growth rate which increasesin every cycle. During the 166th cycle, the openingreaches the critical opening, i.e. �n/δn = 1. We referto this event as ‘fatigue crack initiation’ and terminatethe simulation at this point. Continuation of the simula-tion beyond this point would require additional consti-tutive rules for dislocations wanting to leave the crys-tal through the cohesive surface, which are not knownyet.

The opening response found with the second reali-zation of source strengths is initially similar to that forthe first realization, but then features two ‘jumps’. After38.7 cycles the opening increases rapidly. After thisjump the rate of increase in opening per cycle is initiallythe same as before the jump, but gradually increasesuntil the second rapid increase at 53.9 cycles. Fromthis moment on the opening increases even more rap-idly and reaches the critical value shortly afterwards.Note that both jumps start during the compressive load-ing phase and are finished during the subsequent tensilephase.

We now look into the events which led to the firstrapid increase, which is the origin for the prematurefailure. Prior to the jump, dislocations move chaoti-cally back and forth in the changing energy valleys ofthe potential caused by the other dislocations and theboundary conditions. At almost maximum compressive

µm2

1[

]

ρ dis

ldi

sloc

atio

n de

nsity

cycle

0

50

100

150

200

250

300

350

400

450

500

0 20 40 60 80 100 120 140 160 180

realization 1

realization 2

Fig. 9 Dislocation density, corresponding to Fig. 8, in the2µm×2µm grain for two realizations of dislocation sources

loading the ‘jump’ occurs. The duration of this rapidincrease is roughly 8,000 time increments and thereforenot a single event caused by a numerical instabilitybut a stochastic event. During compression a signif-icant amount of strain energy is stored in the crackprocess zone. Therefore, it is energetically favorablefor dislocations to change to a neighboring valley ofthe potential, thereby changing the global landscape ofthe potential and decreasing the closure in the crackprocess zone. This rapid increase in material separa-tion is what we observe as a jump. The cohesive law(5) induces an exponential traction–separation curve,which leads to the absolute stresses being lower duringthe tensile phase than during compression. This differ-ence leads to a significantly lower stored strain energyduring the tensile phase, which makes reversal of thejump events during the subsequent tensile phase highlyunlikely.

The ‘jumps’ in the opening of the cohesive surfaceare a local event and do not have a strong effect on theoverall response, as is demonstrated in Fig. 9 for theevolution of the overall dislocation density. The dislo-cation density increases rapidly initially, but the aver-age growth rate decreases in later cycles and approachesa slow but steady average growth rate. However, duringa few tens of cycles just prior to fracture initiation, anincrease in the dislocation density is observed for bothrealizations.

Figure 10 shows the deformation in the grain for thefirst realization of dislocation sources after 164 cycles,i.e. a few cycles before crack initiation. From the mis-alignment in the mesh lines we can see a rather well-defined shear band aligned with the 45◦ primary slipdirection, just left of the cohesive elements. Where this

123


Fig. 10 Distribution of themaximum principal tensilestress superposed on thedeformation (magnified by afactor 5) for the firstrealization of dislocationsources after 164 cycles

band intersects the surface, it has produced a significantintrusion, i.e. a crystallographic surface step into thematerial. Near this intrusion the principal stresses areon average 250 MPa higher than in the remainder of thegrain. These high tensile stresses lead to the initiationof the fatigue crack.

It bears emphasis that in the first cohesive element,the tangential opening is smaller than the normalopening. Therefore, this framework predicts thatfatigue initiation is associated with mode I crack open-ing. Furthermore, the crack opening is maximum at thefree surface, i.e. the crack initiates at the free surfaceand grows into the grain.

As in the fatigue crack growth simulations in Desh-pande et al. (2001b, 2002), the strength of the cohe-sive surface, σmax = 600 MPa, is not reflecting actualatomic bond properties even though the normal frac-ture energy �n = 0.82 J/m2 is realistic. The cohesivestrength has been chosen smaller than atomic propertiesin order for the computation to finish in an accessibletime. Therefore, together with our high applied strainrate, the number of cycles to reach fatigue crack initia-tion is much smaller than those found in experiments.With a higher cohesive strength the material is expectedto break later assuming continued growth in the localdislocation-induced normal traction.

In the computations presented above, the evolutionof surface roughness and dislocation plasticity inter-acted in an intricate manner through their stress fields.To study whether surface roughness is a necessarycontribution for fatigue crack initiation, one simulation(using the first realization of sources) is repeated butwithout taking surface roughness into account (i.e. the

( ¯) fields are ignored). Figure 11 shows the effect ofthe presence of surface roughening on the evolution ofthe fatigue crack and dislocation density. While Fig. 8revealed crack initiation within the first 167 cycles,the computation without including surface roughen-ing does not lead to any appreciable accumulation ofthe normal opening at the free surface over the sameperiod. Therefore, the surface roughness is a signifi-cant contribution to fatigue crack initiation. This seemsto be supported by the experimental finding by Hahnand Duquette (1978) that repeated removal of the sur-face roughness leads to a significant delay in fatiguecrack initiation. Moreover, the present study shows thatfatigue crack initiation in the material interior is lesslikely than cracks at the surface because interior crackslack the contribution from surface roughness.

The evolution of dislocation density, shown inFig. 11b, appears not to depend on the surface rough-ness. Thus, the dislocation density depends on theinitial conditions (dislocation source and obstacle dis-tribution, grain geometry) and on the applied load-ing (history). The dislocations that glide to the surfaceand produce surface roughness, have virtually no effecton the dislocation density evolution, but this surfaceroughness is necessary for the initiation of a fatiguecrack.

5 Concluding remarks

We have proposed a new computational frameworkthat combines discrete dislocation plasticity and cohe-sive surfaces with a surface roughness model. While

123


open

ing

/δ

∆ nn

disl

ocat

ion

dens

ity [

m

]µ

−2

(a) (b)

cycle cycle

w/o surface roughness

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100 120 140 160

w/ surface roughness

0

100

200

300

400

500

0 20 40 60 80 100 120 140 160

w/o surface roughness

w/ surface roughness

Fig. 11 Evolution of the normal opening of the first cohesive element �n at the free surface (a) and the evolution of the dislocationdensity (b) for a simulation including the surface roughness and one excluding it

the first two ingredients were already present in thestudies of fatigue crack growth by Deshpande et al.(2001b, 2002), the extension with a description of sur-face roughening is necessary for fatigue initiation.

It bears emphasis that damage evolution in thisapproach is not assumed, but an outcome of the sim-ulation. Damage, in this framework, can be viewed asthe gradual evolution of the dislocation distribution andaccumulation of surface roughness.

This framework was employed to simulate thebehavior of a favorably oriented grain at the free sur-face under remote tension–compression cycles. Fatiguecrack initiation at the free surface is predicted as thecooperative effect of the build-up of local stress by evo-lution of the dislocation distribution and developmentof surface roughness. We here recall the observation inCleveringa et al. (2000) that dislocations play a dualrole in fracture: on the one hand they mediate stressrelaxation by plastic flow, but at the same time theylead to local stress levels of the magnitude of the cohe-sive strength due to discrete dislocations in the vicinityof the crack tip. The opening of a cohesive surface,as a model of atomic separation, in the tensile direc-tion increases slowly in the initial cycles but after acertain number of cycles accelerates till reaching thecritical opening. Even though the crack initiates nextto a persistent slip band, the sliding displacement alongthe cohesive surface is small compared to the normalopening, so that we conclude that fatigue initiation isa mode I feature. There are suggestions in the litera-ture (e.g. Suresh 1998), that fatigue initiation containsa significant mode II component, but it should be kept

in mind that our calculations only pertain to the veryfirst instant of crack initiation, which is probably notobservable experimentally.

Deshpande et al. (2001a) have shown that disloca-tion dynamics is chaotic: arbitrarily small perturbationsof the source positions lead to finitely different dislo-cation structures. They also showed that the perturbeddislocation structure does not need to lead to a differentoverall response, as in the case of the tensile responseof a crystal, but that the response of crack problems isgenerally affected significantly since crack growth iscontrolled by local stress fluctuations. This phenome-non has also been observed in the present study. Thetwo random realizations give rise to crack initiationtimes that differ by at least a factor of two. A randomminor event in one of the realizations triggered a seriesof events which led to a decrease in closure inducedstresses. It should be noted, though, that this variabilitywill not translate into fatigue lifetimes, since this alsorequires a certain amount of crack growth. Also, thereis a much larger ensemble of possible initiation sites inreal polycrystalline materials.

The present work is a first step to shed light onfatigue crack initiation employing the present frame-work. The parameters have been chosen to allow for a‘proof of principle’ within achievable computing times(of many months on a multi-processor computer).Restrictions on computing time, unfortunately, did notenable a more systematic study, while the mesh densityis already questionable. One of the limitations is theuse of size-independent linear elasticity to model thesurface steps, whose unit magnitude is that of a Bur-

123


gers vector. Also, unfortunately, our calculations do notpredict dislocation patterning in ladder-like structuresas observed experimentally in Mughrabi et al. (1979),but the predicted slip bands are comparable to the slipin persistent slip bands. Ladder-like structures mightbe predicted by simulations employing the extendedconstitutive rules of Benzerga et al. (2004), which incor-porate three-dimensional physics of dislocation inter-action. This extended model therebye holds the promiseof being able to predict dislocation patterning, but theeffect of that on crack initiation remains to be investi-gated.

Acknowledgements We gratefully acknowledge the compu-tational resources for the present work, which were providedby the Materials Science Center, University of Groningen, TheNetherlands.

Open Access This article is distributed under the terms of theCreative Commons Attribution Noncommercial License whichpermits any noncommercial use, distribution, and reproductionin any medium, provided the original author(s) and source arecredited.

6 Appendix: Approximation of elastic fields dueto a discrete crystallographic surface step

The elastic solutions for a crystallographic surface stepgiven by Brochard et al. (2000), Marchenko and Parshin(1980) and Kukta and Bhattacharya (2002) are not inagreement with the numerical solution we obtained. Inthis section, we therefore formulate a novel analyticalapproximation, making use of wedge theory.

We consider a single crystallographic surface step,shown schematically in Fig. 12. The elastic fields ateach point scale linearly with the stress σbg appliedparallel to the free surface. The surface steps consid-ered here are produced by dislocations that have left thecrystal. Since the crystal has three slip systems, thereare six possible wedge angles (two for each slip sys-tem, depending on the sign of the dislocations havingmoved out).

To investigate the dependence of the stress field onthe surface step opening angle 2(π − α), a series ofnumerical calculations is executed. The effect of theadditional stress to the uniform background stress, i.e.the singular stress due to the surface step, is shownin Fig. 13 for some point P in the proximity of thewedge tip. When α → π/2, i.e. the flat surface, thesingular stress approaches zero. When α → π , i.e. an

lowerterrace

upperterrace

σbg

σbg

h

step

y

x

Fig. 12 Definitions used for the surface step

infinitely sharp wedge, these stresses increase signif-icantly. Between the extreme configurations, thestresses can be considered independent of α. Sincethe most important of the six surface steps relevant herelie inside the constant region, the stress arising from thesurface step is considered independent of the wedgeangle.

An opening angle of 45◦ is employed as being rep-resentative. For this case, Fig. 14 shows the distributionof the normal stresses in the 6µm×6µm around thestep. This step has a height h = 1µm and is embeddedin a 200µm×200µm block of isotropic material.

We proceed by approximating this field by meansof the asymptotic elastic plane strain field of a wedgein infinite space (Timoshenko and Goodier 1961), asillustrated in Fig. 15a. This means that the traction-freesides of the step itself are accounted for, but the trac-tion-free surface is not. The stresses should thus not beexpected to be accurate at distances h away from theinternal cusp of the step, but this will be checked lateron. The definitions of the local coordinate systems andangles for an infinitely long wedge with an openingangle of 2(π − α) are shown in Fig. 15b.

We use the asymptotic solution for a wedge givenby Timoshenko and Goodier (1961). The associatedstress field is a power expansion of the type rλ−1 withthe λ’s, distinguished here by the subscript k, being thesolution of (λsin 2α)2 = (sin 2αλ)2 with α the wedgeangle. Dimensional analysis of r and normalization bythe background stress σbg (cf. Fig. 12) then leads to thefollowing expression for the stress field:

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Fig. 13 Numericalcalculation: perturbationstresses relative to theuniform background tensilestress σbg in the y-directionat pointP = (3/5h,−3/5h)

(chosen arbitrarily) causedby surface step for differentstep angles

xy

yy

xxσσ

−σ

x

P

5 3 4 61surface step case

y

2

2(π−α)

h

2(π−α)

norm

aliz

ed s

tres

sσσ bg

α

0.2

0.4

0.6

0.8

1.2

1.4

1.0

0.00 π

π π/2

Fig. 14 Normal stresscomponents σxx (a) and σyy

(b) for a representativesurface step in a200 × 200µm blocksubjected to a remote stressσyy = σbg according tofinite element calculation

σ /σyy bg

0.5

0.2

0.0

-0.2

-0.5

σ /σxx bg

−2

−1

1

2

−2

−1

1

2

µm µm

y

x

0 2 3 41−1 2 3 410

(a) (b)

Fig. 15 (a) Approximationof the elastic fields due to asurface step. (b) Definitionsused for the solution of aninfinitely long wedge

t 0=

t 0=

t 0=

t 0=

σbg

π/2

ξy

r

−αα

θη

x

h

(a) (b)

σ =∑

k

rλk−1Kk f ∗k (λk, θ), Kk = akσbgh

1−λk ,

f ∗k = fk√

f 2xx + 2f 2

xy + f 2yy

. (11)

For the representative surface step, i.e. 2(π − α) =π/4 or α = 7/8π , two values of λk can be deter-mined from (λsin 2α)2 = (sin 2αλ)2 (Timoshenko andGoodier 1961), for each of which the vector f ∗

k is cal-culated. This leaves the coefficients ak of Eq. (11) asunknown parameters, which we establish by fitting the

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Fig. 16 Approximate stressdistribution near a surfacestep using the wedgesolution. Stress componentsin the x-direction (a) and inthe y-direction (b)

0.5

0.2

0.0

-0.2

-0.5

σ /σxx bg σ /σyy bg

−2

−1

1

2

−2

−1

1

2

µm µm

y

x

0 2 3 412 3 410

(a) (b)

−1

resulting stress fields to the numerical results (Fig. 14).Figure 16 gives the resulting distribution of σxx andσyy . Comparison with Fig. 14 reveals that there arepoints of agreement and disagreement. The stress com-ponent σxx is matched quite well, except at points closeto the upper terrace (see Fig. 12). These points havea negative x-coordinate and, therefore, lie outside theregion of analysis for the small-strain formulation ofthe dislocation dynamics model. Therefore, the errorin this region is not significant. The stress componentin the y-direction is overestimated by the approxima-tion.

The differences between the approximate analyti-cal and the numerical solution arise because the trac-tion-free conditions along the upper surface terrace(see Fig. 12) are not taken into account by this asymp-totic approach (cf. also dashed line in Fig. 15b). Onlythe boundary conditions on the lower terrace and thestep are accounted for. The influence of the boundaryconditions along the upper surface terrace is strong inthe area shown in Fig. 16 because the surface step heightis on the same order of length as the size of the area ofinterest. Nevertheless, since this approximation yieldsbetter agreement than any of the other solutions in theliterature mentioned in the beginning of this appendix,we employ it in the present study.

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