Discrete dislocation modeling of fatigue crack propagationDiscrete dislocation modeling of fatigue...

Acta Materialia 50 (2002) 831–846www.elsevier.com/locate/actamat

Discrete dislocation modeling of fatigue crack propagation

V.S. Deshpandea, A. Needlemana,*, E. Van der Giessenb

a Division of Engineering, Brown University, Providence, RI 02912-9104, USAb Department of Applied Physics, University of Groningen, Nyenborgh 4, 9747 AG Groningen, The Netherlands

Received 31 August 2001; received in revised form 4 October 2001; accepted 4 October 2001

Abstract

Analyses of the growth of a plane strain crack subject to remote mode I cyclic loading under small-scale yieldingare carried out using discrete dislocation dynamics. Cracks along a metal–rigid substrate interface and in a single crystalare studied. The formulation is the same as that used to analyze crack growth under monotonic loading conditions,differing only in the remote stress intensity factor being a cyclic function of time. Plastic deformation is modeledthrough the motion of edge dislocations in an elastic solid with the lattice resistance to dislocation motion, dislocationnucleation, dislocation interaction with obstacles and dislocation annihilation being incorporated through a set of consti-tutive rules. An irreversible relation is specified between the opening traction and the displacement jump across acohesive surface ahead of the initial crack tip in order to simulate cyclic loading in an oxidizing environment. Thecyclic crack growth rate log(da/dN) versus applied stress intensity factor range log(�KI) curve that emerges naturallyfrom the solution of the boundary value problem shows distinct threshold and Paris law regimes. Paris law exponentsin the range 4 to 8 are obtained for the parameters employed here. Furthermore, rather uniformly spaced slip bandscorresponding to surface striations develop in the wakes of the propagating cracks. 2002 Acta Materialia Inc.Published by Elsevier Science Ltd. All rights reserved.

Keywords: Dislocations; Mechanical properties; Fatigue; Plastic; Computer simulation; Paris law

1. Introduction

For fatigue crack growth in a wide variety ofengineering materials under remote mode I load-ing, there is a threshold value of�KI � Kmax�Kmin below which cracks do not grow at a detect-able rate. Above this threshold value, in the regimewhere the amount of crack growth per cycle,

* Corresponding author. Tel.:+1-401-863-2863; fax:+1-401-863-1157.

E-mail address: [email protected] (A.Needleman).

1359-6454/02/$22.00 2002 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved.PII: S1359 -6454(01 )00377-9

da/dN, is of the order of a few lattice spacings,there is a steep increase in da/dN with �KI. Forlarger values of �KI, the increase in da/dNbecomes less steep and the Paris law regime (Pariset al. [1]) is entered where da / dN�(�KI)m. A criti-cal review of work on fatigue crack growth mech-anisms from the threshold through the Paris lawregimes is given by Davidson and Lankford [2].

There are a number of characteristic features ofthe crack growth behavior in the Paris regime [2].Experimentally, the value of the Paris exponentmranges widely; values of 2–4 are typically reportedfor ductile metals, e.g. [3,4], while values varying

832 V.S. Deshpande et al. / Acta Materialia 50 (2002) 831–846

from 4.5 to as high as 40 have been seen for inter-metallics and nickel-based superalloy single crys-tals [5,6]. Slip striations are commonly observedwith a spacing that is of the order of the amountof crack growth per cycle. In contrast, when stri-ations are observed in the threshold regime, theirspacing is generally much greater than the amountof crack growth per cycle. In some cases, steadycycle-by-cycle crack growth rates have beenobserved, while in other cases crack growth isreported to be intermittent [7] so that a steadygrowth rate is only apparent, arising from averag-ing the growth rate over many cycles.

Many continuum theories have been proposedto rationalize the dependence of the fatigue crackgrowth rate on �KI in the Paris regime. The geo-metrical models, by for example Laird and Smith[8] and McClintock [9], presume that the crackgrowth rate is proportional to the cyclic crackopening displacement which, in turn, is pro-portional to (�KI)2. Hence, geometrical modelspredict a Paris exponent m � 2. Damage accumu-lation models, such as those of McClintock [10],Weertman [11] and Rice [12], give rise to a Parisexponent m � 4. McClintock [10] used a criticalstrain-based failure criterion while Rice [12] useda critical value of the absorbed hysteresis energy.More recently, Nguyen et al. [13] reported numeri-cal calculations of fatigue crack growth in whichthe material was characterized by a conventionalcontinuum plasticity model and the fractureproperties were embedded in a cohesive relation.Non-dissipative cohesive relations resulted in plas-tic shakedown after a few cycles with no furthercrack growth. Thus, Nguyen et al. [13] employeda cohesive law with loading–reloading hysteresisand obtained Paris-like behavior with m�3. Noneof the above continuum models account for thewide range of Paris exponents observed exper-imentally. Furthermore, these models are restrictedto the Paris law regime and do not predict thechange in the dependence on �KI that occurs nearthe fatigue threshold.

Recent literature on dislocation models forfatigue crack growth has been reviewed by Rie-melmoser et al. [14]. These dislocation models, byfor example Pippan and co-workers [15–17] andWilkinson et al. [18], are meant to represent the

deformation-controlled fatigue crack growth mech-anism proposed by Laird and Smith [8] and Neum-ann [19]. In particular, Neumann’s model is basedon a mechanism that accounts for crack growth aswell as striations by an alternating duplex slipmechanism. It is worth noting that striation forma-tion is observed in fatigue crack growth at metal–ceramic interfaces [20–22] even though the kin-ematics of crack growth by an alternating slipmechanism is not clear for a crack growing alongsuch an interface.

Here, we analyze the transition from near-thres-hold to Paris law behavior using a unified frame-work that is applicable in both the near-thresholdand Paris law regimes. Indeed, a key feature of ourapproach is that the material model and the fractureproperties are independent of whether the loadingis monotonic or cyclic. In fact, the material modelis applicable whether or not there is a crack, seee.g. [23]. The fracture properties are embedded ina cohesive surface constitutive relation. As aconsequence, crack initiation and crack growthemerge as natural outcomes of the boundary valueproblem solution. In contrast to the solely defor-mation-controlled crack growth mechanismassumed in [15–18], crack growth in our calcu-lations is stress as well as deformation driven. Thenormal separation of the newly formed crack sur-faces, which gives rise to an open crack, occurswithout requiring a particular crack tip slip mode.

As in our previous study [24] of near-thresholdfatigue crack growth, a plane strain small-scaleyielding boundary value problem is formulated andsolved in the small-strain limit. The deformingmaterial is a planar model crystal with three slipsystems. Plastic flow arises from the collectivemotion of large numbers of edge dislocations,which are represented as line singularities in anelastic solid. Drag during dislocation motion, inter-actions with obstacles, and dislocation nucleationand annihilation are accounted for. The irreversi-bility of dislocation motion is what gives rise tocontinued crack growth under cyclic loading [24].

Most calculations are carried out for a ductilemetal single crystal bonded to a rigid substrate,representing a metal–ceramic interface. In addition,for comparison purposes, some calculations arecarried out for a crack in a homogeneous metal

833V.S. Deshpande et al. / Acta Materialia 50 (2002) 831–846

single crystal. Considering a metal–ceramic inter-face has several advantages over the homogeneousmaterial case: (i) crack growth along interfacesoccurs when the interfacial cohesive strength islower than that of the surrounding homogeneousmaterials; with decreasing cohesive strength theplastic zone size decreases, which reduces thecomputational time and decreasing cohesivestrength increases the effective cohesive length,which improves numerical accuracy for a givenmesh resolution [25]; (ii) along a bimaterial inter-face, crack growth often occurs along the interfaceso that our assumption of straight-ahead crackgrowth is appropriate; and (iii) even thoughstraight-ahead crack growth occurs, the mismatchin material properties allows mixed-mode loadingeffects to be explored. In addition, fatigue crackgrowth along metal–ceramic interfaces is of inter-est in its own right, see [20–22,26].

2. Discrete dislocation formulation

Details of the boundary value problem to besolved are illustrated in Fig. 1. As mentionedabove, most calculations are for a crack along theinterface between a single crystal and a rigid sub-strate. The analyses are two dimensional, underplane strain, and the crystal is taken to have threeslip systems, to mimic the ambiguity of slip thatexists in three-dimensional FCC crystals. Two slipsystems have their slip planes oriented at q �

± 60° from the interface, the third one at q �0°.1 Assuming small-scale yielding, plastic defor-

mation near the crack tip by the motion of discretedislocations is accounted for in a process windowof dimensions Lp � 15 µm by hp � 15 µm. Thesize of the rectangular region analyzed is 1000µm×500 µm.

The dislocations are treated as singularities in anelastic continuum, where for computational con-venience, the crystal is assumed to be elastically

1 In our previous study [24] we analyzed a crystal with twoslip systems at q � ± 60° and found that the qualitative fea-tures of near-threshold fatigue were similar to those of the FCC-like crystal with three slip systems.

Fig. 1. (a) Interface crack problem with the imposed boundaryconditions. (b) Irreversible cohesive law. (c) Schematic of theapplied stress intensity factor as a function of time.

isotropic with shear modulus m � 26.3 GPa andPoisson’s ratio n � 0.33, which are representativevalues for aluminum. Consistent with the planestrain condition, only edge dislocations are con-sidered, all having the same Burgers vector, b �0.25 nm. The potentially active slip planes in the

process window are spaced at 100b. As suggestedby Kubin et al. [27], the following dislocationmechanisms are incorporated: (i) dislocation glide;(ii) annihilation; (iii) nucleation; (iv) obstacle pin-ning. The glide velocity is taken to be linearlyrelated to the Peach–Koehler force with a dragcoefficient B � 10�4 Pa s, a representative valuefor several FCC crystals. Dislocations of oppositesign annihilate when they come within a criticaldistance of Le � 6b. Initially, the three slip systemsare supposed to be dislocation free, but dislo-cations can be generated from discrete sources thatare randomly distributed in the process windowwith a density of 66/µm2. These point sources


mimic Frank–Read sources from pinned segmentson out-of-plane slip systems which are notexplicitly considered. They generate a dipole whenthe Peach–Koehler force exceeds a critical valueof tnucb during a period of time tnuc; here tnuc �50 MPa and tnuc � 10 ns. There is no special dis-

location nucleation from the crack tip. The processwindow also contains a random distribution of 170point obstacles per µm2, which can represent eithersmall precipitates on the slip plane or forest dislo-cations on out-of-plane slip systems. The obstaclespin dislocations as long as the Peach–Koehler forceis below the obstacle strength btobs, where tobs �150 MPa.

Ahead of the crack, a cohesive surface isassumed, with properties specified by relationsbetween the normal (Tn) and tangential (Tt) trac-tions across this surface and the corresponding dis-placement jumps (�n, �t). These relations, whichinclude the mixed-mode failure properties of theinterface between the single crystal and the sub-strate, are derived from a potential f [28] as

Ti � �∂f∂�i

, (i � n,t) (1)

with

f � fn � fn exp��n

dn��1�r �

�n

dn� (2)

1�qr�1

��q � �r�qr�1��n

dn� exp��

�2t

d2t��.

The normal work of separation, fn, and the shearwork of separation, ft, can be expressed as

fn � esmaxdn, ft � �e2tmaxdt (3)

in terms of the normal strength smax and tangentialstrength tmax, and the characteristic lengths dn anddt (e � exp(1)). The coupling between normal andtangential separation is governed by q � ft / fn

and r � �∗n / dn, where �n

* is the value of �n aftercomplete shear separation with Tn � 0. The cohes-ive surface characteristic lengths are taken to bedn � dt � 0.5 nm for the interface. Also, the ratioof tmax to smax is fixed at 2.33 so that fn � ft andwe take r � 0. The cohesive strengths are takenas smax � 0.3 GPa and tmax � 0.699 GPa and thus

fn � ft � 0.408 J m�2. Such strengths are consist-ent with experimental measurements of intrinsicmetal–ceramic interface strengths in [29]. Thework of separation is related to a reference stressintensity factor K0 by the small-scale yieldingrelation

K0 � � Efn

1�n2. (4)

The significance of K0 is that pure mode I crackgrowth in a homogeneous elastic solid with thegiven cohesive properties takes place atKI / K0 � 1 [30]. For the material parameters usedin this interface problem K0 � 0.179 MPa √m.Note that the quantity K0 is a measure of the energyper unit area required to create new free surface.The fracture strength under monotonic loadingconditions, KIc, will in general be significantlygreater than K0 due to plastic dissipation.

The cohesive law described above gives rise toreversible behavior of the interface and can be usedunder monotonic loading and under cyclic loadingin a non-oxidizing environment (vacuum). In oxid-izing environments, oxidation of the new surfacesformed by crack growth inhibits crack closure andhence the cohesive law needs to be irreversible.Our implementation of this is illustrated by the Tn–�n response in Fig. 1b. The relations (1)–(3)describe the loading response as the tractionincreases from A up to the maximum value at B,followed by softening while the formed crackopens further. Unloading from any point C nowis specified to take place along path CD, with thestiffness against normal separation given by

∂Tn

∂�n

� �esmax

dn. (5)

Upon reloading the traction increases along DCand then follows the original softening curve BCE.All other stiffnesses are unchanged from thereversible cohesive relation (i.e. derived from thepotential f). Thus, the irreversible cohesiverelation accounts for premature surface contact dueto oxide formation but neglects any irreversibilityin the tangential tractions. As the normal openinggradually increases with continued cyclic loading,the permanent opening �o grows, but not more


than a predefined value of �s. This represents theasymptotic value of the oxide layer thickness for-med on a metal surface: under ambient conditionsthis is reported to be between 2 and 6 nm foraluminum [31] and in the calculations we take�s � 4 nm.

Apart from the traction-free crack and the cohes-ive surface ahead of the crack, the boundaries ofthe region analyzed are subjected to displacementboundary conditions corresponding to a mode Iisotropic linear elastic crack tip field. Analyticalexpressions are available for two limiting cases.For a crack along a perfectly bonded interface, thenear-tip fields are inherently mixed mode involvingboth mode I (tension) and mode II (shear), evenfor remote tensile loading, with the mode I to modeII ratio varying with distance from the crack tip,e.g. [32,33]. The variation with distance from thecrack tip is oscillatory, with the oscillatory indexdepending on the elastic mismatch, and involvesan arbitrary reference length. On the other hand, ifthe shear cohesive stiffness of the interface is zero,the much simpler homogeneous mode I linear elas-tic crack tip fields (e.g. [3], Chapter 14) prevail.Neither of these is an exact solution for the rela-tively weak interfaces considered. In the calcu-lations here, displacement boundary conditionshaving the form of the homogeneous mode I linearelastic crack tip field are imposed. This has theadvantage, for the rectangular region analyzed, ofavoiding having to specify an arbitrary referencelength. Even though the homogeneous mode I fieldis prescribed on the remote boundary, the stiffnessmismatch between the crystal and the substrategives rise to mixed-mode loading conditions withinthe region analyzed.

The applied stress intensity factor, KI, is pre-scribed to cycle between a maximum value of Kmax

and a minimum value Kmin as shown in Fig. 1c.The loading frequency is chosen such that the load-ing rate |KI| has a constant value of 100 GPa √m/s(this high value is chosen for computationalreasons and is not expected to affect the resultsqualitatively [34]). The duration of a cycle is2�KI / |KI|, and with the interface properties men-tioned above, can be expressed as 3.6 �KI/K0 µs.

The boundary value problem thus defined isnonlinear and is solved in an incremental manner

as discussed in detail in [24,35]. For each timestep, the long-range fields of the dislocations andthe resulting Peach–Koehler forces between themare determined by means of a superpositionmethod [36]. The central idea is to decomposethese fields into the analytical singular fields ofindividual dislocations in a half-space, and imagefields to correct for the boundary conditions. Thelatter are nonsingular and are obtained with a finiteelement method (using a graded mesh of 90 by 90elements in the process window). The singular partof the fields and the discontinuities in the displace-ment fields are represented analytically.

Some computations are carried out for a crackin a metal single crystal, as in our previous studyof the near-threshold regime [24]. The problemsketched in Fig. 1a is then the symmetric half ofthe total problem with �t � Tt � 0. The materialparameters are as specified previously while thecohesive surface properties are taken to besmax � 0.5 GPa, dn � 0.75 nm and �s � 6 nmgiving K0 � 0.283 MPa √m (fn � 1.02 J m�2).This value of smax is about a factor of four smallerthan is representative of the theoretical strength ofaluminum. We employ this small value of smax soas to decrease the plastic zone size and increasethe cohesive length in order to reduce the compu-tation time and increase accuracy. Note that here�n is the total normal separation of the cohesivesurface, �n � 2u2(x1,0), where x2 � 0 gives thecrack plane.

3. Numerical results

In the calculations presented here the appliedstress intensity factor KI was varied between Kmin

and Kmax as shown in Fig. 1c. The ratio R �Kmin / Kmax and the difference �KI � Kmax�Kmin

are used to characterize the cyclic loading. Simula-tions were carried out with a fixed R and theapplied �KI varied to get cycle-by-cycle crackgrowth rates da/dN typically in the range 5×10�4

to 0.1 µm/cycle.

3.1. Fatigue at an interface

We present cyclic loading results for threevalues of the load ratio, R, 0.1, 0.3 and 0.5. First


the case with R � 0.3 is considered. Crackadvance (�a) versus time curves for a threshold,intermediate and high value of applied �KI areshown in Fig. 2a. Here and subsequently the cracktip location is taken to be the point along the cohes-ive surface where �n � 4dn.2 In the three casesshown Kmax is higher than required for crackgrowth to initiate under monotonic loading con-ditions so a “burst” of crack growth occurs duringthe first loading cycle. Subsequently, the incremen-

Fig. 2. (a) Time evolution of the crack growth and (b) timeevolution of the dislocation density for the interface crack(R � 0.3).

2 In all the calculations presented here, the crack advance(�a) identified by a specified amount of normal separationalways exceeded the crack advance based on that amount oftangential separation.

tal cycle-by-cycle crack growth behavior is quali-tatively different in the three cases shown in Fig.2a. Approximately equal amounts of crack advanceper cycle corresponding to “steady” cycle-by-cyclecrack growth rates da / dN�5 × 10�4 and 0.03µm/cycle occur for �KI / K0 � 0.756 and 1.05,respectively. On the other hand, crack growth ismore intermittent at the intermediate value of�KI / K0 � 0.938. At this intermediate �KI value,small amounts of cycle-by-cycle crack growth areintermingled with occasional spurts of relativelylarge crack advance resulting in an averageda / dN�0.02 µm/cycle over the 36 cycles com-puted. No measurable cycle-by-cycle crack growthwas obtained with �KI / K0 � 0.756.

The evolution of the dislocation density rdis (perunit area in the process window) with time isshown in Fig. 2b for the cases discussed above.For �KI / K0 � 0.938 and 1.05 there is a clearincrease in the dislocation density with number ofloading cycles. Consequently, the dislocationstructures at the end and at the beginning of eachcycle differ. This is illustrated in Figs 3a and bwhich show the dislocation structures around thecrack tip after the third and twelfth load peaks,respectively for the case with �KI / K0 � 1.05.The irreversibility of dislocation motion results inan evolving dislocation structure with cyclic load-ing so that the crack can grow to different lengthsduring different loading cycles. On the other hand,the very low crack growth rate with �KI / K0 �0.756 results in nearly no cycle-by-cycle change

in dislocation density indicating that this loadingstate is near the plastic shakedown limit. This isconsistent with the fact that no cyclic crackadvance is observed for �KI / K0 � 0.756.

Experimentally, slip traces in the wake of thepropagating fatigue crack are usually observed inoptical micrographs and are cited as evidence forthe alternating slip mechanism of fatigue crackpropagation, e.g. [8,19]. While the kinematics ofcrack growth along an interface by a duplex slip-type mechanism is unclear, fatigue striations on thefracture surface and discrete slip traces on themetal surface in the wake of the propagating crackare also seen in cyclic crack growth at interfaces[20–22]. In order to compare the predictions of the


Fig. 3. Dislocation structures for interface crack (R � 0.3,�KI / K0 � 1.05) at (a) the 3rd and (b) the 12th load peak asmarked in Fig. 2a. All distances are in µm. The crack openingprofiles (displacements magnified by a factor of 20) are plottedbelow the x1-axis.

discrete dislocation simulations with these experi-mental observations, the deformation pattern pre-dicted by the discrete dislocation calculation isillustrated by plotting slip contours. The calculateddisplacement field is not continuous; dislocationglide gives rise to a displacement jump across theslip plane. However, to visualize the deformations,the values of the displacements ui are evaluated ona uniform rectangular grid with spacing 0.02 µm(this grid is finer than the finite element grid). Thestrain field �ij given by

�ij �12�∂ui

∂xj

�∂uj

∂xi� (6)

is then obtained by numerical differentiation. Theslip g(a) is defined by3

g(a) � s(a)i �ijm(a)

j (7)

where si(a) is the tangent and mj

(a) is the normal toslip plane a.

The total slip over the three slip systems, � �

3

a � 1

|g(a)| is shown at the third and twelfth load

peaks in Figs 4a and b, respectively, for a loadingcharacterized by R � 0.3 and �KI / K0 � 1.05. Atthe third load peak, contours of � in Fig. 4a showstrong evidence of slip traces emanating from theoriginal crack tip on the q � ± 60° slip planes. Bythe twelfth load peak (Fig. 4b) discrete slip traceshave formed at intervals of �0.15 µm on theq � 60° planes in the wake of the propagatingcrack. These are very similar to those seen in theexperimental investigations of [20,22] for cycliccrack advance at Al–Al2O3 interfaces.

The cyclic crack growth rate da/dN in the duplexslip model scales with the slip band spacing.Experimental observations of cyclic crack growthat interfaces [20] and in single crystals [2], how-ever, indicate that this spacing is generally largerthan da/dN and this difference increases withdecreasing �KI. In Fig. 4c contours of � at the 36thload peak are shown for �KI / K0 � 0.938 (R �

3 The quantity g(a) is not the actual slip on slip plane a asit includes contributions from dislocations gliding on all slipsystems. However, it is a convenient quantity for picturing thedeformation pattern.


0.3): the crack advance in this case of �a�0.8 µmis approximately equal to that after 12 cycles with�KI / K0 � 1.05. The slip band spacing in Fig. 4cis less uniform with an average value �0.2 µm.Consistent with experimental observations, the slipband spacing in both Figs 4b and c is greater thanthe respective values of da/dN, with the ratio ofslip band spacing to the average da/dN beingapproximately 2.7 and 8.6 for �KI / K0 � 1.05 and0.938, respectively.

For comparison purposes, a calculation was car-ried out with KI monotonically increasing. Thecurve of KI versus crack advance �a is shown inFig. 5a. Here, as seen by Cleveringa et al. [35],crack growth occurs in “spurts.” Contours of � atthe instant marked in Fig. 5a are shown in Fig. 5b.This corresponds to a crack advance �a�0.8 µmwhich is approximately equal to that achieved after12 and 36 cycles for the cases plotted in Figs 4band c, respectively. Similar to the slip traces in Fig.4, slip bands at discrete intervals also occur in Fig.5b. The major difference here is that the higherapplied KI required to propagate the crack by 0.8µm under monotonic loading results in an increasein the distributed plasticity. The slip bands, whichtend to arrest the crack intermittently, are consist-ent with the sporadic nature of the monotonic frac-ture process [35]. Thus, even though the samecrack growth mechanism operates under both mon-otonic and cyclic loadings, continued crack growthoccurs under cyclic loading at a value of Kmax forwhich the crack would have arrested under mono-tonic loading: the different dislocation structures atthe beginning and end of each loading cycle permitprogressive cyclic crack growth. To illustrate thispoint Kmax/K0 versus the crack extension �a isshown for two cyclic loading cases along with themonotonic KI/K0 versus crack advance curve inFig. 5a. While the �KI / K0 � 0.938 (R � 0.3) cal-culation was carried out for 36 cycles resulting ina crack extension of �0.8 µm, the �KI / K0 �

Fig. 4. Contours of the total slip � � |g(a)| at (a) the 3rd

and (b) the 12th load peak for R � 0.3 and �KI / K0 � 1.05 asmarked in Fig. 2a, and (c) at the 36th load peak for R � 0.3and �KI / K0 � 0.938. All distances are in µm. The crack open-ing profiles (displacements magnified by a factor of 20) areplotted below the x1-axis.


Fig. 5. (a) The applied stress intensity factor KI/K0 as a func-tion of the crack extension �a for monotonic loading of the

interface crack. (b) Contours of the total slip � � |g(a)| at the

point marked in Panel a. All distances are in µm. The crackopening profile (displacements magnified by a factor of 20) isplotted below the x1-axis.

0.972 (R � 0.1) calculation was carried out for 20cycles resulting in �a�0.27 µm. In both cases,continued crack growth occurs under cyclic load-ing at a value of Kmax for which the crack wouldhave arrested under monotonic loading.

Cyclic loading calculations similar to thosedescribed above were carried out for various valuesof R and �KI. The results of those calculations aresummarized in Fig. 6a where we plot da/dN versusthe applied �KI. The da/dN values plotted are aver-ages over the number of cycles computed in eachcase. This was typically 20 cycles in the low �KI

Fig. 6. (a) The cyclic crack growth rate da/dN versus �KI/K0

and �KIeff/K0 for the mode I cyclic loading of the interface

crack. The slopes of the curves marked correspond to the Parislaw exponents for the curves fitted through the numericalresults. (b) The normalized crack-opening stress intensity factorKop/Kmax versus �KI/K0.


regime and at least 10 cycles in the high �KI

regime. Use of other definitions of da/dN wasexplored, e.g. maximum or minimum crack growthin any cycle computed; while such definitionschanged the numerical values of the da/dN versus�KI relation, no change in the qualitative responsewas seen.

The log(da/dN) versus log(�KI/K0) curves inFig. 6a show two distinct stages of fatigue crackgrowth behavior. In the first regime, the averagecrack growth per cycle is smaller than a latticespacing with the crack either remaining dormantor growing at undetectable rates below a thresholdvalue �Kth. Just above the threshold, da/dNincreases sharply with �KI. Subsequently, there isa “knee” in the log(da/dN) versus log(�KI/K0)curve with da/dN increasing more gradually withincreasing �KI. This regime, which is referred toas the Paris law regime, is characterized by apower law relation between da/dN and �KI,

dadN

��KmI . (8)

For the simulations presented here we see fromFig. 6a that the Paris law exponent m lies betweenapproximately 6 and 8 depending on the value ofR. As in [24], the fatigue threshold �Kth decreaseswith increasing R. However, with increasing �KI

the dependence of the crack growth rates on Rdiminishes and the log(da/dN) versus log(�KI/K0)curves for the three R values in Fig. 6a are tendingto converge.

In our previous study [24] we found that thefatigue threshold with an irreversible cohesive lawoccurs at a constant effective threshold stress inten-sity factor range �Kth

eff with the dependence of thecrack growth rates on R a result of crack closure.When the crack faces are in contact, the stressesin the vicinity of the crack tip are much reduced,inhibiting dislocation nucleation and glide as wellas lowering the driving force for separation. As aconsequence, crack propagation tends to take placeonly during the fraction of the fatigue loading cyclein which the crack faces are separated, see forexample [37,38]. We thus proceed to quantify theeffect of crack closure and investigate the originof the dependence of da/dN on R in the post-thres-hold as well in the threshold regime.

The unloading path specified for the normal trac-tions in the irreversible cohesive law simulates sur-face contact due to the formation of oxide layerson the newly formed surfaces; locations along thecohesive surface where the opening �n exceeds 4dn

but are under the action of compressive surfacetractions correspond to points where closure hasoccurred. Closure in these calculations occurred ina similar manner to that observed in [24]. Onunloading from the maximum load, the surfaces atthe original location of the crack tip (x1 � 0) firstcome into contact. Further reduction in the appliedKI results in this contact zone spreading from theoriginal crack tip to the current location of thecrack tip. Complete closure with contact of thecrack faces at the current location of the crack tipoccurs at an applied stress intensity factor KI �Kcl. Upon reloading, the crack faces first separate

at the current location of the crack when KI �Kop. Continued loading results in the contact zone

retracting towards the original location of the cracktip until finally the crack opens completely. In thesimulations here Kcl�Kop. However, consistentwith the intermittent nature of the cyclic crackadvance in most of the calculations, Kop varies sub-stantially from cycle to cycle in any given calcu-lation. The average values of Kop over the numberof cycles computed are shown in Fig. 6b: forR � 0.1 and 0.3 there is a general trend forKop/Kmax to decrease with increasing �KI, which isconsistent with experimental observations of crackclosure in fatigue in aluminum by Ritchie et al. [4].There is no clear trend for Kop for the calculationswith R � 0.5. This may be because computationsfor R � 0.5 were performed only for relatively lowvalues of applied �KI as the large plastic zone sizesassociated with R � 0.5 limited the calculationsthat could be carried out.

The effective stress intensity range �KIeff is

defined as

�KeffI � �Kmax�Kop for Kmin � Kop

�KI for KminKop

(9)

Using this prescription, the da/dN versus �KI dataare replotted in Fig. 6a as da/dN versus �KI

eff. Towithin numerical scatter, all the data collapses ontoa single master curve with no dependence on R.


This indicates that the R dependence of the crackgrowth rate is a crack closure effect and that thecrack growth rate is governed by the effectivevalue of �KI over the range investigated here.

Crack closure as modeled through the irrevers-ible cohesive relation has a profound effect on thecrack growth behavior. The extent of crack closureis governed by �s which represents the asymptoticvalue of the oxide layer thickness formed on thenewly created surface. The effect of �s on thecrack growth behavior is shown in Fig. 7, wherecrack advance versus time curves are plotted for�s � 4dn, 8dn and �s→ for a loading charac-terized by R � 0.1 and �KI / K0 � 1.062. With�s→ the oxide layer thickness is continuouslyincreasing with crack opening displacement. Thisresults in da/dN increasing with the number ofcycles and a “steady” da/dN is not achieved.Decreasing �s reduces the irreversibility in thecohesive relation with the cohesive relation becom-ing completely reversible for �s � 0. Thus, reduc-ing �s has the effect of decreasing the cyclic crackgrowth rates. From Fig. 7 we see that the crackadvance versus time curves are not very sensitiveto the choice of �s in the range 4dn to 8dn with a“steady” da/dN being achieved in both cases. Thisrange of �s is in line with the oxide layer thick-nesses of 2–6 nm reported for aluminum underambient conditions [31]. Decreasing �s below 4dn

Fig. 7. Time evolution of the interface crack growth(R � 0.1, �KI / K0 � 1.062) for three values of �s.

results in very low crack growth rates similar tothose encountered with a fully reversible cohesivelaw while increasing �s above 8dn gives accelerat-ing crack growth.

3.2. Fatigue in a single crystal

The single crystal analyzed is characterized bythe same slip plane geometry and nucleation sourceand obstacle distribution as used to characterize themetal in the metal–ceramic interface problem, thusrepresenting the fatigue behavior of that material.Symmetry about the crack plane is prescribed.Since qualitative features of the mode I crackgrowth behavior in the homogeneous single crystalwere found to be similar to those of the interfacecrack, namely intermittent crack growth at inter-mediate values of �KI and more steady cycle-by-cycle crack growth at threshold and high values of�KI, only results for the da/dN versus �KI behaviorare presented.

The da/dN versus �KI data for R � 0.3 is plot-ted in Fig. 8a. As in Fig. 6a, the da/dN values plot-ted are averages over the number of cycles com-puted. This was generally 20 and 10 for low andhigh values of �KI, respectively. Again, there aretwo distinct regimes of behavior: a steeply risinglog(da/dN) versus log(�KI/K0) curve in the thres-hold regime followed by a more gradual slope inthe Paris regime. The Paris law exponent m in thiscase is �4.4 (m � 2 to 4 is typical for ductile met-als, e.g. [4]). The crack closure stress intensity fac-tor is plotted in Fig. 8b as a function of the applied�KI. As in the interface crack calculations, Kop istaken as the average value over the number ofcycles computed for each loading. The values ofKop/Kmax decrease with �KI in line with the experi-mental findings of Ritchie et al. [4]. The cycliccrack growth rate da/dN is also plotted in Fig. 8aas a function of �KI

eff, with �KIeff calculated using

Eq. (9). The effect of crack closure is more pro-nounced at the lower values of �KI so that �Kth

eff

is much less than �Kth and the cohesive relationmay be overpredicting closure in the near-thres-hold regime. Crack closure also results in anexponent m�2.8 for the modified Paris relation

dadN

�(�KeffI )m. (10)


Fig. 8. The cyclic crack growth rate da/dN versus �KI/K0 and�Keff

I/K0 for the mode I cyclic loading of the single crystal(R � 0.3). The slopes of the curves marked correspond to theParis law exponents for the curves fitted through the numericalresults. (b) The normalized crack opening stress intensity factorKop/Kmax versus �KI/K0 for R � 0.3.

For comparison purposes the “best-fit” da/dNversus �Keff

I/K0 curve for the interface crack fromFig. 6a is replotted in Fig. 8a. The effect of themode mixity at the interface is to increase thefatigue threshold of the interface crack but toreduce its resistance to cyclic crack growth athigher values of applied �KI. These conclusionsare expected to be strongly dependent on thedegree of mode mixity and hence affected by the

cohesive properties and the boundary conditionsapplied.

Rather uniformly spaced slip traces at q � 60°in the wake of the propagating fatigue crack arealso seen for the single crystal. Fig. 9 shows con-tours of � at the twelfth load peak for the casewith R � 0.3 and �KI / K0 � 1.19. These sliptraces closely resemble those for the interfacecrack in Fig. 4. Moreover, this deformation field issimilar to that in the experiments of Laird andSmith [8] and Neumann [19] for cyclic crackgrowth in metals. Note the high values of � parallelto the crack resulting from dislocation activity onthe q � 0° slip planes (Fig. 9).

4. Discussion

In the calculations here and in [24,35], the frac-ture behavior is an outcome of the interplaybetween the cohesive and plastic flow properties.Under monotonic loading conditions, with a suf-ficiently low density of dislocation sources, onlyisolated dislocations are generated and crackpropagation takes place in a brittle manner. Whenample nucleation sources are available and the

Fig. 9. Contours of the total slip � � |g(a)| at the 12th load

peak for a loading characterized by R � 0.3 and �KI / K0 �1.19 in the case of the single crystal. All distances are in µm.

The crack opening profile (displacements magnified by a factorof 20) is plotted below the x1-axis.


obstacle density is sufficiently low, the dislocationsstrongly relax the near-tip stresses, resulting incontinued crack tip blunting without crack propa-gation. In these circumstances, ductile fracturewould eventually take place by a process of voidnucleation, growth and coalescence. The circum-stances considered in this paper fall in an inter-mediate regime where, under monotonic loading,decohesion occurs in the presence of plastic defor-mation because of the high local stress concen-trations ahead of the crack that arise from the near-tip dislocation structures. Under cyclic loadingconditions, the increasing dislocation density nearthe crack tip, with the associated increase in stresslevel, is what allows the decohesion mechanism tooperate at lower imposed driving forces than areneeded under monotonic loading conditions.

Based on the experimental work of Laird andSmith [8] and Neumann [19], fatigue crack growthin ductile metals is often presumed to occur by analternating slip mechanism which is a deformation-controlled phenomenon that does not require highstresses. By contrast, in our framework fracture isboth a deformation- and a stress-governedphenomenon and takes place by a mechanism thatis possible under both monotonic and cyclic load-ing conditions. Crack propagation under mono-tonic loading occurs in “spurts” with slip traces onthe q � 60° slip planes being left behind at ratheruniform intervals in the wake of the propagatingcrack. This is consistent with the experimentalobservations of Kysar [39] for small amounts ofmonotonic crack growth along a copper–sapphireinterface. In our calculations, more or less uni-formly spaced slip traces occur on the q � 60° slipplanes under monotonic loading and in the wakeof a propagating fatigue crack, and the slip bandspacings are about the same in the monotonic andcyclic loading calculations. The present calcu-lations suggest that striations are not generallyobserved under monotonic loading because obser-vations are usually made after fracture when morediffuse plastic deformation has occurred and sme-ared out the slip bands.

Experimentally there is a wealth of evidencerelating to fatigue striations: these striations areripples seen on the fatigue fracture surface of mostductile metals. In our calculations, we obtain fairly

uniformly spaced slip bands at q � 60° which arevery similar to the slip traces seen on metal sur-faces in the wake of a propagating fatigue crack.These slip bands leave slip steps on the crack sur-face, giving rise to a staircase-like fracture surface(the crack profiles in Figs. 4 and 9). The striationsthat emerge in our calculations most closelyresemble type B striations according to the classi-fication scheme of Forsyth [40]. Indeed, experi-mental observations of Nix and Flower [41] showthat type B striations result from atomic separationfollowed by crack tip blunting and dislocationnucleation with the slip band spacing correspond-ing to that of the surface striations. The mechanismof striation formation in our calculations is inremarkable accord with these observations.

Other types of striations (type A according tothe Forsyth [40] classification scheme) are usuallyassociated with off-plane crack growth. While it ispossible in principle to allow off-plane crackgrowth in the cohesive surface framework, that isnot accounted for in the calculations here. Otherlimitations that affect the prediction of striationsinclude: (i) the calculations were carried out for atmost 36 cycles while experimental observationsare usually made after at least a few hundredcycles; and (ii) in our small-strain formulationstress concentrations due to surface steps are notaccounted for, which may affect the evolution ofthe surface profile. Nevertheless, the present calcu-lations provide a possible rational for the minimumstriation spacing of �0.1–0.15 µm seen for a widevariety of materials [42]; namely that the minimumstriation spacing is set by the interaction of thestress fields of the dislocations comprising theslip bands.

In our previous study on near-threshold fatiguebehavior [24] we presented cyclic crack growthcalculations with a reversible cohesive relation asmay occur in vacuum. Crack growth in those cal-culations was solely due to the inherent irreversi-bility of dislocation motion. Striations are typicallynot seen for fatigue loading under high vacuumconditions, e.g. [43]. Fatigue calculations with areversible cohesive relation were not carried outhere as the high �KI values (with correspondinglyhigh dislocations densities) required to get suf-ficient crack growth made such computations pro-


hibitively time-consuming. It is worth noting, how-ever, that crack closure did not occur in the near-threshold calculations in [24] when a reversiblecohesive relation was used and that Nix and Flower[41] suggest that striations are closure induced.

The form of the log(da/dN) versus log(�KI)curve seen experimentally, with a threshold and aParis law regime, is captured in our analysis. Inthe near-threshold regime the log(da/dN) versuslog(�KI) curve is steeply increasing as a significantproportion of the driving energy is going into thework of separation with only small amounts ofplastic dissipation in the bulk material. Withincreasing �KI the plastic zone size increasesresulting in increased plastic dissipation. This givesrise to the knee in the log(da/dN) versus log(�KI)curve with the Paris law exponent m decreasing asthe ratio of the plastic dissipation to the work ofseparation increases. This suggests that when theplastic dissipation is small compared to the workof separation, there is a high Paris law exponentand fatigue is essentially a cleavage process as inintermetallics, e.g. [44,45], and nickel-based super-alloys [6]. On the other hand, in ductile metals theplastic dissipation is typically very large comparedto the work of separation which would tend toreduce the Paris law exponent. Our model is cap-able of capturing the wide range of Paris lawexponents seen experimentally through the interac-tion between the independently specified cohesiveand material properties. It is expected that the Parisexponent m will approach 2 in the limit that plasticdissipation completely dominates, since in thatlimit the only relevant length scales, the size of thecyclic plastic zone and the crack opening displace-ment, scale as (�KI)2.

The calculations presented here were carried outfor relatively low values of �KI and hence maypertain to a regime approaching Paris law behavior.In Fig. 6a the log(da/dN) versus log(�KI) curvestend to converge but there is still an R dependencefor the crack growth rates over the entire range of�KI investigated here. The R dependence of thecrack growth rates in our calculations is a crackclosure effect with all the data for the interfacecrack collapsing onto a single curve when the Parislaw is expressed in terms of �KI

eff (Fig. 6a). Thecrack closure stress intensity factor Kop/Kmax as

modeled through our phenomenological irrevers-ible cohesive relation decreases with increasing�KI in line with the experimental findings of Rit-chie et al. [4]. For a high applied �KI, �Keff

I �Kmax�Kop��KI and thus we expect that the crack

growth rate will have negligible R dependence inline with true Paris law behavior. It is worthemphasizing that while the calculations capture theobserved closure behavior qualitatively, the presentanalysis has several limitations in this regard. Forexample, our phenomenological cohesive relationis not based on a quantitative consideration of oxi-dation kinetics and the small-strain formulationneglects changes in the surface geometry due toslip steps which can give rise to roughness-inducedcrack closure. These limitations can be addressedthrough ab initio calculations such as thoserecently undertaken by Jarvis et al. [46], whichcould provide the basis for a more accurate cohes-ive relation, and through a finite strain discrete dis-location formulation which accounts forgeometry changes.

An indication of how the discrete dislocationpredictions compare with what is observed exper-imentally is shown in Fig. 10 where numericallycomputed and experimental curves of da/dN and

Fig. 10. A comparison between the predicted cyclic crackgrowth rate and experimental measurements of McNaney et al.[20] for the mode I cyclic loading of the interface crack(R � 0.1). The experimental values of �KI from McNaney etal. [20] and the calculated values of �KI are each normalizedby the corresponding threshold value, �Kth. Also shown in thefigure is a comparison between the predicted and measured stri-ation spacings.


striation spacing versus �KI are shown. Theexperimental data are taken from McNaney et al.[20] and, to provide a basis for the comparisonbetween the numerical and the experimental data,each value is normalized by its threshold value,�Kth. The experimental measurements in [20] werecarried out using four-point bend Al2O3–Al–Al2O3

sandwich specimens for a load ratio of R � 0.1and for an applied �KI ranging from approximately1 to 10 MPa √m (only the portion of the data corre-sponding to the computed range is shown in Fig.10). The sandwiched metal layer was made fromhigh-purity (99.999%) polycrystalline Al and itsthickness varied between 100 and 500 µm.McNaney et al. [20] found that the fatigue crackgrowth rates along the Al–Al2O3 interfaces werereasonably independent of the Al layer thickness.The numerical simulations, in contrast, are for Alsingle crystals bonded to a rigid substrate in small-scale yielding. Thus, the comparison is of dissimi-lar situations, but we have not found da/dN versus�KI data for fatigue crack growth along a single-crystal metal–ceramic interface. The predictedParis law exponent of about 6 is greater than themeasured exponent of �1.8. Factors contributingto the more brittle fatigue crack growth in the cal-culations include: (i) the bonding in the McNaneyet al. [20] experiments may be stronger than thatmodeled here; (ii) the material parameters used inthe calculations, i.e. the strength and density of thedislocation sources and obstacles, may not be rep-resentative of the particular alloy in the experi-ments in [20]; (iii) a high loading rate is used inthe calculations to reduce the computing time andthis acts to decrease the amount of plastic dissi-pation; (iv) grain boundary interactions whichretard crack growth and increase plastic dissipationoccur in the experiments but not in the single-crys-tal calculations; (v) crack tip blunting effects arenot taken into account in our small-strain analyses;and (vi) crack closure which plays a significant rolefor R � 0.1 may not be modeled accurately as dis-cussed above. A comparison between the predictedstriation spacings and the measurements ofMcNaney et al. [20] is also shown in Fig. 10. Con-sistent with the experimental findings, the numeri-cal calculations predict striation spacings thatexceed the crack growth increment per cycle and

show less dependence on �KI than the crackgrowth rate. However, the predicted striation spac-ings are greater than the experimental measure-ments which may be due to the reduced plasticityand more brittle nature of fatigue in the calcu-lations.

5. Concluding remarks

We have carried out analyses of fatigue crackgrowth along metal–ceramic interfaces and in sin-gle crystals under remote mode I plane strain con-ditions. Plastic flow arises from the collectivemotion of large numbers of discrete dislocationsand the fracture properties are embedded in acohesive surface constitutive relation. The only dif-ference between the boundary problem formu-lations for monotonic and cyclic loading is the pre-scribed time variation of the remote stressintensity factor.

� Crack growth occurs under cyclic loading con-ditions when the driving force is smaller thanthat needed for the crack to grow under mono-tonic loading conditions.

� Two distinct regimes of behavior emerge nat-urally from the calculations: a steeply risinglog(da/dN) versus log(�KI) curve in the thres-hold regime followed by a more gradual slopein the Paris regime. This change in slope is dueto the increased plastic dissipation at the higher�KI values.

� Steady cycle-by-cycle crack growth occurs fornear-threshold and high values of applied �KI

while crack growth is more intermittent at inter-mediate values of the applied �KI.

� Striations emerge as traces of concentrated slipon the newly created free metal surface forcracks propagating along metal–ceramic inter-faces as well as for cracks in single crystals.

� The striation spacing is of the order of theamount of crack growth per cycle in the Parislaw regime and greater than the amount of crackgrowth per cycle in the near-threshold regime.

� The calculations here and in [24] rationalize awide range of observed fatigue crack growthbehavior within a unified framework.


Acknowledgements

Support from the AFOSR MURI at Brown Uni-versity on Virtual Testing and Design of Materials:A Multiscale Approach (AFOSR Grant F49620-99-1-0272) is gratefully acknowledged. V.S.D.thanks the English Speaking Union for supportthrough the Lindemann Research Fellowship. Theauthors thank Professor Subra Suresh forinsightful comments.

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Discrete dislocation modeling of fatigue crack propagationDiscrete dislocation modeling of fatigue...

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