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Acta Materialia 73 (2014) 167–176

Strain localization and fatigue cracking behaviors of Cu bicrystalwith an inclined twin boundary

L.L. Li a, P. Zhang a, Z.J. Zhang a, H.F. Zhou b, S.X. Qu b, J.B. Yang a, Z.F. Zhang a,⇑

a Shenyang National Laboratory for Materials Science, Institute of Metal Research, Chinese Academy of Sciences, 72 Wenhua Road, Shenyang

110016, People’s Republic of Chinab Department of Engineering Mechanics, Zhejiang University, Hangzhou 310027, People’s Republic of China

Received 20 March 2014; accepted 1 April 2014

Abstract

A Cu bicrystal with a twin boundary (TB) inclined at 45� to the loading direction and its component monocrystal were speciallydesigned to reveal their fatigue cracking mechanisms. Experimental results show that slip bands (SBs) parallel to the TB plane operatedeasily in both component grains and concentrated at the TB, while the SBs distributed more homogeneously in the monocrystal. It wasrevealed by atomistic modeling simulations and anisotropic elasticity theory that the screw dislocations transmitted to the TB coulddissociate into two far-separated partial dislocations. Therefore, these dislocations would be confined in the TB with a high difficultyof cross-slip, leading to strain localization near the TB. With cyclic deformation, the fatigue crack nucleated along the TB preferentiallyin the bicrystal with fewer cycles than that for the SB crack in the monocrystal. Besides, unlike the impingement fatigue crack along high-angle grain boundaries with dislocation impingement, the shear fatigue crack initiates along the TB devoid of dislocation pile-ups. Theexistence of the TB inclined at 45� to the loading direction induces strain localization and early intergranular fatigue cracking in the Cubicrystal. These results could not only expand our knowledge of fatigue cracking mechanisms by clarifying how the boundary behaveswithout the piling up of dislocations, but also provide important implications for future interfacial optimization of materials.� 2014 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Cu bicrystal; Twin boundary; Slip band; Fatigue crack; Molecular statics

1. Introduction

It is well known that the structure-dependent propertiesof grain boundaries (GBs) are important factors influenc-ing the properties of polycrystalline materials [1]. Control-ling the type, frequency and configuration of GBs couldhelp to develop strong and ductile polycrystalline materials[2]. Since the high-angle GBs are particularly effectiveobstacles to dislocation motion, the yield strength ofpolycrystals increases with the grain refinement, which fitsthe well-known Hall–Petch equation [3,4]. However, theblocking effect makes the GB vulnerable to stress concen-

http://dx.doi.org/10.1016/j.actamat.2014.04.004

1359-6454/� 2014 Acta Materialia Inc. Published by Elsevier Ltd. All rights r

⇑ Corresponding author. Tel.: +86 24 23971043.E-mail address: zhfzhang@imr.ac.cn (Z.F. Zhang).

tration and strain incompatibility during cyclic deforma-tion [5–8]. Specifically, the high-angle GBs are always thepreferential fatigue cracking sites regardless of whetherthe GB is parallel, perpendicular or inclined to the loadingdirection in various Cu bicrystals [6]. Recent developmentsof experimental techniques have facilitated the analysis andcontrol of the GB microstructure [9,10]. Notably, the twinboundary (TB) is a special kind of GB with coherent struc-ture and low interfacial energy, which could also work as aslip plane in face-centered cubic (fcc) metals. The TB cannot only provide a barrier to the dislocation motion likea conventional high-angle GB to strengthen the materials,but can also accommodate and transmit dislocations tosustain the plasticity [11–15]. Because GBs constrain thedislocation’s motion, nanograined materials often exhibit

eserved.

168 L.L. Li et al. / Acta Materialia 73 (2014) 167–176

high strength but low ductility [16,17], while the introduc-tion of nanoscale TBs could endow the ultrafine-grainedCu with high strength and moderate ductility under unidi-rectional deformation [14,18,19].

On the other hand, the performance of the TB duringcyclic deformation is significant and has been intenselystudied. It was observed that some TBs are also the fatiguecracking sites in strain-controlled high-cycle fatigue [20].And recently, the fatigue cracking behaviors of the TBsin fcc metals were found to be related to the stacking faultenergy (SFE) [21] and the difference in the Schmid factors(DSF) on both sides of the TBs [22,23]. As the SFEdecreases or the DSF increases, TB cracking becomes eas-ier [22,23], wherein different TB–dislocation interactionsgenerate fatigue damage of different degrees to the TB dur-ing fatigue [24]. The fatigue performance of the TB is dif-ferent from random GBs and the TB can increase thefatigue cracking resistance of the materials [25]. Generally,TB fatigue cracking is interpreted in terms of dislocationimpingement akin to GB fatigue cracking [5,6]. However,the dislocations do not always pile up at but could slip par-allel to the TB plane in many cases. Consequentially, thisraises an intriguing question: how does the TB behavewithout the piling up of dislocations on both sides undercyclic loading? Due to the twinning relationship betweenthe two component grains, the inclination angle of theTB to the loading direction correlates to the grain orienta-tion. When the TB is parallel to the loading direction, theslip planes of the primary slip systems with the highest Sch-mid factors in the two component grains intersect with theTB plane with mirror symmetry and the slip vectors areparallel to each other. The slip bands (SBs) could passthrough the TB and the fatigue crack nucleates along theSB preferentially [26–28]. Furthermore, there are someother remarkable cases, e.g., when the TB is inclined tothe loading direction, perfect and partial dislocations couldslip on the TB plane instead of being blocked [29–34].When the TB is inclined at 45� to the loading direction,the shear stress is the highest along this plane. It is believedthat the active slip systems should be parallel to the TBplane. The fatigue cracking mechanism of the TB in thiscase is intriguing and remains elusive, which will be ofbenefit to the GB design and advance our knowledge ofinterfacial cracking.

Since most previous studies focused on the polycrystal-line aggregates in which the stress state and dislocation–TB interactions are complex, we employed a Cu bicrystalwith a TB as its sole interface to investigate the intrinsicrole of the inclined TB. In the present study, a grown Cubicrystal with a TB inclined at 45� to the loading directionand its component monocrystal were employed. The plasticdeformation and cracking behaviors of the bicrystal andmonocrystal under cyclic loading were investigated andcompared to reveal the intrinsic role of the TB itself. Inaddition, the atomistic dislocation behavior of the bicrystalunder tension was also studied with the help of molecularstatics (MS) simulations.

2. Experimental procedure and atomistic modeling

A bulk Cu bicrystal was grown from oxygen-free high-conductivity Cu of 99.999% purity using the Bridgmanmethod and the TB is the sole interface. The bicrystal andmonocrystal fatigue specimens were cut from the bulk mate-rial with the same shape and size. The shape of the specimenis shown by the inset on the upper left of Fig. 1b, wherein theTB is inclined at 45� to the loading direction. The size of thegauge section of the specimen is �2.5 � 2.5 � 6 mm3. Thecrystallographic orientations were determined by the elec-tron backscatter diffraction (EBSD) method. The {111}pole figure shown in Fig. 1a indicates that it is a coherentTB here. The inverse pole figure (IPF) and GB map of thebicrystal specimen is displayed in Fig. 2b with the color indi-cating the orientation along the X direction, i.e., the loadingdirection of the specimen. The orientation of the monocrys-tal specimen is the same as the twin grain of the bicrystal. Allthe specimens were ground and electropolished carefully toensure that all the surfaces of specimens were strain-free andsmooth before the EBSD detection and cyclic deformation.The fatigue tests were performed under load control atroom temperature in air and a triangle wave with a fre-quency of 1 Hz was applied with an Instron E1000 testingmachine. They were fatigued at increased stress amplitudewith zero mean stress. The shear stress amplitude increasedstepwise from 10 MPa to �32 MPa and after that it waskept at �32 MPa (corresponding to their saturation stress)[35] with little adjustment. The specimens were cyclicallydeformed until fatigue cracks initiated either at the TBs orat SBs; these could be detected by scanning electron micros-copy (SEM). The crack detection was carried out by stop-ping the fatigue test and taking down the specimen severaltimes. At present, the initiated crack is thought to be somehundreds of micrometers long and �1 lm (hundreds ofnanometers) wide. The axial cyclic stress amplitude r couldbe calculated by

r ¼ s=X ð1Þwhere s is the resolved shear stress on the primary slip sys-tem and X is the Schmid factor of the bicrystal, which isequal to the Schmid factor of the soft component grain[6]. After the SEM observation on the surface, the speci-mens were then ground and electropolished again for thecharacterization of dislocation arrangements by the elec-tron channeling contrast (ECC) technique [36]. In addition,transmission electron microscopy (TEM) was also appliedto investigate the dislocation configuration.

Meanwhile, for a better, more detailed understanding ofthe deformation characteristics near the TB, the atomisticdislocation behavior of the bicrystal was also studied withthe help of MS simulations. A bicrystal with a TB is createdin the simulation cell. The three orthogonal axes x, y and zof the simulation box are, respectively, parallel to½1�1 0�; ½11�2� and [111] in the top crystal and ½�11 0�; ½�1�12�and [11 1] in the bottom crystal. Then, a right-handedscrew dislocation a

2½1�10� along x is created in the

Fig. 1. The crystallographic orientations of the Cu bicrystal and its component monocrystal specimen: (a) The {111} pole figure and (b) the IPF + GBmap of the twinned bicrystal with the color key and sample sketch showing as the insets. The orientation of the monocrystal is the same as the twin grain ofthe bicrystal. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 2. SEM images of the fatigued Cu monocrystal: (a) the macroscopic slip morphology with higher magnification shown in (b) and (c) of themonocrystal; (d) the typical SB crack in the fatigued monocrystal.

L.L. Li et al. / Acta Materialia 73 (2014) 167–176 169

simulation cell as Rodney did [37]. The periodic boundaryconditions are applied in x and y, while a maintaining fixedboundary condition along z. Following our previous work[38], the Peierls–Nabarro stress is obtained by applying ashear loading along x to move the partial dislocation.The bicrystal has dimensions of 4.243 � 164.116 �124.708 a3 and contains �34,6000 atoms. The Cu potentialdeveloped by Mendelev et al. [39] is chosen because it canreproduce well both TB energy and SFE. The dissociationbehaviors of screw dislocations in the grain interior andalong the TB were obtained after full relaxation by a com-bination of conjugate gradient energy minimization andquasi-dynamic quenching.

3. Results

3.1. Slip morphology and fatigue cracking

The numbers of cycles required to the initiation offatigue cracks in bicrystals and monocrystals are found

to be �2 � 104 and 1.8 � 105, respectively. It is clear thatthe fatigue performance of the bicrystal is worse than itscomponent monocrystal, which is related to their plasticdeformation behaviors. The slip and cracking morpholo-gies of the fatigued monocrystal are shown in Fig. 2. Asdisplayed in Fig. 2a with the macroscopic morphology,the whole gauge length of the specimen was filled withSBs. The intrusion and extrusion on the surface can be seenmore clearly from Fig. 2b and c with enlarged images of theSBs densely distributed in the center part of the gaugelength. As presented in Fig. 2d, with the cyclic deforma-tion, the SB cracks formed exclusively and preferentiallyin the monocrystal specimen, as was shown in earlyresearch [5].

In contrast, the bicrystal specimen shows quite distinctslip morphology and fatigue cracking behavior from themonocrystal, as manifested in Fig. 3. Apparently, there isstrong strain localization near the TB. The SBs mainly con-centrated in the vicinity of the TB, generating a TB affectedzone (TBAZ), as the macroscopic morphology depicted in

170 L.L. Li et al. / Acta Materialia 73 (2014) 167–176

Fig. 3a. The SBs in both component grains were parallel tothe TB, which could be seen more clearly in Fig. 3b and cwith higher magnification. The strain localization at the TBis also popular in the cyclically deformed polycrystals, butthere were also abundant SBs which might not be parallelto the TB in the grain interior due to the complex stress[20,40]. Unlike the polycrystals, hardly any SBs wereobserved in the interior of the matrix and twin grains faraway from the TB.

Furthermore, the slip morphology of the twinnedbicrystal also differs from that of the bicrystals with ahigh-angle GB [6]. Normally, there are abundant primarySBs in the grain interior and a GB affected zone (GBAZ)near the GB; and some secondary SBs operate in theGBAZ [6,41] due to the stress and strain incompatibility.However, there were hardly any secondary SBs observedin the TBAZ. Generally, the SBs pile up at the high-angleGB and consequently the fatigue crack initiates along theGB preferentially with defects accumulated or stress con-centration [5,6,8,26,28]. However, in the present Cu bicrys-tal, the SBs parallel to the TB plane caused noimpingement damage to the TB during cyclic deformation,but the fatigue crack still nucleated along the TB preferen-tially, as shown in Fig. 3c and d. The SBs parallel to the TBcan bring serious shear damage to the TB and play a cru-cial role in the formation of the TB fatigue crack [21,42].

3.2. Dislocation arrangement

The dislocation arrangements of the monocrystaldetected by the ECC method in SEM are displayed inFig. 4. The dislocation walls, cells and labyrinth flourishedin the fatigued monocrystal as depicted in Fig. 4a and band they were aligned along the direction of the SBs on

Fig. 3. SEM images of the fatigued Cu bicrystal: (a) the macroscopic slip morpthe fatigued bicrystal.

the surface. The SB cracks could still be detected clearlyafter grinding and electropolishing, and it is exhibited asthe bright white region in Fig. 4b. The SB cracks werefound to be surrounded by dislocation cells, which arewell-developed dislocation configurations under cyclicloading.

The dislocation arrangement near and far away from theTB are also different, as illustrated in Fig. 5, which is inaccord with the different surface slip morphologies as sta-ted above. The image in Fig. 5c was obtained from TEMobservation while the other three images were detected bythe ECC method. Some typical dislocation veins formedin the grain interior as displayed in Fig. 5a. However, therewere already some dislocation ladders formed in theTBAZ, as displayed in Fig. 5b, while it still exhibited as dis-location patches in the areas far away from the TB. At theother regions of the TBAZ, some dislocation cells hadformed near the TB in comparison to the dislocation veinsor labyrinth in the grain interior on both sides, as shown inFig. 5c. Some dislocation walls were continuous across theTB, as shown in Fig. 5b and c. The Burgers vectors of thedislocations in the matrix and twin grains are the same andthere might be some dislocations penetrating the TB bycross-slip [12]. The white region with the arrow pointingin Fig. 5d is the TB crack. There were abundant dislocationcells formed around the TB crack while dislocation patchesor ladders distributed in areas far away. It could be seenthat the dislocation arrangement evolved more easily andfaster near the TB, which means that more plastic deforma-tion was accumulated at the TB. Meanwhile, the TBremained straight during the cyclic deformation becausethe dislocations in the matrix and twin grain glided parallelto the TB. In comparison, the high-angle GB becamecurved after cyclic deformation due to the impingement

hology with higher magnification shown in (b) and (c); (d) the TB crack in

Fig. 4. Dislocation arrangement of the fatigued Cu monocrystal: (a) the dislocation arrangements in the grain interior and (b) surrounding a SB crack ofthe fatigued monocrystal detected by ECC method.

Fig. 5. Dislocation arrangement of the fatigued Cu bicrystal: (a) the dislocation arrangements in the grain interior on one side of the TB; (b, c) thedislocation arrangements at different TBAZ along the uncracked TB detected by ECC and TEM method respectively; (d) the dislocation arrangementsaround the cracked TB.

L.L. Li et al. / Acta Materialia 73 (2014) 167–176 171

of SBs from both component grains [26]. The dislocationarrangement also demonstrates that there is strain localiza-tion at the TB mainly caused by the SBs parallel to the TB.

3.3. EBSD detection on the fatigue cracking

The specimens were also examined by the EBSD methodto verify the fatigue cracking sites. The SB crack in themonocrystal and the TB crack in the bicrystal could stillbe detected clearly after grinding and electropolishing, asshown in Fig. 6a and c, respectively. The round edge ofthe SB crack in Figs. 4b and 6a might result from the grind-ing and electropolishing. The corresponding IPF + GBmaps for the two fatigued specimens are exhibited inFig. 6b and d. The step size is 0.1 lm and the hit rate ishigher than 94%. The color of the IPF represents the in-plane orientation parallel to the loading direction of thespecimen. Although some SB cracks were once found toinitiate in the region close to a TB [43], the fatigue crackin this bicrystal is confirmed to be along the TB in the pres-ent resolution scope, as the IPF + GB map shows in

Fig. 6d. The white region (the TB crack) was separatedby the grains of two colors. The white lines represent somelow-angle GBs, which may be in accord with the walls ordislocation cells. It is clear that there are many low-angleGBs surrounding the SB crack in the monocrystal speci-men while there are few low-angle GBs observed near theTB crack. This suggests that the strain is highly localizedat the TB plane while the strain is distributed homoge-neously in a wider region around each SB crack. The obvi-ous difference in the strain localizations around the TB andthe SBs must be responsible for their different fatiguecracking behaviors and will be discussed later.

4. Discussion

4.1. Theoretical primary slip systems

The primary slip systems with the highest Schmid fac-tors for the monocrystal and bicrystal specimens are sche-matized in Fig. 7a and b, respectively. The slip planes andslip vectors of the primary slip systems in the component

Fig. 6. EBSD detection of the fatigued Cu monocrystal and bicrystal: (a) the morphology of the SB crack and (b) the corresponding IPF + GB map ofmonocrystal after grinding and electropolishing; (c) the morphology of the TB crack and (b) the corresponding IPF + GB map of bicrystal after grindingand electropolishing. Black lines indicate the high-angle GBs with misorientation angles higher than 10�, while white lines indicate the low-angle GBs. Thewhite areas indicate the unindexed areas for the crack sites.

172 L.L. Li et al. / Acta Materialia 73 (2014) 167–176

grains on both sides of the TB are exactly the same in thebicrystal specimen. Furthermore, the slip plane and slipvector in the monocrystal specimen are identical with thatin the component grain of the bicrystal specimen. It couldalso be proved by the above SEM observations that there isconsistency between the activated SBs of the monocrystaland bicrystal specimens, which agrees with the theoreticalprediction.

In fact, due to the special crystallographic orientation ofthe twinned bicrystal, the Schmid factors of the 12 slip sys-tems in the component grains could be forecasted. Thegrain orientation is set as [h kl] and the TB plane is set as(111). Since the angle between the TB plane and the load-ing direction is 45�, a set of equations could be written asfollows:

Fig. 7. Simplified sketch of the primary slip systems (a) in the monocrystalspecimen and (b) on both sides of the TB in the bicrystal specimen.

h2 þ k2 þ l2 ¼ 1

ðhþ k þ lÞ=ffiffiffi3p¼ cos 45�

(ð2Þ

where h and k can be obtained when l is defined as anindependent variable. So the grain orientation is deter-mined and all the Schmid factors could be obtained by cal-culation, as displayed in Fig. 8. The values of the Schmidfactors of the 12 slip systems in the matrix grain are dis-played in Fig. 8 for all the bicrystals with a TB inclinedat 45� to the loading direction. And we have confirmed thatit is the same case in the twin grain. The horizontal axisindicates the third index of the loading direction [h k l].The four slip planes are indicated by the letters A–D whilethe six slip directions are indicated by the digits 1–6, respec-tively, as indicated in the label in Fig. 8 [27] and the TBplane is taken as plane A. It could be concluded fromFig. 8 that the slip systems parallel to the TB have the high-est Schmid factors for �80% of all cases. For the rest of thecases, they have the second highest Schmid factors, whichare 0.012 lower than the maximum at the most. In lightof the TB obstruction to other slip systems, it could be con-cluded that the SBs parallel to the TB would be activatedwhen the TB is inclined at 45� to the loading direction,as displayed in Fig. 7b.

There should be little or no difference between the slipmorphologies of the monocrystal and bicrystal specimensbut exceptionally, their SB distributions are quite different.Many investigations attributed the strain localization nearthe TB to the local stress enhancement due to the elasticcompatibility [36,43–46]. It is commonly believed that there

Fig. 8. The Schmid factors of the 12 slip systems in the matrix grain whenthe TB is inclined at 45� to the loading direction in the twinned Cubicrystal. It is the same case in the twin grain.

L.L. Li et al. / Acta Materialia 73 (2014) 167–176 173

is a mismatch in elastic constants between the two compo-nent grains and the incompatibility stresses should beinduced due to the requirement of the stress and displace-ment continuity across the TB. Besides the stress enhance-ment, the dislocation behavior in close proximity to the TBmight be influenced [42,47], which would also contribute tothe strong strain localization near the TB.

4.2. Major cause of strain localization near the twin

boundary

It is known that Cu is a typical wavy-slip material andcross-slip of screw dislocations is quite common in the pres-ent fatigue specimens. The Burgers vectors of the disloca-tions are parallel to the TB on both sides under thepresent crystallographic orientation. The rungs of the SBladders are perpendicular to the TB, as shown in Fig. 5b,and there would be bowed-out screw dislocations glidingalong the channels between the rungs during the cyclicdeformation. Some screw dislocations may also transferto the TB by cross-slip. The perfect dislocations tend to dis-solve into two partial dislocations holding a slice of stack-ing fault and the partials have to be assembled again beforecross-slip. Here MS simulations are performed to show thedissociation of a screw dislocation at different sites in abicrystal containing a TB, since the screw dislocationmotion in the channels plays an important role in the plas-tic deformation of the specimen.

The dissociation of a screw dislocation in the grain inte-rior is illustrated in Fig. 9a and c, while the dissociation onthe TB is shown in Fig. 9b and d. Here the Nye tensor wascalculated according to Hartley and Mishin [48] to deter-mine the core of partials: red atoms represent the a

6½2�1�1�

partial, while blue ones denote a6½1�2 1� partials. The TB is

indicated by a red dashed line. As sketched in Fig. 9aand c, the screw dislocation dissociated into two Shockleypartials with a stacking fault and the separating distancere_bulk is 2.45 a in the grain interior. It is a different casewhen the screw dislocation dissociated along the TB. Assketched in Fig. 9b and d, the screw dislocation dissociated

into two twinning partials. The TB segment between thepartials migrated upward by one {111} layer due to themotion of these two twinning partials. The equilibrium dis-tance re_tb is 59.4 a, which is much higher than re_bulk in thegrain interior.

As to the dissociation in the grain interior, the repulsiveforce between the partials tends to separate them, but theformation of the stacking fault will produce an attractiveforce. An equilibrium distance re_bulk is achieved whenthe attractive force is equal to the repulsive force, whichcould be calculated as [49]:

re bulk ¼a2

16pcðKs � Ke=3Þ ð3Þ

where a, c, Ks and Ke are the lattice parameter, the SFE andthe energy coefficients of screw and edge partial disloca-tions, respectively.

For the dissociation along the TB, there is no attractiveforce when the two partials repel and glide away from eachother, because there is no energy increase with the TBmigration. However, the partials will stop separating whenthe repulsive force becomes equal to the lattice resistanceforce, rp � bp, where rp is the Peierls–Nabarro stress andbp ¼ a=

ffiffiffi6p

is the length of the Burger vectors of the par-tials. By replacing c with rp � bp in Eq. (3) the distancebetween the partials along the TB plane could be obtainedas:

re tb ¼ffiffiffi6p

a16prp

ðKs � Ke=3Þ ð4Þ

From the selected potential, we have a = 3.6 A,c = 0.044 J m�2 and rp = 17 MPa. The calculated elasticconstants are C11 = 174.2 GPa, C12 = 127.3 GPa,C44 = 83.6 GPa, from which we have Ks = 44.3 GPa andKe = 63.5 GPa, according to the equations given by Hirthand Lothe [49]. Then, according to Eqs. (3) and (4), it couldbe obtained that re_bulk = 3.8a and re_tb = 66.2a, which is ingood agreement with the MS simulations as indicated inFig. 9.

It is obvious that the separation distance between thepartials of a perfect screw dislocation is large as it impingeson the TB, as displayed in Fig. 9. The separation distanceon the TB is more than 20 times larger than that in thebulk. It can be concluded that once screw dislocationstransfer to the TB, it is hard for them to leave the TB inview of the two far-separated partial dislocations. After-wards, they could glide along or out of the TB plane[29,33]. Moreover, the energy for dislocation dissociationis lower on the TB than in the grain interior, which isascribed to the increase of the distance between partialsand the disappearance of stacking faults. Therefore, abun-dant screw dislocations can be easily transmitted to andthen trapped by the TB. The high-resolution TEM obser-vation from our fatigued bicrystal specimen also demon-strates that there exist a large number of extensive partialdislocations along the fatigued TB, as shown in Fig. 10.

Fig. 9. The MS simulation results of dislocation dissociation: (a, c) the sketch for the dissociation of a perfect dislocation in the grain interior; (b) and (d)the sketch for the dissociation of a perfect dislocation along the TB.

Fig. 10. High-resolution TEM image of the TB in the fatigued Cubicrystal.

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The cross-slip is randomly distributed and could bereversed in the grain interior. In contrast, it is an irrevers-ible process on the TB. The dislocations transferred to theTB glide along the TB afterwards. Besides, there is ampleroom for the dislocation multiplication and storage in thebicrystal when a large plastic strain is applied along theTB [50]. The TB traps the dislocations and works like a dis-location sink, which causes the strain localization at theTB. There are various simulation results about the disloca-tion behaviors near the TB [51,52] and the discrepancymight come from the difference of the incoming slip vectorsand the stress state.

4.3. Shear fatigue cracking with strain localization

During the cyclic deformation, the positive and negativeedge dislocations which attract each other annihilate toform numerous vacancies [5].

The abundant vacancies will concentrate and precipitatemicro-cracks, which will then grow into fatigue cracks [53].In general, there are various fatigue cracking interfaces,such as SBs, TBs, GBs and so on, with different fatiguecracking modes [5,6,8]. Since the high-angle GB does notpermit the passage of dislocations from one grain intoanother [6], the SBs bringing dislocations and vacancieswould always pile up at the GB as displayed in Fig. 11a.The vacancy would accumulate and grow into the inter-granular crack initiation ultimately with the help of stressconcentration [5,8,26]. Generally, the GB fatigue crackinduced by the piling-up of dislocations is a typicalimpingement crack and by contrast, the SB and TB cracksaccompanied by shear deformation belong to shear cracks[54]. As sketched in Fig. 11b, there is no SB blocked at theSB–matrix interface or the present TB and the vacanciesaccumulated along the SB and TB planes gather to formthe shear crack.

The SB and TB shear cracks occurred with the vacancyaccumulating during the cyclic deformation. And the criti-cal densities for the vacancies to form the TB and SBcracks are deemed to be the same in terms of their closeinterfacial structure and energy, which are indicated bythe black dashed line in Fig. 12c. Under conditions ofsteady state, the increase of the vacancy density Dqv withan increment of shear strain Dc could be estimated as thedecrease of the local density of edge dislocations Dq�e [5]:

Dqv ¼ Dq�e ¼ 2qeyeDc=b ð5Þ

where qe is the dislocation density in the persistent SBwalls, ye is the distance for two edge dislocations to annihi-late and b denotes the Burgers vector. They are constantsexcept for the shear strain increment. Although the exter-nal strain amplitudes of the monocrystal and bicrystal arenearly the same, the local strain at the TB is higher thanthat at the SB, as evidenced in Section 3. Therefore, itcan be concluded from Eq. (5) that the increase of vacancy

Fig. 11. Schematic illustration for (a) the impingement GB crack inducedby the piling up of dislocations and (b) TB and SB cracks induced by shearstrain localization.

Fig. 12. The schematics for the dislocations and vacancies (a) concen-trated at the TB in the Cu bicrystal with a TB crack and (b)homogeneously distributed in the monocrystal with many SB cracks; (c)the qualitative comparison between the vacancy formation rate along theTB and the SB.

L.L. Li et al. / Acta Materialia 73 (2014) 167–176 175

density at each cycle is higher along the TB than that on theSB. In the monocrystal, the dislocations distributed inmore SBs and so did the vacancies, as sketched inFig. 12b. In comparison, the strain as well as vacanciesconcentrated at the TB, as sketched in Fig. 12a. Besides,the vacancies could diffuse along screw dislocations [55].Then the vacancies formed in the vicinity of the TB coulddiffuse to the TB plane along with the screw dislocations’cross-slip. In addition, it was reported that vacancy diffu-sion along the interface may be assisted by the stress con-centration, which is driven by the gradient of strainenergy density. Therefore, it can be concluded that thevacancy formation rate is higher along the TB than thatalong the SB and the vacancy aggregation is accelerated

along the TB plane as demonstrated in Fig. 12c. All the fac-tors help to explain that the TB fatigue crack nucleateswith fewer cycles in the bicrystal than the SB crack in themonocrystal.

Besides, fatigue crack initiation is considered to be anucleation process in which an energy barrier due to theformation of new surfaces must be overcome [56,57].Under cyclic deformation, the plastic strain energy willbe repeatedly input to the materials and there will inevita-bly be an energy increase in the form of increased latticedefects at each cycle on various interfaces such as theSB–matrix interface, GBs, TBs, etc. The energy of the TBincreases faster than the SBs due to the higher local strainamplitude [56,57]. Besides, the interfacial energy of the TBis higher than that of the lattice {111} plane, i.e., the SB–matrix interface. So in light of an energy increase of theinterfaces, it can also be concluded that it occurs earlierfor the TB cracking than for the SB cracking.

Although the TB in Cu polycrystals has been found tobe an intrinsically stronger interface that resists fatiguecracking in some cases [22–24,26,28], the TB may show rel-atively low fatigue cracking resistance despite no slipimpingement. It is found that when the TB is inclined at45� to the loading direction, there would be special disloca-tion behaviors occurring near the TB ascribed to the TBstructure and twinning relationship between the two com-ponent grains. Thus there is strong strain localization atthe TB and, eventually, the TB cracking occurs in a shearmode which is distinct from conventional intergranularcracking. It would be the same case in any Cu bicrystalswith a TB inclined at 45� to the loading direction, whichneeds to be treated with caution.

5. Conclusions

The cyclic deformation and fatigue cracking behaviorsof a grown Cu bicrystal with a TB inclined at 45� to theloading direction were studied. A monocrystal with thesame orientation as its component grain was also investi-gated as a comparison. Based on the results and analysisof experimental, calculation and simulation studies, the fol-lowing conclusions can be drawn:

(1) The SBs parallel to the TB plane are easily operatedin both component grains when the TB is inclinedat 45� to the loading direction, based on the calcula-tion of Schmid factors and stress enhancement. Inaddition, there is obvious strain localization in thebicrystal compared with the homogeneous deforma-tion in the monocrystal. The SBs would concentrateat the TB with the aid of easy and irreversiblecross-slip to the TB plane in comparison with the ran-dom cross-slip in the grain interior.

(2) Though the TB is not impinged by the dislocationwithin the SBs like the conventional high-angleGBs, the shear fatigue cracking still occurs alongthe TB preferentially in the bicrystal. The TB cracks

176 L.L. Li et al. / Acta Materialia 73 (2014) 167–176

nucleate with fewer cycles in the bicrystal than thatfor the primary SB cracks in the monocrystal attrib-uted to the shear strain localization. The existenceof the TB inclined at 45� to the loading directionwould induce higher strain localization and earlyintergranular fatigue cracking of the Cu bicrystal.These results could not only expand the knowledgeof fatigue cracking mechanisms but also provideimportant implications for the future interfacial opti-mization of materials.

Acknowledgements

The authors would like to thank W. Gao, C.H. Li, L.X.Zhang and Q.Q. Duan for the SEM and EBSD observa-tions, sample preparation and fatigue tests. This workwas financially supported by the National Natural ScienceFoundation of China (NSFC) under Grant nos. 50890173,51171194, 51331007, 11172264 and the National BasicResearch Program of China under Grant no.2010CB631006.

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