Materials Science and Engineering A314 (2001) 55–66
Effects of anisotropy and slip geometry on fatigue fracture ofCu/sapphire bicrystals
P. Peralta a, U. Ramamurty b, S. Suresh b, G.H. Campbell c, W.E. King c,T.E. Mitchell d
a Department of Mechanical and Aerospace Engineering, Arizona State Uni�ersity, P.O. Box 876106, Tempe, AZ 85287-6106, USAb Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139-4307, USA
c Lawrence Li�ermore National Laboratory, Mail Stop L-356, Li�ermore, CA 94550, USAd Los Alamos National Laboratory, MST-8, Mail Stop K765, Los Alamos, NM 87545, USA
Abstract
Interfacial fatigue cracks were propagated in Cu/sapphire bicrystals with (110)Cu�(101� 0)Al2O3/[001]Cu�[0001]Al2O3
The fracture behavior at metal–ceramic interfaceshas received considerable attention, since it is highlyrelevant to deformation and failure processes in elec-tronic components and composite materials. Interfacesthat have been studied in detail are those betweenalumina, either poly- or monocrystalline, and metals, asevidenced by the extensive literature available on thetopic [1–12]. These studies have shown a variety ofinteresting phenomena associated with the fracture be-havior at these interfaces.
Turner and Evans [9] showed that the fracture behav-ior of gold/sapphire interfaces under monotonic loadingis influenced by crystallographic slip on the metal side.They found a correlation between features in the frac-ture surface and slip markings on individual grains in
the metal film, which had a strong crystallographictexture. Experiments in Cu/sapphire bicrystals by Beltzand Wang [1] showed that there were strong directionaleffects on the behavior of an interfacial crack undermonotonic loading. Fatigue fracture studies in inter-faces between alumina and aluminum (both polycrys-talline) [6,10], have shown that fatigue cracks tend tostay at the interface for low and medium values of theloading and that striations are indeed formed in themetal side. This implies that plastic blunting of thecrack tip must play an important role in the growth offatigue cracks at these interfaces. In metals, fatiguecracks grow through a double forward-slip mechanism,whereby slip bands produce plastic blunting of thecrack tip at both sides of the fracture plane. In addi-tion, fatigue cracks follow a path that provides theoptimum slip conditions for this plastic blunting pro-cess [13–15]. The double forward slip ahead of thecrack tip needed to produce crack-tip blunting in thisfashion is not possible for cracks along metal/oxideinterfaces, since in most cases only the metal can
P. Peralta et al. / Materials Science and Engineering A314 (2001) 55–6656
provide crystallographic slip. Furthermore, the strongconstraints produced by the sapphire are likely to affectsignificantly these mechanisms for Cu/sapphire inter-faces. This brings out questions about the specificmechanisms for interfacial crack growth in this bimate-rial system and their precise relationship to the slipgeometry in the metal side. Peralta et al. [12], showedthat preferred directions for interfacial fatigue crackpropagation in Cu/sapphire bicrystals are related to thebehavior of the stress fields at the crack tip, in particu-lar to their variation due to elastic anisotropy. In thispaper, additional results related to fatigue crack growthin Cu/sapphire bicrystals are presented. In particular,the elastic analysis presented in [12] is refined to takefull account of the trigonal symmetry of sapphire. Thisrefined analysis is used to find correlations between theobserved behavior in compact tension (CT) specimens
and important variables such as mode I crack tipopening displacement (CTOD). The refined analysis isalso carried out for other Cu/sapphire bicrystal experi-ments described in the literature to verify the validity ofthe found correlations. In addition, the energetics ofdislocation nucleation from the interfacial crack tip isanalyzed following the procedure proposed by Wang[11], with modifications to account for the variations ofthe stress fields due to elastic anisotropy.
2. Experimental
Cu/sapphire bicrystals were prepared by diffusionbonding in the form of cylindrical specimens. Details ofthe procedure were reported in [12]. The crystallogra-phy and dimensions corresponding to each materialand the slip geometry in the metal are shown in Fig. 1.
Fig. 1. (a) Dimensions and crystallography of the cylindrical Cu/sapphire bicrystals, and (b) slip geometry in the Cu side.
P. Peralta et al. / Materials Science and Engineering A314 (2001) 55–66 57
Fig. 2. Dimensions and crystallography for the CT specimens.
Two cylindrical bicrystals with the misorientationshown in Fig. 1 were used to manufacture CT speci-mens with two orientations. The dimensions and crys-tallography of these specimens are shown in Fig. 2. Thefirst orientation, labeled CT1, is such that the crackpropagation direction (x1) is parallel to [11� 0]Cu, and theexpected crack front (x3) is parallel to [001]Cu which isthe first preferred orientation for cracking observed byPeralta et al. [12]. The second orientation, labeled CT2,was chosen at 90° from the previous one, i.e., crackpropagation (x1) along [001]Cu and crack front (x3)parallel to [11� 0]Cu (close to the second preferred direc-tion reported in [12]). The interface was perpendicularto the loading axis. Interfacial fatigue cracks wereinitiated using tension– tension loading (R� �0), whichresulted in an applied �KI of about 2 Mpa m1/2 (farfield).
3. Results
3.1. Compact-tension specimen
The crack in the CT1 sample nucleated at the notchand propagated quickly along the [11� 0]Cu direction, as
expected since this was the preferred crack growthdirection reported in [12]. This crack arrested at amaximum penetration of about 3 mm, and kinked intothe sapphire following the trace of a {101� 1} plane,which can be deduced from the long section of thecrack front that was parallel to the trace of this planejust before the kink, as shown in Fig. 3. The Cufracture surface had large regions almost free of fea-tures, but also showed an area where macroscopicmarks showed the initial crack advance (Fig. 3). Thesemarks are probably crack arrest locations and are notparallel to the traces of the available slip planes. Highermagnification observations did not reveal any striation-like markings between them, which suggests that thefracture was mostly ‘‘brittle’’ in nature. The samplewith the second orientation, CT2, showed a completelydifferent behavior. Heavy slip band activity at thenotch in the metal side, as shown in Fig. 4 for increas-ing number of cycles, seems to have blunted the notchtip and favored cracking in the sapphire. The crackpropagated mostly through the sapphire (Fig. 5), andmade it to the interface in a narrow region locatedaround the middle of the initial unbroken section, onlyto kink back into the sapphire. The interfacial crack leftwell-defined striations on the Cu side. These striations
P. Peralta et al. / Materials Science and Engineering A314 (2001) 55–6658
Fig. 3. Fracture surface of the CT1 sample after the test.
where inclined approximately 28° from the expectedmacroscopic crack front direction (parallel to [11� 0]Cu)as shown in the right side of Fig. 5. Note that they areregularly spaced and they ‘‘wiggle’’ instead of beingcompletely straight. They also run along a direction,approximately parallel to [4� 43]Cu, that is not related tothe traces of any of the four available {111} slip planes,which are also shown in the figure. The crack frontorientation for the CT2 specimen (28°) was close to theorientation of the crack front resulting in a minimummode II mix, according to the calculations presented byPeralta et al. [12]. However, they used hexagonal sym-metry for sapphire in their calculations and did notestablish correlations with either crack opening dis-placements or the energetics of dislocation nucleationfrom the crack tip. In order to explore possible correla-tions with these variables, a refined elastic analysis ofthe crack tip fields is undertaken in the next section.
3.2. Analysis of near-tip fields
It was decided to use the same model adopted byPeralta et al. [12], i.e., the one presented by Qu andBassani [16]. This model is based on the near-tip elasticfields for a plane strain crack between two anisotropicmaterials, as derived from the elastic solution to aGriffith crack along a bimaterial interface. The behav-ior of the near-tip fields for this case, as affected by thevariation of the material properties due to the an-isotropy, should resemble qualitatively that of the ac-tual crack in the CT specimens.
The coordinate system used is shown in Fig. 6. Notethat the direction perpendicular to the crack plane isalways x2= [110]Cu and that the crack growth directionis always x1, whereas the crack front is parallel to x3.The applied tractions in the CT specimens are assumedto be homogeneous and only mode I, i.e., along the x2
direction. The tractions are given unitary value to havethe results normalized per unit of applied stress. Thetractions at the interface at a distance x1 ahead of thecrack tip for the coordinate system used are given by
Fig. 4. Lateral view of the CT2 sample during the test, showingplastic blunting of the notch for increasing cycles.
P. Peralta et al. / Materials Science and Engineering A314 (2001) 55–66 59
Fig. 5. Fracture surface of the CT2 sample after the test.
t(x1)=� a
2x1
�1/2
Y�
(1+2i�)�x1
2a�i�n
qa= [�12 �22 �23]T,
(1)
where a is the crack length, qa is the generalized appliedtraction vector, which is assumed to be equal to[0 1 0]T in this case, and � can be calculated as:
�=1
2�ln�1+�
1−�
n. (2)
The term � is in turn defined as:
�=�
−12
tr(WD−1)n1/2
. (3)
Finally, the matrix function Y is given by
Y[� ]=I−Im[� ]
�D−1W+
(1−Re[� ])�2 (D−1W)2, (4)
where � is the complex argument of Y, and it dependson �, x1, and a (Eq. (1)).
The matrices W and D depend on the elastic proper-ties of the two materials and are obtained using theprocedure shown in [17] via the Barnett–Lothe tensorsL and S. This procedure allows calculation of theelastic fields without solving the non-linear eigenvalueproblem used in [16]. Further details are given by Quand Li [17].
Once the tractions at the interface are obtained, themode mixes of the fields, as functions of x1, are calcu-lated. The mode II mix is defined as
�II[x1]=2�
arc tan��12[x1]
�22[x1]n
, (5)
whereas the mode III mix is defined as
�III[x1]=2�
arc tan� �23[x1]
�(�22[x1])2+ (�12[x1])2
n. (6)
The advantage of using these definitions, which are amodification of the ones used by Bose and Ponte-Cas-taneda [18], is that the angles that result from the
inverse tangents correspond to Euler angles for thetraction vector at the interface, as shown in Fig. 6b.The tractions and the mode mixes change as functionsof the distance from the crack tip due to the oscillatorynature of the fields [16]. These variables were thenobtained along the interface at a very short distanceahead of the tip. In problems related to dislocationemission from crack tips, the length of the Burgersvector of the dislocations in the metal side has been theparameter of choice [19] to evaluate these fields. Theenergy release rate was also obtained in this case usingthe expression given by Qu and Bassani [16], i.e.
Grel=�
4aqa
TDY� 1+4�2
cosh2 ��
nqa. (7)
The value of Grel was normalized by the value of theenergy release rate calculated with the same procedureand loads, but using isotropic elastic properties for thetwo materials (Grel(i)).
Fig. 6. Coordinate system used for the elastic analysis of the crack tipfields: (a) rotation scheme, and (b) Euler angles for the traction vectorand their relationship to the mode mixes.
P. Peralta et al. / Materials Science and Engineering A314 (2001) 55–6660
Table 1Elastic constants for Cu and sapphire [20], in GPa
Constant SapphireCu
C1111 496.7166.1163.4119.9C1122
75.6C1313 147.4119.9C1133 110.7
498.0166.1C3333
0C1123 −23.5
the fourth-order stiffness tensor C through the relation-ships resulting from trigonal symmetry: C2222�C1111,C2233�C1133, C2323�C1313, C1212�(C1111−C1122)/2,C2223�−C1123, and C1312�C1123. The rest of the tensorcan be filled applying the usual symmetries of C, i.e.,Cijkl�Cjikl and Cijkl�Cklij. The resultant tensor is invari-ant with 120° rotations about the c axis, but not forrotations of 180° about the same axis. Note that the[101� 0] direction in sapphire is a 2-fold symmetry axis,and so is [110] in Cu; therefore, negative and positivedirections on the interface should be similar. However,[101� 0] and [1� 010] are not equivalent for the reasonsdiscussed above. Therefore, the analysis is carried outfor both positive and negative �101� 0� directions per-pendicular to the interface to find the best correlationwith the observed results.
The mode II mixes and the normalized energy releaserates calculated one Burgers vector ahead of the cracktip are shown in Fig. 7, assuming that the loading axisis parallel to either [101� 0] (Fig. 7a) or [1� 010] (Fig. 7b).The values reported correspond to a crack length of 6
The elastic properties of the materials are obtained asa function of the angle � (Fig. 6a) using the correspond-ing transformation rule for fourth-order stiffness ten-sors. These rotated tensors are then used to get W andD as functions of �. The values of the elastic propertiesalong the principal axes used for Cu and sapphire weretaken from [20] and are shown in Table 1. Note thatsapphire (�-Alumina) has the R3� c space group, whichresults in six independent elastic constants according toNye [21]. Those six constants can be used to generate
Fig. 7. Mode II mix and normalized energy release rate as functions of the angle of rotation about [110]Cu for positive and negative loading axesin sapphire. (a) [101� 0] axis, and (b) [1� 010] axis.
P. Peralta et al. / Materials Science and Engineering A314 (2001) 55–66 61
Fig. 8. Normalized mode I CTOD and mode III mix as a functions of the angle of rotation about [110]Cu.
mm. Note that the angular location of the local max-ima and minima correlate well with the preferred crackgrowth directions observed in the CT specimens as wellas with those reported in [12]. The lowest mode II mixabsolute values, local maxima in this case, are locatedat about 29 and 151°, and the highest absolute values,local minima, are located at 90° for both cases. Theseresults differ from those presented by Peralta et al. [12]in that they are not symmetric about 90° because of thetrigonal symmetry of sapphire. The effect of changingthe sign of the normal to the interface in the sapphireside is to switch the magnitude of the local maxima.The absolute value of mode II mix for [1� 010] at 29° isabout 7% lower than the corresponding value at 151°.This difference is likely to be higher than the errorderived from the numerical computations used; hence,the difference should be relevant. Fatigue crack growthalong paths with minimum mode II is a well-knowncriterion for crack growth under mixed mode loading[22]. Note also that the maximum energy release occursfor ��90°, which correspond to a crack front parallel to[001]Cu and growth direction parallel to [11� 0]
Cu. Fur-
thermore, the energy release rate at 29° is higher thanthat at 151° for the [1� 010] loading axis in sapphire.These facts favor the choice of the [1� 010] loading axis;therefore, the rest of the fields will be calculated forthat case.
The CTOD vector � was also calculated using theexpression derived in [16]
�(x)= (2ax)1/2DY
� (x/2a)i�
cosh(��)n
qa=�[u1 u2 u3]T, (8)
where x is the distance behind the crack tip. Inter-pene-tration can be neglected for the values of � and loadingphase used here, as discussed in [23]. The mode Iopening, �u2, which will be considered representative ofthe CTOD, was calculated forx= �b� and the resultsnormalized using the values obtained for isotropic elas-
tic properties, �u2(i). The variation of �u2/�u2(i) and themode III mix as functions of � are shown in Fig. 8.Note that the mode I CTOD is also higher at 29° thanat 151°, with a clear maximum at 90°. The mode IIImix was not affected significantly by the choice ofloading axis in the sapphire side. The values of thisvariable for the observed preferred directions, 0.02 at29° and 0 at 90° are low compared to the maximumvalues (�0.14). It is likely that the mode III mix didnot play a significant role in the fracture process.
The presence of striations on the metal fracture sur-face suggests that dislocation activity must play animportant role, perhaps in terms of the energetics ofdislocation nucleation from the interfacial crack tip. Ananalysis to examine this possibility is undertaken in thenext section.
3.3. Dislocation nucleation from the crack tip
Beltz and Wang [1] and Wang [11] have found goodcorrelations between the energetics to nucleate a dislo-cation from an interfacial crack tip and directionaleffects during fracture experiments in gold/sapphire andCu/sapphire bicrystals. Given that the same bimaterialsystem is being used here, the model adopted by Wang[11], based on a modified Rice–Thomson model [19],will be used here to try to find correlations with theobserved behavior. This formulation, which is based onisotropic elasticity, has been modified to try to accountfor anisotropy. The criterion for dislocation nucleationcan be written, according to [11,19,23], as
�K�(S I(�,�,)cos � cos � �+S II(�,�,)sin � cos � �
+S III(�,)sin � �)= fcrit, (9)
where the angles � and � �are defined in Fig. 6b andrepresent the local phase angles. The terms S I, S II andS III take into account the effects of the loading modesand can be expressed as [23]
P. Peralta et al. / Materials Science and Engineering A314 (2001) 55–6662
S I(�, �, )=�sinh �(�−�)cosh ��
sin3�
2+
e−�(�−�)
cosh ��
sin�
2�
cos2 �
2−� sin �
��cos, (10a)
S II(�, �, )
=�cosh �(�−�)
cosh ��cos
3�
2+
e−�(�−�)
cosh ��
cos�
2�
sin2 �
2+� sin �
��cos , (10b)
S III(�, )=cos�
2sin , (10c)
where � is the angle between the slip plane and thefracture surface and is the angle between the Burgersvector and a line contained on the slip plane that isnormal to the crack front, as shown in Fig. 9. Note thatthe functions described in (Eqs. (10a), (10b) and (10c))are strictly valid for the isotropic case. However, thesefunctions have provided good correlations for the be-havior of anisotropic bicrystals [11,19,23]. In addition,the value of � used in this work is that resulting fromthe anisotropic analysis. Similarly, the angles � and � �in Eq. (9) are also related to the mode mixes from theanisotropic solution (Fig. 6b). Note that effects ofmode III have usually been ignored [11,19,23], since inthe isotropic formulation mode III is decoupled frommodes I and II; therefore, if no mode III loads areapplied, no anti-plane tractions should appear at theinterface. In the anisotropic fields, all modes are cou-pled and a mode III component may appear at theinterface, even if only mode I loads are applied. Theformulation proposed here should be able to accountfor these effects, at least qualitatively.
The right-hand side of Eq. (9) can be expressed as[11]
fcrit=0.76A0
b cos
2�mr0
exp��ledgeb cos sin �
�A0
�, (11)
where A0 is a pre-logarithmic factor for the energy of acircular shear loop, m is a factor that arises from theself-energy of a semi-circular shear loop, �ledge is theenergy of the ledge produced at the crack tip due to
dislocation emission and r0 is a dislocation core cut-offradius. Detailed information on these terms can befound in references [11,19,23]. All material properties,slip geometry and Burgers vectors correspond to themetal side [11,19,23]. Representative numerical valuesfor the required variables can be found in [23] forA0�Gb2/10, �ledge/�A0�0.3, and m�1.5. Ref. [19] es-timates r0�2/3b. Therefore, the right-hand side of Eq.(11) can be evaluated.
In isotropic materials, the magnitude of the stressintensity needed to produce dislocation nucleation istaken to be the magnitude of a complex number K suchthat Kb i�= �K �ei� [11]. In this work, the anisotropy hasbeen taken into account using the formulation by Quand Bassani [16] by redefining �K� to be the magnitudeof the vector K�Y(b i�)k. Note that K has the samephase angle as the interface tractions since k is definedas [16]
t(b)= [t1 t2 t3]T=Y(b i�)k
�2�b. (12)
Therefore, the vector K can be obtained as
K= [KII KI KIII]T= �K�[sin � cos � �
cos � cos � � sin � �]T. (13)
The critical value of �K� for dislocation nucleationcan be obtained from equations Eqs. (9), (10a), (10b),(10c) and (11)) and the full vector can be constructedvia (Eq. (13)) using the mode mixes calculated from thetractions. Note that �K� must have units of Pa m1/2. Theenergy to nucleate a dislocation can then be calculatedas [16]
Gdisl=14
kTU−1k, (14)
where k�Y−1(b i�)K and U−1�D+WD−1W.
One limitation of this model is that the trace of theslip plane on the fracture surface must be parallel to thecrack front. Peralta et al. [12] suggested that the non-crystallographic striations observed on the fracture sur-face for the second preferred crack growth directionwere the result of alternating activation of at least twoslip planes. One plane must have a trace at �=0°, with(111) at �=35° or (111� ) at �=145° as two possiblecandidates, and another at �=55°, which is satisfied by(11� 1) at �=90°. This implies that the macroscopiccrack front is actually a ‘‘composite’’ line made of smallsegments parallel to each trace. Evidence for this can beseen in Fig. 5. It is also likely that the arrest marksobserved on the fracture surface of the CT1 specimen(‘brittle’ direction) are made in a similar fashion, bycombining slip traces at 55 and 125°.
The analysis showed that, at �=0°, the minimumenergy for dislocation nucleation corresponds to emit-ting a partial with =0°. At �=55° the least energy isspent nucleating a partial at =30°, whereas at =
Fig. 9. Slip geometry at the crack tip used for the dislocationnucleation analysis.
P. Peralta et al. / Materials Science and Engineering A314 (2001) 55–66 63
Fig. 10. Energy to nucleate a partial dislocation as a function of � fortwo orientations of the growth direction (crack front). Symbolsindicate the energy required for particular values of � correspondingto available slip planes.
This could enhance crack advance to the point that thecrack can grow without excessive plastic blunting.
The observed striations in the CT2 specimen areapproximately located at 28° from [11� 0]Cu. Note thatthis angular range agrees quite well with the angle forminimum mode II mix, which is about 29° (Fig. 7b).The energy release rate for this configuration is notmaximum (it is about 16% lower than the maximum),but the mode II mix is a minimum. This is also acriterion for mixed mode crack propagation [22]. This isin addition to the opening effect produced by thenegative shear discussed above. The results shown hereand in [12] suggest that when the crack can grow alongany direction (as in the cylindrical specimen used in[12]), it prefers to follow the direction of maximumenergy release rate and/or mode I CTOD first and theminimum mode II mix second. If the crack growthdirection is imposed, as in the CT specimens, the crackchooses the closest favorable configuration. This wasthe highest energy release rate and mode I CTOD forsample CT1 and for the CT2 specimen the crack frontrotated to the configuration with the lowest mode IImix. The experiments confirm that the CT1 orientationfavors crack propagation along the interface, probablyby a decohesion mechanism, since the Cu fracturesurface does not show heavy crack blunting. In con-trast, energy is spent producing plastic deformation inthe Cu side for the CT2 orientation. This must increasethe interfacial toughness, since the crack propagatedmostly through the sapphire for this orientation. Thenucleation of dislocations from the crack tip could havea role here. However, the interpretation of the resultsshown in Fig. 10 is complicated because the crackfronts are not parallel to the traces of the available slipplanes for the preferred directions.
Note from Fig. 10 that the energy to nucleate apartial dislocation for �=0° (crack growth along[001]Cu) corresponds to �=145°. The asymmetry of theenergy required to nucleate a dislocation for the twoslip planes at 0° is due to the negative value of themode II loading at the crack tip. This negative shearshould favor slip on the plane inclined backwards withrespect to the crack growth direction, (111� ). Given thatthe ‘‘ductile’’ crack growth in the CT2 sample producedstriations that were a composite of slip lines at 0 and55°, the energy spent to nucleate dislocations along thesecond preferred direction is probably proportional tothe sum of the energy required for each case. The sumof these energies is about 2.1 J m−2. However, theenergy to nucleate a partial dislocation is lower forplanes with traces at �=55°(11� 1) and �=125°(11� 1� ),both with �=90°. Note that the crack arrest marksobserved for growth along �110�Cu are probably acomposite of these two slip traces. The sum of thecorresponding energies for this case is about 1.1 J m−2.A comparison of the energies required would suggest
125° the minimum energy corresponds to a partial at�=30°. A plot of Gdisl as a function of � is shown inFig. 10. Values for the three slip planes mentioned areshown as well. The curve for �=125° is almost identi-cal to the curve for �=55°, so it is not shown.
4. Discussion
The crack that propagated more easily along theinterface grew along a direction, parallel to [11� 0]Cu, thathas the maximum energy release rate (Fig. 7), which isone of the criteria used for mixed mode crack growth[22]. The fact that this is also the orientation with themaximum absolute values of mode II mix complicatesthe interpretation of the results. However, Peralta et al.[12], suggested that the negative mode II mix obtained(negative shear stress) could induce an opening effect atthe crack tip. Bose and Ponte-Castaneda [18] discussedthis effect for a similar shear configuration at an inter-face between a ductile and a rigid material, whichshould make a good approximation to the Cu/sapphirecombination used here.
Note that the preferred crack growth direction alsocorresponds to that with the maximum mode I CTODfor the given misorientation and applied loads. Mostkinematical models for fatigue crack propagation agreeon establishing a proportionality between the mode ICTOD and the crack advance [13–15,24], i.e., thehigher the mode I CTOD the higher the crack advanceper cycle. Therefore, the ‘‘brittle’’ behavior observed forthe �110� growth direction could simply indicate that,for a given level of loading, a crack growing along thatdirection can advance a longer distance per cycle thanalong any other direction on the plane of the interface.
P. Peralta et al. / Materials Science and Engineering A314 (2001) 55–6664
that the ‘‘brittle’’ and ‘‘ductile’’ crack growth directionsshould be the reverse of those observed experimentally.A way to explain the discrepancy is by turning to thekinematics of fatigue crack propagation. In the modelproposed by Peralta and Laird [15], the ‘‘efficiency’’ ofa slip plane to produce fatigue crack advance via plasticblunting is given, approximately, by cos � sin2 �. Theplanes likely to produce blunting for the ‘‘brittle’’ direc-tion, i.e., those with �=90°, have zero efficiency. Onthe other hand, the planes with �=35 and 145°, whichcan produce blunting along the ‘‘ductile’’ direction,have efficiencies different from zero.
In order to verify if the correlations found apply toother misorientations, the Cu/sapphire bicrystal testedby Beltz and Wang [1] is analyzed. The geometry of thespecimen they used is shown in Fig. 11. The crystallog-raphy of the Cu crystal was shown in [1], but thecrystallography of the sapphire was not fully defined,and it was only reported that the interface was parallelto the basal plane. It was assumed that the x3, directionwas parallel to [101� 0] (Fig. 11). In addition, the fieldswere evaluated using h=1 mm, instead of the cracklength, as suggested by the geometry of the specimen[1]. The sample used in reference [1] was loaded infour-point bending, which resulted in a phase angle of−52° for the applied loads. In order to simplify theanalysis, it was assumed that the near-tip fields ob-tained by Qu and Bassani [16] also apply to this case.The applied traction vector qa. was taken as[−sin 52o cos 52o 0]T, to have a unitary traction vec-tor with the right phase angle. A rotation schemesimilar to that shown in Fig. 6 was implemented, withthe rotation axis parallel to [2� 21� ]Cu for this specimen.�=0° corresponds to growth along [11� 4� ]Cu, which wastermed the ‘‘ductile’’ direction, whereas �=180° corre-sponds to growth along [1� 14]Cu, the ‘‘brittle’’ direction.The calculations resulted in a local phase angle �between −72 and −76°, depending on the value of �,for a distance of one Burgers vector ahead of the cracktip. This agrees well with the value of −79° predictedby the isotropic model used in [1]. The relevant fieldquantities are shown in Fig. 12.
Not all the plots in Fig. 12 have the same values at�=0° and �=180°. This should be expected, since theplanes at the interface do not have twofold symmetry.The same correlations established before apply in thiscase. The direction for which interfacial cracking is
preferred, [1� 14]Cu, indeed shows a higher energy releaserate than the ‘‘ductile’’ direction. There is, however,only a 5% difference for this variable between the twodirections. The most significant difference can be ob-served in the crack opening. The mode I CTOD for the‘‘brittle’’ direction, �=180°, is almost 20% higher thanthat for the ‘ductile’ direction, �=0°, for the same levelof applied stress. Hence, crack propagation should befavored along the ‘brittle’ direction. Regarding the ef-fect of plastic blunting, the slip plane capable of blunt-ing the crack tip along the brittle direction has �=15°[1], which means that the ‘efficiency’ for that plane iscos 15° sin2 15°=0.065. The slip plane for the ‘duc-tile’ direction has �=55° and the ‘‘efficiency’’ is 0.385.Therefore, the ‘brittle’ direction has a low plastic blunt-ing efficiency compared to that of the ‘ductile’direction.
5. Conclusions
(1) The results obtained show that there are twopreferred directions for crack propagation along thestudied Cu/sapphire interface, with one of these direc-tions taking preference over the other. The first pre-ferred crack direction corresponds to growth along the[11� 0]Cu direction and the second is at 28° from [001]Cu.
(2) The propagation of the fatigue crack at theinterface studied here is probably due to a combinationof plastic blunting via crystallographic slip and brittledebonding processes, since there are large regions at thefracture surface that are free of features and the crackarrest markings did not correspond to the traces of the{111} slip planes with ‘‘forward’’ configurations, whichis usually the case during fatigue crack growth in singlecrystals and bicrystals of ductile metals [13–15], wherethis configuration leads to enhanced fatigue crackgrowth.
(3) The preferred directions for crack propagationobserved for the particular misorientation studied cor-related fairly well with the results of an anisotropicelastic analysis of the variation of the fields at the cracktip as functions of the orientation of the crack front onthe plane of the interface. It was found that the firstpreferred direction coincides with an orientation wherethe energy release rate and the mode I CTOD aremaximized. The second preferred direction correspondsto a minimum absolute value of the mode II mix.
(4) Analysis of interfacial fatigue crack propagationin Cu/sapphire bicrystals reported in the literatureconfirmed that directions with higher energy releaserates and mode I CTODs correlate with observations of‘brittle’ fracture. In addition, ‘brittle’ directions for thetwo cases analyzed had low values of the plastic blunt-ing efficiency factor proposed in [15]. These facts sug-gest that directions with high mode I CTOD and energy
Fig. 11. Geometry and crystallography of the Cu/sapphire sampleanalyzed by Beltz and Wang [1].
P. Peralta et al. / Materials Science and Engineering A314 (2001) 55–66 65
Fig. 12. Crack tip variables as a function of the angle of rotation about [2� 21� ]Cu for the sample analyzed in [1]. (a) Mode II mix, (b) Normalizedenergy release rate, (c) Normalized mode I CTOD, and (d) Mode III mix.
release rates, as predicted by the elastic analysis, andunfavorable kinematical configurations for crack ad-vance via plastic blunting would result in brittle fracture.These variables then need to be considered while studyingfracture along metal/oxide interfaces in addition to theenergetics to nucleate dislocations from the crack tip.
Acknowledgements
This research was supported by the Department ofEnergy (OBES), a Faculty Grant in Aid from ArizonaState University and by the National Science Founda-tion, under grant no. DMR-9984633.
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