I. INTRODUCTION
Thermal barrier coatings (TBCs) spallation and delamination are well-known problems that have been addressed by a number of experimental and theoretical studies.Reference Evans, Mumm, Hutchinson, Meier and Pettit1–Reference Guo and Shang8 The service lifetime of the TBCs is strongly affected by thermal cycling of the engine, which eventually causes the coating to spall. Often the failure appears to occur along the bond coat/TGO interface, by nucleation of small cracks and voids at defects, which then coalesce, leading eventually to large-scale buckling and spallation.Reference Evans, Mumm, Hutchinson, Meier and Pettit1,Reference Padture, Gell and Jordan2 To avoid spallation and failure of the TBC, it is important to improve the adhesion between the bond coat and the TGO since the bond coat/TGO interface appears to be the weak link. Considering the selections of the bond coat (typically made of a Ni-based alloy) and the TGO (predominantly α-alumina) for a TBC system and their fabrications/formations, the exact microstructure and composition of the real interfaces introduced could be very complicated.Reference Meltzman, Mordehai and Kaplan9–Reference Shi, Tanaka and Kohyama12 In this study, focus is put on the modeling of the Ni(111)/α-Al2O3(0001) interface. The reasons and properness of choosing this particular Ni/α-Al2O3 interface were elaborated as well in our previous papers.Reference Guo and Shang6,Reference Guo and Shang8
To make a quantitative evaluation of the interfacial adhesion, our previous worksReference Guo and Shang6,Reference Guo and Shang8 have analyzed the basic mechanical strength and failure behaviors of the Ni/α-Al2O3 interface under tension (mode I) and shear (mode II) loadings. From the precise first-principle calculations, we obtained some of the characteristic parameters to describe the mechanical property of the interface, including the theoretical tensile strength and the work of separation for mode I case, as well as the theoretical shear strength and the unstable stacking energy for mode II case. These are useful information in constructing the widely used cohesive zone models in continuum mechanics framework.Reference Barenblatt13–Reference Needleman15 Considering the actual fracture and failure problem, the effect of mixed-mode loadings is fundamental to investigate. Among technologically important adhesion challenges, the loading angle with respect to the interface could range from 0° (i.e., mode II, shear) to 90° (i.e., mode I, tension), and many intermediates (so-called mixed mode)Reference Hutchinson and Suo16 where the lattices are locally subject to stress conditions involving not only shear but also normal stresses. Thus, studies should not limit to the loading cases of pure tension and pure shear for the considered interfaces, although these two are equally important and relatively easier to address.
In fact, Krenn et al.Reference Krenn, Roundy, Cohen, Chrzan and Morris17 pointed out that the effect of compression on shear stress played a significant role in evaluating the critical shear stress at the onset of plastic deformation observed in nanoindentation. The mechanical responses of many metals and covalent crystals have been investigated under mixed-mode loading condition.Reference Černý and Pokluda18–Reference Umeno and Černý20 According to the authors' knowledge, a few theoretical work have been done, however, on the Ni/α-Al2O3 interface under mixed-mode loadings so far. Solely, Jiang et al.Reference Jiang, Wei, Smith, Hutchinson and Evans5 conducted a mixed-mode first-principle calculations on the Al-terminated O-site γ-Ni(Al)/α-Al2O3 interface model for 45° orientation with shear along the <112> direction to check the extended potential energy function of this interface. They demonstrate that the analytical traction–displacement curve agrees well with the computational results. However, a detailed examination on the mechanical responses for this Ni/α-Al2O3 interface is lacking in their paper. To investigate the mechanical strength, the nature of the loadings as well as the failure mechanisms of this interface under excitations, it is essential to perform a more systematic mixed-mode first-principle calculations.
Following the description of Sun et al.,Reference Sun, Beltz and Rice21 some researchers derived and revised the analytical expression of tension–shear coupling potential energy for different materials.Reference Jiang, Wei, Smith, Hutchinson and Evans5,Reference da Silva, Beltz and Machová22,Reference Lazar and Podloucky23 The potentials can be calibrated by the results from first-principle calculations for the cases of tension and shear conditions, wherein the tensile and shear stress–displacement relationships are well approximated by the universal binding relationReference Rose, Ferrante and Smith24 and the sinusoidal force law of Frenkel,Reference Frenkel25 respectively. According to the original derivation process of Sun et al., we further attempt to derive out the analytical expression of the Ni/Al2O3 interfacial potential and to verify the properness of that by compared with the results of the mixed-mode first-principle calculations.
In this study, we perform the first-principle mixed-mode simulations for the Al-terminated O-site Ni(111)/α-Al2O3(0001) interface that displaces along 22.5, 45, and 67.5° orientations with respect to the interface. The computational methodology is first described in Sec. II. In Sec. III, the results of mixed-mode mechanical responses from the first-principle calculations are elaborated. The divergence of the potential function between the analytical results and computational ones are discussed in particular. Section IV concludes the whole study.
II. METHODOLOGY
For the first-principle mixed-mode calculations, we employ the Vienna ab initio simulation packageReference Kresse and Hafner26,Reference Kresse and Furthmuller27 of density functional theory within a plane wave basis.Reference Hohenberg and Kohn28,Reference Kohn and Sham29 The interactions between the valence electrons and the nuclei plus core electrons are described by projector augmented wave potentialsReference Blöchl30,Reference Kresse and Joubert31 for all elements. The Perdew and WangReference Perdew and Wang32 version of the generalized gradient approximation for electron exchange and correlation was used. The k-point sampling was attained upon using 5 × 5 × 1 points in a regular Monkhorst–Pack scheme.Reference Monkhorst and Pack33 The plane wave cutoff energy of 500 eV and an energy convergence criterion of 10−5 eV for self-consistency are adopted throughout all the calculations. Relaxation is done until the Hellman–Feynman forces are less than 0.01 eV/Å using a conjugate gradient algorithm.
The coherent Al-terminated O-site Ni(111)(√3 × √3)/α-Al2O3(0001)(1 × 1) interface model given in Ref. Reference Guo and Shang6 is dealt with in the present work (in short, Al–O model). Figure 1 presents the atomic structure of the typical simulation cell and the settings for the applied mixed loading. The supercell is constructed by a slab of α-Al2O3(0001) layers consisting of five O atomic layers and 10 Al atomic layers for this model, sandwiched between two Ni slabs [each consisting of five Ni (111) atomic layers]. Shear calculations suggested that relaxation has been sufficient for two interfacial rows of Ni atoms and six-layer atoms of α-Al2O3 for this model.Reference Guo and Shang8 Hence, the atomic layers out of these rows are considered as rigid regions, which are indicated by the two shadow rectangles in Fig. 1. In the present simulations, the supercell is split into two blocks, namely, Ni block and Al2O3 block. The Ni block translates by a displacement (r, θ) relative to the Al2O3 block. Here r is the horizontal shift of the Ni block relative to the Al2O3 block along the <110> or <112> direction of the Ni lattice. The θ opening results in the two blocks being pulled apart. By adjusting r and θ values according to 
$\tan {\rm{\varphi }} = {\textstyle{{\rm{\theta }} \over r}}$, the loading angle ϕ is displaced along 22.5, 45, and 67.5° orientations with respect to the interface, respectively, inducing δ displacement along the traction direction. At each displacement level, the atoms out of the rectangles are fully relaxed until the Hellman–Feynman forces are less than 0.01 eV/Å. This cycle is repeated with increasing displacement until the stress level reached the point where the bonds in the model started to deform and break.
Pictorial representation showing the simulation cell for the Al-terminated O-site Ni(111)/α-Al2O3(0001) interface model and the settings for the applied mixed loading. The Ni, Al, and O atoms are represented by blue, purple, and red spheres, respectively. The initial interlayer distance between Ni and Al2O3 is defined by h = 1.403 Å. Note that the models have been duplicated in the lateral direction to better show the system.
III. RESULTS AND DISCUSSION
A. Mechanical strength parameters
Figures 2 and 3 show the variations in the tensile and shear stresses obtained from first-principle calculations with respect to the tensile and shear displacements for the <110> and <112> slip cases along different loading angles, respectively. It is seen from Fig. 2(a) that the tensile stress of the <110> slip case increases as the tensile displacement increases until that reaches to the maximum value of 10.65, 10.03, and 9.82 GPa at the critical tensile displacement 1.10, 1.10, and 1.20 Å for 22.5, 45, and 67.5° loading angles, respectively. These stress amplitudes are very close to the theoretical tensile strengths (9.92 GPa in the uniaxial tension, 11.10 GPa in the uniaxial extension, according to Ref. Reference Guo and Shang6). The maximum tensile stress for the <112> slip case is 8.84, 9.71, and 9.72 GPa for 22.5, 45, and 67.5° loading angles, respectively, which is slightly smaller than the theoretical tensile strength [see Fig. 2(b)]. Moreover, for both the slip cases, the tensile stress–displacement curves from first-principle calculations are similar in their overall shapes, which infer that the tensile stress–displacement relationships are insensitive to the mixed-mode loading angle.
Tensile stress σ from first-principle calculations of the Ni/Al2O3 interface as a function of the tensile displacement θ for different loading angles: (a) the <110> case and (b) <112> case.
Shear stress τ from first-principle calculations of the Ni/Al2O3 interface as a function of the shear displacement r for different loading angles: (a) the <110> case and (b) <112> case.
In Fig. 3(a), the maximum shear stress, τmax, is 3.05, 2.39, and 1.72 GPa at the critical shear displacement, δc, 0.97, 0.5, and 0.41 Å for 22.5, 45, and 67.5° loading angle, respectively. These values are much smaller than the theoretical shear strength 5.77 GPa at the critical shear displacement 1.21 Å for the <110> slip case.Reference Guo and Shang8 For the <112> slip case, Fig. 3(b), the similar distinction appears, i.e., the peak shear stress is 2.94, 2.51, and 1.69 GPa at 0.72, 0.6, and 0.37 Å for 22.5, 45, and 67.5° loading angle, respectively, whereas the theoretical shear strength is 5.49 GPa at δc = 1.02 Å.Reference Guo and Shang8 These indicate that the shear strength of the interface decreases as the mixed-mode loading angle increases, which suggests that the tensile stress in the direction perpendicular to the slip plane has a substantial effect on the shear strength [see Fig. 4]. Figure 4 shows the shear strength τmax and critical shear displacement δc as a function of tensile stress σ and loading angle ϕ for the Ni/Al2O3 interface. It is found that the shear strength decreases almost linearly with the increase of the tensile stress as shown in Fig. 4(a). A similar tendency is seen for the critical shear displacement δc. This finding is identical to that reported for other metals as well, see, e.g., the detailed results of the ideal shear strength under the excitation of uniaxial stress by Černý and PokludaReference Černý and Pokluda18 and Černý et al.Reference Černý, Sesták and Pokluda19 They found that the ideal shear strength is an increasing (decreasing) function of the applied compressive (tensile) stress for all studied fcc and bcc metallic perfect crystals, whereas the covalent crystals except of carbon often exhibit a trend opposite to that for metals.Reference Umeno and Černý20 It thus appears that our results for the Ni/Al2O3 interface are more analogue to the former case.
The shear strength τmax and critical shear displacement δc as a function of tensile stress σ and loading angle ϕ for the Ni/Al2O3 interface: (a) tensile stress and (b) loading angle.
Moreover, reduction of shear strength due to influence of tensile stress also demonstrates the effect of tension softening, i.e., the tension tends to soften the shear strength of the Ni/Al2O3 interface. As shown in Fig. 4(b), the larger the loading angle is, the smaller the shear strength is, and the greater the effect of tension softening becomes. Similar effect of tension softening has also been found in other materials.Reference Lazar and Podloucky23 In the present study, mixed-mode loading situations to the Ni/Al2O3 interface produce both the tensile and shear stress fields, and the tensile stress actually lowers the interfacial shear strength. In other words, if one needs to estimate the theoretical shear strength of some materials, any calculation that does not relax the tensile stress in the direction perpendicular to the slip plane might strongly overestimate the values of shear strength. This finding is consistent with that obtained by Umeno and Kitamura,Reference Umeno and Kitamura34 where they concluded that the normal stresses strongly affect the ideal shear strength of silicon.
As mentioned above, the maximum shear stresses of the interface under any mixed-mode loading condition are much smaller than its theoretical shear strength for both the slip cases. When the Ni/Al2O3 interface is under 0° loading angle (pure shear), the shearing condition in fact corresponds to so-called constrained shear coupling with the zero tensile/compressive displacement. A nonzero compressive stress, which is inevitable to be induced under mixed-mode loading cases, see Fig. 4(a), improves the shear strength compared with that under other loading angles. According to the study of Jiang et al.,Reference Jiang, Wei, Smith, Hutchinson and Evans5 however, even if the compressive stress is relaxed to zero, which corresponds to another shearing method, i.e., unconstrained shear, the theoretical shear strength will vary by only about 5% from that in constrained shear. Therefore, the effect of compressive stress on the shear strength could be not remarkable, but rather the tensile stress strongly influences it.
The other important mechanical parameters are so-called interfacial potential energy Ψ and unstable stacking energy γus. We define Ψ as the energy of supercell of two blocks separated by the tensile displacement θ and slipped by the displacement r. Figure 5 displays the first-principle calculations results of Ψ. It is seen that Ψ increases with the increase of the displacement δ, except for the 0° loading angle (pure shear), and the general tendency is similar for all the mixed-mode loading angles in both the slip cases. It is proper to conclude from these results that the interfacial potential energy Ψ is much insensitive to the loading angle under mixed-mode loading. In addition, the unstable stacking energy γus was first developed by Rice,Reference Rice35 which is identified as the area under the shear stress τ versus shear displacement r curve between r = 0 and r = b/2, where b equal to the Burgers vector (more generally, γus is the area between r = 0 and the first r = r t at which τ = 0 again). The first-principle calculations results of γus are shown in Figs. 6(a) and 6(b) for the <110> and <112> slip cases, respectively. The maximum values of γus under mixed-mode loadings along the nonzero angle direction are generally lower than that under constrained shear, i.e., the 0° direction slip (θ = 0) in the figures, and the maximum value of unstable stacking energy decreases as the mixed-mode loading angle increases. In short, two facts are obtained from the above results, i.e., Ψ is not much dependent on the loading angle and γus reduces with the loading angle. These demonstrate that the tension contribution to the potential energy becomes more notable as the loading angle increases. These results suggest that tensile stress has an essential influence on the unstable stacking energy. Recalling that γus is the energy barrier to be overcome in the block-like shear of one-half of a crystal relative to the other, we made use of γus to measure the resistance to sliding of the Ni/Al2O3 interface.Reference Guo and Shang8 As the loading angle increases, the tensile stress tends to reduce the resistance to sliding of the Ni/Al2O3 interface.
Potential energy Ψ from first-principle calculations of the Ni/Al2O3 interface: (a) the <110> slip case and (b) <112> slip case.
Unstable stacking energy γus from first-principle calculations of the Ni/Al2O3 interface: (a) the <110> slip case and (b) <112> slip case.
B. Failure mechanism
To better understand the failure process of the Ni/α-Al2O3 interface under the mixed-mode loading conditions, the atomic bonds near the interface are checked in detail. Figures 7 and 8 show a series of snap shot for notable changes in atomic bonds during the mixed-mode loading process for both the <110> and <112> slip cases, respectively.
Pictorial representation showing the evolution of the interfacial atomic bonds for the <110> slip case under mixed-mode loading angle: (a) 22.5°, (b) 45°, and (c) 67.5°. Blue and red/black ring represents the Ni–O bond and Al–Ni bond, respectively; green ring represents the Ni atomic reconfiguration.
Pictorial representation showing the evolution of the interfacial atomic bonds for the <112> slip case under mixed-mode loading angle: (a) 22.5°, (b) 45°, and (c) 67.5°. Blue and black ring represents the Ni–O bond and Al–Ni bond, respectively; green ring represents the Ni atomic reconfiguration.
For the <110> slip case, see Fig. 7, when the tensile displacement reaches 0.3 Å, the weaker Ni–O bonds (blue ring) are broken first for each loading angle case. With the increase of the applied load, a new Al–Ni bond (red ring) emerges resulting in appearance of two Al–Ni bonds (red and black rings) in the interface. After that, the applied load continues to increase until one of the Al–Ni bonds becomes broken. When the tensile displacement reaches to the critical value, the other Al–Ni bonds across the Ni/Al2O3 interface also breaks. The generation of a new Al–Ni bond and the premature breakage of one Al–Ni bond have effectively delayed the final failure of the Ni/Al2O3 interface. Noticing the gradual drop of the tensile stresses after the critical tensile displacement in Fig. 2, the failure of this interface under any mixed-mode loading case takes place by breaking the Al–Ni bond in a ductile fracturing manner. This failure feature is similar to that of the Ni/Al2O3 interface under uniaxial extension or uniaxial tension for all the Al-terminated interface models.Reference Guo and Shang6 It should be mentioned that when the tensile displacement θ exceeds 0.5 Å, as can be observed in the 22.5° loading angle during the <110> sliping process, Fig. 7(a), the atomic rearrangement of two Ni atomic layers (green ring) near the interface takes place distinctly, coupling with the rebonding of the Al–Ni bond near the interface. With the increase of the applied load, the tensile stress continues to increase so as to overcome the generation of a new Al–Ni bond until the Ni/Al2O3 interface fractures; meanwhile, the shear stresses become from positive to negative, see Fig. 3(a), which correspondingly characterizes the reconfiguration of two Ni atomic layers after the shear displacement r = 1.21 Å [i.e., θ > 0.5 Å in Fig. 7(a)].
For the <112> slip case, see Fig. 8, the early breakage of the weaker Ni–O bonds (blue ring) are also observed at 0.3 Å tensile displacement for all the loading angles, along with the reconfiguration of the third Ni atomic layer (green ring) near the interface. Before the failure of the interface, the skew Al–Ni bond (black ring) tends to gradually unbend and extend under the mixed-mode loading conditions. This corresponds well to the tensile stress platforms observed in Fig. 2(b). Finally, the interface fails by breaking the Al–Ni bond in a ductile manner. It should be note that although the tensile stress reaches to its peak value at 0.7 Å tensile displacement along 22.5° loading angle, the fracture of the interface does not happen [see Fig. 8(a)]. This is because that when the tensile stress reaches to its maximum, the third Ni atomic layer (green ring) has taken place a reconfiguration, resulting in that the shear stresses become from positive to negative as shown in Fig. 3(b).
C. A discussion on the potential function and its development
On the topic of potential energy Ψ, one has to refer to the pioneering work by Sun et al.Reference Sun, Beltz and Rice21 suggested an analytical form of Ψ for the combined opening and slip displacements between two atomic planes with an initial separation h, wherein the tensile and shear stress–displacement relationships are well approximated by the universal binding relationReference Rose, Ferrante and Smith24 and the sinusoidal force law of Frenkel.Reference Frenkel25 Based on this excellent development, some researchers derived the analytical expression of tension–shear coupling potential energy for different materials.Reference Jiang, Wei, Smith, Hutchinson and Evans5,Reference da Silva, Beltz and Machová22,Reference Lazar and Podloucky23 For the Ni/Al2O3 interface, our previous studiesReference Guo and Shang6,Reference Guo and Shang8 have successively obtained the tensile stress–displacement relationships along the <001> direction and shear stress–displacement relationships slipping along <110> and <112> directions under constrained shear. According to the original derivation process of Sun et al., the following formulation for Ψ(r, θ) can be proposed in terms of Ψ associated with the shear displacement r and the tensile one θ for the <110> slip case:
$$\Psi (r,{\rm{\theta }}) = W_{{\rm{sep}}} \left\{ {1 - [1 + \left( {{{\rm{\theta }} \mathord{\left/ {\vphantom {{\rm{\theta }} L}} \right. \kern-\nulldelimiterspace} L}} \right)\exp ( - {{\rm{\theta }} \mathord{\left/ {\vphantom {{\rm{\theta }} L}} \right. \kern-\nulldelimiterspace} L})] + \sin ^2 \left( {{{{\rm{\pi }}r} \over b}} \right) \times \left[ {q + {{q - p} \over {1 - p}}({{\rm{\theta }} \mathord{\left/ {\vphantom {{\rm{\theta }} L}} \right. \kern-\nulldelimiterspace} L})} \right]\exp ( - {{\rm{\theta }} \mathord{\left/ {\vphantom {{\rm{\theta }} L}} \right. \kern-\nulldelimiterspace} L})} \right\}\quad ,$$and for the <112> slip case:
$$\Psi (r,{\rm{\theta }}) = W_{{\rm{sep}}} \left\{ \matrix{ 1 - \left[ {1 + \left( {{{\rm{\theta }} \mathord{\left/ {\vphantom {{\rm{\theta }} L}} \right. \kern-\nulldelimiterspace} L}} \right)} \right]\exp ( - {{\rm{\theta }} \mathord{\left/ {\vphantom {{\rm{\theta }} L}} \right. \kern-\nulldelimiterspace} L}) \hfill \cr + \left[ {\sin ^2 \left( {{{{\rm{\pi }}r} \over b}} \right) + {{0.65} \over 2}\sin ^2 \left( {{{{\rm{2\pi }}r} \over b}} \right)} \right] \times \left[ {q + {{q - p} \over {1 - p}}({{\rm{\theta }} \mathord{\left/ {\vphantom {{\rm{\theta }} L}} \right. \kern-\nulldelimiterspace} L})} \right]\exp ( - {{\rm{\theta }} \mathord{\left/ {\vphantom {{\rm{\theta }} L}} \right. \kern-\nulldelimiterspace} L}) \hfill \cr} \right\}\quad ,$$where W sep is the work of separation for cleaving the two blocks, b is the Burgers vector, L is the characteristic length of the decohesion process (i.e., tensile stress undergoes a maximum at θ = L in the absence of any shear deformation), and the parameters p and q are the two dimensionless material parameters that quantify the extent of tension–shear coupling. Specifically, q is defined as q = γus/W sep (here, γus refers to the unstable stacking energy), and p is related to the dilatation undergone by the crystal when it shears to its unstable stacking position at zero normal stress. Through some calculations and fittings, all the characteristic parameters of the tension–shear coupling in Eqs. (1) and (2) are summarized in Table I. The potential energy function Ψ(r, θ) above is related to stress components through its derivatives as follows:
$${\rm{\tau }}(r) = {{\partial \Psi } \over {\partial r}};\;{\rm{\sigma }}({\rm{\theta }}) = {{\partial \Psi } \over {\partial {\rm{\theta }}}}\quad .$$Parameters describing tension–shear coupling for the Ni(111)/α-Al2O3(0001) interface, including the work of separation (W sep), the unstable stacking energy (γus), the critical length (L), and the materials parameters p and q.
In Figs. 9 and 10, we show the variations in the tensile and shear stresses obtained from Eq. (3) with respect to the tensile and shear displacements for both the <110> and <112> slip cases along different loading angles. Compared Fig. 9 with the tensile stress–displacement curves from first-principle results (Fig. 2), it appears that the analytical expression (3) could reasonably reproduce the peak tensile stresses in terms of magnitudes for the cases of loading angle 45 and 67.5°, whereas for the case of 22.5°, the predictions are not much satisfactory. In addition, we found that they are not robust enough to reproduce the particular values of θ at which the maximum tensile stresses occur. Moreover, as shown in Fig. 10, the analytical expression (3) does not fully reproduce the shear stress–displacement curves when compared with the results from first-principle calculations for the different loading angles (Fig. 3). The derived shear strengths of the interface are overestimated to some extent by comparing with the first-principle results as shown in the inset (Fig. 10) except for 22.5° in the <112> slip case. And the derived shear strengths and the critical shear displacements increase with the increase of loading angle, which are in contrast to the tendency obtained from first-principle calculations.
Tensile stress σ from analytical expression of the Ni/Al2O3 interface for different loading angles: (a) the <110> slip case and (b) <112> slip case.
Shear stress τ from analytical expression of the Ni/Al2O3 interface for different loading angles: (a) the <110> slip case and (b) <112> slip case. The derived shear strengths and critical shear displacements comparing with the results from first-principle calculations are shown in the inset.
Another pioneering work on this topic by Jiang et al.Reference Jiang, Wei, Smith, Hutchinson and Evans5 has to be mentioned here. In Ref. Reference Jiang, Wei, Smith, Hutchinson and Evans5, they have extended the potential function to predict the macroscopic toughness of the Al-terminated O-site γ-Ni(Al)/α-Al2O3 interface model. They indicate that the analytical traction–displacement curve for 45° orientation with shear along <112> is consistent well with the result from the mixed-mode first-principle calculations. However, the derived results from our present calculations do not agree well with the potential function extended by them. By examining their process of deriving the analytic function, a novel parameter, β, is introduced in the potential expression representing a dilatation coefficient. Although β corresponds to 
${\textstyle{{p - {p \mathord{\left/ {\vphantom {p q}} \right. \kern-\nulldelimiterspace} q}} \over {1 - p}}}$ in original expression of Sun et al. in form, they are confirmed by two different processes, respectively. β is determined from a best fit to normal stress–shear displacement curves in the paper of Jiang et al., although q = γus/W sep and p = θ*/L in the paper of Sun et al., where θ* is the relaxed opening displacement corresponding to tensile stress σ = 0 when the shear displacement r = 0.5b, i.e., the value of r at which τ = 0 at unstable stacking. Apparently, β reflects the effect of normal stress on the shear behavior to some extent, leading to a suitable extended potential function from Jiang et al. In the work of Jiang et al., unfortunately, only 45° mixed-mode loading angle along the <112> slip case was displayed and used to check the consistency of that with the results from first-principle calculations. Quite possibly, the traction–displacement curve obtained merely from 45° loading angle along the <112> slip case might not capture the sufficient information in tensile and shear strengths as well as the interfacial failure mechanisms.
From the above findings, we conclude that the derived stresses–displacement relationships are partly consistent with the results from first-principle calculations. These divergence can be attributed to two reasons. One reason is that a pronounced relaxation effect exists in the mixed-mode first-principle calculations, which was totally neglected in the calculations of Sun et al. The atoms out of the shadow rectangles (Fig. 1) were fully relaxed and the Ni, Al, and O atoms near the interface interacted with each other. The relaxation effect will induce the inner displacements of atoms. As already confirmed by the study,Reference Umeno and Kitamura34 the ideal shear strength can be strongly affected by such inner displacements. The inner displacements of atoms under the mixed-mode loading condition will reduce the shear strengths of the Ni/Al2O3 interface, which are smaller than that reproduced by the analytical formations for different loading angles. Whereas the actual tensile direction of atoms is exactly along the <001> direction, the tensile stress–displacement relationships can be reasonably derived out. The other reason is that the analytical potentials do not fully take into account of the facts observed in the Ni/Al2O3 interface under mixed-mode loading conditions. As mentioned above, the reconfiguration of Ni atomic layers and the generation of a new Al–Ni bond all have influences on the slip behavior during the mixed-mode loading. However, these particular physical phenomena are not characterized in the analytical formations yet. These imply that further modification should be done in constructing the constitutive relation of the Ni/Al2O3 interface to best match the data obtained from numerical simulations or experimental measurements. This will be the topic for future research.
IV. CONCLUSIONS
In this study, the mixed-mode simulations of the Ni(111)/α-Al2O3(0001) interface by first-principle approach are performed to investigate the mechanical responses and also to verify the analytical relations for the interfacial potential energy that is suggested by Sun et al. Two types of slip systems that displace along 22.5, 45, and 67.5° orientations with respect to the interface are considered, i.e., the shear displacements are applied along the <110> and <112> directions of the Ni lattice. The mechanical responses under mixed-mode loading conditions are carefully examined, and special attention is paid to the effect of tensile stress on the shear behavior and the failure mechanism of the interface. And the deviations of interfacial potential energy from the results of first-principle calculations are investigated as well, and the properness of the analytical forms of the potential energy functions proposed in the literature is discussed. From our computational results and analytical developments, the following conclusions can be reached:
(1) For both the slip cases, the tensile stress–displacement relationships and the potential energy obtained from the first-principle calculations are not much sensitive to the mixed-mode loading angles, and the maximum tensile stresses are close to the theoretical tensile strength of the Ni/Al2O3 interface. The failure of this interface under any mixed-mode loading is taken place by breaking the Al–Ni bond in a ductile manner along the interlayer between the Ni and Al layers, which is similar to that under uniaxial extension/tension.
(2) Due to the substantial effect of the tensile stress, the shear strength, critical shear displacement, and unstable stacking energy obtained from the first-principle calculations decrease with the increase of loading angle for both the slip cases, and they are much smaller than the predicted values of 0° loading angle, respectively. Tension softening effect can be also observed in this Ni/Al2O3 interface, and the tension contribution to the potential energy becomes more notable as the loading angle increases.
(3) Analytical relations based on the formulation of Sun et al. could partially provide a reliable description of the tensile stress–displacement relationships for the Ni/Al2O3 interface under the mixed-mode loading conditions; however, the analytical shear stress–displacement curves are not consistent with the first-principle results. This is owing to no consideration of the relaxation effect and particular physical phenomenon in the analytical expressions.
ACKNOWLEDGMENTS
This work was supported by NSAF through Grant No. U1330116 and Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20110201110019).
