Interfacial effect on deformation and failure of Al/Cu nanolaminates under shear loading

To investigate the response of Al/Cu nanolaminates to shear loading, four typical stress–strain curves with different interface orientations are analyzed here, and the repeat layer spacing λ is 5 nm. On the basis of the inflection points on the shear stress–strain ($\tau-\gamma$) curves as well as dislocation evolutions, different deformation stages can be identified for each case of in-plane shear loading (figure 2).

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For the Al(1 0 0)/Cu(1 0 0) and Al(1 1 1)/Cu(1 0 0) nanolaminates (figure 2), these stages are (I) elastic, (II) yield and (III) failure stages. In stage I (γ is below the yield strain $\gamma_{\rm Y}$), τ increases with γ. Since no dislocations are generated in Al or Cu layers as well as at the Al/Cu interfaces, $\tau= G^{\rm e}\gamma$. Here $G^{\rm e}$ is the effective elastic shear modulus of the nanolaminates. The yield strain $\gamma_{\rm Y}$ for the Al(1 0 0)/Cu(1 0 0) and Al(1 1 1)/Cu(1 0 0) nanolaminates are 0.038 and 0.075, respectively. Since the shear strength of Cu(1 0 0) is higher than that of Al(1 0 0) and Al(1 1 1) [37], in-layer dislocations are generated in the Al layers, leading to the first inflection point on the $\gamma-\tau$ curves between stages I and II. This phenomenon is also consistent with the results of metal nanolaminates under tensile loading [22]. $\gamma_{\rm Y}$ and failure stress $\tau_{\rm F}$ at stage II are different for the Al(1 0 0)/Cu(1 0 0) and Al(1 1 1)/Cu(1 0 0) nanolaminates; this will be discussed in the following dislocation analysis. With the increase in γ, τ increases to $\tau_{\rm F}$, which corresponds to the second inflection point on the $\gamma-\tau$ curves between stages II and III. The failure strains for Al(1 0 0)/Cu(1 0 0) and Al(1 1 1)/Cu(1 0 0) nanolaminates are $\gamma_{\rm F}=0.085$ and 0.095, respectively. Meanwhile, there is a remarkable difference in $\tau_{\rm F}$ between these two types of nanolaminates (see more discussion in dislocation analysis). At stage III, the nanolaminates undergo plastic flow with $\tau\sim1.0$ GPa, and the slip bands appear in the Cu layers. The τ-values of these nanolaminates in plastic flow are close to that of pure Cu and higher than that of pure Al [37].

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Compared with the Cu(1 0 0)-stacking nanolaminates (figure 2), there are only two stages (the yield stage is absent) for Al(1 0 0)/Cu(1 1 1) and Al(1 1 1)/Cu(1 1 1) nanolaminates (figure 3). Similar to the Cu(1 0 0)-stacking nanolaminates, stage I also represents elastic deformation. The difference in failure strain $\gamma_{\rm F}$ is minor between Al(1 0 0)/Cu(1 1 1) and Al(1 1 1)/Cu(1 1 1) nanolaminates, but drastic in failure stress $\tau_{\rm F}$: the $\gamma_{\rm F}$-values for Al(1 0 0)/Cu(1 1 1) and Al(1 1 1)/Cu(1 1 1) are 0.09 and 0.085, and the $\tau_F$-values are 2.5 GPa and 1.9 GPa, respectively. Especially, there is approximately a stage of periodic evolution for the Al(1 1 1)/Cu(1 1 1) nanolaminates, a common phenomenon for FCC metal under shear loading in the (1 1 1) plane [38].

In general, the repeat layer spacing λ, as well as interface orientations, affects the mechanical properties of metallic nanolaminates [14, 22]. Thus, it is necessary to study the effect of λ on yield and failure of Al/Cu nanolaminates. Here, λ is varied from 4 to 100 nm, a common range for metallic nanolaminates [16–20].

According to the stress superposition principle for composites with interfaces [38], the effective shear stress in nanolaminates $\tau^{\rm e}$ is given as:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{equ2} \tau^{\rm e}=\tau^{\rm Al}\eta^{\rm Al}+\tau^{\rm Cu}\eta^{\rm Cu}+\tau^{\rm i}, \nonumber \end{align} \tag{ 2 }$$

where $\tau^{\rm Al}$ and $\tau^{\rm Cu}$ denote shear stresses in Al and Cu layers, $\newcommand{\e}{{\rm e}} \eta^{\rm Al} = \eta^{\rm Cu} = 0.5$ are the volume fractions of Al and Cu layers, and $\tau^{\rm i}$ is the shear stress at Al/Cu interfaces. Equation (2) can be rewritten in terms of λ as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{equ3} \tau^{\rm e}(\lambda)=\left\{\begin{array}{@{}ll@{}}\frac{1}{2}\tau^{\rm Al}+\frac{1}{2}\tau^{\rm Cu}+\tau^{\rm i}(\lambda)\qquad\quad {\rm for}~ \gamma < \gamma^{\rm Al}_{\rm F} \nonumber \\ \frac{1}{2}\overline{\tau}^{\rm Al}+\frac{1}{2}\tau^{\rm Cu}+\tau^{\rm i}(\lambda)\quad\, \quad\quad{\rm for}~\gamma^{\rm Al}_{\rm F} < \gamma < \gamma^{\rm Cu}_{\rm F} \nonumber \\ \frac{1}{2}\overline{\tau}^{\rm Al}+\frac{1}{2}\overline{\tau}^{\rm Cu}+\tau^{\rm i}(\lambda)\qquad\quad {\rm for}~\gamma > \gamma^{\rm Cu}_{\rm F}\end{array}\right.. \nonumber \end{align} \tag{ 3 }$$

Here, the $\overline{\tau}^{\rm Al}$ and $\overline{\tau}^{\rm Cu}$ are the shear stresses when Al and Cu layers are in the plastic flow stage, and $\gamma^{\rm Al}_{\rm F}$ and $\gamma^{\rm Cu}_{\rm F}$ are the failure strains of Al and Cu layers, respectively. In general, there are three stages of deformation for Al/Cu nanolaminates. When $\gamma < \gamma^{\rm Al}_{\rm F}$, the $\tau-\gamma$ curve is elastic for both Al and Cu layers. When $\gamma^{\rm Al}_{\rm F} < \gamma < \gamma^{\rm Cu}_{\rm F}$, plastic deformation occurs in Al layers but not in Cu layers. In this semi-plastic situation, the stress in Al layers in plastic flow becomes $\overline{\tau}^{\rm Al}$, while Cu layers are still in the elastic stage. When $\gamma > \gamma^{\rm Cu}_{\rm F}$, both Al and Cu layers undergo plastic flow. The critical stresses and strains for pure Al and Cu [37, 38] are listed in table 1 for comparison. For shearing of FCC metals in (1 1 1) plane, the flow stress periodically increases and decreases due to recovery and generation of slip bands [36–38]. Thus, the $\tau_{\rm F}$ and $\overline{\tau}$ for (1 1 1) shearing are identical in table 1. The $\gamma_{\rm F}$-values of Al and Cu layers are reduced due to the Al/Cu interfaces compared to their counterparts for pure Al and Cu.

Table 1. Critical stress and strain for pure Al and Cu under shear loading.

Metal	Shear plane	Failure strain ($\gamma_{\rm F}$)	Failure stress ($\tau_{\rm F}$)/GPa	Stress in plasticity flow ($\overline{\tau}$)/GPa
Al	(1 0 0)	0.128	3.25	0.50
Al	(1 1 1)	0.095	1.79	1.79
Cu	(1 0 0)	0.142	8.91	1.00
Cu	(1 1 1)	0.106	2.54	2.54

However, as shown in section 3.1, the semi-plastic stage ($\gamma^{\rm Al}_{\rm F} <\gamma < \gamma^{\rm Cu}_{\rm F}$) only exists in the Cu(1 0 0)-stacking nanolaminates. Although the $\tau_{\rm F}$-value of Cu(1 1 1) is higher than those of Al(1 0 0) and Al(1 1 1), the semi-plastic stage is absent for the Cu(1 1 1)-stacking nanolaminates. Thus, the interfacial stress plays an important role in the shear response of nanolaminates, which depends on interface orientations and repeat layer spacing λ.

The results for λ ranging from 4 to 20 nm are illustrated in figure 4. $\lambda= 50$ and 100 nm are also explored, and the results are consistent. If we assume $\tau^{\rm i}(\lambda)=0$, the yield strain of these nanolaminates should be equal to that of pure Al. However, as shown in figure 4(a), $\gamma_{\rm Y}$ is less than that of pure Al, especially for the Al(1 0 0)/Cu(1 0 0) nanolaminates. In addition, as shown in figure 4(b), $\tau_{\rm F}$ of Al(1 1 1)/Cu(1 0 0) nanolaminates is higher than $\tau^{\rm Al(1\, 1\, 1)}_{\rm F}$, while $\tau_{\rm F}$ of Al(1 0 0)/Cu(1 0 0) nanolaminates is less than $\tau^{\rm Al(1\, 0\, 0)}_{\rm F}$. Following equation (3), when $\gamma < \gamma_{\rm Y}$,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{equ4} \tau^{\rm e}=\frac{1}{2}G^{\rm Al}\gamma+\frac{1}{2}G^{\rm Cu}\gamma+\tau^{\rm i}. \nonumber \end{align} \tag{ 4 }$$

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Here, $G^{\rm Al}$ and $G^{\rm Cu}$ are the shear modulus of Al and Cu, respectively. At 300 K, the shear moduli of Al in the (1 0 0) and (1 1 1) planes are 26 GPa and 25 GPa, while the shear moduli of Cu in these two planes are 68 GPa and 28 GPa, respectively [36–38]. We found that the shear modulus of nanolaminates was determined by interfacial combinations and independent of repeat layer spacing. According to our MD simulations, the shear moduli $G^{\rm Al(1\, 0\, 0)/Cu(1\, 0\, 0)}$, $G^{\rm Al(1\, 0\, 0)/Cu(1\, 1\, 1)}$, $G^{\rm Al(1\, 1\, 1)/Cu(1\, 0\, 0)}$ and $G^{\rm Al(1\, 1\, 1)/Cu(1\, 1\, 1)}$ are about 45 GPa, 26 GPa, 41 GPa and 24 GPa, respectively. Assuming $\tau^{\rm i}$ is negligible in equation (4), the effective shear modulus of A/B nanolaminates is approximatively given as $G^{\rm A/B}=(G^{\rm A}+G^{\rm B})/2$, which is in agreement with our MD results. For the Cu(1 0 0)-stacking nanolaminates, $G^{\rm Cu(1\, 0\, 0)}$ is much larger than $G^{\rm Al(1\, 0\, 0)}$ and $G^{\rm Al(1\, 1\, 1)}$. Thus, $\tau^{\rm e}$ is generally larger than $\tau^{\rm Al}$ at the same γ, and $\tau_{\rm Y}$ of the Al(1 1 1)/Cu(1 0 0) nanolaminates is higher than $\tau^{\rm Al(1\, 1\, 1)}_{\rm F}$. However, the $\tau-\gamma$ curves (figure 2) show a remarkable premature yield for the Al(1 0 0)/Cu(1 0 0) nanolaminates; $\gamma_{\rm Y}$ of the Al(1 1 1)/Cu(1 0 0) nanolaminates is close to $\gamma_{\rm F}$, resulting in the anomalous relationship of $\tau^{\rm Al(1\, 0\, 0)/Cu(1\, 0\, 0)}_{\rm Y} < \tau^{\rm Al(1\, 0\, 0)}_{\rm F}$. This phenomenon is explained through dislocation analysis in the following section. Moreover, both $\gamma_{\rm Y}$ and $\tau_{\rm Y}$ are weakly dependent on λ. This indicates that the interfacial stress plays the dominant role in yield, which is in turn determined by the interface orientations instead of λ.

As mentioned in section 3.1, with increased shear strain, plastic deformation occurs in Cu layers, which leads to the structural failure of nanolaminates immediately. In this situation, both Al and Cu layers undergo plastic flow. Before structural failure, the effective stress of nanolaminates is

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{equ5} \tau^{\rm e}=\frac{1}{2}\overline{\tau}^{\rm Al}+\frac{1}{2}\tau^{\rm Cu}+\tau^{\rm i}. \nonumber \end{align} \tag{ 5 }$$

$\gamma_{\rm F}$ of Al/Cu nanolaminates ranges from 0.075 to 0.1, and is still independent of λ (figure 5(a)). Since there is a slight difference in $\gamma_F$ for these nanolaminates, the interface orientations do not have a strong effect on $\gamma_{\rm F}$. On the other hand, although $\tau_{\rm F}$ of the nanolaminates is also independent of λ (figure 5(b)), there is a remarkable difference in $\tau_{\rm F}$ among nanolaminates with different interface orientations: $\tau^{\rm Al(1\, 1\, 1)/Cu(1\, 0\, 0)}_{\rm F} \approx \tau^{\rm Al(1\, 0\, 0)/Cu(1\, 0\, 0)} > \tau^{\rm Al(1\, 0\, 0)/Cu(1\, 1\, 1)}_{\rm F} >$$\tau^{\rm Al(1\, 1\, 1)/Cu(1\, 1\, 1)}_{\rm F}$. The results for $\lambda =50$ and 100 nm are also consistent with figure 5. However, for the Al(1 0 0)/Cu(1 0 0) nanolaminates with $\lambda <12$ nm, $\tau_{\rm F}$ increases with increasing λ; thus, the interfacial stress $\tau^{\rm i}$ is influenced by λ in such cases.

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From equation (5), we can obtain shear strength of nanolaminates, i.e. failure shear stress $\tau^{\rm e}_{\rm F}$, as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{equ6} \tau^{\rm e}_{\rm F}=\frac{1}{2}\overline{\tau}^{\rm Al}+\frac{1}{2}\tau^{\rm Cu}_{\rm F}+\tau^{\rm i}=\tau^{\rm l}+\tau^{\rm i}, \nonumber \end{align} \tag{ 6 }$$

where $\tau^{\rm l}=\frac{1}{2}\overline{\tau}^{\rm Al}+\frac{1}{2}\tau^{\rm Cu}_{\rm F}$ is the average stress in metal layers. As shown in figure 5(b), the shear strength of these nanolaminates are less than $\tau^{\rm l}$, but higher than $\overline{\tau}$ of pure Al and Cu. Moreover, $\tau_{\rm F}$ of Cu(1 0 0)-stacking nanolaminates is considerably higher than that of Cu(1 1 1)-stacking nanolaminates, and $\tau_{\rm F}$ of Al(1 1 1)/Cu(1 1 1) nanolaminates is the lowest. Along with the previous analysis of yield, we demonstrate that both yield and failure of Al/Cu nanolaminates are mainly controlled by interface orientations. The effect of repeat layer spacing λ can be neglected in most situations except for Al(1 0 0)/Cu(1 0 0) nanolaminates.

To reveal the mechanisms underlying macroscopic phenomena of plasticity and failure observed above, we perform atomic-level dislocation analysis below.

During the yield stage of Cu(1 0 0)-stacking nanolaminates, there is a remarkable premature yield for Al(1 0 0)/Cu(1 0 0) nanolaminates, which results in an anomalous relationship of $\tau^{\rm Al(1\, 0\, 0)/Cu(1\, 0\, 0)}_{\rm Y} \ll \tau^{\rm Al(1\, 0\, 0)}_{\rm Y}$. For Al(1 0 0)/Cu(1 0 0) nanolaminates (figure 6(a)), semi-coherent interfaces are generated due to large mismatch (12%) of lattice parameters, which lead to the misfit network (perfect dislocations, blue lines) between Al and Cu layers [21, 52, 53]. The perfect dislocations may transform into new dislocations even in low shear strains [22] in two ways: (i) one interfacial misfit dislocation transforms to one stair-rod ($\frac16\langle 1\, 1\, 0 \rangle$) and two leading Shockley partials ($\frac16\langle 1\, 1\, 2 \rangle$) dislocations [22, 52]; and (ii) one interfacial misfit dislocation changes to one Frank partial ($\frac13\langle 1\, 1\, 1 \rangle$) and one Shockley partial ($\frac16\langle 1\, 1\, 2 \rangle$) dislocation [54]. These Shockley dislocations extend into Al layers, while the stair-rod dislocations remain at the interfaces. These dislocations merge under a small shear strain of about 0.038, and this eventually leads to the premature yield of Al(1 0 0)/Cu(1 0 0) nanolaminates. In contrast, the interfaces in Al(1 1 1)/Cu(1 0 0) nanolaminates are incoherent. As shown in figure 6(b), the Shockley dislocations can extend into Al layers only when shear strain reaches a value close to $\gamma^{\rm Al(1\, 1\, 1)}_{\rm Y}$, and the plastic deformation of Al layers is induced by trailing Shockley dislocations emitted from neighboring interfaces. These processes are completely determined by the interface orientations, and yield of Cu(1 0 0)-stacking nanolaminates is independent of repeat layer spacing λ.

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The stress–strain curves show that the yield stage is absent for the Cu(1 1 1)-stacking nanolaminates. As shown in figure 7, there are no stacking faults before plastic deformation occurs in Al layers, since $\gamma^{\rm Cu(1\, 1\, 1)}_{\rm Y}$ is close to $\gamma^{\rm Al(1\, 1\, 1)}_{\rm Y}$ and less than $\gamma_{\rm Y}^{\rm Al(1\, 0\, 0)}$. However, when shear strain reach the yield shear strain of Cu(1 1 1)-stacking nanolaminates, dislocations are generated from interfaces and extend into Al and Cu layers simultaneously, and both Al and Cu layers undergo plastic flow, leading to a structural failure. Thus, failure occurs in the nanolaminates immediately after the initial dislocation activities. This is consistent with the effect of yield strain difference on dislocation evolution of Cu/Ta nanolaminates [55]. Moreover, according to equation (3), the interfacial dislocation evolution results in strengthening for Al layers but softening for Cu layers. This effect is also in agreement with a previous study [56]. Since the interfacial stress is determined by interface orientations (rather than λ), the failure of nanolaminates is also independent of λ.

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In particular, $\tau_{\rm F}$ of Al(1 0 0)/Cu(1 0 0) nanolaminates decreases with decreasing λ for $\lambda <12$ nm, while $\gamma_{\rm F}$ is not influenced by λ (figure 5). Due to stress concentration at interfaces, misfit dislocations are transformed and extend into metal layers [52, 57, 58]. Dislocations transformation leads to a premature yield for Al(1 0 0)/Cu(1 0 0) nanolaminates, and there are only leading Shockley dislocations generated without the emission of trailing Shockley dislocations. Thus, stacking faults do not disappear and even become longer [52, 59]. As shown in figure 8, the two types of stacking faults generated through Shockley partial dislocations, are at 54.7° and 125.3° with the interfaces, respectively. Thus, these stacking faults are not parallel, and there must be interactions between the stacking faults and dislocations. The stacking faults impede the extension of dislocation, which leads to the increase of strain and stress [60]. With increasing λ, the length of stacking faults increases as shown in figure 8. Thus, since the probability of dislocation interactions increases, $\tau_{\rm F}$ of Al(1 0 0)/Cu(1 0 0) nanolaminates increases with λ. However, with increasing λ, the probability of dislocation interactions reaches a limit, and the related shear stress is about 3.0 GPa [36, 37]. Thus, $\tau_{\rm Y}$ of Al(1 0 0)/Cu(1 0 0) nanolaminates increases with increasing λ for $\lambda <12$ nm, while it becomes a constant when $\lambda \geqslant 12$ nm. The low strength of Al/Cu nanolaminates with $\lambda <12$ nm should be considered in fabrication, since it is easier to tailor λ [32] than interface orientations [61, 62].

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