Introduction

The nanoscale and interface effects crucially influence plastic deformation mechanisms/modes operating in nanocrystalline (NC) and ultrafine-grained (UFG) bulk materials, ultrathin films, and nanowires showing excellent mechanical properties (see, e.g., [1–11]). For instance, in NC and UFG metallic materials, conventional lattice slip is hampered by large amounts of grain boundaries (GBs), in which case plastic flow often occurs through alternative deformation modes mediated by GBs (see, e.g., [4]). In particular, the stress-driven athermal migration of GBs is recognized as one of the alternative deformation modes effectively operating in NC and UFG metals in wide range of their structural parameters [4, 12–34]. The stress-driven GB migration leads to grain growth that destroys the NC/UFG structure of metals and thus results in degradation of their properties attributed to such a structure.

In general, conventional thermally activated grain growth is viewed as the key negative factor that can come into play in NC/UFG metals during their fabrication and thus destroy their NC/UFG structure [4]. In order to inhibit the thermally activated grain growth, several approaches are utilized which are also typical for microcrystalline and coarse-grained polycrystals [35, 36]. Among the most effective approaches is the use of nanoinclusions (like nanoscale precipitates of the second metallic phase in metallic alloys and second-phase ceramic nanoparticles in metal matrix nanocomposites) as obstacles for GB migration (see, e.g., [4, 37, 38]). In addition to their role as inhibitors of thermally activated grain growth, nanoinclusions serve as reinforcing structural elements that hinder lattice dislocation slip (see, e.g., [8, 39, 40]) and affect crack growth [41–43].

At the same time, the research efforts focused on the effects exhibited by nanoinclusions on the stress-driven GB migration in NC and UFG metallic materials are limited. In particular, very recently, the effects of Al2O3 ceramic nanoinclusions on both stress-driven GB migration and grain growth processes in nanostructured Al matrix and Al alloy matrix nanocomposites have been experimentally examined by Lin with co-workers [44, 45]. In their works, nanocrystalline Al having an initial grain size of 79 nm and containing 13 vol% of Al2O3 nanoparticles with an average size of 27 nm, has been extruded at a temperature of 400 °C with the strain rate of 0.6 s−1 until an area reduction of 9–10. Hot extrusion of nanocrystalline Al containing Al2O3 nanoparticles resulted in significant grain growth (up to grain size of 177 nm). In the situation without extrusion, essential grain growth has not occurred even after annealing at a higher temperature of 600 °C. In another experiment, Dám et al. [46] observed stress-driven migration of low-angle GBs in the ultrafine-grained (with a grain size of 200–300 nm) Al-3 Mg-0.2Sc (wt%) alloy, containing coherent nanoscale (3–5 nm in diameter) Al3Sc precipitates, during tensile deformation at 300 °C. At the same time, in the experiment [46], the migration of high-angle GBs has not been observed.

Also, a model of stress-driven grain growth hampered by ceramic nanoinclusions in nanostructured Al matrix nanocomposite was suggested [45, 47]. Also, stress-driven migration of low-angle grain boundaries in the presence of rigid impenetrable precipitates has been simulated using 3D discrete dislocation dynamics [48]. However, models [45, 47, 48] are too simple because they do not involve in examining the role of GB junctions in stress-driven GB migration. At the same time, the effects of GB junctions on thermally activated and especially stress-driven GB migration processes are critical in NC and UFG materials specified by extremely large amounts of triple junctions of GBs (see, e.g., [14, 21, 23, 28, 49–51]). The main aim of this paper is to theoretically describe the stress-driven GB migration as a plastic deformation mode in metal matrix nanocomposites containing incoherent reinforcing (ceramic or metallic) nanoinclusions, with the role of GB junctions taken into consideration. The focuses of our examination will be placed on the effects of such nanoinclusions on both the critical stresses for the GB migration processes and profiles of migrating GBs in these metal matrix nanocomposites. In particular, we will reveal the dependences of the critical stress on nanoinclusion sizes.

Stress-driven migration of low-angle grain boundaries in metal–ceramic and metal–metal nanocomposites: geometric aspects

Let us consider a nanocomposite solid consisting of an NC/UFG metal matrix and incoherent reinforcing (metallic or ceramic) nanoinclusions (Fig. 1). We examine the situation where a mechanical load is applied to the nanocomposite and drives GB migration in the metal matrix of the nanocomposite. For definiteness and simplicity, our consideration will be focused on stress-driven migration of low-angle tilt boundaries conventionally represented/modeled as walls of edge lattice dislocations. Due to the high Peierls barrier for dislocation slip in the ceramic nanoinclusions and the difference in the types of the crystal lattices between the matrix and the metallic nanoinclusions, migrating GBs cannot penetrate the nanoinclusions. Instead, GBs are bent around the incoherent nanoinclusions. In doing so, the migrating GBs can either stop at nanoinclusions or move further, depending on the applied load as well as both the geometry and the size of the GBs and nanoinclusions.

Fig. 1
figure 1

(Color figure online) Stress-driven migration of a grain boundary in a nanocomposite solid containing hard (ceramic or metallic) nanoinclusions inside grains of a metallic matrix. a A nanocomposite specimen is under a mechanical load (a two-dimensional general view.). b The magnified region [bounded by an ellipse in (a)] highlights the initial state of the nanocomposite, before the grain boundary migration. c The magnified region [bounded by an ellipse in (a)] highlights the final state of the nanocomposite, after the stress-driven grain boundary migration

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In order to calculate the critical parameters at which a migrating GB can bypass a nanoinclusion, we consider a symmetric low-angle tilt boundary terminated at triple junctions, A and B, of GBs in a composite solid in its initial state before the migration (Fig. 2a). Although, in general, migrating low-angle GBs can be of an arbitrary (tilt, twist, or mixed) type, we focus our consideration on the most widespread tilt boundaries. Each such boundary consists of edge dislocations and can easily migrate, as compared to twist and mixed GBs. In the initial state, in the absence of a mechanical load, the low-angle boundary AB is represented as a straight wall of periodically arranged edge dislocations having the same Burgers vector b. The low-angle boundary is characterized by the tilt misorientation angle θ being in the Frank relationship \( \left[ {{ \sin } \left( {\theta / 2} \right) = b/\left( { 2h} \right)} \right] \) [52] with the parameters (period h and Burgers vector magnitude b) of the dislocation arrangement in the boundary.

Fig. 2
figure 2

(Color online) Migration of a low-angle tilt boundary near an incoherent nanoinclusion impenetrable for dislocations

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In the initial state before the migration process, other GBs (AC, AD, BE, and BF) adjacent to the GB AB are assumed to be symmetric tilt boundaries that form the geometrically compensated triple junctions A and B with the boundary AB (Fig. 2a). The triple junctions A and B are geometrically compensated in the sense that there are no angle gaps at these junctions. In other words, the sum of tilt misorientation angles of all GBs joining at each of these junctions is equal to zero, where summation of the angles is performed clockwise along a circuit surrounding a triple junction [53, 54]: \( \theta + \theta_{\text{AC}} + \theta_{\text{AD}} = 0 \) and \( - \theta + \theta_{\text{BF}} + \theta_{\text{BE}} = 0 \). Here θ AC, θ AD, θ BF, and θ BE are the tilt misorientation parameters of the GBs AC, AD, BF, and BE, respectively (Fig. 2a). In the situation under examination, we will consider the dislocation structures of the GBs AC, AD, BE, and BF to be unchanged during the stress-driven migration of the GB AB (Fig. 2). In this situation, the GBs AC, AD, BE, and BF cooperatively serve as constant stress sources located at triple junctions and balanced in the initial state (Fig. 2a). More precisely, following the approach [14, 21, 23], the GBs AC and AD are modeled as a wedge disclination located at the triple junction A and characterized by the disclination strength \( - \omega = - \theta \) (Fig. 2a). The GBs BF and BE are modeled as a wedge disclination located at the triple junction B and characterized by the disclination strength \( \omega = \theta \) (Fig. 2a).

We now consider migration of the GB AB under the applied shear stress \( \tau \) in the nanocomposite. When the shear stress \( \tau \) acts in slip planes of the lattice edge dislocations belonging to the GB AB, these dislocations cooperatively glide from their initial positions (Fig. 2a) to the new positions (Fig. 2b). These stress-driven cooperative displacements of the dislocations result in the migration of the GB AB (Fig. 2b). Within our two-dimensional model, we assume that the migrating GB is retarded by a wire nanoinclusion having both a square cross section and the long axis normal to the plane of Fig. 2 (and parallel with dislocations lines that form the migrating GB). Within the model, the sides of the square that forms the nanoinclusion cross section have the length \( d \), and one of the square sides makes the angle \( \alpha \) with the normal to the GB plane (Fig. 2). Also, since the inclusion is assumed to be incoherent, it is not strained to adjust the matrix and does not create any elastic stresses. Besides, since the inclusion is incoherent, it is assumed to be impenetrable for dislocations that form the dislocation wall. The effect of coherent inclusions (that both create elastic stresses and are penetrable for dislocations) on GB migration is considered elsewhere [55]. As the main role of the inclusion is in retarding dislocations, we believe that its exact shape is not important. Therefore, although here, for definiteness, we have modeled the inclusion as a square, we believe that its effect on GB migration will be qualitatively the same for other (e.g., elliptic) shapes of the inclusion. In general, the inclusion–matrix interface can exert elastic forces on dislocations if the matrix and inclusions have vastly different elastic moduli (due to the dislocation attraction to the softer phase). However, in our manuscript, we focus our consideration on the situation where the elastic moduli of the matrix and inclusions are sufficiently close to each other, and their difference can be neglected.

Stress-driven migration of low-angle grain boundaries in metal–ceramic and metal–metal nanocomposites: stress characteristics and profile of a migrating grain boundary

Let us examine the process of the stress-driven migration of the low-angle GB AB meeting an incoherent nanoinclusion in a metal matrix nanocomposite. For definiteness, we focus our examination on the case of low homological temperatures. In this case, one can neglect the effects associated with the thermally assisted GB migration, in particular, diffusional accommodation of the stresses produced by triple junction disclinations. Besides, in this case, due to the low diffusion rate, the inclusion is immobile and cannot move together with the migrating GB. In our analysis, we will exploit the methods of the two-dimensional discrete dislocation dynamics approach employed previously for description of the formation, decay or evolution of GBs (see, e.g., [56–59]). The two-dimensional dislocation dynamics approach is a partial case of the 3D discrete dislocation dynamics method (see, e.g., [60–63]). Although 3D simulations are most effective for a detailed comprehensive analysis of migration of low-angle GBs in solids containing inclusions/precipitates, the 2D approach definitely catches essential physics of the GB migration process and, at the same time, does not need excessively complicated analytical calculations and/or computer simulations. Within the approach in question, each dislocation at the low-angle GB AB is under the combined actions of the forces created by the shear stress, other dislocations belonging to the boundary, and the disclination dipole.

We now calculate these forces and write the corresponding equations for dislocation motion. To do so, we assume that dislocations can move along one slip plane (along the x-axis in the coordinate system shown in Fig. 2), and, therefore, only the projections of the forces on the x-axis matter. In these circumstances, the solution of the system of equations, describing one-dimensional motion of dislocations, will be expressed as dependences \( x_{i} (t) \), where \( x_{i} \) is the coordinate of the ith dislocation (\( i = 1, \ldots ,N \)), and \( t \) is time. Also, in our examination, in a first approximation, we neglect the difference in the elastic moduli between the nanoinclusions and the matrix and model the nanocomposite as an elastically isotropic, homogeneous solid characterized by the shear modulus \( G \) and Poisson’s ratio \( \nu \). Also, in our two-dimensional model, we assume that if a dislocation approaches the inclusion, it stops. We, therefore, neglect the possibility for dislocations to pass the inclusion by bowing out or by double cross-slip.

With the assumption under examination, our theoretical analysis can be extended to the situation with metal matrix nanocomposites containing incoherent nanoparticles in a NC matrix (Fig. 3). In this situation, the length of moving dislocation segments is small, which prevents these segments from bypassing isolated particle-like nanoinclusions in nanograins. At the same time, if some dislocations can bypass nanoinclusion(s), the above assumption can lead to an overestimate of the effect of nanoinclusions on the critical stress for GB migration.

Fig. 3
figure 3

(Color online) Fragment of a composite solid containing a low-angle grain boundary ABCD (with an array of dislocation segments) and spherical inclusions (whose size is comparable to the width of the grain boundary), illustrating the situation where grain boundary dislocations cannot bypass inclusions

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Within the approach under discussion, the projection \( F_{i} \) of the total force acting on the ith dislocation belonging to the GB AB is written as follows [56]:

$$ F_{i} = b\tau + Db^{2} \sum\limits_{\begin{subarray}{l} k = 1 \\ k \ne i \end{subarray} }^{N} {\frac{{(x_{i} - x_{k} )[(x_{i} - x_{k} )^{2} - (y_{i} - y_{k} )^{2} ]}}{{[(x_{i} - x_{k} )^{2} + (y_{i} - y_{k} )^{2} ]^{2} }}} - Db\omega \left( {\frac{{x_{i} (y_{i} + L/2)}}{{x_{i}^{2} + (y_{i} + L/2)^{2} }} - \frac{{x_{i} (y_{i} - L/2)}}{{x_{i}^{2} + (y_{i} - L/2)^{2} }}} \right), $$
(1)

where \( D = G/[2\pi (1 - \nu )] \), \( L \) denotes the GB length (the distance between the triple junction disclinations that form the dipole (see Fig. 2)), while \( x_{i} \) and \( y_{i} = h(i - 1/2) - L/2 \) are the coordinates of the ith dislocation. The first term on the right-hand side of formula (1) describes the force created by the shear stress \( \tau \), the second term describes the force of the interaction of the ith dislocation with the other dislocations of the boundary, and the third term describes the force of the interaction of the ith dislocation with the disclination dipole. In other words, the third term characterizes the role of GB junctions in the stress-driven GB migration.

The equations for the motion of dislocations composing the dislocation wall (low-angle tilt boundary) AB have the following form:

$$ m\frac{{d^{2} x_{i} }}{{dt^{2} }} + \beta \frac{{dx_{i} }}{dt} = F_{i} ,\;\;\;\;i = 1, \ldots ,N . $$
(2)

The first derivatives dx i /dt in these equations take into account the dislocation motion friction (associated with the dynamic retardation of the crystalline lattice to the dislocation glide), and \( \beta \) is the viscosity coefficient. The dislocation mass \( m \) is given by the standard approximation [55] as \( m = \rho b^{2} /2 \), where \( \rho \) is the material density.

With Eqs. (1) and (2), we simulated the motion of the low-angle tilt boundary AB in the presence of a nanoinclusion. In the simulations, it was assumed that if a dislocation approaches the boundary of the nanoinclusion, it stops. At the same time, any \( i \) th dislocation can move back from the nanoinclusion boundary if the projection \( F_{i} \) of the total force acting on this dislocation becomes negative. Our analysis has demonstrated that similar to the case of GB migration in single-phase metallic solids [14], GB motion in the nanocomposite can occur in two different modes. In the first mode, all the dislocations eventually approach their equilibrium positions, and we refer to this mode as the limited GB migration. In the second mode, some dislocations stop at the nanoinclusion boundary, while others move unrestrictedly far away from it. We define this mode as the unlimited GB migration (Obviously, in reality, the unlimited migration of a GB is eventually stopped when this boundary meets a neighboring GB). Thus, in the simulations of dislocation motion, with Eqs. (1) and (2), we stop the simulations in the two following situations. In the first situation, the displacements of all the dislocations at a given simulation step is smaller than a preset value, that is, all the dislocations have reached their equilibrium positions corresponding to the limited mode of GB migration. In the second situation, corresponding to the unlimited mode of GB migration, some dislocations are retarded by the nanoinclusion boundary, whereas other dislocations move far enough from their initial positions.

With the calculated positions of the dislocations, we revealed the profiles of the migrating GB AB in various cases. Figures 4 and 5 show these profiles in the case of a composite solid with the titanium matrix, for the following values of parameters: \( G = 44 \) GPa, \( \nu = 0.32 \), \( b = 0.295 \) nm and \( \rho = 4506 \) kg m−3. Also, following [64], we take the value of \( \beta \) as \( \beta = 5 \times 10^{ - 5} \) Pa × s as well as put the values of other parameters as follows: α = 45°, ω = 5°, and \( N = 30 \). The latter number of dislocations at the GB AB corresponds to the GB length of \( L \approx 101 \) nm. Note that, according to our examination (that is, according to numerical computation of the equilibrium GB profiles at various ratios of \( d/L \)), for a given value of the applied shear stress, the equilibrium profiles of the GB in the normalized coordinates (\( x/d \),\( y/d \)) do not change, when the number \( N \) of dislocations changes, but the ratio \( d/L \) is constant. This means that the profiles of the migrating GBs are not sensitive to both the GB size \( L \) and the nanoparticle size \( d \) as separate parameters, but they are influenced by the ratio \( d/L \).

Fig. 4
figure 4

Geometry of a migrating grain boundary that bends around a nanoinclusion at various values of applied stress. a, b Equilibrium profiles of a migrating grain boundary. c Nonequilibrium profiles of a migrating grain boundary at successive time moments

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Fig. 5
figure 5

Geometry of a migrating grain boundary that bends around a nanoinclusion at different values of applied stress τ. a Equilibrium profile of a migrating grain boundary at τ = 0.75 GPa. b Nonequilibrium profile of a migrating grain boundary at τ = 0.85 GPa

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Figure 4a, b present the simulated equilibrium profiles of the migrating GB, for \( d/L = 0.3 \) and two different values of the applies shear stress \( \tau \). It is seen that the migrating GB is bent around the inclusion. Also, from Fig. 4, it follows that the GB migration length grows with the increasing applied stress \( \tau \). When the stress \( \tau \) reaches its critical value \( \tau_{c} \), the limited GB migration switches to the mode of unlimited migration, in which case the middle segment of the migrating GB is stopped by the nanoinclusion and separates from the upper and lower GB segments that move in the unlimited way (Fig. 4c). In other words, the GB splits into three GB segments. The middle segment is stopped by the inclusion, while the upper and lower segments (that move above and below the inclusion, respectively) are mobile. Obviously, the size of the middle GB segment (which is bent around the nanoinclusion) is smaller than the size of the whole migrating GB in Fig. 4a, b. Since the characteristic equilibrium migration length of a GB decreases with the decreasing GB length [14], the equilibrium migration length for the middle GB segment is smaller than that for the whole GB. As a result, with time, this GB segment moves back toward the initial position of the migrating GB (Fig. 4c).

It should be noted that there is an analogy between the presence of the critical stress for unlimited GB migration and that for dislocation interactions with obstacles. Below some critical stress, the force acting on the migrating GB due to the applied stress can be balanced by both the returning force created by the disclination dipole and the capillary force of the curved boundary (which bows out between the neighboring GBs or between a GB and a precipitate), and the motion stops. Above the critical stress, even the maximum GB curvature (together with the disclination dipole) cannot balance the stress and the boundary moves unlimitedly.

Figure 5 shows (a) equilibrium and (b) nonequilibrium profiles of the migrating GB, for \( d/L = 0.6 \) and two different values of the applies stress \( \tau \). As it follows from Fig. 5b, if the lengths of the upper and lower segments of the migrating GB are small, their separation weakly influences the profile of the GB segment which is bent around the nanoinclusion. In this case, the length of the middle GB segment is close to that of the entire GB, and the equilibrium migration length for this GB segment is close to the equilibrium migration length of the whole GB in the absence of the nanoinclusion. As a result, a considerable reverse motion of this GB segment does not occur, in contrast to the situation illustrated in Fig. 3c.

By comparing Figs. 4b, c, and 5a, b, one can conclude that the transition from limited to unlimited GB migration occurs at some critical applied stress \( \tau_{c} \). Previous calculations [14] have demonstrated that in the absence of a nanoinclusion, such a transition occurs at the critical stress \( \tau_{c} \approx 0.8D\omega \), for any materials parameters. Our analysis has demonstrated that in the examined case of a nanocomposite containing nanoinclusions, the ratio \( \tau_{c} /(D\omega ) \) also does not depend on the material parameters, but is sensitive to both the ratio \( d/L \) and the angle \( \alpha \). The calculated dependence of \( \tau_{c} /(D\omega ) \) on \( d/L \), for α = 45°, is presented by the dashed line in Fig. 6. Surprisingly, it appears that for small values of \( d/L \), the critical stress \( \tau_{c} \) decreases with an increase in the nanoinclusion size \( d \) (from \( 0.81D\omega \) at \( d/L = 0 \) down to the minimum value of \( 0.67D\omega \) at \( d/L = 0.09 \)). At the same time, for comparatively large values of \( d/L \), the critical stress \( \tau_{c} \) increases with increase in the ratio \( d/L \), as one could expect.

Fig. 6
figure 6

Dependence of the normalized critical stress \( \tau_{c} /(D\omega ) \) for unlimited migration of a low-angle grain boundary on the ratio d/L of the nanoinclusion size d to the grain boundary length L

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Moreover, Fig. 6 demonstrates that the critical stress \( \tau_{c} \) for the unlimited GB migration in the presence of a small nanoinclusion is lower than that in the absence of nanoinclusions. In order to explain this intuitively unexpected effect of nanoinclusions on the stress-driven GB migration, we calculated the profile of a migrating GB in the case of a small nanoinclusion specified by \( d/L = 0.1 \) and α = 45° (Fig. 7). In the case under discussion, Fig. 7 demonstrates that due to the small size of the nanoinclusion, the equilibrium migration distance of the upper and lower parts of the GB can be much larger than the nanoinclusion size. In other words, in this situation, the lower and upper GB segments separate from the immobile middle GB segment before the GB migration switches to its unlimited mode. In this case, the distance between the middle GB segment and its upper and lower segments is large. According to the theory of dislocations [65], two edge dislocations with the same Burgers vectors directed along the x-axis (see Fig. 2b) and the coordinates (\( x_{i} \), \( y_{i} \)) and (\( x_{k} \), \( y_{k} \)) repel, if their lateral distance \( |x_{i} - x_{k} | \) is larger than their vertical distance \( |y_{i} - y_{k} | \). Thus, in the discussed case, the dislocations at the middle GB segment (stopped at the nanoinclusion boundary) repel the dislocations in the upper and lower GB segments. (It is contrasted to the case of a straight GB whose dislocations do not exert forces along the x-axis on each other). As a result, the force \( F_{i} \) acting on these dislocations becomes larger than it would be in the case where the straight GB migrates as a whole. Therefore, the critical stress \( \tau_{c} \) for the unlimited GB migration in the presence of a small nanoinclusion is lower than that in the situation without a nanoinclusion.

Fig. 7
figure 7

Equilibrium profile of a migrating grain boundary that bends around a small nanoinclusion

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At the same time, for large enough values of the ratio \( d/L \), the transition from the limited migration mode to the unlimited migration mode occurs at the stress level at which the upper and lower GB segments start separating from the middle one. In this situation, due to small distances between the dislocations in the middle GB segment and the dislocations in the upper and lower GB segment, the dislocations in the middle part of the GB attract the dislocations in the upper and lower GB segments (along the x-axis). As a corollary, the force \( F_{\text{i}} \) acting on the latter dislocations decreases, compared to the case where the straight GB migrates as a whole, and the critical stress \( \tau_{\text{c}} \) increases.

Figure 6 demonstrates that for \( d/L < 0.2 \), individual inclusions can promote GB migration. In the examined two-dimensional model, this range of d/L corresponds to the range r < 0.04, where r is the ratio of the inclusion volume to the volume of the grain. Although the estimate r < 0.04 for the normalized volume of inclusions which can promote GB migration is valid in the two-dimensional model, it is logical to assume that this estimate can also be extended to the case of three-dimensional grains and inclusions. At the same time, even small inclusions can hinder migration of low-angle GBs, if the concentration of inclusions is high, so that each grain contains a large number of nanoinclusions. In this situation, the total length of moving GB segments can be significantly reduced by nanoinclusions, which should increase the critical stress for unlimited GB migration and thus hamper grain growth. The exact relation between the critical parameters of the inclusions (their critical volume fraction critical size) below which inclusions can promote migration of low-angle GBs cannot, however, be found within our model.

Finally, let us consider the stress-driven GB migration as a plastic deformation mode in metal matrix nanocomposites containing incoherent reinforcing nanoinclusions. To do so, we roughly estimate the average plastic strain \( \varepsilon \) produced by migrating GBs. In the first approximation, this plastic strain can be presented as \( \varepsilon = (\alpha \,\bar{\omega }\,\bar{s}/d)f \), where \( \alpha \) is the geometric factor (of the order of unity) arising as a result of the averaging of plastic strain over various orientations of migrating low-angle GBs, \( \bar{\omega } \) is the average misorientation angle of the migrating low-angle GBs, \( \bar{s} \) is their average migration distance, \( d \) is average grain size, and \( f \) is the fraction of low-angle GBs. In order to estimate the average plastic strain, let us put \( \alpha = 0.5 \), ω = 10°, and \( f = 0.3 \). Then, in the limited case of a very high applied stress at which all GBs move unlimitedly (in reality, across one grain), we obtain \( \bar{s}/d = 1 \), which yields \( \varepsilon = 0.026 \). In the case where the average migrating distance is much smaller than the grain size (\( \bar{s}/d = 0.1 \)), we obtain \( \varepsilon = 0.0026 \). The above estimates demonstrate that in the discussed cases, the macroscopic plastic strain produced by pre-existent migrating low-angle GBs is small. Therefore, stress-driven migration of low-angle GBs should occur in parallel with other deformation mechanisms. At the same time, note that the above estimates describe the contributions of pre-existent low-angle GBs, while plastic deformation can be accompanied by the formation of new low-angle GBs as a result of accumulation of lattice dislocations in dislocation walls [10] and/or due to splitting of the existing GBs [59]. Also, if migration of high-angle GBs can occur in parallel with the migration of low-angle ones, such migration can significantly increase the contribution of GB migration to the total plastic strain. Thus, migration of GBs contributes to the total plastic deformation and, therefore, can be considered as a particular deformation mode.

Concluding remarks

To summarize, in metal matrix nanocomposites with reinforcing (metallic or ceramic) incoherent nanoinclusions, stress-driven migration of GBs represents a special deformation mode significantly influenced by these nanoinclusions. In this paper, we theoretically examined a rather widespread situation where low-angle tilt boundaries migrate under a shear stress, and nanoinclusions cannot be penetrated by migrating GBs. In this situation, according to the results of our theoretical analysis, the stress-driven GB migration in the nanocomposite can occur in the limited and unlimited modes. In the limited migration mode, migrating GBs eventually approach their equilibrium positions corresponding to a given value of the applied stress (Figs. 4a, b, 5a, 7). In the unlimited migration mode, some segments of a migrating GB stop at the nanoinclusion boundary, while others move unrestrictedly far away from the nanoinclusion (Figs. 4c, 5b). (In real materials, GBs exhibiting the unlimited migration mode are stopped by their neighboring GBs.) The transition from limited to unlimited GB migration occurs at some critical value of the applied shear stress. This critical stress serves as the key characteristic for the effects exhibited by nanoinclusions on the stress-driven GB migration in nanocomposites. Our analysis demonstrated that in the examined case of a nanocomposite containing nanoinclusions, the critical stress significantly depends on such geometric parameters as the ratio \( d/L \). In particular, surprisingly, it has been found that for small values of \( d/L \), the critical stress decreases with the rise in nanoparticle size \( d \) (Fig. 6). This means that small nanoparticles in a moderate concentration can promote migration of low-angle GBs. At the same time, for comparatively large values of \( d/L \), the critical stress grows with an increase in the ratio \( d/L \), as one could expect (Fig. 6). This implies that large enough nanoinclusions hinder GB migration.

In general, it would be interesting to compare the presented theoretical results with experimental data. However, since experimental research in this area is in its infancy, comparison of such a kind is very limited. So, as to our knowledge, there are only experiments [44, 45] concerning the effects of ceramic incoherent nanoinclusions on the stress-driven GB migration in nanostructured metal matrix composites. These experiments showed that (i) thermally activated grain growth is effectively suppressed by Al2O3 ceramic nanoinclusions in these composites having nanostructured metal (Al or Al-based alloy) matrixes under thermal treatment for several hours at elevated temperature (600 °C); (ii) grain growth effectively occurs through both stress-driven GB migration and grain rotation processes in nanostructured metal–Al2O3 nanocomposites under mechanical treatment. These experimental data are consistent with our theoretical prediction that the stress-driven GB migration in metal matrix nanocomposites reinforced by (ceramic or metallic) incoherent nanoinclusions occurs at some critical stress. That is, in agreement with the experimental data [44, 45], in the absence of a mechanical load, GB migration is suppressed. Also, in agreement with the same experiments, we theoretically predict that the stress-driven GB migration occurs when the external critical stress is applied. However, other theoretical predictions presented here still wait for comparison with future experiments in this new area of nanomaterials science. Besides, the theoretical model formulated in this paper can be modified for the scientifically interesting case of high-angle GBs migrating in metal matrix nanocomposites reinforced by nanoinclusions of the second phase. This modification will be the subject of our future examinations.