Plastic deformation of micro- and nanoscale samples differs from macroscopic plasticity in two respects: (i) the flow stress of small samples depends on their size, and (ii) the scatter of plastic deformation behavior increases significantly. In this work, we focus on the scatter of plastic behavior. We statistically characterize the deformation process of micropillars using results from discrete dislocation dynamics (DDD) simulations. We then propose a stochastic microplasticity model that uses the extracted information to make statistical predictions regarding the micropillar stress-strain curves. This model aims to map the complex dynamics of interacting dislocations onto stochastic processes involving the continuum variables of stress and strain. Therefore, it combines a classical continuum description of the elastic-plastic problem with a stochastic description of plastic flow. We compare the model predictions with the underlying DDD simulations and outline potential future applications of the same modeling approach.
The miniaturization of systems and devices creates the need to address the mechanical properties of materials on smaller and smaller scales. Figure 1 illustrates the differences between the stress-strain curve of a macroscopic Mo single crystal specimen and the stress-strain curves of micropillars of the same material. We observe that microplasticity differs from macroplasticity in two important aspects. The stress-strain curves of the micropillar samples exhibit strong fluctuations, and on average, the micropillar specimens are much stronger than the macroscopic sample.
Although a lot of effort has gone into understanding and modeling the size-dependent strength of small samples for both face-centered cubic (fcc) and body-centered cubic (bcc) materials (for recent reviews, see [4–6]), the question of fluctuations in strength has been less investigated. It is clear that such an investigation ought to be based on studying the dynamics of dislocations as the main carriers of plastic deformation in crystalline materials. The collective motion of dislocations occurs in an intrinsically jerky and intermittent manner. Even in macroscopic specimens, acoustic emission measurements reveal intermittent fluctuations of the energy release rate (“dislocation avalanches”) whose magnitudes span over 6 decades in energy release . Although these fluctuations are not directly visible on the stress-strain curves in macroscopic specimens, with decreasing sample size, the intermittent avalanche-like dynamics of dislocations becomes directly visible in the form of stress drops or strain bursts punctuating the stress-strain curves.
A significant amount of papers have discussed the question how we should understand the term “plastic yielding” in small samples. Some studies argue that the yield stress corresponds to the occurrence of the first large avalanche . However, given that deformation bursts in microplasticity tend to follow power law distributions [8–11], it is not quite clear how to define a threshold for “large” avalanches in any meaningful manner. Maaß et al.  suggested to associate yielding with the first observation of lattice rotations. However, because any dislocation activity is associated with microscale lattice rotations, the problem of defining a threshold is not solved by this definition. Other studies refer to concepts drawn from statistical physics and envisage yielding as a depinning-like phase transition [8, 9, 12, 13], although this idea has been recently questioned . Despite the differences in interpretation, there is some consensus in the literature that the statistics of strain bursts in microplasticity can be meaningfully described by (truncated) power law distributions. In the present paper, we refrain from entering controversies regarding interpretation; we simply determine the parameters of these distributions in a phenomenological manner to best reproduce stress-strain curves obtained from discrete dislocation dynamics (DDD) simulations. The same is done for the yield stress.
Theoretical approaches to microplasticity have mainly focused on the modeling of size effects, by including length scales into constitutive equations of plasticity [15–17] or more recently by formulating the dynamics of dislocations within a continuum framework [18, 19]. Finally, an alternative approach to plasticity of micron-scale samples is provided by the method of DDD simulation , which, although computationally demanding, provides complete information about stresses and strains on the dislocation scale and thus gives natural access to both size effects [21–24] and fluctuation phenomena [11, 22].
Our proposition in the present manuscript is to generalize continuum theories by an appropriate stochastic description of the deformation process to include local variability. Following the ideas expressed by Zaiser , we construct a stochastic model for the deformation behavior based on the statistical analysis of DDD simulations. The paper is organized as follows. Section 2 provides a description of the details of the 3-D DDD simulations and illustrates the statistical analysis of the DDD data. Section 3 describes the stochastic model and evaluates its performance for different degrees of complexity of the statistical model. General conclusions are given in Section 4.
2 Statistical analysis of DDD simulations
For this work, we simulated the strain-controlled tensile deformation of cubic samples with dimensions of 0.50×0.50×0.50 μm3. The monocrystalline samples have fcc lattice structure, and their edges are oriented along the cubic axes of the crystal lattice. We impose a constant displacement rate on the upper sample surface, corresponding to an imposed strain rate (displacement velocity divided by specimen height) of 5000 s-1. The bottom surface of the specimens remains fixed, and the side surfaces are free. The initial dislocation microstructures consist of 48 randomly distributed Frank-Read sources. On each slip system, there are four sources of 0.22 μm length leading to an initial dislocation density of approximately 8×1013 m-2. The material is assumed to have Young’s modulus E=72.7 GPa (close to Al). Results of 22 different simulations with different initial source positions are shown in Figure 2.
As the dislocation lines are randomly distributed in the sample at the beginning of the simulation, their interaction in the initial configuration leads to a certain amount of dislocation motion even in the absence of external action on the system. The concomitant plastic relaxation strain, which may be either positive or negative, is offset to ensure that all DDD stress-strain curves start in the origin.
The simulated deformation curves can be divided into an initial, quasi-elastic part and a subsequent regimen of plastic flow. Analyzing the quasi-elastic part, we find that the slopes of the stress-strain curves in this regimen are less than the value of E=72.7 GPa expected from the material’s elastic constants. This can be readily understood by observing that, even before the yield stress is reached, dislocations undergo quasi-reversible (in the terminology of Zaiser and Seeger : “inversive”) motion. Such motion reverses upon unloading in such a manner that the dislocation arrangement reaches its initial configuration and no permanent strain is produced. It is however thermodynamically irreversible because the loading-unloading curve encloses a finite area in the stress versus strain plane. For illustration, we consider the subcritical bowing out of a Frank-Read source of length l. Assuming an isotropic line tension T=Gb2 for simplicity, the critical configuration of the source (a semi-circle of a radius l/2 and area πl2/8) is reached at a stress of τ=2Gb/l. For an ensemble of Frank-Read sources of volume density n (dislocation density ρ=nl), the corresponding “inversive” strain is given by
where M is the Schmid factor and the effective elastic modulus can be estimated as Eeff=E/(1+εinv/εel). With ρl2≈1 in our simulations, we find an effective elastic modulus of approximately 60 MPa, in good agreement with the simulated stress-strain curves. Of course, our estimate based on a consideration of single sources is an oversimplification because dislocation-dislocation interactions affect the stress response of the initial dislocation configuration. As a consequence, we find a significant scatter of the initial slopes of the stress-strain curves (cf. Section 3.3).
Above a critical stress, the samples enter a plastic deformation regimen where dislocation motion becomes irreversible (the initial configuration is not restored upon unloading). Again as a consequence of dislocation-dislocation interactions, the corresponding critical stresses fall significantly below the estimate of τ=2Gb/l for a single Frank-Read source. The ensuing plastic deformation regimens are characterized by strongly intermittent behavior. Deformation proceeds as a discrete sequence of “deformation events”, the so-called avalanches  during which the plastic deformation rate increases significantly. During an avalanche, the plastic strain rapidly increases and the stress decreases [Equation (3)].
Figure 3 demonstrates the correlation between stress and plastic strain rate and the correlation between plastic strain and plastic strain rate, respectively. Clearly, we are dealing with two different types of dynamic behavior: the avalanches/strain bursts and the intervals in between. The first step toward the statistical characterization of the stress-strain curves is therefore to separate our time records into active and inactive parts. The active parts are the time intervals during which avalanches occur. The inactive parts are the intervals between the avalanches. First, we smooth all the time records by an averaging process of adjacent points. This serves to eliminate the rapid oscillations that stem from the discrete time stepping of the DDD code and are thus numerical artifacts. We note that an analogous procedure may be needed in analyzing experimental data where comparable oscillations may arise from the mechanical action and electronic control of the microdeformation rig .
Excluding the initial elastic loading part, we separate the DDD simulation records into “active” and “inactive” time intervals by imposing a threshold value on the plastic strain rate. By choosing this threshold to equal the imposed strain rate, the former correspond to decreasing and the latter to increasing parts of the stress versus total strain curve. Thus, in the present analysis, a strain burst or dislocation avalanche is simply defined as a time interval over which the plastic strain rate exceeds the imposed value. Subsequently, we determine the changes in stress and in plastic strain, which occur during the active and inactive time intervals. The resulting records can be statistically characterized in terms of probability distributions of the respective variables. To determine these probability distributions, we use rank-order statistics .
We make an important simplification: Figure 3 (right) indicates that, during an inactive time interval, the plastic strain is approximately constant. Similarly, Figure 3 (left) shows that the duration of an “active” time interval is much less than the duration of an “inactive” one. Accordingly, we assume the plastic strain change during an inactive interval and the total strain change during an active interval to be equal to zero. This implies that the entire sequence can be characterized in terms of two sequences of variables only: the plastic strain increments Δεa during the “active” time intervals and the stress increments Δσi during the “inactive” intervals. Because the deformation is assumed to be instantaneous during the active intervals and purely elastic during the inactive intervals, Δεa also corresponds to a stress drop Δσa≈-EΔεa and Δσi to a total strain increment Δσi/E, where E is the Young’s modulus of the simulated samples.
We collected data from multiple simulations and evaluated by rank ordering the probability distributions of the plastic strain differences
where M is the total number of entries in the list. The resulting probabilities p(Δεa) and p(Δσi) are shown in double logarithmic plots in Figure 4. The black curves present the data and the red curves the respective fitting functions. Comparing the two graphs, we note that there is a remarkable degree of similarity between the statistics of both stress and plastic strain increments.
It is well established that plastic strain increments produced by slip avalanches can be characterized by truncated power law distributions [8, 10]. In our case, both p(Δεa) and p(Δσi) seem to be well described by truncated power laws,
The fitting parameters (red curves) are given in Table 1.
3 Stochastic model
3.1 Naive model: uncorrelated avalanche sequence
The aim of a stochastic microplasticity model is to map the complex dynamics of interacting dislocations onto stochastic processes involving the continuum variables of stress and strain. Using statistical information extracted from DDD, our stochastic model is constructed to reproduce the essential statistical features of the deformation processes in small volumes of a material. The simplest conceivable model is to assume that the variables characterizing the “active” and “inactive” intervals, which alternate above the yield stress, represent stationary, uncorrelated stochastic point processes, with probability distributions given by Equation (3).
In a strain-controlled tension stochastic simulation, the stress-strain curve then consists of an initial elastic part up to a yield stress σ=60 MPa, which is chosen to optimally match, on average, the DDD simulations. Afterward, the deformation curves consist of alternating segments. During an active interval, the total strain remains constant, the plastic strain increases by an amount Δεa randomly drawn from the distribution p(Δεa), Equation (2), and the stress decreases by EΔεa. During the subsequent inactive interval, stress increases by an amount Δσi randomly drawn from the distribution p(Δσi), Equation (3). The plastic strain remains constant and the total strain increases by Δσi/E. This process is repeated until the desired end strain is reached. No correlation is assumed between Δσi and Δεa, or between the sequential values of either of these variables. A stress-strain curve simulated using this simple algorithm is shown in Figure 5. (We note that the model has close similarities to the SUDTS algorithm proposed by Kugiumtzis and Aifantis  for constructing random surrogates to stress-strain curves in macroscopically jerky plastic flow.)
To quantitatively compare results obtained from the stochastic model with those from 3-D DDD simulation, we consider the mean and the standard deviation of stress calculated as functions of total strain for ensembles of both DDD and of stochastic simulations.
Figure 6 illustrates the results of 1000 different stochastic simulations as compared with the results of the 22 3-D DDD simulations.
Figure 7 compares the mean values and standard deviations of stresses obtained from the DDD and stochastic simulation ensembles. The fact that the curves for the stochastic ensemble are much smoother is simply due to the much larger number of simulations (1000 stochastic vs. 22 DDD simulations). We first focus on the mean stress-strain curves. The average stress and plastic strain reached after n active-inactive cycles are given by
It follows that the ensemble-averaged hardening coefficient is given by
The calculation shows that our stochastic model should produce, on average, linear hardening above yield. As can be seen from Figure 7, this result is in line with the stochastic simulation data. It is also seen that the averaged stress-strain curves from the stochastic model and the DDD simulations are in reasonable agreement.
This is not true for the statistical scatter of flow stresses, which is very significantly overestimated by the present, simplistic model. The longtime behavior of the stress in this model can be envisaged as a random walk superimposed on a linear trend. For small strains, the parabolic growth of the stress scatter (diffusion-like behavior of the random walker, see Figure 7, right) predominates to such an extent that, in a nonnegligible fraction of all realizations of the stochastic model, negative flow stresses are reached (Figure 6). Of course, this is completely unphysical. We may thus conclude that the assumption of a sequence of active and inactive periods with uncorrelated stress and strain increments does not adequately represent the DDD simulations.
3.2 Correlated stochastic model
To improve the stochastic model, we more closely examine the assumption that the stress changes during “active” and “inactive” parts are uncorrelated random variables. Consider a single DDD simulation as shown in Figure 8 (right). We plot the stress changes during the active and inactive time intervals versus the index n, which indicates the position of an interval in the respective record (i.e. the first stress drop and the subsequent stress increase both have index n=1, the second stress drop and its subsequent stress increase have index n=2, etc.). We can distinguish two regimens: at the onset of deformation, we see pronounced stress increases but only very small stress drops. This corresponds to the microplastic regimen before yield, where we essentially see elastic loading punctuated by few and small stress drops. Above the yield stress, the picture changes. In this regimen, we see that large stress drops tend to be followed by large stress increases; hence, there is a significant correlation between the stress drop and the stress increase corresponding to a given “active-inactive cycle”.
These observations lead to the definition of an improved stochastic model where, in the plastic regimen, the variables characterizing a stress drop/strain burst and the subsequent elastic stress increase within an active-inactive cycle possess some degree of correlation. The elastic part of the stress-strain curve, on the other hand, is still envisaged as a straight line up to the yield point. As previously, the yield stress has a fixed value, which is chosen to best match the DDD simulations.
In constructing correlated random variables
From these, we obtain correlated values of Δεa and Δσi by setting p(Δεa)=Y1 and p(Δσi)=Y2.
The stress-strain curve of a stochastic simulation with q=1 is shown in Figure 9 and exhibits an interesting shape. The stress decreases during the active part and the stress increases during the inactive part are now always of the same order of magnitude. Basically, stress drops due to the plastic strain increment are mostly reversed during the subsequent quasi-elastic stress increase. This behavior is expected if we consider that, in the fully developed plastic regimen, the dislocation microstructures before and after a strain burst are statistically equivalent to a large degree; hence, we expect the strain burst initiation stresses to be not too different. The same effect also prevents the stress-strain curves in different simulations from drifting too far apart and prevents the simulations from straying into the unphysical regimen of negative flow stresses (compare Figure 9 [left] with Figure 6).
To investigate the performance of the modified model, we again compare the mean and the standard deviation of stress as functions of strain as obtained from ensembles of DDD simulations and of correlated stochastic simulations, now for different correlation factors.
Figure 10 (right) shows the mean stress as a function of strain. Here there is no influence of the correlation factor. This is to be expected because Equations (4) and (5) do not depend on the presence or absence of correlations. However, correlations have a significant impact on the scatter of the stress-strain curves, as seen in Figure 1 (left). As the correlation between stress drops and stress increases becomes more pronounced, the scatter of the stress-strain curves obtained from the stochastic model decreases and approaches the scatter of the DDD simulations. In other words, as the correlation increases between the active and the inactive intervals, the model becomes more reliable in reproducing the fluctuations around the mean stress level. Still, our model is amenable to improvements because the assumption of fully deterministic behavior up to a uniform yield point is clearly unrealistic. In the DDD simulations, we see a gradual rather than an abrupt onset of scatter in the stress-strain curves. In the following section, we explore the possibility of including statistical scatter in dislocation behavior before yield.
3.3 Stochastic model with fluctuations in dislocation behavior before yield
As discussed above in Section 2, the initial slope of the stress-strain curves is influenced by “inversive” dislocation motions. These motions do not lead to slip avalanches but can rather be envisaged as a quasi-reversible polarization of the initial dislocation configuration that occurs once a stress is applied. Owing to the randomness of the initial configuration, the strain produced by such motions – and accordingly the effective elastic modulus – exhibits statistical scatter. This is shown in Figure 11, which shows the distribution of effective elastic moduli E in the DDD simulations, defined as the ratio between axial stress and total axial strain at an axial strain of 0.001, which is below the strain where large-scale plastic yielding occurs (see Figure 6, left). We see that the average value E∼58 MPa agrees well with the estimate provided in Section 2. At the same time, there is some statistical scatter around this value.
We now construct a simple stochastic model as follows: We draw an effective elastic modulus from the distribution shown in Figure 11, which we approximate by a Weibull distribution
where E0=52 GPa and E1=5.5 GPa. We then apply a purely elastic deformation up to a yield strain of 0.13 chosen to match the average behavior of the DDD curves. Above this yield strain, we continue with a simulation as in Section 3.2. Note that this procedure has the consequence that yield stresses in our simulation are Weibull distributed with an exponent of approximately 3.5, as previously proposed by other authors for interpreting micropillar experiments and simulations (see, e.g. [31, 32]).
Comparing the mean and scatter of the flow stress versus strain curves obtained from this model with those obtained from DDD demonstrates that the behavior both before and after yield is now well represented. In particular, the gradual increase of the scatter before yield is now well represented (Figure 12).
4 Summary and conclusions
We have explored some simple models describing stress-strain curves in microplastic deformation as stochastic processes consisting of quasi-elastic stress increases and sudden plastic strain increments/stress drops. Such models can provide a reasonably good approximation of the behavior found in DDD simulations, provided that a strong correlation is assumed between a stress drop/plastic strain burst and the following stress increase. However, even in this case, a certain overestimation of the fluctuations persists. This indicates that longer-term correlations are present in the dislocation dynamics, which encompass multiple slip events and delimit fluctuations.
A strong limitation of the model consists in the assumption that the stress-strain curves can be modeled in terms of stationary stochastic processes. In materials undergoing significant strain hardening, this approximation cannot be sustained in general because the dislocation microstructure may undergo significant changes in the course of strain hardening. In the DDD simulations considered here, however, no very significant increase in dislocation density is observed, such that over the limited strain ranges attainable in the DDD simulations, the assumption of stationarity after yield may be considered acceptable.
We acknowledge support by Deutsche Forschungsgemeinschaft DFG-FOR1650, grant WE3544/5-1, and by EPSRC under grant no. Ep/J003387/1.
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