Statistical Analysis and Stochastic Dislocation-Based Modelling of Microplasticity

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1 Statistical Analysis and Stochastic Dislocation-Based Modelling of Microplasticity O. Kapetanou 1, V. Koutsos 1, E. Theotokoglou 2, D. Weygand 3, M. Zaiser 1,4 1 School of Engineering, Institute for Materials and Processes, The King s Buildings, Sanderson Building, Edinburgh EH93JL, UK 2 School of Applied Mathematical and Physical Sciences Department of Mechanics Laboratory of Testing and Materials National Technical University of Athens Zographou Campus, Theocaris Bld GR , Athens, Greece 3 IAM-ZBS, Karlsruhe Institute of Technology, Karlsruhe, Germany 4 Department of Materials Science, Institute for Materials Modelling, University of Erlangen-Nürnberg, Dr.-Mack-Strasse 77, 90762, Fürth, Germany Abstract. Plastic deformation in microscale differs from the macroscopic plasticity in two respects: (i) the flow stress of small samples depends on their size (ii) the scatter of plasticity increases significantly. In this work we focus on the scatter of plasticity. We statistically characterize the deformation process of micropillars under tension using results from discrete dislocation dynamics (DDD) simulations. We then propose a stochastic microplasticity model which uses the extracted information from the above statistical characterization to make statistical predictions regarding the micropillar stress-strain curves. This model aims to map the complex dynamics of interacting dislocations onto a 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 the discrete dislocation dynamics. We compare the model predictions with the underlying DDD simulations and outline potential future applications of the same modelling approach. 1. Introduction The miniaturisation 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.

2 Fehler! Verweisquelle konnte nicht gefunden werden. Figure 1: Top: stress-strain curves of [100] oriented Mo micropillars, mean diameter d=0.3µm [4, 5]; Bottom: room-temperature stress-strain curve of macroscopic [100] oriented Mo single crysta[6]. While a lot of effort has gone into understanding and modelling the size-dependent strength of small samples (for recent reviews see [1-3]), the question of fluctuations in strength has been less investigated. It is clear that such an investigation ought to be based upon 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 [6], While, in macroscopic specimens, these fluctuations are not directly visible on the stress-strain curves, 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 [7] but, 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. suggest to associate yielding with the first observation of lattice rotations [8] but again, since any dislocation activity is associated to some degree 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], though this idea has been recently questioned [14]. 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 the controversies regarding interpretation - we simply determine the parameters of these distributions in a phenomenological manner to best reproduce stress strain curves obtained from DDD simulations. The same is done for the yield stress. Theoretical approaches to micro-plasticity have mainly focused on the modelling 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 discrete

3 dislocation dynamics (DDD) simulation [20] which, while computationally demanding, provides complete information about stresses and strains on the dislocation scale and thus gives natural access to both size effects [21,22] and fluctuation phenomena [11,22]. Our proposition in the present manuscript is to generalise continuum theories by an appropriate stochastic description of the deformation process in order to include local variability. Following the ideas expressed in [23], we construct a stochastic model for the deformation behaviour based upon the statistical analysis of DDD simulations. The paper is organised as follows. Section 2 provides a description of the details of 3D 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 Statistical Analysis of DDD Simulations For this work, we simulated strain-controlled tensile deformation of cubic samples with dimensions of 0.50 x 0.50 x The monocrystalline samples have face centered cubic (fcc) lattice structure, and their edges are oriented along the cubic axes of the fcc lattice. We impose a constant displacement rate to the upper sample surface, corresponding to an imposed strain rate (displacement velocity divided by specimen height) of The bottom surface of the specimens remains fixed, and the side surfaces are free. The initial dislocation microstructure consists of 48 randomly distributed Frank- Read sources. On each slip system there are 4 sources of 0.22 µm length. 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 1. Figure 1: Stochastic nature of plastic flow as illustrated by superimposing the stress-strain curves of 22 DDD simulations. For details see text. Tensile straining deformation curves of micropillars can be characterized by strongly intermittent behaviour. Deformation proceeds as a discrete sequence of deformation events, so called avalanches[11] during which the plastic deformation rate increases significantly. During an avalanche the plastic strain rapidly increases and the stress decreases (Figure 2).

4 Figure 2: Stress, strain rate and plastic strain vs. time signals in a DDD simulation of uniaxial compression. Left, plastic strain vs. time and strain rate vs. time; right, stress vs. time and strain rate vs. time Figure 2 demonstrates the correlation between stress and plastic strain rate and the correlation between plastic strain and plastic strain rate, respectively. Clearly we were dealing with two different processes the avalanches and the intervals in between. The first step toward the statistical characterization of stress-strain curves consisted therefore of separating our time records into active and inactive parts. The active parts were the time intervals which include the avalanches. The inactive parts were the intervals between the avalanches. Firstly, we smoothened all the time record by an averaging process of adjacent points. This serves to eliminate the rapid oscillations which stem from the discrete timestepping of the DDD code and are thus numerical artefacts. We note that an analogous procedure is needed in analysing experimental data where comparable oscillations arise from the mechanical action and electronic control of the microdeformation rig [4]. Then we exclude the elastic part which was approximately chosen from DDD simulations and imposed a threshold value on the plastic strain rate. By choosing this threshold to equal the imposed strain rate, we separated the stress strain curve into decreasing (active) and increasing (inactive) parts. Therefore, the avalanche was defined in our case as a time interval over which the strain rate exceeds the imposed value. Subsequently, we defined the stress difference of the active and inactive parts. The resulting series of stress jumps can be statistically characterized in terms of probability distributions of stress decreases and strain increases at the active and inactive parts respectively. In order to define these probability distributions we use rank ordering statistics [18]. Supposing strain differences of the active and stress differences of the inactive parts, respectively as the random variables for the rank ordering.

5 Figure 3: Left, distribution of plastic strain increments between strain bursts-active parts; right, distribution of stress increments during inactive parts in DDD simulations of uniaxial compression. Black line corresponds to simulation data; red line corresponds to fitting function Therefore we collected the results from each simulation and we merged them. We evaluated by rank ordering the probability distribution of: the strain difference of an avalanche and the stress difference of the inactive part shown in a double logarithmic plot in Figure 3. The black curves present the data and the red curve the fitting function.on the left hand site the results of the plastic strain during the active part are shown while on the right hand site the probability distribution of the stress difference at the inactive parts. Comparing the two graphs we note that there is a remarkable degree of similarity. It is well established that plastic strain increments produced by slip avalanches follow a power law distribution [9]. In our case both and seem to be well described by a truncated power law, 1 More specifically, the fitting function(red curve) for the probability distribution of the is: 1, $5,84 10 )! * +, -./ 0, ,00 * +,,1 -./,70 3 6, 48 95,84 10 ) 2) Where 5,84 10) and 1, for the second part of the fitting function, while for the first part of the fitting function is equal to unit as illustrated in the Figure 3. Similarly for the stress differences of the inactive part where the fitting function is: 1, $3, ; * += >?-./.0 3 A6.30 B +=.0 >?-./ 3 C, 5 93, )

6 We note that there is a remarkable degree of similarity in p( and p( ) (Figure 3). 3. Stochastic Model 3.1 Naïve model: Uncorrelated avalanche sequence The aim of this 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. Consequently, we demonstrate the stochastic simulation of a strain-controlled tension experiment on a cubic sample 0.50 x 0.50 x The sample is restrained at the bottom surface and loaded at the top by imposing a constant displacement rate, corresponding to a strain rate of The simulation is terminated once the total strain exceeds The stochastic simulations envisage the stress strain curve as a sequence of uncorrelated deformation steps which correspond to the alternating active and inactive parts in the DDD simulation. The statistics of these are taken from the statistical analysis of the DDD curves. In a strain controlled tension stochastic simulation the stress strain curve consists of an initial elastic part up to a stress 6 10 ) Pa and a stain ε = The yield point is approximately chosen from the DDD simulations. Afterwards, the plastic part consists of a segment of stress decrease distributed according to equation 2 (after multiplication with E) and then another segment of stress increase distributed according to equation 3 and at the same time a strain increase. This strain increase comes from the stress distributed according to equation 3 divided by E. The first segment corresponds to the active part as demonstrated at the above statistical analysis, while the second one corresponds to the inactive part. To generate the stress increase/decrease, we choose a random number uniformly distributed on the closed interval [0,1] and determine the corresponding stress difference of the active or inactive part by inverting Equation 2 or 3, depending on the case. The sequence of active and inactive segments is continued until the stopping condition of our stochastic simulation completes the plastic part of the stress strain curve. More specifically this stochastic model is structured as follows: 1 Elastic part 2 Plastic part Execution of the following iterative process Active part a b Decrease the stress by a random amount D drown from the distribution of equation 2 Keep the same strain Inactive part a Increase the stress by a factor of drown from the distribution equation 3 b Increase the strain by a factor of /E.

7 Repeat until the total strain exceeds the value of A stochastic simulation is shown in Figure 4. Figure 4 :A stress-strain curve calculated from the stochastic model Comparing results which occur from the stochastic model with those from 3D DDD simulation will give an overview of the effectiveness of the stochastic model. For that reason we calculate me mean and the standard deviation of stress as a function of total strain for both DDD and stochastic simulations. Figure 5:Stress strain curves of 22 DDD simulations after initial relaxation Figure 6:Stress strain curves of 1000 stochastic simulations Running the stochastic model does not require computational resources. Therefore, we can easily have many stochastic simulations for better statistical analysis. Therefore, we run our model 1000 times and the results are illustrated in Figure 6. Figure 5 indicates the results of the 22 3D DDD simulations after an initial relaxation. As has already mentioned the dislocation lines are randomly distributed in the sample at the beginning of the simulation, which means that they may start interact each other and move even without any external loading. Thus, in the initial configuration the dislocations produce a small plastic strain which may be either positive or negative. If this 'instant strain' is positive, it creates a negative stress, otherwise if it is negative it creates a rapid stress rise to a positive value. The inverse

8 proportional relation of plastic strain and stress is easily explained by the inverse proportional relation of plastic and elastic strain during a strain controlled DDD simulation. The stress behaves proportional to the elastic strain. Therefore during an avalanche when the plastic strain increases rapidly while the elastic strain decrease the stress decreases as well. While in the intermediate region elastic strain increases much more than plastic strain so the stress. Each statistical aggregate has been presented in the same graph(figure 7) for DDD and stochastic simulations. Figure 7: Top, average stress of DDD simulations (blue line) and stochastic simulations (green line); Bottom, stress standard deviation of DDD simulations (blue line) and stochastic simulations(green line) In Figure 7(top) is demonstrated the mean of stress while in Figure 7(bottom) the standard deviation of stress. The blue line illustrates the results from the statistical analysis of the DDD simulations while the green one the results from the statistical analysis of the stochastic simulations. Firstly, we observe that the statistical analysis of the stochastic simulations produces almost smooth mean and mean scatter. This is an outcome of the large number of simulations (1000) which used for the statistical analysis. On the other hand both mean and scatter of the DDD simulations fluctuate because we had only 22 results to elaborate. The results of the statistical analysis for elastic part should not be considered as a criterion for the effectiveness of our model since they are produced directly from the results of the DDD simulations and do not contain any stochastic parameter. Focusing on the plastic part we observe a small stress decrease right after the yield point (Figure 7(top)). This stress drop comes from the fact that the plastic part starts with an active part, namely a stress drop. Subsequently, we observe that the model produces a linear hardening during the plastic part which does not seem that representative at the initial plastic part but overall give a better approximation of the DDD simulations. On the other hand the stress scatter of the DDD simulations remains low but the model s stress scatter increases rapidly at the beginning of the plastic strain and rises until the end of the simulations fact which derives from the random nature of the model(figure 6). From the comparison it is clear that the stochastic model greatly over-estimates the scatter of the stress values. Thus, the stochastic model does not represent the DDD simulations adequately and needs improvements in order to avoid the rapid increase of stress scatter during the plastic part and eliminate the phenomena of stress drops under the yield point.

9 3.2 Correlated Stochastic Model To improve the stochastic model we examine the assumption that the stress changes during active and inactive parts are uncorrelated random variables. For that reason we check the stress difference during each interval at the same graph. Lets assume a single deformation where Figure 8 is shown the stress difference of the active and the inactive part versus an index which indicates the recorded order. The stress difference of the active part corresponds to an avalanche, a stress drop as explained previously, while the stress difference of the inactive part exactly the opposite. We observe that the initial stress drops are almost zero while the stress increases are not negligible. This fact derives from the different behaviour of the sample during the elastic and the plastic part. respective stress strain curve Figure 8: right, black line: stress difference versus Index during the active part- zoom in the first simulation; red line: stress difference versus Index during the inactive part- zoom in the first simulation During the elastic part there is not a significant dislocation activity which means that the stress increases almost strictly and the small dislocation interactions correspond to unimportant stress drops. Conversely, during the plastic part the stress drops and increases match. The referred observations lead to the definition of an advanced stochastic model, where a correlation factor is established between the active and the inactive part of the plastic part while the elastic part remains a straight line up to the yield point which is approximately chosen from the DDD simulations. For that reason we establish a correlation factor q between sequential active and inactive parts. We still choose active and inactive stress changes from the respective distributions by choosing two uniformly distributed random numbers between 0 and 1 and then determining the corresponding stress values from the cumulative distributions(figure 3,Formulas?). However, now these two random numbers are correlated. The correlation factor q controls the degree of correlation between the active and the inactive part. The correlation factor lies in the closed interval [0,1]. When q=1 the two random numbers are identical while when q=0 the random variables are completely independent. Each intermediate value of the correlation factor leads to a result between the two referred situations. To construct correlated uniformly distributed random numbers, we generate two independent random variables R 1 and R 2 drawn from the standard normal distribution. Using the fact that the sum of two independent Gaussian random variables is also a Gaussian random variable we create a pair of Gaussian random variables. Assuming a constant value F GH*1 F 6 where q is the correlation factor we use the referred property and generate the pair of Gaussian random variables I,F J K 1 F J F correlated with correlation factor q. Then we convert the to uniformly distributed variables using the probability integral transform [19]. Thus L~N(0,1) and Y=Φ(L) is uniformly distributed, where: L,F MI F N 4/O O P Q1GRS * T F T V F 6U,I IR

10 In our model Y1, Y2 are uniformly distributed probabilities which correspond to a strain difference according to the probability distribution of the strain difference and to a stress difference according to the probability distribution of the stress difference (equation??, Figure 3) respectively. The definition of the correlated stochastic model is exactly the same as that of the simple stochastic model with the only difference that now subsequent active and inactive parts (but not vice versa!) are correlated through the correlation factor q. Figure 9:Left, stress strain curve calculated from the correlated stochastic model for correlation factor equal to 1;right, 1000 stress strain curves from correlated stochastic simulations with correlation factor q=1, the elastic part coincides for all the simulations The stress strain curve of a stochastic simulation with correlation factor equal q=1 is shown in Figure 9 and exhibits an interesting shape. The stress decrease during the active part and the stress increase during the inactive part are of the same order of magnitude. This fact is rational if we consider that the two sequence intervals (active-inactive) are strongly correlated and their probability distributions display a remarkable similarity. The contribution of the correlation factor is obvious if we compare Figure 9(left) with Figure 6. The scatter is narrower and the downdrafts under the yield point have been significantly reduced when the correlation factor equals to one. To investigate thoroughly the impact of the correlation factor on the results of the stochastic model we implement the same statistical analysis as for the simple stochastic model and we illustrate on the same graph the mean and the standard deviation of stress as a function of strain for DDD simulations and correlated stochastic simulations for different correlation factors q=0-1 running for each factor 1000 simulations. The results are demonstrated in Figure 9(right).

11 Figure 10:Top, average stress of DDD simulations (blue line) and correlated stochastic simulations (coloured lines according to the correlation factor); right, average stress of DDD simulations (blue line) and correlated stochastic simulations (coloured lines according to the correlation factor) Figure 10 is shown the mean stress as a function of strain. We should highlight at this point that the mean stress is not significantly affected by the correlation factor. Moreover, we observe that the stochastic simulations produce linear hardening with hardening rate equal to 2.9GPa similar to the hardening rate observed from the simple stochastic model. In order to extract more information about our results we consider the standard deviation of mean stress as a function of strain as shown in the bottom part of Figure 10. Other than with the mean stress, the scatter of stress is affected by the different values of the correlation factor, as observed to Figure 10 as well. More specifically, for correlation factors from 0 to 0.4 we cannot see large differences between the scatter that they produce, on the other hand for correlation factors from 0.5 to 1 we observe a significant decrease of the scatter. While the factor increases from 0.5 to 1 the stress scatter decreases and approaches the scatter of the DDD simulations. In other words, as the correlation increases between the active and inactive intervals the model becomes more reliable in reproducing the fluctuations around the mean stress level. 4. Summary and Conclusions Much effort has been devoted to study-understand the deformation process in microscale during the past decades. After pointing out the weakness of previous research in the introduction we end up with a promising probabilistic approach in order to predict the local variability observed in microplasticity. In order to implement our model we used results from 3D DDD simulations, we statistically characterize them in order to extract useful information for the strain burst and the intermittent region and therefore use them to build our model. The model used the probabilistic variables of stress and strain, which as presented follow a truncated power law, to build a stress strain curve of a similar micropillar under the same loading conditions with the one of a 3D DDD simulation. Initially an uncorrelated stochastic model was build which didn t give a sufficient approximation of DDD results. Afterwards, a correlation factor between active and inactive part was included to the model which provided an adequate approximation for fully correlated sequential parts. Therefore, we ended up with a well-defined approach which overcomes the conduction difficulties of

12 an experiment and reduces tremendously the computational time of a simulation of micropillars under tension. In the future in order to improve our model we can address the distribution of the yield point or use non-stationary probability distributions to predict the plastic part in order to avoid linear hardening. Use as input results from different models or even 3D DDD simulations with different dimensions would probably help to avoid that much hardening during the plastic part. Acknowledgement: We acknowledge support by Deutsche Forschungsgemeinschaft DFG-FG1650, grant WE3544/5-1, co-funded by EPSRC under grant no. Ep/J003387/1. References: 1. Greer, J.R. and J.T.M. De Hosson, Plasticity in small-sized metallic systems: Intrinsic versus extrinsic size effect. Progress in Materials Science, 2011, 56, Kraft, O., P.A. Gruber, R. Mönig and D. Weygand, Plasticity in confined dimensions. Annual Review of Materials Research, 2010, 40, Uchic, M.D., P.A. Shade, and D.M. Dimiduk, Plasticity of Micrometer-Scale Single Crystals in Compression. Annual Review of Materials Research, 2009, 39, Schneider, A.S., et al., Effect of orientation and loading rate on compression behavior of smallscale Mo pillars. Materials Science & Engineering A, 2009, 508: Hollang, L., D. Brunner, and A. Seeger, Work hardening and flow stress of ultrapure molybdenum single crystals. Materials Science & Engineering A, 2001, 319: Miguel, M.C., A. Vespignani, S. Zapperi, J. Weiss, and J.R. Grasso, Intermittent dislocation flow in viscoplastic deformation, Nature, 2001, 410, Dimiduk, D.M., M.D. Uchic, and T.A. Parthasarathy, Size-affected single-slip behavior of pure nickel microcrystals. Acta Materialia, 2005, 53, Nikitas, N. and M. Zaiser, Slip avalanches in crystal plasticity: scaling of the avalanche cutoff. J. Stat. Mech: Theory and Experiment, 2007, P Zaiser, M. and P. Moretti, Fluctuation phenomena in crystal plasticity - a continuum model. J. Stat. Mech: Theory and Experiment, 2005, P Csikor, F.F., et al., Dislocation avalanches, strain bursts, and the problem of plastic forming at the micrometer scale. Science, , Maaß, R., et al., Smaller is stronger: The effect of strain hardening. Acta Materialia, , Zaiser, M., Scale invariance in plastic flow of crystalline solids. Adv. Phys., 2006, 54, Chan, P. Y. et. al., Plasticity and dislocation dynamics in a phase field crystal model. Phys. Rev. Letters, 2010, 105, Ispánovity, P. D. et. al., Avalanches in 2D Dislocation Systems: Plastic Yielding Is Not Depinning. Physical review letters, 2014, 112, Aifantis, E., On the microstructural origin of certain inelastic models p Fleck, N. and J. Hutchinson, Strain gradient plasticity. Advances in applied mechanics, : p Gurtin, M.E., On the plasticity of single crystals: free energy, microforces, plastic-strain gradients. Journal of the Mechanics and Physics of Solids, (5): p Zaiser, M., et al., Modelling size effects using 3D density-based dislocation dynamics. Philosophical Magazine, (8-9): p Hochrainer, T., S. Sandfeld, M. Zaiser and P. Gumbsch, Continuum dislocation dynamics: towards a physical theory of crystal plasticity. Journal of the Mechanics and Physics of Solids, 2014, 63, Weygand, D., et al., Aspects of boundary-value problem solutions with three-dimensional

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