The Universality Class of Nano-Crystal Plasticity: Self-Organization and Localization in Discrete Dislocation Dynamics
TThe Universality Class of Nano-Crystal Plasticity: Self-Organization and Localizationin Discrete Dislocation Dynamics
Hengxu Song,
1, 2
Dennis Dimiduk, and Stefanos Papanikolaou ∗
1, 2 Department of Mechanical and Aerospace Engineering,West Virginia University, Morgantown, WV26506, United States. Department of Mechanical Engineering, Johns Hopkins University, Baltimore, MD21218, United States. Department of Materials Science Engineering, Ohio State University, Columbus, OH43210, United States. (Dated: October 30, 2018)The universality class of the avalanche behavior in nanocrystals under uniaxial compression hasbeen typically described as being statistically similar to the plastic response of amorphous solids,polycrystals, frictional contacts and earthquakes, despite the vast differences of the microscopicplasticity defects’ character. A characteristic outcome of the crystal dislocations’ character is thatin macroscopic crystals under uniaxial compression, the flow stress is known to dramatically in-crease at high loading rates. We investigate the effect of loading rate by performing simulations ofa two-dimensional discrete dislocation dynamics model that minimally captures the phenomenologyof nanocrystalline deformation. In the context of this model, we demonstrate that a classic rate-dependence of dislocation plasticity at large rates ( > /s ), fundamentally controls the system’sstatistical character as it competes with dislocation nucleation: At small rates, plasticity localizationdominates in small volumes. At large rates, the behavior is statistically dominated by long-rangecorrelations of “dragged” mobile dislocations. The resulting behavior suggests that the experimen-tally relevant quasi-static loading limit of crystal plasticity in small volumes belongs in a uniqueuniversality class that is characterized by a spatial integration of avalanche behaviors at variousdistances from a parent critical point. Crystal plasticity in small volumes has been inves-tigated in the last two decades through the compres-sion of micro and nanopillars [1–6]. In these small vol-umes, the material strength is size-dependent due tostrain gradients[7–17] generated due to the absence oftypical gradient-free dislocation motion and multiplica-tion mechanisms. Furthermore, macroscopic work hard-ening [18, 19] is replaced by a wealth of abrupt plas-tic events [20–24] that originate in both the presence ofdislocation correlations, as well as the dramatic smallvolume effect of mobile dislocations forming geometricsteps on free pillar surfaces [21, 22, 25–27]. Abruptplastic events are common in avalanche phenomena ofvarious disordered non-equilibrium systems across lengthscales [28–32], especially elastic interface depinning phe-nomena, with which crystals share similar, but not iden-tical, avalanche statistics [33]. However, in typical crys-tal “depinning” modeling attempts [34–38], avalanchesare caused by a direct competition of elastic loadingand long-range elastic interactions with quenched disor-der, without temporal bursts in the number of elasticdegrees of freedom. In contrast, dislocations in nano-crystals can also nucleate, multiply and deposit on freeboundaries [5, 39, 40], naturally causing additional frus-tration that may influence the statistical avalanche be-havior [37, 41–43]. Here, in the context of an explicitdislocation dynamics model, we show that the competi-tion between two different ways to mediate plastic slip –dislocation nucleation and over-damped dislocation mo-bility ( ie. dislocation drag) – leads to a distinct rate ef-fect on the avalanche statistics that becomes more pro-nounced for stress-controlled loading conditions. We in- terpret the phenomenon in terms of a spatial integrationof avalanche behaviors across slip planes [41]. This is ageneric mechanism in bifurcation processes such as theFrank-Read nucleation of a single dislocation, and thuswe argue that the proposed effect should extend to 3D-DDD models [32, 44].Dislocation avalanches [45] have been observed ex-perimentally in diameter- D micro and nano pillar com-pression studies [21, 46, 47] where power law statis-tics for the sizes S of the form P ( S ) = S − τ P ( S/S )has been established, where τ ∈ (1 . , . S ∼ D and P resembles an exponential cutoff function [48].Two [33, 49–61] and three [6, 27, 49, 62–65] dimensionalmodels of atomic displacements or/and discrete dislo-cations simulations[33, 49, 51, 66, 67] have establishedthat τ ∼ . τ ∼ . τ may takemuch larger values, with possible reasoning focused oninternal disorder or/and thermal relaxation mechanismssuch as cross-slip.The effect of loading protocols on the statistical behav-ior of nanopillar compression response has been studiedrecently [71, 72], even though the connection betweenstress rate ˙ σ (in SC) and strain rate ˙ (cid:15) (in displacement-controlled loading (DC)) has been lacking at small rates. a r X i v : . [ c ond - m a t . m t r l - s c i ] O c t n contrast, at large loading rates ( > /s ) and in themacroscale, it is well known that crystals exhibit a sharpincrease of the flow stress due to viscoplastic dislocationdrag effects when strain rate is higher than ∼ /s [73–76]. This fact has been well verified in DDD simula-tions [77–79] and originates in the natural competitionbetween the timescale for a dislocation to move a Burg-ers vector distance at terminal speed and the timescalefor dislocation “nucleation” at a source (for example, ata pinned bulk segment – Frank-Read source) [80]. Howdoes this competition translate to the statistical behaviorof plasticity avalanches in small volumes at rates smallerthan 10 /s ?In this paper, we consider a minimal model of crystalplasticity for uniaxial compression in small volumes. By“minimal”, we imply a model that respects: i) the ener-getics of room temperature crystal deformation being me-diated by dislocations gliding along slip planes of at leastone slip system ii) the fact that small-volume crystallineplastic deformation originates in nucleation, iii) openboundaries absorb dislocations. In order to maximizestatistical sampling and computational efficiency, we per-form simulations of 2D samples using a benchmarked dis-location dynamics model [81, 82] that displays the basicphenomenology of nanocrystalline compression: Size ef-fects in the material yield strength and emergent crack-ling noise. For pure elasticity, SC and DC loading modescan be compared by using ˙ σ = E ∗ ˙ (cid:15) , where E ∗ = E − ν is the equivalent modulus for plane strain applicationsand ν is the Poisson’s ratio. The loading strain-rate ˙ (cid:15) is varied from 10/s to 10 /s. The model crystal is ini-tially stress and mobile-dislocation free. This is analo-gous to a well-annealed sample, yet with pinned disloca-tion segments that can act either as dislocation sources( eg. Frank Read sources) or as obstacles. Dislocations aregenerated from randomly distributed point sources whenthe resolved shear stress crosses a random threshold fora finite time 10 ns [91]. The nucleated dislocation pairis placed at a distance L nuc = E/ (4 π (1 − ν )) b/τ nuc andfor our system parameters, it is 35 nm on average [92].Randomly distributed point obstacles account for pre-cipitates and forest dislocations on out-of-plane slip sys-tems. Microstructural parameters are chosen based ona previous study [81] that matches various experimentalfacts.The timescale competition in this model is generic andpresent not only in all dislocation dynamics models, butalso in generic non-equilibrium processes [83]. Its basicorigin can be distilled by considering an imperfect pitch-fork bifurcation: d(cid:15)/dt = σ + µ(cid:15) − (cid:15) , where (cid:15), σ arescalars resembling strain and stress variables, and µ is amobility parameter. Neglecting dislocation interactions,on a slip plane without sources but a mobile dislocation, µ = µ drift is negative and the relaxation timescale for ev-ery incremental step of σ is τ drift = | µ drift | − . However,if a dislocation source is present without any mobile dis- locations, then µ = µ nuc > τ nuc = µ − nuc . Typically, τ nuc (cid:29) τ drift , so incrementsof σ will typically be accomodated by nucleation events.However, if a system of such possible bifurcations interact(if multiple dislocation sources are present), then mutualdislocation interactions may cause a frustrating situationwhere the disparity of relaxation times may cause a com-plexity in the evolution dynamics. In our DDD model,the two timescales are concerned with the nucleation andpropagation of single dislocations, where the timescalefor a dislocation to move by a Burgers vector distancewhen the applied stress is near the dislocation nucleationstress B/τ nuc ∼ × − ns where B is the linear dragcoefficient.Driven by local stress-induced forces [80], dislocationsfollow athermal dynamics with mobility µ d . Sample lat-eral surfaces are free for dislocations to escape from thesurfaces. Samples (aspect ratio h/w = 4) are assumed tocarry single slip plasticity oriented at 30 ◦ ( cf. Fig. 1(a)).Dislocation sources (red dots) and obstacles (blue dots)are located on slip planes, spaced 10 b apart, with b =0 . E = 70 GPa and ν = 0 .
33. As it maybe seen in Fig. 1(b), a significant difference between twoloading rates (SC) can be seen through strain pattern-ing at the same final strain (5%): plasticity is localized(Fig. 1(c)) for small loading rates while it is relativelyuniform for a high loading rate (Fig. 1(d)).As expected and shown in Fig. 2 (a), SC leads to hard-ening while DC to softening, with the discrepancy be-coming dramatic as system size decreases to sub-microndimensions. Typical size effects ( σ Y ∼ w − . − . ) areseen for both loading protocols ( cf. Fig. 2(b)), despitethe fact that dislocation density at 2% strain, shown inthe inset, increases with increasing w in different waysdepending on the loading conditions. In addition, theflow stress shows a rate dependence for both loading con-ditions (see Fig. 2(c)), even though DC shows weakerdependence. The dislocation density and flow stress de-pendences on the rate suggest that SC rates statisticallyresemble larger DC rates. This conclusion is also sup-plemented by avalanche statistics ( cf. Fig. 2(d)): In SC,event size is defined as S = (cid:80) i ∈ { δ(cid:15) i >(cid:15) threshold } δ(cid:15) i ; in DC,an event is characterized by stress drops δσ which leadto temporary displacement overshoots – thus, in order tocompare the two loading conditions, a DC strain burstevent size is defined as S = (cid:80) i ∈ {− δσ i >σ threshold } δ(cid:15) i [44].In this model, dislocation plasticity is loading rate de-pendent as there are two intrinsic time scales [77]: First,the dislocation nucleation time t nuc , which is chosen as10 ns and can be associated to the dislocation multi-plication timescale in other models. Second, the ratiobetween dislocation mobility and material Young’s mod-ulus B/E which is chosen as 10 − ns. These parameters2 -0.0416667-0.0833333-0.125-0.166667-0.208333-0.25-0.291667-0.333333-0.375-0.416667-0.458333-0.5 h w yy yy y x (a) (b)(c) (d) FIG. 1:
The model. (a)
The pillar has width w and aspectratio h/w =4. Single slip system which oriented at 30 ◦ rela-tive to y axis is used. Distance between planes is 10 b where b = 0 . (b) Sample stress strain curves of com-pression at high (10 / s) and low (10 / s) stress rates ˙ σ . (c) strain pattern after deformation at low ˙ σ , (d) strain patternafter deformation at high ˙ σ . are consistent with recent single-crystal thin film experi-ments [84, 85]. Phenomenology in metallurgy [75, 86, 87],suggests that at low rates the flow stress is controlledby dislocation nucleation while above a certain strainrate ( ∼ − / s), it is mainly controlled by dis-location drag. Fig. 2(c) shows the rate effect under SCand DC conditions. For DC and at strain rates higherthan 5000/s, there is a strong flow stress rate depen-dence. In SC, the drag regime starts when stress rateis E ∗ ∗ /s. The origin of this strain-rate crossover isclear in this model: It is clear that during the nucleationevents, strain-increments are necessarily mediated by dis-location drag; for mobile dislocation density ρ (cid:39) /m and flow stress τ f (cid:39) M P a , the strain generated bymoving dislocations in time-intervals of nucleation timecan be up to 0 . ρτ f b /B ∗ t nuc (cid:39) × − . Thus, for strain-rates greater than 10 /s , the strain increment requiredduring τ nuc is ˙ (cid:15) × τ nuc > − , which implies that thenucleation-induced strain is not adequate. Thus, it isplausible that for ˙ (cid:15) > − /s , dislocation drag takes overthe dynamics of dislocations.While both DC and SC display a flow stress rate effect,their statistical noise behavior is very different: As shownin Fig. 2(d), the plastic events statistics based on stressstrain curves shown in Fig. 2(a), have different τ expo-nents: While plastic events show power law behavior, τ is close to 3 . (a) (b)(c) (d) ( ) ˙ /E ⇤
Effect of loading protocol: Stress-Controlled(SC) vs. Displacement-Controlled (DC). (a)
Stress-strain curves of different w using two different loading pro-tocols. The strain rate ˙ (cid:15) is 10 /s in DC and stress rate ˙ σ is E ∗ ∗ /s. A particular strain burst is shown; (b) Size effectof flow stress at 2% strain (blue stands for DC and red standfor SC, results are based on 50 realizations). The inset showsthe dependence of dislocation density on w at 2% strain fordifferent loading protocols. (c) Dependence of flow stress (for w = 1 µ m) on rate. Strain rate ˙ (cid:15) is used in DC (blue curve)while the elastic corresponding stress rate ˙ σ = E ∗ ˙ (cid:15) is usedin SC (red curve). (d) : Events (strain jumps) statistics fordifferent loading protocols, different point size represents dif-ferent w . blue: DC, red: SC. Strain jump in DC mode iscalculated according to the method in [44]. (a) (b) increasing Lower Larger FIG. 3:
SC Rate Effect Crossover. (a):
Event statisticsfor different ˙ σ using SC. The effective τ changes from ∼ − . σ = E ∗ ∗ ∼ − . σ = E ∗ ∗ /s. (b): Effect ofdislocation source density ρ nuc and mobility B on power lawexponent: when ˙ σ = E ∗ ∗ /s, changing ρ nuc from 60 µ m (purple curve) to 15 µ m (blue curve) leads to the exponentchanging from -2.5 to -2.1. Increasing B from 10 − Pa.s to10 − Pa.s results in the change of exponent from -2.5 to -2.2. between SC and DC disappears at high stress load-ing rates: Fig. 3(a) shows avalanche statistics for dif-ferent stress rate which varies from ˙ σ = E ∗ ∗ σ = E ∗ ∗ /s. Power law events distribution appear forall stress rates, yet with different exponent which changesfrom -3.5 for ˙ σ = E ∗ ∗ σ = E ∗ ∗ /s.The dependence of the exponent on the stress rate clearly3 a) (b)(c) (d) Increasing
FIG. 4:
Spatial and temporal event distribution inSC.
Event distribution on all slip planes during the loadingup to 10% strain for small ˙ σ ( (a) ) and for large ˙ σ ( (b) ): n is the total number of slip planes in the model. The colorchanges from dark purple to yellow with increasing loadingstrain. (c) : Average avalanche size for small ˙ σ in a singlesample. (d) : Average avalanche size for large ˙ σ in a singlesample. indicates that there is an intrinsic connection betweenevent statistics and dislocation drag. In order to verifythe connection, in Fig. 3(b) red curve, we increase thedislocation mobility B by which the drag effect is en-hanced, it is seen that the exponent changes from -2.5to -2.2. Dislocation drag effect will also magnify whendislocation nucleation effect is weakened due possibly todislocation cross-slip and other mechanisms (since themain source of plasticity will be the moving of disloca-tions instead of nucleations of new dislocations). Thiscan be seen in Fig. 3(b) blue curve, when lower disloca-tion source density is used, the exponent changes from-2.5 to -2.1.Power law avalanche behavior in the elastic responseof disordered systems has been well established [29, 34–36, 88]. In the context of nanopillars, the dislocationensemble should be the homogeneously disordered elas-tic system and in this case, the spatial distribution ofevents on all slip planes should be on average flat or dis-play relatively small fractal exponents [89] in the absenceof localization. However, crystal plasticity is known to beunstable to strain localization [90]. In Fig. 4(a) and (b),we plot events spacial distribution along all slip planesfor the whole loading process (from small (cid:15) to large (cid:15) which is represented by the color map from purple toyellow). Fig. 4(a) shows the event spatial distributionfor a smaller loading rate. It can be seen that eventsare localized around certain slip planes, moreover, eventsdo not always happen at the same slip planes. By con-trast, the event distribution shown in Fig. 4(b) for higherloading rate is more uniform among slip planes, further- more, events always happen at the same active slip planeswhich is similar to having an propagating interface. Ad-ditionally, we plot event size with increasing strain (S vs.time). Very clear oscillatory-like behaviour emerges forsmall stress rate shown in Fig. 4(c) while no periodicityis observed for higher stress rate. These results are strik-ingly similar to the avalanche oscillator found in [41].The onset of quasi-periodic response at small rates, inthe absence of overall weakening in this model, is theoutcome of the interplay between a timescale competi-tion (as in other elasticity models [37]) and a distinctfeature of small volumes: ie. Free boundaries that mayabsorb propagating dislocations. Due to this property, itis natural to expect an integration of avalanche behav-iors, dependent on the resetting behavior that emergesfrom absorption and re-nucleation of dislocations at var-ious slip planes. The overall effect can be thought ofas originating within a relaxation process (nucleation)that contributes to slip, in addition to mobile dislocationmotion. This is the type of coarse-grained dislocationmodeling that was pursued in Ref. [41] and its analy-sis leads to critical power law exponents that are higherthan typical ones ( ∼ / P ( S, k ) ∼ S − τ e − kS ,with k a cutoff parameter then this spatiotemporal inte-gration leads to an effective probability distribution: P int ( S ) = (cid:90) ∞ g ( k (cid:48) ) P ( S, k (cid:48) ) dk (cid:48) (1)where g ( k (cid:48) ) is the weight factor that characterizes thecontribution of various sub-critical, quasi-localized spa-tial contributions to slip events and depends on the ap-plied loading rate. This weight factor g ( k (cid:48) ) contains anatural k (cid:48) → g ( k (cid:48) ) ∼ k (cid:48) α . Thus,for the critical aspect of P int ( S ) ∼ S − τ − α − , with theultimate avalanche size exponent being, τ = τ + α + 1 (2)For the current model, by the analysis of Figs. 4(a, b),we can estimate α : If we assume that each 3 nearby slipplanes are locally independent from the rest of the sys-tem, then the max event size in that area can providean estimate of the cutoff scale ( k ∼ /S ). Then, thedistribution of k ’s provides the exponent. We find that α (cid:39) τ (cid:39) .
5. However, the statistics has not been ex-haustive enough to justify a precise identification of these4xponents.In conclusion, we provided strong evidence through anexplicit model of crystal plasticity for nanopillar com-pression, that the statistical behavior of nanocrystal plas-ticity forms a novel universality class that is distinct fromother plasticity behaviors such as amorphous BMGs andgranular systems [32]. We find that the free nanoscaleboundaries and the competition between dislocation nu-cleation and drag conspire to cause the emergence of un-conventional quasi-periodic avalanche bursts and highercritical exponents as strain rate decreases. While theinvestigated strain-rates and the associated transitionemerges at relatively high loading rates, the experimen-tally relevant quasi-static response may be controlled bythe same qualitative behavior [72], or more timescalesmight be in competition. Plasticity is locally heteroge-neous, both spatially and temporally, and this reason liesbehind the rate dependence of the avalanche distributionexponent.We would like to thank I. Groma, P. Ispanovity, R.Maass, L. Ponson for encouraging and insightful com-ments. This work is supported through awards DOC -No. 1007294R (SP) and DOE-BES de-sc0014109. Thiswork benefited greatly from the facilities and staff of theSuper Computing System (Spruce Knob) at West Vir-ginia University. [1] J. R. Greer and J. T. M. De Hosson, Progress in MaterialsScience , 654 (2011).[2] M. D. Uchic, P. A. Shade, and D. M. Dimiduk, AnnualReview of Materials Research , 361 (2009).[3] J. R. Greer, W. C. Oliver, and W. D. Nix, Acta Materi-alia , 1821 (2005).[4] J. R. Greer and W. D. Nix, Physical Review B , 245410(2006).[5] I. Ryu, W. Cai, W. D. Nix, and H. Gao, Acta Materialia , 176 (2015).[6] J. Krebs, S. Rao, S. Verheyden, C. Miko, R. Goodall,W. Curtin, and A. Mortensen, Nature materials (2017).[7] M. F. Doerner and W. D. Nix, Journal of Materials re-search , 601 (1986).[8] W. D. Nix and H. Gao, Journal of the Mechanics andPhysics of Solids , 411 (1998).[9] J. J. Vlassak and W. Nix, Philosophical Magazine A ,1045 (1993).[10] M. R. Begley and J. W. Hutchinson, Journal of the Me-chanics and Physics of Solids , 2049 (1998).[11] Y. Wei and J. W. Hutchinson, Journal of the Mechanicsand Physics of Solids , 2037 (2003).[12] J. W. Hutchinson, International Journal of Solids andStructures , 225 (2000).[13] M. D. Uchic, D. M. Dimiduk, J. N. Florando, and W. D.Nix, Science , 986 (2004).[14] M. D. Uchic, D. M. Dimiduk, J. N. Florando, andW. D. Nix, MRS Online Proceedings Library Archive (2002). [15] N. Fleck and J. Hutchinson, Journal of the Mechanicsand Physics of Solids , 2245 (2001).[16] E. C. Aifantis, Journal of Engineering Materials and tech-nology , 326 (1984).[17] E. C. Aifantis, Strain gradient interpretation of size ef-fects (Springer, 1999), pp. 299–314.[18] P. M. Anderson, J. P. Hirth, and J. Lothe,
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