Quasi-periodic events in crystal plasticity and the self-organized avalanche oscillator

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When external stresses in a system—physical, social or virtual—are relieved through impulsive events, it is natural to focus on the attributes of these avalanches1, 2. However, during the quiescent periods between them3, stresses may be relieved through competing processes, such as slowly flowing water between earthquakes4 or thermally activated dislocation flow5 between plastic bursts in crystals6, 7, 8. Such smooth responses can in turn have marked effects on the avalanche properties9. Here we report an experimental investigation of slowly compressed nickel microcrystals, covering three orders of magnitude in nominal strain rate, in which we observe unconventional quasi-periodic avalanche bursts and higher critical exponents as the strain rate is decreased. Our experiments are faithfully reproduced by analytic and computational dislocation avalanche modelling10, 11 that we have extended to incorporate dislocation relaxation, revealing the emergence of the self-organized avalanche oscillator: a novel critical state exhibiting oscillatory approaches towards a depinning critical point12. This theory suggests that whenever avalanches compete with slow relaxation—in settings ranging from crystal microplasticity to earthquakes—dynamical quasi-periodic scale invariance ought to emerge.

At a glance


  1. Dislocation motion and several slow relaxation processes during the waiting intervals between avalanches.
    Figure 1: Dislocation motion and several slow relaxation processes during the waiting intervals between avalanches.

    a, Diagram of typical unit dislocation motions. Lighter to darker indicates time evolution. Under stress, a dislocation loop nucleates and grows until it gets pinned on its slip plane, which is a common and fast glide-slip burst unit process. Then, a screw dislocation segment undergoes double cross-slip to a parallel slip plane, bypassing glide barriers. Finally, the dislocation glides and ultimately, may climb. These unit processes underlie the dislocation ensemble dynamics (not shown). b, Strain jump rate time series of a nickel sample at a strain rate of 10−6s−1. The avalanche phenomenon not only involves fast and violent scale-invariant bursts24, but also long waiting times3 between glide events. During those times, slow relaxation events happen that are typically hard to experimentally distinguish due to the external noise levels (inset). c, Black data points show the estimated strain percentage accumulated in slow relaxation events (generally called ‘creep’), with the threshold set by the event size distribution (Fig. 2): this percentage (relaxation strain/total final strain) strongly increases as the rate decreases. Experimental noise contributes to the relaxation strain measured. Red data points show that a non-trivial quasi-period (see Fig. 2) of avalanche behaviour emerges and increases dramatically as the nominal rate decreases. Error bars indicate the size of systematic variability due to thresholding.

  2. Comparison between microplasticity experiments and theoretical modelling.
    Figure 2: Comparison between microplasticity experiments and theoretical modelling.

    a, Average avalanche size in 400-s windows versus time for different strain rates (decreasing top to bottom: 10−4, 10−5, 10−6s−1, corresponding to respective velocities of the microcrystal top surfaces of 4, 0.4, 0.04nms−1). Time axes are rescaled by the nominal strain rate, aligning the ‘strain scales’. Quasi-periodic avalanche behaviour emerges as the nominal strain rate decreases. The period is similar in ‘strain scale’—a key prediction of our theory. b, Stick-slip oscillations observed experimentally in a show typical characteristics of the model of equation (1) (using 400δ-long averaging windows). The relaxation rate is fixed (R = 2) and the strain-rate is varied (by modifying c: top to bottom; 10−2/δ, 10−3/δ, 10−4/δ), following the experiments. The unit of strain is 2×10−6 and the fast timescale δ = 0.5s. We show the actual avalanche events without the distortion that appears due to the strain coming from slow relaxation; this difference gives the overall scale mismatch of a and b. c, The probability distribution P(S) is shown. There is a marked increase in the critical exponent for the size of the displacement jumps as the strain rate decreases. The highest (10−4) and lowest (10−6) strain rates are fitted to power laws S−1.5f(S/S0) and S−1.9f(S/S0), respectively. The key shows strain rate and corresponding velocity for data points. d, In the model of equation (1), the variation of the rate c shows similar behaviour to that observed in c, with exponent drift from ~1.5 (ref. 8) to ~2.1, with fitting error ~0.2, consistent with the discussion in the text and with fitting cut-off functional forms f(S/S0) that are discussed in Methods.

  3. The avalanche oscillator mechanism and stochastic modelling of the slip susceptibility.
    Figure 3: The avalanche oscillator mechanism and stochastic modelling of the slip susceptibility.

    a, As the rate cd increases (key), for = 0.1, large noise causes ρ(t) of equation (4) to oscillate between ρ1 and small ρ, causing larger exponents. b, The probability distribution of ρ. As cd increases (stronger relaxation, slower strain rate), any ρ becomes equiprobable. In the inset, we use left fenceSright fence50 calculated for equation (1), averaged with a running window of size 50δ. The simulations of equation (1) have the same parameters as in Fig. 2. The histograms, shown in the appropriate scale (11/left fenceSright fenceρ for the kernel used) shows qualitatively similar flattening behaviour as equation (4). c, The novel regime (‘integrated quasi-periodic’) with large ρ fluctuations is separated from the traditional regime ρρ0. The line cd1/ shown, as described in the text. was calculated using equation (3) at equidistant points with a final Gaussian interpolation for the colour background.


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Author information


  1. Department of Mechanical Engineering and Materials Science and Department of Physics, Yale University, New Haven, Connecticut 06520-8286, USA

    • Stefanos Papanikolaou
  2. Air Force Research Laboratory, Materials and Manufacturing Directorate, AFRL/RXCM, Wright-Patterson AFB, Ohio 45433, USA

    • Dennis M. Dimiduk,
    • Michael D. Uchic &
    • Christopher F. Woodward
  3. Laboratory of Atomic and Solid State Physics, Department of Physics, Clark Hall, Cornell University, Ithaca, New York 14853-2501, USA

    • Woosong Choi &
    • James P. Sethna
  4. CNR–Consiglio Nazionale delle Ricerche, IENI, Via R. Cozzi 53, 20125 Milano, Italy

    • Stefano Zapperi
  5. ISI Foundation, Via Alassio 11/c, 10126 Torino, Italy

    • Stefano Zapperi


D.M.D., M.D.U. and C.F.W. designed and performed the experiments. S.P., D.M.D. and C.F.W. performed the experimental data analysis. S.P., W.C., J.P.S. and S.Z. developed the theoretical modelling, performed the numerical simulations and carried out the data analysis. S.P. wrote the first draft of the manuscript and then all authors contributed equally to improve the manuscript.

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