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Edited by: Abram H. Clark, Naval Postgraduate School, United States

Reviewed by: Martin Kröger, ETH Zürich, Switzerland; Paul A. Johnson, Los Alamos National Laboratory (DOE), United States; Jonathan Barés, Université de Montpellier, France

This article was submitted to Soft Matter Physics, a section of the journal Frontiers in Physics

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Recent experiments show that the deformation properties of a wide range of solid materials are surprisingly similar. When slowly pushed, they deform via intermittent slips, similar to earthquakes. The statistics of these slips agree across vastly different structures and scales. A simple analytical model explains why this is the case. The model also predicts which statistical quantities are independent of the microscopic details (i.e., they are “universal”), and which ones are not. The model provides physical intuition for the deformation mechanism and new ways to organize experimental data. It also shows how to transfer results from one scale to another. The model predictions agree with experiments. The results are expected to be relevant for failure prediction, hazard prevention, and the design of next-generation materials.

Different types of solids can have vastly different structures at the microscopic scale. Their atoms may be arranged on a lattice, as in crystals, or sit in random positions, as in metallic glasses. At longer length scales, solids can comprise densely packed grains, powders, or porous structures (e.g., in the case of many rocks). Despite these widely different structures, it appears these solids have surprisingly similar deformation properties: under slow compression many deform via intermittent slips, similar to earthquakes. Most earthquakes are small, some are of intermediate size, and relatively few are large. Histograms of earthquake size follow the famous Gutenberg-Richter law, which is a simple power-law function that extends over many decades in size. Lab-scale solids behave in the same way. If you were to bend your fork and could listen to the acoustic emission, you would hear crackling noise, which is the manifestation of small slips in the metal [

The aforementioned observations suggest that a universality class may exist for solids that show intermittent discrete deformation events (often referred to as

The premise of the model is that most solids have

The model assumes each weak spot is stuck until the local force or stress exceeds a random threshold. For ductile materials [such as nanoscale single crystals [

Details of the model equations are given, for example, in Dahmen et al. [_{l} at a lattice point _{l} = (_{m} (_{m} –_{l})] + _{m} is the accumulated local slip distance along the slip plane at site _{f, l} = τ_{s, l} (or τ_{d, l}). Here τ_{s, l} is a static failure stress, and τ_{d, l} (τ_{d, l} < τ_{s, l}) is a weakened dynamic failure stress. A site can then slip by a random amount Δ_{l} resulting in a stress reduction τ_{f, l} – τ_{a, l} ~ 2_{l} where _{a, l} is the local random arrest stress or sticking stress, at which the site resticks. For zero weakening (i.e., weakening parameter ε = 0), the failure stress takes a static value τ_{f, l} = τ_{s, l.} For stress-controlled deformation starting from a relaxed (zero stress) state, the external stress _{c}, which has been studied previously in other contexts [_{s, l} – τ_{d, l})/(τ_{s, l} – τ_{a, l}) > 0) after a slip at the static failure stress τ_{s, l}, the slip stress is reduced to the lower dynamic threshold τ_{f, l} = τ_{d, l} and it remains at that reduced value until the ensuing slip avalanche is completed. Only afterward does it reheal to its static threshold stress τ_{f, l} = τ_{s, l}.

For the slow deformation of ductile solids, like copper, the model predicts broad distributions of slip avalanche sizes, following a power law distribution with a cutoff of a size that grows with applied stress up to a maximum value that is set by either the sample size or the work-hardening of the material [

In experiments, the slip-size distributions can be extracted from measurements of time series of the stress or strain or in some cases from acoustic emission or heat pulses. The emission experiments use sound waves or heat pulses to extract information about the size/energy and location of the slips [

For ductile materials the model predictions can be used to extract the failure stress from the slip statistics measured at lower stresses.

Nanopillar compression tests from Friedman et al. [

Stress-binned complementary cumulative distribution C(S) of slip sizes S for different bins of applied stress (note that “s” on the horizontal axis in the main figure represents the slip size denoted “S” in the inset and also in this review). The data was obtained from the compression of seven Mo nanopillars of approximate diameter 800 nm, compressed at a 0.1 nm/s nominal displacement rate. The inset shows a scaling collapse of the data using the mean-field model predictions κ = 1.5 and σ = 0.5 onto the predicted mean-field scaling function shown in gray. For details, see [

The model has no intrinsic length scale nor any information about the material's microstructure. We therefore expect it to apply equally well to amorphous materials, such as bulk metallic glasses (BMGs), as to crystals. BMGs are metallic alloys with amorphous structure and excellent strength properties.

Experimental setup and stress drops.

This particular experiment was performed at exceptionally high time-resolution (100 kHz); a high data acquisition rate is required to conduct a detailed study of the dynamics of individual slips [

The stress drop rate Ṡ vs. time for a single avalanche in the small avalanche regime after Wiener filtering is performed. The unfiltered and Wiener-filtered data for stress vs. time are also shown. The axes scales are different for the plots.

The individual avalanche profiles (in red) as well as the averaged profile (in black) of those “large” avalanches within the bin 7 – 0.7 ms ≤ T ≤ 7 + 0.7 ms and S ≥ 10 MPa, i.e., avalanches with duration of 7 ms ± 0.7 ms in the large avalanche regime [Reprinted from Wright et al. [

Average avalanche profiles of “small” avalanches. ^{−1/2}). The main figure shows the agreement of the collapse with the predicted collapse scaling function (black curve, for non-universal values of the constants A = 3.98 × 10^{11} and B = 2.18 × 10^{11}) [Reprinted with permission from Antonaglia et al. [

Similar scaling has been found for experiments at different stresses and strain-rates [

Sketch of size scales of samples showing the same slip avalanche statistics and spanning 12–13 decades in length [Reprinted from Uhl et al. [

Scaling collapse of the slip statistics (size CCDF) from different stress-windows of five different materials on scales spanning 12 decades in length, onto each other and onto the predicted scaling function of the mean-field model (see [

Similar agreement with the model predictions has been observed in ferroelastics with twin boundaries [

Tools from the theory of phase transitions, such as the renormalization group [

These results tell us that simple models are useful for describing far-from-equilibrium avalanche statistics and dynamics. In fact, the parameter range where critical scaling is seen is often much larger in far-from equilibrium systems than in typical equilibrium systems. In plasticity the power-law slip-size distributions are often observed over a wide range of stresses and strain rates. For both science and engineering, many important insights and applications may emerge from these results:

We have already seen that the noise created by the slips at low stresses contains information about the stress at which the material will break.

The simple model provides intuition for the deformation process. It describes material failure as a non-equilibrium phase transition, and identifies the relevant tuning parameters, such as stress and strain-rate, that affect the size of the largest slip avalanches [

The model predictions for universal behavior can be used to transfer results from one materials study to another and from one scale to another. Thus, the model tells us how to use laboratory experiments on bulk metallic glasses to interpret the slip statistics and dynamics in granular materials and even earthquakes.

The model tells us how to organize experimental data. It predicts which quantities are expected to be the same for different materials and on different scales and which ones are not. For example, the scaling exponents and collapse functions are predicted to be the same for many different material structures and scales, while the exact value of the failure stress should depend on the microscopic details, and thus be material specific.

The model predicts how the statistical properties of the slip avalanches are related. For example it gives exponent relations that predict how the scaling behavior of the slip-size distributions, the slip-duration distributions, and the power spectra of acoustic emission during the deformation are related. Knowledge about these connections is useful for comparing experiments that use different experimental measurement methods. For example, the model enables us to predict the scaling behavior obtained from acoustic emission experiments from stress drop measurements and vice versa.

The model provides an understanding of the deformation mechanism from the slip statistics and the slip

Potential deviations from the mean-field predictions that may be discovered in future experiments can be used to provide new insights into differences between avalanche dynamics in various contexts. For example, neuron firing avalanches in the brain have different slip statistics than the slip avalanches in solid materials, even though both result from coupled threshold processes. The reason is that in the brain the individual cells are coupled through a complicated network that affects the scaling behavior of the avalanche statistics [

Similar modeling ideas and data analysis tools can be applied to many other systems with avalanches, including magnetic materials, neuron firing avalanches in the brain, decision making processes, sociological conflicts, imbibition, fracture front propagation, ferroelastics, power-grid failures, and maybe even the stock-market [

KD and JU wrote the first draft of the manuscript. All authors contributed to manuscript revision, read, and approved the submitted version.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Many people have contributed to this perspective; we especially thank the scientists who kindly shared their experimental and observational data with us: Thorsten Becker, Robert Behringer, Rachel Byer, Dmitry Denisov, Georg Dresen, Thomas Goebel, Julia Greer, Xiaojun Gu, Todd Hufnagel, Andrew Jennings, Ju-Young Kim, Peter Liaw, K. A. Lörincz, Robert Maaß, Junwei Qiao, Steven Robare, Peter Schall, Danijel Schorlemmer, Molei Tao, Katherine Van Ness, Xie Xie, Jien-Wei Yeh, Yong Zhang, and Jien-Min Zuo, and the students who analyzed the data and compared them to model predictions, including James Antonaglia, Braden Brinkman, Nir Friedman, Michael LeBlanc, Xin Liu, Yun Liu, Aya Nawano, Shivesh Pathak, Ryan Swindeman, Gregory Schwarz, Li Shu, Georgios Tsekenis, and Matthew Wraith. We also thank Jim Wolfe for very helpful comments.