Propagating bands of plastic deformation in a metal alloy as critical avalanches

Portevin–Le Chatelier deformation bands obey predictions of a simple mean-field model of critical avalanche dynamics.


Introduction
Complexity in materials deformation is important for engineering and involves fundamental non-equilibrium physics. Such phenomena are encountered when samples are loaded beyond the regime of linear, elastic response. Then, metals yield and the plastic deformation prior to failure is now known to exhibit very complex properties on various scales in time and space (1)(2)(3)(4). The challenges this brings up range from avalanches of plastic deformation to the statistical fluctuations of the yield stress in finite samples to deformation localization. A typical manifestation of localization is the appearance of shear bands and here we study the Portevin-Le Chatelier effect (5,6).
The PLC effect implies the creation of deformation bands in a sample ( Fig. 1) when it is loaded beyond the yield point: such bands nucleate, and may or may not propagate depending on the class of PLC instability present (7,8) (in the common classification type A denotes propagating and types B and C nonpropagating bands). The deformation bands are accompanied by material instabilities, in the case of tensile tests stress-drops which then produce serrated stress-strain curves (Fig. 1b). This kind of Strain-Rate Sensitivity (9, 10) (SRS) arises as a strain-rate dependent phenomenon; moreover its character and presence are dependent on the temperature. The PLC effect is attributed to Dynamic Strain Aging (11)(12)(13) (DSA), and the crucial physics is in the interaction of the dislocations as the fundamental carriers of plastic deformation with the solute atoms in the alloy (14)(15)(16). On the mesoscopic level, theories of increasing complexity have been proposed such that they would account for the necessary dislocation physics: elementary classes of immobile and "aging", solute bound dislocations, and mobile ones producing plastic deformation. Such models and a multitude of experiments have been recently introduced to explore the physics of the PLC effect: phases in the band nucleation (17)(18)(19)(20) and dynamics including serrations in the stress-strain curves (8,10,(21)(22)(23)(24)(25), acoustic emission (25)(26)(27)(28) from the effect and so forth.

Results
Here we take a fundamentally different approach of coarse-graining, where the bands are reduced to zero-dimensional "particles". This amounts to studying the propagation velocity signals v b (t) of each individual propagating ('type A') band during a deformation experiment. Our high-resolution experiments based on speckle imaging of the deforming sample (see Fig. 1 and Methods for details) reveal that the v b (t) signals are reminiscent of crackling noise bursts found in numerous driven systems ranging from propagating cracks (29) and fluid fronts invading porous media (30) to the jerky field-driven motion of domain walls in ferromagnets (31)(32)(33) (see Fig. 1c). This is in contrast to the traditional viewpoint where one would characterize the movement of the bands only via their average velocity v b . Time-averaging each of the fluctuating v b (t)-signals we recover the known phenomenology in that v b is found to decrease with the strain and increase with the strain-rate˙ (34-37) (Fig. 2). In this case we found a powerlaw increase with the strain-rate and an exponential decrease with strain so that they can be summarized as and with the data set at hand, we find p = 0.6 and 0 = 0.16.
In order to characterize the properties of the v b (t) signals/velocity bursts corresponding to individual bands, we start by considering their average shapes v b t−t 0 T at a fixed duration/band lifetime T (where t 0 is the start of the band propagation); this is one of the standard quantities used to characterize crackling noise bursts. We find that short-lived bands exhibit an approximately parabolic shape, while considering bands with a longer T results in v b t−t 0 T displaying an increasingly flattened profile (red symbols in Fig. 3).
How can one theoretically understand the origin and properties of the crackling noise -like v b (t) band propagation velocity signals, exhibiting such average temporal velocity profiles?
The starting point of our analysis is the empirical observation that the bands tend to propagate essentially as 'rigid bodies', and hence a description based on a single degree of freedom, the band position x b , is appropriate. This rigid body then moves via overdamped dynamics due to the forces acting on it. As the sample is strained with a constant strain-rate˙ , it is natural to assume that the band position is driven at a rate c ∝˙ . This is countered by a stiffness term k which includes the hardening of the sample which can be incorporated in the simplest form as a linear dependence to the strain k ∝ . As the band propagates along the long axis of the specimen, it samples the random dislocation microstructure it encounters during motion, resulting in a position-dependent random force W (x b ), with Brownian correlations, Collecting these terms, one arrives at an equation of motion for x b which has the same form as the Alessandro-Beatrice-Bertotti-Montorsi (ABBM) model (38) used as the mean-field description of domain wall depinning in disordered ferromagnets, i.e., where D is the disorder strength. The ABBM model (Eq. 2) is known to produce crackling noise or avalanches with power-law distributed sizes and durations (31,32), charaterized by c-dependent exponents; for instance, the size distribution scales as P (S) ∼ S −(3−c)/2 where c = c/D is the normalized driving rate. Following Ref. (39), Eq. 2 can be transformed to a form including a time-dependent noise term, with ξ being a white noise term with unit variance ξ(t)ξ(t ) = δ(t − t ). This Eq. 3 has the advantage of allowing one to solve analytically quantities like the average burst shape in the k = 0, c = 0 limit, resulting in an inverted parabola, while a finite k gives rise to a flattening of the shape for long avalanches (39).
Comparing the model and the experiments To compare this model with experimental data, we simulate it by "nucleating bands" at random initial positions x i b ∈ (0, L) within a sample of length L, and let them propagate in a random direction according to Eq. 3. To mimic effects due to the finite length of the sample, we consider only bands that stop before the end of the sample.
This leads to a L-dependent cutoff to the "avalanche" distribution; in addition, a cutoff could in principle be due to k in Eq. 3, but here k is sufficiently small such that the L-dependent cutoff dominates. This then results in a scaling form for the avalanche size distribution (see Methods for details) To connect the predictions of the ABBM model to the stress-strain curve one can also study the the scaling of the average stress-drop size ∆σ from the stress-strain curves with duration.
The one-to-one correspondence between deformation bands and stress-drops is broken by the observed multiple simultaneously propagating bands ( Fig. 1) but the average scaling seems to be similar for the avalanche sizes and stress-drop sizes (Fig. 4a). The shorter stress-drop du-rations are due to the simultaneous bands and the short S < 2 mm bands that are otherwise neglected from the analysis.
The prediction for the avalanche size distribution in a finite-size sample (Eq. 4) has a driving rate dependent exponent (3−c)/2. However as we are again observing bands at different strains and strain-rates it is easier to consider a distribution P (S) ∝ 1 − S L S −α with some exponent α. The experimental data seems to follow this distribution quite well (Fig. 4b) and maximum likelihood estimation gives α = 0.99. The same is true for the ABBM model where a slightly lower estimate of α = 0.73 is obtained. This disparity in the exponent values is likely due to the simplicity of the model parameters (linearity of c and k in˙ and ) and the fact that the parameter values were fitted just to reproduce the behavior with increasing strain and strain-rate.
The instantaneous (band) velocity distribution in the ABBM model is known to be of the form (33,38,39,41) wherek = k/D is the normalized stiffness term and Γ represents the Gamma function. As the observed size distribution suggestsc to be around unity one would expect an exponential band show that both the average PLC band propagation distances and ABBM avalanche sizes scale similarly with duration. The same scaling can also be seen for the stress-drop sizes in the stressstrain curve. Although the one-to-one correspondence between bands and stress-drops is lost with multiple simultaneous bands, the average scaling remains the same.
We analytically show that the finite size of the sample introduces a 1 − S L -cutoff to the known power-law avalanche size distribution. Both the PLC band propagation distances and the simulated bands from the ABBM model follow this P (S) ∼ 1 − S L S −α distribution with exponents α close to unity. Based on this one would then expect the instantaneous band velocity distribution to follow an exponential distribution, which is what we see in the experiments and in the simulations. The ABBM model is commonly studied close to the quasistatic limit c → 0, however we have showed here that it can be also used to explain the behavior of fronts under strong drive, here the deformation bands. Looking at these results from the viewpoint of classical theories of the PLC, it is an important question how to modify and adapt such models of DSA (42) so that they reproduce correctly the kind of stochasticity seen in the band dynamics.
This may be restated so that the "correct" model one should be able to reduce to the ABBM used here.
What our results show is that interacting, mobile dislocations create avalanches of deformation in metal alloys. Here, the necessary conditions for this are temperature and strain-rate values within a specific window such that propagating, or type A PLC bands are observed.
Given this, the avalanches follow the paradigm of the mean-field-like ABBM model. The eventual stopping of the band is a random fluctuation, and depends on the local, heterogeneous material properties. Thus the physics of these bands arises from a mixture of external drive, local randomness, and the coarse-grained, collective response of many dislocations. More work is needed in understanding the implications to other PLC band types, and what the practical predictions or consequences are for alloys with different composition ("disorder") and for samples of different sizes. It is likely that the ABBM exponentc is material-dependent. A wider look suggests to consider the eventual interaction physics of multiple bands present in the sample, where their interaction with others and with the sample or disorder would be crucial (43). In the same vein, propagating bands of deformation with serrations of the stress-strain curve are also seen in the plastic deformation of amorphous materials (44,45). An obvious question would be if these also can be shown to follow ABBM-like dynamics with a careful study, but then again if such bands do not follow this simplest paradigm that is also of profound interest.

Materials and Methods
Experimental methods The laser speckle technique (35) was used to observe the bands in a commercial aluminum alloy AW-5754 sample. The samples were lasercut to a flat dogbone shape with the dimensions 28 mm × 4 mm × 0.5 mm for the gauge volume. The samples have a polycrystalline structure with an average grain size of 38 ± 14 µm. The experimental setup is illustrated in Fig. 5.
The samples were tensile loaded with Instron ElectroPuls E1000 using an Instron Dynacell load cell with a constant displacement rate. The stress and strain were calculated from the displacement and force data provided by the machine. These were recorded with an acquisition rate of 500 Hz and the samples were held using an initial force of 4 N.
The speckle pattern was recorded with ProtoRhino FlexRHINO DynaMat system which includes a high speed camera, a laser and a FPGA-chip based unit for data acquisition and storage. The camera had an electronic freeze-frame shutter and a Navitar MVL7000 objective with a macro zoom lens, an aperture of f/2.5 and a spatial resolution of 54 µm. The laser used was a collimated laser diode with a wavelength of 638 nm and a power of 200 mW. The acquisition rates varied around 0.5-2.0 kHz.
The speckle images were analyzed using the equal interval subtracting method (similar to Ref. (35)) where the subtraction was done for consecutive images or with the highest acquisition rate for every other image. A 1D projection was taken from these subtracted images in direction perpendicular to the band with the two different band inclinations. This provides two different effective strain-rate maps where the measured quantity˙ spec corresponds to the time derivative of the speckle image intensity.
As the band angles and widths were observed to remain very close to constant (the band widths are 0.9 ± 0.1 mm which is of the order of the sample thickness 0.5 mm) these effective strain-rate maps were used to track the band movement as a 1D rigid body. The maximum value of˙ spec around the visible band corresponds to the leading edge of the band and this was used as the band position x b therefore also determining the propagation distance S and band duration T . As we are considering type A band dynamics a propagation distance cutoff of 2 mm was imposed. Bands that propagate less than 2 mm correspond more to the type B regime of nonpropagating bands and were excluded from the analysis. The band velocity signal v b was then obtained by numerically differentiating the band position signal and the average band velocity was calculated simply as v b = S/T . There are sometimes multiple simultaneous bands present in the sample (see Fig. 1a and and one can get the distribution of the travel distance S by calculating two convolutions The conditional probability is handled most simply by splitting it into two portions. After starting the band goes in either direction with equal probability and as is known for the ABBM model travels a distance that is power-law distributed. The joint distribution is then and the convolutions give Normalizing this distribution (from a minimum value S 0 to L) gives the full functional form or in the special case of α = 1 For the special case of k = 0 one can obtain an analytic solution for the exponent in the    Figure 5: The experimental setup. The sample is tensile loaded and simultaneously imaged using the laser speckle technique. Here the sample is illuminated by a diffuse laser at a slight angle and the produced speckle pattern on the sample surface is imaged using a high speed camera. Photo credit: Tero Mäkinen