Analysis of serrations and shear bands fractality in UFGs

1 Introduction

Four classes of polycrystalline materials can be identified based on grain size (d): (i) large-grained materials (d>20 μm); (ii) micro-grained (d≈1–10 μm); (iii) ultrafine-grained (UFG) (d≈300–900 nm); and (iv) nanocrystalline (NC) (d<100–200 nm) [1]. The cases (i) and (ii) correspond to coarse- and fine-grained (d>1 μm) materials. In coarse-grained materials, the deformation mechanism involves the motion, interaction, and production/annihilation of dislocations in the grain interior, whereas the grain boundaries (GBs) act as obstacles to plastic flow. These microscopic processes often manifest at the macroscale through the appearance of plastic instabilities (necking/shear banding, dislocation density patterns) (see, for example, [2] and references quoted therein) defining the interplay between significant work hardening and ductility. On the contrary, in UFG and NC materials, the deformation mechanisms are different, due to small grain sizes which localize plastic activity mainly in GBs, which now act as facilitators rather than obstacles of plastic flow. As a result, the usual dislocation processes in the grain interior responsible for plastic flow in conventional grain-size materials are replaced by GB processes in UFGs and NCs. Such processes involve grain rotation and GBs sliding, dislocation nucleation, emission and absorption in GBs, twinning and faulting as well as diffusional creep and GB migration ([3] and refs therein). UFG and NC materials have enhanced properties such as significant higher strength, superior superplasticity, low friction coefficient, high wear resistance, enhanced high cycle fatigue life, and good corrosion resistance in comparison with conventional polycrystalline materials [1].

One significant feature of the deformation behavior of UFG materials is the manifestation of serrated flow in stress strain/graphs. For example, serrated flow was observed during the tensile test of bulk cryomilled Al-7.5% Mg alloy [4], in typical compression stress-strain curves of an UFG 5083 Al alloy [5], as well as in consolidated powder specimens of Al-5% Mg-1.2% Cr alloys processed by equal channel angular pressing (ECAP) tested in tension [6]. Possible reasons for the observed serrated flow are Lüders band formation, dynamic strain aging (DSA), and, if the magnitude of the observed stress drops is much larger than those typically associated with DSA, the governing deformation mechanism is attributed to twinning [4]. However, most commonly, serrations are associated with the DSA mechanism, that is, the Portevin-Le Chatelier (PLC) effect, due to negative strain rate sensitivity. The DSA mechanism, which is also commonly observed in some coarse-grained alloys, is associated with the interactions between mobile dislocations and diffusing solute atoms, generating a recurrent process of pinning and unpinning dislocations from solute atoms [5].

Another significant feature observed in UFGs is (multiple) shear band formation. In this case, “massive” shear localization appears to be the basic failure mechanism instead of intragrain dislocation motion [7]. The particular characteristics, such as shear band thickness/spacing and morphology depend on sample microstructure, preparation routine, and testing conditions. Experimentally, compression tests for most NC/UFG metals showed perfect plasticity deformation curves, in contrast to their conventional counterparts exhibiting the usual work (dislocation) hardening behavior [8]. The small grain sizes enhance the propensity for the grains to orient themselves along the direction of maximum shear stress, ultimately leading to the failure of such NC/UFG materials through the formation of one or several dominant shear bands. Typical examples of multiple shear banding observations were considered in a pioneer work on compressed UFG Fe-10%Cu alloys [9] and in follow-up compression tests in bcc Fe and its alloys ([10] and refs therein).

It follows from the above discussion that describing the stress serrations and shear banding characteristics of UFG and NC materials is an important issue since it is related with strength and ductility properties of these materials. With a few exceptions for some of these characteristics (e.g., [8, 11, 12]) differential equations to model serrated flow and shear band formation in these materials are not available. This is not surprising since the nature of plastic deformation in these materials is complex, involving long-range interactions and power law scaling commonly observed in far from thermodynamic equilibrium physical processes. Plastic flow evolves simultaneously at micro-, meso-, and macro-levels resulting to an active multiscale hierarchical system. Nevertheless, in the absence of evolution equations to model these systems, a quantitative description of the experimentally observed statistical features should be useful. Along these lines, we focus in this paper on the description of statistical features as observed experimentally through the underlying deformation mechanisms of UFG materials, namely bimodal Al-Mg alloys and UFG Fe-Cu alloys. As already mentioned, the analysis is concerned with (i) serrations and (ii) shear bands. The results show that the experimentally recorded statistical spatio-temporal features cannot be fitted with the usual Boltzmann-Gibbs entropy statistics but can be interpreted through Tsallis q-entropy statistics [12]. This indicates that the emergence of these deformation-induced spatio-temporal instabilities is associated with highly dissipative irreversible processes, driven far from equilibrium, where the maximization of Tsallis entropy gives rise to stress drops and optimum fractal structures connected with Tsallis q-Gaussian distributions, requisite for further functioning of the system as a whole. The plan of the paper is as follows: in Section 2, a summary of Tsallis nonextensive thermodynamics and associated methodology is provided; in Section 3, this formalism is applied to describe the statistics of serrations; in Section 4, a corresponding analysis for shear bands is provided, along with a fractal analysis for their spatial arrangement; and finally in Section 5, a discussion of the results is given and future directions are outlined.

2 Tsallis Nonextensive statistics and fractals

The basic building block of Tsallis nonextensive thermodynamics is his generalization of Boltzann-Gibbs (BG) statistical entropy expression in the form

where k is Boltzmann’s constant, W a set of discrete states, and q is the degree of nonextensivity [13]. For two probabilistically independent systems (A, B), the relation given by Eq. (1) transforms into

The first part of S_q(A+B) is additive S_q(A)+S_q(B), whereas the second part is multiplicative. This second part is responsible for nonextensivity and describes the long-range interactions between the two systems. For q<1, q=1, q>1 we have respectively, superadditivity, additivity, and subadditivity. When q=1, Eq. (2) corresponds to the entropy of the usual BG statistical mechanics. The Tsallis entropy S_q measures the complexity of the system, with the parameter q indicating the degree of nonextensivity of the system. From the maximization of S_q given by Eq. (1) subjected to specific constraints occupational probabilities are generated following a q-exponential distribution or q-Gaussian distributions. These probability distributions (q-distributions) exhibit heavy-tails modeling power law phenomena – a typical characteristic of complex systems [14].

The estimation of Tsallis q entropic index, which is also called stationary q=q_stat and is related to the size of the distribution tail, is usually based on q-Gaussian distributions which describe metastable or stationary states of the system. In particular, Tsallis q-Gaussian distribution is given by

where e_q=[1+(1-q)x]^1/(1-q) is referred to as q-exponential, β is a positive number and C_q is a normalization constant, namely Cq=∫-∞∞eq-x2dx. For 1<q<3, C_q has the following form:

The q entropic index, can then be estimated using the probability density function (PDF) computed from the experimental data X={x_t; t=1, 2, …, N}. The statistical analysis is based on the algorithm described in [15]. We construct the PDF of the input time series (obtained from the original or the detrended data if we want to study the rapid fluctuations) as follows:

The interval {min(X), max(X)} range is subdivided into bins of width δx, centered at x_i so that we can assess the frequency X-values falling within each bin. The resultant histogram is properly normalized (the sum of all probabilities is equal to unity) yielding the stationary PDF {p(xi)}i=1N. Thus, p_i is the probability of a X-value to fall into the ith-bin centered at x_i. For the estimation of q value, we vary q within the interval [1, 3] with a step δq=0.005 and the best q-value corresponds to the best linear fit (maximum correlation coefficient, cc) of the graph ln_q(p(x_i)) vs. xi2, where the function lnq(x)=x1-q-11-q corresponds to the q-logarithm (inverse of the q-exponential). Then, with the obtained q-value=q_stat we compute the q_stat-Gaussian given by the Eq. (3) for different β-values. After selecting the β-value minimizing the quantity ∑i[Gqstat(β;xi)-p(xi)]2, we compare the experimental distribution to the theoretical q-Gaussian and the normal Gaussian PDFs, in a log[p(x_i)] vs. x_i graph.

In order to study shear bands in terms of Tsallis nonextensive statistics we analyzed available scanning electron microscopy (SEM) images. We proceed by producing an image series corresponding to the intensity of pixels values. For this we use a procedure based on space-filling curves methods. A space-filling curve is a way of mapping a multidimensional space into the 1-D space. It acts like a thread that passes through every cell element (or pixel) in the D-dimensional space so that every cell is visited exactly once [16]. In particular, we used the Z-order transformation (Morton order) [17] as a function to map the 2D data (in this case the image pixels’ matrix) to one dimension preserving the locality and the histogram of the data points (pixels’ intensity). The Morton curve can be best described by recursive algorithm. The algorithm successively fills squares of sizes (2ⁱ×2ⁱ), i=0, ..., n. A pattern of size (2ⁱ×2ⁱ) always consists of four subpatterns of size (2^i-1×2^i-1). In the Morton curve, the orientation of subpatterns is fixed but does not provide a continuous path through the squares. Because the basic pattern of the Morton curve resembles the shape of a “Z”, it is sometimes referred to as “Z”-curve. Thus, an “image” series is produced to be used for estimating the Tsallis q exponent as an indicator of the pixel’s distributions.

Evidence for Tsallis nonextensive statistics is often indicated by the presence of a (multi)fractal hierarchical geometry. Thus, in order to verify the above hypothesis, we need to estimate the fractal dimension of shear band SEM images. For such estimation we used an improved variation of box counting method, called box merge [18]. In standard box counting methods, scans are made to the data set with a box sized 1/s=1/2, 1/4, ..., 1/s_max of the size of the box containing the data set. In each scan the number n of the nonempty boxes is counted. The fractal dimension is calculated from the slope of the linear part of the log(n)-log(s) plot. In box merging, the first iteration is executed with the finest possible partitioning (s=s_max). If the positions of the nonempty boxes are known, the nonempty boxes belonging to the immediately coarser partitioning (s_max/2) can be found without further scanning. A table is thus constructed with the coordinates of all partitions which contain at least one element of the data set. The partition table of a new iteration can easily be produced by an integer division by 2 of all contents of the preceding table by merging identical rows, so that no scanning is needed any more. The fractal dimension is then calculated from the slope of the linear part of the log(n)-log(s) plot

3 Tsallis statistical features of serrations

In this section, the analysis is focused on temporal stress discontinuities associated with deformation behavior of an UFG Al-Mg alloy. This alloy shows a bimodal grain-size distribution, with elongated large (coarse) grains with sizes around 5 μm surrounded by ultrafine grains with sizes around 120 nm, and was synthesized by the consolidation of cryomilled powders and subsequent hot extrusion [19]. At room temperature, compression tests at low strain rates of 10^-4 and 10^-3 s^-1, produce serrated stress-strain curves without evidence of “microscopic” shear banding (e.g., curve A in Figure 1A), whereas at higher strain rates of 10^-2 and 10^-1 s^-1 (e.g., curve C in Figure 1A), shear band formation is observed, without evidence of “macroscopic” serrated flow.

Figure 1:

(A) Compressive true stress-strain curves [19] for an as-extruded Al-Mg alloy at room temperature at different strain rates. (B) Serrations of the true stress-strain curve (curve A) shown in Figure 1A.

Dynamical systems evolving nonlinearly generate complex fluctuations in their output signals that reflect the underlying dynamics. Thus, examination of serrations can reveal statistical features that may provide insights into the details of the underlying complex deformation processes. Following this line, we analyzed data concerning the stress serrations of curve A shown in Figure 1A, obtained by Fan et al. [19]. An enlarged view of the stress serrations shown in Figure 1B reveals that serrations clearly have an irregular, nonperiodic, fluctuating character. Such nonstationary character is connected with a general increase (drift) of the stress level (background) due to strain hardening, which is irrelevant to the dynamical aspects of serrations. This drift can be removed either by fitting the stress serrations time series with a low order polynomial or by constructing a moving average over a large number of points and subtracting the background value [20]. Next, we apply a parametric detrending by employing a polynomial of third degree. Initially, we consider that the stress time series σ_t is the sum of a trend μ and the residuals S

and that the trend in the data can be fitted by a polynomial of order p (p=3)

Then, we remove the trend and the residuals comprise the detrended time series [21].

The resulting stationary detrended serration series (S) is shown in Figure 2A. It is this series that we used for the analysis of Tsallis q – entropic index. In Figure 2B, we present the best linear correlation between ln_q[p(s_i)](open blue circles) and (s)i2. The best fitting was found for the value of q=1.29±0.05, with a correlation coefficient (cc) cc=0.9302. This value was used to estimate the q-Gaussian distribution presented in Figure 2C by the solid red line, where the difference between the q-Gaussian and the Gaussian PDF (green line) in long tails is clearly pictured in a log[p(s_i)] vs. s_i graph. The open blue circles correspond to the experimental detrended time series. The value q>1 suggests the presence of long-range interactions (a distinctive property of open and nonequilibrium systems) in the underlying dynamics which is characterized by non-Gaussian (q-Gaussian) distributions. It is the maximization of Tsallis entropy which leads to the observed q-Gaussian distribution which has a power-law tail of the form PDF(|s|)~|s|-2q-1 [14], in contrast to the BG formalism which yields exponential equilibrium distributions. Thus, the system dynamics manifests through a series of metastable or nonequilibrium stationary states which are described by a Tsallis q-Gaussian distribution with an entropic index q=1.29>1.

Figure 2:

(A) Detrended serrations (S) corresponding to Figure 1B. (B) Best linear correlation fitting between ln_q[p(s_i)] and (s_i)² for the detrended stress signal. (C) log(p(s_i) vs. s_i for the detrended stress time series S (blue circles), the theoretical q-Gaussian (red line), and the normal Gaussian (green line).

4 Tsallis statistical features and fractality for shear bands

Another aspect for the previously considered UFG Al-Mg alloys is the occurrence of multiple shear bands at high strain rates [19]. The shear band image analysis based on Tsallis q index and fractal dimension estimation is considered in this section. Figure 3A shows a SEM image of multiple shear band formation on the surface of the sample compressed at strain rates 10^-2s^-1 for ε=0.2, at low magnification 100 μm. The average size of the coarse grains is approximately 3.1 μm and of the nanocrystalling grains around 197 nm. In this case, plastic deformation is accompanied with a nearly constant flow stress with no serrations. Corresponding image analysis is conducted to estimate Tsallis q exponent and the results are presented in Figure 3B, C, D. We generated a pixel intensity series I, by utilizing the observed shear band spatial arrangement, through the method described earlier, and we detrended this series by a first difference filter, that is, a high frequency pass filter. In this way, we focus on the rapid fluctuations of intensity values, which in image analysis correspond to high spatial frequencies enhancement, emphasizing fine detail and edges [23]. Thus, the first-difference intensity series Z is the intensity series I at a value j minus the series at a value j-1

Figure 3:

(A) SEM image showing the shear band formation on the surface of the sample compressed at strain rates 10^-2 s^-1 for ε=0.2, at low magnification 100 μm, [22]. (B) Detrended intensity series. (C) Best linear correlation between ln_q[p(z_i)] and (z_i)² for the image series. (D) log[p(z_i)] vs. z_i for the detrended image series (blue circles), the theoretical q-Gaussian (red line), and the normal Gaussian (green line).

In particular, Figure 3B presents the detrended intensity image series (Z) generated and Figure 3C presents the best linear correlation between ln_q[p(z_i)] (open blue circles) and (z)i2. The best fitting was found for the value of q=1.222±0.042, with a correlation coefficient cc=0.9446. This value was used to estimate the q-Gaussian distribution presented in Figure 3D by the solid red line, with the difference between the q-Gaussian and the Gaussian PDF (green line) in long tails being clearly pictured in a log[p(z_i)] vs. z_i graph. The open blue circles correspond to the detrended intensity series. The estimation of Tsallis q exponent is used here as a statistical index of the image complexity. The estimated value of q=1.222>1 suggests the presence of q-Gaussian pixel distributions indicating the presence of nonrandom long-range correlations between the pixels, connected with nonrandom, hierarchical, and fractal geometry of the shear band network.

Next, we present results concerning the structure of intense shear banding in the deformation behavior a bulk Fe-Cu UFG iron alloy. In this case, shear banding was also the only mechanism for plastic deformation, as mentioned earlier [7]. An example of observed shear bands is shown in the photo Figure 4A and an enlarged view of the upper half of the photo is shown in Figure 4B, corresponding to the area denoted by the red square in Figure 4A. This area is chosen for the analysis of shear bands. Figure 4C, D present the detrended intensity image series (Z) and the best linear correlation between ln_q[p(z_i)] (open blue circles) and (z)i2. The best fitting was found for the value of q=1.235±0.039, with a correlation coefficient (cc) cc=0.9546. This value was used to estimate the q-Gaussian distribution presented in Figure 4E by the solid red line, with the difference between the q-Gaussian and the Gaussian PDF (green line) in long tails clearly pictured in a log[p(z_i)] vs. z_i graph. The open blue circles correspond to the detrended intensity series. The estimation of Tsallis q exponent is used here as a statistical index of the image complexity. The estimate of q=1.235>1 suggests the presence of q-Gaussian pixel distributions indicating the presence of nonrandom long-range correlations between the pixels, connected with nonrandom, hierarchical, and fractal geometry of the shear band network.

Figure 4:

(A) A photograph showing shear band formation in the nanostuctured Fe-10% Cu alloy. (B) Enlarged view of shear banding corresponding to the area denoted in the red square in Figure 4A. (C) Detrended intensity series Z. (D) Best linear correlation between ln_q[p(z_i)] and (z_i)² for the image series. (E) log(p(z_i)) vs. z_i for the image series of curve B (blue circles), the theoretical q-Gaussian (red line), and the normal Gaussian (green line).

In order to explore shear band fractality, we estimated the fractal dimension of the shear band SEM image. In particular, the results, estimated by the method described earlier, are presented in Figure 5. This figure depicts the logarithm of the number of boxes log(n) against the logarithm of the box size value of log(s). The slope which was used for the estimation of the fractal dimension corresponds to the linear scaling regime of the interval log(s)=0.5–5.5. Above the value of log(s)=6, a plateau is reached and this is due to the finite amount of data available. The presence of a small degree of multifractality is noted as seen in the interval log(s)=1–3. The fractal dimension was found to be D=2.3091. Thus, the spatial evolution of multiple shear bands forms eventually a complex fractal (slightly multifractal) network, clearly different from periodic or quasi-periodic structures (e.g., structures alternate in a more or less regular way) and from random-homogeneous structures [24]. This result is also in accordance with the results presented previously concerning Tsallis statistics.

Figure 5:

The logarithm of the number of boxes log(n) against the logarithm of the box size value of log(s) (blue line). The estimation of the fractal dimension D of Figure 3A corresponds to the best linear fitting of the slope (red line).

Corresponding results concerning the photo shown in Figure 4B are shown also. The results are presented in Figure 6, which depicts the logarithm of the number of boxes log(n) against the logarithm of the box size value of log(s). The slope which was used for the estimation of the fractal dimension corresponds to the linear scaling regime of the interval log(s)=1–6. Above the value of log(s)=6.5, a plateau is reached and this is due to finite amount of data. The presence of a small degree of multifractality is also noted as seen in the interval log(s)=2–5. The fractal dimension was found to be D=2.72. Thus, the spatial evolution of multiple shear bands forms eventually a complex fractal (slightly multifractal) network. This result is also in accordance with the results presented previously concerning Tsallis statistics.

Figure 6:

The logarithm of the number of boxes log(n) against the logarithm of the box size value of log(s) (blue line). The estimation of the fractal dimension D of Figure 4B corresponds to the best linear fitting of the slope (red line).

5 Summary and discussion

We analyzed statistical features of serrations, which are observed in low strain rates, as well as the spatial distribution of extensive shear bands which are formed in high strain rates of an UFG bimodal Al-Mg alloy. We also studied the spatial arrangement of multiple shear bands in an UFG Fe-10% Cu alloy. The analysis was based on Tsallis nonextensive statistics and fractal theory. The results demonstrate that plastic flow occurs simultaneously at various levels giving rise to the formation of serrations with a Tsallis q-Gaussian distributions or fractal multiple shear banding, also characterized by q statistics. In particular, we showed that

• The distribution of the detrended serration time series for the UFG bimodal Al-Mg alloy is characterized by a Tsallis q-Gaussian distribution with a Tsallis entropic index q=1.29±0.05, indicating a nonextensive character of the underlying dynamics.

• The shear band distribution, studied through the pixel detrended intensity signals of SEM images for the UFG bimodal Al-Mg alloy also corresponds to a Tsallis q-Gaussian distribution with q=1.222±0.042. This indicates the presence of nonrandom long-range correlations between the pixels.

• The fractal dimension of the shear band SEM image of the UFG bimodal Al-Mg alloy was found to be D=2.3091, indicating that shear bands form a complex fractal network.

• The shear band distribution for the UFG Fe-10 Cu alloy also corresponds to a Tsallis q-Gaussian distribution with q=1.235±0.039, indicating the presence of nonrandom long-range correlations between the pixels.

• The fractal dimension of the shear band photo image for the UFG Fe-10 Cu alloy was found to be D=2.72, indicating that shear bands form an intense complex fractal network.

In order to further substantiate the aforementioned results, the analysis should be extended for estimating Tsallis q-triplet: (q_sen, q_stat, q_ref) [15]. The q-triplet triplet values characterize the attractor set of the dynamics in the phase space, providing further information about multifractality and relaxation times characterizing spatio-temporal deformation pattering in UFG materials. In addition, other indexes which are connected with Tsallis statistics and intermittent turbulence, such as flatness coefficient and generalized structure function [25] may need to be explored.