Modeling the complexity of acoustic emission during intermittent plastic deformation: Power laws and multifractal spectra

Scale-invariant power-law distributions for acoustic emission signals are ubiquitous in several plastically deforming materials. However, power-law distributions for acoustic emission energies are reported in distinctly different plastically deforming situations such as hcp and fcc single and polycrystalline samples exhibiting smooth stress-strain curves and in dilute metallic alloys exhibiting discontinuous flow. This is surprising since the underlying dislocation mechanisms in these two types of deformations are very different. So far, there have been no models that predict the power-law statistics for discontinuous flow. Furthermore, the statistics of the acoustic emission signals in jerky flow is even more complex, requiring multifractal measures for a proper characterization. There has been no model that explains the complex statistics either. Here we address the problem of statistical characterization of the acoustic emission signals associated with the three types of the Portevin–Le Chatelier bands. Following our recently proposed general framework for calculating acoustic emission, we set up a wave equation for the elastic degrees of freedom with a plastic strain rate as a source term. The energy dissipated during acoustic emission is represented by the Rayleigh-dissipation function. Using the plastic strain rate obtained from the Ananthakrishna model for the Portevin–Le Chatelier effect, we compute the acoustic emission signals associated with the three Portevin–Le Chatelier bands and the Lüders-like band. The so-calculated acoustic emission signals are used for further statistical characterization. Our results show that the model predicts power-law statistics for all the acoustic emission signals associated with the three types of Portevin–Le Chatelier bands with the exponent values increasing with increasing strain rate. The calculated multifractal spectra corresponding to the acoustic emission signals associated with the three band types have a maximum spread for the type C bands and decreasing with types B and A. We further show that the acoustic emission signals associated with Lüders-like band also exhibit a power-law distribution and multifractality.

Figure 1

Model acoustic energy $R_{AE}$ associated with (a) the uncorrelated static type C bands for ${\dot{\epsilon }}_a=1.125\times {10}^{-5}{\text{s}}^{-1}$, (b) the partially propagating type B bands for ${\dot{\epsilon }}_a=3.0\times {10}^{-5}{\text{s}}^{-1}$, and (c) the fully propagating type A bands for ${\dot{\epsilon }}_a=5.5\times {10}^{-5}{\text{s}}^{-1}$. The inset in (a) shows the nonoverlapping nature of the bursts for the type C bands. The amplitude decreases in an oscillatory manner, not visible on this scale. The inset in (b) shows the increasing background AE level caused by SASs for the type B bands. The inset in (c) shows the increased level of background AE is due to the long stretches of SASs for the type A bands.

Figure 2

Power-law distributions for the AE events $R_{AE}\left(p\right)$ for (a) the type C bands for ${\dot{\epsilon }}_a=1.125\times {10}^{-5}{\text{s}}^{-1}$, (b) the partially propagating type B band for ${\dot{\epsilon }}_a=3.0\times {10}^{-5}{\text{s}}^{-1}$, and (c) the type A band for ${\dot{\epsilon }}_a=5.5\times {10}^{-5}{\text{s}}^{-1}$.

Figure 3

(a) Model acoustic energy $R_{AE}\left(p\right)$ for the Lüders-like propagating band along with the stress-strain curve. (b) Corresponding power-law distribution for the AE signals ${\dot{\epsilon }}_a=1.67\times {10}^{-6}{\text{s}}^{-1}$.

Figure 4

Plots of respective subsegments of the AE signals shown in (a) Fig. 1 for the type C band, (b) Fig. 1 for the type B band, and (c) Fig. 1 for the type A band.

Figure 5

Multifractal spectrum $f\left(\alpha \right)$ associated with the AE signals accompanying the type C, B, and A bands for ${\dot{\epsilon }}_a=1.125$, $3.0$, and $5.5\times {10}^{-5}{\text{s}}^{-1}$, respectively.

Figure 6

Generalized dimensions for the AE events associated with the type C, B, and A bands for ${\dot{\epsilon }}_a=1.125$, $3.0$ and $5.5\times {10}^{-5}{\text{s}}^{-1}$.

Figure 7

Multifractal spectrum $f\left(\alpha \right)$ for the Lüders-like propagating band for ${\dot{\epsilon }}_a=1.67\times {10}^{-6}{\text{s}}^{-1}$.