Signatures of Plastic Instabilities and Strain Localization in Acoustic Emission Time-Series

Abstract

Acoustic emission (AE) is a powerful tool for investigating the intermittency of plastic flow by capturing elastic waves generated by dislocation rearrangements under load. This study explores the correlation between AE and plastic instabilities, such as Lüders bands, the Portevin–Le Chatelier (PLC) effect, and necking, each showing distinct AE signatures. Lüders and PLC bands generate significant AE during discontinuous yielding, with a sharp rise in AE levels and a shift in the spectrum to lower frequencies—characteristic of localized deformation. In contrast, necking exhibits limited AE activity, due to reduced strain hardening and dislocation mobility during late-stage deformation. A phenomenological model, based on dislocation dynamics and initially devised for uniform deformation, is discussed to explain the observed AE spectral features during localized plastic flow. This study underscores AE’s potential for non-destructive evaluation and failure prediction in structural metals, emphasizing its sensitivity to microstructural changes and instabilities. Understanding AE behavior across deformation stages offers valuable insights into improving material reliability and predicting failure.

1. Introduction

1.1. Intermittency of Plastic Flow and Acoustic Emission

Plastic deformation is often regarded as a necessary precursor to failure in structural metallic materials from the weakest single crystals to the strongest metallic glasses unless they are tested at extremely low temperatures or high strain rates. “Ideally brittle fracture” is an abstraction adopted in elastic fracture mechanics, originally developed by A.A. Griffith in 1921 [1], as a fundamental concept applicable to brittle solids, such as ceramics and glasses [2], that fracture through the propagation of pre-existing sharp cracks, which start to advance as a certain criterion is met. However, the fracture of structural metals deviates from this idealization due to the plasticity inherent in materials with metallic bonding. Thus, the fracture of most metals is not an instantaneous catastrophic event but rather a multi-stage process evolving with time, strain and stress. This process occurs on many scales and involves an accumulation and interplay of a variety of structural defects with significantly different properties and topologies, including point atomic-size defects, linear dislocations and disclinations, planar stacking faults and grain boundaries, volumetric inclusions, voids and, finally, cracks—all evolving and often coexisting as a result of plastic deformation.

The manifestation of plastic deformation and its microscopic mechanisms vary broadly depending on the microstructure and loading conditions [3,4]. However, despite all variations, the ubiquitous feature of plastic deformation is that it is inherently spatially heterogeneous and temporally intermittent, primarily due to the discrete nature of primary carriers of plasticity, with dislocations of various kinds characterized by a unique figure: the magnitude of the Burgers vector related to a crystal lattice parameter [5]. Being a linear defect, in crystalline solids, a dislocation line is localized at an atomistic scale. The motion of the dislocation is confined to specific crystallographic planes. When it escapes to a free surface, a discrete step of the size of the Burgers vector is formed, and averaging overall distinct slip events over the deforming volume and time gives a value of the total plastic strain incurred to the loaded body. This essentially discrete behavior of dislocations causes temporal and spatial fluctuations in stress and strain: in a simplified picture, as a dislocation leaves a crystal, stress relaxation occurs, and the internal energy of the crystal drops abruptly, whereas when the dislocation stops inside the crystal, the local stress along its line increases. The stress (force) drops, and the corresponding displacement jumps due to individual dislocation slip events can be directly resolved in micropillar experiments [6,7] (cf. also the in-depth review by Papanikolaou et al. [5]). From these experiments, the velocities of the slip events can be calculated. Concurrently recorded AE events [8,9] are correlated with the strain bursts, thus offering an extra dimension in which to analyze collective dislocation dynamics. The overlapping and averaging of a vast number of slip events occurring at the same time in the deforming macroscopic specimens not only smoothens the stress–strain curves obtained using conventional equipment, as illustrated schematically in Figure 1a, but this smoothness is also influenced by extrinsic metrological factors. These include the inherent softness of testing machines (specifically, the load cell acts as a spring between the specimen and the testing frame), and the low-pass filtering of electric signals representing the stress and strain data. Altogether, these factors impede the resolution of individual or collective slip effects constituting the stress–strain data and mask the intermittent nature of plastic flow appearing on a macroscopic scale unless macroscopic plastic instabilities, such as Lüders bands, as shown in Figure 1b, or Portevin–Le Chatelier (PLC) bands [10], as shown in Figure 1c, come into the stage. The tool capable of unveiling the intermittency of plastic flow is widely known as acoustic emission (AE) associated with elastic waves arising from rapid local stress relaxations in solids under external stimulus [11,12]. Due to its sensitivity to dynamic processes, AE is particularly effective in reflecting fluctuations in dislocation velocity and local density [13]. Notably, when dislocations escape a crystal at a free surface, they can generate significant AE due to the sharp local elastic energy release [14], with attendant slip lines producing transient acoustic emissions, albeit of a very low magnitude [15] as displayed in Figure 2a. Overlapping of the large number of AE bursts stemming from quasi-simultaneously moving dislocation segments results in a continuous noise-like signal [16,17] instead of a pulse-train of distinct bursts such as those observed commonly due to mechanical twinning [18,19,20,21,22]. By analysing the AE time series, waveforms and related features of individual events, one can gain insights into the dynamic and statistical properties of sound emission events, including dislocation movement, mechanical twinning or deformation-induced phase transformations in solids, collectively referred to as “AE sources” [12,23].

Traditionally, the deformation behavior of metals has been explained by assuming uncorrelated, uniform motion of independent dislocation segments or loops. While this assumption has its merits, it is increasingly questioned in light of our current understanding of dislocation behavior and their interactions with other lattice defects across various length scales. This complexity arises from the long-range elastic stress fields of dislocations and from short-range nonlinear interactions among dislocations and impurities. Furthermore, plastic flow in a deforming crystal is a dissipative, non-equilibrium process driven by external stresses, leading to the self-arrangement of dislocation ensembles into distinct patterns at different length scales [24,25,26]. These patterns exhibit significant spatial and temporal variations in dislocation density. In the mature deformation stage, plastic deformation deviates from uniformity and tends to localize prior to crack initiation and fracture. The most common type of strain localization preceding crack nucleation is known as necking, which manifests as localized strain in the continuously shrinking cross-section and which is virtually omnipresent in the tension of ductile structural metals and alloys, as displayed in Figure 1a–c.

Predicting catastrophes before they occur is a significant challenge. Several data-driven approaches have been proposed to address this issue in nonlinear dynamical systems, assuming that the governing equations are unknown, cf. [27]. These methods rely solely on random time series data reflecting the evolution of the system’s dynamical variables, which is notably relevant in the context of AE studies. Identifying the nonlinear dynamics of systems from AE data is a critical, yet particularly challenging task, even when the equations governing microstructural evolution can be reasonably assumed; see [28] for an example. Moreover, interpreting AE during strain localization is complicated not only by the intricacies of AE technology—often involving numerous parameters that are difficult to correlate with dynamic processes—but also by the multiscale complexities driving plastic flow localization.

Beyond academic interest, understanding strain localization has significant practical implications. Since plastic strain localization precedes crack nucleation, identifying its onset is crucial for non-destructive evaluation routines and for advancing AE as a valuable tool for failure predictions. While many aspects of AE during uniform deformation of metals and alloys have been explained through dislocation evolution and interactions with impurities, the AE accompanying strain localization processes under load remains less comprehensively understood from both phenomenological and microscopic perspectives. Although the manifestations of plastic instabilities under loading are diverse and influenced by numerous internal and external factors, I am optimistic that general tendencies in AE behavior due to strain localization can be identified and classified at least phenomenologically, based on the results accumulated to date. This optimism motivated me to write the present review, which includes a critical analysis of the existing literature as well as some previously unpublished data.

This review is intended not only for a knowledgeable audience in the fields of AE, non-destructive testing and physical metallurgy but also for novices, such as students eager to delve into the details of the multifaceted problem of strain localization using AE as a study tool. The literature spanning 5–6 decades of intensive research is extensive, and I have endeavoured to maintain a balance by presenting both classical early perspectives and modern concepts. However, given the breadth of the field, it is unavoidable that certain relevant topics fall outside the scope of this focused survey, as will be explicitly noted in the sections that follow.

1.2. Classifications of Plastic Instabilities

A simple phenomenological classification of plastic instabilities associated with stress–strain behavior has been proposed by Estrin and Kubin [29,30,31] by conducting linear stability analysis of a generic constitutive relation between the flow stress σ, plastic strain ε (here and in what follows we refer to true stress and true (logarithmic) strain unless stated otherwise) and plastic strain rate $\dot{\epsilon }$, which, in a differential form, reads as

$$d\sigma =\theta d\epsilon +Sd\ln \dot{\epsilon }$$

(1)

where $\theta ={\left(\partial \sigma /\partial \epsilon \right)}_{\dot{\epsilon }}$ and $S={\left(\partial \sigma /\partial \ln \dot{\epsilon }\right)}_{\epsilon }$ stand for the strain-hardening coefficient and the strain-rate sensitivity, respectively. Apparently, this simplistic constitutive formulation does not include any internal microstructural variables but rather lumps various contributions into the phenomenological material’s characteristics, such as $\theta$ and S, which are easily measured experimentally and which are representative of the material’s mechanical response. Expressing a local strain perturbation in a conventional exponential form $\delta \epsilon ={\left(\delta \epsilon \right)}_0\exp \left(\lambda t\right)$ with initial perturbation ${\left(\delta \epsilon \right)}_0$ and growth parameter λ, the stability criterion was formulated as

$$\lambda ={\displaystyle \frac{\sigma -\theta }{S}}\dot{\epsilon }$$

(2)

Obviously, the positive value of $\lambda$ signifies instability, i.e., the onset of strain localization that occurs according to one of the following two conditions:

$$\begin{array}{l}\theta <\sigma (S>0)\\ S<0 (\theta >\sigma )\end{array}$$

(3)

The corresponding two types of instabilities are commonly referred to as $\theta$ and S-type instabilities. While necking and the Lüders phenomenon belong to the strain-hardening $\theta$-type of instabilities, the Portevin-Le Chatelier effect is a common example of the type S instability associated with apparently negative values of the strain-rate sensitivity resulting from dynamic strain aging. All these commonly faced instabilities received attention in the AE community, albeit to a different extent, as will be reviewed in the corresponding sections below. A more general constitutive description accounting for stress transients arising in response to a strain-rate change (the point that is missing in the simplified expression (1)) was proposed by Zaiser and Hähner in [32].

Unlike Lüders bands or PLC bands, which propagate along the material, necking is a form of instability that remains localized to a specific region of the material. It is typically referred to as a geometrical instability because the reduction in the cross-sectional area increases the local stress, accelerating deformation in that region and not along the entire length of the specimen. While other instabilities, like Lüders bands, PLC bands and shear banding are material-specific, necking is more universally observed and widely studied in the context of tensile deformation. Buckling and wrinkling represent other types of macroscopic instabilities commonly faced in metal forming [33], and are not considered in the present review.

2. Intermittency of “Continuous” Acoustic Emission

A key characteristic of acoustic emission (AE) waveforms recorded during homogeneous plastic deformation in many pure fcc metals is their common manifestation as continuous noise-like signals, as illustrated in Figure 2. This continuous signal arises from the vast number of dislocation segments that are generated throughout the loaded solid and that move nearly concurrently across the grains. However, on a microscopic scale, plastic deformation in metals is spatially heterogeneous and temporally intermittent due to the discrete nature of the carriers of plasticity, lattice dislocations [34,35,36], creating clearly separated fine slip lines on the surface during deformation, cf. Figure 3d. The discrete behavior of dislocations leads to local fluctuations in stress and strain, which produce measurable AE. The detectability of AE induced by small crystalline defects, such as dislocations, is inherently limited [11], as it strongly depends on the sensitivity of the transducer and the thermal electric noise generated primarily by the electronic circuits in the preamplifier. While this thermal noise is unavoidable, it is also temperature- and frequency-dependent. Nevertheless, modern signal-processing techniques can significantly enhance the signal-to-noise ratio and reliability of phase picking by exploiting the stationary nature of electric noise [37,38,39,40,41]. The measured waveform represents an additive random process comprising both the AE component and noise. Various denoising methods can alleviate the stationary spectral characteristics of the noise. One widely used algorithm is known as spectral subtraction [42], which restores the power spectrum or magnitude spectrum of a signal observed in additive noise by subtracting an estimate of the average noise spectrum. To unveil the intermittent structure of the continuous AE signal detected during scratch testing on well-annealed copper polycrystal, as displayed in Figure 3, we applied the spectral-noise-gating algorithm [43] described in [44]. Compared to conventional macroscopic testing/compression or micropillar testing, scratching with a sharp diamond indenter offers distinct advantages for studying fundamental processes governing plastic flow in small volumes. Notably, the moving sharp indenter stimulates sources in close proximity to its tip, sequentially generating signals from the undeformed region, thus forming a stationary time series and effectively eliminating strain hardening effects. Prior Electron Backscatter Diffraction (EBSD) analysis, as illustrated in Figure 3a, enables a comprehensive characterization of the grain structure in the tested samples, allowing for comparisons with results obtained from single crystals of varying orientations across different grains. This approach also reveals the influence of grain boundaries. The interactions between the indenter and the boundaries are reflected in the measured force response, slip patterns, and corresponding AE signals.

Figure 3a depicts the indenter’s footprint superimposed with the EBSD orientation map alongside the raw AE waveform, where the continuous low-amplitude signal is barely discernible amidst electrical noise. However, the discrete nature of the AE time series becomes apparent following noise-gating filtering. The occurrence of AE bursts can be directly correlated with the observed slip patterns. As slip lines emerge sequentially, one-by-one, while the stylus traverses the grain, this method enables the assessment of the statistics of time intervals, $\mathsf{\Delta }{t}_i=t_{i+1}-t_i$, between successive arrivals. This approach was first proposed for the analysis of the AE time series in the early work of Erlenkämper [45] in which he studied the fracture mechanics of annealed carbon tool steel CK45 and aluminum alloy 7075-T651. He concluded that the Poisson-type activity, caused by independent sources, could be associated with AE recorded in the steel samples, whereas a flux of clustered AE events obeying the Polya distribution better described the AE behavior in the Al-alloy. Later Braginskii et al. [46] applied a similar statistical approach to study the appearance of strongly localised shear bands in metallic glasses under uniaxial tension. This approach was further developed and refined in a mathematically more robust and conclusive manner in [47,48]. The burst AE time series (either raw or denoised as shown in Figure 3c) is transformed to a point “shapeless” process $\left\{A(t)\right\}={\displaystyle \sum _{i=1}^N{U}_k\delta (t-t_i)}$ represented by the arrival times $t_i$ of the amplitude $U_i$ as exemplified in a smaller scale fragment shown in Figure 4. Then, the experimentally observed histogram of waiting time is plotted, as displayed in Figure 5, together with the expected distribution for the Poisson process [49,50] having the probability density function for the interarrival times $\rho (\mathsf{\Delta }t)={\displaystyle \frac{1}{\overline{\mathsf{\Delta }t}}}\exp (-\mathsf{\Delta }t/\overline{\mathsf{\Delta }t})$ determined by a single parameter $\overline{\mathsf{\Delta }t}$–the average time interval between the pulses. In the case illustrated in Figure 5, the calculated test ${\chi }^2$ statistics equals ${\chi }^2=4.60<{\chi }_{12,0.95}^2=21.03$, which is smaller than the upper-tail critical value for the ${\chi }^2$ distribution with 12 degrees of freedom at both the 0.05 and 0.0005 confidence levels. (See [51,52], for example, about hypothesis testing.) This result provides strong evidence against the null hypothesis that the two distributions are significantly different. Thus, we may conclude that the observed distribution is possibly exponential and could be generated by the Poisson process. Several fragments of the AE streams were examined yielding similar results supporting the independent nature of AE sources activated during scratching tests. However, it is important to note that the simple Chi-square method cannot serve as definitive proof of such a conclusion. The substantially more rigorous analysis of AE time series involving independent statistical methods has been methodologically justified and conducted, as demonstrated in the studies of plastically deformed Mg alloys, by Agletdinov et al. [47,48]. Notably, this more elaborate approach led us to conclude that dislocation slip can be regarded as a Poisson process (similarly to that reported in [45] albeit in a more compelling way), while twinning exemplifies a self-exciting Hawkes process of clustered closely-spaced events with memory on a sub-millisecond time scale. The computer simulation by Devincre et al. [53] using discrete dislocation dynamics corroborates this conclusion for dislocation slip over shorter observations times up to 1 μs, as illustrated in Figure 6.

It should be noted that accurate phase picking is crucial for this analysis. If the simple spectral-noise gating technique falls short of providing the desired precision in arrival time detection, alternative methods can be utilized. Specifically, many algorithms focusing on detecting signals with low signal-to-noise ratios (SNR) have been proven useful. For example, Agletdinov et al. [41] introduced an approach utilizing the advantages of the logarithmic derivative of the AE power spectral density, cf. Figure 4. Pomponi et al. [39] employed a discrete wavelet transform technique tailored to the specifics of the AE acquisition chain. Figure 7 illustrates the results of this algorithm applied to very low SNR AE signals deliberately masked by background noise. It showcases the algorithm’s ability to detect the arrival of low-amplitude transient events with high confidence and accuracy. The effectiveness of this method for capturing intermittent plastic flow from local scratching was demonstrated in [41].

Thus, we have demonstrated that even when acoustic emissions appear as a continuous random signal, they still retain essential features related to their fundamentally intermittent origin. With this in mind, we will proceed to a comparative analysis of AE behavior recorded during various types of plastic instabilities outlined in the Introduction. Before delving into this analysis based on the measurements of AE spectral features, it is essential to briefly survey a recently proposed phenomenological AE model that is grounded on basic dislocation-mediated processes under load. This will facilitate a better understanding of both the measurement settings and the interpretation that follows.

3. Model of Dislocation-Mediated AE During Uniform Plastic Deformation

Recently, in a series of publications [13,17,54,55] we proposed a phenomenological model capable of describing the findings of AE behavior commonly seen during uniform plastic deformation of metals and alloys. The model is based on the following three key assumptions. (1) The raw AE signal is generated by concurrently operating randomly distributed dislocation sources forming a continuous quasi-stationary random process $\left.\left\{U\left(t\right)\right.\right\}$ that can be represented by an Ornstein–Uhlenbeck-type [56] model obeying the stochastic differential Equation:

$${\displaystyle \frac{dU}{dt}}=-{\displaystyle \frac{U}{{\tau }_r}}+\tilde{\xi }(t)$$

(4)

with the first term $-U/{\tau }_r$ on the right hand side representing the relaxation process with the intrinsic characteristic time ${\tau }_r$, and $\tilde{\xi }(t)$–the random exciting function given by δ-correlated white noise. This represents a good approximation, since the correlation time is small compared to the time scale at which the evolution of dislocation system becomes discernible as strain hardening proceeds. The relaxation time ${\tau }_r$ serves as a universal measure of the temporal interrelation among individual events within the time series. The $\tilde{\xi }(t)$ term captures the stochastic dislocation behavior, which arises from the fundamentally probabilistic nature of all dislocation life-time stages including thermally activated production, moving through randomly distributed obstacles of widely distributed strengths and the probabilistic process of dislocation annihilation. Let us reiterate that in this purely phenomenological approach, there is no expectation to link the entering variables–$U$ and ${\tau }_r$–with the properties of individual dislocations as it would be natural for microscopic formulations. Such microscopic models have been proposed in the elastodynamic continuous mechanical approach, e.g., [57,58,59,60,61], building on the fundamental analogy between AE and the seismic sources [62]. The exceptional significance of these models lies in their ability to establish a general relation between the size of the source (e.g., the diameter of the expanding dislocation loop $a$), its velocity $v$ and the surface displacement ${\delta }_s$ at the sensor location. Within the approximations (unavoidable for analytical solutions), ${\delta }_s$ appears to be proportional to the product of $a$ and $v$: ${\delta }_s\sim av$. This elastic displacement, in turn, can be related, ideally linearly, to the electric voltage $U$ measured at the sensor output, representing the AE signal. Equation (4), however, provides a distinct pathway to assess the statistical properties of the emitting collectives of sources—dislocations (as ensembles, not individual ones)—in the present study, as will be made explicit below. (2) The total density of dislocations $\rho$ is an intrinsic microstructural state variable governing the macro-mechanical response in terms of extrinsic stress $\sigma$ and strain $\epsilon$. The population of dislocations evolves in line with first-order kinetics laws accounting for the dislocation production and annihilation through an appropriate differential equation. While the exact form of this equation can vary significantly, depending on the dislocation reactions involved, in its simplest form it reads as

$${\displaystyle \frac{d\rho }{d\gamma }}={\displaystyle \frac{d\rho }{Md\epsilon }}={\displaystyle \frac{k_1}{b}}\sqrt{\rho }-k_2\rho$$

(5)

as proposed by Kocks and Mecking [63,64], and, similarly, in earlier works by Bergstrom [65] and Klepaczko [66]). The meaning of the dimensionless parameters $k_1$ and $k_2$ is transparent and it has been discussed in the literature in great detail: in brief, $k_1$ controls the dislocation multiplication whereas $k_2$ is responsible for dynamic recovery. (3) The flow stress scales with $\sqrt{\rho }$ as described by the Taylor equation:

$$\sigma \left(\epsilon \right)=M\tau =M\alpha \mu b\sqrt{\rho }$$

(6)

where $M$ is the orientation factor that converts shear stress $\tau$ and strain $\gamma$ to corresponding axial parameters as $\tau =\sigma /M, \gamma =\epsilon M$; $b$ is the magnitude of the Burgers vector of the dislocation, $\mu$ is the shear modulus, and $\alpha =\alpha (\dot{\epsilon },T)$ is a numerical factor depending on the dislocation arrangement, strain rate $\dot{\epsilon }$ and temperature $T$, varying typically between 0.1 and 0.5 [67]. When solving Equations (5) and (6) together with the initial condition $\sigma \left(\epsilon =0\right)={\sigma }_0$), the familiar stress–strain relation is obtained as:

$$\sigma \left(\epsilon \right)={\sigma }_0+\left({\sigma }_S-{\sigma }_0\right)\left(1-\exp \left(-{\displaystyle \frac{\epsilon }{\tilde{\epsilon }}}\right)\right)$$

(7)

where ${\sigma }_S=k_{1}M\alpha \mu /k_2$ determines the saturation stress, and $\tilde{\epsilon }=2/k_{2}M$ is the parameter that determines the necking strain [68,69,70].

The strain-hardening rate $\mathsf{\theta }=d\sigma /d\epsilon$ is obtained from Equation (7) as:

$$\theta \left(\epsilon \right)={\displaystyle \frac{1}{\tilde{\epsilon }}}\left({\sigma }_S-{\sigma }_0\right)\exp \left(-{\displaystyle \frac{\epsilon }{\tilde{\epsilon }}}\right)={\displaystyle \frac{1}{\tilde{\epsilon }}}\left({\sigma }_S-\sigma \left(\epsilon \right)\right)$$

(8)

The exact form of the evolutionary Equation (5) is not crucially important for the conclusion drawn for the AE analysis. Without loss of generality, it can be replaced with any equation appropriately describing the synergistically competing processes of dislocation production and disappearance under load. The collective AE sources associated with the transformation in the microstructure due to the motion and annihilation of mobile dislocations appear to be inherently related to the fundamental solutions described above. The experimentally obtained AE parameters, such as AE power and the median frequency of the spectral density, can be linked to the development of the defect microstructure under increasing load. Drawing inspiration from the concepts articulated by the AE pioneers in the 1970s [45,57,62,71,72,73,74,75,76,77], we treat AE as a result of fluctuating intermittent dislocation motion, generating the continuous random Ornstein–Uhlenbeck process in accordance with Equation (4) [13,54,78]. Stochastic properties of this process are well-documented; notably, its power spectral density is given by a familiar Lorentz-type function [50,52]:

$$G\left(f\right)\sim P_{\mathrm{AE}}{\displaystyle \frac{f_c}{f_{\mathrm{c}}^2+4{\pi }^2{f}^2}}$$

(9)

where $P_{\mathrm{AE}}$ is the power of the random AE signal determined through its squared variance, and $f_c=1/{\tau }_r$ is a characteristic frequency that determines the effective bandwidth of the spectral density function; as ${\tau }_{\mathrm{r}}$ increases, $f_c$ decreases and so does the width of the spectrum tending to concentrate at lower frequencies, and vice versa. The median AE frequency $f_{\mathrm{m}}$ is obtained from the definition ${\displaystyle {\int }_0^{f_{\mathrm{m}}}G(f)df}={\displaystyle {\int }_{f_{\mathrm{m}}}^{\infty }G(f)df}$. For the Lorentz spectrum defined by Equation (9), $f_c$ and $f_{\mathrm{m}}$ are simply interrelated as $f_{\mathrm{c}}=2\pi f_{\mathrm{m}}$ [79]. For this reason, we have chosen the median frequency and power as independent descriptive AE variables. Furthermore, the relaxation time ${\tau }_{\mathrm{r}}$ in the dislocation system can be associated with the mean dislocation lifetime before being stored or annihilated in the microstructure as defined by the scaling law ${\tau }_{\mathrm{r}}=〈\mathsf{\Lambda }〉/〈V〉$, with $〈\mathsf{\Lambda }〉$–the dislocation mean free path and $〈V〉$–the average dislocation velocity (as argued by Hähner [80], and see [13] for details). Using Orowan’s expression for the plastic strain rate $\dot{\epsilon }=M{\rho }_{\mathrm{m}}b〈V〉$ (with ${\rho }_{\mathrm{m}}$–the mobile dislocation density), and recalling that $〈\mathsf{\Lambda }〉$ scales with the inverse dislocation density as $〈\mathsf{\Lambda }〉\sim 1/\sqrt{\rho }$ [81], one can obtain that the $f_{\mathrm{m}}$ value is proportional to $\sqrt{\rho }\sim \sigma$, and inversely related to ${\rho }_{\mathrm{m}}$ as:

$$f_{\mathrm{m}}\sim {\displaystyle \frac{\dot{\epsilon }}{Mb{\rho }_{\mathrm{m}}}}\sqrt{\rho }=K_{\mathrm{f}}{\displaystyle \frac{\dot{\epsilon }}{{\rho }_{\mathrm{m}}}}\sigma$$

(10)

Here, the $K_{\mathrm{f}}$ is a numeric factor comprising related materials’ constants and spectral features of the AE acquisition channel. As a corollary the model predicts that the AE median frequency $f_m$ scales linearly with the flow stress $f_{\mathrm{m}}~\sqrt{\rho }~\sigma$ and strain rate $f_m~\dot{\epsilon }$ during homogeneous plastic deformation.

Assuming the dislocation segments move by a Peach–Köhler force through a crystalline lattice, and relating the average elastic energy dissipated per unit time to acoustic emission, we derived the following phenomenological expression for the AE power [54]

$$P_{\mathrm{AE}}=K_{\mathrm{AE}}{\rho }_{\mathrm{m}}{\displaystyle \frac{\theta }{\sigma }}\dot{\epsilon }=K_{\mathrm{AE}}{\rho }_{\mathrm{m}}{\displaystyle \frac{\dot{\sigma }}{\sigma }}$$

(11)

In this expression $P_{\mathrm{AE}}$ is related to the mobile dislocation density and the strain-hardening process. A linear relationship between $P_{\mathrm{AE}}$ and the reduced hardening rate $\dot{\sigma }/\sigma$ has been observed in many fcc metals and alloys including Cu, Ag, Ni, Al [55] and CuZn alloys [17], etc., thus suggesting that the ${\rho }_{\mathrm{m}}$ value does not vary significantly during the homogeneous deformation stage, the strong hypothesis that has been well-received in many insightful discussions, e.g., [82,83,84]. The numeric factor $K_{\mathrm{AE}}$, similarly to $K_{\mathrm{f}}$, integrates all relevant model parameters into a single proportionality constant.

4. Overview of AE Spectral Features Due to Plastic Instabilities

4.1. Yield Point and Lüders Band

The dislocation avalanches driving discontinuous yielding though nucleation and propagation of deformation bands [85] are easily detectable by AE equipment and have been the focus of many investigations; the early studies have been reviewed in [86]. The AE features reflecting the Lüders bands have been studied across different materials such as AlMg alloys [87], α-Iron [88], CuZn alloys [17,78], low [89,90] or medium carbon steel [91] as well as nuclear grade steel A533B [58,92], and can be complemented by the digital image correlation technique unveiling the local distribution of strain and its gradients along the gauge part of the loaded specimen [93,94]. Notably, Petit et al. [90] examined both the Lüders band propagation and the progress of necking in low-alloyed and low-carbon steels by using speckle interferometry (ESPI) and AE. The AE activity accompanying the Lüders bands was related to the front of the band, and was correlated with the tensile force fluctuations. Prior to necking, AE decreased regularly through the homogeneous plasticity stage and then increased when the plasticity stared to localize in the neck. From my perspective, the most prominent, yet less-examined, AE features lie in the spectral domain, and these features are outlined in what follows.

Figure 8 demonstrates a fragment of AE measured during the early deformation stage of the tensile tested well-annealed Cu30Zn α-brass (See [17,95,96] for the microstructure and experimental details). The AE power and medial frequency are synchronized with the stress–strain curve. The snapshots selected from the continuously preformed DIC (digital image correlation, see the review [93], and [95] for specific details) analysis represent the local distribution of the strain rate along the gauge part of the specimen. The typical behavior of the Lüders band initiating at the upper shoulder and propagating towards the lower grip can be noticed, followed by a homogeneous strain-hardening stage after approximately 0.01 strain.

The AE power peaks twice—at the onset of plastic yielding corresponding to the initiation of the deformation band and again as the band propagates with visible “thickening” of the high-strain-rate region. The AE level then drops sharply and subsequently decreases steadily as it is commonly observed during uniform elongation. Concurrently, in response to unstable behavior of the deformation band unveiled by DIC, the AE spectral density exhibits highly a fluctuating evolution on par with the results reported in [90]. The median frequency drops at the onset of the deformation band (compare results in [88]) and increases before the band stops and the homogeneous deformation stage begins. As strain hardening commences, the $f_m$ value initially decreases and then rises in parallel with the external stress as is also commonly observed for a wide range of fcc metals and alloys [55].

4.2. Serrated Plastic Flow–The Portevin–Le Chatelier (PLC) Effect

The PLC effect, a phenomenon observed in many solid solutions (see the reviews by Lebyodkin et al. [97,98]), manifests as a series of discontinuous yielding events in the stress–time or strain curve, which can be either quasi-periodic or random [30,99,100]. Each stress drop corresponds to the nucleation of a localized plastic deformation band that propagates through the material, causing jerky plastic flow [101] and showcasing intriguing spatiotemporal behavior, which has triggered intensive interdisciplinary research gathering scholars in physics, mathematics and materials science. When a PLC band encounters a boundary, such as the edge of a specimen, it can be reflected in the same way a wave reflects off a wall. While the analogy between PLC bands and solitary waves [102] provides a useful conceptual framework, it is important to note that the underlying physics of these phenomena may be more complex since significant microstructural changes occur both within the bands themselves as well as in the surrounding matrix experiencing strain hardening.

The microscopic origin of the PLC effect is well understood and is widely attributed to the dynamic strain aging of the material, caused by the interaction between mobile dislocations and diffusible solute atoms [103,104,105]. In specific ranges of strain rates and temperatures, the diffusion time of solute atoms aligns closely with the waiting time of dislocations temporarily trapped at obstacles. As solute atoms diffuse and effectively pin the dislocations, the stress required to break away increases with longer waiting times and decreasing strain rates. This relationship leads to an apparently negative sensitivity of flow stress to strain rate, leading to plastic instabilities depicted as jerky flow [100]. A thorough understanding of the evolution kinetics of dislocation densities and the mechanisms of dislocation–solute interactions provides the necessary foundation for comprehending this complex phenomenon [30,32,106].

As dislocation avalanches drive discontinuous PLC yielding, they can be effectively examined by the AE technique [107,108], capable of unveiling a wide variety of spatiotemporal patterns emerging during the initiation and propagation of the deformation bands. Research across various materials and strain rates—though the effect of temperature has been explored to a lesser extent [109]—has consistently shown a strong correlation between AE transients and the degree of heterogeneous deformation. The comparative analysis of AE features documented in multiple studies over several decades for various solid solutions, such as AlMg, CuAl, MgZr, CuZn and steels, reveals generally similar behaviors across systems exhibiting the PLC effect, e.g., [97,107,108,110,111,112,113,114,115,116,117,118].

Three distinct types of PLC instabilities, type A, B and C, can be identified based on their qualitative appearance on the stress–strain curves, quantitative statistics of stress serrations and the kinetics of PLC band propagation [99,100]. Type A serrations are characterized by a quasi-continuous stable propagation of PLC bands, with stress drops preceded by stress rises above an envelope stress–strain curve (This type of behavior is illustrated in Figure 9, and will be discussed in more detail below). Type B instability exhibits “hopping” bands and sharp stress oscillations around the envelope curve, revealing more erratic band propagation that can significantly impact material stability. Type C behavior is associated with randomly emerging bands and nearly normally distributed acoustic emissions and stress drops occurring below the envelope curve [115], reflecting a highly disordered state in the deformation process.

Recent quantitative analyses of stress serrations, utilizing methods grounded in nonlinear dynamic systems theory, have meticulously elucidated dynamic regimes specific to each type of PLC effect [98]. Notably, C-type serrations are often regarded as uncorrelated, while a transition to low-dimensional deterministic chaos occurs with increasing strain rates when the emergence of type B serrations is observed [119,120]. Conversely, type A serrations exhibit power-law statistics in stress jump parameters [119,120] and corresponding acoustic emissions [121,122], indicating self-organized criticality (SOC) in the externally driven dissipative system with multiple degrees of freedom [106,123,124,125]. Power-law distributions have been emphasized in a vast amount of research on dislocation avalanches, both through numerical simulations using discrete dislocation dynamics and in the statistical analysis of stress drops [5]. Acoustic emissions accompanying these phenomena have been represented by various authors using different metrics, including amplitudes, energies and power, as well as different methods for evaluating these quantities [126,127,128,129,130,131,132,133,134,135,136,137,138]. However, in my view, significant work remains to be completed before experimental acoustic emission studies on metallic materials can be considered “well-controlled” in order to draw unambiguous conclusions about the dynamics involved, whether SOC or otherwise. Therefore, this intriguing and challenging area, due to its complexity, deserves special attention but is not addressed in the present review. Readers interested in this topic are encouraged to consult the cited literature for further insights. Instead, following the methodology adopted in the previous sections, we focus on the less-investigated question of how the AE spectrum is influenced by abrupt yield events during the PLC phenomenon.

Figure 9 presents a fragment of the AE data accompanying the serrated flow of the tensile-tested Cu30Zn alloy, highlighting the key features of the PLC effect. The experimental details are provided in [17], which also comprehensively discusses the microstructure and its evolution alongside the entire AE stream and its characteristics during the uniform strain hardening stage. The enlarged view of the AE parameters reveals that intermittent plastic flow in the high strain-rate regime, corresponding to A-type serrations, is accompanied by a continuous AE signal that reflects the “smooth” dynamics of collective sources on a macroscale. This finding aligns with earlier work on a similar alloy with a different experimental setup [78], which reported no distinct bursts at various time scales. This behavior contrasts sharply with that observed for B- and C-type serrations [111] accompanied by distinct AE transients.

Simultaneous video recording followed by high-resolution sub-pixel DIC analysis indicates a clear transition from uniform to non-uniform deformation. PLC bands typically initiate at one edge of the specimen and propagate pseudo-continuously through the gauge section in a highly fluctuating manner, exhibiting phases of expansion and contraction, as well as splitting and merging. These dynamic changes are reflected in significant systematic variations in both AE power and median frequency. A noteworthy observation is that while the AE power sharply increases at the onset of band initiation, the median frequency shows the opposite trend (cf. [78]). High AE levels are not necessarily indicative of localized deformation. As Figure 9 illustrates, as the band propagates between successive serrations, the degree of strain localization remains high, while the average AE power tends to decrease in response to the ongoing strain hardening process. This behavior contrasts with that of the Lüders band discussed in the preceding section (cf., Figure 8).

Similarly, the $f_m$ value recovers after its initial drop at the onset of strain localization and increases alongside the flow stress as the band progresses through the specimen. Thus, the AE spectrum and its characteristics—power and median frequency—primarily respond to changes in the strain-hardening rate. Furthermore, although the aforementioned AE model was developed under the assumption of macroscopic uniformity in plastic flow, the observed behavior remains consistent with this model if we assume that the region within the band experiences a sharp local increase in mobile dislocation density. This concept aligns with the strain aging PLC model proposed by Kubin and Estrin, which incorporates two coupled intrinsic variables—mobile and immobile dislocation densities [103]. AE monitoring thus provides indirect information about the evolution of local dislocation density, influenced by both mobile and immobile sub-systems, as reflected in Equations (10) and (11).

4.3. Necking

Necking refers to the localized reduction in the cross-sectional area that occurs in a material under tensile stress [4]. Beyond the necking point, deformation becomes localized in a small region, leading to the shrinking of the cross-section area. The necking instability condition is commonly obtained in terms of extrinsic variables, the true stress and true strain, for a given plastic strain rate $\dot{\epsilon }$ according to the Considère criterion [139]:

$${\left.{\displaystyle \frac{\theta }{\sigma }}\right|}_{\dot{\epsilon }}<1$$

(12)

In this criterion, the strain rate sensitivity of the flow stress is neglected, which is acceptable for many fcc and hcp metals and alloys. The more general Hart instability condition [140], which does account for the strain rate sensitivity reads as

$${\left.{\displaystyle \frac{\theta }{\sigma }}\right|}_{\dot{\epsilon }}+S<1$$

(13)

Noticing that neither mechanical stress nor strain can be regarded as state variables, it has been demonstrated that both the Considère and Hart conditions follow from the evolution laws for the principal internal variable—the total dislocation density [68,69,141]. While the conditions (12) or (13) predict the onset of necking, precise calculation of the strain and stress distribution within a necked region is difficult due to the substantially triaxial stresses formed beyond the instability point. P.W. Bridgman dealt with this issue in his classical work [142] and proposed a simple correction scheme to approximate the neck shape in a nominally uniaxial tensile deformation. Recently, Tu et al. [143] reviewed this topic carefully and developed an elaborate mechanistic approach describing the propagation of deformation in the post-necking regime. Necking is thus regarded as the most prevalent and extensively studied form of instability in the tension of ductile structural metals and alloys, given its frequent occurrence and crucial role in deformation and failure processes. However, investigations into acoustic companions to this instability are surprisingly scarce.

Furthermore, inconsistent or even conflicting results have been reported regarding the AE appearance during necking. Lazarev and Vinogradov [144] carried out the analysis of the AE spectral density during Lüders band propagation and necking in α-Iron. The similarity of AE appearance was noticed in the occurrence of these two phenomena. The conceptual similarity between them has been discussed in the Introduction, cf. also [145]. Consistent with these results, the use of speckle interferometry [90] or local extensometry [146] combined with simultaneous AE measurements during tensile deformation of low carbon steel or α-Ti, respectively, revealed an increase in acoustic activity concurrent with strain concentration in the progressively narrowing necked region. Baral et al. [147] carried out a series of experiments aiming at the monitoring of necking during uniaxial tension and cup drawing of AA6013-T4 aluminum sheets in parallel with the DIC analysis of the strain localization process. These authors observed the maximum AE amplitude corresponding to the onset of diffuse necking. As opposed to these observations, many investigators ([148]–Nimonic PE16 alloy, [149]–pure Al, [150]–6082-T6 aluminum alloy, [151]–A533B grade Mn-Mo-Ni steel, [152]–M250 grade maraging steel, [153]–316 nuclear grade austenitic stainless steel, [154]–304LN austenitic stainless steel, [155]–plain carbon steel, [156]–AISI 4140 steel, [58]–A533B, and [151]–Mn-Mo-Ni steel) failed to find any clear evidence of strain localization in the AE signal at the onset of necking in respective materials, at least until the microcracking (or intensive shear banding [157]) started to dominate the failure process in the necked region. The same applies to a more recent comprehensive study by Sendrowicz et al. [17] performed on CuZn alloys with the different Zn content, as illustrated in Figure 10.

Figure 11 presents five successive images of the rapidly developing neck in pure copper during the final second before fracture. These images can be compared to the late stage depicted in Figure 10a, where rapid cross-sectional shrinking leads to a reduction in the assessed stress value (apparently miscalculated, as previously discussed), although the images originate from a separate test. The formation of ductile microcracks is clearly visible, illustrating the onset of the ductile fracture process after significant strain localization in the necked region. This process involves the nucleation, growth and coalescence of microvoids. As plastic deformation intensifies at increasing strain rates, the microvoids expand and merge, tearing the material ligaments between them, ultimately leading to fracture and significantly influencing the AE signal.

In my view, the observed discrepancies can be largely attributed to experimental issues and differences in acquisition setups among researchers. The silent, yet most tangible, instrumental difficulty in measuring AE during the mature deformation stage is the notably reduced acoustic activity at this stage of strain hardening. This commonly observed reduction, predicted by the model outlined in Equation (11), is associated with low $\dot{\sigma }$ and high $\sigma$ values when the neck sets in. At that point, both true stress and the strain-hardening rate do not exhibit any features and remain changing steadily in sharp contrast with the other instabilities discussed above. This behavior leads to conventional threshold-based techniques often failing to detect AE. Moreover, even when continuous thresholdless acquisition methods are employed, deeper challenges remain. Hartman’s early study on serrated plastic flow [158] highlighted the substantial influence of transducer bandwidth on AE output and the conclusions derived from signals captured by either resonant or broadband sensors.

The recorded AE signals are heavily influenced by the quality of the sensor, including factors such as sensitivity and the number, distribution and quality factors of resonances. Figure 12 illustrates conflicting results obtained from two different sensors attached to the same tensile-tested specimen of Cu20Zn alloy: Micro-F30 (150–750 kHz) and Nano-30 (125–750 kHz) transducers, which differed in terms of their frequency response and sensitivity, from MISTRAS Ltd., Princeton, NJ, USA, were used to highlight the challenge. While the bandwidth is comparable for both sensors, our study shows that the Nano-30 possesses a pronounced resonant peak around 300 kHz; this is also confirmed by the certificate provided by the manufacturer. Notably, there are no discernible features in the AE power around the point of macroscopic instability; the $P_{AE}$ level decreases smoothly and steadily, as is also seen in Figure 10. However, beyond the necking point, the AE spectra behave quite differently. The higher-sensitivity Micro-F30 sensor with a more regular (yet not flat) response shows an increasing trend in the median frequency value $f_m$, while the Nano-30 sensor with its spectrum dominated by the 300 kHz resonance displays the opposite trend.

The highlighted discrepancy adds further ambiguity regarding the reliability of the AE technique in detecting the necking process. The key (yet trivial) takeaway from these observations is that AE analysis results are significantly enhanced by using sensors with higher sensitivity, wider frequency bands and better signal-to-noise ratios during the mature deformation stage.

In principle, to gain a better understanding of the discrepancy discussed here, a simplified modeling of the AE response can be performed. The AE sensor transfer function can be represented as a multiresonant filter, as outlined by Ono [159,160], which, in the first order approximation, can be used alongside the source model presented earlier (cf. Equation (9)) to simulate the AE sensor’s response to evolving random processes. However, the details of these numerical simulations fall outside the scope of this work and will be presented elsewhere. In brief, these simulations yield the following deductions: (i) conclusions drawn from a transducer exhibiting a single sharp resonance can be misleading regarding the evolution of the source; and (ii) the AE spectral density reasonably shifts from the low-frequency to the high-frequency domain, in line with the model predictions, eventually resembling electrical background noise as the AE signal diminishes and noise contributions increase. This is clearly illustrated in Figure 12, where the behavior of the AE median frequency in the post-necking regime demonstrates a shift in the spectrum towards characteristic noise patterns, determined by the sensor’s frequency response in the absence of the AE signal. The higher-frequency noise spectrum corresponds to the Micro-F30 sensor, while the Nano-30 sensor tends to shift the measured spectrum to lower frequencies, representing the limiting background noise for the given sensor. (iii) Ideally, the noise spectral density should be subtracted from the AE signal, although implementing this procedure is complex and requires careful mathematical consideration.

Taken together with the above notes, these observations highlight the advantages of broadband, thresholdless acquisition by a high-sensitivity sensor with a well-calibrated response. Such a setup facilitates monitoring spectral changes in materials under load, particularly when strain localization occurs or is anticipated. This applies to all forms of strain localization discussed in the preceding sections. In fact, the issue is broader than these recommendations. Ideally, the acquisition setup should not only include the hardware specifications mentioned above but should also be harmonized with the data processing scheme. Effective analysis requires integrating the raw AE data with advanced signal-processing techniques to extract meaningful patterns and insights. Furthermore, these efforts should be accompanied by microstructure-informed modeling to link the observed AE phenomena to the underlying physical mechanisms of plastic instabilities. This integrative approach is crucial for accurately interpreting AE data and advancing our understanding of strain localization effects.

Although the results in this section may initially seem concerning, the positive aspect is that they are fundamentally consistent with the considerations made in the preceding sections. The necking point, while signaling the onset of macroscopic plastic instability, does not indicate an anomaly or discontinuity in the stress–strain curve; both the true flow stress and strain hardening rate continue to change smoothly. This contrasts sharply with discontinuous yielding associated with the upper and lower yield points, as well as Lüders or PLC band initiation and propagation. If we accept the rationale behind Equations (10) and (11) central to the outlined model in Section 3, we should not anticipate drastic changes in the AE signal at the onset of necking. Instead, a low, steadily diminishing AE signal is plausible unless substantial changes occur in the mobile dislocation density or their velocity in the neck. In this latter scenario, which is material-specific, we could anticipate an increase in the AE signal alongside a shift of the spectrum to lower frequencies as it is commonly seen in other plastic instabilities emerging during uniaxial monotonic tension.

In the context of the present discussion highlighting the challenges of detecting the onset of necking by non-destructive means, it is timely to revisit a key insight from the proposed modeling framework. By applying the analytical solution of the Kocks–Mecking model, or similar models, e.g., Equation (7), into the Considère criterion (12), it can be immediately deduced that the necking strain ${\epsilon }_{\mathrm{N}}$ is controlled by the rate of dynamic dislocation recovery governed by the parameter $k_2$ in the evolution Equation (5). A more rigorous derivation, based on the linear stability analysis of the total dislocation density’s kinetic equations, is provided in [68,69,161], yielding a similar criterion to Considère’s or Hart’s instability conditions, and demonstrating the reciprocal relation between ${\epsilon }_{\mathrm{N}}$ and $k_2$: ${\epsilon }_{\mathrm{N}}~\tilde{\epsilon }=2/k_{2}M$. As noted in our recent works [17,55], the most uncertain model parameter—the mobile dislocation density that is still hard to assess quantitatively by any means—can be eliminated simply by multiplying Equations (10) and (11) as

$$P_{\mathrm{AE}}{f}_{\mathrm{m}}=K_{\mathrm{AE}}{K}_{\mathrm{f}}\theta {\dot{\epsilon }}^2\approx K\exp \left(-{\displaystyle \frac{k_{2}M\epsilon }{2}}\right)$$

(14)

with $K=0.5K_{\mathrm{AE}}{K}_{\mathrm{f}}{M}^2{k}_1\alpha \mu {\dot{\epsilon }}^2$ being constant under constant strain rate. Therefore, the $\ln \left(P_{\mathrm{AE}}\cdot f_{\mathrm{m}}\right)$ function is expected to exhibit a negative linear relationship with strain. This behavior has been indeed demonstrated in [17] for several CuZn alloys, and in [47] for pure fcc metals like Cu, Al, Ag and Ni with different stacking fault energies. The most captivating result is that the slope of $\ln \left(P_{\mathrm{AE}}\cdot f_{\mathrm{m}}\right)$ vs. true strain $\epsilon$ is proportional to $k_2$, meaning that this coefficient can be recovered solely from AE signals. Consequently, the onset of necking can be predicted from AE measurements in the early stages of deformation, without relying on stress–strain data. While proof of this concept has been presented in the cited works [17,55], and the linear $\ln \left(P_{\mathrm{AE}}\cdot f_{\mathrm{m}}\right)$ vs. $\epsilon$ relation has been observed in other materials, such as Inconel 625 alloy and 316L austenitic stainless steel, some discrepancy remains between $k_2$ values derived from AE and stress–strain data. However, this seems to be a technical issue that will be addressed in future studies.

Before drawing final remarks, it is essential to reiterate that high AE levels are not necessarily indicative of localized deformation. We have demonstrated this throughout our work on monotonic tension. Similar observations were reported during the cyclic deformation of copper single crystals oriented for single slip, with the longitudinal axis aligned with the [123] direction, as well as in copper polycrystals with coarse or fine grains [79,162]. In those experiments, the cyclic hardening stage was accompanied by a regular reduction in AE power and an increase in AE median frequency, consistent with the predictions of model Equations (10) and (11) for monotonic deformation. Following this initial hardening, a shift of the AE spectrum towards lower frequencies was systematically observed during the formation of persistent slip bands—regions with highly ordered dislocation arrangements where plastic strains strongly localize in many fcc metals and alloys under certain loading conditions (e.g., at a specific range of strain amplitudes). Meanwhile, the AE power (amplitude or energy) did not show any notable changes at the onset of cyclic strain localization, unlike during PLC band propagation or necking. This is further supported by the author’s extensive early studies on Cu single crystals with other axial orientations, such as [100], [111], [211], [110], and [210], which exhibited diverse self-organizing patterns emerging during cyclic deformation in response to various dislocation reactions determined by the crystallographic orientations. The results of this unpublished work will be presented elsewhere.

5. Summary and Future Outlook

In this overview, we have focused on the utility of acoustic emission (AE) as a powerful tool for understanding the physical mechanisms underlying plastic instabilities within the broader context of strain-hardening in structural metals and alloys. We have intentionally minimized the discussion of experimental details and signal processing, unless deemed essential. Mathematical derivations have also been reduced, opting instead to use the outcomes as a framework for exploring the underlying features of AE behavior. We encourage interested readers to delve into the extensive literature cited for a deeper understanding of these aspects.

The present study reviews the current understanding of AE as a tool for investigating plastic flow and deformation in structural metals. Future work should focus on refining phenomenological AE models to better capture the complexities of dislocation ensembles involving different types of dislocations and localized deformation. These models should link dislocation mechanisms with nonlinear dynamics. Extending these models to account for multiscale material behaviors will enhance predictive accuracy, especially for real-world applications.

Improvements in AE metrology are essential to minimize the influence of the acquisition chain and signal processing on data interpretation. This applies to conventional amplitude distributions and spectral/correlation estimates, aiming to reveal the chaotic or self-organized critical behavior in plastic deformation.

Integrating AE data with modern machine learning techniques (beyond the scope of this review) offers a promising direction for developing more robust, non-destructive evaluation methods, particularly in AE-signal categorization, pattern recognition, and anomaly-detection in the AE time series. However, fully data-driven approaches may not deepen our understanding of the microstructure-sensitive behavior of dislocations and their role in plastic instabilities. Microstructurally informed, model-based approaches promise higher predictive power and better decision-making support, potentially enabling real-time failure prediction under various loading conditions by accounting for microstructural variations.

Continued research on AE responses across a broader range of alloys and microstructures, combined with advanced signal processing, could further expand AE’s industrial utility, from material design to safety-critical monitoring. Ultimately, deeper understanding of AE behavior during plastic deformation offers opportunities to enhance material reliability, extend service life and develop early-warning systems for structural failure.