Microsoft Word - numero_72_art_13_5357.docx S. K. Kourkoulis et al., Fracture and Structural Integrity, 72 (2025) 179-192; DOI: 10.3221/IGF-ESIS.72.13 179 Comparative assessment of the acoustic activity and the Pressure Stimulated Voltage in marble specimens under compression Stavros K. Kourkoulis, Ermioni D. Pasiou National Technical University of Athens, School of Applied Mathematical and Physical Sciences, Department of Mechanics, Zografou Campus, 5 Heroes of Polytechneion Avenue, 157 73, Attiki, Greece stakkour@central.ntua.gr, http://orcid.org/0000-0003-3246-9308 (SKK) epasiou@central.ntua.gr, https://orcid.org/0000-0003-1580-3415 (EDP) Dimos Triantis, Ilias Stavrakas University of West Attica, Department of Electrical & Electronics Engineering, Electronic Devices and Materials Laboratory, Ancient Olive Grove Campus, 250 Thivon Avenue, 122 44, Athens, Greece triantis@uniwa.gr, http://orcid.org/0000-0003-4219-8687 (DT) ilias@uniwa.gr, https://orcid.org/0000-0001-8484-8751 (IS) KEYWORDS. Pressure Stimulated Voltage; Acoustic Emissions; Pressure Stimulated Currents; F-function; Marble. Citation: Kourkoulis, S.K., Pasiou E.D., Triantis D., Stavrakas I., Comparative assessment of the acoustic activity and the Pressure Stimulated Voltage in marble specimens under compression, Fracture and Structural Integrity, 72 (2025) 179-192. Received: 23.01.2025 Accepted: 01.03.2025 Published: 03.03.2025 Issue: 04.2025 Copyright: © 2025 This is an open access article under the terms of the CC-BY 4.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. INTRODUCTION rediction of upcoming disastrous failures of engineering structures concerns the community of Structural Engineers from the dawn of civilization. Especially for the discipline of Earthquake Engineering, the detection of signals that could early warn about upcoming seismic events is equivalent to finding the Holy Grail. The respective research is continued uninterrupted and intensively long ago and this is, perhaps, the reason that most of the theories and techniques P https://youtu.be/wFVq_Hnxh-I S. K. Kourkoulis et al., Fracture and Structural Integrity, 72 (2025) 179-192; DOI: 10.3221/IGF-ESIS.72.13 180 related to the detection of pre-failure indices were firstly developed and applied in the direction of predicting upcoming strong earthquakes. Although, for the moment being, the specific target is not achieved, things are, at least, encouraging in the laboratory environment (which can be relatively easily “sterilized” from “external noises”): the techniques developed have indeed highlighted the existence of signals that can be considered as pre-failure indices. Among these techniques the one based on the detection and exploitation of Acoustic Emissions [1] is the most mature and successfully applied worldwide one for Structural Health Monitoring (SHM) purposes. The specific technique takes advantage of the mechanical waves generated within the bulk of a loaded system (either it is a specimen or a structural member or a structure or, even, the crust of the Earth) due to the sudden redistribution of local stress fields and the consequential abrupt release of elastic strain energy, caused, for example, by the generation of micro-cracks, or the activation of other damage mechanisms. However, the process of micro-cracking is responsible for additional geophysical emissions, detected in the form of micro- seismic excitations [2], infrared radiation [3], electromagnetic radiation [4], and, also, electric emissions, quantified either in the form of Pressure Stimulated Currents (PSC) [5] or in the form of Pressure Stimulated Voltage (PSV) (sometimes denoted as Electric Potential, EP) [6]. The PSC and the PSV are due to the same damage mechanisms, which are analogous to the mechanisms responsible for seismic excitations. A variety of theories have been proposed to explain the mechanisms that are responsible for the generation of electric emissions. One could mention, indicatively only, the Moving Charged Dislocations (MCD) model [7], and the theory of outflow of positive holes [8]. Nowadays, interesting monitoring techniques have been developed based on both PSC and PSV, which are considered as flexible SHM tools, complementary to those based on the Acoustic Emissions. They have been successfully applied for the study of the mechanical response and the evolution of damage within specimens made of concrete [5, 9, 10], marble [4, 5, 11], granite [4, 12], sandstone [4, 13], coal [14, 15] and other rocks [16, 17]. The results of these studies converge to that the electric signals are of very low intensity and of smooth (or imperceptible) increase for load levels within the elastic region of the loaded material. They start increasing gradually, according to an accelerating manner, only after the load exceeds the threshold designating onset of thermodynamically irreversible deformations. As the load approaches its peak value the electric signals exhibit a global maximum. The specific response provides a series of signs that can be considered as safe pre-failure indices [5, 18-20]. Very good correlation between the outcomes of the Acoustic Emissions technique and the techniques based on electric emissions has been highlighted in a series of research papers, especially concerning the pre- failure indices detected by each technique [5, 6, 13, 14]. It is very interesting to emphasize that according to recent studies the indices provided by exploiting the electric signals precede slightly the ones provided by the acoustic activity [5, 21]. The potential correlation between AE and PSV (or, more generally, between acoustic and electric emissions) attracted the interest of the engineering community rather recently, and, as a result, the relative works are scarce. Indicatively only one could mention the pioneering study by Archer at al. [6] who definitely concluded that their “… experiments demonstrate a very strong positive correlation between AE and PSV in both the non-quartzose halite and quartz rich granite ...” specimens. In addition, they suggested that “… the strong correlation between AE and PSV emissions provides good evidence that Electric Potential Signals could be used as a cost-effective complementary technology, and possibly even an alternative, to piezo transducers” [6]. Along the same line Zhang et al. [13] concluded that “… In different failure stages of sandstone specimens, the Electric Potential Signal shows a periodic change characteristic, which has a good corresponding relationship with AE counts …”. In addition, Zhang et al. [13] attempted to arrange in chronological order the pre-failure signals provided by the Acoustic Emissions and the Electric Potential Signals. They arrived to the conclusion that “… Compared with the Electric Potential Signal time series, the precursory point of the AE count time series also appears during the accelerated damage development stage of sandstone samples, with a lag time of 7.3 s. In terms of monitoring and early warning, the Electric Potential precursory point has more advantages” [13]. Within the frame of the above argumentation, the electric activity developed in marble specimens submitted to uniaxial compression, will be quantified in the present study in terms of the PSV recorded. The electric activity will be considered in juxtaposition to the respective acoustic one, which will be quantified in terms of the F-function [5], and, also, in terms of the energy content of the acoustic signals. The latter, i.e., the fact that the correlation between the acoustic activity and the respective electric one is attempted in quantitative terms using parameters with either temporal (F-function) or energy characteristics, is among the innovative aspects of the present study. Moreover, the temporal evolution of the PSV will be analyzed in juxtaposition to the respective one of the PSC filling a knowledge gap of the literature, since the intrinsic relation between these two different manifestations of the electric activity developed while loading brittle materials is not as yet properly highlighted. Another novel aspect of the study is that the potential influence of the loading rate on the correlation between the acoustic and electric activities is explored (although at a preliminary stage). Finally, following Zhang et al. [13] additional experimental evidence will be added on the time sequence of appearance of the pre-failure indices provided by the Acoustic Emissions and Electric Signals, in an attempt to further assess the potentialities of the Electric Signals (either quantified in terms of the PSV or the PSC) to be used as SHM tools, complementary to those based on Acoustic Emissions. S. K. Kourkoulis et al., Fracture and Structural Integrity, 72 (2025) 179-192; DOI: 10.3221/IGF-ESIS.72.13 181 EXPERIMENTAL PROTOCOL: THE MATERIAL, THE SPECIMENS AND THE EXPERIMENTAL SET-UP he specimens of the present experimental protocol were made of a fine white marble, quarried from mountain Dionysos in the Attiki region, Greece. Its main physico-mechanical properties are close to those of Pentelic marble, the building stone used by ancient Greeks for the erection of the Athenian Acropolis monuments. Due to this, Dionysos marble is used extensively to cover the needs of the ongoing restoration/conservation project of the monuments of the Acropolis of Athens, implemented by the “Acropolis Restoration Service” under the auspices of the “Committee for the Conservation of the Acropolis Monuments”, an interdisciplinary committee of the Greek Ministry of Culture. To the best authors’ knowledge there are not prior studies investigating the correlation between AE and PSV for this material. Dionysos marble is an anisotropic rock. Its mechanical properties, as they are obtained from the relative literature, exhibit strong scattering. It could be mentioned characteristically that for its tensile strength the values reported vary in the interval from about 2.5 MPa to about 19.5 MPa. Similarly, for its modulus of elasticity experimental values ranging from 23 GPa to about 90 GPa have been reported [22]. The above scattering (besides the inherent variability of the mechanical response of rock-like materials, the mechanical properties of which depend strongly on the exact quarrying position and depth) is well attributed to the anisotropic nature of Dionysos marble, which is in fact an orthotropic material characterized by three distinct directions of anisotropy. Vardoulakis and Kourkoulis [23], after long series of uniaxial compression and direct tension tests, with specimens cut along the three anisotropy axes, concluded that the mechanical properties of Dionysos marble along the two of the above anisotropy directions are very similar to each other, as it can be seen in Tab. 1. Taking into account the data of Tab. 1, it is concluded that, in a first approximation, Dionysos marble could be modelled as a transversely isotropic material, with only two independent anisotropy directions. Property → Direction↓ Modulus of Elasticity [GPa] Poisson’s ratio [---] Tensile strength [MPa] Compressive strength [MPa] Strong 84.5 0.26 10.8 82.5 Intermediate 79.5 0.26 9.5 78.2 Weak 50.0 0.11 5.3 48.2 Table 1: Numerical values of some critical mechanical properties of Dionysos marble along the three characteristic anisotropy directions [23] as they were obtained from quasi-static experiments under displacement-controlled conditions (loading rate equal to 10-6 m/min). In addition, Vardoulakis and Kourkoulis [23] pointed out that that the specific variety of rock material is bimodular: The modulus of elasticity under compression exceeds slightly that under tension. Another worth-mentioning feature of Dionysos marble is that it exhibits (according to a very pronounced manner) the size effect, namely the dependence of its mechanical properties on the size of the specimens used for the laboratory experiments. Especially for the Uniaxial Compressive Strength (UCS), this dependence is not monotonous but rather a global maximum appears for cylindrical specimens of length equal to about 125 mm (and length-to-diameter ratio equal to 2). The specimens of the present experimental protocol were shaped in the form of orthogonal prisms. The dimensions of their cross section were equal to 40 mm x 40 mm, while their length (height) was equal to 90 mm. Attention was paid for the specimens of the same class to be cut from the same core, since (as it was already mentioned) the properties of rocks and rock-like materials depend, among others, on the exact quarry point and depth. Comparing the results concerning the mechanical properties of the material, as they were obtained from the present protocol, against those mentioned by Vardoulakis et al. [23], it appears that the specific cores were oriented along the weak anisotropy axis. The compressive load was applied along the longitudinal axis of the specimens by means of a very stiff servo-hydraulic INSTRON SATEC loading frame of capacity 300 kN. A representative axial stress-axial strain plot, from a typical experiment is shown in Fig.1a. Slight discrepancies for the mechanical properties depicted in Fig.1a and those in Tab. 1 are attributed to the different loading mode (load- versus displacement-controlled) and, also, to the “shape effect” (recall that the specimens used in the protocol realized by Vardoulakis and Kourkoulis [23] were cylindrical while those used in the present protocol were shaped in the form of orthogonal prisms. It is seen from Fig.1a that the response of Dionysos marble is linear elastic for the major portion of the interval of the applied stresses. It is only for very low loading levels and, also, when the load approaches the Uniaxial Compressive Strength (UCS) that the response deviates from linearity. The initial deviation from linearity is attributed to bedding errors and, also, to a kind of stiffening due to elastic closure of pre-existing defects and pores [24]. The second one is attributed to the onset of generation of thermodynamically irreversible phenomena, correlated strongly to significant accumulation of internal damage. In general, the second nonlinear region starts at load levels ranging from about 85% to about 92% of the UCS (at least for the specimens of the present protocol). T S. K. Kourkoulis et al., Fracture and Structural Integrity, 72 (2025) 179-192; DOI: 10.3221/IGF-ESIS.72.13 182 (a) (b) Figure 1: (a) A typical axial stress - axial strain plot of Dionysos marble under uniaxial compression. (b) Draft sketch of the specimens: The position of the acoustic sensor (AS), mounted on the center of the front lateral surface of the specimens, is depicted, together with the position of the two electrodes (electric sensors, ES), properly attached at the center of two opposite lateral surfaces and that of the electrical strain gauge SG, attached at the rear lateral surface. The acoustic activity was recorded with the aid of an R6α acoustic sensor (mounted at the geometric center of the front lateral surface of the specimens, Fig.1b) with resonance frequency equal to 55 kHz and function bandwidth zone ranging in the interval from 35 kHz to 100 kHz. A preamplifier (gain 40 dB) was used, combined with analogue filters (pass zone in the range from 20 kHz to 400 kHz). The cut-off threshold was set to 40 dB. The equipment and software for detecting and recording the acoustic signals were provided by Mistras Group, Inc., Physical Acoustics Corp., New Jersey, USA. Given that only one sensor was used to capture the acoustic signals, it was deemed reasonable for the sensor to be placed as far as possible from the bases of the specimens in order to avoid recording signals due to friction effects between the loading platens and the bases of the specimens (although Teflon sheets were interposed between both bases of the specimens and the respective loading platens to minimize friction effects, see Fig.1b). In the direction of further reducing the role of any parasitic signals, proper timing parameters were selected regarding the AE system. This effort focuses mainly on the proper selection of parameters like the minimum inter-hit time (Hit Definition Time - HDT) and the attenuation factor of the specific variety of marble. These parameters were properly chosen after series of preliminary tests. In this context, when successive hits were recorded within time intervals approaching the time required for the AE to travel from the one side of the specimen to the other (matching the respective attenuation factor) it was straight forwardly concluded that the source of the specific signal is the same with that of the previous one and, thus, the specific signal is ignored as an outlier. The respective electric activity, was quantified in terms of the PSV, using an ultra-sensitive electrometer (Keithley 6517A) of resolution equal to 1 fA and input impedance 200 TΩ using a pair of circular copper gold-plated electrodes that were attached at the geometric centers of two opposite lateral surfaces of the specimens. To achieve optimum electric contact, a thin layer of conductive silver paste was smeared at the points where the electrodes were to be attached. The backside of the electrodes was supported through additional thick Teflon sheets by springs that were used to keep the electrodes in position and ensure the uninterrupted contact during the loading process. Obviously, the Teflon sheets interposed to minimize friction effects (mentioned in previous paragraph) offer, also, electric insulation between the specimen and the loading platens. Finally, in order to assure that no electromagnetic noise distorted the recordings of the electric activity, the whole experimental set-up was placed in a Faraday cage. It is emphasized here that the background noise of the PSV signal recorded before the onset of the loading procedure was always lower than 0.1 mV for all the experiments of the present experimental protocol, without any exception at all. Concerning the position of the electric sensor, it should be mentioned that it does influence the outcomes of any relative experiment, since the micro-fracturing processes responsible for the generation of the electric signals are not uniformly distributed all over the mass of the specimen. This was clearly highlighted by Stavrakas et al. [25], who used an integrated grid of five sensors (instead of a single pair of electrodes) to detect the electrical signals, that were emitted while submitting marble prisms to uniaxial compression. They clearly pointed out that the “… use of such a grid of sensors does indeed enhance the 0 20 40 60 0.0000 0.0005 0.0010 0.0015 A xi al s tr es s [M P a] Axial strain E≈52 GPa UCS≈56.4 MPa Linearity limit≈49.0 MPa 90 mm 40 mm 40 mm AS F F Teflon (PTFE) Sheets SG S. K. Kourkoulis et al., Fracture and Structural Integrity, 72 (2025) 179-192; DOI: 10.3221/IGF-ESIS.72.13 183 capabilities of the PSC technique providing valuable information about the initiation and propagation of micro-fracturing at the interior of the specimens”. In the present study the position of the single electric sensor used was decided taking into account that attention is focused to the comparative study of the acoustic and electric activities and, therefore, it was of critical importance for the two sensors (the electric and the acoustic one) to be placed as close to each other as possible. In addition, the specific position chosen (the central area of the specimen) limits possible effects of edge charges or any triboelectric or electrokinetic effects originating from the immediate vicinity of the loaded surfaces. In any case, it must be emphasized that the impact of the above-mentioned effects is rather negligible, since marble exhibits low porosity (non-electrokinetic), low quartz and high resistivity (non-triboelectric). Finally, it must be mentioned at this point that no cross-talking between the acoustic sensor and the electric voltage electrodes is possible and, therefore, placing them in the same position of the specimen has no impact on the measurements recorded. The axial strains developed were recorded by means of an electric strain gauge attached at the specimens’ rear lateral surface. Special attention was paid for the synchronization of all recording systems (i.e., the systems used to record the acoustic activity, the electric emissions, the axial strain and the applied load), an issue of extreme importance for proper interpretation of the experimental data [5, 26], which, if disregarded leads inevitably to erroneous results. In this direction, the PCI-2 (Mistras Group, Inc – Physical Acoustics Corp.) AE data acquisition card was used, equipped with parametric inputs, as it is analytically described in the user’s manual of the manufacturer. These parametric inputs are used as an extra input to the AE system that can be used to measure any other external quantity from other transducers. In the present protocol the PCI- 2 was used to capture the AE activity and the parametric data were used to record the mechanical load from the loading frame (which is equipped with analogue signal outputs that can be used to share the values of the load applied to other systems). Then, the numerical data for the load recorded were shared to all the measuring systems of the complete set-up. Regarding the AE recordings it is, also, noted that the software used to control the AE systems was the AEWin ® software (provided, also, by Mistras Group, Inc – Physical Acoustics Corp.). Parameterizing this software enables recording both hit driven and time driven data (in the specific protocol the AE hits including the corresponding load level as shared from the loading frame). Furthermore, and at an independent sampling rate of 0.1 s additional load values were recorded to ensure the ability to reconstruct the complete load curve from the AE system for accurate data synchronization. Concerning the synchronization of the PSV recording system, it was achieved using the shared load level from the loading frame. Specifically, the PSV and the axial strain recording subsystem includes a data acquisition card (National Instruments Corp. ® 16 bit data acquisition PCI card module) an analogue input of which is dedicated to the load proportional voltage output of the loading frame. The data are recorded in the same environment prepared using the Agilent Vee ® software for visualization and data storing. Considering the above-mentioned precautions and measures taken, it is safely stated that all recording systems employed in the present protocol were properly synchronized. The specimens were submitted to uniaxial compression, adopting a load-controlled scheme. In an attempt to take into account, also, the potential influence of the loading rate on the correlation between the electric and the acoustic activities, two classes of experiments were implemented, one with loading rate equal to 34 kPa/s and a second one with loading rate equal to 140 kPa/s. Unfortunately, there is not a standard dictating numerical values for the specific parameter, and, also, there is a lack of relative studies in the literature. It could be anticipated at this point that the choice of these two loading rates is more or less an arbitrary decision of the authors. Indeed, this is the case, since an exhaustive analysis of the role of this parameter was not the core target of the present study. This is just a first approach towards the specific issue, and, in this context, a very low loading rate was chosen (approaching quasi-static conditions) together with a relatively higher one (almost four times higher), which, however, is by no means considered as an actual dynamic loading scheme. The loading rate was kept constant throughout the duration of the experiments of each class. The load was applied monotonically from the onset of the tests until the final macroscopic distraction of the specimens. Three specimens were tested for each class. The scattering of the results between the three specimens of each class was relatively low, concerning both the mechanical response and, also, the parameters of the electric and acoustic activities. EXPERIMENTAL RESULTS Criticality by means of the electric activity quantified in terms of the Pressure Stimulated Voltage (PSV) (Electric Potential - EP) n this section the results obtained from two representative experiments (one from each class of specimens) are analyzed. For the first one, in which the specimen was loaded at a constant rate of 34 kPa/s, the fracture stress (or, equivalently, the Uniaxial Compressive Strength, UCS) was equal to about 56.3 MPa. In Fig.2 the temporal evolution of the PSV is plotted, in juxtaposition to that of the applied axial stress (the latter is, obviously, a perfectly linear segment given that load-controlled conditions were adopted). The sampling rate for the PSV was set equal to 1 sample/s, while the I S. K. Kourkoulis et al., Fracture and Structural Integrity, 72 (2025) 179-192; DOI: 10.3221/IGF-ESIS.72.13 184 background level of the respective signal before the onset of the loading procedure was equal to about 90 μV. It was deemed necessary to subtract this background level from the values of the PSV, and, therefore the data plotted in Fig.2 correspond to the “net” PSV developed during loading. Figure 2: The temporal evolution of the “net” PSV and that of the applied axial stress for two characteristic experiments, one from each class of specimens tested (red lines and lettering correspond to loading rate equal to 34 kPa/s, while blue ones to 140 kPa/s. It is observed from Fig.2 that the “net” PSV starts increasing abruptly at the time instant tc≈1520 s, at which the respective axial stress applied is equal to about 92% of the respective peak level attained. Recalling that the stress level at which Dionysos marble ceases responding elastically (i.e., the limit after which the deformations become “permanent” or, equivalently, thermodynamically irreversible) was determined in the interval between 85% and 92% of the UCS, it could be concluded that the PSV starts increasing abruptly as soon as the material enters into its stage of non-reversible deformations. About 5.5 s before fracture the PSV attains a global peak, which for the specific experiment is equal to about 2.86 mV, and, then, it starts decreasing quite rapidly. The above-described response for the temporal evolution of the PSV is quite similar to the respective one of the PSC recorded in a series of experimental protocols with various loading schemes and specimens of various geometries made of a large variety of materials [5, 11, 12, 20, 27]. For the second experiment, the specimen was loaded at a constant rate equal to 140 kPa/s. The UCS was slightly lower from that of the previous experiment, and it was equal to about 52 MPa. This is, perhaps, unexpected, taking into account that, in general, increased loading rate corresponds to increased peak stress. Indeed, as it is mentioned by Rae et al. [28] “… in general, results of mechanical testing demonstrate dynamic strength increase under uniaxial compression at large strain rates”. The specific inconsistency is clearly attributed to the fact that the two loading rates used in the present protocol are relatively close to each other, and, also, to the fact that the specimens of the two classes were cut from different cores. The temporal evolution of the PSV and that of the axial stress applied, are plotted, also, in Fig.2, so that a direct comparison between the two classes of specimens to be clear. The sampling rate was again 1 sample/s. The background level of the electric signal, before starting the loading procedure, was equal to 65 μV and it was, again, subtracted from the values plotted in Fig.2. From a quantitative point of view the overall response of this specimen is similar to that of the previous one. There are, obviously, some quantitative differences: The PSV starts increasing abruptly at the instant tc≈317 s, an instant at which the respective axial stress is equal to about 86% of the UCS. It seems, therefore, that the increased loading rate triggers the mechanisms responsible for the electric emissions at lower stress levels with respect to the loading at lower rate. Similarly to the previous specimen, 4.7 s before macroscopic fracture the PSV exhibits a global peak (higher than that of the lower loading rate) equal to about 3.9 mV. In order to achieve a direct comparison of the electric activity generated in the two specimens, the “net” PSV recorded is plotted in Fig.3 for both experiments, versus the normalized time. The normalization is realized over the time instant of fracture, tf, of each specimen, so that the horizontal axis of the two graphs to be common for the two specimens (i.e., 0(UCS)/2. In order, now, to describe quantitatively the intensity of the acoustic activity during the loading procedure, advantage is taken of the temporal evolution of the respective F-function, which represents the average frequency of generation of acoustic signals, within a given “window” of n successive signals [5]. The F-function is calculated as the “sliding” average of the instantaneous values fi=1/(δti) of the inverse interevent time intervals, (δti), between any two successive acoustic signals. The procedure for the determination of the F-function starts by considering a first “window” of acoustic signals, which includes the first n signals, where n is a relatively small fraction of the overall number N of the signals recorded. Although n is more or less arbitrarily chosen, its influence on the final conclusions is rather insignificant as long as n is small with respect to N. Further details about the role of n can be found in ref. [5]. For this first “window”, one determines the 0.0 1.0 2.0 3.0 4.0 0.00 0.25 0.50 0.75 1.00 P S V [ m V ] t/tf 34 kPa/sec 140 kPa/sec Loading rate: tc tc 0 20 40 60 1E+01 1E+03 1E+05 1E+07 0 500 1000 1500 2000 A xi al s tr es s [M P a] E ne rg y of A E s [a J] Time [s] energy AE stress Loading rate: 34 kPa/s Energy of AEs Axial stress 0 20 40 60 1E+01 1E+03 1E+05 1E+07 0 100 200 300 400 energy AE stress A xi al s tr es s [M P a] Time [s] E ne rg y of A E s [a J] Loading rate: 140 kPa/s Energy of AEs Axial stress S. K. Kourkoulis et al., Fracture and Structural Integrity, 72 (2025) 179-192; DOI: 10.3221/IGF-ESIS.72.13 186 (n-1) interevent time intervals (δti) and the respective fi-values. The mean value of these (n-1) fi-values corresponds to the first value F1 of the F-function. “Sliding” then the first “window” by one signal (i.e., considering the “window” including the signals from the 2nd one up to the (n+1)th one) the second value F2 of the F-function is obtained and so on (concerning the “sliding” step, its value is defined taking into account the total number of acoustic signals recorded, and, also, the level of resolution that is to be achieved). For the temporal evolution of the F-function to be plotted, each F-value is paired to the respective “average time”, τ, which is calculated as the mean value of the instants, ti, at which each one of the signals of the specific “window” was recorded. The main advantage of the description of the acoustic activity in terms of the F- function is that it provides very fine “resolution”, which is of crucial importance, especially during the very last loading stage (i.e., as the load approaches its peak value), since the number of signals recorded during this stage increases dramatically. It has been verified that an abrupt increasing trend characterizes the temporal evolution of the F-function a few seconds before the final fracture. Series of experimental studies have proven that the onset of this abrupt increase of the numerical values of the F-function is a flexible (and safe) pre-failure index, early warning about entrance of the mechanically loaded system (the specimen in this study) into its critical stage, namely the stage of impending macroscopic fracture [5]. For the experimental protocol of the present study, the number of the signals of each “window” was set equal to n=30. The temporal evolution of the F-functions determined for the two experiments analyzed in previous sections is plotted against the “average time” parameter τ (adopting logarithmic scale for the F-function) in Fig.5a for the specimen loaded at a rate of 34 kPa/s and in Fig.5b for the specimen loaded at a rate of 140 kPa/s. The acoustic activity in the specimen loaded under higher rate appears much more intense. Such a response could be expected taking into account that almost the same number of hits (with respect to the number of hits recorded in the experiment loaded under low rate) was recorded in a much shorter time interval. For both experiments the F-functions exhibit quite abrupt increase tendency, as the load approaches the respective peak value. The onset of this abrupt increase is detected at a stress level equal to about 94% of the UCS for the specimens loaded under low rate. The respective ratio for the specimen loaded under high rate is equal to about 88% of the respective UCS. It is recalled that the onset of abrupt increase of the PSV was detected at ratios equal to 92% and 86% of the UCS, respectively. It can be, therefore, concluded that the agreement between the critical time instants (rapid increase of either the acoustic or the electric activities) as obtained from the temporal evolution of the PSV and that of the F-function is quite satisfactory, at least for the material tested in this protocol. (a) (b) Figure 5: The temporal evolution of the F-function for the two representative experiments analyzed in this study plotted versus the “average time” parameter τ, along a semi-logarithmic system of axes (the values of the F-functions are plotted along a logarithmic scale). (a) Loading rate equal to 34 kPa/s; (b) Loading rate equal to 140 kPa/s. To highlight further, comparatively, the temporal evolution of the acoustic activity generated during the two representative experiments discussed, the respective F-functions are plotted, in juxtaposition to each other, in Fig.6 versus a common axis, i.e., the normalized “average time” parameter, τ/tf, where the normalization is implemented over the duration of each experiment. The huge quantitative differences between the two F-functions and, also, their qualitative similarities, are vividly seen: Both F-functions increase smoothly and almost linearly for the major portion of the loading process. As the load approaches its peak, the two F-functions exhibit an abrupt increasing trend. Moreover, it is observed that the acoustic activity in the specimen loaded under higher rate is systematically stronger, with respect to the one loaded under lower rate, for the overall duration of the experiments. A few seconds before fracture the signal drops abruptly for both specimens. A xi al s tr es s [M P a] F-function Axial stress Loading rate: 34 kPa/s 0.94 σmax 0 20 40 60 0.01 0.1 1 10 100 1000 0 500 1000 1500 2000 F [ s-1 ] Time, τ [s] F function stress F-function Axial stress 0 20 40 60 1 10 100 1000 0 100 200 300 400 F function stress Time, τ [s] F [ s-1 ] A xi al s tr es s [M P a] Loading rate: 140 kPa/s 0.88 UCS F-function Axial stress A xi al s tr es s [M P a] Loading rate: 34 kPa/s 0.94 UCS S. K. Kourkoulis et al., Fracture and Structural Integrity, 72 (2025) 179-192; DOI: 10.3221/IGF-ESIS.72.13 187 Figure 6: The F-functions of the two representative experiments discussed, plotted in juxtaposition to each other along a logarithmic scale, versus the normalized (over the duration of each experiment) “average time” parameter, τ/tf. Criticality by means of the acoustic activity in terms of the Cumulative Energy of the Acoustic Emissions An additional Acoustic Emissions parameter, which is quite often used in the direction of detecting signs warning about impending approach of the loaded system (specimen) to the stage of accelerated damage processes (which will lead to the catastrophic fracture), is the Cumulative Energy of the Acoustic Emissions (CE)AE. An abrupt increase of the temporal rate of the (CE)AE is usually conceived as a pre-failure index designating entrance into the stage of impending failure [29]. The temporal evolution of the (CE)AE for the two experiments analyzed here are plotted in Fig.7a, in juxtaposition to each other, in terms of the normalized (over the duration of each test) time. Attention is paid to the time interval t/tf > 0.7, i.e., after a significant amount of acoustic energy is accumulated in the specimens. In Fig.7a the Cumulative Energies are plotted along logarithmic scale. It is observed from this figure that during the early loading steps (i.e., while the response of the specimens’ material is still linear) the (CE)AE increases obeying, for both specimens, a power law of the following form:   μ AE f t CE β t        (1) In Eq.(1) β is an energy constant and μ is an exponent related mainly to the level of the cumulative energy of the Acoustic Emissions of each test and the respective background level of the cumulative energy at lower stress levels. It is seen from Fig.7 that for the specimen loaded at high rate the values of the (CE)AE are systematically higher. This is well attributed to the (a) (b) Figure 7: (a) The Cumulative Energy of the Acoustic Emissions, (CE)AE, plotted in juxtaposition to each other along a logarithmic scale, versus the normalized (over the duration of each experiment) “average time” parameter, t/tf. (b) The ratio of the two (CE)AE. 0.01 0.1 1 10 100 1000 0.00 0.25 0.50 0.75 1.00 34 kPa/sec 140 kPa/sec F [ s-1 ] τ/tf 0.94 0.88 Loading rate: 0.0 1.0 2.0 3.0 4.0 0.70 0.80 0.90 1.00 t/tf R = ( C E ) A E ,1 40 / (C E ) A E ,3 4 1E+03 1E+04 1E+05 1E+06 1E+07 0.70 0.80 0.90 1.00 Loading rate Exp. data Power law 34 [kPa/s] C(t/tf)5.6 140 [kPa/s] C(t/tf)2.3   0.87 0.90 C um ul at iv e E ne rg y of A E s [a J] t/tf S. K. Kourkoulis et al., Fracture and Structural Integrity, 72 (2025) 179-192; DOI: 10.3221/IGF-ESIS.72.13 188 fact that for this specimen an increased number of acoustic hits had been recorded during the early load levels, in accordance to the conclusions drawn from the temporal evolution of the rate of generation of the acoustic signals in terms of the F- function (see Fig. 6). In this context, the exponent μ is equal to μ=2.3 for the specimen with loading rate of 140 kPa/s while for that with loading rate equal to 34 kPa/s the exponent μ is equal to μ=5.6. The above exponential law (Eq.(1)) governs the temporal evolution of the (CE)AE until the critical (normalized) instants t/tf,34=0.90 and t/tf,140=0.87, for the specimen loaded under low and high strain rates, respectively. From these instants on, the temporal rate of the (CE)AE increases more abruptly, according to an accelerating manner, until the instant of fracture. At this point, it is quite interesting to note that as the applied load tends to its peak value (i.e., while t/tf →1), the difference between the two (CE)AE decreases. This is clearly seen in Fig.7b, in which the ratio, R, of the two (CE)AE, namely, the ratio R=[(CE)AE,140 kPa/s/(CE)AE,34 kPa/s] is plotted versus the normalized time, t/tf, for the interval 0.70