Acoustic Emission Analysis of Fracture and Size Effect in Cementitious Mortars

Abstract

The size effect is a phenomenon where the strength and the ductility of a material depend on the size of the structure. Investigating size effects and related crack formation in brittle materials requires advanced monitoring methods. The aim of this research is to experimentally investigate the impact of size effect with the acoustic emission (AE) technique. Brazilian splitting tests with AE monitoring were performed on cement-based mortar cylinders of three sizes. It was found that in addition to the size, the boundary condition affects the final strength. When adopting similar boundary conditions in samples with different sizes, the larger samples had the lowest tensile splitting strength. For the larger samples, initially, there were fewer AE activities. However, there was a surge of high-amplitude AE events near the peak load. This indicates that as size increases, there is a lack of micro-cracking before macro-crack propagation, and the material fails in a more brittle manner. The width of the fracture process zone was quantified with AE and increased with sample size. A further analysis of the AE amplitude distribution demonstrated a change in the distribution in the pre-peak phase for the larger samples and for the smaller samples in the post-peak phase, signifying the brittle to ductile failure transition that occurs as size decreases.

1. Introduction

The existing methods of analyzing and designing structures using strength-based criteria often lack the ability to explain structural phenomena, such as the size effect. Size effect is the phenomenon where the tensile strength and ductility of a material depend on the structure’s geometrical size [1,2,3]. According to the strength failure criteria, the tensile strength and the ductility of a structure are size-independent. However, the existence of this effect in materials has been identified since the early 1900s by Griffith and Taylor [4] and it has been the subject of many studies [1,2,5,6,7,8]. Structural engineers design new infrastructures using material strength determined from tests conducted on small- and medium-scale samples in a laboratory environment. Hence, utilizing these values without regard to the size effect can result in a possible erroneous estimation of the real-life structural performance [1,6]. Recently, the application of advanced fracture monitoring techniques has provided new insights in this fundamental issue [9,10,11,12,13,14,15]. Acoustic emission (AE) [9,10,11,12,13,14,15], digital image correlation (DIC) [9,10,15], and X-ray imaging techniques [13] have been used to study the effect of specimen size on the energy release rate, the time of crack initiation, and the size of the fracture process zone (FPZ).

In addition, for existing structures, researchers are developing AE-based crack monitoring methods [16]. Aggelis [17] and Colombo et al. [18] correlated damage levels to specific values of AE parameters on concrete prisms and reinforced concrete beams, respectively. Bayane and Brühwiler [19] then used these reference values of AE parameters to monitor the level of damage of a concrete bridge. However, since fracture behavior evolves with the size of the structure, the evolution of AE parameters with size requires further investigation. Hence, this paper seeks to understand if the geometrical size of a structure affects the AE-based fracture analysis.

Size effect originates from two leading causes. The first one is the well-known Weibull statistical size effect, which stipulates that increasing the volume increases the probability of encountering micro-scale flaws that can initiate fracture. However, even for a purely deterministic (non-statistical) material, the second cause, being the mechanical size effect, would still exist [1,2,6]. This effect is due to the magnitude of the stored energy released at the crack tip. At the same nominal stress, larger structures release higher energy for a unit extension of a fracture. This means, for a larger structure, that the required fracture energy will be met at a lower nominal stress [8]. As the sample size increases, in addition to the decrease in the ultimate nominal stress, the failure also becomes more brittle [1].

Strength and ductility of a quasi-brittle material is related to the size of the fracture process zone (FPZ), which also varies with specimen size and geometry [12]. The FPZ is a region ahead of the main crack where micro-cracking accumulates. The size of this zone is wider than the material heterogeneity and is on the same order of magnitude as the sample size [7]. The FPZ is a region where tension softening occurs [1]. Hence, it is the region where fracture energy is released. Researchers have used direct observation [13], image-based techniques such as X-ray visualization [13], and digital image correlation [9] to measure the size of FPZ. AE-based techniques have been used by several researchers [9,11,12,13,14,15]. The AE technique allows capturing the high-frequency elastic waves that are emitted upon micro-fracture [20,21]. It can monitor the damage through the medium’s three-dimensional space, and it can also track the damage in real time. In this paper, AE monitoring will be used to track the fracture progress, as well as to determine the FPZ.

On a three-point bending test of notched beams, the size of FPZ was found to increase with sample size [12,15]. Otsuka and Date [13] performed tensile tests on notched concrete samples with varying sizes. Additionally in their work, the size of FPZ was found to increase with sample size, but the rate of growth was less than the sample size. Zietlow and Labuz [14] observed that the size of the FPZ was more affected by the type of rock used than the sample size increase rate. Labuz et al. [11] also found sample size to have no significant effect on the width of the located AE clusters (FPZ) when performing Brazilian splitting tests on quasi-brittle materials. Hence, in this paper, this ambiguous relation between sample size, AE source locations, and the size of FPZ is investigated.

A straightforward test that can induce a brittle failure is a Brazilian splitting test [8]. In this test, a cylindrical sample is diametrically compressed. For a linear elastic brittle material, this will generate a nearly uniform tensile stress at the center plane that passes along the loading direction [2]. The tensile stress at failure will split the sample, and this stress is used to measure the tensile splitting strength ($f_{\mathrm{ct}}$) [22]. The relation between the $f_{\mathrm{ct}}$ and the sample diameter (D) is often contradictory in the literature. Hondros [23] (for D = 150 mm to 600 mm) observed the $f_{\mathrm{ct}}$ to increase with diameter. However, Rocco et al. [24] (for D = 37 mm to 300 mm) and Bažant Zdeněk et al. [8] (for D = 19 mm to 508 mm) found the $f_{\mathrm{ct}}$ to decrease as sample size increases. Yet, in the study of Bažant Zdeněk et al. [8], this effect was reversed when passing above a specific diameter, while Rocco et al. [24] observed the $f_{\mathrm{ct}}$ to reach asymptotically a minimum value as the diameter increases. From these literature analyses, it can be inferred that the correlation between sample diameter and the tensile splitting strength depends on the range of the compared specimens’ diameter. In addition to the sample size, the boundary condition was found to affect the $f_{\mathrm{ct}}$ [24]. In the current paper, the Brazilian splitting test method is adopted not only to clarify the impact of size on strength, but also on fracture initiation and development. The focus will be on the smaller diameter range (27–106 mm) and constant boundary conditions.

This paper aims at studying the causes of size effect phenomena with the AE technique. It is clear from the literature study that researchers have investigated the size effect in the Brazilian splitting test. However, to the authors’ knowledge, simultaneous direct and derived AE parameters and waveform analyses have not been applied for size effect investigation in the Brazilian splitting test of cementitious mortars. Hence, the AE analysis is used to validate prior extensive theoretical and experimental analyses found in the literature. The size effect is analyzed by comparing the failure strength and the AE-based fracture analysis. The paper is organized as follows: the second section describes the experimental test program and the overall methodology. The third section presents observations on mechanical behavior of cementitious mortars in a Brazilian splitting test in relation to sample size and boundary conditions. In the fourth section, the size effect is investigated by means of AE analysis: AE amplitude, AE hit count, AE-based FPZ, and b-value analysis are reported and linked to the fracture process.

2. Experimental Program and AE Analysis Methods

This section describes the experimental methods used to test different-sized cementitious mortar cylinders in the Brazilian splitting test. In addition, the AE setup and processing techniques are elaborated.

2.1. Materials and Test Specimens

Mortar cylinders with varying sample sizes were made in accordance with EN 196-1 [25]. The cement mortar is prepared with a Portland cement (CEM II/B (S-L) 32.5 N) and river sand with a maximum aggregate size of 4 mm and 0/2 grain size curve distribution according to EN 13139 [26]. The sand:water:cement ratios were 3:0.5:1 by weight. According to EN 196-1 [25], three prisms (40 mm × 40 mm × 160 mm) were prepared from the same batch. After casting, the prisms were covered by a glass plate and kept in a room at ($20\pm 1){}^{°}\mathrm{C}$. The cylinders were stored in a curing room controlled at ($20\pm 1){}^{°}\mathrm{C}$ and ($95\pm 3)\%$ relative humidity. The molds were removed after 24 h and all the specimens (prisms and cylinders) were placed in the curing room for 28 days. At the age of 28 days, the prisms were tested according to EN 196-1 and resulted in a mean compressive strength of 20.65 MPa with a standard deviation of 0.71 MPa and a mean flexural strength of 3.56 MPa with a standard deviation of 0.33 MPa.

2.2. Test Setup and Loading Condition

The Brazilian splitting test setup is shown in Figure 1. For this test, cylinders of three different sizes were prepared, with a diameter of 27, 46, and 106 mm. The load is distributed in small strips at the top and bottom contact points to avoid excessive stress concentration and crushing at load application points. These load-bearing strips are made of hardboard with 3 mm thickness. The widths of these strips $\left(B\right)$ were adapted according to the sample diameter $\left(D\right)$ to give different boundary conditions ($\beta =B/D$). A standard value of $\beta =0.15$ was taken, and then variations on this value were tested for each sample size. The geometry and the number of samples for the different boundary conditions are summarized in

Table 1. Specimen geometry (thickness (t)).

Sample Size	D [mm]	t [mm]	No of Samples
			Total	$\mathit{\beta }$ = 0.15	$\mathit{\beta }$ ≠ 0.15	with AE
Small	27	27	11	8	3	8
Medium	46	47	7	4	3	7
Large	106	106	5	5	-	5

. In addition, in Table 1, the number of samples monitored with AE technique is also indicated. All of the tests were conducted according to EN 12390-6 [27], in a load control test with a stress rate of 0.04 MPa/s. The loading is applied using a universal testbench (Shimadzu AG-X), with a maximum capacity of 100 kN. During the test, the load and stroke displacement of the crosshead are recorded.

2.3. AE Test Setup and Post-Processing

AE monitoring during the tests was carried out using six piezoelectric broadband sensors (Digital Wave B1025) with a 50 to 2000 kHz frequency range. The sensors are chosen mainly due to their miniature size (9.3 mm diameter × 12.7 mm depth) that allows them to be easily attached to the smaller sample surface. In addition, a frequency analysis of the sensors revealed that these sensors are capable of capturing higher frequency content of the AE signals in the limited sample sizes used in the laboratory environment. These sensors were mounted on the sample surface with a thin layer of hot-melt glue. The sensor layout is shown in Figure 2 and is designed such that the three-dimensional information of AE generated across the entire sample can be captured. The AE sensors were connected to preamplifiers with 34 dB gain (AEP5, Vallen) and subsequently to a Vallen AMSY-6 acquisition system with six AE channels. The acquisition system digital frequency filter was set between 25 kHz to 850 kHz. The amplitude threshold was set at 40 dB, a value determined by monitoring the environmental noise of the laboratory. AE data are recorded at a sampling rate of 10 MHz, with 4096 samples per signal and 409.6 μs in resulting page length. The rearm and the duration discrimination time for AE hit definition are both taken as 200 μs. Real-time data is visualized using Vallen Visual AE software. Further post-processing analysis was conducted with an in-house developed AE toolbox in Matlab.

2.4. AE Analysis Methods

In this study, AE hit count and amplitude are the main parameters used to investigate fracture behavior. In addition, localization plots and b-value analysis are used for more advanced analysis.

2.4.1. AE Source Localization

Localization of AE sources was computed using Geiger’s method [28]. This is an iterative source localization method that tries to reduce the difference between the calculated and the observed arrival time with each iteration [28]. The observed arrival time is determined using the Akaike information criterion (AIC) picker [29]. The AIC picker was found to improve the accuracy of the arrival-time estimation [30,31]. In addition, a single value of wave velocity is determined from the average P-wave velocity in UPV (ultrasonic pulse velocity) tests. The measured representative wave velocity was 3650 m/s. The 3D AE source locations were computed using the wave velocity, the arrival time, the center of the sample as the trial hypocenter, and the locations of all six sensors.

2.4.2. b-Value Analysis

AE peak amplitude is related to the magnitude of fracture [32]. The formation of micro-cracks will initiate a high number of low-amplitude AE events, while macro-cracks will generate a low number of high-amplitude events. Similarly, in seismology, the frequency of earthquake events decreases as the magnitude of the shocks increases [33,34]. The magnitude is directly proportional to the logarithm of the rupture area. This makes the magnitude and its level of re-occurrence a vital tool to assess the physical damage level [35,36]. In seismology, the frequency–magnitude relation is quantified by the Gutenberg–Richter empirical formula:

$$lo{g}_{10}N=a-b{M}_L$$

(1)

in which $M_L$ is the Richter magnitude of the event, N is the number of events within a certain defined magnitude range, a and b are empirical constants [18,32,34,35,37].

Hence, in AE analysis the damage stages can also be quantified with the slope of the AE amplitude distribution [18,20]. This slope is the same as the b-value in the modified version of the Gutenberg–Richter empirical equation:

$$lo{g}_{10}N=a-b^{\prime }{A}_{dB}$$

(2)

in which N is the number of events within a certain defined magnitude range, a and $b^{\prime }(b^{\prime }=20\times b)$ are empirical constants, and $A_{dB}$ is the peak amplitude of the AE events in decibels (dB) [18,32,35].

In the early stage of material life, the dominance of micro-cracks will give rise to a higher b-value. As macro-cracks develop, more high-amplitude events start to occur, resulting in a lower b-value. Cox and Meredith [38] investigated micro-crack formation in rock materials and found that at earlier loading stages, b-value remains high. However, this value lowers when approaching the peak load, finally dropping to a minimum value at the stress drop or at the highest event rate. In addition, the b-value was found to be linearly related to the fractal dimension of a crack length, meaning that a lower b-value indicates a more localized fracture (less distributed damage) that results in a greater crack length. Colombo et al. [18] performed b-value analysis on AE data gathered from simply supported reinforced concrete beams subjected to four-point bending. They observed that the b-value correlates to the fracture in the concrete and the degree of damage localization. Carpinteri et al. [39], Sagar [40] and Colombo et al. [18] related the fracture stages with a specific b-value: a b-value of 1.5 or higher is typically related to distributed micro-cracking, and a b-value of 1 or lower indicates macro-cracking.

3. Analysis of Mechanical Behavior

In this section, the resulting tensile splitting strength ($f_{\mathrm{ct}}$) and effects of the sample size (diameter D) and the boundary condition (strip width B) are discussed. This is achieved by conducting Brazilian splitting tests on samples with varying strip widths and sizes.

Rocco et al. [24] reported that $f_{\mathrm{ct}}$ is not affected by one of these parameters individually (size or strip width), but by their ratio $\beta$ ($\beta =B/D$). However, EN 12390-6 [27] recommends a fixed strip width of 15 mm without referencing a specific sample size. In addition, the equation used in EN 12390-6 [27] for calculating the $f_{\mathrm{ct}}$ does not take into account the strip width. In the literature, various equations are found for calculating $f_{\mathrm{ct}}$ as a function of $\beta$. They are summarized in

Table 2. Expression to determine the tensile splitting strength ($f_{\mathrm{ct}}$) according to the literature.

EN 12390-6 [27]	$f_{\mathrm{ct}}=\frac{2P}{\pi tD}$	P is the applied load; t is the sample thickness.
Tang [41] at r = 0	$f_{\mathrm{ct}}=\frac{2P}{\pi tD}{(1-{\beta }^2)}^{\frac{3}{2}}$	r is the radial distance from the center of the disc.
Hondros [23] at r = 0	$f_{\mathrm{ct}}=\frac{2P}{\pi Dt\alpha }(sin2\alpha -\alpha )$	$\alpha$ is the angle made at the sample center by the load, $\alpha ={tan}^{-1}\left(2\beta \right)={tan}^{-1}\left(\frac{B}{R}\right)$; R is the sample radius.

.

The initial development of the equations shown in Table 2 began in the 1890s [42,43]. Hertz [44] developed an exact mathematical solution for the stress state of an elastic homogeneous isotropic disk under a diametrical compression line load. This solution was later modified by Hondros [23] for a load distributed over an arc of contact [22,43,45,46]. By superimposing the elastic solutions for opposite compression point loads and circumferential tension load on a circular disk, Timoshenko et al. [47] arrived at a similar conclusion as Hertz [44]. Tang [41] modified this solution to account for a load distributed over a strip. However, unlike Hondros [23], the distribution was not over an arc but instead on a horizontal projection plane of the arc (the distributed load on the strip is assumed to be only vertical). The equation used in EN 12390-6 [27] is derived similarly, but with the assumption that the load is applied as a line load ($\beta$ = 0). All of the equations in Table 2 are under the assumption that failure occurs at the point of maximum tensile stress (at the center of the disk).

Figure 3 shows the resulting $f_{\mathrm{ct}}$ as a function of $\beta$. The tensile splitting strength is calculated from the experimentally obtained maximum load (P) according to EN 12390-6 [27]. It is observed that the $f_{\mathrm{ct}}$ computed with EN 12390-6 [27] increases with the strip width. This is because, as the strip width increases, the failure load increases, and the maximum tensile stress decreases. However, the equation in EN 12390-6 [27] only considers the increment in load and not the contribution of $\beta$ in stress reduction. This results in an overestimation of $f_{\mathrm{ct}}$. However, as seen in Table 2, the equations proposed by both Tang [41] and Hondros [23] account for this stress reduction. Accordingly, a more accurate estimation of $f_{\mathrm{ct}}$ that allows it to be considered as a material property can be obtained with these two equations. In Figure 4, a comparison is made between the different equations by calculating the ratio F between the average tensile splitting strength ($f_{\mathrm{ct}}$) for varying $\beta$ values. An F ratio of one hence indicates no effect of $\beta$ on the calculated $f_{\mathrm{ct}}$. As seen in Figure 4, the change in $f_{\mathrm{ct}}$ with $\beta$ raised to 25% for the medium and 75% for the small samples for the EN 12390-6 [27] equation; however, this change is significantly reduced to 11% and 9% (close to one) for the medium and the small samples, respectively, when applying Hondros [23] and Tang [41] equations.

When investigating the role of the strip width in the Brazilian splitting test of brittle materials, Rocco et al. [24] have found the strip width to affect not only the $f_{\mathrm{ct}}$ but also the extent of the size effect. The research concluded that as this width approaches zero (from $\beta =0.16$ to $\beta =0.04$), the size effect disappears, and $f_{\mathrm{ct}}$ will become an intrinsic material property. However, as described in Section 2.2, distributing the applied load over a small width is essential to avoid local compression failure [48]. In addition, this width can also affect the failure initiation [22]. Even though the effect of the strip width on $f_{\mathrm{ct}}$ can be reduced by using appropriate equations, its effect on fracture behavior is still not clear. Hence, to isolate and investigate the size effect alone, a test series with constant $\beta$ value on varying sample sizes was designed (see Figure 5a).

Figure 5b presents the resulting $f_{\mathrm{ct}}$ computed using Tang’s equation [41]. It can be inferred that there is no significant difference between small and medium samples. This is due to the high standard deviation in the smaller samples, which can be attributed to the strip thickness not being adopted to the sample size and/or the lack of accuracy in symmetrical strip attachment. However, the larger samples had the lowest $f_{\mathrm{ct}}$, that is 14% lower than the the small samples $f_{\mathrm{ct}}$. More importantly, the larger samples also showed a much more brittle failure during testing. This is because, for a large sample, the rate of strain energy growth with stress is high. At the same stress level, as the sample size increases, the stored strain energy becomes higher. Hence, for the larger samples, enough fracture energy can develop at a relatively low stress level.

4. AE Analysis: Results and Discussion

4.1. AE Amplitude and Hit Count Analysis

The cumulative AE hit count for a representative large, medium, and small sample are shown in Figure 6b,d,f respectively. The cumulative AE hit counts are presented for sensors 1 and 3 (referred to as channels Ch-1 and Ch-3), which are positioned on the side of the sample, towards the front and back of the sample, respectively (see also Figure 2b for the sensor locations). These channels are chosen to investigate a possible unsymmetrical nature of the crack propagation (crack initiates on one side and propagates towards the other side of the sample). As there is a small difference observed between the results of both sensors, it can be assumed that there is indeed some asymmetric effect. Hence, the hit-count is captured using these opposite-sided sensors, whichever side the crack initiates.

Generally, all samples show a steep rise in AE hit count when reaching the peak load. For the larger sample, there is negligible AE activity until the load approaches the peak load (see Figure 6b). On the other hand, for the medium and the small samples, there are relatively more AE hits occurring early in the test (see Figure 6d,f). From Figure 6a,c,e, it can be seen that these AE hits are low-amplitude hits that can probably be attributed to micro-cracking. Conversely, the AE hits occurring near the peak load have a higher amplitude. There are more of these high-amplitude AE events at the peak load for the larger samples than for the other two sizes.

These observations support the theory on size effect due to variations in the release of strain energy. This variation in damage progress can be best explained using the axial bars shown in Figure 7a. The three bars are all made from the same material but have different lengths and cross-sectional areas. When subjected to an axial load, the peak stress and the ductility evolves with sample size (see Figure 7b). Ductility decreases with sample size because the stiffness decreases with length. Hence in the elastic phase, the longer sample deforms more (more elastic energy is stored). The sudden release of this stored energy will cause unstable and more brittle failure. Meanwhile, for the smaller samples, there is no high elastic energy build-up. Instead, there are micro-damages that progressively dampen the energy. Eventually, these micro-damages coalesce to induce failure.

Hence, the early occurrence of low-amplitude AE events in the smaller sample can be linked to the sample size. As the sample size decreases, the sample cannot accommodate large elastic deformation. Instead, it starts to form micro-cracks early on in the test (see Figure 6e). The lack of early AE activity on the larger samples indicates the occurrence of more elastic deformation (storage of energy) and less micro-damage (see Figure 6a). The sudden rise of high-amplitude AE in the larger sample indicates a high release of energy at once (at peak load) for the larger samples. Hence, in quasi-brittle material, the failure becomes more brittle as the sample size increases.

Similar results were observed by Alam et al. [10] when investigating the evolution of AE energy on three different sizes of concrete beams. It was found that there was little to no AE activity for the larger samples for the majority of the test duration. However, there was a growth of high-energy AE activity near failure. On the other hand, there was an early occurrence of AE activities related to micro-cracking for the medium and the small samples.

4.2. Localization of AE Sources

The accuracy of the AE source localization is initially investigated using pencil lead break (PLB) tests. These tests were part of the calibration test conducted on all samples. Six to eight locations of the PLBs are used on each sample, and the average localization error for the x-, y-, and z-axis of the larger, medium, and small samples are (2.59 mm, 4.32 mm, 4.2 mm), (1.74 mm, 1.77 mm, 2.73 mm), and (0.71 mm, 1.52 mm, 1.13 mm) respectively. Also note that for the fracture process zone, AE sources are located with respect to the X-direction, which has the lowest error for all sample sizes.

In Figure 8, the located AE sources during the Brazilian splitting tests are shown in three directions for a representative large sample. The color of the located AE events indicates the load ratio at which the event was detected, and the size of the dots is relative to the AE amplitude.

Initially, AE sources are located at the bottom contact surface, occurring at around 20% of the peak load (see Figure 8). Despite using packing strips, local crushing due to concentrated compressive stresses and a non-fully smooth sample surface occurred. This causes early friction events that are observed at the start of the test (see Figure 6a). As the load reaches the peak value, clusters of AE sources are detected along the vertical center line of the sample (at X = 53 mm) that goes along the direction of the loading (from Z = −53 to Z = 53 mm).

The unsymmetrical nature of crack propagation is captured and is shown in Figure 8. This is observed in the top view; the splitting crack initiated at one end of the sample (at Y = 100 mm) and propagated to the other side (Y = 0). This type of crack propagation has long been identified, and for this reason, most researchers working with splitting tests monitor crack propagation from both directions [49,50,51,52]. Finck et al. [53] also observed similar crack propagation patterns using AE localization plot on a splitting test of concrete cubes. This can be due to the compression surfaces not being fully parallel and/or one end of the sample being slightly weaker due to the casting process. Casting mold imperfections that make the sample slightly conical (one side diameter larger than the other) can also cause the crack to initiate on the side with a larger diameter. Overall, the 3D located AE sources clearly represent the cracking process during the splitting tests.

The localization plots for three representative, different-sized samples are presented for subsequent loading steps in Figure 9. For the larger sample, it can be observed that there is a lack of AE activity in the post-peak period. This is because, at peak load, there is a rapid crack propagation that immediately splits the sample apart. During the test, the larger sample failed in a very brittle (almost “explosive”) manner, making a loud noise. On the other hand, the smaller sample failure was hardly noticeable, and only a small crack was formed, allowing for post-peak crack bridging and friction events. In addition, there are hardly any low-amplitude AE events located for the larger samples. This was attributed to the lack of micro-cracking before the development and propagation of the main macro-crack.

4.3. AE-Based Fracture Process Zone (FPZ) Analysis

Several researchers have proposed a method of quantifying the size of FPZ based on the AE technique. According to Alam et al. [9], Otsuka and Date [13] and Keerthana and Kishen [15], the energy in the FPZ accounts for 95% of total AE energy. The histogram of located AE events along the direction of interest is used to measure the size of the FPZ. For a splitting crack, this direction of interest is usually perpendicular to the direction of crack propagation (x-axis). The size of the FPZ is measured at 20% of the maximum magnitude of the histogram. Hadjab et al. [54] defined the FPZ as the region around the macro-crack where high-amplitude AE events are located. Ohno [12] measured the width of high-energy AE event clusters.

Here, the width of an FPZ is computed using similar methods to Otsuka and Date [13]. This method had previously also been used by Deresse et al. [55] to compare the size of the AE-based FPZ for mortars loaded in monotonic and fatigue loads in a Brazilian splitting test. In this method, the circular surface of the sample is divided into 4 mm (the estimated localization error) width of regions. AE events located in each of these regions are counted and connected to give the histogram of the AE events (N) line; see Figure 10. Here, it is assumed that only the AE events that occur after 20% and up to 100% of peak load cause the final fracture, and thus, these are taken for the FPZ analysis. The method and the computed FPZ width for three representative, different-sized samples are shown in Figure 10. As can be seen from Figure 10a–c, the width of FPZ increases with sample size. It was also noticed that most of the located events occur between 80% to 100% of the peak load for the larger sample.

The width of the FPZ for all of the samples tested in this research is summarized in Figure 10d. This plot shows that FPZ width increases with sample size. However, similar to Otsuka and Date [13], the rate of increment was slower than the sample size enlargement. The dependence of FPZ width on sample size is mainly because at macro-scale, strain energy is size-dependent. The stored strain energy increases with sample size.

Fracture initiates when this stored energy reaches the required fracture energy level at the region where stress concentration occurs (at existing micro-cracks, voids, or pores). As stated before, the FPZ is a region where the fracture energy is released. Hence, for the larger samples, the release of this enormous stored energy at the location of stress concentration results in the broader FPZ. This observation is also supported by the experiments conducted by Ohno [12]. The investigation was on a three-point bending test of notched beams. The fracture energy was found to be directly related to the size of the FPZ.

4.4. b-Value Analysis

The method of b-value analysis was introduced in Section 2.4.2. In the current section, b-value analysis is used to investigate the fracture process in different sample sizes to link this with previous observations. In this work, the b-value is calculated on a window of 100 AE hits moving at a step of 25 hits (with 75% overlap). In each window, the hits are grouped into 5 dB sets. This is similar to the approach used by Lei [56] for a study of fracture in granite.

4.4.1. Role of Sensor–Source Distance in the b-Value Analysis

In the Brazilian splitting test, cracking may start at one side of the sample and quickly propagate to the other side (y-direction). Hence, the b-value of AE hits captured by sensors located close to the fracturing region and those positioned on the other end are compared in this section to verify the effect of source–sensor distance. When approaching the peak load, micro-cracks will localize and coalesce to form macro-cracks. These macro-cracks typically start to form between 70 to 95% of the peak load. This is shown on the localization plot in Figure 11.

The resulting b-value evolution for AE hits captured by separate channels is presented in Figure 12. From Figure 12a,b it can be seen that, for hits coming from channels closer to the fracturing region (Ch-1 and Ch-4), the b-value drops below 1 before reaching the peak load. This drop occurred when the load was between 70 to 95% of the peak load, indicating the macro-crack formation observed at the side of channels 1 and 4 in the localization plot. This clearly shows the capability of b-value analysis in identifying the fracture progression.

On the other hand, this b-value drop was not captured less distinctively for hits coming from the sensors positioned at the other end of the sample, further from the fracturing region. This is shown in Figure 12c,d for channels 2 and 3, respectively. A drop in b-value requires the occurrence of high-amplitude hits. Due to the attenuation of signals when traveling to reach these sensors, this drop is not clearly observed. Hence, quantifying fracture behavior using AE b-value analysis requires signals that have not been subjected to a significant attenuation. The sensitivity of b-value analysis to AE amplitude attenuation has been observed by Weiss [57], on AE data acquired during a compression creep test on ice. Thus, designing an AE sensor layout requires a preliminary investigation on the possible location of fracture (initial damaged zone, a region with high-stress concentration or notches).

In the current work, to avoid the impact of attenuation, only the sensor close to the fracturing region is used for further b-value analysis. However, the difference in sample size can also play a role in the variation of AE attenuation. This is because compared to the smallest sample, the AE signals within the larger sample can travel 35 to 50 mm more. The attenuation coefficient for the material was determined using the PLB test and was found to be 0.1 dB/mm. Thus, to avoid the impact of attenuation in the b-value analysis, only the AE hits captured by the sensors closer to the fracture region were used. Hence, the signals in the larger sample can attenuate by 3.5 to 5 dB more than the smallest sample. This can be assumed as sufficiently lower than the possible difference in AE due to size variation. As the sample size increases, the fracture energy is also expected to increase. Hence, AE of higher amplitudes are also expected. There is a tradeoff between attenuation and this increment in amplitude. Hence, this analysis is within the assumption that, for this sample size and the material attenuation coefficient, there is more increment in amplitude than the possible attenuation. This assumption is corroborated with the fact that in the larger samples, a similar amount of AE signals with higher amplitudes were still detected compared to the smaller samples, as exemplified in Figure 13.

4.4.2. Fracture Analysis Using b-Value

This section investigates the fracture process near the peak load using the b-value analysis. For the reasons discussed in the previous section, only the AE hits captured by the sensors closer to the fracture region are used for this analysis. Hereby, the location of the fracture region is determined by 3D localization of AE sources (see Section 4.2).

The role of sample size in the fracture progress is investigated using the evolution of the b-value in different sample sizes. Two representative samples are shown for the large and small sample sizes in Figure 14. Generally, it is observed that the b-value indeed decreases when approaching the peak load. As discussed in Section 2.4.2, micro-cracking is related to a b-value of 1.5 or higher, whereas macro-cracking is assumed to occur when the b-value is 1 or lower. The limits of b-values are also indicated in Figure 14. Figure 14a,b show that for the larger samples, the b-value is below or close to 1 even in the pre-peak period. This would indicate the formation of macro-cracks. However, the amount of AE events in this period is too small for the large samples to reliably determine a b-value (see also Figure 6b). On the other hand, in this period the b-value for the smaller samples remains above 1 (see Figure 14c,d). This implies that for the smaller sample, micro-cracking dominates in the pre-peak period.

To smoothen the b-value curves, a moving mean with a five data point moving window is indicated in Figure 14. In general, from the moving mean, it can be seen that the decreasing tail of the b-value curve starts at a relatively earlier time for the larger samples, and the drop almost coincides with the peak load. However, for the smaller samples, the major decrease in the b-value occurs in the post-peak period. This indicates the premature fracture that occurs in larger samples. This premature failure also explains why larger samples would fail before reaching the intrinsic material strength.

For the larger samples, the b-value rises before it falls, which indicates the micro-cracking that develops and extends the FPZ (due to the release of fracture energy) just before the main macro-crack develops. For the smaller samples, the less gradual decrease in b-value indicates the formation of a macro-crack due to the coalescence of micro-cracks without a significant energy release during the macro-crack growth.

5. Conclusions

The effect of sample size and boundary condition on strength and fracture progress of cementitious mortar samples was investigated in Brazilian splitting tests. Size effects were analyzed and explained by means of AE analysis, and the following conclusions were drawn:

1.

Tensile splitting strength computed with EN 12390-6 [27] was found to be dependent on the load-bearing strip width. This tensile splitting strength changed by 25% and 72% for the medium and the small samples, respectively, as strip width varied. The increase in the strip width caused the failure load to increase and the maximum stress to decrease. EN 12390-6 [27] disregards the role of this strip in stress reduction, resulting in an overestimated tensile splitting strength. However, when using equations that properly account for the boundary condition, the variation of tensile splitting strength with strip width is reduced to 11% and 9% for the medium and the small samples, respectively.

2.

The size effect was investigated on samples with varying sizes but similar boundary conditions (strip width). It was observed that, similar to Rocco et al. [24] and Bažant Zdeněk et al. [8], the larger sample gave the lowest tensile splitting strength, giving 14% lower tensile splitting strength than the small sample. The manner of fracture energy dissipation was identified as the cause for this effect. The smaller sample had the highest standard deviation. Hence, there was no significant difference between small and medium samples. In addition, the larger samples failed explosively (by making a loud noise), while the failure in the smaller samples was hardly noticeable, indicating the failure becoming more brittle as the sample size increases.

3.

The AE hit count and the AE amplitude evolution for samples with different sizes were investigated. It was found that for larger samples, there was a lack of AE activity before approaching the peak load. In addition, near the peak load, there were more high-amplitude AE events for the larger samples compared to the other samples. This shows that as sample size increases, less distributed micro-cracks form before macro-crack propagation, and the macro-fracture is more brittle.

4.

The AE source location plots at subsequent loading stages indicated the unsymmetric nature of crack propagation in the Brazilian splitting test. It also showed that the amplitude of the localized events increases with sample size. In addition, it was observed that there are hardly any located AE events in the post-peak period for the larger samples. This was explained by the more brittle failure as the sample size increases.

5.

The width of the fracture process zone (FPZ) was calculated from located AE events and was found to correlate to the sample size. The size of the FPZ was found to increase with sample size, which is in agreement with the literature. In addition, it was noticed that the growth rate in FPZ was slower than the sample size increase rate. The wider FPZ indicates the higher amount of fracturing energy consumed by the larger sample.

6.

Fracture analysis through b-value analysis was affected by the sensor–source distance. It was found that b-value analysis of AE hits detected by sensors close to the cracking region resulted in a better interpretation of the fracturing phenomena. This was due to signal amplitude attenuation. The b-value analysis revealed that for the larger samples, there is a short period of pre-peak micro-cracking and sudden macro-cracking at peak load. On the other hand, the macro-crack formation occurs at the post-peak period for the smaller samples. Again, this is an indication of the ductile to brittle failure transition that occurs when sample size increases.

The research validated the impact of boundary conditions on the tensile splitting strength and was able to explain the occurrence and causes of size effect phenomena through advanced AE analysis. In future work, this research will be extended by investigating the impact of sample size on the ductility and tensile splitting strength of brittle materials in fatigue loading. In fatigue loading, micro-fracture growth and coalescence occur progressively under loads that remain well below the material strength. As many real-life structures are subjected to fatigue loading conditions, it should be investigated whether current conclusions can be extended and how the variation of energy release rate with sample size manifests itself in fatigue loading.