# Nondestructive Monitoring Techniques for Crack Detection and Localization in RC Elements
## 7 AE Measurements
### The \(b\)-Value Analysis  
It is reasonable that as the load increases, larger-scale fracture events occur giving rise to AE signals of larger amplitude. The signal amplitudes are converted in a magnitude scale by using the equation \(M=\log_{10}\left(V/V_{0}\right)\), where \(V\) is the peak of voltage signal expressed in microvolts, and \(V_{0}\) = 1 \(\upmu V\) is the reference voltage (conversion to decibels is possible through \(M=A_{\text{dB}}\)20). The signal magnitudes are studied by their cumulative distribution, using the \(b\)-value analysis. For a population of \(N\) events, the \(b\)-value is calculated as:  
\[\log_{10}N(M)=a-bM, \tag{2}\]  
where \(N(M)\) is the number of signals with magnitude higher than \(M\), and \(a\) and \(b\) are fitting constants. Figure 5: Step load (blue line) and cumulated AE signals time history (red line). Dashed lines correspond to the 3 rd, 6 th and 8 th load stages. Figure 6: \(Cont.\)  
The \(b\)-value expresses the absolute gradient of the cumulative magnitude distribution as illustrated in Figure 7. As the fracture process becomes more intense, the percentage of high-amplitude events increases relative to the low-amplitude ones in the total population, resulting in drops of the \(b\)-value [25]. Figure 6: (**a**) AE features relationships and distributions of received signals by sensors AE0 and AE1. The mean values of distributions represented by histograms from left to right are: \(<\)\(A\)\(>\) = 3.3 mV, \(<\)\(f\)\(>\) = 17 kHz, and \(<\)\(\Delta\)\(t\)\(>\) = 0.8 ms. (**b**) AE features relationships and distributions of received signals by sensors AE4 and AE5. The mean values of distributions represented by histograms from left to right are: \(<\)\(A\)\(>\) = 4.5 mV, \(<\)\(f\)\(>\) = 31 kHz, and \(<\)\(\Delta\)\(t\)\(>\) = 0.4 ms. Taking into account the number \(N\) of recent events to calculate every single \(b\)-value, possible decreasing trends can be identified and associated with damage severity and change in structural performance state, especially during laboratory loading tests where a single minimum is observed just prior to specimen failure. The recommended value number of events for computation (see e.g., [24]) is typically \(N=100\). Smaller values would yield too strong oscillations, masking the actual trend. Instead, using too large \(N\), any small occurrence producing a small number of AE signals would be undermined in the average of the large population. Here, four partitioning criteria are followed to calculate the evolving \(b\)-values during the test, namely using disjoint subsets of \(N=100\) and \(150\) events (Figure 8 (left)), or overlapping subsets of \(N=100\) events with step \(25\) and a loading stage-based partition (Figure 8 (right)) [25; 26]. The most important parameters to be considered in a \(b\)-value graph are the general trend and the minima of the curve. Comparing all the time series, nothing more than an oscillating trend can be identified, with \(b\)-values mainly ranging between \(1.2\) and \(1.6\) (with an isolated minimum at \(1.1\) reported in two cases) as descriptive of minor to moderate damage levels reached during the test. As herein illustrated--and widely recognized--the \(b\)-value, although being sensitive to damage, does not necessarily decrease monotonically. That is due to the presence of both healthy material encountered during crack propagation and steel reinforcements that limit crack growth in concrete matrix. Namely, fracture propagates during steel deformation or concrete sliding with respect to steel. Fracture generates signals related to early damage, which are mixed with those of the already heavily damaged part, as stated in [27; 28; 29; 30]. Figure 8: \(b\)-value time histories with dashed lines referring to the load stages of Figure 5. Using disjoint subsets of \(N=100\) (red dots) and \(N=150\) signals (cyan dots) **(left)**. \(lb\)-value with overlapping subsets of \(N=100\) events with step \(25\) (green dots) and \(b\)-value by a stage-based partition (blue dots) **(right)**.