# Damage Evolution Analysis in a \"Spaghetti\" Bridge Model Using the Acoustic Emission Technique
## 2 Theoretical Foundations
### Acoustic Emission Technique  
When a mechanical system is excited by an external source or undergoes changes in its internal structure, it presents local instabilities (events) that propagate as mechanical waves. These Acoustic Emission (AE) signals are usually accelerations with frequencies ranging from \(10^{4}\) to \(10^{7}\) Hz [13]. These signals are detected by accelerometers that are mounted on the structure's surface, as depicted in Figure 1, where the excitation is usually a force or prescribed displacement in time. Figure 2 illustrates a typical AE signal. Several parameters can be extracted from these signals [13], but the ones of interest here are: maximum Amplitude (\(A_{p}\)), signal  
Figure 1: Basic setup for acquiring AE signals, with two AE sensors. Although both signals refer to the same event, they differ according to their positions along the structure. threshold (\(A_{th}\)), initial time (\(t_{i}\)), and final time (\(t_{f}\)), where both times are functions of the fixed threshold level. The Rise Time (\(R_{T}\)) is defined as \(R_{T}=t_{p}-t_{i}\), i.e., the difference between the instant of maximum amplitude (\(t_{p}\)) and the moment \(t_{i}\) when amplitude rise from threshold levels was first detected. From these data, the Rise Angle (\(R_{A}\)) is given by \(R_{A}=R_{T}/A_{p}\). Finally, the area under the signal is also of interest, because it bears a direct relation with the acoustic energy emitted during the event, as explained in detail in [14]. Besides having their own meaning regarding both spatial and temporal distributions of the events in AE tests, these parameters can also be combined, generating reliable indicators regarding the possibility of system collapse. Such indicators are:  
1. **Relation between the number of events \(N\) and the signal amplitude \(A\):** this relation has been long used in seismological applications, as illustrated by the classic Gutenberg & Ritcher law [4], which is of universal nature and it does not depend on the scale of the distribution [11; 15; 16]: \[N(\geq A)\propto A^{-b},\] (1)  
where \(N\) is the cumulative number of signals and \(A\) is the signal amplitude. The physical meaning is discussed in [17; 18; 19]. It is hypothetically related, according to the expression \(\mathcal{D}=2b\), to the fractal dimension \(\mathcal{D}\) of the material domain from which the signals that are generated by cracks are emitted. When the damage process begins within a structure, signals are emitted from a micro-cracks roughly evenly distributed in the material volume, i.e., \(\mathcal{D}=3\) and \(b=1.5\). Thus, according to Equation (1), most events produce small-amplitude signals. As damage advances, localization effects take place, and the signals are preferentially emitted from micro-cracks that distribute on preferential surfaces, which results in macro-crack nucleation. Therefore, in this last phase, the values for \(\mathcal{D}\) and \(b\) become 2 and 1, respectively, and the application of Equation (1) yields an increase in the number of large-amplitude events. Thus, the evaluation of \(b\) and how it changes with time allows for one to keep track of damage processes. The procedure for computing \(b\) is schematically described in Figure 3a. The amplitudes due to each signals are collected and organized in a histogram. Subsequently, a bi-log diagram is built to illustrate the cumulative number \(N\) of signals with amplitude \(\geq A\). Finally, \(b\) is the angular coefficient of the fitting line. For a more detailed discussion about this computing procedure, see [20]. 2. **Relation between \(N\) and the signal energy emission \(E_{s}\):** the energy that is carried in the signal is also related to \(N\) in a form similar to that of the amplitude \(A\), using \(\epsilon\) a fitting coefficient analogous to \(b\): \[N(\geq E_{s})\propto{E_{s}}^{-\epsilon}.\] (2)  
Figure 2: Typical AE signal with its parameters of interest. The calculation of \(\epsilon\) is analogous to the one described for \(b\) in case (a). It is also described in Figure 3a.