# Damage Evolution Analysis in a \"Spaghetti\" Bridge Model Using the Acoustic Emission Technique
## 2 Theoretical Foundations
### Acoustic Emission Technique
Because the emitted AE signal energy is proportional to the squared maximum amplitude (\(E_{s}\propto A^{2}\)), it is apparent that the expected interval for \(b\) [1.0, 1.5] translates to [0.5, 0.75] for \(\epsilon\), as discussed in [2] and shown by numerical simulation in [21]. One can also compute the energy emission from the area under the signal envelope, as also indicated in Figure 3b. This approach, referred to here as the RILEM method, was proposed in [13; 22]. Finally, it is also possible to calculate energy emission from the Root Mean Square of the signal. (c) **Relation between \(N\) and the characteristic signal frequency \(f_{s}\):** this parameter was introduced by the same research team involved in this work as a reliable indicator for avalanches during a damage process [23]. This newly introduced coefficient c is also obtained by analogous means to those that are given for Equations (1) and (2), but focusing on the frequency distribution of the AE signals, i.e.,:  
\[N(\geq f_{s})\propto{f_{s}}^{-c} \tag{3}\]  
where \(N\) is the cumulative number of signals with frequencies greater or equal to \(f_{s}\). The value of \(c\) can be similarly calculated to that used for \(b\), as indicated in Figure 3c: it is the slope line of the signals distribution during the damage process as a function of the frequency that characterizes the signals. As in the case of the \(b\)-value for amplitudes, the \(c\)-value indicates changes in the damage process and the imminence of collapse by keeping track of the acquired AE signals' frequencies. For instance, if the number of events with lower characteristic frequencies increases as compared to the higher ones, a change in the damage process has occurred. Still, regarding Figure 3c, note that there are two ways to calculate the characteristic frequency \(f_{s}\) of the AE signal. The first is taking the ratio between the number of cycles np and the signal time interval (\(t_{f}-t_{i}\)), as proposed by the RILEM [22] and referred to here as RILEM frequency. An alternative definition is by determining  
Figure 3: Precursors from AE tests, (**a**) obtaining \(b\) from Equation (1); (**b**) calculation of \(\epsilon\) from Equation (2); and, (**c**) determining \(c\) from Equation (3). the spectral distribution of the AE signal and taking the frequency with the highest peak, i.e., the FFT method. 4. **Frequency fluctuations during the damage process:** a well-known measure of energy fluctuations in AE signals relies on their dependence on signal frequency, as described by the spectral density function (SDF), as mentioned by [24]. The first observations regarding this dependence are reported by [25], who coined the term \(1/f\)-noise or Flicker Noise when studying the noise effects in electronic circuits. According to [26], the dependence of noise energy distribution with respect to frequency is given by: \[E(f)=a\ 1/f^{\gamma},\] (4) where \(E(f)\) is the energy emission, \(f\) is the signal frequency, while \(\gamma\) and \(a\) are the scalar fitting coefficients. Taking the logarithm of both sides in Equation (4), the best fitting line leads to a linear law, where gamma is the angular coefficient. Its calculation is schematically described in Figure 4.  
This type of fluctuation is widely observed in many different scientific fields, such as geology [27], finances [28], bioengineering [29], and even music [30,31], as observed in [32] and many other works. Moreover, the distributions described in Equation (4) is also observed to hold for frequencies ranging from fractions of hertz (in seismology) to gigahertz (microelectronics), which clearly illustrates the fractal character of this distribution and the phenomena to which it applies [32]. In the specific case of Acoustic Emission applications, the study of damage in historic buildings by [33] is an interesting example, where the exponent changes in the imminence of (either local or global) instabilities that are associated with structural collapse.