## 3 Finite Element Simulation of Acoustic Emission (AE) Signal
### Sensor Effect  
The sensors are assumed to be solely sensitive to displacement normal to the surface of the sample. The quantitative analysis of the AE data involves understanding of the sensors response in reception [48; 49; 50; 51; 52]. The reciprocity method is used for sensor's calibration [50; 51; 52]. The determination of the sensitivity needs three transducers alternatively working as transmitters and receivers. The reception  
Figure 3: Scheme of the numerical specimen with the boundary conditions. wave sensitivity is measured on a steel block [51; 52] for Rayleigh waves. Figure 4a shows the Rayleigh wave reception sensitivity of a \(\upmu\)80 sensor, nano30, R15 and picoHF sensor obtained with the reciprocity method. For the WD sensor, [PERSON] et al. [39] obtained the calibration curve used in this study on a composite material by a modelling approach. WD is wideband transducers with a bandwidth of 100-1000 kHz and a resonant frequency at 650 kHz. R15 is a general-purpose 150 kHz resonant sensor, nano30 with nominal operating frequency of 200 kHz-400 kHz. In order to simulate sensors, axial wave velocities at nodal locations \(\
u_{z}\) are calculated on the sensor surface (Figure 4b). The normal vibration velocity at each point of its surface is not uniform. An average \(\
u_{Z}\) axial wave velocity is numerically calculated by Equation (3):  
\[V_{z}(t)\ =\ \frac{1}{A_{Sensor}}\iint k.v_{z}(x,y,t)dxdy \tag{3}\]  
\(A_{sensor}\) represents the area of the sensor. The radius for the sensors are equal to R\({}_{nano30}\) = 3.2 mm, R\({}_{picoHF}\) = 2.3 mm and R\({}_{R15}\) = 7.5 mm. In order to take into account the aperture effect, we introduce the k parameter, which is a real number between 0 and 1. It represents the sensitivity of a point at the distance r from the centre of the sensor surface. Figure 5 represents the evolution of the parameter k. This evolution is deduced from the work of [PERSON] [42]. The aperture function depends not only on the geometry of the sensor and the spatial evolutions of the sensitivity of sensor surface but also on the frequency. In this study, we do not take into account the effect of the frequency. Figure 4: Response in reception (**a**) on steel block for the sensor nano30, PicoHF, R15 alpha and micro80 sensors obtained with the reciprocity method; (**b**) visualisation of the surface of the nano30 and picoHF sensors. For the WD sensors, we use the calculation proposed by [PERSON] et al. [39], following Equation (4). The signal is obtained from the average at the surface of the disc and ring sensor elements:  
\[V_{z}(t)\ =\ \frac{1}{A_{Disc}}\iint v_{z}(x,y,t)dxdy+\frac{1}{A_{ring}}\iint v_{z}(x, y,t)dxdy \tag{4}\]  
where \(A_{Disc}\) is the area of the disc and \(A_{ring}\) is the area of the second element in the sensor, the ring (R\({}_{\text{Disc}}\) = 4.1 mm and the thickness of the ring is equal to 1.6 mm). This transfer function is added in the post-processing phase in the Fourier domain, following Equation (5):  
\[V_{surface}(\omega)\times S_{sensor}(\omega)\ =\ V_{sensor}(\omega) \tag{5}\]  
where \(V_{sensor}\) is the velocity calculated with considering sensor, \(V_{surface}\) (\(\omega\)) represents the mean of out of plane velocity on the surface and \(S_{sensor}\) (\(\omega\)) the sensitivity function. In this study, the coupling medium is not taken into account. All numerical velocity signals are filtered with a Butterworth 20 kHz to 1200 kHz band pass filter. As it has been pointed out in literature [39; 52; 53], the propagation medium used for calibration will not only disturb the signals due to modifications in the waves velocity but will also modify the sensor response due to change in acoustic impedance. In this study, we do not take into account such change of the sensor response. As reported by [42; 53; 54], the couplings show slight influence of the sensors reception sensitivity, but this is negligible compared to the other contributions.