PSI - Issue 52

Donato Perfetto et al. / Procedia Structural Integrity 52 (2024) 418–423 Donato Perfetto / Structural Integrity Procedia 00 (2019) 000–000

421

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Table 2. Details different FE approaches. Approach Component element type*

Element average length (mm) # of elements

# of nodes 7212005

# of d.o.f. 21636015

Plate PZTs Plate PZTs

SC8R

0.5 0.3 0.5 0.3

1440000

3D

C3D8R

2808

5828

17484

S4R

573588

1442401

8654406

2D

C3D8R

2808

5828

17484

*from Abaqus® CAE elements library.

Concerning the wave propagation, the excitation source was modelled as a 5-cycle tone burst Hanning windowed signal with a central frequency of 300 kHz and an amplitude of 1V. Starting by the piezoelectric properties of the PZTs, the radial displacement field equivalent to the actuation signal was obtained through a MATLAB code and it was applied on the nodes of the actuator upper edge. The average length of the elements was chosen to ensure a discretization of 20 nodes per wavelength (NPW) of the implemented signal. This value is above the minimum threshold, equal to 8, that ensures a satisfactory evaluation of GW propagation [15]. Since the explicit formulation scheme is conditionally stable, a time step equals to 8.3953×10 −8 was selected, taking into account the wave speed and the smallest element length. A node-to-surface contact formulation was then used at the "tied" interfaces to simulate the adhesive layer between sensors and plate in order to assure the contact between the two parts. Finally, the four corners of the plate were pinned. A total time for the analysis equals to 210 µs was set, large enough to completely inspect the wave behaviour. 3. Results For the analysis of the results, the recorded signals at the PZT 2 location for each modelling technique were compared in terms of time of flight, calculated group velocity, and percentage of cross correlation. The normalized signals at the carrier frequency of 300 kHz are plotted in Figure 4. As visible at first glance, the signals recorded in the 2D-Shell analyses do not match the trend of the 3D-Shell ones: a slight mismatch in phase can be easily distinguished for all the three investigated offset representations (refer to Figure 2). However, to carry out a preliminary study about the possibility to model the geometrical discontinuity using the 2D-Shell formulation (which is the aim of this study), the aforementioned signal features were further investigated, and herein detailed (see also Table 3): • time of flight (ToF): it is the time necessary for an emitted wave packet to travel the distance between the two transducers. It was calculated by an algorithm developed in Matlab® and used for the signals post-processing; • group velocity: it is the velocity with which the overall envelope shape of the wave's amplitudes propagates through space. As for the ToF. the group velocity was automatically calculated by the algorithm given the schematic geometry of the plate, the positions of the PZT, and the extracted ToF values of the 0-order symmetric mode (S0). • cross correlation: it is a measure of similarity between the recorded signal and the one obtained in the 3D Shell approach, expressed as a percentage. It was evaluated through Matlab®. By analysing these features, the 2D-Shell technique based on the modelling of the two different thickness regions sharing the same middle plane, represented in Figure 2b, appears to be the one that best matches the 3D-Shell approach. In this case, indeed, the highest cross correlation value and the closest group velocity with respect of the reference modelling approach were obtained. Furthermore, the 3D-Shell analysis, performed with an Intel® Xeon ® Gold 6248R CPU with a total of 24 cores and 48 threads, took about 7 hours, whereas the 2D-Shell ones required a computational cost of about just 3 hours leading to a time saving of about 57%, further justifying the need to properly employ the 2D formulation for the modelling of the UGW-based SHM system on more complex components.