PSI - Issue 42

Diogo Montalvão et al. / Procedia Structural Integrity 42 (2022) 1215–1222 Montalvão, Hekim, Costa, Reis, Freitas / Structural Integrity Procedia 00 (2019) 000 – 000

1219

5

{ = (1+ ) (1−2 ) [(1 − ) + ( + )] = (1+ ) (1−2 ) [(1 − ) + ( + )] = (1+ ) (1−2 ) [(1 − ) + ( + )] { = 1 [ − ( + )] = 1 [ − ( + )] = 1 [ − ( + )]

(8)

(9)

Cruciform test specimens are moderately thin throughout, with a considerably thinner central area (Fig. 4). Maximum stresses and strains will occur at the centre, where will have a situation of plane stress with ≃ and ≠ . In other words, stresses develop in the plane only, but strains are developing in all three cartesian directions, so it is not difficult to understand that does not necessarily has to be equal to .

Fig. 4. Cruciform test specimen’s general drawing (Montalvão and Wren, 2017) (symmetric equibiaxial case for illustration purposes only).

3. Methodology To better understand how the 4 biaxiality ratios , , and Δ change and compare with each another, Hooke’s law equations ( 8) and (9) are used, as well as Finite Element Analysis (FEA) using ANSYS Workbench 2021 R2. Initially, specimens are modelled in free-free configuration and a simulation of the type Modal is run. The objective is to slightly adjust dimensions so that specimens are tuned to have either a CT or a TT mode shape at 20±0.5 kHz, which is the operational range available at the machine’s at both the University of Lisbon in Portugal and the ADDISONIC lab at Bournemouth University in the UK. Similar simulations are then run but with the specimen assembled at the machine, as depicted in Fig. 2, to have a better representation of reality and further refine the specimens’ design. Hex dominant elements were used with attention to distortion of the elements to produce results with an acceptable level of accuracy. This was determined after mesh convergence was achieved when refining both the global mesh and local mesh at the centre. The relation between displacements at the tips of the specimen and stress and strain at the centre, which is needed in the case of experimental testing as stress and strain are monitored from the displacement at the tips from a contactless laser sensor, is determined from a simulation of the type Harmonic (used to get the frequency response functions). Excitation forces of 0.25N, 0.5N and 1N were applied to the model at one of the anti-nodes (i.e., specimen’s tips). The different excitation forces were used to assess the system’s response against increasing loading conditions. Three different loads were used to check linearity of the model. The analyses settings are set up to test the specimen at the exact value of the natural frequency in the vicinity of 20 kHz.