PSI - Issue 66

Daniele Gaetano et al. / Procedia Structural Integrity 66 (2024) 478–485 Author name / Structural Integrity Procedia 00 (2025) 000 – 000

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can be noted that, for higher values of the percentage of the mode II displacement jump, denoted as u IIf , the critical mode shapes exhibit a longer wavelength than the RVE length, thus showing a transition from micro to macro instability.

Fig. 2. Numerical outcomes obtained by the proposed formulation: (A) Microscopic load factors evaluated with the Exact stability functional (Red line), the Simplified I (Green line), and the Simplified II (Blu line) formulations; (B) Critical mode shapes associated with the exact solution. Moreover, Table 1 reports the percentages of variation between the critical load factors evaluated with the simplified formulations and the exact ones. As expected, the simplified formulations lead to an underestimation of the microscopic critical load factor as the percentage of the maximum displacement jump increases. Table 1. Percentages of variation between the critical load factors evaluated with the simplified formulations and the exact ones. Percentages of mode II critical displacement jump [%] Simplified I [%] Simplified II [%] 0 0.79 10.17 5 1.57 10.84 30 2.95 14.89 80 11.93 33.05 90 31.05 55.41 95 39.61 62.90 97 37.05 64.77 The results show a decreasing trend regarding the load factor, with a marked decrease in the microscopic critical load for percentages of the critical tangential displacement jump greater than 80%. It can be noted that, for percentages of maximum displacement jump less than 80%, the proposed formulations show a smoother decreasing trend, with a percentage of variation with respect to the exact one less than 30%. In particular, the simplified formulation 1 shows a percentage of variation less than 10%. However, it is important to note that, unlike what has been observed for a uniaxial loading condition (Greco et al., 2024), the shear effect leads to a notable decrease in the critical load level and to consistent differences between the exact and the simplified formulations, even for small percentages of the maximum tangential displacement jump with respect to the critical one.