The critical neck spacing in ductile plates subjected to dynamic biaxial loading: On the interplay between loading path and inertia effects

A three-pronged approach to predict the effect of plastic orthotropy on the formability of thin sheets subjected to dynamic biaxial stretching

“…The stability analysis also predicted in agreement with the finite element calculations the orientation of the necking bands that form the localization pattern, which are aligned with the direction of zero stretch for loading paths between uniaxial tension and plane strain (Hill, 1952), and they are perpendicular to the principal loading direction when the loading path lies between plane strain and equibiaxial tension. In addition, the finite element calculations indicated that the formability of metal sheets increases as the loading rate increases, in agreement with the experimental evidence discussed above (see also Rodríguez-Martínez et al (2017)).…”

“…Using the concept of effective instability introduced by Dudzinski and Molinari (1991), which states that a perturbation mode turns into a necking mode when the grow rate of the perturbation reaches a critical value, Zaera et al (2015) used the results of the stability analysis to construct dynamic forming limit diagrams. The predictions of the stability analysis were in qualitative agreement with finite element results which showed that the average spacing between necks increases as the loading path moves away from uniaxial tension and approaches equibiaxial tension (see also Rodríguez-Martínez et al (2017)). The stability analysis also predicted in agreement with the finite element calculations the orientation of the necking bands that form the localization pattern, which are aligned with the direction of zero stretch for loading paths between uniaxial tension and plane strain (Hill, 1952), and they are perpendicular to the principal loading direction when the loading path lies between plane strain and equibiaxial tension.…”

“…Moreover, we have checked that the same overall trends and conclusions are obtained for loading paths other than plane strain, however, we do not show additional plots for the sake of brevity. On the other hand, it is worth mentioning that, as shown in the works of Rodríguez-Martínez et al (2017) and Jacques (2020), the increase of the loading parameter decreases the influence of inertia on neck formation. For instance, for ψ = 0 • and greater loading parameter χ = 0.5, the two-zone model predicts that the increase of the critical major necking…”

“…Similarly, in the finite element calculations, the major necking strain corresponds to the major logarithmic strain measured in the finite element located at the upper right corner of the thickest section of the cell (see the yellow marker in Figure 5) when the axial strain rate equals 10 −3 s −1 in that specific element (similar procedure was applied in Rodríguez-Martínez et al (2017) and N'souglo et al (2020)).…”

“…A mesh convergence study has been performed, in which the time evolution of different output variables, namely stress, strain and necking inception, were compared for different mesh sizes. The regularizing effects of inertia and strain-rate sensitivity minimize the mesh sensitivity of the numerical calculations (see Needleman, 1988;Jacques et al, 2012;Czarnota et al, 2017;Rodríguez-Martínez et al, 2017).…”