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

Abstract

In this paper, we have investigated the effect of material orthotropy on the formability of metallic sheets subjected to dynamic biaxial stretching. For that purpose, we have devised an original three-pronged methodology which includes a linear stability analysis, a nonlinear two-zone model and finite element calculations. We have studied 5 different materials whose mechanical behavior is described with an elastic isotropic, plastic anisotropic constitutive model with yielding based on Hill (1948) criterion. The linear stability analysis and the nonlinear two-zone model are extensions of the formulations developed by Zaera et al. (2015) and Jacques (2020), respectively, to consider Hill (1948) plasticity. The finite element calculations are performed with ABAQUS/Explicit (2016) using the unit-cell model developed by Rodríguez-Martínez et al. (2017), which includes a sinusoidal spatial imperfection to favor necking localization. The predictions of the stability analysis and the two-zone model are systematically compared against the finite element results -which are considered as the reference approach to validate the theoretical models-for loading paths ranging from plane strain stretching to equibiaxial stretching, and for different strain rates ranging from 100 s −1 to 50000 s −1 . The stability analysis and the two-zone model yield the same overall trends obtained with the finite element simulations for the 5 materials investigated, and for most of the strain rates and loading paths the agreement for the necking strains is also quantitative. Notably, the differences between the finite element results and the two-zone model rarely go beyond 5%. Altogether, the results presented in this work provide new insights into the mechanisms which control dynamic formability of anisotropic metallic sheets.

Effect of the third invariant on the formation of necking instabilities in ductile plates subjected to plane strain tension

“…The results obtained with the three approaches -two-zone model, stability analysis and unit-cell finite element calculations-showed qualitative agreement for the five materials considered, for the whole range of strain rates investigated, and for loading paths ranging from plane strain tension to equibiaxial stretching. Consistent with the quasi-static forming limit diagrams obtained by Sowerby and Duncan [43], Parmar and Mellor [37], Barlat [3] and Butuc et al [5] using the two-zone approach of Marciniak and Kuczyński [29], N'souglo et al [34] showed that if inertial effects are considered, just like in the quasi-static case, the influence of the shape of the yield surface on the forming limit strains is small for plane strain stretching while being important for biaxial stretching.…”

“…On the other hand, while the same rationale has been employed by other authors, e.g. Zaera et al [52] and N'souglo et al [35], we are aware that it is also customary in the literature to impose the stretch rate as the loading condition [49,34], since it may bear closer resemblance to the actual boundary conditions in dynamic experiments like the expansion of hemispherical shells (see Mercier et al [30]). Hence, stability analysis results and finite element calculations obtained imposing the stretching rate, namelyε 0 xx = 4000 s −1 andε 0 xx = 80000 s −1 , are reported in Appendix A.…”

“…Excellent agreement was found between theoretical model predictions and finite element calculations, for all the loading paths considered, and strain rates ranging from 5000 s −1 to 50000 s −1 . Moreover, N'souglo et al [34] have extended the non-linear two-zone model developed by Jacques [24] to consider materials described with the anisotropic yield criterion of Hill [20], and constructed dynamic forming limit diagrams for strain rates ranging from 100 s −1 to s −1 . The formability limits at necking obtained with the non-linear two-zone model were compared with unit-cell finite element calculations and with a linear stability analysis specifically developed to model dynamic necking in anisotropic thin plates subjected to biaxial stretching.…”

“…The formability limits at necking obtained with the non-linear two-zone model were compared with unit-cell finite element calculations and with a linear stability analysis specifically developed to model dynamic necking in anisotropic thin plates subjected to biaxial stretching. The study of N'souglo et al [34] encompassed five different materials, two of them were model materials with mechanical properties specifically tailored to bring to light the roles of inertia and orthotropy in dynamic formability, and the other three were actual materials, namely, TRIP-780 steel, AA 5182-O and AA 6016-T4.…”

“…refs. [17,31,50,26,27,36,34]), and therefore they do not account for the effect of the third invariant of the stress deviator on necking localization. An exception is the recent work of N'souglo et al [35], who developed a linear stability analysis and performed unit-cell finite element calculations to model the formation of dynamic necks in plates subjected to plane strain tension and made of isotropic ductile materials displaying tension-compression asymmetry.…”