Extension of 1D linear stability analysis based on the Bridgman assumption. Applications to the dynamic stretching of a plate and expansion of a ring

Abstract

Abstract The analysis of multiple necking phenomenon during dynamic stretching of a metallic plate is investigated. For that purpose, a new 1D linear stability analysis is proposed to capture the perturbation evolution. The multidimensional aspect of the stress field within the neck region is taken into account with a Bridgman correction factor. The novelty of the 1D approach is to let the time evolution of the perturbation determined by the linearized field equations. So, no predefined time dependency is assumed as in classical 1D linear stability analysis of the literature, see Zhou et al., An elastic-visco-plastic analysis of ductile expanding ring, Int. J. Impact Eng., 2006. The proposed model, named hereafter 1D-XLSA (standing for 1D-eXtended Linear Stability Analysis), can also be viewed as the restriction to 1D of the 2D-XLSA model developed in Xavier et al., Extension of linear stability analysis for the dynamic stretching of plates: Spatio-temporal evolution of the perturbation, European Journal of Mechanics-A/Solids, 2020. A comparison for a thermo-viscoplastic material with strain hardening is proposed for three possible routes : a 1D model based on the frozen coefficient theory named 1D-CLSA (standing for 1D-Classical Linear Stability Analysis), the proposed 1D-XLSA and 2D-XLSA models. It is shown that while growth rates in late deformation stage are similar, a strong difference in amplitude exists due to large discrepancies in the early deformation stage. The comparison also illustrates the importance of the modeling of multiaxiality of the stress field within the neck region and of initial defects. While differences between models are limited for perturbations with small wavenumbers, large discrepancies are observed for perturbations with large wavenumbers. Our model is also derived for the case of the extension of a cylindrical bar which is representative of the ring expansion during dynamic loading.