Extension of linear stability analysis for the dynamic stretching of plates: Spatio-temporal evolution of the perturbation

Abstract

Being able to predict fragment distributions in terms of speed and size, following the fracture of thin metallic shells subjected to dynamic expansion, is of major importance for civil and military applications. For ductile metals, this fracture process is initiated by plastic flow instability resulting in necking, i.e. the occurrence of local thinnings where the plastic deformation is localized. Since decades, linear stability analyses have been carried out to study the multiple necking formation via a perturbation of the fundamental state. The underlying assumption related to the linear stability analyses developed so far is the time scale separation (meaning that the development of the instability is much faster than the evolution of the fundamental state), see Fressengeas and Molinari (1994) or Shenoy and Freund (1999). The aim of the work is to propose an extended linear stability analysis which can tackle situations where the time scale separation hypothesis is no more satisfied (i.e. at very large strain rates). The proposed methodology is exemplified by considering the dynamic extension of a plate under plane strain condition; the material behavior being modeled adopting various constitutive laws from rate insensitive to thermo-viscoplastic ones. The role of initial perturbation (or defect) is discussed. While the role of the initial conditions is important at the early stage of the deformation process, their influence on the growth rate and on the dominant mode are negligible at large strain for moderate loading rate. One main feature of the proposed model is the estimation of the amplitude development of each mode. A strong difference in the amplitude predictions is revealed between the new model and the classical linear stability analysis of the literature, even if the growth rates are comparable for both approaches at late deformation stage. However, history effect related to the amplitude of a given mode, originating from the early stage process, may lead to strong amplitude differences which remain visible at the late stage. The benefit of the new theory is the fact that the spatio-temporal evolution of the mechanical and thermal parts of the perturbation can be captured naturally, even for cases where the plastic flow is initially stable and becomes unstable as the deformation progresses.