Ring expansion experiments demonstrate (Zhang & Ravi-Chandar, 2008) that fragmentation of thin structures is triggered inside strain localization loci (necks for ductile materials). In the literature, authors have analyzed strain localization with analytical approaches based on equivalent configurations: bars in dynamic extension, representative of rings in expansion. In our presentation, we focus on linear stability analysis (LSA) [see (Mercier & Molinari, 2003) or (N’souglo, et al., 2018) among others]. LSA has proved its interest to evaluate mean size of fragments by determining the fastest growing mode. However, different limitations of this approach can be mentioned. In LSA, perturbations are considered as cosine functions of the spatial coordinates. So, in order to compare with real experimental profiles, (El Maï, et al., 2014) have introduced a multimodal description of the surface roughness with recourse to a Fourier decomposition with random phases. In previous works, the growth rate of the perturbation was determined based on a frozen coefficients assumption. This assumption has been relaxed recently in (Xavier, et al., 2021) where a 1D eXtended LSA (XLSA) unimodal approach has been proposed. Nevertheless, even for those recent approaches based on LSA, determining the time to failure or describing the transition between diffused necking to localized necking is still a challenge.
In this work, we propose a 1D Weakly Non Linear Stability Analysis (WNLSA) based on the development of the perturbations with second order perturbation terms, which extends the 1D XLSA approach of (Xavier, et al., 2021). Firstly, comparisons with numerical simulations show that this approach can describe the development of the principal harmonics when unimodal geometric perturbations are considered. Secondly, for the realistic case of an initial multimodal random surface roughness, it is shown that the onset time of localized necking can be captured.