On the propagation of localization in the plasticity collapse of hardening–softening beams

This paper is focused on the propagation of localization in hardening–softening plasticity media. Using a piecewise linear plasticity hardening–softening constitutive law, we look at the 1D propagation of plastic strains along a bending beam. Such simplified models can be useful for the understanding of plastic buckling of tubes in bending, the bending response of thin-walled members experiencing softening induced by the local buckling phenomenon, or the bending of composite structures at the ultimate state (reinforced concrete members, timber beams, composite members, etc.). The cantilever beam is considered as a structural paradigm associated to generalized stress gradient. An integral-based non-local plasticity model is developed, in order to overcome Wood’s paradox when softening prevails. This plasticity model is derived from a variational principle, leading to meaningful boundary conditions. The need to introduce some non-locality in the hardening regime is also discussed. We show that the non-local plastic variable during the softening process has to be strictly defined within the localized softening domain. The propagation of localization is theoretically highlighted, and the softening region grows during the softening process until a finite length region. The pre-hardening response has no influence on the propagation law of localization in the softening regime. It is also shown that the “material” time derivative and the “partial” time derivative have to be explicitly distinguished, especially for moving elastoplastic boundaries. It is recommended to use the “material” time derivative in the rate-format of the boundary value problem.