Shakedown at Finite Elasto-Plastic Strains

Abstract

For structures undergoing small elastic-plastic deformations (i. e. small rotations and small strains), shakedown or non-shakedown can be determined by applying the well-known statical shakedown theorem of Melan [1], which leads to a lower bound of the load factor. Dual to the Melan’s theorem Koiter [2] formulated a kinematical shakedown theorem yielding an upper bound of the load factor. While many publications deal with shakedown of structures at small deformations taking into account linear and non-linear hardening material behaviour (e. g. Stein et al. [3]) the search for generalisations of the shakedown theorem for large deformations was initiated by the paper of Maier [4]. Weichert [5, 6], Gros-Weege [7], Saczuk and Stumpf [8], Tritsch and Weichert [9] and Weichert and Hachemi [10] presented generalisations of the Melan’s theorem for structures undergoing large deformations with large plastic strains and moderate or large rotations. The underlying idea is to determine a moderately or finitely deformed configuration of the structure as new reference and then to investigate the shakedown behaviour for superposed small deformations. Typical applications for these methods are thin plate and shell structures. While the shakedown analysis for arbitrary superposed small deformation histories can be performed with optimisation technique, the reference configuration has to be determined by using an appropriate shell finite element for moderate or large deformations.