Path-following methods for unstable structural responses induced by strain softening: a critical review

Abstract

Path-following methods for describing unstable structural responses induced by strain-softening are discussed. The main ingredients of the formalisms introduced by Riks and Crisfield for arc-length methods for geometrical non-linearities are presented. A link between two ways (monolithic and partitioned) of solving the resulting augmented equilibrium problem is discussed based on the Sherman-Morrison formula. The original monolithic approach assumes that the path-following constraint equation is differentiable with respect to the unknown displacement field and load factor. However, when dealing with material nonlinearities, it is often preferred to consider constraint equations controlling the maximum of a field defined on the computational domain (e.g., a scalar strain measure, the rate of variation of an internal variable of the constitutive model). In that case, differentiability cannot be guaranteed due to the presence of the maximum operator. This makes only the partitioned formulation usable. Several path-following constraint equations from the literature are presented, and the corresponding implementations in the finite element method are discussed. The different formulations are compared based on a simple two-dimensional test case of damage localization in a beam submitted to tension. A test case involving multiple snap-backs is illustrated, finally, to show the robustness of the considered formulations.Résumé. Des techniques de pilotage indirect du chargement pour décrire des réponses structurelles instables induites par l'adoucissement en déformation sont discutées. Après avoir rappelé les principaux ingrédients des formalismes introduits par Riks et Crisfield pour les méthodes de longueur d'arc pour le traitement de

One-dimensional study of boundary effects and damage diffusion for regularized damage models

“…Jirásek and Desmorat [40] proposed two schemes to perform the integral calculations: a simple trapezoidal rule such that 1 − D is element-wise constant and a modified approach assuming that damage is linear between two adjacent integration points. As shown in [47], the first approach has better localization properties, especially when combined with appropriate path-following algorithms [50].…”