A Hysteretic Model with Damage Based on Bouc-Wen Formulation

Hysteresis is a well-known phenomenon involving history-dependent response of many structural (and non-structural) materials. Among the various models that can be adopted to describe it, the Bouc-Wen model is certainly one of the most widely used, and it is considered the forefather of an entire class of hysteresis models. During the last decades, the original formulation was modified to take into account the effects of strength and stiffness degradation, pinching and non symmetric response. Moreover, issues related to its thermodynamic admissibility were tackled. This research presents a discussion on the acknowledged limits of the original Bouc-Wen model, focusing on thermodynamic admissibility and Drucker’s postulate for hardening materials. The influence of the parameters \(\beta \) and \(\gamma \), governing the shape of hysteresis cycles, is highlighted and a revised formulation of the dissipated energy is proposed. Then, an enriched version of the model is introduced, in which degradation of strength and stiffness is accounted for by means of a single scalar damage variable. The effect of pinching can also be included resorting to a parallel arrangement of a nonlinear elastic element. Some considerations on the thermodynamic admissibility of this model are briefly presented and some simple validation examples are reported to show its main features. The proposed damage model is implemented in a 2D beam finite element within the framework of equivalent frame models to reproduce, through lumped plastic hinges, the shear and flexural mechanisms characterizing the response of masonry panels under cyclic in-plane horizontal loads. The repeated excursions into the inelastic range are in fact responsible for both strength and stiffness degradation accompanied by energy dissipation. Some numerical analyses are performed and the results here presented show the capability of this model to describe, at the macroscopic scale, damaging behaviour characterizing masonry structures under dynamic loading conditions.