A general framework for continuum damage models. II. Integration algorithms, with applications to the numerical simulation of porous metals

In this paper, we develop numerical algorithms for the integration of the continuum plastic damage models formulated in the general framework identified in Part I of this work. More specifically, we focus our attention on a particular plastic damage model of porous metals, involving a classical von Mises yield criterion coupled with a pressure dependent damage surface to model the nucleation and growth of voids in the metallic matrix. Unilateral damage leading to a sudden change of stiffness in the material's response due to the closing/opening of these voids is also incorporated through the imposition of the unilateral constraint of a positive void fraction, thus, illustrating the clear physical significance added by this framework in the resulting constitutive models. The proposed integration algorithms fully use the modularity of the identified framework, leading in this way to independent integration algorithms for the elastoplastic part and each damage mechanism. Remarkably, all these individual integration schemes share the same formal structure as the classical return mapping algorithms employed in the numerical integration of elastoplastic models, namely an operator split structure consisting of a trial state and the return map imposing the plastic and damage consistency, respectively. A Newton iterative scheme imposes the equilibrium (equal stresses) among the different mechanisms of the response of the material. This modular structure allows to obtain the closed-form consistent linearization, involving in a simple form the algorithmic consistent tangents corresponding to each independent mechanism, thus resulting in a very modular and efficient computational implementation. The performance of the proposed algorithms is illustrated in several representative numerical simulations