Failure-Oriented Multi-scale Variational Formulation: Micro-structures with nucleation and evolution of softening bands

Abstract

This contribution presents the theoretical foundations of a Formulation (FOMF) for modeling heterogeneous softening-based materials undergoing strain localization phenomena. The multi-scale model considers two coupled mechanical problems at different physical length scales, denoted as macro and micro scales, respectively. Every point, at the macro scale, is linked to a Representative Volume Element (RVE), and its constitutive response emerges from a consistent homogenization of the micro-mechanical problem. At the macroscopic level, the initially continuum medium admits the nucleation and evolution of cohesive cracks due to progressive strain localization phenomena taking place at the microscopic level and caused by shear bands, damage or any other possible failure mechanism. A cohesive crack is introduced in the macro model once a specific macroscopic failure criterion is fulfilled. The novelty of the present Failure-Oriented Multi-scale Formulation is based on a proper kinematical information transference from the macro-to-micro scales during the complete loading history, even in those points where macro cracks evolve. In fact, the proposed FOMF includes two multi-scale sub-models consistently coupled: (i) a Classical Multi-scale Model (ClaMM) valid for the stable macro-scale constitutive response.ii) A novel Cohesive Multi-scale Model (CohMM) valid, once a macro-discontinuity surface is nucleated, for modeling the macro-crack evolution. When a macro-crack is activated, two important kinematical assumptions are introduced: (i) a change in the rule that defines how the increments of generalized macro-strains are inserted into the micro-scale and (ii) the Kinematical Admissibility concept, from where proper Strain Homogenization Procedures are obtained. Then, as a consequence of the Hill-Mandel Variational Principle and the proposed kinematical assumptions, the FOMF provides an adequate homogenization formula for the stresses in the continuum part of the body, as well as, for the traction acting on the macro-discontinuity surface. The assumed macro-to-micro mechanism of kinematical coupling defines a specific admissible RVE-displacement space, which is obtained by incorporating additional boundary conditions, Non-Standard Boundary Conditions (NSBC), in the new model. A consequence of introducing these Non-Standard Boundary Conditions is that they guarantee the existence of a physically admissible RVE-size, a concept that we call through the paper objectivity of the homogenized constitutive response.Several numerical examples are presented showing the objectivity of the formulation, as well as, the capabilities of the new multi-scale approach to model material failure problems.