An objective multi-scale model with hybrid injection

This paper presents a new semi-concurrent multi-scale model to study the behaviour of composite materials in softening regime. A mixed formulation is used to simulate discontinuities in both scales. The traction over the crack is included as a unknown field in the equations system of the problem, and the jump displacement across the discontinuity is obtained with a cohesive constitutive relation (traction-separation law). Axiomatic variational principles and the injection concept are used and formulated to get an objective model with respect to the representative volume element size (RVE). The projected stress over the normal vector of the macro discontinuity is injected in the localized domain in the RVE, obtaining as a dual variable the jump of the displacement field in the macro structure. In this way, during the stable phase of the behaviour, the scale transition is performed in the classical way injecting the strain tensor and obtaining the stress tensor as a dual variable. At the beginning of the unstable regime, the transition between the scales is defined by injecting the traction (stress projection on the normal vector to the crack) in the localization domain in the micro scale and the displacement jump at the macro scale is obtained as a dual variable. This new concept leads to a new multi-scale approach with an hybrid injection. The basic equations of the model are derived, and finally some numerical examples are developed, showing the objectivity of the homogenized response of composite material problems that involve strain localization at the macro-scale.