Damage and localization of engineering materials: modelling across length scales

Introduction: The past decade was marked by the emergence of a large number of so-called regularized models, which are dedicated to the analysis of damage and softening in engineering materials. Whereas many of these models include gradients in the weakly nonlocal sense, some models are built on a strongly nonlocal concept, which provides a robust numerical scheme and a well-established physical response. The strongly nonlocal concept (based on a gradient formulation) was recently extended to softening plasticity for small deformations and large deformations, enabling its use in the analysis of manufacturing processes. Such models can be easily interpreted within a multi-scale methodology, aiming to predict, describe, quantify or qualify the 'macroscopic' behaviour of engineering materials and composites through the consistent modelling of the mechanics and physics of the heterogeneous, multi-phase microstructure. This contribution focuses on such macroscopically enhanced constitutive models (macroscopic models enriched with microstructural information) in direct relation to micro-macro coupled computational homogenisation of the underlying micromechanics. The presentation sketches an overview of these various classes of multi-scale models, with a special emphasis on their mutual links and applicability to damage and localization. Special attention is given to computational homogenisation schemes, which are based on the hierarchical solution of two coupled boundary value problems. Most important features of such computational homogenisation schemes are: no constitutive assumption on the macro level; large deformations and rotations on the micro and macro level; arbitrary physically nonlinear and timedependent material behaviour on the micro level; any modelling technique on the micro level can be used; evolving and transforming microstructures can be easily dealt with. Recently, computational homogenisation has been improved to deal with localization and size e ects in engineering materials. Higher-order continua are naturally retrieved through the incorporation of size and localisation e ects in the presented computational multi-scale model. Special emphasis is given on inherent nonlocal properties of these various multi-scale techniques. The role and relevance of this nonlocality is further discussed in order to properly understand the limitations and range of applicability of higher-order phenomenological models and homogenization schemes. A few practical examples will be given to illustrate and to support the di erent issues raised. Finally, conclusions with respect to the practical use of these techniques for the adequate modelling of damage and localization will be drawn