A multi-scale computational approach for the fracture behaviour of quasi-brittle materials

Increasingly advanced numerical techniques are nowadays used to represent damage and failure in quasi-brittle materials. Such numerical methods may be used for the analysis of structures if they are able to account realistically for the possible failure modes of the materials, which strongly depend on their microstructure, i.e. their constituents. During the failure process, complex microstructural evolutions take place, giving rise to the overall damage evolution in the material. In their averaged behaviour, the localisation associated to intrinsic softening and the resulting damage-induced anisotropy (potentially interacting with the initial anisotropy) are typical results thereof.The formulation of closed-form constitutive relations which account for such mechanical effects is complicated, and strong assumptions are therefore often required in order to render such frameworks tractable. Furthermore, their experimental identification is mostly troublesome. Despite the intensive research dedicated to this field, the representation of general damage-induced anisotropy effects by means of closed-form constitutive laws remains far from established, even for initially isotropic materials. Existing frameworks accounting for cracking-induced anisotropy make use of tensorial damage variables of order two for orthotropic damage or of higher order for a more complex anisotropy evolution. This results in elegant but complex frameworks, featuring large numbers of parameters and/or model relations. The identification of material specific relations and parameters in such models poses a substantial difficulty, which is to be repeated for each new material. The main mechanism for the degradation of quasi-brittle materials is damage. This phenomenon crosses all length scales, and the quasi-brittle nature of the material therefore requires an approach which reflects the interplay between several length scales in the damaging process. The damage-induced anisotropy is also a key feature for a proper representation of the quasi-brittle behaviour. As a result, multi-scale schemes based on a computational homogenisation scheme can be used to handle this type of problem. These methods were developed in recent years for the characterisation of metallic and polymer materials. In these developments, the coarse scale behaviour is modelled using a classical continuum description, but the constitutive behaviour of the material is determined on-line by fine scale analyses. For this purpose, the local macroscopic strain is applied in an average sense to a representative volume element (macro-micro scale transition); and the resulting mesostructural stresses are determined by a finite element analysis, and are averaged to derive the macroscopic material response (micro-macro scale transition). The application of such an approach to quasi-brittle heterogeneous materials however requires proper adaptations. After a short recall of computational homogenisation concepts, the paper reviews some of the existing multi-scale approaches based on computational homogenisation for the behaviour of quasi-brittle materials, enlightening their main assumptions. This classification allows to identify the key issues to handle in order to properly formulate a multi-scale approach for quasi-brittle materials. Among these, the most salient adaptations are related to: (i) the choice of a proper representative volume element size, depending on the typical microstructure of the material (structured or periodic materials), (ii) a proper treatment of damage localisation at both the coarse and fine scales and its impacts on the formulation of scale transitions, (iii) a proper representation of the energy dissipation at the coarse scale, by explicitly taking into account the volume on which the fine scale dissipation occurs, and (iv) the set-up of multi-scale path following techniques to trace quasi-brittle responses on finite volumes. These adaptations are discussed in this paper, together with some potential solutions.Next, the discussion is focused on the case of textured (periodic) quasi-brittle heterogeneous materials for which the first adaptation (choice of a representative element) is more easily solved. The case of masonry is used as an illustration, for which a coupled two-scale (mesoscopic-macroscopic) framework is discussed. In order to deal with localisation at the coarse scale, embedded localisation bands surrounded by unloading material are introduced in a standard first order continuum description (adaptation (ii)). A material bifurcation analysis based on the homogenised acoustic tensor is used to detect localisation and to deduce band orientations which are consistent with the underlying mesostructural damage modes. The width of these localization band is directly deduced from the initial periodicity of the material and used in the coarse scale description, which ensures a correct amount of energy dissipated (adaptation (iii)).Also, the use of homogenization techniques on finite volumes containing quasi-brittle constituents leads to snap-back effects in the homogenised material response deduced by the scale transition. A path-following methodology to introduce this type of response in the originally strain driven multi-scale technique is proposed (adaptation (iv)). The paper is closed with some remarks on the potential improvements or extensions of existing frameworks.Keynote Lecture