Computational & Technology Resources
an online resource for computational,
engineering & technology publications 

Computational Science, Engineering & Technology Series
ISSN 17593158 CSETS: 14
INNOVATION IN COMPUTATIONAL STRUCTURES TECHNOLOGY Edited by: B.H.V. Topping, G. Montero, R. Montenegro
Chapter 15
A MultiScale Computational Approach for the Fracture Behaviour of QuasiBrittle Materials T.J. Massart*, R.H.J. Peerlings+, M.G.D. Geers+ and Ph. Bouillard*
*Structural and Material Computational Mechanics Department, Université Libre de Bruxelles, Brussels, Belgium T.J. Massart, R.H.J. Peerlings, M.G.D. Geers, Ph. Bouillard, "A MultiScale Computational Approach for the Fracture Behaviour of QuasiBrittle Materials", in B.H.V. Topping, G. Montero, R. Montenegro, (Editors), "Innovation in Computational Structures Technology", SaxeCoburg Publications, Stirlingshire, UK, Chapter 15, pp 303324, 2006. doi:10.4203/csets.14.15
Keywords: multiscale modelling, damage, quasibrittle behaviour, localisation, energy dissipation, masonry.
Summary
Increasingly advanced numerical techniques are nowadays used to represent damage
and failure in quasibrittle 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,
that is, their constituents. During the failure process, complex microstructural evolutions
take place, giving rise to overall damage evolution in the material. On
average, the localisation associated with intrinsic softening and the
resulting damageinduced anisotropy (potentially interacting with the initial
anisotropy) are typical results thereof.
The formulation of closedform 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 damageinduced anisotropy effects by means of closedform constitutive laws remains far from established, even for initially isotropic materials. Existing frameworks accounting for crackinginduced 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 quasibrittle materials is damage. This phenomenon crosses all length scales, and the quasibrittle nature of the material therefore requires an approach which reflects the interplay between several length scales in the damaging process. Damageinduced anisotropy is also a key feature for a proper representation of quasibrittle behaviour. As a result, multiscale 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 online by fine scale analyses. For this purpose, the local macroscopic strain is applied in an average sense to a representative volume element (macromicro scale transition) and the resulting mesostructural stresses are determined by a finite element analysis. These are averaged to derive a macroscopic material response (micromacro scale transition). The application of such an approach to quasibrittle heterogeneous materials, however, requires proper adaptations. After a short recall of computational homogenisation concepts, the paper reviews some of the existing multiscale approaches based on computational homogenisation for the behaviour of quasibrittle materials, highlighting their main assumptions. This classification allows us to identify the key issues to be handled in order to properly formulate a multiscale approach for quasibrittle 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 impact 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 setup of multiscale path following techniques to trace quasibrittle 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) quasibrittle 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 twoscale (mesoscopicmacroscopic) 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 localisation 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 homogenisation techniques on finite volumes containing quasibrittle constituents leads to snapback effects in the homogenised material response deduced by the scale transition. A pathfollowing methodology to introduce this type of response in the originally strain driven multiscale technique is proposed (adaptation (iv)). The paper is closed with some remarks on the potential improvements or extensions of existing frameworks. purchase the fulltext of this chapter (price £20)
go to the previous chapter 
