Keywords: multi-scale modelling, computational homogenisation, localisation, flexural effects, beams.

Numerous materials like composites or masonry have a strongly heterogeneous microstructure which influences their overall mechanical behaviour in a considerable way. In their usual structural use, such materials may be subjected to important flexural effects, such as for instance laminate composites undergoing delamination or masonry walls with cracking. Unfortunately, the modelling of heterogeneous structures with an explicit representation of heterogeneities by the means of classical finite element method unfortunately requires too great a computational effort. The multi-scale methods help handling this issue by deducing a homogeneous response at the macroscopic level [1,2] based on constituents properties and averaging theorems. The material behaviour of the basic constituents of their microstructure is postulated with closed-form laws, and is used for stress calculation at integration points through a representative volume element (RVE) of the microstructure.

At present, the multi-scale methods based on homogenisation concepts are mainly used for macroscopic plane state or three-dimensional descriptions and do not take into account flexural effects, except for homogenisation towards higher order continua [1]. As a result, there is a need to extend the multi-scale methods to use them in beam or shell macroscopic descriptions for which this paper represents a first step. The introduction of the flexural effects is first studied for the case of a beam formulation. An averaged axial deformation and a curvature are imposed to the RVE during the scale transition. A normal effort and a bending moment, i.e. the generalised stresses at the macroscopic level, are deduced from the stress integration at the microscopic level. For this approach, a RVE is represented by a stacking of bars, which prescribes the use of different types of kinematics at both the macroscopic and microscopic scales.

In order to model localisation of inelastic phenomena (damage or plasticity) without loss of the well-posedness of the boundary value problem, the localisation of the degradation has to be detected and treated at both the microscopic and macroscopic levels [3]. For the former case, existing closed-form solutions can be used. For the latter, the detection of macroscopic localisation is based on the computationally homogenised tangent stiffness which is obtained from the scale transition. An inelastic hinge formulation proposed in [4,5] is used at the macroscopic scale to account for the localisation at this scale. The behaviour of this inelastic hinge is directly deduced from a RVE using the scale transition. This approach is illustrated for the case of a beam formulation. A comparison with a closed form evolution law for the hinge behaviour is conducted. Its extension to plate and shell descriptions is also discussed.

1

V. Kouznetsova, "Computational homogenisation for the multi-scale analysis of multi-phase materials", PhD thesis, Eindhoven University of Technology, 2002

2

R.J.M. Smit, "Toughness of heterogeneous polymeric systems", PhD thesis, Eindhoven University of Technology, 1998

3

T.J. Massart, "Multi-scale modeling of damage in masonry structures", PhD thesis, Eindhoven University of Technology and Université Libre de Bruxelles, 2003

4

F. Armero, D. Ehrlich, "Numerical modeling of softening hinges in thin Euler-Bernoulli beams", Computers and Structures 84 (2006) 641-656, Elsevier Ltd doi:10.1016/j.compstruc.2005.11.010

5

F. Armero, D. Ehrlich, "Finite element methods for the multi-scale modeling of softening hinge lines in plates at failure", Comput. Methods Appl. Mech. Engrg. 195 (2006) 1283-1324, Elsevier B.V. doi:10.1016/j.cma.2005.05.040

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