Keywords: multi-level finite element method, masonry structures, damage, macroscopic localization, mesostructural snap-back.

Ensuring the safety of historical buildings requires a careful analysis of the residual strength of the structures and of the effect of repair operations. Conventional finite element analyses require a constitutive model of the building material. For masonry, however, the formulation of closed-form constitutive continuum relations which can accurately describe the collective degradation behaviour of bricks and mortar joints is made difficult as a result of the variety of possible failure mechanisms [1]. These failure modes and the mechanical responses associated with them are dominated by the mesostructure of the material (geometric arrangement and properties of the constituents), as cracks often follow the mortar joints. Realistic predictions of strength and failure modes of masonry may be obtained from mesoscopic modelling, in which the geometry of the bricks and mortar joints is explicitly modelled [2]. Modelling the full mesostructure of entire walls or structures, however, may quickly become prohibitively expensive. A compromise between computational cost and mesostructural detail can be obtained by using a coupled mesoscopic-macroscopic modelling approach. This means that walls are modelled using an homogenised continuum description, but the constitutive behaviour of the masonry material is determined on-line by mesoscopic analyses. In a finite element context, in each Gauss point of the macroscopic finite element discretisation a discretised sample of the mesostructure is used to determine this material response. For this purpose the local macroscopic strain is applied in an average sense to the mesostructure and the resulting mesostructural stresses are determined by a finite element analysis. The averaging of these mesostructural stresses and the condensation of the mesostructural tangent stiffness to the homogenised tangent stiffness then furnish the macroscopic material response associated with the Gauss point. This concept is also known as multilevel-FEM, FE or computational homogenisation [3]. Its added value in the context of masonry resides in the fact that no complex closed-form constitutive relation needs to be postulated for the representation of the overall material behaviour.

The purpose of this paper is to present how such a multi-scale framework can be adapted for masonry structural computations. This paper addresses the following specificities related to the behaviour of masonry structures: (i) the choice of a macroscopic continuum representation, and (ii) the set-up of scale transitions linking macroscopic and mesoscopic quantities. These features have to be carefully selected in order to allow a proper incorporation of the localisation behaviour at the macroscopic scale. Based on the periodicity of the initial mesostructure of masonry, periodic homogenisation concepts are used in order to build scale transitions between the mesoscopic and macroscopic scales. In this contribution, the smallest periodic mesostructural sample (a unit cell) is selected as the representative volume element in order to limit the computational effort at the mesoscopic scale. In order to deal with localisation at the macroscopic scale, embedded localisation bands surrounded by unloading material are introduced in a standard continuum description. These embedded bands are introduced upon localisation detection obtained from a material bifurcation analysis. The band width is deduced from this orientation and the initial periodicity of the mesostructure. Moreover, mesoscopic damage concentration in zones of the order of a thin mortar joint may lead to snap-backs in the retrieved homogenised material response used at the macroscopic scale as a result of the finite size of the used mesoscopic sample. Since the mesostructural problem is deformation-driven in 'classical' multi-scale schemes, an adaptation of the framework is introduced to handle such mesostructural snap-backs. This enhancement consists in steering the mesostructural computation on the snap-back path. This is achieved where needed by forcing further mesoscopic energy dissipation through the imposition on the mesostructural unit cell problem of properly selected mesoscopic quantities (crack opening displacements) by the macroscopic solution procedure. These controlling mesoscopic quantities are transferred to the macroscopic solution procedure as unknowns together with a conjugated mesoscopic residual equation. The proposed multilevel scheme is implemented using parallel computing facilities. The capacities of the proposed approach are illustrated by means of structural computations. A typical structural application is treated, consisting of the shearing of a confined wall with opening, showing the ability of the model to account for the strong coupling between the structural response of the material and its underlying mesostructural features.

1

M. Dhanasekar, A.W. Page, P.W. Kleeman, "The failure of brick masonry under biaxial stresses", Proceedings of Institution of Civil Engineers, Part 2, 295-313, 1985. doi:10.1680/iicep.1985.992

2

T.J. Massart, R.H.J. Peerlings, M.G.D. Geers, "Mesoscopic modelling of damage-induced anisotropy in brick masonry", Submitted to European Journal of Mechanics A/Solids, 2004. doi:10.1016/j.euromechsol.2004.05.003

3

R.J.M. Smit, "Toughness of heterogeneous polymeric systems. A modelling approach", PhD Thesis, Eindhoven University of Technology, The Netherlands, 1998.

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