Masonry is a composite material made of units (brick, blocks, etc.) and mortar. For periodic arrangements of the units, the homogenisation techniques represent a powerful tool for structural analysis. The main problem pending is the errors introduced in the homogenisation process when large difference in stiffness are expected for the two components. This issue is obvious in the case of non-linear analysis, where the tangent stiffness of one component or the tangent stiffness of the two components tends to zero with increasing inelastic behaviour. The paper itself does not concentrate on the issue of non-linear homogenisation. But as the accuracy of the model is assessed for an increasing ratio between the stiffness of the two components, the benefits of adopting the proposed method for non-linear analysis are demonstrated. Therefore, the proposed model represents a major step in the application of homogenisation techniques for masonry structures. The micro-mechanical model presented has been derived from the actual deformations of the basic cell and includes additional internal deformation modes, with regard to the standard two-step homogenisation procedure. These mechanisms, which result from the staggered alignment of the units in the composite, are of capital importance for the global response. For the proposed model, it is shown that, up to a stiffness ratio of one thousand, the maximum error in the calculation of the homogenised Young's moduli is lower than five percent. It is also shown that the anisotropic failure surface obtained from the homogenised model seems to represent well experimental results available in the literature.

Cracking is responsible for the vast majority of masonry non-linear behaviour, due to the low tensile strength of the material. Masonry features orthotropic behaviour with material axes normal and parallel to the bed joints, being the response straightforward for tension normal to the bed joints and rather complex for tension parallel to the bed joints. This paper addresses the formulation and implementation of coupling between a micro-mechanical homogenisation model and an isotropic damage model for the masonry components. The non-linear homogenisation formulation requires an improved internal deformation mode of the masonry basic cell, with respect to previous works. Finally, the model is validated with a comparison with numerical results available in the literature, using interface modelling.

a b s t r a c tAn improved micro-mechanical model for masonry homogenisation in the non-linear domain, is proposed and validated by comparison with experimental and numerical results available in the literature. Suitably chosen deformation mechanisms, coupled with damage and plasticity models, can simulate the behaviour of a basic periodic cell up to complete degradation and failure. The micro-mechanical model can be implemented in any standard finite element program as a user supplied subroutine defining the mechanical behaviour of an equivalent homogenised material. This work shows that, with the proposed model, it is possible to capture and reproduce the fundamental features of a masonry shear wall up to collapse with a coarse finite element mesh. The main advantage of such homogenisation approach is obviously the possibility to simulate real complex structures while taking into consideration the arrangement of units and mortar, which would otherwise require impractical amount of finite elements and computer resources.

Despite considerable experimental and analytical research in the past, modern regulations still adopt very conservative simplified formulas for the compressive strength of masonry. The present paper contributes to the understanding of masonry under compression, using a novel non-linear homogenisation tool that includes the possibility of tensile and compressive progressive damage, both in the unit and mortar. The simplified homogenised model uses an iterative procedure and a few ingenious micro-deformation mechanisms, being able to accurately reproduce complex simulations carried out with non-linear continuum finite element analysis, at a marginal cost of CPU time and with no convergence difficulties. In addition, the comparison of the model with experimental results available in the literature indicates that an estimation of the compressive strength of masonry better than the one provided by the codes is possible, using the mechanical and geometrical properties of the masonry components.

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