Keywords: masonry, damage, creep, experimental tests, numerical simulations.

During the last decades material models based on damage mechanics concepts have been developed and adapted to masonry, possibly in combination with plasticity laws. A correct modeling for masonry cannot forget that it is an aggregate of bricks with interposed mortar joints. Consequently, brick masonry is a heterogeneous material with orthotropic mechanical characteristics, which depend not only on the properties of each phase (brick and mortar), but also on their interaction.

Starting from these considerations, at least two length scales of interest may be identified: mesoscopic scale and macroscopic scale. The first one is the characteristic size of the constituents of the masonry. Bricks and mortar are separately discretized and consequently spatial variations of the stress, strain and damage fields over the bricks and mortar can be identified. The macroscopic scale corresponds to the typical size of masonry structures. An equivalent homogeneous material, namely a fictitious material having mechanical properties that are equivalent to the overall, average properties of the given, non-homogeneous material, is defined. Accordingly the macroscopic variables are related to the average stress, strain and damage fields over any Representative Volume Element. Within each category, the damage and failure models can be based on a discrete or smeared crack approach.

In the first two Sections of the paper, a brief survey of the available literature regarding mesoscopic and macroscopic constitutive laws for masonry is presented. Then, two models based on Damage Mechanics and suitable for the analysis of masonry structures are presented.

The first one belongs to the class of macro-models for which the homogenization process is applied just once, when the material is in its initial, virgin state. It has been conceived to describe the behaviour of masonry subjected to cyclic in-plane excitation. Masonry is considered as a macroscopically orthotropic material, whose average physical and mechanical properties are obtained from experimental results on brickworks. Starting from test data on masonry walls subjected to multi-axial loads, a piecewise-linear surface that bounds the elastic domain is introduced. Its evolution law is defined by introducing piecewise linear functions, that provide hardening and softening rules. During the non-linear finite element analysis, the plastic strain rate vector is obtained by solving a Linear Complementarity Problem (LCP). At the end of each time step, this vector is split into a plastic (non-reversible) and a reversible part. The material damage is taken into account by considering the change in the stiffness matrix associated with the latter contribution of the inelastic deformation. The model has been checked by comparing numerical results and data from experimental tests on masonry walls (with and without openings) subjected to plane stress states [1].

The second model has been proposed for the description of the creep behaviour of masonry, at relatively high stress levels. The model is based on the introduction of suitable damage variables in a rheological model. In this way it is possible to predict not only the deformation of the material under sustained loading, but also its behaviour up to failure under stresses either increasing or constant in time. The capability of the model to describe the material response under different stress histories is checked through correlations with experimental data obtained from tests performed on masonry and concrete specimens. The reliability of this model has been assessed referring to a masonry tower recently collapsed in Pavia (Italy): the predicted time to failure turned out to be close to the real one. Then the model has been used to reproduce the behaviour of the belltower of the Cathedral of Monza, (Italy).The numerical analyses point out the dangerousness of the tower conditions: indeed, a time to failure of about two centuries starting from today is predicted [2].

1

Callerio A., Papa E., Nappi A., "Analysis of masonry structures subject to variable loads: a numerical approach based on damage mechanics", in "Inelastic Analysis of Structures under Variable Loads. Theory and Engineering Applications", Weichert, D. and Maier, G., (Editors), Kluwer Ac. Publ. 121-134, 2000.

2

Papa, E., Taliercio, A., Binda, L., "Safety assessment of ancient masonry towers", Proc. of 2nd Int. Conf. on Studies in Ancient Structures, Istanbul, 345-354, 2001.

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