Keywords: constitutive models, masonry, finite element modelling, non-linear analysis, crack-band model, plane stress.

This paper presents a constitutive model for masonry. This model is aimed to be more precise than the linear elastic models that still are used in the building industry but it is also simplier and easier to use than the currently available complex approaches, such as given in references [3,2].

We have decided to develop a compromise-based constitutive model that should respect the non-linear and non-homogenous nature of a masonry but that should be less complicated than the complex constitutive models.

The proposed model uses different constitutive models for both materials of masonry. It permits the creation of computation models that will respect the real geometry and positions of individual bricks (or stones) and the areas of mortar in a masonry structure. The proposed constitutive model is intended for use on two-dimensional problems.

The model for a mortar is based mostly on already proven approaches. The model is controlled by an equivalent uniaxial stress-strain relationship. A smeared crack approach [1] is used for modelling of a material behaviour in tension. Principal stresses are used for the equivalent stress-strain relationship.

The limits for the equivalent uniaxial stress-strain relation are computed from a two-dimensional stress state. The Kupfer's failure condition for a concrete is used for computation of the limits.

The bricks are defined in a different way. It is possible to use the previously described model as a constitutive model for bricks, too. But it has some disadvantages because the typical bricks are usually relatively brittle and the quassibrittle material model is not always ideal. However, the use of an approach based on linear elastic fracture mechanics is not very easy here. This is because a model (and also an original structure) includes a large number of bricks.

In this paper we propose a different material model. It is based on a simple kind of a non-local material model. We detect the cracking or crushing of the material by a stress size in an area around the material point where the failure condition is tested. The behaviour that we have to obtain is that the brick should crack at once. Thus we set the size of the area to be comparable with a typical height of the brick. The paper discusses the current status of the model and provides numerical examples to illustrate its behaviour. For the implementation and for the testing of the proposed model we use our in-house developed computer code called uFEM. The code uses a four-node serendipity family isoparametric finite element for two-dimensional problems [4].

1

Z.P. Bazant, J. Planas, "Fracture and Size Effect in Concrete and Other Quasibrittle Materials", CRC Press, Boca Raton, 1998.

2

S. Pietruszczak, R. Ushaksaraei "Description of inelastic behaviour of structural masonry", International Journal of Solids and Structures Volume 40, Issue 15, July 2003. doi:10.1016/S0020-7683(03)00174-4

3

J. Sýkora, J. Vorel, J. Zeman, M. Šejnoha, J. Šejnoha, "Multi-scale Modeling of Masonry Structures - Synthesis of Constitutive Models and Scale Transition", The 5th International Congress of Croatian Society of Mechanics - Proceedings [CD-ROM] Zagreb, Croatian Society of Mechanics, 2006.

4

O.C. Zienkiewicz "The Finite Element Method in Engineering Science", London, McGraw-Hill, 1971

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