Keywords: masonry, first-order homogenization, fracture energy, unit cell definition.

It is now being widely accepted fact that hierarchical or multi-scale homogenization techniques offer a reliable route in numerical investigation of deformation and failure processes in complex modern materials. Owing to the high computational complexity of fully coupled homogenization schemes, uncoupled (or first-order) homogenization schemes still remain the most appealing ones in multi-level modeling of heterogeneous materials and structures. Here, the main objective is to define a model on the macroscopic level of observation almost independently of the analysis on smaller length scales. This clearly allows a substantial reduction of computational cost together with very efficient parallelization of the whole procedure [2].

Nevertheless, for realistic real-world large structures, such as the rehabilitation of the Charles Bridge in Prague [1], this approach is still too demanding. Moreover, we believe that from the practical point of view it is sufficient to capture "only" the most important source of damage in a structure on the basis of slightly simplified models. Hence, the goal of the present contribution is to define objective macroscopic parameters on the macro-scale, which would eventually allow the introduction of a "homogenized" material law for the macroscopic analysis. In this modeling framework such analysis can be run as a separate task, which now becomes a manageable problem [1].

In doing so, the analysis on a meso-scale usually draws upon the existence of a periodic heterogeneous medium. When such a medium is loaded in the elastic or plastic regime with the limit to small strains and more or less uniform fields, it is well-known that the analysis on the meso-scale can be reduced to the smallest periodic unit cell (PUC). However, even for linearly elastic but geometrically non-linear deformation modes more than one smallest unit cell is necessary in order to correctly capture, e.g., a proper buckling mode [3]. The situation is even more complex when strain-softening materials are taken into account. It is still an open question in the homogenization community whether a concept of periodic unit cell is in such a case still applicable in a meso-macro approach [4].

In view of the above comments, it has been suggested in the literature that the macroscopic stress-strain path may depend on the size of the PUC. This conclusion, however, has not been confirmed by the present study. When assuming a perfect periodic structure with periodic boundary conditions, it is clear that up to the onset of damage, the homologous points in a periodic medium has to experience the same stresses and strains. Therefore, the damage initiation will occur at these points simultaneously. Provided that the finite element meshes are equivalent from the point of view of element sizes and number of elements in the smallest PUC, the damage pattern remains the same. Therefore, from the macroscopic point of view, the material response is identical for any number of unit cells, which results in a perfect match of the macroscopic stress-strain paths derived from periodic unit cells of various sizes. Such a conclusion promotes the use of only the smallest unit cell even for strain softening media. Note, however, that the periodicity conditions do not allow a direct scale up of the results into macro-scale with expected localization so that a special treatment on that level must be introduced when applying the concept of meso-macro approach. A suitable method of attack is presented in [5]. Nevertheless, since the main objective of the presented analysis is to extract only the basic macroscopic properties such as homogenized tensile and compressive strengths and homogenized fracture energy, we believe that this approach is still acceptable.

To conclude, we recall that all numerical simulations were performed with the commercial software ATENA 2D [6] using a plastic-fracturing NonLinearCementitious model exploiting the mesh-adjusted softening modulus in the smeared-crack approach to address the behavior of individual stone blocks and mortar phase.

1

J. Novak, J. Zeman and M. Sejnoha, "Multiscale-based constitutive modeling of regular natural stone masonry", In Proceedings of The Seventh International Conference on Computational Structures Technology, Edited by B.H.V. Topping and C.A. Mota Soares, Libon-Portugal, 7-9 September, paper No. 197, 2004. doi:10.4203/ccp.79.197

2

K. Matsui, K. Terada and K. Yuge, "Two-scale finite element analysis of heterogeneous solids with periodic microstructures", Computers & Structures, 82, 593-606, 2004. doi:10.1016/j.compstruc.2004.01.004

3

I. Saiki, K. Terada, K. Ikeda and M. Hori, "Appropriate number of unit cells in a representative volume element for micro-structural bifurcation encountered in a multi-scale modeling", Computer Methods in Applied Mechanics and Engineering, 191, 2561-2585, 2002. doi:10.1016/S0045-7825(01)00413-3

4

M. Stroeven, H. Askes and L. J. Sluys, "Numerical determination of representative volumes for granular materials", Computer Methods in Applied Mechanics and Engineering, 193, 3221-3238, 2004. doi:10.1016/j.cma.2003.09.023

5

T. J. Massart, "Multi-scale modeling of damage in masonry structures", Ph.D. thesis, Technische Universiteit Eindhoven, 2003.

6

V. Cervenka, L. Jendele and J. Cervenka, "ATENA Program Documentation: Part I: Theory", Cervenka Consulting Company, Prague, 2002.

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