Keywords: masonry homogenization, irregular structure, random morphology, statistically optimized periodic unit cells, global stochastic optimization, FFT-based method.

Two-scale homogenization of masonry structures has attracted a considerable attention in the last decade, especially in connection with reconstruction and rehabilitation of historical structures; see, e.g., [1, and references therein]. A particular application of this methodology to numerical assessment of quarry and regular natural stone masonry of the Charles Bridge in Prague has been presented in contributions [5] and [3], respectively. In both cases, the obtained results allow us to conclude that the multiscale modelling strategy is fully capable of delivering at least qualitatively correct answer once the geometry and material properties of individual constituents are specified.

Recall that for the quarry masonry [5], geometry of the unit cell was estimated on the basis of results of dug and core holes analysis. In this case, such a simplification appears to be necessary due to rather limited information available. A rather different situation appears for regular stone masonry [3] where geometry of the unit cell can be directly specified by in-situ measured parameters. A visual inspection of the Charles Bridge reveals that stone masonry in certain locations of the structure exhibits significant irregularities. For these parts, however, an exhaustive characterization of a geometrical configuration of individual constituents can be obtained from digitized photographs of the structure. Nevertheless, the definition of the periodic unit cell is not very straightforward even from these detailed data. One possible option is the numerical analysis of the whole micrograph as one unit cell by, e.g., the finite element method. This is, however, a rather complex task mainly due to complicated mesh generation and expensive non-linear analysis of the unit cell. Other approach, originated in the work of Povirk [4] and further extended in [6], builds upon replacing the complex and possibly non-periodic structure by a simpler periodic unit cell, which still optimally resembles the original material in sense of selected statistical descriptors.

The key stone of the considered framework is a selection of proper description of heterogeneous materials with random structure. Here, instead of relying on detailed specification of positions of individual constituents, an attention is paid to the determination of proper statistical descriptors which contain non-trivial information about the shape and topology of individual phases. In the context of present work, we restrict our attention to two specific descriptors - one- and two-point probability functions and , see, e.g. [7, and references therein] for more details. The particular choice is related to the fact that they can be easily evaluated even for high-resolution digitized images of real-world materials.

Once geometry of the disordered material has been quantified, the model of an idealized periodic unit cell needs to be defined. In this contribution, a two-layer unit cell with running bed joints was constructed. It incorporates different widths and heights of individual blocks as well as thicknesses of bed and head joints and a relative horizontal shift of individual layers. Note that the choice of the unit cell was motivated by similar analysis performed for textile composites [7] that revealed necessity of at least two-layer unit cells for realistic description of real-world heterogenous materials. The optimal parameters of the unit cell (in particular, the selected unit cell is determined by twelve values), then follow from minimization of the least square error where is the two-point probability function related to the original microstructure while stands for the two-point probability function of the idealized unit cell. A closer inspection suggests that the objective function is non-convex, multimodal and discontinuous due to the effect of limited bitmap resolution. Based on our previous works, a stochastic global optimization algorithm based on combination of real-valued genetic algorithms and the simulated annealing method, see [6], was employed to solve this optimization problem. The quality of the resulting unit cells is then demonstrated by comparing effective elastic properties computed by FFT-based numerical algorithm proposed in [2].

1

T.J. Massart, "Multi-scale modeling of damage in masonry structures", Ph.D. thesis, Technische Universiteit Eindhoven, 152 pp., 2003.

2

H. Moulinec, P. Suquet, "A fast numerical method for computing the linear and nonlinear mechanical properties of composites", Comptes Rendus de l'Academie des Sciences Serie II, 318(11), 1417-1423, 1994.

3

J. Novák, J. Zeman, M. Šejnoha, "Multiscale-based constitutive modeling of regular natural stone masonry", Proceedings of CST 2004 conference, 2004. doi:10.4203/ccp.79.197

4

G.L. Povirk, "Incorporation of microstructural information into models of two-phase materials" Acta Metallurgica et Materialia, 43(8), 3199-3206, 1995. doi:10.1016/0956-7151(94)00487-3

5

M. Šejnoha, V. Blazek, J. Zeman, J. Šejnoha, "Non-linear homogenization of quarry masonry", Proceedings of CST 2004 conference, 2004. doi:10.4203/ccp.79.196

6

J. Zeman, M. Šejnoha, "Numerical evaluation of effective properties of graphite fiber tow impregnated by polymer matrix", Journal of the Mechanics and Physics of Solids, 49(1), 69-90, 2001. doi:10.1016/S0022-5096(00)00027-2

7

J. Zeman, M. Šejnoha, "Homogenization of plain weave composites with imperfect microstructure: Part I-Theoretical formulation", International Journal of Solids and Structures, accepted for publication, 2004. doi:10.1016/j.ijsolstr.2004.05.011

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