Keywords: quarry masonry structures, first-order homogenization scheme, damage and plasticity models, damage-induced anisotropy.

The quarry masonry as a structural material has been extensively used in the history, mainly due to wide availability of the material as well as its good mechanical properties. Therefore, in the context of repair and preservation of historical structures, a reliable and realistic constitutive model of material behavior is needed for a proper planning of necessary technological operations. In the past, design and construction of these structures has been based on a balanced combination of experience and trial-and-error methods, which compensated for limited knowledge of material behavior.

Even nowadays, in spite of substantial progress in constitutive and numerical modeling, the engineering analysis of these structures builds upon a number of simplifying assumptions and phenomenological relations. This can be primarily attributed to the fact that quarry masonry is a heterogenous material with very complex structure consisting of phases that exhibit quasi-brittle behavior. In the last decade, techniques of numerical first-order homogenization have acquired an increasing attention in realistic modeling of regular masonry structures, both in the elastic [1] and inelastic range [5]. The most attractive features of the homogenization-based approach is that the non-linear behavior of the material, its initial anisotropy together with its evolution due to progressive failure, observed for real-world structures [3], directly follow from the analysis of a unit cell of a masonry structure [5] rather than from a particular format of the constitutive model. The present contribution addresses a specific application of these techniques to the homogenization of quarry masonry forming the filling of the Charles Bridge in Prague.

The first step of a successful implementation of the numerical homogenization analysis is the definition of geometry of the periodic unit cell that represents the morphology of a real material. In the context of the present problem, the only morphology data available are provided by in-situ observation. Based on the analysis of core and dug holes, the volume fraction of arenaceous marl blocks, average joint thickness and representative shapes of individual stone blocks were estimated and used to construct an eight-stone periodic unit cell with dimensions 0,64 m0,4 m and the stone volume fraction being 80%. The unit cell was discretized by triangular elements using the mesh generation code T3D [6]. The periodicity tyings were implemented to ensure the periodicity of stress and strain fields in the unit cell, see, e.g., [4] for more details. The behavior of individual phases was described by the plastic-fracturing model implemented in ATENA 2D program [2]. The main input data of the current model are the tensile and compressive strength, the elastic modulus and specific fracture energy of individual phases. Note that, due to unavailability of these parameters for the present material system, two representative sets of material parameters were used to reliably simulate the response of the material. Finally, the unit cell was subjected to incremental "prescribed-strain" load path to study the overall behavior.

The obtained response of the unit cell confirmed the ability of the model to (at least qualitatively) represent the complex behavior of irregular quarry masonry. For a loading dominated by the macroscopic strains and , a number of aligned distributed cracks appear first followed by the closing of some existing cracks and opening of new ones. Finally, a magistral crack forms, which leads to the final failure of the material. The similar behavior can be observed for both compressive and tension loading. For the shear-strain driven response, the basic characteristics of deformation process remain the same. Contrary to the previous case, however, the overall behavior shows the strain hardening instead of softening observed for and loading for both sets of material parameters. This seems to comply well with the experimental results.

In summary, the proposed methodology appears to be able to obtain quantitatively correct behavior of the material. Moreover, the damage-induced anisotropy evolution, neglected in simplified engineering approaches, will be captured by the present numerical simulation. Once the missing material data for individual phases are provided by experiments, the effects of repair and strengthening operations on analyzed structure can be realistically assessed.

1

A. Anthoine, "Derivation of in-plane elastic characteristics of masonry through homogenization theory", International Journal of Solids and Structures, 32(3): 137-163, 1995. doi:10.1016/0020-7683(94)00140-R

2

V. Cervenka, L. Jendele and J. Cervenka, "ATENA Program Documentation - Part I : Theory", Cervenka Consulting Company, 117 pp., 2002.

3

K. De Proft, "Combined experimental-computational study to discrete fracture of brittle materials", Ph.D. thesis, Vrije Universiteit Brussel, 2003.

4

V.G. Kouznetsova, W.A.M. Breckelmans, F.T.P. Baaijens, "An approach to micro-macro modeling in heterogeneous materials", Computational Mechanics, 27, 37-48, 2001. doi:10.1007/s004660000212

5

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

6

D. Rypl, "Sequential and parallel generation of unstructured 3D meshes", CTU Reports, 2(3), 164 pp., 1998.

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