Keywords: masonry structures, first-order homogenization, failure envelopes, meso-to-macro transition.

Masonry structures have been extensively used in the whole history of mankind. In the current engineering practice, the design and assessment of masonry structures typically relies on traditional phenomenological approaches that incorporate a number of simplifying assumptions such as isotropy of the material or its perfectly brittle behavior. The results of experimental tests, however, demonstrate inadequacy of such assumptions and hence advanced constitutive models are needed to simulate the response of these structures; see, e.g., [3], where a two-dimensional orthotropic model was proposed to predict the behavior of masonry structures. Such constitutive models, however, consistently take into account neither properties of constituents nor their geometrical arrangement. Therefore, they require a new parameter fitting for each configuration of the analyzed structure.

An alternative approach to the constitutive modeling of masonry builds on the techniques of two-scale numerical homogenization, see, e.g. [6, and references therein]. In this framework, two different scales of the analysis are assumed to exist - the mesoscale level where a typical dimension is related to the size of a block and the macroscale level, which is the level of traditional engineering analysis. Linking the macroscale behavior to deformation processes taking place on mesoscale then yields a substitute for phenomenological closed-form constitutive laws. In the context of simulation of masonry structures, this approach was pioneered by Anthoine [1] for linear elastic behavior. An extension to nonlinear regime including macro-micro localization analysis can be found in recent collection of works [4].

In contradiction to the traditional application of homogenization (fibrous and textile composites), the definition of the geometry on mesoscale of regular natural stone masonry structures is relatively straightforward. In particular, the geometry is fully described by specifying the width and height of basic blocks, the thickness of head and bed joints and the type of a bond of a given structure. Such information can be easily obtained, e.g., by in-situ measurements or analysis of digital photographs of a structure.

Then, thanks to the assumed periodicity of the structure, the displacement field inside the unit cell admits the decomposition where is the prescribed overall strain and is the -periodic fluctuating displacement due to heterogeneities. In the present work, the commercial non-linear finite element code ATENA 2D [2] is used to solve the pertinent boundary value problem on mesoscale. The periodicity of the fluctuating displacement is ensured by tying the degrees of freedom related to displacements of the boundary of  (see [6] for more detailed discussion). The plastic-fracturing model [2] is used to describe the behavior of individual phases.

Typical output of the proposed homogenization procedure is the average stress vs average strain curves together with the simulation of initiation and evolution of cracks in the structure. In the two-scale modeling framework these results need to be properly transferred to mesoscale. Although the fully discrete approach, where the average stresses together with the algorithmic tangent stiffness matrix are extracted from the mesoscale analysis, has been recently successfully implemented [4], it is extremely computationally demanding even for modern distributed computer architectures. Hence, proper simplifying approaches are needed to decrease the cost of this step. The most straightforward application is the construction of failure envelopes as a design aid for a quick assessment of the load-bearing capacity. Another possibility is to interpret the homogenization results as "virtual tests" and extract parameters of the selected constitutive law from the computed curves. Finally, the recently introduced nonuniform transformation field analysis [5], which is based on extraction of typical deformation modes obtained during the analysis, can be employed as an efficient scale transfer technique.

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

3

P.B. Lourenço, R. de Borst, J.G. Rots, "A plane stress softening plasticity model for orthotropic materials", International Journal for Numerical Methods in Engineering, 40(21), 4033-4057, 1997. doi:10.1002/(SICI)1097-0207(19971115)40:21<4033::AID-NME248>3.0.CO;2-0

4

T.J. Massart, "Multi-scale modeling of damage in masonry structures", Ph.D. thesis, Technische Universiteit Eindhoven, 152 pp., 2003. doi:10.1007/s004660000212

5

J.C. Michel, P. Suquet, "Nonuniform transformation field analysis", International Journal of Solids and Structures, 40(25), 6937-6955, 2003. doi:10.1016/S0020-7683(03)00346-9

6

R.J.M. Smit, W.A.M. Breckelmans, H.E.H. Meijer, "Prediction of the mechanical behaviour of nonlinear heterogeneous systems by multi-level finite element modeling", Computer Methods in Applied Mechanics and Engineering, 155, 181-192, 1998. doi:10.1016/S0045-7825(97)00139-4

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