Keywords: periodic homogenization, failure, masonry, plasticity, interface.

Introduction

The strength of masonry material is the result of the interaction of complex phenomena. Expensive experimental procedures are usually needed to evaluate the strength of masonry. Its complex response is dictated by phenomena at the scale of its constituents, including a strong dependence on the loading path. Computational tools for the study of masonry structures are nowadays the subject of intensive research. Such tools are mostly based on a single scale representation, i.e. either on macroscopic or on mesoscopic models. The former consider the material at a structural scale with constitutive laws linking average stresses and strains in an equivalent homogeneous material. The latter consider the material at the scale of the basic constituents of the masonry structure, i.e. bricks and mortar joints, which prevents their use in computations of large structures [1]. From an identification point of view, macroscopic models are based on failure envelopes and on expensive experiments, while mesoscale tests are cheaper and more reliable. An alternative approach thus consists in identifying the macroscopic parameters of masonry with homogenization techniques. In this case, a mesoscopic model is used to quantify these parameters on a Representative Volume Element. This technique was used for elastic behaviour [2] and for failure prediction based on mesoscopic scalar damage for specific load cases, or for failure envelopes [3]. The objective of the paper is to build failure envelopes based on periodic homogenization, with more accurate mesomodels than hitherto, using for instance the model proposed in Lourenço [1].

Homogenization

The identification of equivalent continuum properties is based on averaging relations carried out on a RVE. The macroscopic strain is required to be identical to the average of the mesoscopic strain field of the RVE. The same requirement is used to relate mesoscopic and macroscopic internal works. The strain averaging relation is classically obtained by the selection of a mesoscopic displacement field of the following form where a mesostructure induced fluctuation is added to the linear field:

is the constant overall strain tensor, is the position vector of a point in the RVE. Using this displacement field, the strain averaging requirement leads to:

In the frame of periodic homogenization, the field is unknown inside the unit cell and is forced to be periodic at the boundaries. In such a case, the integral of equation (64) indeed vanishes. Efficient mesoscopic models have been developed [4], but have not been investigated yet to build full envelopes with homogenization methods. The aim of the paper is to build such an envelope with the multisurface plasticity model of Lourenço [1], which is based on an interface formulation representing potential cracks. This model can incorporate complex responses in mode I and mode II fractures of mortar joints, with independent fracture energies. These envelopes are built based on proportional loading in the macroscopic stress space and are assessed in terms of load carrying capacity and of failure mode predictions. The calculation is done over a unit cell, which is the smallest periodic RVE possible. The load carrying capacity is assumed to coincide with the peak of the load displacement curve of the unit cell under proportional loading. To ensure the identification of a true failure mechanism, the computation is carried out using an adaptive path following method as presented in [5]. The failure patterns are analyzed for some of the loading paths to ensure a complete failure of the brick.

Acknowledgement

This research is supported by the Région Wallone de Belgique in the frame of the Objective III programme, under grant number 215089 (HOMERE)

1

P.B. Lourenço, "Computational Strategies for Masonry Structures", PhD thesis, Delft University of Technology, 1996.

2

A. Anthoine, "Derivation of the in-plane elastic characteristics of masonry through homogenization theory", International Journal of Solids and Structures, 32(2), 137-163, 1995. doi:10.1016/0020-7683(94)00140-R

3

T.J. Massart, "Multi-scale modeling of damage in masonry structure", PhD thesis, Universit´e Libre de Bruxelles and Eindhoven University of Technology, 2003.

4

P.B. Lourencço, J G. Rots, "A multi-surface interface model for the analysis of masonry structures". SCE Journal of Engineering Mechanics, 123(7), 660-668, 1997. doi:10.1061/(ASCE)0733-9399(1997)123:7(660)

5

M.G.D. Geers, "Enhanced solution control for physically and geometrically non-linear problems. Part I - The subplane control approach", International Journal for Numerical Methods in Engineering, 46, 177-204, 1999. doi:10.1002/(SICI)1097-0207(19990920)46:2<177::AID-NME668>3.0.CO;2-L

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