Homogenization and multiscale analysis

This chapter presents a survey of the homogenization and multiscale analysis of periodic, masonry walls subjected to in-plane loading conditions. The scale of the structure, macroscale, and that of the material, microscale, are introduced, invoking the assumption of the separation of the scales. Two-dimensional (2D) Cauchy and Cosserat models are adopted at the macroscale to describe the response of the wall. A unit cell (UC) is introduced and analyzed adopting three-dimensional, 2D, and enriched plane state Cauchy models. A damage-plastic model is considered for the mortar joints, accounting for the unilateral effect due to reclosure of the mortar joint and friction occurring when the microcracks are closed. A different damage model is introduced for bricks capable of properly describing their compressive failure. The homogenization for deriving the overall nonlinear response of the UC is performed by developing a piecewise transformation field analysis technique. Several numerical applications are presented to assess the material modeling and the effectiveness of the adopted homogenization technique and that of the multiscale approach. These concern the response of the UC to verify, on the one hand, the ability of the constitutive laws proposed for brick and mortar to describe the response of the masonry and, on the other hand, the performance of the homogenization technique. Then, structural analyses of masonry elements are presented, comparing the results obtained using different modeling approaches among them and with available experimental evidence. Finally, concluding remarks are given.