This paper presents a multi-scale model for the analysis of the in-plane structural response of regular masonry. It is based on a computational periodic homogenization technique and is characterized by the adoption of the Cosserat continuum model at the macroscopic structural level, taking into account the influence of the microstructure on the global response and correctly describing the localization phenomena; at the microscopic representative volume element (RVE) level, where the nonlinear constitutive behavior, geometry, and arrangement of the masonry constituents are modeled in detail, a standard Cauchy model is employed. An isotropic nonsymmetric damage model is adopted for the bricks and mortar joints. The solution algorithm is based on a parallelization strategy and on the finite-element method. Some numerical applications on typical masonry structures are reported, showing both the global response curves and the stress and damage distributions on the RVEs.

Different scale approaches, micromechanical, multiscale and macromechanical or phenomenological, are presented to study the structural response of masonry elements. First, a micromechanical model is introduced and the masonry is considered to be a heterogeneous material, made of mortar and bricks joined by interfaces, where the mortarbrick decohesion mechanisms occur. To this end, a special interface model combining damage and friction is proposed. Then, two multiscale procedures are presented, that consider regular arrangements of bricks and mortar, modeled by nonlinear constitutive laws which account for damage and friction effects. A homogenization technique is developed to derive two different equivalent continuum models at the macro-level, a micropolar Cosserat continuum and a nonlocal Cauchy model. Finally, a macromechanical model, based on the adoption of a classical No-Tension Material (NTM) model, and on the presence of irreversible crushing strains, is proposed. A zero tensile strength is assumed, thus fracture strains arise when the stress is zero. Moreover, an elastoplastic model is considered for the material response in compression. Numerical applications are performed on a masonry arch and two masonry panels, by adopting the three approaches presented. Comparisons with experimental outcomes, published elsewhere, are performed.