Issue 51

M. Pepe et alii, Frattura ed Integrità Strutturale, 51 (2020) 504-516; DOI: 10.3221/IGF-ESIS.51.38 505 I NTRODUCTION asonry is a composite (heterogeneous) material obtained by assembling natural or artificial blocks by means of mortar layers or dry joints and it is one of the more common structural materials adopted for centuries for ordinary or monumental constructions. The investigation of its mechanical behavior plays a fundamental role in view of the protection and conservation of architectures of historical and archaeological interest. However, to deal with the structural response of historical masonry structures is a complex task. In the last decades a large variety of numerical models and approaches have been proposed in literature, but no one can be applied in a general manner regardless the constructive typology. The selection of the most appropriate modelling strategy is indeed strictly related with the nature of the object to analyze. Depending on the adopted model for analysis it is possible introduce three distinct categories: micro-mechanical, macro- mechanical models and multiscale models. The choice of a micro-mechanical model involves a distinct representation of masonry constituents (units, mortar and unit/mortar interface) whose properties are obtained from experimental tests on small masonry specimens. A micro-mechanical model is suitable for a very detailed response [1-5], but this approach has a limitation represented by the great computational effort due to the high number of degrees of freedom connected to each unit and joint in case of real masonry structures, characterized by considerable number of units. Macro-mechanical models describe masonry as homogeneous continuum, use phenomenological constitutive laws for constituents, including also some inner variables for damage, and friction coefficients. The material parameters are derived by means of experimental tests on small masonry specimens or directly on the single constituents. Macro-mechanical models are characterized by high computational efficiency, as they do not provide an accurate description of the internal structure of masonry material [6-8]. Multiscale continuum models represent a very promising approach for the analysis of masonry structures since they can accurately retain memory of the mechanical and geometrical properties of the material (micro-structure) together with the capability to contain the computational effort compared to a fully micro-mechanical model [9-12]. These models are often derived by considering only two material scales: a micro-scale where, after deducing the mechanical properties of the components, preferably through experimental tests, a material representative volume element (RVE) is defined and a macro-scale continuum model, is obtained by performing a homogenization procedure in most cases based on the solution of notable boundary conditions problems for the RVE. Among these approaches, both concurrent and semi- concurrent multiscale models have been proposed in the literature for masonry-like materials, the first ones referring to a strong coupling between the micro- and macro-levels [13-16], the latter ones, often referred to as computational homogenization models, characterized by an only weak (although two-way) coupling between them [17-20]. Most of the existing approaches are devoted to the mechanical behavior of periodic (i.e. regular) masonries, for which a suitably defined unit cell plays the role of RVE, but there exist also different homogenization techniques for both linear and nonlinear analyses of random microstructures, already applied or directly applicable to irregular masonry structures [21-23]. Other multiscale strategies have been proposed that exploit different homogenization techniques based on the so-called Cauchy rule, and its, generalizations [24] that allowed the derivation of both classical and generalized continua able to properly represent scale effects, that in masonry materials are prominent [25-29]. In this work the attention is mainly focused on the category of micromodels particularly focusing on Limit Analysis, which represents a very effective tool to estimate the collapse load and collapse mechanism for one-leaf masonry structures [30-35]. In particular the model here presented considers an associative flow-rule connected to a dilatant behaviour of the joints. This hypothesis has been successfully adopted to study the behaviour of historical masonry structures [36-38] even if it may occur that the flow rule is quasi-associated under shear actions, depending by the level of pre-compression of masonry. It must be pointed out that a limitation of Limit Analysis (both associated and not associated) concerns ductility that is assumed infinite, and this hypothesis not always fits well the softening behaviour of masonry [39]. A validation of the proposed model is provided, via suitable comparisons with two models that finely describe the microstructure and already adopted to the evaluation of the in-plane failure behavior of masonry panels: (i) a discrete model based on a combined Finite/Discrete Element method (FEM/DEM) [40, 41], and (ii) a heterogeneous nonlinear Finite Element model (FEM), derived from the works [13, 15, 16]. The three models are used to reproduce the analysis performed in [42] - here regarded as a benchmark - on several masonry panels with different height-to-width ratio and with or without openings. The comparison of the numerical M