Issue 51

M. Pepe et alii, Frattura ed Integrità Strutturale, 51 (2020) 504-516; DOI: 10.3221/IGF-ESIS.51.38 508 models considered fall in the field of discrete or distinct element methods, which have been proved to be particularly suitable for the study of masonry structures [45]. DEM model is based on the original numerical method formulated by [46], and recently developed by [43]. The model is based on the assumption of rigid block and mortar joints modeled as zero-thickness elastic-plastic interfaces, adopting a Mohr-Coulomb yield criterion. Masonry is seen as a system of rigid blocks, whose interactions are represented by forces and moments depending on their relative displacements and rotations. FEM/DEM method is a combination of discrete elements, originally formulated by [47], and developed by [48], it consists in a discrete element method in which the individual elements are meshed into finite elements, adopting a triangular discretization of the domain with embedded crack elements that activate whenever the peak strength is reached. The method, initially developed in the field of geo-mechanics, has been adopted to study the behavior of historical masonry [49]. Differently from the DEM described above, blocks can be assumed to behave as rigid or elastic bodies. Mortar joints might be idealized as elastic or elastic–plastic zero-thickness Mohr–Coulomb interfaces. Blocks are modeled by means of finite elements while interfaces are modeled as discrete elements. In this work, discrete models are realized adopting the FEM/DEM method. H ETEROGENEOUS FEM A NALYSIS he second comparison model is a heterogeneous finite element model, by which masonry is described as a system of n deformable blocks and m dry joints in which all the nonlinearities due to unilateral contact together with decohesion and friction phenomena are lumped. In detail, masonry blocks are made by bulk finite elements with linearly elastic and isotropic material properties, whereas dry joints are represented as zero-thickness damage-based cohesive interface elements placed in between, equipped with an intrinsic mixed-mode displacement-based traction-separation law (TSL). The variational formulation for the equilibrium problem at the microscopic scale is rather classic and, thus, not reported here for the sake of brevity. However, with the aim to find more theoretical and computational details, the reader is referred to the works [13,16,17], in which this formulation is embedded within a more general multiscale framework. In this paper the following mixed-mode TSL of the type   = T T  , proposed by [50], is chosen (see Fig. 2), incorporating unilateral contact with friction in an approximated manner, exploiting a penalty approach to enforce the highly nonlinear kinematical constraint in the normal direction to the joint surface:         max n n nc n p n n max s n sc s max s s p n n sc 1 f , 0 T K , 0 1 f , 0 T 1 f sign K , 0                                (1.4) where   , T n s T T  T and   = , T n s    denote the traction and displacement jump vectors, written in the local coordinate system   , n s attached to the interfaces; nc  and sc  are the critical values of the normal and tangential displacement jump components, corresponding to the total decohesion in pure mode I and mode II, respectively; p K is the penalty stiffness constant for contact enforcement;   max f  is a damage function, defined as follows:     2 max max max max max max 27 1 2 , 1 4 0, 1 T f               (1.5) T