Issue 29

G. Gianbanco et alii, Frattura ed Integrità Strutturale, 29 (2014) 150-166; DOI: 10.3221/IGF-ESIS.29.14 151 degradation, loss of adhesion and unilateral contact with related frictional phenomena. The contribution of internal stresses and strains at the interface level has been finally inserted by Giambanco et al. [6] to enrich the interface model. The mesoscopic approach is a rigorous method of analysis but many difficulties arise in the mesh creation and a fine discretization of the structure has to be used, which lead to prohibitive computational costs for large structures. The macroscopic approach considers the structure as constituted by a fictitious homogeneous and continuous material. In this approach simplified constitutive models are formulated in a phenomenological manner, intentionally neglecting the actual material composition but smearing the effects of the heterogeneities presence by appropriate mechanical assumptions [7]. The multiscale techniques belong to the second approach and couple different scales of interest by means of specific transition laws capable to exchange information between different consecutive scales [8-10]. According to [11] three homogenization techniques exist: the analytical homogenization, the numerical homogenization, and the computational homogenization (CH). In particular the CH consists in the following steps. A macroscopic strain or stress is used to apply kinematic or static boundary conditions to the UC ( macro-meso scale transition ). The equilibrium of the UC is obtained by solving a BVP at the mesoscale. Lastly, the macro-strains or macro-stresses are assumed to be the average on the UC of the corresponding meso-strains or meso-stresses ( meso-macro scale transition ). In a strain driven process, the aforementioned boundary conditions are of Dirichlet type and are chosen linear or periodic. The latter imply the repetition of inelastic phenomena for adjacent UCs. Despite it is known in literature that periodic conditions offer a better estimate of stiffness matrix with respect to linear conditions, in this paper linear conditions are used, with the aim to verify the difference in future works. Two different CH methods are distinguished. In the first order CH method Cauchy models are used at all scales. This method is based on the satisfaction of the Principle of Separation of Scales , which asserts that the characteristic size of the UC is much smaller then the size of the structure [12]. Massart [13] proposed an enhanced multiscale model for masonry structures using nonlocal implicit gradient isotropic damage models for both constituents of the UC. A second CH method employs Cosserat or higher order continua to consider those cases in which strong strain or stress gradients arise at the macroscopic level or the mesoscopic dimensions of the UC constituents are comparable with the ones at the macro-level. Addessi and Sacco [14] analyzed masonry panels in the framework of transformation field analysis by using a two-dimensional Cosserat continuum for macro-scale and a non-linear damage contact-friction model for the mortar joints at the meso-scale. Many authors have also faced the problem of localization of deformations. For ductile materials with an elastic-plastic behavior sufficiently deformed into the plastic range, the abrupt changes in the deformation field occur across narrow zones (shear bands). This phenomenon is called localization of plastic deformation. Rice [17] and Ottosen and Runesson [18] studied the problem and found the conditions under which the localization occurs. Massart et al [16] inserted the localization procedure into a multi-scale problem, considering a gradient damage model at the mesoscale. In this paper a multiscale computational strategy for the analysis of masonry structures is presented. The applied technique belongs to the first order CH methods. The UC chosen at the meso-scale is composed by a brick and half of its surrounding joints. Brick is assumed elastic while mortar joints are modeled by means of zero-thickness elasto-plastic interface elements. The BVP is solved applying the Meshless Method (MM) and the UC is subjected to linear boundary displacements. Unlike FEM, in MM the shape functions are only approximants and not interpolants. These are obtained by defining the influence of nodes on a fixed point by means of weight functions, which have a compact support. The support size is dependent by the so-called dilatation parameter or smoothing length. The smoothing length is critical for the solution accuracy and stability, it plays the role of the element size in the finite element method. The MM is employed to overcome the problem of computational cost associated with the use of a classical FEM analysis at the meso-scale. This method is also able to easily treat the problem of crack formation and propagation. It will therefore be applied to describe crack formation and propagation in the brick, that is one of the future developments of the present work. Another advantage is related to the possibility to use higher-order continuous weight functions. The macro-scale consistent tangent stiffness matrix and localization of deformation are derived. Numerical examples show the behavior of the UC and results of localization for two different cases, i.e. when boundary displacements enslave the UC to normal and shear strains. T HE COMPUTATIONAL HOMOGENIZATION ( CH ) SCHEME et us consider, in the Euclidean space  3 referred to the orthonormal frame 1 2 3 ( , , , ) O i i i , a structure  constituted by an heterogeneous material, at the mesoscale (Fig. 1). L