Issue 29

G. Gianbanco et alii, Frattura ed Integrità Strutturale, 29 (2014) 150-165; DOI: 10.3221/IGF-ESIS.29.14 164 the BVP. Also, the consistent macroscopic tangent stiffness matrix is evaluated and a localization procedure to find discontinuity bands at the macroscale is taken into account. Two numerical examples focusing on the UC response only are reported. The results of the BVP are analyzed to confirm the goodness of the Meshless Method to be employed to solve the problem at the mesoscale. The localization procedure for the two examples correctly identifies the failure mode type. Future developments regards the extension of this strategy to other heterogeneous periodic or non-periodic materials also for 3D analyses and the introduction of a non linear behavior for bricks. Particular attention will be paid on mesh- dependency for the response at the macroscale level. A CKNOWLEDGEMENTS he authors acknowledge the financial support given by the Italian Ministry of Education, University and Research (MIUR) under the PRIN09 project 2009XWLFKW_005, “A multiscale approach for the analysis of decohesion processes and fracture propagation”. R EFERENCES [1] Giambanco, G., Rizzo, S., Spallino, R., Numerical analysis of masonry structures via interface models, Comput. Methods Appl. Mech. Eng., 190 (49) (2001) 6493-6511. [2] Lourenço, P.B., Rots, J.G., Multisurface interface model for analysis of masonry structures, J. Eng. Mech. (ASCE), 123 (7) (1997) 660-668. [3] Alfano, G., Sacco, E., Combining interface damage and friction in a cohesive zone model, Int. J. Numer. Methods Engrg., 68 (2006) 542–582. [4] Spada, A., Giambanco, G., Rizzo, P., Damage and plasticity at the interfaces in composite materials and structures., Comput. Methods Appl. Mech. Eng., 198 (49) (2009), 3884-3901. [5] Sacco, E., Lebon, F., A damage-friction interface model derived from micromechanical approach., Int. J. Solids and Struct., 49 (26) (2012) 3666-3680. [6] Giambanco, G., Fileccia Scimemi, G., Spada, A., The interphase finite element, Comput. Mech., 50 (2012), 353-366. [7] Fuschi, P., Giambanco, G., Rizzo, S., Nonlinear finite element analysis of no-tension masonry structures, Meccanica, 30 (3) (1995) 233-249. [8] Kouznetsova, V., Geers, M.G., Brekelmans, W.M., Multi-scale constitutive modelling of heterogeneous materials with a gradient-enhanced computational homogenization scheme, Int. J. Numer. Meth. Eng., 54 (2002) 1235-1260. [9] Miehe, C., Koch, A., Computational micro-to-macro transitions of discretized microsctructures undergoing small strains, Applied Mech., 72 (2002) 300-317. [10] Feyel, F., Chaboche, J.L., FE 2 multiscale approach for modelling the elasto-viscoplastic behavior of long fibre SiC/Ti composite materials, Comput. Methods Appl. Mech. Eng., 183 (3) (2000) 309-330. [11] Nguyen, V.P., Stroeven, M., Sluys, L.J., Multiscale continuous and discontinuous modeling of heterogeneous materials: A review on recent developments, J. Multiscale Model., 3 (04) (2011) 229-270. [12] Geers, M.G.D., Kouznetsova, V.G., Brekelmans, W.A.M., Multiscale computational homogenization: Trends and challenges, J. Comput. Appl. Math., 234 (7) (2010) 2175-2182. [13] Massart, T.J., Multi-scale modeling of damage in masonry structures, Ph. D. Thesis, University of Technology, Eindhoven, (2003). [14] Addessi, D., Sacco, E., A multi-scale enriched model for the analysis of masonry panels, Int. J. Solids and Struct., 49 (6) (2012) 865-880. [15] Jirásek, M., Mathemathical analysis of strain localization, REGC – Damage and fracture in geomaterials, 11 (2007) 977-991. [16] Massart, T.J., Peerlings, R.H.J., Geers, M.G.D., An enhanced multi-scale approach for masonry wall computations with localization of damage, Int. J. Numer. Meth. Engrg, 69 (2007) 1022-1059. [17] Rice, J.R., The localization of plastic deformation, Theoretical and Applied Mech., Proc. 14 th IUTAM Congr., Delft, The Netherlands, 30 Aug-4 Sept (1976) 207-220. [18] Ottosen, N.S., Runesson, K., Properties of discontinuous bifurcation solutions in elasto-plasticity, Int. J. Solids and Struct., 4 (1991) 401-421. T