Journal for Multiscale Computational Engineering, 9 (5): 565–578 (2011) NON-LOCAL COMPUTATIONAL HOMOGENIZATION OF PERIODIC MASONRY Andrea Bacigalupo & Luigi Gambarotta * Department of Civil, Environmental and Architectural Engineering, University of Genova, via Montallegro, 1–16145 Genova, Italy * Address all correspondence to Luigi Gambarotta, E-mail: gambarotta@dicat.unige.it Micro-polar and second-order homogenization procedures for periodic elastic masonry have been implemented to include geometric and material length scales in the constitutive equation. From the evaluation of the numerical response of the unit cell representative of the masonry to properly prescribed displacement boundary conditions related to homogeneous macro-strain fields, the elastic moduli of the higher-order continua are obtained on the basis of an extended Hill–Mandel macro-homogeneity condition. Elastic moduli and internal lengths for the running bond masonry are obtained in the case of Cosserat and second-order homogenization. To evaluate these results, a shear layer problem representative of a masonry wall subjected to a uniform horizontal displacement at points on the top is analyzed as a micro-polar and a second-order continuum and the results are compared to those corresponding with the reference heterogeneous model. From this analysis the second-order homogenization appears to provide better results in comparison with the micro-polar homogenization. KEY WORDS: computational homogenization, micro-polar continuum, second-order continuum, periodic micro-structure, masonry, characteristic length, boundary shear layer 1. INTRODUCTION Classical homogenization methods (i.e., first-order or Cauchy homogenization procedures) have been proposed and applied to derive the average properties of periodic masonry that depend on the properties of the constituents (brick/ blocks, mortar) and their arrangements (see Mistler et al., 2007). However, it is well known that first-order homoge- nization methods have disadvantages because the size of the micro-structure is considered to be irrelevant and, hence, micro-structural and geometrical size effects are not taken into account in the resulting overall constitutive equations. To overcome the limitations of the classical approach, nonlocal continuum theories, like multi-field and higher-order equivalent continua, are usually considered and proper homogenization techniques are developed, with particular reference to material with a periodic micro-structure. Overall constitutive equations for a two-dimensional Cosserat continuum representative of some periodic ma- sonry have been obtained by Trovalusci and Masiani (2005), Sulem and M¨ uhlhaus (1997), and Salerno and de Felice (2009) through a homogenization procedure based on an idealization of the masonry as an assemblage of rigid blocks represented as a Lagrangian system composed of bodies interacting through linear elastic interfaces. Although these approaches provide overall elastic moduli that depend, in closed form, on the mechanical and geometric characteris- tics of the components, the rigid-block assumption seems rather restrictive in many cases when brick compliance is comparable to that of mortar. To include this effect, a Cosserat computational homogenization of a representative unit cell extracted from the periodic masonry made up of elastic brick units and mortar joints has been proposed by Ca- solo (2006), where the rotational degrees of freedom of the micro-polar homogenized continuum have been identified through a heuristic evaluation of the mean local rotation of the brick units. Higher-order homogenized equations for periodic, linear elastic heterogeneous media have been deduced by Bakhvalov and Panasenko (1984), Boutin (1996), Triantafyllidis and Bardenhagen (1996), Smyshlyaev and Chered- 1543–1649/11/$35.00 c ⃝ 2011 by Begell House, Inc. 565