SUMMARYThe damage model for mortar joints proposed in the companion paper (Reference 1) is here applied to an extended approach for the evaluation of the lateral response of in-plane loaded brick masonry shear walls. The continuum model considered here is based on the simplifying assumption of an equivalent stratified medium made up of layers representative of the mortar bed joints and of the brick units and head joints, respectively. The constitutive equations for the brick masonry are obtained through a homogenization procedure involving the damage model proposed in the companion paper and simple damage constitutive equations for the brick layer. The constitutive model is used in a finite element analysis of the lateral response of brick masonry shear walls in-plane loaded either by cyclic horizontal actions superimposed on vertical loads or by dynamic loads, which are representative of the seismic actions. The capabilities and the validity limits of the finite element analysis obtained by the continuum approach are indicated from the simulation of experimental results concerning rectangular slender and squat shear walls and also by the comparison with the theoretical results from the composite model proposed in the companion paper. Moreover, simulations of the experimental results from large-scale brick masonry shear walls carried out at the University of Pavia are presented. Finally the same shear wall has been analysed under dynamic strong motion at the base from which the suitability of the approach for the evaluation of the seismic vulnerability of masonry buildings emerges.1997 by John Wiley & Sons, Ltd.

SUMMARYThe response of brick masonry walls to in-plane horizontal cyclic loads analogous to those induced during seismic events is analysed by applying constitutive models which take into account the mechanical behaviour of each component and its interfaces, i.e. decohesion and slipping in the mortar joints and failure in bricks. To this end, a damage model for mortar joints is proposed and then applied in two different approaches to the analysis of brick masonry walls which are described both in the present paper and in the companion paper. The response of the mortar joint model to varying stress and strain is here analysed and applied to the simulation of experimental results from shear tests on mortar-brick assemblages. From the description of the mortar joints and assuming brittle constitutive equations for the brick units, a composite model based on a finite element approach is here developed and applied to the lateral analysis of rectangular shear walls. Even if this model turns out to be computationally burdensome, it may give information on the inelastic mechanisms and related strains by means of a local description of the element motion. Some simulations of the lateral response of experimented walls under cyclic horizontal actions superimposed on vertical loads are carried out and an interpretation of the influence of the wall geometry on the lateral stiffness degradation and on the hysteretic energy dissipation is given.1997 by John Wiley & Sons, Ltd.

A procedure for second-order computational homogenization of heterogeneous materials is derived from the unit cell homogenization, in which an appropriate representation of the micro-displacement field is assumed as the superposition of a local macroscopic displacement field, expressed in a polynomial form related to the macro-displacement field, and an unknown micro-fluctuation field accounting for the effects of the heterogeneities. This second contribution is represented as the superposition of two unknown functions each of which related to the first-order and to the second-order strain, respectively. This kinematical micro-macro framework guarantees that the micro-displacement field is continuous across the interfaces between adjacent unit cells and implies a computationally efficient procedure that applies in two steps. The first step corresponds to the standard homogenization, while the second step is based on the results of the first step and completes the second-order homogenization. Two multi-phase composites, a three-phase and a laminated composite, are analysed in the examples to assess the reliability of the homogenization techniques. The computational homogenization is carried out by a FE analysis of the unit cell; the overall elastic moduli and the characteristic lengths of the second order equivalent continuum model are obtained. Finally, the simple shear of a constrained heterogeneous two-dimensional strip made up of the composites considered is analysed by considering a heterogeneous continuum and a homogenized second-order continuum; the corresponding results are compared and discussed in order to identify the validity limits of the proposed technique

The homogenization of periodic hexachiral and tetrachiral honeycombs is dealt with two different\ud techniques. The first is based on a micropolar homogenization. The second approach, developed to \ud analyse two-dimensional periodic cells consisting of deformable portions such as the ring, the \ud ligaments and possibly a filling material, is based on a second gradient homogenization developed by \ud the authors. The obtained elastic moduli depend on the parameter of chirality, namely the angle of \ud inclination of the ligaments with respect to the grid of lines connecting the centers of the rings. \ud For hexachiral cells the aux- etic property of the lattice together with the elastic coupling \ud modulus between the normal and the asym- metric strains is obtained; a property that has been \ud confirmed here for the tetrachiral lattice. Unlike the hexagonal lattice, the classical constitutive \ud equations of the tetragonal lattice turns out to be characterized by the coupling between the \ud normal and shear strains through an elastic modulus that is an odd function of the parameter of \ud chirality. Moreover, this lattice is found to exhibit a remarkable variability of the Young’s \ud modulus and of the Poisson’s ratio with the direction of the applied uniaxial stress. Finally, a \ud simulation of\ud experimental results is carried out

The paper is focused on a homogenization procedure for the analysis of wave propagation in materials with periodic microstructure. By a reformulation of the variational-asymptotic homogenization technique recently proposed by Bacigalupo and Gambarotta (2012a), a second-gradient continuum model\ud is derived, which provides a sufficiently accurate approximation of the lowest (acoustic) branch of the dispersion curves obtained by the Floquet–Bloch theory and may be a useful tool for the wave propagation analysis in bounded domains. The multi-scale kinematics is described through micro-fluctuation functions of the displacement field, which are derived by the solution of a recurrent sequence of cell BVPs and obtained as the superposition of a static and dynamic contribution. The latters are proportional to the even powers of the phase velocity and consequently the micro-fluctuation functions also depend on the direction of propagation. Therefore, both the higher order elastic moduli and the inertial terms result to\ud depend by the dynamic correctors. This approach is applied to the study of wave propagation in layered bi-materials with orthotropic phases, having an axis of orthotropy parallel to the direction of layering, in which case, the overall elastic and inertial constants can be determined analytically. The reliability of the proposed procedure is analysed by comparing the obtained dispersion functions with those derived by the Floquet–Bloch theory