High continuity second-order homogenization of in-plane loaded periodic masonry

In this paper the second-order homogenization of periodic masonry based on a computational analysis of the unit cell representative of the masonry wall is derived. The multi-scale approach is based on an appropriate representation of the micro-displacement field as the superposition of a local macroscopic displacement field, represented in a polynomial form related to the macro-displacement field, and an unknown micro-fluctuation field accounting for the effects of the heterogeneities. By this approach a continuous micro-displacement field is obtained, i.e. in each unit cell and across the interfaces between adjacent unit cells. The computational procedure is applied in two steps: the first one corresponds to the standard homogenization, while the second step is a second-order homogenization based on the results of the first step. Two numerical examples are presented concerning running bond and English bond masonry. For both the masonry patterns the overall elastic moduli of the second-order model and the corresponding characteristic lengths are obtained; the effects on the characteristic lengths of the stiffness mismatch between the brick phase and the mortar phase are considered. Moreover, the wave propagation in the homogenized medium is considered and dispersive waves are obtained. It is shown that remarkable differences in the phase and group velocities between the first-order and the second-order homogenized models are obtained for wavelengths shorter than ten times the average brick unit size