Computational two-scale homogenization of periodic masonry: Characteristic lengths and dispersive waves

The equations of motion of a second-order continuum equivalent to the periodic masonry made of deformable bricks and mortar are obtained and the overall elastic moduli and the inertial properties are evaluated through a homogenization technique derived from the variational-asymptotic approach proposed by Smyshlyaev and Cherednichenko [23]. The computational method consists in solving two sequences of cell problems in the standard format of vanishing body forces and prescribed boundary dis- placements. In the first step the classical first-order homogenization is carried out by solving four cell problems; the second step concerns the second-order homogenization and involves the solution of six additional cell problems. The equations of motion and the wave equation are specialized to the case of centro-symmetric periodic cells and orthotropic material at the macro-scale, conditions that are common in brick masonry. The characteristic lengths and dispersive elastic waves are obtained. The special cases of characteristic lengths and wave propagation along the orthotropy axes are studied. In the examples running bond and English bond masonry are analyzed by varying the stiffness mis- match between the brick and the mortar. In all cases, the obtained characteristic lengths associated to the shear and extensional strains result to be a fraction of the periodic cell size and become zero for van- ishing stiffness mismatch between the brick and the mortar. For both the masonry bonds here consid- ered, the characteristic lengths associated to the shear strain are higher by about an order of magnitude than those associated to the extensional strain. The characteristic lengths along the direction parallel to the mortar joints are prevailing on those along the normal direction. In particular, small char- acteristic lengths are obtained along the direction normal to the bed mortar joints for both the running bond and the English bond masonry. The wave propagation along the orthotropy axes in both the running bond and English bond masonry is analyzed by considering wave-lengths multiple of periodic cell size. Dispersive waves propagating along the orthotropy direction parallel to the mortar joints are characterized by velocities that differ quite markedly from the corresponding ones in the classical continuum and this difference increases with the increase of the stiffness mismatch between the brick and mortar. Conversely, along the direction per- pendicular to the mortar joints the velocity of the shear waves is approximately equal to that in the clas- sical equivalent continuum. These findings show the qualitative similarity of the mechanical behavior of masonry with layered materials.