Dispersive wave propagation in two-dimensional rigid periodic blocky materials with elastic interfaces

摘要

Dispersive waves in two-dimensional blocky materials with periodic microstructure made\udup of equal rigid units, having polygonal centro-symmetric shape with mass and gyro- scopic \udinertia, connected with each other through homogeneous linear interfaces, have been analyzed. \udThe acoustic behavior of the resulting discrete Lagrangian model has been obtained through a \udFloquet–Bloch approach. From the resulting eigenproblem derived by the Euler–Lagrange equations for \udharmonic wave propagation, two acoustic branches and an optical branch are obtained in the \udfrequency spectrum. A micropolar continuum model to approximate the Lagrangian model has been \udderived based on a second-order Taylor expansion of the generalized macro-displacement field. \udThe constitutive equations of the equivalent micropolar continuum have been obtained, with the \udpeculiarity that the posi- tive definiteness of the second-order symmetric tensor associated to the \udcurvature vector is not guaranteed and depends both on the ratio between the local tangent and \udnormal stiffness and on the block shape. The same results have been obtained through an ex- \udtended Hamiltonian derivation of the equations of motion for the equivalent continuum that is \udrelated to the Hill-Mandel macro homogeneity condition. Moreover, it is shown that the \udhermitian matrix governing the eigenproblem of harmonic wave propagation in the micropolar model is \udexact up to the second order in the norm of the wave vector with respect to the same matrix from \udthe discrete model. To appreciate the acoustic behavior of some relevant blocky materials and to \udunderstand the reliability and the validity limits of the micropolar continuum model, some blocky \udpatterns have been analyzed: rhombic and hexagonal assemblages and running bond masonry. From the \udresults obtained in the examples, the obtained micropolar model turns out to be particularly \udaccurate to describe dispersive functions for wavelengths greater than 3-4 times the characteristic \uddimension of the block. Finally, in consideration that the positive definiteness of the second order \udelas- tic tensor of the micropolar model is not guaranteed, the hyperbolicity of the equation of \udmotion has been investigated by considering the Legendre–Hadamard ellipticity conditions\udrequiring real values for the wave velocity.