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

TL;DR

This paper analyzes dispersive wave propagation in two-dimensional periodic blocky materials with elastic interfaces, deriving a discrete Lagrangian model and an approximate micropolar continuum model, and compares their predictions for various block arrangements.

Contribution

It introduces a novel micropolar continuum approximation for 2D blocky materials with elastic interfaces, including conditions affecting its positive definiteness and validity.

Findings

Eigenproblem reveals two acoustic and one optical branch.

Micropolar model accurately approximates the discrete model up to second order in wave vector.

Validation on rhombic, hexagonal, and masonry patterns demonstrates model reliability.

Abstract

Dispersive waves in two-dimensional blocky materials with periodic microstructure made up of equal rigid units having polygonal centro-symmetric shape with mass and gyroscopic inertia, connected each other through homogeneous linear interfaces, have been analysed. The acoustic behavior of the resulting discrete Lagrangian model has been obtained through a Floquet-Bloch approach. From the resulting eigenproblem derived by the Euler-Lagrange equations for harmonic wave propagation, two acoustic branches and an optical branch are obtained in the frequency spectrum. A micropolar continuum model to approximate the Lagrangian model has been derived based on a second-order Taylor expansion of the generalized macro-displacement field. The constitutive equations of the equivalent micropolar continuum have been obtained, with the peculiarity that the positive definiteness of the second-order symmetric tensor associated to the curvature vector is not guaranteed and depends both on the ratio between the local tangent and normal stiffness and on the block shape. The same results has been obtained through an extended Hamiltonian derivation of the equations of motion for the equivalent continuum that is related to the Hill-Mandel macro homogeneity condition. Moreover, it is shown that the hermitian matrix governing the eigenproblem of harmonic wave propagation in the micropolar model is exact up to the second order in the norm of the wave vector with respect to the same matrix from the discrete model. To appreciate the acoustic behavior of some relevant blocky materials and to understand the reliability and the validity limits of the micropolar continuum model, some blocky patterns have been analysed: rhombic and hexagonal assemblages and running bond masonry.