MICROMORPHIC MODELLING OF PERIODIC BLOCKY MATERIALS WITH ELASTIC JOINTS: OVERALL CONSTITUTIVE TENSORS AND INERTIAL TERMS

It is well known that in continuum modelling of blocky structures made up of equal rigid units (representative of granules, brick or rock units, nanoparticles etc.), non-local constitutive models may be required to appreciate the size effects. In a simple two-dimensional approach, the granules or the blocks may be assumed to have polygonal or circular shape endowed with mass and gyroscopic inertia, connected with each other through homogeneous linear interfaces. Under this hypothesis, the deformability of the composite material is localized at the interfaces and a Lagrangian model can be assumed to simulate the structural behavior of the blocky structure. The acoustic behaviour of the resulting discrete Lagrangian model is obtained through a Floquet-Bloch approach. It is shown that the resulting eigenproblem derived by the Euler-Lagrange equations for harmonic wave propagation provides two acoustic branches and an optical branch in the frequency spectrum [1]. A continuum description of the discrete system has been proposed in [1] based on a micropolar approach relying on a continualization of the Christoffel equations in terms of translations and rotations that simulate with a good agreement both the acoustic and the optical branches in the frequency spectrum (see also [2]). Nevertheless, the resulting micropolar continuum model suffers some inconsistencies in the static response that requires a deepening on the homogenization procedure. The most noticeable point concerns the rotational behaviour of the discrete system that exhibits saw-tooth oscillations of the rotational field in the boundary layer of a blocky chain [3]. Such behaviour, that turns out to be hardly represented by continuum models obtained via standard continualization technique, is here modelled through a quasi-continuum approximation of the Hamiltonian that is similar to that proposed in [4,5]. The resulting enhanced micromorphic model is analysed and its validity limits are given by comparing the static response and the frequency spectrum of the Lagrangian and continuum.