Generalized micropolar continualization of 1D beam lattices

Abstract

The enhanced continualization approach proposed in this paper is aimed to overcome some drawbacks observed in the homogenization of beam lattices. To this end an enhanced homogenization technique is proposed and formulated to obtain consistent micropolar continuum models of the beam lattices and able to simulate with good approximation the boundary layer effects and the Floquet-Bloch spectrum of the Lagrangian model. The continualization technique here proposed is based on a transformation of the difference equation of motion of the discrete system via a proper down-scaling law into a pseudodifferential problem; a further McLaurin approximation is applied to obtain a higher order differential problem. The formulation is carried out for simple one-dimensional beam lattices that are, nevertheless, characterized by a rather wide variety of static and dynamic behaviors: the rod lattice, the beam lattice with node rotations and a 1D beam lattice model with generalized displacements. Higher order models may be obtained which are characterized by differential problems involving non-local inertia terms together with spatial high gradient terms. Moreover, the homogenized models obtained by the proposed enhanced continualization technique turn out to be energetically consistent and provide a good simulation of both the static response and of the acoustic spectrum of the original discrete models. The proposed homogenization procedure is first presented for the simple case of monoatomic axial chains. The beam lattice with node rotations and displacement prevented exhibits, in the static regime, decaying oscillations of the nodal rotation in the boundary layer which is well simulated by the homogenized model obtained by the proposed approach. Similar good results are obtained in the simulation of the optical spectrum. It is worth to note that in this case the homogenized model obtained via Padé approximation turns out to be energetically non-consistent. The analysis of the homogenized model derived from the beam lattice with transverse displacement and rotation of the nodes with elastic supports has shown that both the static and the dynamic response are strongly variable on the parameters of the Lagrangian model. Finally, several different cases have been considered and good simulations have been obtained both in describing the static response to prescribed displacements at the end nodes and in representing the Floquet-Bloch spectrum and the polarization vectors.

Acoustic waveguide filters made up of rigid stacked materials with elastic joints

“…To obtain an equivalent continuum model able to approximate finely the dynamic behavior of the Lagrangian one, an enhanced continualization approach proposed by Bacigalupo and Gambarotta, 2018…”

“…Moreover, a convenient fine approximation of the dispersion functions of the Lagrangian model is pursued through the formulation of an equivalent continuum model. This continuum model is based on an enhanced continualization approach for discrete models proposed by Bacigalupo and Gambarotta, 2018 in order to circumvent some drawbacks emerging by applying classical continualization approaches (Bacigalupo and Gambarotta, 2017). This procedure provides different equivalent continua with increasing order of the leading derivative with non-local constitutive and inertial terms.…”

Nonstandard continualization of 1D lattice with next-nearest interactions. Low order ODEs and enhanced prediction of the dispersive behavior

“…Nonstandard continualization techniques have been also used [6][7][8][9]. These techniques consist in transforming the discrete equations into pseudodifferential equations, which are later expanded by Taylor series [6,7,9] or by Pad e approximants [8].…”

“…Nonstandard continualization techniques have been also used [6][7][8][9]. These techniques consist in transforming the discrete equations into pseudodifferential equations, which are later expanded by Taylor series [6,7,9] or by Pad e approximants [8]. Bacigalupo and Gambarotta [7] proposed an enhanced continualization scheme in which, besides using pseudodifferential operators, a central difference scheme is employed to discretize the first spatial derivative of the continuous displacement, achieving a continuous governing equation of higher order than that corresponding to the classical one.…”

Mechanical Metamaterials Foams with Tunable Negative Poisson’s Ratio for Enhanced Energy Absorption and Damage Resistance

“…In recent years, the groups of Prof. Gambarotta and Prof. Bacigalupo systematically studied the overall mechanical properties (e.g., equivalent elastic properties and wave propagation characteristics etc.) of several kinds of metamaterials, such as hexachiral [ 26 , 27 ], tetrachiral [ 26 , 28 , 29 , 30 ] and anti-tetrachiral [ 31 , 32 , 33 ] materials, periodic beam lattice materials [ 34 , 35 , 36 ], rigid periodic blocky materials [ 37 ], and so on.…”