Atypical dynamics of materials with periodic microstructure and local resonance

Invited paper

Stephane Hans


Wednesday 3 june, 2015, 15:20 - 15:40

0.7 Lisbon (47)

This work investigates the dynamic behavior of periodic unbraced frame structures made up of interconnected beams or plates. Such structures can represent an idealization of numerous reticulated systems, natural as foams, plants, bones or man-made as sandwich panels, trusses and buildings. Two types of microstructures are especially studied in this paper: non- orthogonal unbraced frame and honeycombs. In theses cases, the unbraced framed microstructure is much stiffer in compression than in shear, what generates a variety of behaviors more important than in filled materials. Assuming the condition of scale separation is respected, that means the size of system or the wavelength is larger than the size of the cell, the dynamical behaviors at the leading order are approached by the homogenization method of periodic discrete media. The main advantages of this method are the analytical formulation and the possibility to understand the behavior of the elements at the local scale. In the studied ranges, the local elements behave ever in quasi-statics, ever in dynamics. For studied materials, the elastic law are given in function of the elements properties. These laws correspond to upgraded materials as double gradient media or meta-material. To illustrate their atypical properties, propagations of ‘shear’ and ‘compression’ waves are studied. For example, in the case of inclined lattice, only two directions of propagation are possible for shear waves, whereas the compression can propagate in all directions, but with a dispersive and anisotropic character. In the presence of the local resonance, the form of the equations is unchanged but some macroscopic parameters depend on the frequency. In particular, this applies to the mass leading to a generalization of the Newtonian mechanics. As a result, frequency band gaps appear.

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