Atypical dynamics of materials with periodic microstructure and local resonance
Invited paper
ENTPE - LGCB/LTDS CNRS UMR 5513
Wednesday 3 june, 2015, 15:20 - 15:40
0.7 Lisbon (47)
Abstract:
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.