Keywords: phononic materials, plate models, homogenization, band gaps, wave dispersion.

The problem of wave propagation in periodically heterogeneous plates with high contrasts in elastic coefficients is considered in this paper. Following the approach of [1,3] the unfolding method of homogenization is applied to obtain limit plate models which as a result of the high contrast ansatz in scaling the elasticity coefficients of inclusions retain the dispersion properties in the limit when the scale (the characteristic size) of the microstructure tends to zero.

Two plate modelsare studied: 1) according to the Reissner-Mindlin (R-M) theory the plate deformation is described by the mid-plane deflections and by rotations of the plate cross-sections which account for the shear stress effects; 2) using the Kirchhoff- Love (K-L) theory, the plate deflections are described by the bi-harmonic operator, thus neglecting the shear effects. In both cases heterogeneities having a form of cylindrical inclusions orthogonal to the mid-plate coordinates are assumed. As a result of the high contrast ansatz in scaling the elasticity coefficients of inclusions, as employed in [1], dispersion properties are retained in the homogenized plates: the frequency-dependent mass coefficients associated with the inertia can be negative over the band-gaps [2], consequently propagation of elastic waves is suppressed. The phononic effect, in general, is associated with vibration modes excited at the "microscopic" level [4]; in [1] the positivity, or negativity of the "homogenized masses" is described and how it can be employed, to predict band gaps in the three-dimensional phononic crystals. Using a numerical example the existence of band gaps is demonstrated in a guided wave propagation in an infinite homogenized R-M plate. For the K-L plate a simple calculation shows that there is always a propagating wave (arising from the deflection modes) even if the mass tensor associated with the plate rotations is negative definite. For standing waves the band gap phenomenon will be studied separately, whereby influence of boundary conditions must be respected.

1

A. Avila, G. Griso, B. Miara, E. Rohan, "Multi-scale modelling of elastic waves. Theoretical justification and numerical simulation of band gaps", Multiscale Model. Simul., 7(1), 121, 2008.

2

E. Rohan, B. Miara, "Band gaps and vibration of strongly heterogeneous Reissner Mindlin elastic plates", C. R. Acad. Sci. Paris, Ser. I, 349, 777-781, 2011. doi:10.1016/j.crma.2011.05.013

3

E. Rohan, B. Miara, F. Seifrt, "Numerical simulation of acoustic band gaps in homogenized elastic composites", International Journal of Engineering Science, 47, 573-594, 2009. doi:10.1016/j.ijengsci.2008.12.003

4

T.-T. Wu, J.-Ch. Hsu, J.-H. Sun, "Phononic plate waves", Ultrasonics Symposium (IUS), IEEE, 145-154, 2010. doi:10.1109/ULTSYM.2010.5935874

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