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CivilComp Proceedings
ISSN 17593433 CCP: 99
PROCEEDINGS OF THE ELEVENTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY Edited by: B.H.V. Topping
Paper 209
Two Scale Modelling of Acoustic Waves in Phononic Plates using Homogenization of HighContrast Media E. Rohan^{1}, R. Cimrman^{2} and B. Miara^{3}
^{1}Department of Mechanics, Faculty of Applied Sciences, ^{2}New Technologies Research Centre,
E. Rohan, R. Cimrman, B. Miara, "Two Scale Modelling of Acoustic Waves in Phononic Plates using Homogenization of HighContrast Media", in B.H.V. Topping, (Editor), "Proceedings of the Eleventh International Conference on Computational Structures Technology", CivilComp Press, Stirlingshire, UK, Paper 209, 2012. doi:10.4203/ccp.99.209
Keywords: phononic materials, plate models, homogenization, band gaps, wave dispersion.
Summary
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 ReissnerMindlin (RM) theory the plate deformation is described by the midplane deflections and by rotations of the plate crosssections which account for the shear stress effects; 2) using the Kirchhoff Love (KL) theory, the plate deflections are described by the biharmonic operator, thus neglecting the shear effects. In both cases heterogeneities having a form of cylindrical inclusions orthogonal to the midplate 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 frequencydependent mass coefficients associated with the inertia can be negative over the bandgaps [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 threedimensional phononic crystals. Using a numerical example the existence of band gaps is demonstrated in a guided wave propagation in an infinite homogenized RM plate. For the KL 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. References
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