Application of Multi-Scale Modelling to Some Elastic, Piezoelectric and Electromagnetic Composites

One way to obtain “new materials” is to mix classical ones with periodic heterogeneous microstructures. Using the homogenization method, we rigorously justify the limit models obtained (when the size of the microstructures goes to zero) for composites with very promising properties. We present here three different examples: a new bio-compatible piezoelectric/living cells material designed to improve bone regeneration, a bianisotropic electromagnetic material in a first attempt of photonic crystals modelling and a strongly heterogeneous elastic material exhibiting acoustic band-gaps. We also provide numerical simulations in order to illustrate the influence of the change of intrinsic parameters such as shape and physical characteristics of the micro structures.

Keywords:

• homogenization
• piezoelectricity
• electromagnetic composites
• bio-material
• strong heterogenetics
• elastic waves
• band gaps

6. ACKNOWLEDGMENT

This work has been supported by the European project “Smart Systems” HPRN-CT-2002-00284 by the French project Mathstic “Métamatériaux” and by the Chilean projects Fondecyt 1020298 et 7040209.

Notes

^a See the definition of {.} in Section 2.

²Latin indices and exponents take their values in the set {1,2,3} and Einstein's summation rule is used. Boldface symbols represent vectors or vector spaces.

⁴Recall the {· } notation given in Section 2.

³In this case the microstructure is of size α instead of ϵ in order to avoid any confusion with the permittivity parameter.