Homogenization and Shape Sensitivity of Microstructures for Design of Piezoelectric Bio-Materials

Abstract

Shape sensitivity of effective constitutive parameters is studied for homogenized piezoelectric composite which formerly was intended for bio-material application. It consists of the piezoelectric matrix in which elastic inclusions are distributed periodically. The microstructures are assumed to be parametrized in terms of the shape of the inclusions. Microstructures with elliptic inclusions are considered in numerical examples which indicate strong influence of the geometry on the homogenized piezoelectric properties. As the main theoretical result of the paper, the shape sensitivity formulae of the homogenized elastic, dielectric and piezoelectric coefficients are derived using the domain method of the material derivative.

Keywords:

• piezoelectric materials
• homogenization
• sensitivity analysis
• bio-materials

Acknowledgments

This work has been supported by the European Community's Human Potential Programme under contract “Smart Systems” number HPRN-CT-2002-00284 and in part also by project MŠM 49777513 03 of the Czech Republic.

Notes

¹In the standard way, by H ₀ ¹(Ω) we denote the Sobolev space of square-integrable functions up to their 1st generalized derivative with vanishing function value on ∂ Ω.

²The Y-periodic functions attain the same function values at each couple corresponding points on the opposite sides of ∂ Y, see e.g. [16].

³As these figures are displayed rather for better comprehension of the two-scale nature of the modelling, we give no details on the specific macroscopic problem considered, either on the “post-processing” calculation—it was the subject of the authors' publication in [12] and it is not relevant to the merit of this paper.

⁴We were motivated by the possible application in the design of a bio-material, see [12].

⁵However, in the context of the homogenization problem the variation of |Y| is senseless.