Calculation of effective coefficients for piezoelectric fiber composites based on a general numerical homogenization technique Harald Berger a, * , Sreedhar Kari a , Ulrich Gabbert a , Reinaldo Rodriguez-Ramos b , Julian Bravo-Castillero b ,Rau ´l Guinovart-Diaz b a Institute of Mechanics, University of Magdeburg, Universitaetsplatz 2, D-39106 Magdeburg, Germany b Facultad de Matema ´tica y Computacio ´n, Universidad de La Habana, San La ´zaro y L, CP 10400 Vedado, Habana 4, Cuba Abstract Numericalunitcellmodelsfor1–3periodiccompositesmadeofpiezoceramicunidirectionalcylindricalfibersembeddedinasoftnon- piezoelectric matrix are developed. The unit cell is used for prediction of the effective coefficients of the periodic transversely isotropic piezoelectric cylindrical fiber composite. Special emphasis is placed on the formulation of the boundary conditions that allows the sim- ulationofallmodesofoveralldeformationarisingfromanyarbitrarycombinationofmechanicalandelectricalloading.Thenumerical approach is based on the finite element method (FEM) and it allows the extension to composites with arbitrary geometrical inclusion configurations, providing a powerful tool for fast calculation of their effective properties. For verification the effective coefficients are evaluatedforsquareandhexagonalarrangementsofunidirectionalpiezoelectriccylindricalfibercomposites.Theresultsobtainedfrom the numerical technique are compared with those obtained by means of the analytical asymptotic homogenization method (AHM) for different fiber volume fractions. Ó 2005 Elsevier Ltd. All rights reserved. Keywords: Composite materials; Piezoelectricity: Homogenization; Finite element method 1. Introduction Piezoelectric materials have the property of converting electrical energy into mechanical energy, and vice versa. But bulk piezoelectric materials have several drawbacks, hence composite materials are often a better technological solution in the case of a lot of applications such as ultra- sonic transducers, medical imaging, sensors, actuators and damping. Recently, due to miniaturization of the piezocompositesandtheuseofPZTfibersinsteadofpiezo- electric bars, new applications toward electro-mechanical sensors and actuators have become possible. To design such structures it is of interest to know the overall elec- tro-mechanical behavior of the composite. Different homogenizationtechniqueshavebeendevelopedtopredict effective material constants of piezocomposites. Severalanalyticalmethodsallowthepredictionofover- all moduli for composites with regular fiber arrangements [1,2].However,numericalmethodsgivethebasisfortreat- mentofmorecomplexfiberconfigurations [3–5].Basedon a unit cell model with appropriate periodic boundary con- ditions and load cases the finite element method can be used to calculate the full set of effective coefficients of the composite.Inthispaper,ageneralnumericalhomogeniza- tion concept is introduced which can be extended to com- posites with arbitrary piezoelectric inclusions. 2. Constitutive equations and unit cell models For a transversely isotropic piezoelectric composite the materialbehaviorisbasedonelevenindependentconstants. Usingeffectivecoefficients(C eff ij —elastic, e eff ij —piezoelectric, 0263-8223/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2005.09.038 * Correspondingauthor.Tel.:+493916712406;fax:+493916712439. E-mail address: berger@mb.uni-magdeburg.de (H. Berger). www.elsevier.com/locate/compstruct Composite Structures 71 (2005) 397–400