Multiscale thermopiezoelectric analysis of laminated plates with integrated piezoelectric fiber composites Alden C. Cook, Senthil S. Vel * Department of Mechanical Engineering, University of Maine, Orono, ME 04469, USA article info Article history: Received 22 November 2011 Accepted 27 December 2012 Available online 11 January 2013 Keywords: Multiscale analysis Effective properties Thermal stress Active fiber composite Asymptotic expansion homogenization abstract This study presents a comprehensive multiscale analysis of laminated plates with integrated piezo- electric fiber composite actuators. A detailed framework based on the asymptotic expansion homoge- nization method is developed to couple the microscale and macroscale field variables. The microscale fluctuations in temperature, mechanical displacement and electric potential are related to the macroscale temperature change, mechanical strain and electric fields through 43 distinct characteristic functions. The local thermal, mechanical and charge equilibrium equations yield a system of partial differential equations for the characteristic functions that are solved using standard finite element techniques. The homogenized thermoelectroelastic properties of a representative material element are computed using the characteristic functions and the constituent material properties. The EshelbyeStroh formalism is used to analytically solve the three-dimensional macroscopic equilibrium equations for thick and thin laminated piezoelectric plates with arbitrary boundary conditions at the edges. Interscale transfer op- erators emerging from the asymptotic expansion homogenization method relate the macroscale fields to the microscale heat flux, stress and electric displacement in the individual fibers and matrix. The present multiscale analysis procedure is demonstrated by considering two model problems. In the first problem, a simply-supported sandwich plate consisting of a piezoceramic fiber composite shear actuator embedded between two graphite/epoxy layers is studied. The second problem concerns a simply- supported graphite/epoxy substrate with piezoceramic fiber composite extension actuators attached to its top and bottom surfaces. In both model problems, the responses of the laminated plates under thermal and electrical loading conditions are examined. Results are presented for the homogenized material properties, macroscale deformation, macroscale average stresses and microscale stress distributions. Ó 2013 Elsevier Masson SAS. All rights reserved. 1. Introduction Over the past several decades, there have been significant ad- vancements in the field of smart structures which integrate adap- tive piezoelectric components (see, e.g., Crawley and de Luis, 1987; Chopra, 2002). Depending on the desired function, piezoelectric materials can be used as actuators or sensors. Numerous applica- tions have been proposed and conceived experimentally such as active vibration suppression, structural health monitoring, noise cancellation and energy harvesting (see, e.g., Bailey and Hubbard, 1985; Crawley and de Luis, 1987; Fuller et al., 1987; Clark and Fuller, 1992; Zou et al., 2000; Caccese et al., 2004; Baillargeon and Vel, 2005a; Vel and Baillargeon, 2005; Krishna and Harursampath, 2007; Erturk and Inman, 2009). Traditionally, monolithic piezoceramic wafers were used as active components in smart structures applications, despite their brittle nature and limited capacity to conform to curved surfaces. In an effort to overcome these deficiencies, researchers have developed flexible piezoelectric fiber composites, such as Active Fiber Composites (AFC) (see Bent et al., 1995; Bent and Hagood, 1997) and Macro- fiber Composites (MFC) (see Wilkie et al., 2000; High and Wilkie, 2003), that consist of aligned, axially-poled, piezoceramic fibers that are embedded within a polymer matrix. AFCs are made with circular piezoceramic fibers while MFCs contain piezoceramic rib- bons of rectangular cross section (see Williams et al., 2002). MFC actuators have been successfully used in a variety of applications (see, e.g., Sodano et al., 2004; Bilgen et al., 2009, 2010). In the present work, we consider laminated fiber-reinforced composite plates with integrated piezoelectric fiber composite (PFC) actuators consisting of finescale unidirectional circular fibers * Corresponding author. Tel.: þ1 207 581 2777. E-mail address: senthil.vel@maine.edu (S.S. Vel). Contents lists available at SciVerse ScienceDirect European Journal of Mechanics A/Solids journal homepage: www.elsevier.com/locate/ejmsol 0997-7538/$ e see front matter Ó 2013 Elsevier Masson SAS. All rights reserved. http://dx.doi.org/10.1016/j.euromechsol.2012.12.011 European Journal of Mechanics A/Solids 40 (2013) 11e33