Geometrically nonlinear analysis of composite laminated structures with multiple macro-fiber composite (MFC) actuators Shun-Qi Zhang a,b,⇑ , Zhan-Xi Wang a , Xian-Sheng Qin a , Guo-Zhong Zhao b , Rüdiger Schmidt c a School of Mechanical Engineering, Northwestern Polytechnical University, West Youyi Street 127, 710072 Xi’an, PR China b State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, 116024 Dalian, PR China c Institute of General Mechanics, RWTH Aachen University, Templergraben 64, 52062 Aachen, Germany article info Article history: Received 9 December 2015 Revised 26 March 2016 Accepted 24 April 2016 Available online 28 April 2016 Keywords: Geometrically nonlinear Macro-fiber composite Piezoelectric Smart structures Laminated structures abstract The paper develops geometrically nonlinear finite element (FE) models for macro-fiber composite (MFC) bonded thin-walled smart structures using various nonlinear shell theories based on the Reissner– Mindlin hypothesis. The nonlinear FE model considers arbitrary piezoelectric fiber orientation in MFC patches, which influences the structural response significantly. Two different kinds of MFCs are consid- ered, namely MFC-d31 and MFC-d33, in which the former one is dominated by the d 31 effect, while the latter one is by the d 33 effect. The mathematical model is first validated by a cantilevered plate with the simulation and experimental results in the literature. Afterwards, a cantilevered plate and a semicircular shell both bonded with multi-MFC patches are analyzed by the present nonlinear FE model. Ó 2016 Elsevier Ltd. All rights reserved. 1. Introduction Thin-walled structures bonded with smart materials, e.g. piezo- electric materials, magnetostrictive materials, which are called smart structures, have a great potential in the application of shape control, vibration suppression and health monitoring. In the last three decades, there were published a number of scientific papers on modeling, computation and application of smart structures bonded with commonly used piezo-ceramics or polymers, which are isotropic materials. From the application of conventional piezoelectric materials, it has been found that piezoceramics, like lead zirconium titanate (PZT), have relatively strong actuation forces, but the brittle nature of ceramics makes them being easily damaged during handling and bonding process [1]. Piezoelectric polymers, like polyvinylidene fluoride (PVDF), are much more flexible compared to piezoceram- ics, but it has lower actuation forces, which makes them being used frequently in sensing problems rather than actuation problems. To overcome the limitations existing in conventional piezoelec- tric materials, piezo composite materials were proposed and developed by some researchers in the 1990’s. The first type of piezo composite is referred to as 1–3 composite invented at the Fraun- hofer Research Facility in Germany [1], the second one is an active fiber composite (AFC) initially developed by MIT [1] and the third one is a macro-fiber composite (MFC) proposed by NASA Langley Research Center [2]. Because of many advantages of MFC patches, e.g. thin film, in-plane piezo fiber orientation, large actuation forces and flexibility, they are used frequently in sensing and actu- ation problems. The MFC consists of an active layer sandwiched between layers of adhesive, electrodes and polyimide film. The active layer is constructed by rectangular piezo-ceramic rods embedded in epoxy matrix. The electrodes in an interdigitated pat- tern of MFC are attached to the active layer, which transfers the applied voltage directly to and from the piezoceramic rods. The special structural arrangement of MFCs makes them diffi- cult in modeling of structures bonded with such MFC patches. To give overall material parameters for MFC patches, Park and Kim [3], Deraemaeker et al. [4–6], Biscani et al. [7] developed various methods for homogenization of material properties. Based on the homogenized material properties, there are only some papers available in the literature which developed geometrically linear simulation models using commercial software, e.g. ANSYS [8,9], ABAQUS [10,11], and compared the results with those obtained from experiments. Furthermore, an analytical model was devel- oped by Bilgen et al. [12] for frequency response analysis of an MFC bonded clamped-free thin beam. Recently, Zhang et al. [13] developed a linear finite element model of MFC bonded smart http://dx.doi.org/10.1016/j.compstruct.2016.04.037 0263-8223/Ó 2016 Elsevier Ltd. All rights reserved. ⇑ Corresponding author at: School of Mechanical Engineering, Northwestern Polytechnical University, West Youyi Street 127, Postbox 554, 710072 Xi’an, PR China. E-mail addresses: sqzhang@nwpu.edu.cn, shunqi.zhang@hotmail.com (S.-Q. Zhang). Composite Structures 150 (2016) 62–72 Contents lists available at ScienceDirect Composite Structures journal homepage: www.elsevier.com/locate/compstruct