A nonlinear finite element model using a unified formulation for dynamic analysis of multilayer composite plate embedded with SMA wires S.M.R. Khalili a,b,⇑ , M. Botshekanan Dehkordi a , E. Carrera c a Centre of Excellence for Research in Advanced Materials and Structures, Faculty of Mechanical Engineering, K.N. Toosi University of Technology, Tehran, Iran b Faculty of Engineering, Kingston University of Technology, London, UK c Department of Mechanics and Aerospace Engineering, Politecnico di Torino, Torino, Italy article info Article history: Available online 11 July 2013 Keywords: Vibration damping Shape memory alloys Material nonlinearity Nonlinear finite element Composite plate Advanced plate theories abstract In this study, a new nonlinear finite element model is presented in the frame work of Carrera’s Unified Formulation (CUF) for the dynamic analysis of SMA hybrid composite considering the instantaneous phase transformation and material nonlinearity effects, for every point on the plate. The CUF unify many theories in a unified form which can be differed by the order of expansion and definition of the variables in the thickness direction. The Brinson’s SMA constitutive equation is used to model the behavior of SMA wires. The governing equations are derived using the Reissner Mixed Variational Theorem (RMVT) in order to enforce the interlaminar continuity of transverse shear and normal stresses between two adja- cent layers. A transient finite-element-based method beside an iterative incremental procedure is pre- sented to study the dynamic response of multilayered composite plate embedded with SMA wires. A suppressed vibration of the plate is observed, which is due to the energy dissipation of SMA wires. The parametric effects like length-to-thickness ratio, plate aspect ratio and also the effect of different bound- ary conditions, upon the loss factors are investigated. Results show that as the length-to-thickness ratio and also the plate aspect ratio increases, the loss factor decreases. Ó 2013 Elsevier Ltd. All rights reserved. 1. Introduction Vibration damping consists of a challenging task to increase fa- tigue life and comfort of advanced structures. The variation of the properties of the material composing the structure provides a valu- able approach to this task. Only, very limited options are available for such changes in the material properties. The present work pro- poses the use of NiTi shape memory alloys for this purpose. Pre- senting an exhaustive mathematical model for these structures is suffering from the nonlinearity behavior which is due to the phase transformation. Therefore, in the most cases, the proposed models are accompanied with a lot of simplifying assumptions. In the study by Jafari and Ghiasvand [1] dynamic analysis of SMA beam under a moving load was investigated. In their study, the effect of hysteretic loops was modeled by substitution of an equivalent damping ratio in the equation of motion. The dynamic analysis of SMA beam was studied by Hashemi and Khadem [2]. They consid- ered the effect of the phase transformation. But In their research they assumed that, the beam behaves like a one-degree-of freedom system. Zbiciak [3] investigated the response history of SMA beam under impulse loading. He employed the rheological scheme for modeling the behavior of SMA material. He assumed that the material properties of SMA are constant. Recently, many considerations have been focused on the improvement in the properties of the composite structures by shape memory alloys. In the study by Rogers and Barker [4] SMA wires were used to control the frequency of a graphite/epoxy mul- tilayered beam. Upon the heating of SMA wires, an axial force was generated in the beam because of the shape memory effect. They showed that, the fundamental frequency of the beam was in- creased significantly by utilizing 15% volume fraction of SMA wires. Baz et al. [5] demonstrated that the SMA wires embedded to composite beams have a capability to control their natural fre- quencies. The effect of pre-strain, and also the effect of tempera- ture on the SMA wires, was taken to account in their study. The effect of SMA wires on the controlling of the buckling and fre- quency analysis of composite beam was studied by Baz et al. [6]. They found that the buckling load of a flexible composite beam was increased up to three times the uncontrolled beam. Epps and 0263-8223/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compstruct.2013.07.006 ⇑ Corresponding author. Address: Centre of Excellence for Research in Advanced Materials and Structures, Faculty of Mechanical Engineering, K.N. Toosi University of Technology, Pardis Street, Molasadra Avenue, Vanak Square, Tehran, Iran. Tel.: +98 2188674747; fax: +98 2188674748. E-mail addresses: smrkhalili2005@gmail.com (S.M.R. Khalili), mbd_dehkordi@ yahoo.com (M. Botshekanan Dehkordi), erasmo.carrera@polito.it (E. Carrera). Composite Structures 106 (2013) 635–645 Contents lists available at SciVerse ScienceDirect Composite Structures journal homepage: www.elsevier.com/locate/compstruct