Dimitris I. Chortis Research Assistant Dimitris S. Varelis Postdoctoral Fellow Dimitris A. Saravanos 1 Professor e-mail: saravanos@mech.upatras.gr Department of Mechanical Engineering and Aeronautics, University of Patras, Patras 26500, Greece Linearized Frequencies and Damping in Composite Laminated Beams Subject to Buckling This paper considers the damped small-amplitude free-vibration of composite laminated strips subject to large in-plane forces and rotations. A theoretical framework is formu- lated for the prediction of the nonlinear damping of composite laminates subject to large Green–Lagrange axial strains and assuming a Kelvin viscoelastic solid. An extended beam finite element is developed capable of providing the nonlinear stiffness and damp- ing matrices of the system. The linearized damped free-vibration equations associated with the deflected strip shape in the pre- and postbuckling region are presented. Numeri- cal results quantify the strong geometric nonlinear effect of compressive in-plane loads on the modal damping and frequencies of composite strips. Measurements of the modal damping of a cross-ply glass/epoxy beam subject to buckling were also conducted and correlate well with the finite element predictions. [DOI: 10.1115/1.4023051] Keywords: nonlinear damping, composite, laminates, buckling, beams, finite element 1 Introduction Fiber reinforced composite materials are known to provide damping, which is beneficial for the passive control of vibratory, aeroelastic, and acoustic loads, in a variety of structural applica- tions. Many of such structures are exposed to large deformations and high compressive loads, leading to nonlinear vibrations, loss of stability, and buckling. The characterization and prediction of structural composite damping parameters in such cases exceed the range of known linear models, and the inclusion of nonlinear damping effects into analytical and numerical formulations is essential for describing and robustly predicting the complex struc- tural dynamics of large-scale composite structures, such as pre- stressed fuselage panels and large wind-turbine blades. Various analytical models have been developed to predict the linear damping of composite materials and structures [1–6], including micromechanical computational approaches [7–10] as well as finite element formulations for the damped free-vibration analysis of composite beams and plates [11–14]. Important works also focus on the experimental characterization of damping, which is further correlated with theoretical models predictions [15–18]. Plagianakos and Saravanos [19] presented a finite element method for predicting the damping of doubly curved laminated shells of composite structures based on first order shear theory assump- tions. Damping mechanics and specialty finite elements based on high-order layerwise kinematic assumptions capable of predicting damping in thick composite plates, and sandwich laminates and damped plates with constrained damping layers were subse- quently reported by Saravanos and coworkers [20,21]. A theoreti- cal framework for predicting the linear section damping matrices of composite blades with hollow laminated cross-sections was reported by Saravanos et al. [22] and incorporated into a shear beam finite element of the so-called DAMPBEAM blade analysis code. The work was extended to include material coupling effects on the damping of composite blade structures [23]. Significant works have been reported for the prediction of the undamped nonlinear dynamic behavior of composite beams sub- ject to buckling loading [24–28]. Varelis and Saravanos [29,30] presented coupled mechanics and finite element formulations for analyzing the buckling and postbuckling response of active piezo- composite laminated beams, plates, and shells, including their undamped free-vibration response [31]. Only recently, works have been reported on the damping of composite structures exceeding the range of linear theory. Kosmatka [32] studied both analytically and experimentally the damping response of a beam subject to an initial axial force. His work included an analytical solution for tension and compression of prestressed Euler–Bernoulli beams composed of a homogenous material with viscous damping and also experimental results for the damping of various composite beams under the same loading conditions. Kosmatka [33] presented a discrete model with geometrically-imperfect spring and damper for the vibration of postbuckled carbon/epoxy beams. Lesieutre [34–37] reported an analytical solution based on classical laminate theory, which included the effect of membrane loads on the modal damping of composite structures. Chortis et al. [38] presented a damping mechanics formulation and a beam finite element model for the prediction of nonlinear damping of laminated composite strips under large in-plane tension loads. This work set the basis of the theoretical formulation but was restricted to the case of pre- stressed composite beams, taking into account only first-order nonlinear stiffness and damping terms. The present paper aims to analyze and characterize the damping behavior of laminated strips subject to large in-plane compressive stresses and buckling conditions. In the following sections, the theoretical framework and the beam finite element [38] are extended to include new second-order nonlinear damping and stiffness terms required to fully describe the buckling response of composite laminated beams. Kinematic equations based on first order shear deformation theory and Green–Lagrange strains are implied, and the effective and tangential (linearized) laminate stiffness and damping matrices are formulated assuming a Kelvin viscoelastic material. A new damped beam finite element provides the linearized free-vibration of composite strips subject to buckling. Numerical results evaluate the capability of the present models to predict the modal damping and frequencies of composite plate-strips in the pre- and the postbuckling regime. Experiments conducted on glass/epoxy [0 2 /90 2 ] s cross-ply com- posite beam specimens subject to buckling are also presented. Measured modal damping and frequency values are correlated with numerical predictions, which quantify the accuracy of the 1 Corresponding author. Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received November 15, 2011; final manuscript received October 3, 2012; published online February 25, 2013. Assoc. Editor: Marco Amabili. Journal of Vibration and Acoustics APRIL 2013, Vol. 135 / 021006-1 Copyright V C 2013 by ASME Downloaded From: http://asmedigitalcollection.asme.org/ on 10/15/2013 Terms of Use: http://asme.org/terms