Nonlinear vibrations of functionally graded cylindrical shells Matteo Strozzi, Francesco Pellicano n Department of Engineering ‘‘Enzo Ferrari’’, University of Modena and Reggio Emilia, Via Vignolese 905, 41125 Modena, Italy article info Article history: Received 17 February 2012 Received in revised form 8 January 2013 Accepted 24 January 2013 Keywords: Circular cylindrical shells Functionally graded materials Nonlinear vibrations abstract In this paper, the nonlinear vibrations of functionally graded (FGM) circular cylindrical shells are analysed. The Sanders–Koiter theory is applied to model the nonlinear dynamics of the system in the case of finite amplitude of vibration. The shell deformation is described in terms of longitudinal, circumferential and radial displacement fields. Simply supported, clamped and free boundary condi- tions are considered. The displacement fields are expanded by means of a double mixed series based on Chebyshev orthogonal polynomials for the longitudinal variable and harmonic functions for the circumferential variable. Both driven and companion modes are considered; this allows the travelling- wave response of the shell to be modelled. The model is validated in the linear field by means of data retrieved from the pertinent literature. Numerical analyses are carried out in order to characterise the nonlinear response when the shell is subjected to a harmonic external load; a convergence analysis is carried out by considering a variety of axisymmetric and asymmetric modes. The present study is focused on determining the nonlinear character of the shell dynamics as the geometry (thickness, radius, length) and material properties (constituent volume fractions and configurations of the constituent materials) vary. & 2013 Elsevier Ltd. All rights reserved. 1. Introduction Functionally graded materials (FGMs) are composite materials obtained by combining and mixing two or more different con- stituent materials, which are distributed along the thickness in accordance with a volume fraction law. Most of the FGMs are employed in the high-temperature environments because of their heat shielding capacity. The idea of FGMs was first introduced in 1984/1987 by a group of Japanese material scientists [1]. They studied many different physical aspects, such as temperature and thermal stress distri- butions, static and dynamic responses. Loy et al. [2] analysed the vibrations of the cylindrical shells made of FGM, considering simply supported boundary conditions. They found that the natural frequencies are affected by the constituent volume fractions and configurations of the constitu- ent materials. Pradhan et al. [3] studied the vibration characteristics of FGM cylindrical shells made of stainless steel and zirconia, under different boundary conditions. They found that the natural frequencies depend on the material distributions and boundary conditions. Arshad et al. [4] and Naeem et al. [5] analysed the effect of the boundary conditions on the frequency spectra of isotropic and FGM cylindrical shells; Darabi et al. [6] and Ng et al. [7] carried out a nonlinear analysis of the dynamic stability under periodic axial loading. Wu et al. [8] and Haddadpour et al. [9] studied the thermo elastic stability and the thermal effects. Iqbal et al. [10] considered vibrations by using a wave propagation approach. Shah et al. [11] analysed different exponential volume laws. Amabili et al. [12,13] studied the nonlinear vibrations of FGM doubly curved shallow shells; they considered the thermal effect and used a higher order shear deformation theory. Readers interested in deepening the knowledge on shells behaviour are suggested to refer to the works of Leissa [14] and Yamaki [15]. The first one is mainly concerned with linear dynamics of shells, exhibiting different topologies, materials and boundary conditions. The second one is focused on buckling and post-buckling of the shells in linear and nonlinear fields. In Refs. [14,15] one can find the most important shell theories, such as Donnell, Reissner, Flugge, Sanders–Koiter, as well as solution methods, numerical and experimental results. A modern treatise on the shells dynamics and stability can be found in Ref. [16], where also FGMs are considered. Refs. [17,18] are strictly related with the present work. In Ref. [17] the effect of the geometry on the nonlinear vibrations of isotropic shells was investigated, leading to conclusions similar to those of the present work. The method of solution used in the present work was developed in Ref. [18]. Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/tws Thin-Walled Structures 0263-8231/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.tws.2013.01.009 n Corresponding author. Tel.: þ39 059 205 6154; fax: þ39 059 205 6126. E-mail addresses: francesco.pellicano@unimore.it, pellif2000@yahoo.it (F. Pellicano). Thin-Walled Structures 67 (2013) 63–77