Variable kinematic models applied to free-vibration analysis of functionally graded material shells
Introduction
Functionally graded materials (FGMs) have been the subject of numerous studies in the recent past. FGMs are classified as a family of composite materials; they are characterized by a functional variation in the composition of the material properties through an assigned direction, which often coincides with the thickness direction. Their particular feature is that they have the characteristic behavior of composite materials, but they do not show material discontinuities at the interfaces of a classical laminate. Their composition, and therefore their manufacturing, are designed to optimize the use of materials, mostly by reducing the weight of the structure. This is obtained by modifying the constituent phases through grading (mathematical) laws. FGMs are able to offer benefits compared to traditional laminates. The constituent phases, which may be more than two, are adjusted by varying the volume fractions whose rates are in turn related to the mathematical law that is used. The benefits associated with the presence of such materials are significant: the possibility of reducing the interlaminar discontinuities which are the main cause of delamination and the consequent failure of classical laminates.
Various FGM power laws have been used in the open literature, some of which have been provided by Mori and Tanaka, 1973, Kashtalyan, 2004 and Zenkour (2006). Among the various topics related to FGM, reference can be made to the review articles by Birman and Bird (2007). The present work is focused on refined shell models for accurate free-vibration analysis of layered shell with FGM layers. Several works concerning FG shell vibration have been presented over the last year. A short review, that is useful for our purpose, is given below.
Loy et al. (1999) have studied the vibrations of functionally graded cylindrical shells. The results show that the frequency characteristics are similar to those observed for homogeneous isotropic cylindrical shells and that the frequencies are affected by the constituent volume fractions and the configurations of the constituent materials. The analysis was carried out with strain-displacement relations from the Love shell theory and the eigenvalue governing equation was obtained using the Rayleigh–Ritz method. Pradhan et al. (2000) have examined the vibration characteristics of functionally graded cylindrical shells under various boundary conditions. The Rayleigh method was used to derive the governing equations. The effects of boundary conditions and volume fractions (power law exponent) on the natural frequencies were studied; it has been shown that the frequency characteristics of the FG shell are found to be similar to those of isotropic cylindrical shells. Chen et al. (2004) have proposed a three-dimensional vibration analysis of fluid-filled orthotropic FGM cylindrical shells. A state equation, with variable coefficients, was derived in a unified matrix form on the basis of the three-dimensional fundamental equations of anisotropic elasticity. A laminate approximate model, which is suitable for an arbitrary variation of the material constants along the radial direction was employed; numerical examples are presented and compared with existing results. Tornabene (2009) has conducted a free-vibration analysis of functionally graded conical, cylindrical shell and annular plate structures with proposed a four-parameter power law distribution. Based on the First-order Shear Deformation Theory (FSDT), the discretization of the system equations was made by means of the Generalized Differential Quadrature. Few works are available on the free vibration of FGM shells and none of these quotes the assessment of advanced or classical theories.
The present work deals with variable kinematic models in the framework of Carrera’s Unified Formulation (CUF) for the analysis of FGM shells. CUF was originally developed for classical layered structures (Carrera, 1998a, Carrera, 2003); it has has been extended to FGM structure by Carrera et al. (2008). The generalized expansion, upon which the CUF is based, relies on a set of functions which are indicated as thickness functions. CUF reduces a three-dimensional problem to a bi-dimensional one and the order of expansion along the thickness of the plate is taken as a free parameter of the problem. As a result, an exhaustive variable kinematic model is obtained.
The principle of virtual displacements (PVD) has been employed in Carrera et al. (2008), while Reissner’s mixed variational theorem (RMVT) has been extended to FGM in Brischetto and Carrera (in press). RMVT permits one to assume both displacement and transverse shear/normal stress variables. Related plate/shell bending problems have recently been analyzed in Carrera et al. (in press). Application to FGM beams has been given in Giunta et al. (in press) where several refined beam theories were applied to the linear static analysis of beams made of FGM. The extension of CUF to vibration analysis for plates has been discussed in Cinefra and Soave (in press).
In the present work the variable kinematic model is extended to the dynamic analysis of FGM shells, and the natural frequencies of single-layered shells are compared to other available solutions. Both PVD and RMVT are employed to compare classical and mixed shell theories.
The article has been organized as follows: the shell geometry is given in Section 2; the used variational statements and constitutive equations are given in Sections 3 Variational statements, 3.3 Hooke Law for PVD and RMVT, respectively; the considered shell theories are described in Sections 4 Considered shell theories, 5 Governing equations; the closed-form solution for the considered free-vibration problem is described in Section 6; the numerical discussion is conducted in Section 7.
Section snippets
Geometry
Shells are two-dimensional structures with one dimension, in general the thickness in the z direction, negligible with respect to the other two in the plane directions. The shells present radii of curvature Rα and Rβ along the two in-plane directions α and β, respectively. A curvilinear reference system (α, β, z) for shells is indicated in Fig. 1. In the case of layered shells, the reference surface of the k-layer is denoted by Ωk, and the curvilinear coordinates are αk and βk. The following
Principle of Virtual Displacement – PVD
Classical displacements formulations consider displacements u as primary variables. The following two-dimensional approximation is introduced in a very general form:whereu are displacements in each point P(α, β, z), uτ are displacements in each point PΩ(α, β) on the reference surface Ω. Fτ are the introduced thickness functions.
The Principle of Virtual Displacements (PVD) states (Washizu, 1968):where T indicates
Considered shell theories
CUF permits to introduce several two-dimensional models for shells. The governing equations are written, in a unified form, in terms of few fundamental nuclei which form do not formally depend on the order of expansion N that is used in the z direction as well as on the variables description used in the multilayered structure (Layer-Wise (LW) or Equivalent Single Layer (ESL)).
The generic variable a(α, β, z) and its variation δa(α, β, z) can be written according to the following general
Governing equations
This section presents the dynamic governing equations based on the variational statements of Section 2. The derivation of governing equation permits to obtain the so-called fundamental nuclei. These consist of [3 × 3] arrays that represent the basic items from which the stiffness matrix of the whole structure can be computed. Detailed derivation procedure is given in the previous works (Carrera, 1999a, Carrera, 1999b).
For a laminate with Nl layers, the PVD (Eq. (5)) for pure mechanical analysis,
Closed-form solution for free-vibration problem
For the derived boundary value problem, for particular geometry, material symmetry and boundary conditions, an analytical solution can be derived. For simply supported shells, a Navier-type closed-form solution may be found with the following harmonic assumptions for the field variables:where ak and bk are
Numerical results
The developed shell theories have first been applied to compare the results with the available reference solutions (Matsunaga, 2009) in which higher order ESL type shell theories were developed (N = 1,2,3,4). The adopted grading law distribution is that of Zenkour (2006); Young’s Modulus, shear modulus and density, vary continuously through the thickness, accordingly to the change in volume fractions of the materials, which takes place according to the law:where ℘m is the
Conclusions
A free-vibration problem of multilayered shells embedding FGM layers has been considered in this work by referring to variable kinematic shell theories developed in the framework of the Carrera Unified Formulation. Attention has been restricted to orthotropic, simply supported, shells. Classical theories are compared to higher order ones as well as to layer-wise kinematic and mixed shell formulations based on the Reissner Mixed Variational Theorem. The conducted numerical analysis has shown
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