Free Vibrations of Doubly Curved Functionally Graded Shells

Functionally graded materials are composite materials with material properties varying in one or more directions. Most studies consider isotropic FGM that often consist of a metal matrix such as stainless steel or aluminium and spherical ceramic particles. The volume fraction of the particles then varies through the shell thickness. Anisotropic FGM have been considered more recently. These are fiber reinforced composite materials with through-the- thickness variation of the fiber volume fraction and/or the fiber orientation. Previous studies [1,2] have shown that plates made out of isotropic FGM behave like homogeneous plates. Elastic coupling between in-plane and out of plane deformations can be eliminated from the equations of motion if the neutral surface is taken as the reference surface instead of the mid-plane. Thus, no new method need be developed and the natural frequencies of such functionally graded plates are directly proportional to those homogeneous plates with the same shape and support conditions. Based on results published by previous authors, it was shown that these results also hold for isotropic functionally graded shells [3] and for very thick shells [4]. For anisotropic FGMs the same relationship exists between graded and shells with constant fiber orientation and fiber volume fraction through the thickness. The First-order Shear Deformation Theory (FSDT) is used to analyze moderately thick structural elements. In order to include the effect of the initial curvature a generalization of the Reissner-Mindlin theory [5] is adopted. The governing equations of motion, written in terms of stress resultants, are expressed as functions of five kinematic parameters, by using the constitutive and kinematic relationships. The solution is obtained by using the Generalized Differential Quadrature (GDQ) method [6] which leads to a standard linear eigenvalue problem. This simple and direct procedure can be applied to a large number of cases [5,7-10] to circumvent the difficulties of programming complex algorithms for the computer, as well as the excessive use of storage and computer time. Results are obtained taking the meridional and circumferential co-ordinates into account, without using the Fourier modal expansion methodology. GDQ results are validated with those obtained by using commercial programs. Natural frequencies of panels made of isotropic FGMs with various shapes (cylindrical, spherical, hyperbolical, catenoidal) and boundary conditions were determined using the GDQ method. It was found that the natural frequencies of the graded shells were directly proportional to those of the same shell made out of isotropic materials and that ratio is independent of shape and boundary conditions. In spite of the asymmetric properties of the material with respect to the mid-plane, these shells behave as if they were homogeneous and isotropic. Using the neutral surface as the reference surface, elastic coupling between the in-plane and transverse deformations due to material properties varying through the thickness can be removed from the equations of motion. Non-dimensional parameters were developed to assess a number of complicating factors such as: the effect of transverse shear deformation and rotary inertia, aspect ratio and curvature. This allows to determine when these factors are important and when certain levels of refinement in the analysis are required.