Free Vibrations of Functionally Graded Cracked Plates of Arbitrary Shape via GDQFEM

This paper focuses on the dynamical behavior of moderately thick functionally graded flat plates with geometric discontinuities, such as cracks, and arbitrary curved boundaries. The problem of elastic plates has been deeply studied over the past decades [1-3]. Recently, the application of smart materials to every kind of structure has received great attention in the structural mechanics research field [4]. Functionally Graded Materials (FGMs) are composites with the mechanical properties varying along the thickness of the plate. Most studies consider isotropic FGMs consisting of a metal matrix, such as stainless steel or aluminum and spherical ceramic particles [5]. The volume fraction of the ceramic material varies through the plate thickness. The theory of thick flat plates, known as First-order Shear Deformation Theory (FSDT), is considered in this study. This paper applies the Generalized Differential Quadrature Finite Element Method (GDQFEM) [6,7] to analyze the free vibrations of arbitrarily shaped flat plates with internal cracks [8]. GDQFEM numerical method is based on the Generalized Differential Quadrature (GDQ) Method [6], as far as the derivative calculation is considered. The accuracy of GDQ method has been already presented in literature by the authors [5,9-12]. If the physical boundary is generally curved, the DQ method cannot be directly applied. So, the GDQFEM firstly decomposes the whole domain into several sub-domains of a generic shape and the DQ discretization is applied locally within the coordinate transformation approach. The irregular physical domain in Cartesian coordinates is transformed into a regular domain in natural coordinates, also known as parent domain. It appears that GDQFEM is analogous to the well-known Finite Element Method (FEM). It should be noted that, with reference to the proposed technique, the governing equations are solved in their strong formulation and the connections between the elements are imposed with the inter-element compatibility conditions. The validity of the proposed methodology is confirmed throughout several convergence studies involving natural frequencies of arbitrary shape plates with generally oriented cracks and mixed boundary conditions. The frequency results show excellent agreement with other numerical solutions obtained by FEM. Furthermore, FGM plate natural frequencies are computed using a four-parameter power-law distribution [5]. Most of the considered configurations are not analyzed in any previous work.