Static and free vibration analyses of functionally graded plates based on an isogeometric scaled boundary finite element method

Abstract

A novel semi-analytical plate formulation based on the isogeometric analysis (IGA) and scaled boundary element method (SBFEM) is proposed in this paper, and aims to solving the static bending and free vibration problems of functionally graded material (FGM) plates. The modulus of elasticity and density of the material are assumed to vary continuously along the thickness direction with a power-law distribution. The proposed formulation is derived on the basis of 3D theory of elasticity, while only the 2D in-plane dimension of reference surface needed to be discretized with three degrees of freedom (Dofs) per control points, and the non-uniform rational B-spline (NURBS) basis functions are adopted for both the representation of exact geometry and mechanical analysis. The system equations are obtained by using the principle of virtual work. A set of dual variables involving the displacements and nodal forces are introduced to reduce the system equations into first order ordinary differential equations (ODEs), which can be solved by conventional methods. The proposed approach possesses many advantages, such as the problem dimension is successfully reduced by one due to that the physical fields in the thickness direction are expressed analytically as Padé expansion associated with the material property distributed in gradient form, which are able to predict the stress through the thickness more precisely then traditional formulations. The IGSBFEM meshes can maintain exact geometry at any level without further communication with the computer-aided design (CAD) model, which enables the proposed method a flexible method in simulation of FGM plates with complex geometries. The NURBS basis functions can reach to C∞ continuous within a knot span, which means that the discontinuity between adjacent elements on the same patch will vanish naturally. Numerous numerical examples are performed to evaluate the superior accuracy, convergence, locking-free, grid adaptability based on the comparison with other references. Effects of gradient index and several aspect ratios on the static and free vibration responses of FGM plates are investigated.