Keywords: isogeometric analysis, NURBS, free vibration, buckling, laminated composite plates, Lagrange multiplier method.

As a result of the higher strength to weight and stiffness to weight ratios of composite materials compared with the conventional isotropic materials, laminated composites have been increasingly and widely used in a wide range of engineering structures and modern industries. Therefore, studies involving the stability behaviour and natural vibrations of those structures are of great importance and essential in predicting their structural response.

Isogeometric analysis (IGA) is a novel computational approach which was introduced by Hughes and co-workers in 2005 [1] with the aim of integrating the conventional finite element method (FEM) and computer aided design (CAD). The basic idea of IGA is to utilize the basis functions that are able to model accurately the exact geometries from the CAD point of view for numerical simulations of physical phenomena. A distinct advantage over the FEM is that the mesh refinement is simply accomplished by re-indexing the parametric space without interaction with the CAD system. An intriguing trait of these functions is that they are typically smooth beyond the classical C⁰ continuity of the standard FEM. As a result of the good inherent characteristics of the IGA method, developing an isogeometric finite element method associated with the NURBS shape functions for free vibration and buckling analysis of thin symmetrically laminated composite plates is presented in this paper. The present formulation follows the classical plate theory (CPT). The finite element formulation based on the classical plate theory (Kirchhoff theory of plates) requires elements with at least C¹ inter-element continuity, which has many difficulties to achieve the free-form geometries when using the standard Lagrangian polynomials as basis functions. As in the isogeometric analysis, higher order NURBS basis functions with an increased inter-element continuity can be easily obtained, the NURBS are well suited for the Kirchhoff elements. The approximation of the solution space for the deflection of the plate and the parameterization of the geometry are performed using the NURBS-based approach. The essential boundary conditions are formulated separately from the discrete system equations with the aid of the Lagrange multiplier method, while an orthogonal transformation technique is also applied to impose the essential boundary conditions in the discrete eigenvalue equation. The accuracy and the efficiency of the proposed method are thus demonstrated through a series of numerical experiments of laminated composite plates with different boundary conditions, fibre orientations, lay-up number, etc. The numerical results obtained are then compared with either the analytical solutions or other available numerical methods, and excellent agreements are found.

1

T.J.R. Hughes, J.A. Cottrell, Y. Bazilevs, "Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement", Computer Methods in Applied Mechanics and Engineering, 194, 4135-4195, 2005. doi:10.1016/j.cma.2004.10.008

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