Radial basis functions–finite differences collocation and a Unified Formulation for bending, vibration and buckling analysis of laminated plates, according to Murakami’s zig-zag theory J.D. Rodrigues a , C.M.C. Roque a,⇑ , A.J.M. Ferreira b , E. Carrera c , M. Cinefra c a INEGI, Faculdade de Engenharia, Universidade do Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal b Departamento de Engenharia Mecânica, Faculdade de Engenharia, Universidade do Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal c Department of Aeronautics and Aerospace Engineering, Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129 Torino, Italy article info Article history: Available online 20 January 2011 Keywords: Radial basis functions Finite differences Zig-zag theory Composites abstract In this paper, we propose to use the Murakami’s zig-zag theory for the static and vibration analysis of laminated plates, by local collocation with radial basis functions in a finite differences framework. The equations of motion and the boundary conditions are obtained by the Carrera’s Unified Formulation, and further interpolated by a local collocation with radial basis functions and finite differences. This paper considers the analysis of static deformations, free vibrations and buckling loads on laminated com- posite plates. Ó 2011 Elsevier Ltd All rights reserved. 1. Introduction Multilayered structures show a piece-wise continuous displace- ment field in the thickness plate/shell direction. This change in slope between two adjacent layers, that are considered to be per- fectly bonded together, is known as the zig-zag (ZZ) effect, see Fig. 1. The different transverse (both shear and normal compo- nents) deformability of the layers is the source of the ZZ effect. Fur- thermore, these transverse strains are linked with transverse shear and normal stresses that, for equilibrium reasons, are continuous at the each layer interface. These equilibrium conditions are known as interlaminar continuity (IC) for transverse stresses. There are several possibilities to take into account ZZ and IC in multilayered structures [1–6]. Some of these have been developed in the framework of layer-wise (LW) models, in which the number of the unknown variables depend on the number of layers, but they could result computational expensive, for laminates with large number of layers. Other theories have been formulated in the framework of equivalent single-layer (ESL) models, in which the unknown variables are the same for the whole laminate. The resulting theories are often denoted as zig-zag theories (ZZT). Among the ZZT, three independent approaches are known. These have been denoted in [7] as Lekhnitskii multilayered theory, Ambartsuniam multilayered theory and Reissner multilayered the- ory. LMT and AMT describe the ZZ effect by enforcing IC via consti- tutive equations of the layer along with strain-displacement relations. Independent assumptions for displacement and trans- verse stresses are instead made in the RMT applications. In the framework of RMT applications, Murakami [8] introduced a function of the thickness coordinate able to emulate the ZZ effect. In [9], such a function was denoted as ‘Murakami zig-zag function’ (MZZF). MZZF has been used in [8–13] to analyse static response of layered plates and shells in conjunction of RMT applications. Mixed finite elements for plates and shells have been developed in [14– 18]. MZZF has been also applied in the framework of plate/shell theories using only displacement variables [13,19]. From imple- mentation point of view, the inclusion of MZZF in existing plate models requires the same efforts that are required by the inclusion of an additional degree of freedom. On the other hand, from numerical point of view, as it will be clear in this paper, inclusion of MZZF leads to significant improvements of the existing plate theories; however, these improvements are difficult to be obtained by the use of other functions which differ by MZZF. An extensive evaluation of the use of MZZF has made in [20,21] using an analyt- ical formulation and in [22] using the finite element method. In the present work the attention is restricted to the application of ZZF to bending, vibration and buckling analysis of laminated plates by a local collocation scheme with radial basis functions and finite differences. A new displacement theory is used, introduc- ing a quadratic variation of the transverse displacements. This can be seen as a variation of the Murakami’s ZZ displacement field [8]. The use of global collocation based on radial basis functions (RBF) has been proposed by Kansa [23] who introduced the con- cept of solving PDEs by an unsymmetric RBF collocation method based upon the MQ interpolation functions. Unfortunately, this 0263-8223/$ - see front matter Ó 2011 Elsevier Ltd All rights reserved. doi:10.1016/j.compstruct.2011.01.009 ⇑ Corresponding author. E-mail address: croque@fe.up.pt (C.M.C. Roque). Composite Structures 93 (2011) 1613–1620 Contents lists available at ScienceDirect Composite Structures journal homepage: www.elsevier.com/locate/compstruct