PSI - Issue 5

Behzad V. Farahani et al. / Procedia Structural Integrity 5 (2017) 584–591 Behzad V. Farahani et al./ Structural Integrity Procedia 00 (2017) 000 – 000

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1. Introduction A novel methodology for solving partial differential equations is based on multiquadric radial basis functions (RBFs). The meshless multiquadric method was firstly introduced by Kansa (Kansa 1990a; Kansa 1990b). It is a well- known approach to solve the system of partial differential equations with superior accuracy. An important feature of the meshless RBF method is that is does not require a predefined grid. Nevertheless, the method quality is extremely dependent on an adequate choice of a parameter used in the multiquadric interpolation function, Φ( , ) = ( , ) = √ 2 + 2 , where r is the distance between the points x discretizing the problem domain (usually, but not necessarily, an Euclidean distance) and is a user defined parameter (commonly known as shape parameter). The method can be used with several other RBFs, such as classical RBFs; Gaussians , Inverse multiquadrics, thin plate spline and recently a popular family of Wendland’s Compactly Supported Functions (Schaback & Wendland 2001;Wendland 2005). The applications of RBFs for two-dimensional solids have been projected by Liu et al. [see e.g. (Liu & Gu 2001; J.G. Wang & Liu 2002; J. G. Wang & Liu 2002)] and by Ferreira et. al (Ferreira 2003a; Ferreira 2003b) for composite laminated plates and beams using the first-order shear deformation theory. This work focuses on the analysis of a single-layer isotropic Reissner- Mindlin plate by the use of Wendland’s RBF (Wendland 2005) and using the Reddy’s first -order deformation theory (Reddy 1997). 2. First-order Shear Deformation Theory The classical laminate plate theory and the first-order shear deformation theories describe with reasonable accuracy the kinematics of most thin laminates. These classical theories were originally developed for single-layered isotropic structures. These theories might be divided into two main categories: Love first-approximation theories (LFAT) and Love second-approximation theories (LSAT). LFATs rely on the well-known Cauchy – Poisson – Kirchhoff – Love thin shell assumptions (Cauchy 1828; Poisson 1829; Kirchhoff 1850; Love 1906): normals to the reference surface Ω remain normal in the deformed states and do not change in length. It means that transverse shear and transverse normal strains are negligible with respect to the other strains. When one or more of these LFAT postulates are removed, we obtain the so-called LSAT (Koiter 1960), which means the effects of transverse shear and/or transverse normal stresses can be taken into account. As a consequence of the introduction of multi-layered structures, several LFAT and LSAT were extended to multi-layered plates and shells. However, these extensions are part of the framework of equivalent single layer (ESL) theories: the layers in the multi-layered structure are seen as only one equivalent plate or shell, and the 2D approximation does not consider dependency on the index layer k (Carrera et al. 2011). The unified formulation (UF) proposed by Carrera (Carrera 1996; Carrera 1998), also known as CUF, is a powerful framework for the analysis of beams, plates and shells, because it permits to obtain the governing equations, irrespective of the shear deformation theory being considered. This formulation has been applied on several finite element analyses. Besides, Ferreira (Ferreira 2003a; Ferreira 2003b) has extensively applied the RBF collocation to the static deformations of composite beams and plates. One of the typical LSAT for the case of multi-layered structures is the first-order shear deformation theory (FSDT). The third part of the Kirchhoff hypotheses (Kirchhoff 1850) is removed, therefore the transverse normals do not remain perpendicular to the mid-surface after deformation. In this way, transverse shear strains and are included in this theory. Nevertheless, the inextensibility of the transverse normal remains, therefore displacement is constant in the thickness direction . Therefore, FSDT is an extension of the so-called Reissner – Mindlin model (Reissner 1945; Mindlin 1951) to multi-layered structures. In FSDT, the Kirchhoff hypothesis is relaxed by removing the third part; i.e., the transverse normals do not remain perpendicular to the mid surface after deformation (Reddy 1997)(Carrera et al. 2011). This amounts to including transverse shear strains in the theory. The inextensibility of transverse normals requires that w not be a function of the thickness coordinate, z. The Mindlin plate theory or first-order shear deformation theory for plates includes the effect of transverse shear deformations (Reddy 1997). It may be considered an extension of the Timoshenko theory for beams in bending. The main difference for thin, Kirchhoff-type theories is that in the Mindlin theory the normals to the undeformed middle plane of the plate remain straight, but not normal to the deformed middle surface. In the case of FSDT, under the same assumptions and restrictions as in the classical laminate theory, the displacement field of the first-order theory is of the form (Reissner 1945; Mindlin 1951):