Elsevier

Composite Structures

Volume 96, February 2013, Pages 545-553
Composite Structures

Finite element formulation of a generalized higher order shear deformation theory for advanced composite plates

https://doi.org/10.1016/j.compstruct.2012.08.004Get rights and content

Abstract

This paper presents a generalized higher order shear deformation theory (HSDT) and its finite element formulation for the bending analysis of advanced composite plates such as functionally graded plates (FGPs). New shear strain shape functions are presented. The generalized HSDT accounts for non-linear and constant variation of in-plane and transverse displacement respectively through the plate thickness, complies with plate surface boundary conditions and do not require shear correction factors. The generalized finite element code is base on a continuous isoparametric Lagrangian finite element with seven degrees of freedom per node. Numerical results for different side-to-thickness ratio, aspect ratios, volume fraction, and simply supported boundary conditions are compared. Results show that new non-polynomial HSDTs solved by the proposed generalized finite element technique are more accurate than, for example, the well-known trigonometric HSDT, having the same DOFs. It is concluded that some non-polynomial shear strain shape functions are more effective than the polynomials counterparts.

Introduction

Functionally graded materials (FGMs) are advanced composite materials that have been proposed, developed and successfully used in industrial applications since 1984 [1]. FGMs, initially used in the aerospace as thermal barrier material, nowadays have extensive application in nuclear reactors, chemical plants, and other applications such as biomechanical, optical, automotive, electronic, mechanical, civil and shipbuilding industries. FGMs are both macroscopically and microscopically heterogeneous advanced composites which are for example made from a mixture of ceramics and metals with continuous composition gradation from pure ceramic on one surface to full metal on the other. Such gradation leads to a smooth change in the material profile as well as the effective physical properties. In addition, FGMs have interesting thermo-mechanical properties that can alleviate or eliminate the deformation of structural components [2]. Its counterpart, classical composite materials, suffers from discontinuity of material properties at the interface of the layers and constituents of the composite. Therefore the stress fields in these regions create interface problems and thermal stress concentrations under high temperature environments. Furthermore, large plastic deformation of the interface may trigger the initiation and propagation of cracks in the material [3]. These problems can be decreased by gradually changing the volume fraction of constituent materials and tailoring the material for the desired application, as in FGMs.

Literature survey shows that many papers dealing with static and dynamic behaviour of functionally graded materials (FGMs) have been published recently. An interesting literature review of this work may be found in the paper of Birman and Byrd [4]. More recent work on 3D exact and closed-form analytical solutions and an updated review can be found in Carrera et al. [5] and Mantari and Guedes Soares [6]. Thus the following analysis addresses mainly the finite element formulation.

It can be said that not much work has been done for the finite element (FE) analyses of advanced composites structures such as functionally graded plates (FGPs) as for classical composites. Often, the studies on FGPs are largely devoted to thermal stress analysis [7], [8], [9] and fracture analysis FG plates and shells [10], [11]. In addition, some FE models have been already proposed for the study of FG plates and shells [7], [8], [12], [13]. Recently, Chinosi and Della Croce [14] used a mixed interpolated finite element to study cylindrical shells made of functionally graded materials. Della Croce and Venini [15] previously developed a finite element for functionally graded Reissner–Mindlin plates. Carrera et al. [16] employed the concept of virtual displacements to obtain finite element solutions of functionally graded plates subjected to transverse mechanical loadings. Later Reissner’s Mixed Variational Theorem (RMVT) and CUF were adapted to use in multilayered structures embedding FG layers in [17], [18]. Talha and Singh [19] investigated the free vibration and static analysis of functionally graded plates using the finite element method by employing a quasi-3D higher order shear deformation theory.

Carrera et al. [5] studied the effects of thickness stretching in FG plates and shells. The importance of the transverse normal strain effects in mechanical prediction of stresses for FG plates was pointed out. In fact, this work is an extension of other several FGM papers published by using Carrera’s Unified Formulation (CUF) [16], [17], [18], [20], [21]. Other refined higher-order models reproduced by CUF for example adopting the Mixed Interpolation of Tensorial Components (MITCs) finite element scheme for the approximation of FGPs was given by Cinefra et al. [22], [23], [24].

In the context of generalized or unified formulations in classical composites, it is important to notice the work done by Soldatos [25], Carrera (CUF) [26] and Demasi (GUF) [27], [28]. As mentioned, Carrera et al. [16], extended the Carrera’s Unified Formulation to advanced composites plates. Besides the powerful CUF there exists a generalized formulation proposed by Zenkour [29], which were extended to cover the stretching effect in Zenkour [30]. Also the shear deformation theory proposed by Matusanga [31], [32], [33] based on polynomial shape strain functions, needs to be mentioned in this group. The generalized shear deformation theory of FGP presented by Zenkour [29], [30] and the one proposed here are similar to the one formulated by Soldatos [25] for laminated composites. Normally non-polynomial shear strain shape functions, such as trigonometric, trigonometric hyperbolic, exponential, etc., can be used in this type of generalized formulation. However, new shear strain shape functions are introduced in this paper.

Recently, Neves et al. [34], [35] and Ferreira et al. [36] developed a quasi-3D hybrid shear deformation theory for the static and free vibration analysis of functionally graded plates by using collocation with radial basis functions. Their formulation can be seen as a generalization of the original CUF, by introducing different non-polynomial displacement fields for in-plane displacements and polynomial displacement field for the out-of-plane displacement.

Mantari and Guedes Soares [6], [37] presented bending results of FGP by using a new non-polynomial HSDT. In [37] the stretching effect was included, while in Mantari and Guedes Soares [38] different non-polynomial shear strain shape functions were considered in this context. Therefore, shape strain functions need to be further explored because they are more effective than polynomial ones [5], [39].

In the present paper, a set of new HSDTs not including the stretching effect is discussed in a generalized way by using only non-polynomial HSDTs. The present generalized theory allows the inclusion of an odd shear strain shape function, f(z). Then, adequate distribution of the inplane and transverse shear strains and the tangential stress-free boundary conditions is guaranteed, and thus a shear correction factor is not required. The generalized finite element code is based on a continuous isoparametric Lagrangian finite element with seven degrees of freedom per node. Numerical results for different side-to-thickness ratio, aspect ratios, volume fraction, and simply supported boundary conditions are compared. Results show that some of the new HSDT solved by the proposed generalized finite element technique are more accurate than, for example, the well-known trigonometric HSDT, having the same DOFs. It is concluded that some non-polynomial shear strain shape functions are richer than polynomials counterparts.

Section snippets

Generalized displacement field

A FGP of uniform thickness h is shown in Fig. 1. The rectangular Cartesian coordinate system x, y, z, has the plane z = 0, coinciding with the mid-surface of the plate. The material is inhomogeneous and the material properties vary through the thickness with a simple power-law distribution, which is given below:P(z)=(Pt-Pb)g(z)+Pb,g(z)=zh+12n,where P denotes the effective material property, Pt and Pb denote the property of the top and bottom faces of the panel, respectively, and n is the

Finite element formulation

In the present work, a four-nodded quadrilateral C0 continuous isoparametric element with seven-degrees-of-freedom per node is employed. The generalized displacements included in the present theory can be expressed as follows:δ=i=1kNiδi,where δ = {uo, vo, wo, w/∂x, w/∂y, θ1, θ2}T, δi is the displacement vector corresponding to node i, Ni is the shape function associated with the node i and k is the number of nodes per element, which is four in the present study. Considering the Eq. (7), the strain

Numerical results and discussion

In this paper a set of non-polynomial HSDTs and finite element formulation for functionally graded plates is presented. However, polynomial HSDTs can be also created. Different HSDTs are obtained or implemented by using the present generalized formulation; see the shear strain shape functions in Table 1. In this paper, only non-polynomial HSDTs are discussed. Notice that Touratier’s HSDT [44] (see also Zenkour [29]) and one developed by Karama et al. [39] can be derived as special cases of the

Conclusions

A generalized higher order shear deformation theory (HSDT) and its non-conforming finite element solution for the bending analysis of functionally graded plates (FGPs) are presented. The generalized HSDT accounts for non-linear and constant variation of in-plane and transverse displacement respectively through the plate thickness, complies with plate surface boundary conditions and do not require shear correction factors. The generalized finite element code is based on a continuous

Acknowledgment

The first author has been financed by the Portuguese Foundation of Science and Technology under the contract number SFRH/BD/66847/2009.

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