Generalized hybrid quasi-3D shear deformation theory for the static analysis of advanced composite plates

Abstract

This paper presents a generalized hybrid quasi-3D shear deformation theory for the bending analysis of advanced composite plates such as functionally graded plates (FGPs). Many 6DOF hybrid shear deformation theories with stretching effect included, can be derived from the present generalized formulation. All these theories account for an adequate distribution of transverse shear strains through the plate thickness and tangential stress-free boundary conditions on the plate boundary surfaces not requiring thus a shear correction factor. The generalized governing equations of a functionally graded (FG) plate and boundary conditions are derived by employing the principle of virtual work. Navier-type analytical solution is obtained for FGP subjected to transverse load for simply supported boundary conditions. Numerical examples of the new quasi-3D HSDTs (non-polynomial, polynomial and hybrid) derived by using the present generalized formulation are compared with 3D exact solutions and with other HSDTs. Results show that some of the new HSDTs are more accurate than, for example, the well-known trigonometric HSDT, having the same 6DOF.

Introduction

Functionally graded plates (FGPs) are made of advanced composite materials such as functionally graded materials (FGMs) which have been proposed, developed and successfully used in industrial applications since 1984 [1]. Currently FGMs are alternative materials widely used in aerospace, nuclear reactor, energy sources, biomechanical, optical, civil, automotive, electronic, chemical, mechanical, and shipbuilding industries. FGMs are both macroscopically and microscopically heterogeneous advanced composites which are made for example 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.

Classical composite structures suffer 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 [2]. These problems can be decreased by gradually changing the volume fraction of constituent materials and tailoring the material for the desired application.

In fact, FGMs are materials with spatial variation of the material properties. However, in most of the applications available in the literature, as in the present work, the variation is through the thickness only. Therefore, the early state development of improved production techniques, new applications, introduction to effective micromechanical models and the development of theoretical methodologies for accurate structural predictions have encouraged researchers and opened several research topics in this field.

Literature survey shows that many papers dealing with static and dynamic behavior of functionally graded materials (FGMs) have been published recently. An interesting literature review of abovementioned work may be found in the paper of Birman and Byrd [3]. Therefore, for completeness, in the present article, the relevant work from 2007 up to now is described. For research articles perhaps not included in the above mentioned review paper, readers may consult Carrera et al. [4] and Mantari et al. [5].

Zenkour [6] investigated the static problem of transverse load acting on exponentially graded (EG) rectangular plates using both 2D trigonometric plate theory (TPT) and 3D elasticity solution. The 2D trigonometric plate theory presented in this paper includes the stretching effect, ε_zz ≠ 0. Sladek et al. [7] presented the static and dynamic analysis of functionally graded plates by the meshless local Petrov–Galerkin method. The Reissner–Mindlin plate bending theory was utilized to describe the plate deformation. Numerical results were presented for simply supported and clamped plates.

Bo et al. [8] presented the elasticity solutions for the static analysis of functionally graded plates for different boundary conditions. Stress analysis due to thermal and mechanical loads was given by Matsunaga [9] by using a two-dimensional higher-order theory. A power law distribution for the volume fractions of constituents was assumed for the calculation of modulus of elasticity. Navier solution of a simply supported functionally graded plate was provided for stress and deflections. Khabbaz et al. [10] provided a nonlinear solution of FGPs using the first and third-order shear deformation theories. More recently, Aghdam et al. [11] presented a static analysis of fully clamped functionally graded plates and doubly curved panels by using the extended Kantorovich method.

Wu and Li [12] used a RMVT-based third-order shear deformation theory of multilayered FGPs under mechanical loads. The exponent-law distributions through the thickness and the power-law distributions of the volume fractions of the constituents were used to obtain the effective properties. Talha and Singh [13] 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. Vaghefi et al. [14] presented a three-dimensional static solution for thick functionally graded plates by utilizing a meshless Petrov–Galerkin method. An exponential function was assumed for the variation of Young’s modulus through the thickness of the plate, while the Poison’s ratio was assumed to be constant. RMVT-based meshless collocation and element-free Galerkin methods for the quasi-3D analysis of multilayered composite and FGPs were presented by Wu and Li [15].

Carrera et al. [4] studied the effects of thickness stretching in FGPs plates and shells. The importance of the transverse normal strain effects in mechanical prediction of stresses for FGPs was pointed out. In fact, this work is an extension of several FGM papers published by using Carrera’s Unified Formulation (CUF), as described in Carrera et al. [16], Brischetto [17] and Brischetto and Carrera [18], [19]. Recently, Neves et al. [20], [21] and Ferreira et al. [22] presented a quasi-3D hybrid polynomial and trigonometric 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 et al. [23], [24] presented bending results of FGPs by using a new non-polynomial HSDT. In the authors last paper [24] the stretching effect was included, and therefore it can be reproduced also by the generalized formulation proposed here.

Regarding to generalized formulations in classical composites, it is important to remark the work done by Soldatos [26], Carrera (CUF)[27] and Demasi (GUF)[28], [29]. As mentioned above, Carrera et al. [16], recently extend his unified formulation to advanced composites (FGM). Besides the powerful CUF there exists a generalized formulation proposed by Zenkour [30], which were extended to cover the stretching effect in Zenkour [6]. Also the shear deformation theory proposed by Matusanga [31] (based on polynomial shear strain shape functions) needs to be mentioned in this group.

The generalized shear deformation theory of FGPs presented by Zenkour [6], [30] is similar to the one formulated by Soldatos for laminated composites [26]. Normally non-polynomial shear strain shape functions, such as trigonometric, trigonometric hyperbolic, exponential, etc., can be used in this type of generalized formulation. However, the thickness expansion modeling is conditioned by the in-plane displacement model (the transverse non-linear function in the modeling of the thickness expansion is an even function which is the derivative of the in-plane non-linear shear strain shape function, g(z) = f′(z)). Therefore, it is not free to choose shear strain shape the thickness displacement field. The present formulation has that freedom, and infinite hybrid type shear deformation theories (polynomial or non-polynomial or hybrid type) can be created just having 6DOF.

As mentioned above, the unified formulation of Carrera applied to advanced composite plates by using meshless methods was recently extended to include non-polynomial shear strain shape functions in their formulation [20], [21], [22], and the need of new non-polynomial shear strain shape functions, which can be adapted to this advanced generalized formulation, is demanding. In addition to the main contribution of the present paper, new non-polynomial shear strain shape functions are presented, which can be of interest of researchers on this field, and in particular to the ones that present generalized formulations like the present one.

In the present paper, a generalized formulation for the static analysis of functionally graded plates is developed using a 6DOF generalized displacement field. In fact, many hybrid (polynomial and/or non-polynomial) HSDTs, including the so-called “stretching effect”, can be derived by using the present generalized formulation. The theory allows the inclusion of an even shear strain shape function, g(z), for the adequate distribution of the transverse shear strains through the plate thickness and in this way complies the tangential stress-free boundary conditions on the plate boundary surface, and thus a shear correction factor is not required. The plate governing equations and its boundary conditions are derived by employing the principle of virtual work. Navier-type analytical solution is obtained for plates subjected to transverse load for simply supported boundary conditions. Benchmark results for the displacement and stresses of functionally graded rectangular plates are obtained. The results of some new hybrid HSDTs are compared with 3D exact solution and with other HSDTs available in the literature. New simple and accuracy non-polynomial HSDTs were created.