Elsevier

Aerospace Science and Technology

Volume 58, November 2016, Pages 559-570
Aerospace Science and Technology

General recommendations to develop 4-unknowns quasi-3D HSDTs to study FGMs

https://doi.org/10.1016/j.ast.2016.09.007Get rights and content

Abstract

In this paper a generalized non-polynomial quasi-3D shear deformation theory for advanced composite plates is presented and further discussed. The beauty of this generalized theory relies on its 4-unknowns displacement field, which is even less than the classical first order shear deformation theory (FSDT). Moreover, this generalized theory models the stretching effect due to its quasi-3D nature. The main conclusions after solving the governing equations of functionally graded plates (derived by employing the principle of virtual work and solved via Navier-type closed-form solution) are the following: (a) This theory was not much explored and needs further research, (b) the theory performs very well for non-polynomial shear strain shape functions f(z) and g(z), but not for a hybrid case (non-polynomial and polynomial), (c) an optimization procedure is mandatory to select the parameters directly related to f(z) and g(z), (d) bending results strongly depend on the selected shear strain shape functions and the case dependent problem can be verified.

Introduction

Functionally graded materials (FGMs) are considered a kind of advanced composite material which was originally developed in Japan [1], [2]. The interesting feature of this kind of composites is the continuity along a desired direction through a structural element (shell, plate or beam). Some types of classical composites suffer in continuity through the thickness direction; such discontinuity can be alleviated by a gradual and smooth change of mechanical properties through the thickness of the structural element as in FGMs. Moreover, FGMs allow obtaining high thermal and toughness mechanical properties, as a result of mixing for example ceramic and steel.

General remarks on FGMs can be found in the review paper by Jha et al. [3]. However, a detailed review that focuses on the stress, vibration and Buckling analysis of FGPs were recently carried out by Swaminathan and coauthors [4]. From this paper a very large list of contributions on shear deformation theories of plates made of FGMs can be obtained. Both analytical and numerical solution of the shear deformation plate theories subjected to several types of loads were listed but without taking into account the mathematical implication of the used methodologies and contributions.

From the review made in Refs. [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22] and considering the analytical point of view, it can be observed that there exist two well-described variational statements (Principle of virtual displacement (PVD) and Reissner mixed variational theorem (RMVT)). The contributions are substantially more representative on PVD than in RVMT. Regarding PVD, there are two ways to model the displacement field, using axiomatic expansion or asymptotic expansion. In this group, it can be said the most utilized expansion is the axiomatic approach. Since this paper and most of the contributions of this author and colleagues are on PVD and axiomatic approach, and the recently developed quasi-3D HSDT with four-unknowns with thickness stretching effect is also based on this approach [23], a review of these theories is made in what follows.

It can be observed that there are two kinds of displacement fields or shear deformation theories based on PVD and axiomatic approach. In one hand, the shear deformation theories with displacement fields that require only displacement continuity across the interelement boundaries (C0, ui=f(z)f(u), where i can be any of the directions x, y and z; f(z) is a priori defined function and u is to be determined through analytical or numerical procedures), which is suitable for the formulation of general finite elements, see for example the FSDT or Carrera Unified Formulation (CUF). On the other hand, the HSDTs (C1 continuity, ui=f(z)f(u,u)) that account for approximately parabolic transverse shear deformation and satisfy transverse shear traction free conditions on the top and the bottom surfaces of the plate and do not require any shear correction factor. However these shear deformation theories have all positive features except one drawback, which is found in a situation when finite element analysis is applied to this theory.

Researchers in this field reach to these points, and a normal question is: why does one could be interested on PVD, axiomatic, and C1 approach? or why a HSDT (C1 continuity) should be selected to model a physical problem? These questions could be answered with two thoughts in mind: first of all, it was verified that theories with C1 continuity and a given number of unknowns with non-polynomial expansion are more precise than polynomial ones (see Karama et al. [24] and Mantari et al. [25], [26], [27], [28]). For details readers may consult the paper by Karama et al. [24]. The second idea is that, theories with reduced number of unknowns and optimized to get closed to 3D solutions can give as good accuracy as advanced theories with higher number of unknowns [29], [30]. Consequently, a comparative study on optimized non-polynomial shear deformation theories with C1 and C0 continuity should be carried out. Additionally, further research work should be also performed to study shear deformation theories with non-polynomial asymptotic expansion, because this topic is in its infancy, i.e. very few works in this topic exist [31].

The first non-polynomial HSDT was developed by Levy [32] in 1877 [33] and after more than one century the sinusoidal HSDT were studied and assessed by Stain [34] in 1986. Currently, this HSDT was extensively used by Touratier [35], Vidal and Polit [36], [37], [38], [39], Ghugal [33], Zenkour [40], [41], etc. Then, Soldatos [42] proposed a hyperbolic shear strain shape function, Karama et al. [24] introduced an exponential expansion, etc. Others non-polynomial shear strain shape functions can be found in the paper by Viola et al. [43], [44]. The first author that optimized a non-polynomial HSDT was Aydogdu [45]. The author in an elegant way optimized the exponential HSDT without thickness stretching effect proposed by Karama by changing the number e of the shear strain shape function for alpha which needed to be optimized. Based on this idea, this author and colleagues proposed new optimized shear strain shape functions for both HSDT with and without thickness stretching effect in Refs. [25], [26], [27] and Ref. [28], respectively.

A refined and generalized self-consistent theory was considered by Bian et al. [46]. This is an extension of Soldatos' HSDT [42] to study the cylindrical bending behavior of FGPs. Zenkour [47] extended the Levy's HSDT (sinusoidal in its nature) to develop a generalized formulation for FG sandwich plates (layered structures). The author further used the sinusoidal plate theory to study different case problems in Ref. [48]. Mantari et al. [25], [49], [50], Mantari and Guedes Soares [51] and Mantari and Granados [52], [53] developed several refined non-polynomial HSDTs to study the static, vibration and thermoelasticity problems of classical composites and FGPs. In most of the papers abovementioned the results obtained were compared with those obtained using 3D elasticity solution and the sinusoidal HSDT. It was concluded that the refined and optimized theories give better results compared with the sinusoidal HSDT. In fact, the authors introduced original and very accurate shear strain shape functions which permitted to obtain good results. Daouadji et al. [54] used an unusual hyperbolic trigonometric HSDT with only four unknowns without including the thickness stretching effect to study the static and dynamic analysis of FGPs. Mechab et al. [55] used a new hyperbolic function for static and dynamic analyses of FGPs using the four-variable refined plate theory but without including the thickness stretching effect. Thai et al. [56] and Hebali et al. [57] developed a quasi-3D hyperbolic shear deformation theory for bending and vibration analysis of FGPs. Belabed et al. [58] presented an efficient and simple HSDT for bending and free vibration analyses of FGPs. The number of unknowns and governing equations were reduced to five instead of six by splitting the transverse displacement into bending, shear and thickness stretching parts.

It should be remarked that in this decade Zenkour [59] introduced for the first time a very interesting HSDT with just four unknowns and thickness stretching effect. A refined trigonometric plate theory with four-unknowns and thickness stretching effect was proposed by Zenkour [60] for FG sandwich plates. The accuracy of the obtained results was demonstrated by comparing with those already available in the literature. In Mantari and Guedes Soares [61] further assessment of the 4-unknown shear deformation theory with thickness stretching effect was performed.

Overall, it is important to remark that the amount of contributions on 4-unknown HSDTs compared with other types of quasi-3D HSDTs with more than 4 unknowns is very few. Even, the quasi-3D HDST with 5-unknowns was not much explored. More importantly, this theory has special features and particularities that researchers should know before the further implementation of such kind of theories. The goal of this paper is to demonstrate that some theories are good for a given case problem but not for others, i.e. the case independent problem is shown. Moreover, this paper shows that is possible to gain accuracy by keeping constant the number of unknown but using non-polynomial quasi-3D HSDT with 4-unknowns. Those key features will be discussed in this paper for the first time. Moreover, interesting comparison analysis with data available in the literature is performed.

The paper is organized as follow. Section 2 outlines the mathematical modeling of the present quasi-3D HSDT. In view of the fact that details of this theory for the bending problem can be also found in Ref. [61], short version of the solution methodology is presented in Section 3. In section 4 the results and discussions are given. Finally, further general considerations of this kind of theories are given in the conclusions.

Section snippets

Analytical modeling

In a FGP the mechanical properties can be smoothly graded from different directions and considering different shapes. This paper considers the well-known across the thickness gradation modeling of mechanical properties of FGPs.

Solution procedure

Exact solutions of the partial differential equations (6a), (6b), (6c), (6d) on arbitrary domain and for general boundary conditions are difficult. Although the Navier type solutions can be used to validate the present theory, more general boundary conditions will require solution strategies involving, e.g., boundary discontinuous double Fourier series approach.

Solution functions that completely satisfy the boundary conditions are assumed as follows:u(x,y)=r=1s=1Urscos(αx)sin(βy),0xa;0

Numerical results and discussion

The bending analysis of FGPs is presented in what follows. Several quasi-3D hybrid type HSDT, i.e. theories with thickness stretching effect and just 4-unknowns are presented in this paper for the first time. The goal of this paper is to demonstrate that is possible to gain accuracy by keeping constant the number of unknown variables and by using non-polynomial quasi-3D HSDT with 4-unknowns; the case dependent problem is also verified.

One of the key features of this paper is the proper

Conclusions

A generalized quasi-3D hybrid type HSDT with 4 unknowns has been discusses and several HSDTs of this kind are presented here for the very first time. Two interesting case problems are discussed as well as their governing equations and boundary conditions are derived by employing the principle of virtual work. A Navier-type closed-form solution is obtained for functionally graded single and sandwich plates for simply supported boundary conditions. Results show that the present generalized

Conflict of interest statement

None declared.

Acknowledgement

The author wants to dedicate this work to Franz Williams who pass away during the supervision of my bachelor thesis. I remember their words “Job, family (Cielo and Lizbeth) and thesis is not easy but not impossible”. I also appreciate a lot his bibliographic support on classical composites.

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