Elsevier

Composites Part B: Engineering

Volume 60, April 2014, Pages 319-328
Composites Part B: Engineering

A new tangential-exponential higher order shear deformation theory for advanced composite plates

https://doi.org/10.1016/j.compositesb.2013.12.001Get rights and content

Abstract

This paper presents the static response of advanced composite plates by using a new non-polynomial higher order shear deformation theory (HSDT). The present theory accounts for non-linear in plane displacement and constant transverse displacement through the plate thickness, complies with plate surface boundary conditions, and in this manner a shear correction factor is not required. Navier closed-form solution is obtained for functionally grade plates (FGPs) subjected to transverse loads for simply supported boundary conditions. The optimization of the shear strain function and bi-sinusoidal load is adopted in this publication. The novelty of this work is the geometry used, stretching effect is not applied and the amount of unknown displacement functions are 5. In addition to the optimization, the inclusion of an exponential function to the tangent function is an interesting feature in this paper. The accuracy of the present HSDT is discussed by comparing the results with an existing quasi-3D exact solution and several HSDTs results. It is concluded that the present non-polynomial HSDT, is more effective than the well-known trigonometric HSDT for well-known example problems available in the literature.

Introduction

Functionally graded materials (FGMs) are a mixture of ceramic and metal in such way that the ceramic can resist high temperature in thermal environments, whereas the metal can decrease the tensile stresses that are produced on the ceramic surface at the earlier state of cooling. The application of FGMs are diverse, such as in spacecraft heat shields, heat exchanger tubes, biomedical implants, flywheels, plasma facings for fusion reactors, chemical plants, solar energy generators, heat exchangers, nuclear reactors and high efficiency combustion systems. An advantage of FGMs is that there are not discontinuities through the thickness and nor development of failures due to interfacial stress concentrations. An interesting review by Jha et al. [1] pointed out the application of the advanced composite materials in several fields and described the contributions on FGMs up to 2012; they were denominated as novel composite materials proposed by Bever and Duwez [2] and then developed and successfully used in a lot of industrial applications nowadays [3].

In a standard functionally grade plates (FGPs); the material properties vary on the thickness direction. However, this is the simplest model usually used in the open literature. In fact, FGPs can vary in a 3D way. In the engineering field there is a tremendous need to develop efficient manufacturing techniques, economical and effective repair techniques, as well as methods to predict the short and long-term behavior of advanced composite structures under a variety of loading and environmental conditions. Therefore, it is important to introduce effective macro, micro or perhaps nano mechanical models, and the development of new theoretical methodologies for simple and accurate predictions of the structural response.

The literature for the bending analysis of FGPs is limited when compared to isotropic and laminated plates. The reason behind that fact is because FGMs are applied in high temperature environments and most behavioral studies of plates were focused on thermo-mechanical aspects [4], [5], [6] and fracture analysis of the same way for shells [7], [8].

The study on FGPs started with the analysis of thin plates using Classical Plate Theory (CPT). Then the first order shear deformation theory (FSDT) was utilized but as is well-known their accuracy depends on the shear correction factor which may be difficult to compute on advanced composites. Fortunately, the higher order shear deformation theories (HSDTs) do not have such inconvenience. However, most of these theories do not account for transverse extensibility by neglecting the stretching effects.

During the last forty years, a significant number of higher order shear deformation theories (HSDTs) for classical composites (plates and shells) have been presented. These theories can be classified in different models, such as equivalent single layer (ESL), quasi-layerwise and layerwise models in [9], [10]. Another important point in the analysis of composite structures is the variational statement used to derive the necessary governing equations and the boundary conditions. Approximate solutions can be either developed using displacement-based theories (when the principle of virtual displacement is used), stress-based theories or displacement-stress-based theories (when Reissner mixed variational theorem is used), for more details see [9], [10], [11], [12], [13], [14].

Several studies have been done to analyses the thermal, mechanical, and dynamic responses of functionally graded (FG) beams, plates and shells. Tauchert [15] gave a nice overview of thermally induced flexure, buckling and vibration applied to classical composite plates described by the Kirchhoff theory. Based on the knowledge gained in classical composites and the importance of FGMs’ thermal behavior, the studies on advanced composites were largely devoted to thermal stress analysis [16], [17], [18]. Tanigawa [19] presented an extensive review of thermo elastic of FG structures. An excellent introduction to the fundamentals of FGMs was provided by Suresh [20]. Main and Spencer [21] calculated 3D exact solutions for FG plates with traction-free surfaces.

Reddy [6] presented Navier’s closed-form solutions and finite element formulations including also geometric non-linearity based on the polynomial shear deformation theory for the analysis of FGM plates. Cheng and Batra [22] derived the governing equations for a functionally graded plate by utilizing the shear deformation theories of plates and simplified them for a simply supported polygonal plate. An exact relationship was established between the deflection of the functionally graded plate and an equivalent homogeneous Kirchhoff plate. Batra and Vel [23] presented a 3D solution for the cylindrical bending vibration of simply supported FG thick plates using displacement fields that identically satisfy boundary conditions to reduce the governing equations to a set of coupled ordinary differential equations.

Vel and Batra [24], [25] developed a three-dimensional analysis of the transient thermal stresses, and the free and forced vibration of a simply supported FGM rectangular plate. Pan [26], proposed an exact solution for functionally anisotropic elastic composite laminates based on Pagano’s exact solution [27]. A 3D elasticity solution was proposed by Kashtalyan [28] for a functionally graded simply supported plate under transversely distributed load.

Ferreira et al. [29] studied the static characteristics of functionally graded plates using third-order shear deformation theory and meshless method based on the multiquadrics radial basis function. The natural frequencies of functionally graded plates using multiquadrics radial basis function were calculated by Ferreira et al. [30]. A generalized shear deformation theory for bending analysis of functionally graded plates was developed by Zenkour [31]. Large deflection analysis of rectangular FG plates was studied by Ghannad Pour and Alinia [32]. An exact and approximate solution for the bending problem of FGPs was presented by Zenkour [33]. Ferreira et al. [34] studied the static deformations of functionally graded plates using the radial basis function collocation method and a higher-order shear deformation theory.

Carrera et al. [35] presented the static analysis of functionally graded material plates subjected to transverse mechanical loadings. The natural frequencies and buckling stresses of plates made of functionally graded materials (FGMs) using a 2-D higher-order deformation theory was calculated by Matsunaga [36]. The effect of thickness stretching in plate and shell structures was studied by Carrera et al. [37]. An interesting literature review of above mentioned work may be also found in the paper of Birman and Byrd [38].

This paper presents the static response of advanced composite plates by using a new trigonometric-exponential higher order shear deformation theory (HSDT). Previous HSDTs [39], [40], [41] demonstrated that the inclusion of exponential function in the shear strain function produces results with good accuracy. Based on this experience, the exponential function and the tangential shear strain shape [42] is combined here to obtain a new, improved and accurate HSDT. With this foundation, the elasticity theory and the principle of virtual work are utilized to obtain the strain field, derivative the governing equations and boundary conditions, respectively. Navier’s closed-form solutions for different FGPs under bi-sinusoidal and distributed loads are presented. An optimization process is required for the correct selection of a shear strain function, and it was performed in the same way as in Mantari and Guedes Soares [43], which is a common point between the two papers, besides that both studies adopt a bi-sinusoidal load. However the following differences should me mentioned: (a) The analysis conducted in this paper is of a rectangular plate and the other work is a rectangular doubly-curved shell; (b) The displacement functions have five unknowns, being six unknown displacement functions in [43]; (c) in the present publication the stretching effect is not considered, being present in [43].

As a novelty, the present work includes a new exponential function added to the tangent function, forming a non-polynomial function which improves the results obtained by others HSDTs for plates, having as reference the quasi-exact 3D solution provided by Brischetto and Carrera [44].

The accuracy of the present code is verified by comparing it with the quasi-exact 3D solution by Brischetto and Carrera [44] and other HSDTs such us Reddy’s HSDT [6]. It is concluded that the present non-polynomial HSDT, is more effective than the well-known trigonometric HSDT for well-known example problems available in the literature.

Section snippets

Theoretical formulation

A rectangular plate of uniform thickness h made of a functionally graded material 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)V+Pb,V=zh+12n,V=g(z¯)=z¯+12n,where P denotes the effective material property, Pt and Pb denote the property of the top and

Analytical solution

Exact solutions of the partial differential Eqs. (10a–e) on a 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 (see Oktem and Chaudhuri [45] Oktem and Guedes Soares [46]).

Solution functions that completely satisfy the boundary conditions in Eqs. (12a), (12b), (12c)

Bending analysis of FGM (Al/Al2O3 plates): a validation procedure

The results are presented for the simply supported plate under bi-sinusoidal and distributed transverse loads of intensity q. The static analysis was conducted using aluminium (bottom, Al) and alumina (top, Al2O3).

The following material properties are used for computing the numerical results [33]:Et=380GPa,vt=0.3;Eb=70GPa,vb=0.3

The following non-dimensional quantities are used:u¯(z)=100h3Etqa4ua2,b2,z,w¯(z)=10h3Etqa4wa2,b2,z,σ¯xx(z)=σxxhqaa2,b2,z,σ¯xy(z)=σxyhqa(0,0,z),σ¯xz(z)=σxzhqa0,b2,z.where

Conclusions

The static responses of FGPs are discussed using a new and several HSDTs. The HSDT models represent the displacement field in a proper manner. Elasticity theory is utilized to obtain the strain field and the principle of virtual work is adopted to obtain the governing equations and boundary conditions. Navier’s closed-form solutions for different FGPs under bi-sinusoidal and distributed loads are presented. The optimization of shear strain function and bi-sinusoidal load used in this

Acknowledgment

The first author has been financed by the Portuguese Foundation of Science and Technology under the Contract No. SFRH/BD/66847/2009.

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