An efficient and simple higher order shear and normal deformation theory for functionally graded material (FGM) plates
Introduction
With the ever-increasing demand for high strength-to-weight ratio materials, engineering materials are continuously replacing the conventional materials in the automotive, nuclear and aerospace industries. In general, most engineered materials are inspired from nature. A new class of materials was introduced by Japan scientists [1] to decrease the thermal stresses in propulsion and airframe structural systems of astronautical flight vehicles. This class of engineered materials was designated as functional graded material (FGM). FGMs are a class of composites that exhibit continuous variation of material properties from one surface to another and thus eliminate the stress concentration generally encountered in laminated composites. A typical FGM is synthesized from a mixture of ceramic and metal. These materials are often isotropic but non-homogeneous. The reason for sustained interest in FGMs is that it may be possible to create certain types of FGM structures capable of adapting to operating conditions. The increase in FGM applications requires accurate mathematical models to predict their responses.
In the past three decades, researches on functionally graded material (FGM) plates have received substantial attention, and an extensive spectrum of plate theories has been introduced based on the classical plate theory and shear deformation plate theory. The classical plate theory (CPT) neglects shear deformations and can lead to inaccurate results for moderately thick plates. This theory has been implemented for buckling analysis of FGM plates by Feldman and Aboudi [2], Abrate [3], Mahdavian [4], and Mohammadi et al. [5]. However CPT under-predicts deflections and over-predicts frequencies as well as buckling loads of moderately thick plate [6]. First-order shear deformation theory (FSDT) [7], [8] considers the transverse shear deformation effects, but needs a shear correction factor in order to satisfy the zero transverse shear stress boundary conditions at the top and bottom of the plate. Many studies of the mechanical behavior of plates have been carried out using FSDT [9], [10], [11], [12], [13], [14], [15]. To avoid the use of shear correction factors, several higher-order shear deformation plate theories (HSDTs) have been proposed such as the theory propounded by Nelson and Lorch [16] with nine unknowns, Lo et al. [17] with eleven unknowns, Bhimaraddi and Stevens [18] with five unknowns, Reddy [19] with five unknowns, Kant and Pandya [20] with seven unknowns, Kant and Khare [21] with nine unknowns and Talha and Singh [22] with eleven unknowns. Some higher order theories based on Carrera’s Unified Formulation (CUF) such as proposed in Refs. [23], [24], [25], [26], [27], [28], [29], [30], [31] have been used also to study FGM structures.
The majority of HSDTs used to investigate FGM plate mechanics have the same five unknowns (e.g., third-order shear deformation theory [32], sinusoidal shear deformation theory [33], hyperbolic shear deformation theory [34], [35]). The resulting equations of motion are much more complicated than those yielded with FSDT. In addition, it should be noted that the above-mentioned two-dimensional plate theories (i.e., CPT, FSDT, and HSDT) discard the thickness stretching effect (i.e. they assume ) as they consider a constant transverse displacement through the thickness. This assumption is appropriate for thin or moderately thick FGM plates, but is inadequate for thick FGM plates [36]. The importance of the thickness stretching effect in FGM plates has been identified succinctly in the work of Carrera et al. [37]. This effect plays a significant role in thick FGM plates and should be taken into consideration.
In general, higher order shear and normal deformation theories which consider thickness stretching effect can be implemented using the unified formulation initially proposed by Carrera [38], [39], [40] and recently extended by Demasi [41], [42], [43], [44], [45]. More detailed information and applications of the unified formulation can be found in the recent books by Carrera et al. [46], [47]. Many higher order shear and normal deformation theories have been proposed in the literature [48], [49], [50]. These theories are cumbersome and computationally expensive since they invariably generate a host of unknowns (e.g., theories by Talha and Singh [49] with thirteen unknowns; Reddy [50] with eleven unknowns; and Neves et al. [24], [25], [26] with nine unknowns). Although some well-known quasi-three-dimensional theories developed by Zenkour [51] and recently by Mantari and Guedes Soares [52], [53] have six unknowns, they are still more complicated than the FSDT. Thus, there is a scope to develop an accurate higher order shear and normal deformation theory, which is relatively simple to use and simultaneously retains important physical characteristics. Indeed, Huu and Seung [54] presented recently a quasi-3D sinusoidal shear deformation theory with only five unknowns for bending behavior of FGM plates.
In the present article, a new higher order shear and normal deformation theory with only five unknowns is developed for FGM plates. Contrary to the four-variable refined theories elaborated in [55], [56], [57], [58], [59], [60], [61], [62], [63], where the stretching effect is neglected, in the current investigation this so-called “stretching effect” is taken into consideration. The displacement field is chosen based on a hyperbolic variation of in-plane and transverse displacements through the thickness. Partitioning the transverse displacement into the bending, shear and thickness stretching components leads to a reduction in the number of unknowns, and consequently, makes the new theory much more amenable to mathematical implementation. The equations of motion and boundary conditions are derived from Hamilton’s principle. Analytical solutions for bending and free vibration are obtained. Numerical examples are presented to verify the accuracy of the present theory. The current study is relevant to aero-structures.
Section snippets
Kinematics
The displacement field of the present theory is chosen based on the following assumptions: (1) The transverse displacements are partitioned into bending, shear and stretching components; (2) the in-plane displacement is partitioned into extension, bending and shear components; (3) the bending parts of the in-plane displacements are similar to those given by CPT; and (4) the shear parts of the in-plane displacements give rise to the hyperbolic variations of shear strains and hence to shear
Analytical solutions
Consider a simply supported rectangular plate with length a and width b under transverse load q. Based on Navier solution method, the following expansions of displacements (, ) are assumed as:where and unknown parameters must be determined, is the eigenfrequency associated with (m, n)th eigenmode, and and .
The transverse
Numerical results
In this section, various numerical examples are presented and discussed to verify the accuracy of the present theory in predicting the natural frequency of simply supported plates. Two types of FGM plates of and [69], [70], [71] are used in this study, and their corresponding material properties are listed in Table 1. The Young’s modulus and density of FGM plates (unless otherwise stated) are evaluated using the power law distribution (see Eq. (14)). The effective density
Conclusions
An efficient and simple higher order shear and normal deformation theory is presented for bending and free vibration analyses of FGM plates. The accuracy of the present theory has been demonstrated for both bending and vibration analyses of simply supported FGM plates. The theory accounts for the stretching and shear deformation effects without requiring a shear correction factor. By dividing the transverse displacement into bending, shear and stretching components, the number of unknowns and
References (78)
- et al.
Buckling analysis of functionally graded plates subjected to uniaxial loading
Compos Struct
(1997) Functionally graded plates behave like homogeneous plates
Composites Part B
(2008)- et al.
Finite elements for functionally graded Reissner–Mindlin plates
Comput Meth Appl Mech Eng
(2004) - et al.
Second-order statistics of the elastic buckling of functionally graded rectangular plates
Compos Sci Technol
(2005) - et al.
Mechanical and thermal buckling analysis of functionally graded plates
Compos Struct
(2009) - et al.
Homotopy perturbation study of nonlinear vibration of Von Kármán rectangular plates
Comput Struct
(2012) - et al.
A simple finite element formulation of a higher-order theory for unsymmetrically laminated composite plates
Compos Struct
(1988) - et al.
Static response and free vibration analysis of FGM plates using higher order shear deformation theory
Appl Math Modell
(2010) Free vibration analysis of functionally graded conical, cylindrical shell and annular plate structures with a four-parameter power-law distribution
Comput Meth Appl Mech Eng
(2009)- et al.
A quasi-3D sinusoidal shear deformation theory for the static and free vibration analysis of functionally graded plates
Compos Part B: Eng
(2012)
A quasi-3D hyperbolic shear deformation theory for the static and free vibration analysis of functionally graded plates
Compos Struct
free vibration and buckling analysis of isotropic and sandwich functionally graded plates using a quasi-3D higher-order shear deformation theory and a meshless technique
Compos Part B: Eng
Radial basis function method applied to doubly-curved laminated composite shells and panels with a general higher-order equivalent single layer formulation
Compos Part B: Eng
General higher-order shear deformation theories for the free vibration analysis of completely doubly-curved laminated shells and panels
Compos Struct
Static analysis of completely doubly-curved laminated shells and panels using general higher-order shear deformation theories
Compos Struct
Free vibration analysis of sandwich plates with anisotropic face sheets in thermal environment by using the hierarchical trigonometric Ritz formulation
Compos Part B Eng
Static analysis of doubly-curved anisotropic shells and panels using CUF approach, differential geometry and differential quadrature method
Compos Struct
Generalized shear deformation theory for bending analysis of functionally graded plates
Appl Math Modell
Static and dynamic deformations of thick functionally graded elastic plates by using higher-order shear and normal deformable plate theory and meshless local Petrov–Galerkin method
Comp Part B: Eng
A unified formulation to assess theories of multilayered plates for various bending problems
Compos Struct
6 Mixed plate theories based on the generalized unified formulation. Part I: governing equations
Compos Struct
6 Mixed plate theories based on the generalized unified formulation. Part III: advanced mixed high order shear deformation theories
Compos Struct
6 Mixed plate theories based on the generalized unified formulation. Part V: results
Compos Struct
Stress analysis of functionally graded plates subjected to thermal and mechanical loadings
Compos Struct
Static response and free vibration analysis of FGM plates using higher order shear deformation theory
Appl Math Modell
Generalized hybrid quasi-3D shear deformation theory for the static analysis of advanced composite plates
Compos Struct
A novel higher-order shear deformation theory with stretching effect for functionally graded plates
Compos Part B: Eng
A simple quasi-3D sinusoidal shear deformation theory for functionally graded plates
Compos Struct
A new hyperbolic shear deformation theory for buckling and vibration of functionally graded sandwich plate
Int J Mech Sci
Static analysis of functionally graded sandwich plates using an efficient and simple refined theory
Chin J Aerosp
A refined trigonometric shear deformation theory for thermoelastic bending of functionally graded sandwich plates
Aerosp Sci Tech
Finite element formulation of a generalized higher order shear deformation theory for advanced composite plates
Compos Struct
A new approach to the application of Mori–Tanaka’s theory in composite materials
Mech Mat
Average stress in matrix and average elastic energy of materials with misfitting inclusions
Acta Metall
Stochastic analysis of compositionally graded plates with system randomness under static loading
Int J Mech Sci
Random vibration of the functionally graded laminates in thermal environments
Comp Meth Appl Mech Eng
Study on the free vibration of thick functionally graded rectangular plates according to a new exact closed-form procedure
Compos Struct
A new exact analytical approach for free vibration of Reissner–Mindlin functionally graded rectangular plates
Int J Mech Sci
A simple higher-order shear deformation theory for bending and free vibration analysis of functionally graded plates
Compos Struct
Cited by (583)
Vibration analysis of tri-directionally coated plate via thickness-stretching and microstructure-dependent modeling
2024, Mechanics Research CommunicationsThermoelastic free vibration analysis of functionally graded conical shell based on trigonometric higher-order shear deformation theory
2023, International Journal of Solids and StructuresNonlinear aerodynamic analysis of functional graded plates using an HSDT-based isogeometric approach
2023, Thin-Walled Structures