An assessment of a non-polynomial based higher order shear and normal deformation theory for vibration response of gradient plates with initial geometric imperfections
Introduction
Due to extensive research efforts during the last three decades, great varieties of advanced technology materials are available today. Functionally graded materials (FGMs) are considered as one of them. The concept of functionally graded material was first suggested by Bever and Duwez [1] in 1972. FGMs are high performance, microscopically heterogeneous materials having engineered gradients of composition with specific properties in the preferred orientation [2]. Gradual changes of the material properties are achieved by controlling the volume fractions of constituent materials which make it able to be tailored as per the working environment and requirement. In the recent years, FGMs have been adequately explored in a diversity of areas, including optoelectronics, chemical science, biomedicine, nuclear science, aeronautical, mechanical and civil industries [3].
The analysis of composite structures is one of the most promising research areas since last three decades. Accurate prediction of structural and dynamic responses is prerequisite to design various components of aerospace, mechanical as well as civil assemblies. Numerous structural kinematics have been developed by the researchers for accurate prediction of the response of plates and shells. In this context, the earliest plate theory suggested for the plate was Classical plate theory (CPT) in 1888 by Love [3]. This theory can be applied to get the response of thin plate up to a fine degree of accuracy. In CPT, transverse shear effects are neglected, and it leads to delivers irrational results for moderately thick plates. This theory has been implemented by several authors [4], [5], [6], [7], [8], [9] and found that it under-predicts the deflections and over predicts the frequencies as well as buckling loads of moderately thick plate. To avoid this problem, first-order shear deformation theory (FSDT) was proposed by Reissner [10] and Mindlin [11] in which the transverse shear deformation is considered. In FSDT, a shear correction factor is required to accommodate the traction free boundary conditions at the top and bottom surface of the plate. This shear correction factor depends on various parameters like geometric configuration, loading conditions, etc. To circumvent the calculation of shear correction factor, higher-order shear deformation theories (HSDTs) have been developed. These theories are computationally more difficult but provide better kinematics to predict the structural response of plate and shell. The HSDTs can be categorized as polynomial based HSDTs [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28] in which Taylor's series expansion is used to formulate the higher order terms of displacement field, and non-polynomial based HSDTs in which shear-strain function is employed to accommodate the transverse shear deformation [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43]. A review of various shear strain function which has been used for the development of non-polynomial theories is presented in Appendix A.
Chattopadhyay and Seeley [23] proposed a new polynomial refined higher order laminate theory with five unknowns to investigate the static response of advanced composite materials. Thai and Choi [24] developed a third-order shear deformation theory which comprises shear and bending components in the transverse displacement to analyse the static and stability response of FGM plate. Cho and Oh [27] used Heaviside unit step function to develop a zig-zag theory with seven independent unknowns for smart materials. Mantari and Soares [37] employed Navier solution for the assessment of newly developed trigonometric plate theory with five unknowns. Nguyen [38] proposed inverse hyperbolic function in the development of displacement field to investigate vibration and bending response of FGM plate using Navier solution. Belabed et al. [39] used the combination of hyperbolic and exponential function to establish a new higher order shear deformation theory for static and vibration response of FGM plate. Akavci and Tanrikulu [43] presented a new quasi two and three-dimensional shear deformation theories using hyperbolic shape function to examine the vibration and static characteristics of FGM plate.
Chen and Kitipornchai [44] used CPT and FSDT kinematics to obtain the stability and dynamic characteristics of FGM plates via analogy with membrane vibrations. Reddy and Cheng [45] used an asymptotic formulation and transfer matrix method to present three-dimensional vibration analyses of FGM rectangular plates. Ferreira et al. [46] showed meshless methods to investigate natural frequencies of functionally graded plates. Roque et al. [47] utilized the multiquadric radial basis function method and HSDT for free vibration analysis of FGM plate with various boundary conditions. Thai and Vo [48] developed a new sinusoidal shear deformation theory for bending, buckling and vibration of functionally graded plates. Neves et al. [49] used meshless technique and higher order shear deformation theory to investigate static, free vibration and buckling analyses of sandwich functionally graded plates. Talha and Singh [16] modified the displacement field to obtain static and vibration characteristics of FGM plate. Zhu and Liew [50] presented a meshless method to investigate the vibration analysis of FGM plate under thermal loading. The author assumed temperature-dependent material properties which vary as a continuous function along the transverse direction of the plate. Woo et al. [51] presented the influence of material properties, boundary conditions and loading conditions on the dynamic characteristics of the plates. Vel and Batra [52] proposed a three-dimensional exact solution for free and forced vibrations of FGM plates by using suitable displacement functions to reduce equations governing steady state vibration of the plate.
In the aforesaid studies, the only perfect plate is used to investigate the vibration characteristics. During the manufacturing process of FGM plate, it is quite possible that the plate can possess an unintentional geometric imperfection (the difference between the required shape and actual shape). This unavoidable imperfection influences the vibration responses of the plates [53]. A limited literature is available in which such types of problems have been dealt with. Hui and Leissa [54] investigated the effects of geometric imperfections on the vibration response of flat plate with sinusoidal wave type geometric imperfection in both the in-plane directions. The author continued the research on the geometric imperfection sensitivity on the static and dynamic response of plates and panels [55], [56]. Liu and Lam [57] investigated the semi-analytical finite strip analysis of laminated plates based on general initial imperfection. Yang and Huang [53] presented the nonlinear vibration analysis of imperfect FGM plate using HSDT with Von-Karman type nonlinearity. Fung and Chen [58] investigated the nonlinear vibration of initially stressed FGM plate with geometric imperfection using perturbation technique.
It has been observed from the comprehensive review that in most of the two-dimensional plate theories (i.e. CPT, FSDT, and HSDT) thickness stretching effect is neglected due to the constant transverse displacement through the thickness. This supposition is appropriate for thin or moderately thick FGM plates, but it leads to erroneous results for thick FGM plates [59]. Carrera et al. [59] investigated the effect of thickness stretching effect in FGM plates and suggested that increase in the order of expansion for in-plane displacements is pointless if the thickness stretching effect is ignored in the plate theories.
By retaining the above viewpoint in mind, the aim of this study is to develop a non-polynomial based structural kinematics for graded plates which incorporate thickness stretching effects and provide analogous results for thin and thick plates. The proposed theory is based on inverse hyperbolic and hyperbolic function as shear strain function in the in-plane and transverse displacement, respectively. It also satisfies the traction free boundary conditions at the top and bottom surfaces of the plate without using any shear correction factor. Two micromechanics models (Mori-Tanaka and Voigt) have been used to determine the effective properties of the plate. Power law and exponential law are used to incorporate the gradation in the effective material properties of the FGM plate. Convergence and validation studies have been performed to ascertain the efficacy of the proposed theory. The effect of the plate parameters such as imperfection size, aspect ratios, side to thickness ratios, and volume fraction index on the natural frequencies of FGM plates is presented for various combinations of boundary conditions.
The paper is organized as follows. Section 2 presents the theoretical formulation of proposed structural kinematics. Section 3 addresses the finite element formulation, energy calculation, and governing equation. Section 4 demonstrates the result and discussion section of the analysis. Influence of various parameters on vibration characteristics is shown in this section. Finally, Section 5 presents the conclusions of the present study.
Section snippets
Homogenization and material properties of the FGMs plate
The Voigt model and Mori-Tanaka are used in the present study to determine the effective material properties of FGM plate.
Solution methodology
In the present finite element formulation, a C0-continuous nine nodded isoparametric finite element with eight degrees of freedom per node (Fig. 4) is employed to discretize the plate geometry. Subsequently, the generalized displacement vector and element geometry of the model at any point can be expressed in terms of shape functions as follows,where,{Ni} and are{R}{R}{R} the shape function and displacement vector of ith node respectively. ‘nn’ is
Convergence and comparison studies
Numerous examples are discussed to establish the accuracy and applicability of the proposed theory in order to predict the realistic vibration response of the FGM plate. Convergence and comparative studies have been carried out with the available existing results in the literature. The description of material properties used in the analysis is provided in Table 1. The effective material properties of FGM plates are assumed to be graded continuously through the thickness according to a simple
Conclusions
In the present study, imperfection sensitivity in the linear vibration of FGM plate is examined using a new non-polynomial based higher order shear and normal deformation theory. The outline of the present study is summarized as follows:
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The present theory accounts for nonlinear in-plane and transverse displacement through the plate thickness.
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The present theory consists only four unknowns in the displacement field and also accounts for thickness stretching effect.
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The theory is variationally
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