Analysis of laminated composite plates using wavelet finite element method and higher-order plate theory
Introduction
Laminated composites are in the category of crucial parts in the field of new materials. In recent years, laminated composites have been applied extensively in the areas of aircraft, aerospace, defense and transportation industries for their excellent properties as high strength, high stiffness-to-weight ratio, high damping and outstanding design ability. Different from conventional metal materials, mechanical behaviors of laminated composites depend strongly on the matrix material and reinforced material. The complex structure of laminated composites may result in matrix cracking, delamination, fiber breakage and interface debonding [1] which can severely reduce the system reliability and induce serious disasters. Therefore, knowledge of deflections, stresses and frequencies in laminated composite structures are of great practical importance and significance for optimal design of laminated composites in industrial applications.
Various theories related to the effects of transverse shear deformation and rotary inertia have been proposed for laminated composite plates. The simplest classical Kirchhoff plate theory, named classical plate theory (CPT), has been employed to deal with the static and free vibration problems of laminated composite plates [2], [3], [4]. Unfortunately, CPT neglects the transverse shear deformation and rotary inertia, and these studies merely give acceptable results for thin laminated composite plates. Due to low transverse shear stiffness, transverse shear deformation becomes remarkable and CPT underestimates deflections and overestimates natural frequencies for moderately thick laminated composite plates.
Then, the first-order shear deformation theory (FSDT) is proposed for moderately thick plate to overcome the limitation of CPT. This theory takes into account the effect of transverse deformation and improves the calculation accuracy of moderately thick plate [5], [6]. A number of remarkable investigations based on FSDT for laminated composite plates have been investigated by Civalek using highly efficient discrete singular convolution method (DSC) which reduces partial differential equations into a standard eigenvalue problem [7], [8], [9]. FSDT has been proved to be effective and accurate for moderately thick laminated composite plates [10], [11], [12]. However, this theory does not yield to the stress-free boundary conditions on the surface of plates and a shear correction factor is required to correct variation of transverse shear stress and shear strain through thickness. Although shear correction factor is essential, it is difficult to accurately determine shear correction factor for laminated composite plate, as it depends strongly on material properties, lamination schemes, geometric parameters and boundary conditions [13].
To avoid using shear correction factor, different higher-order shear deformation theories (HSDTs) including non-linear distributions of shear stress through thickness have been widely preferred for thick plates [14]. The well-known Reddy’s HSDT [15] provides a parabolic distribution of shear stress through thickness and satisfies the zero transverse shear stress condition on the top and bottom surfaces of plates. This theory has been further applied to free vibration of laminated composite and sandwich plates by Khdeir [16], [17], [18]. After then, Ferreira and his co-workers have carried out a lot of remarkable investigations on laminated composite plates using Reddy’s HSDT [19], [20]. The radial basis function method is used for interpolation of geographical scattered data in their investigations. Besides Reddy’s HSDT, there are some other higher-order theories developed for laminated composite plates. Matsunaga has presented a global HSDT for the free vibration of cross-ply [21] and angle-ply [22] laminated composite plates considering the effects of shear deformations, length-to-thickness ratios and rotary inertia. A set of excellent solutions based on a higher-order refined theory for laminated plates has been given by Kant and Swaminathan [23], [24], [25]. The proposed theory takes into account 3D Hooke’s law and incorporates the effect of transverse normal strain as well as transverse shear deformations. The trigonometric shear deformation theory has been successfully applied to composite plates and shells by Ferreira [26], [27]. Kim has developed an accurate and simple method using two variable refined plate theory (RPT) for bending [28], free vibration [29] and buckling [30] analysis of laminated composite plates. Recently, Singh has developed zigzag theory [31], [32] and a new inverse hyperbolic shear deformation theory [33] for free vibration analysis of laminated composite and sandwich plates. As for higher terms, these theories can give more accurate solutions than FSDT for both thick and thin laminated composite plates.
3D elasticity theory always serves as benchmark solutions for comparisons [34], [35], [36]. Though achievements based on this theory have been reported for laminated composite plates, the accurate solutions are often mathematical complicated and computationally expensive for complicated stacking sequences or larger numbers of plies. Therefore, high efficient numerical methods, such as FEM [3], [37], p-Ritz method [38], [39], mesh-free method [19], [40], etc, have been developed for static and free vibration analysis of laminated composite plates. However, these numerical methods still have some drawbacks and limitations in application. For complicated problems with high gradient or nonlinearity in engineering fields, traditional FEM has disadvantages of low efficiency, insufficient accuracy and slow convergence. In p-Ritz method, it is difficult to choose appropriate trial functions. This greatly influences the accuracy and convergence of solutions for complicated problems. Although the mesh-free method improves the continuity of interpolation, the complex approximation space greatly increases the computational cost.
Wavelet finite element method (WFEM) is a powerful numerical computation method in engineering fields. This method can be regard as a combination of wavelet and traditional FEM. It employs a series of scaling functions as approximating functions instead of polynomials interpolation functions used in traditional FEM. Attributing to the desirable advantages of wavelet for multi-resolution property and various basis functions, WFEM adopting different wavelet basis functions has been successfully applied to structural analysis. Chen [41] has successfully constructed the Daubechies wavelet-based method for isotropic plate analysis. Xiang [42] proposed a multi-scale wavelet-based method for shaft analysis by using Hermite cubic spline wavelet. However, lacking explicit expressions or sufficient regularity greatly limits applications of these wavelet basis functions. As scaling functions of spline wavelet, the B-spline wavelet on the interval (BSWI) basis inherits superiorities of compact support, smoothness in addition to the multi-resolution analysis. What’s more, the explicit expressions of BSWI make it convenient and easy to obtain coefficient integrations and differentiations. Based on the excellent characteristics of BSWI basis, WFEM has been successfully applied to structural analysis [43], [44], damage identification [45] and wave propagation [46] in recent years. In previous studies, WFEM mainly focuses on isotropic material structure and investigations for laminated composite plates using wavelet-based BSWI finite element method are not available in literatures. Therefore, WFEM adopting BSWI basis is extended to laminated composite plates in this paper.
The main purpose of this paper is to employ WFEM to investigate the static and free vibration problems of laminated composite plates. The two-dimensional BSWI scaling functions are, instead of polynomials interpolation functions used in FEM, directly used to approximate the unknown field functions. Based on HSDT, the wavelet-based BSWI element is constructed according to the principle of minimum total potential energy. Numerical examples are given to illustrate the effectiveness, accuracy and stability for laminated composite plates.
Section snippets
B-spline wavelet on the interval [0, 1]
BSWI scaling functions are employed as approximating functions of WFEM. Here a brief introduction of BSWI is given. The mth order B-spline in knot sequence can be easily constructed by (m-1)th order piecewise polynomial and their derivatives up to (m-2)th order are continuous. For any one-dimensional functions on the interval can be transformed to the standard interval [0, 1] by a simple linear mapping , it only needs to construct B-spline functions on the interval [0,
Theory and formulation of wavelet-based laminated plate element
An n-layered laminated composite rectangular plate of length a, width b and thickness h is considered and shown in Fig. 2. with Cartesian coordinate system at the center of the mid-plane. The coordinate system (1, 2) denotes the principal material coordinate system of each lamina. Besides, each lamina is assumed to possess the same thickness, orthotropic material properties.
Upon the effects of transverse shear deformation and rotary inertia for different plate theories (such as CPT,
Numerical examples
The wavelet-based BSWI element for static and free vibration problems of laminated composite plates via higher-order plate theory has been formulated in Section 3. Here several numerical examples are carried out to illustrate and demonstrate the efficiency and accuracy of proposed wavelet-based method through comparison with exact results or referential results available in literatures. In addition, the similar wavelet-based BSWI element based on FSDT has also been developed to compare with
Conclusions
In this present paper, a simple but accurate wavelet-based BSWI finite element method has been developed for static and free vibration analysis of laminated composite plates. Equations of motion for static and free vibration are derived by principle of minimum total potential energy based on HSDT. The wavelet-based BSWI element of laminated composite plates is constructed by using two-dimensional BSWI scaling functions as approximating functions. Thanks to the excellent characteristics of BSWI
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Nos. 51225501 and 51405369), the China Postdoctoral Science Foundation (No. 2014M560766), the Fundamental Research Funds for the Central Universities (No. xjj2014107), and the National Key Basic Research Program of China (No. 2015CB057400).
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