Non-linear analysis of functionally graded fiber reinforced composite laminated plates, Part II: Numerical results

https://doi.org/10.1016/j.ijnonlinmec.2012.03.003Get rights and content

Abstract

In this Part, the extensive parametric studies performed are reported and numerical results are presented for the non-linear vibration, non-linear bending and compressive postbuckling of uniformly distributed and functionally graded fiber reinforced unsymmetric cross-ply and/or antisymmetric angle-ply laminated plates resting on Pasternak elastic foundations under different hygrothermal environmental conditions. The numerical results show that the functionally graded fiber reinforcement has a significant effect on the postbuckling response and load-bending moment curves of plate bending, whereas this effect is less pronounced on the load-deflection curves of plate bending and the linear and non-linear frequencies of the same plate.

Highlights

► FG fiber reinforcement has a significant effect on the postbuckling response of the plate. ► FG fiber reinforcement has a significant effect on the bending moment of the plate. ► FG fiber reinforcement has a small effect on the bending deflection of the plate. ► FG fiber reinforcement has a small effect on the linear and non-linear frequencies of the plate.

Introduction

The solution methodology is described with sufficient detail in Part I. Results are presented herein for non-linear free vibration, non-linear bending and postbuckling of (0/90)S symmetric cross-ply, (0/90)2T unsymmetric cross-ply and (45/−45)2T antisymmetric angle-ply laminated plates resting on an elastic foundation in hygrothermal environments. The plate geometric parameters a/b=1, b/h=10, the thickness of each ply is identical and the total thickness of the plate h=5 mm. Four types of functionally graded fiber reinforced composite (FG-FRC) laminated plates are configurated. For Type V, the fiber volume fractions are assumed to have graded distribution [0.75/0.65/0.55/0.45] for four plies, referred to as FG-V. For Type Λ, the distribution of fiber reinforcements is inversed, i.e. [0.45/0.55/0.65/0.75], referred to as FG-Λ. For Type X1, a mid-plane symmetric graded distribution of fiber reinforcements is achieved, i.e. [0.75/0.45/0.45/0.75], and for type X2 the fiber volume fractions are assumed to have [0.45/0.75/0.75/0.45], referred to as FG-X1 and FG-X2, respectively. A uniformly distributed fiber reinforced composite (UD-FRC) laminated plate with the same thickness is also considered as a comparator for which the fiber volume fraction of each ply is identical and Vf=0.6. In such a way, the two cases of UD- and FG-FRC laminated plates will have the same value of total fraction of fiber.

For all cases discussed below, graphite/epoxy composites are selected. Unlike in [1], [2], [3], [4], the material properties of fibers are assumed to be anisotropic and are taken to be [5] E11f=233.05GPa, E22f=23.1GPa, G12f=8.96GPa, νf=0.2, α11f=0.54×106/°C, α22f=10.08×106/°C, ρf=1750 kg/m3. The material properties of matrix are assumed to be cfm=0, νm=0.34, αm=45.0×10−6/°C, ρm=1200 kg/m3, βm=2.68×10−3/wt percent H2O, and Em=(3.51−0.003T−0.142C) GPa, in which T=T0T and T0=25 C (room temperature), and C=C0C and C0=0 wt percent H2O.

Two foundation models are considered. The stiffnesses are (k1, k2)=(100, 10) for the Pasternak elastic foundation, (k1, k2)=(100, 0) for the Winkler elastic foundation and (k1, k2)=(0, 0) for the plate without any elastic foundation. The in-plane boundary condition is assumed to be immovable (case 2) except for the Table 7 in Section 2 and Fig. 7 in Section 3, whereas in Section 4 the in-plane boundary condition is assumed to be movable (case 1).

Section snippets

Non-linear vibration of FG-FRC laminated plates

Before generating extensive results, a few check cases are considered in order to test the derived solutions.

As first example, the first four dimensionless natural frequencies of a (0/90)S symmetric cross-ply laminated plate at C=0.1% and T=325 K are calculated and compared in Table 1 with Ritz method results of Whitney and Ashton [6], finite element method (FEM) results of Ram and Sinha [7] and Parhi et al. [8], and perturbation solutions of Huang et al. [4]. The geometric parameters and

Non-linear bending of FG-FRC laminated plates

As part of the validation of the present method, the load-deflection curves of a (0/90)T unsymmetric cross-ply laminated square plate with b/h=10 subjected to a transverse uniform load are compared in Fig. 1 with author's previous results [12], and FEM results of Singh et al. [13]. The material properties adopted are: E11/E22=25, G12/E22=G13/E22=0.5, G23/E22=0.2 and ν12=0.25. Excellent agreement is observed between these two results.

Fig. 2 compares the load-deflection curves of a (45/−45)2T

Postbuckling of FG-FRC laminated plates

To validate the present approach, the buckling loads (Ncra2/E22h3) for symmetric cross-ply laminated plates as function of modulus ratio are calculated and compared in Table 8 with the 3-D linear elasticity solutions of Noor [15] by using a finite difference scheme, different kinds of HSDPT solutions of Putcha and Reddy [16], Khdeir [17] and Khdeir and Librescu [18], and the wavelet collocation method results of Ferreira et al. [19] based on the first order shear deformation plate theory

Concluding remarks

The large amplitude vibration, the non-linear bending and the compressive postbuckling analyses for a FG-FRC laminated plate have been presented on the basis of a micro-to-macro analytical model and multi-scale approach. Numerical calculations have been made for UD and FG unsymmetric cross-ply and antisymmetric angle-ply laminated plates resting on an elastic foundation in different set of hygrothermal environmental conditions. Perturbation solutions are obtained in an explicit form that can be

Acknowledgments

This work is supported in part by the National Basic Research Program of China under Grant 2010CB631005. The authors are grateful for this financial supports.

References (21)

There are more references available in the full text version of this article.

Cited by (16)

  • On the free vibration behavior of nanocomposite laminated plates contained piece-wise functionally graded graphene-reinforced composite plies

    2022, Engineering Structures
    Citation Excerpt :

    However, recently, an increasing number of researchers have investigated the mechanical behavior of different nanocomposite structures reinforced with graphene, in which the so-called extended Halpin-Tsai model proposed by Lin et al. [52] is employed to estimate the effective material properties of the mixture. In this approach, it should be noted that the studies cited used a piece-wise technique to make the graphene nanofiller distributed in a functionally graded manner across the thickness of the nanocomposite structures; this technique was firstly proposed by Shen and his co-authors [53,54] when they carried out a nonlinear analysis of FG fiber-reinforced composite laminated plates. Among the research that used the extended Halpin-Tsai model and the piecewise technique, we mention, Fan et al. [55] who investigated the non-linear forced vibration behavior of piece-wise FG-GRC plates resting on visco-Pasternak foundations.

  • Free vibration analysis of functionally graded porous plate using 3-D degenerated shell element

    2021, Composites Part C: Open Access
    Citation Excerpt :

    Matsunaga [13] adopted two-dimensional (2-D) higher order deformation theory to investigate the buckling and free vibration analysis of FG plates. Nonlinear vibration, bending and postbuckling analyses of FG fiber reinforced composite laminated plates resting on Pasternak and Winkler elastic foundation were studied Shen and Zhang [14] [15] under hygro-thermal environment. Jha et al. [16] reviewed the recent works on the static, vibration and buckling analyses of FG plates.

  • Buckling and postbuckling of functionally graded graphene-reinforced composite laminated plates in thermal environments

    2017, Composites Part B: Engineering
    Citation Excerpt :

    The equivalent isotropic Young's modulus of the nanocomposite is obtained by using the modified Halpin–Tsai model, and the material properties are assumed to be independent of temperature. The concept of functionally graded materials can also be utilized for the fiber reinforced composite laminates by non-homogeneous distribution of fiber reinforcements into the matrix along the thickness direction with a piece-wise type [37,38]. The present work focuses attention on the postbuckling analysis of GRC laminated plates subject to uniaxial compression in thermal environments through careful selection of material properties of graphene sheets and a novel reinforcing scheme.

  • Nonlinear bending of functionally graded graphene-reinforced composite laminated plates resting on elastic foundations in thermal environments

    2017, Composite Structures
    Citation Excerpt :

    The equivalent isotropic Young’s modulus of the nanocomposite is obtained by using the modified Halpin–Tsai model, and the material properties are assumed to be independent of temperature. The concept of functionally graded material can also be utilized for the fiber reinforced composite laminates by non-homogeneous distribution of fiber reinforcements into the matrix along the thickness direction with a piece-wise type [33,34]. In the present work, we focus our attention on the nonlinear bending of nanocomposite plates reinforced by graphene sheets with low graphene volume fractions.

  • Initial imperfection effects on postbuckling response of laminated plates under end-shortening strain using Chebyshev techniques

    2016, Thin-Walled Structures
    Citation Excerpt :

    Ovesy et al. [17,18] used both spline and semi-analytical finite strip methods for predicting the postbuckling response of rectangular composite laminated plates with initial imperfections, when subjected to progressive end shortening. In other researches, Shen et al. [19,20] investigated the postbuckling analyses of composite and functionally graded plates under thermal and mechanical loads. He also [21,22] analysed simply supported plates with initial imperfection in thermal environment.

View all citing articles on Scopus
View full text