Elsevier

Composite Structures

Volume 143, 20 May 2016, Pages 324-335
Composite Structures

Laminated composite plates subject to thermal load using trigonometrical theory based on Carrera Unified Formulation

https://doi.org/10.1016/j.compstruct.2016.02.020Get rights and content

Abstract

In the present work, an analytical solution for the thermoelastic static problem of simply supported laminated composite plates is presented. The present mathematical model uses a unified new trigonometric displacement field expansion under Carrera Unified Formulation (CUF). The equivalent single layer (ESL) governing equations are written using CUF notation for static thermal stress analysis employing the Principle of Virtual Displacement (PVD). The highly coupled partial differential equations are solved using Navier solution method. Normalized and non-normalized unified trigonometric shear strain shape functions are introduced for the first time. Shear deformation results are compared with the classical polynomial ones, which is usually adopted in several refined plate theories under CUF. Linear temperature profile and non-linear temperature profile obtained by solving heat conduction problem are taken into account. Good agreements with 3D solution for several order of expansion are reached, but instabilities are shown for some particular order of expansion even when an exact through the thickness integration technique was adopted. Similar values are presented between polynomial and non-polynomial displacement fields. However, non-polynomial functions can be optimized by changing the arguments of such functions in order to improve the results. Future studies are necessary in this direction.

Introduction

Nowadays, laminated composite materials are used in several construction fields, including aerospace, marine, civil, biomedical and other areas. Because of their attractive mechanical properties (high specific stiffness, excellent fatigue strength and resistance to corrosion) the demand of these kinds of materials in the industries is increasing. However, it is not simple to study the behavior of the laminated structures. Therefore, various theories and different variational statements were developed in order to study the mechanical behavior of the laminated composite materials under different loading conditions. In what follows some theoretical model are discussed.

In order to study composite materials, the classical plate theory (CPT) for metallic structures, based on the Kirchhoff’s assumptions, was extended to laminated plates. Nonetheless, only give acceptable results for thin plates, because the shear deformation effect is ignored. To overcome this problem the first order shear deformation theory (FSDT) based on Reissner [1] and Mindlin [2] was introduced. A constant transverse shear strain component is taken into account with their respectively shear correction factor. This value depends on the material coefficients, geometry, boundary conditions and loading conditions, which is difficult to calculate. Therefore, higher-order shear deformation theories (HSDTs) were introduced to avoid the shear deformation effect. The HSDTs can be developed using polynomial shape functions, for example Reddy [3], [4], Levinson [5] and Librescu [6]. Further, non-polynomial shape functions are developed by Touratier [7], Soldatos [8], Karama [9], Mantari and Guedes [10], Zenkour [11], [12], [13], [14] and Mantari et al. [15], [16], [17], [18].

The thermoelastic problem for laminated cross-ply shells were studied by Khdeir et al. [19] using a third-order HSDT which was compared with the CPT and FSDT. In the HSDT presented by Zhen and Wanji [20] the shear stress continuity conditions of composite laminated and sandwich plates under thermo-mechanical load are satisfied. Zenkour and Alghamdi [21], [22], [23] have presented the thermoelastic bending analysis of functionally graded sandwich plates. The thermoelastic bending response is presented for a simply-supported cross-ply composite laminated plate subjected to a thermal field by Zenkour and Maturi [24].

Carrera Unified Formulation (CUF) was developed for composite laminated plates and shells [25], [26], [27] using originally Taylor’s expansions of N-order. A sinusoidal shear deformation theory (SSDT) within the CUF framework was developed by Ferreira et al. [28] for static and free vibration analysis of laminated shells. In addition, a static analysis and free vibration analysis for several theories based on polynomial, trigonometric, hyperbolic, exponential and zig–zag function were developed by Carrera et al. [29], [30] for laminated beam. The effect of the normal strain effect was analyzed for static thermostatic bending problem for multilayer plates by Carrera [31]. ESL and Layerwise (LW) theories were presented based on the Principle of the Virtual Displacement (PVD) and Reissner’s Mixed Variational Theorem (RMVT) in Carrera et al. [32]. Free vibration analysis for laminated and sandwich plates using the Hierarchical Trigonometric Ritz Formulation (HTRF) were developed by Fazzolari and Carrera [33]. Furthermore, a coupled thermo-mechanical analysis for multilayer plates was studied by Brischetto and Carrera [34]. Thermal buckling analysis based on HTRF for multilayer plates was carried out by Fazzolari et al. [35], [36] and extended to study sandwich plates made of FGM by Fazzolari et al. [37].

This paper proposes a modified non-polynomial displacement field under CUF to study the thermoelastic analysis of simply-supported laminated plates under bisinusoidal thermal load. Normalized shear strain function (sin1) based on trigonometric theory are developed for several order of expansion (N = 4, 5, 6, 7, 8, 9, 10, 15, 30) and are compared with trigonometric functions in non-normalized form (sin2) and the polynomial kinematic (pol). A linear temperature field through the thickness is taken into account in this work. In addition, the temperature profile obtained by solving heat conduction problem was carried out in order to get proper results. The governing equations for the static analysis are obtained through PVD, and solved using Navier solution method.

The paper is organized as following: The analytical modeling is presented in Section 2. The geometry of the composite laminated plate, kinematic, the principle of virtual works and the governing equations under CUF platform is presented in this section. The analytical solution methodology is given in Section 3. The results are presented and discussed in Section 4. Finally, conclusions and the references list are given.

Section snippets

Plate geometry

This paper considers a simply-support cross-ply laminated plate with a uniform thickness h, length a, and width b on the plane xy at z=0 in a Cartesian coordinate system as shown in Fig. 1. The relationship between the plate coordinates x, y, z and the material coordinates 1, 2, 3 are represented in Fig. 2, where θ is the fiber orientation.

Model kinematics

CUF states that displacement field for plates, u(x,y,z), is modeled in the following manner:u(x,y,z)=Fs(z)us(x,y),s=0,1,,N,δu(x,y,z)=Fτ(z)δuτ(x,y),τ=0,1,,N,

Analytical solution

Navier type closed form solution are possible for simply-supported cross-ply laminated plates such displacement variables and the transverse distributed load can be expressed in the following Fourier seriesuxskuyskuzskTθk=m,nUxskcos(λxk)sin(βyk)Uysksin(λxk)cos(βyk)Uzsksin(λxk)sin(βyk)Tθksin(λxk)sin(βyk),0xa;0yb,where λ=mπ/a, β=nπ/b and Uxsk, Uysk, Uzsk and Tθk are amplitudes, m and n are the number of waves and a and b are the dimensions of the plate.

Therefore, the governing equations

Numerical results and discussions

The thermoelastic bending analysis of simply-supported laminated plate is presented in what follows. Trigonometric theory within the CUF framework is developed and compared with the unified polynomial theory for several orders of expansion. Three layers symmetric cross-ply plate (0°/90°/0°) under a bisinusoidal thermal load according to Eq. (9). Linear temperature profile with T=T0(z/2h)sin(λx)sin(βy) and nonlinear temperature profile, T=T1eszsin(λx)sin(βy), obtained by solving heat conduction

Conclusions

This paper presents an analytical solution for the thermoelastic bending analysis of simply-supported laminated plates using Carrera’s Unified Formulation with trigonometric shear strain functions. A classical PVD and ESL model without interlaminar continuity are used in the present work.

These new trigonometric displacement fields have the capability to reproduce the same results of the classical Taylor expansion for thin and thick laminated cross-layer plates. Furthermore, good results

Acknowledgement

The authors want to dedicate this work to Professor J.N. Reddy for his 70 anniversary.

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