Elsevier

European Journal of Mechanics - A/Solids

Volume 36, November–December 2012, Pages 163-172
European Journal of Mechanics - A/Solids

Static response of functionally graded plates and doubly-curved shells based on a higher order shear deformation theory

https://doi.org/10.1016/j.euromechsol.2012.03.002Get rights and content

Abstract

An analytical solution to the static analysis of functionally graded plates and doubly-curved shells, modeled using a higher order shear deformation theory (HSDT), is presented. A solution methodology, based on boundary-discontinuous generalized double Fourier series approach is used to solve a system of five highly coupled linear partial differential equations, generated by the higher order-based laminated shell analysis with the fully simple supported boundary condition prescribed at all edges. The mechanical properties of the panels are assumed to vary in the thickness direction according to a power-law distribution in terms of the volume fractions of the constituents. In order to verify the present solution, a comparison of the present results is made with the finite element solutions to verify the present solution with the homogeneous (isotropic) and functionally graded plates. Important numerical results are presented to show the effect of inhomogeneities, thickness and membrane effects, as well as their interactions.

Highlights

► Higher order-based laminated shell analysis is conducted. ► Fully simple supported boundary conditions are prescribed at all the edges. ► The solution is based on boundary-discontinuous generalized double Fourier series. ► The mechanical properties are assumed to vary in the thickness direction. ► Results show the effect of inhomogeneities, thickness and membrane effects.

Introduction

Advanced composite materials have been extensively used in high performance industrial applications since many years. Among composite materials, functionally graded materials (FGMs) obtained considerable attention for their superior features such as resistance to high temperature environments (better thermal resistance), high wear resistance, among others.

Classical composites suffer from discontinuity of material properties at the interface of layers and constituents of composite. Therefore the stress fields in these regions create interface problems and thermal stress concentrations particularly under high temperature environments. These problems can be reduced by gradually changing the volume fraction of constituent materials and tailor the material for desired application. For example, the composition of ceramic and metal can be varied from a ceramic rich surface to a metal rich surface and hence the thermal resistance of the material is increased due to low thermal conductivity of the ceramic and low toughness problem of ceramic is eliminated by using the metal.

Several researchers investigated the static and dynamic behavior of functionally graded materials (FGMs). Reddy (2000) presented Navier's solutions, and finite element models including geometric non-linearity based on the third-order shear deformation theory for the analysis of through-thickness FGM plates. Cheng and Batra (2000) derived the field equations for a functionally graded plate by utilizing the first order shear deformation theory or the third-order shear deformation theory and simplified them for a simply supported polygonal plate. An exact relationship was established between the deflection of the functionally graded plate and that of an equivalent homogeneous Kirchhoff plate. A three-dimensional elasticity solution was presented by Kashtalyan (2004) for a functionally graded simply supported plate under transversely distributed load. Exponential variation of the Young's modulus through the thickness was assumed and the Poisson's ratio of the plate was taken constant. Ferreira et al. (2005) used a meshless method for the static analysis of a simply supported functionally graded plate by a third-order shear deformation theory. The effective material properties were calculated by using the rule of mixtures and the Mori-Tanaka scheme. Elishakoff, 2005 developed a three-dimensional elasticity solution using the Ritz method for the static response of a clamped rectangular functionally graded plate. Material properties were calculated using a power-law distribution. The static response of functionally graded plates was presented by Zenkour (2006) using the generalized shear deformation theory developed by the author. Effective material properties were calculated by assuming a power-law. A discrete layer model in conjunction with the Ritz method developed by Ramirez et al. (2006) for the approximate solution of a static analysis for the two types of functionally graded plates. In one type, an exponential variation of the material properties through the thickness was assumed, while in the second type, the variation was the function of the fiber orientation. Sladek et al. (2007) presented the static and dynamic analysis of functionally graded plates by the meshless local Petrov–Galerkin method. The Reissner–Mindlin plate bending theory was utilized to describe the plate deformation. Numerical results were presented for simply supported and clamped plates. Bo et al. (2008) presented the elasticity solutions for the static analysis of functionally graded plates for different boundary conditions. Stress analysis due to thermal and mechanical loads was given by Matsunaga (2009) by using a two-dimensional higher order theory. A power-law distribution for the volume fractions of constituents was assumed for the calculation of modulus of elasticity. Navier solution of a simply supported functionally graded plate was provided for stress and deflections. Khabbaz et al. (2009) provided a nonlinear solution of FGM plates using the first and third-order shear deformation theories. For the through the thickness variation of properties, a power-law was considered.

Recently, the extended Kantorovich method successfully applied for the static analysis of fully clamped functionally graded plates and doubly-curved panels by Aghdam et al. (2010). Talha and Singh (2010) investigated the free vibration and static analysis of functionally graded plates using the finite element method by employing a higher order shear deformation theory. Wu and Li (2010) used a third-order theory for the static analysis of simply supported, multilayered functionally graded material plates under mechanical loads. The exponent-law distributions through the thickness and the power-law distributions of the volume fractions of the constituents were used to obtain the effective properties. The mechanical properties were assumed to vary continuously through the thickness by a simple power-law distribution in terms of the volume fractions of the constituents. Vaghefi et al. (2010) presented a three-dimensional static solution for thick functionally graded plates by utilizing a meshless Petrov–Galerkin method. An exponential function was assumed for the variation of Young's modulus through the thickness of the plate, while the Poison's ratio was assumed to be constant. More information may be found on the review paper given by Birman and Byrd (2007) for the manufacturing, design, modeling, testing methods and applications of functionally graded materials between the years 2000 and 2007.

In the present paper, an analytical solution to the static analysis of functionally graded plates and doubly-curved shells is developed using a higher order shear deformation theory (HSDT). A solution methodology, based on boundary-discontinuous generalized double Fourier series approach which is effectively used to solve the laminated composite shell and plate problems lately by Oktem and Chaudhuri (2007a,b; 2009a,b), Oktem and Guedes Soares (2011) is used to solve highly coupled linear partial differential equations, generated by the higher order-based theory with the fully simply supported boundary condition prescribed at all the edges. A simple power-law distribution in terms of the volume fractions of the constituents for the calculation of mechanical properties of the panels is assumed. Hitherto unavailable important numerical results presented to show the effect of inhomogeneities, thickness and membrane effects, as well as their interactions. The results are verified with the finite element counterparts for homogeneous (isotropic) and functionally graded plates by utilizing commercially available software (ANSYS).

Section snippets

Theoretical formulation

The rectangular doubly-curved shell made of FGM of uniform thickness of h is shown in Fig. 1. The ξ1 and ξ2 curves are lines of curvature on the shell mid-surface, ξ3 = ζ = 0, while ξ3 = ζ is a straight line normal to the mid-surface. The principal radii of normal curvature of the reference (middle) surface are denoted by R1 and R2. The displacement field by considering the cubic terms and satisfying the conditions of transverse shear stresses (and hence strains) vanishing at a point (ξ1, ξ2, ±h

Solution procedure

The solution of the partial differential equations Eq. (11a) in conjunction with the admissible boundary conditions given by Eqs. (12b), (12a) is assumed as follows:u1(x1,x2)=m=0n=1Umncos(αx1)sin(βx2),0x1a;0<x2<bu2(x1,x2)=m=1n=0Vmnsin(αx1)cos(βx2),0<x1<a;0x2bu3(x1,x2)=m=1n=1Wmnsin(αx1)sin(βx2),0x1a;0x2bϕ1(x1,x2)=m=0n=1Xmncos(αx1)sin(βx2),0x1a;0x2bϕ2(x1,x2)=m=1n=0Ymnsin(αx1)cos(βx2),0x1a;0x2bwhereα=mπa,β=nπb.

Introduction of assumed displacement functions into

Numerical results and discussions

The results are presented for a fully simply supported square spherical panels and plates under uniformly distributed transverse load of intensity qo. The following material properties are used for computing the numerical values:Et=151GPa,vt=0.3;Eb=70GPa,vb=0.3

The following non-dimensional quantities are used:w=D(2,2)q0a4u3(a2,b2),M1=M1q0a2(a2,b2),σxx=σxxhq0a2(a2,b2,h2),σxy=σxyhq0a2(0,0),σxz=σxzhq0a2(0,b2,0).where q0 denotes the transverse load.

Fig. 2 presents the through the thickness

Conclusions

The static response of functionally graded plates and doubly-curved shells is analyzed using a higher order shear deformation theory (HSDT). A simple power-law distribution in terms of the volume fractions of the constituents for the calculation of mechanical properties of the panels is assumed. A solution methodology, based on a boundary-discontinuous generalized double Fourier series approach is used to solve the highly coupled linear partial differential equations. Important numerical

Acknowledgment

The first author and the second author have been financed by the Portuguese Foundation of Science and Technology under the contract numbers SFRH/BPD/47687/2008 and SFRH/BD/66847/2009, respectively.

References (24)

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