Dynamic responses of functionally graded magneto-electro-elastic shells with open-circuit surface conditions

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Abstract

Three-dimensional (3D) free-vibration analysis of simply supported, doubly curved functionally graded (FG) magneto-electro-elastic shells with open-circuit surface conditions is studied using an asymptotic approach. The material properties of the FG shells are regarded as heterogeneous through the thickness coordinate. The 29 basic equations of 3D magneto-electro-elasticity are firstly reduced to a system of 10 state space vector equations in terms of 10 primary variables in elastic, electric and magnetic fields. Apart from the regular asymptotic expansion in the early paper on static analysis, the method of multiple time scales is used to eliminate the secular terms and to make the asymptotic expansion feasible. Through a straightforward derivation, we finally decompose the present 3D problem as recursive sets of two-dimensional (2D) problems with motion equations of the coupled classical shell theory (CCST). The orthonormality and solvability conditions for various order problems are derived. With these conditions, it is shown that the 3D asymptotic solutions can be obtained by repeatedly solving the CCST-type motion equations order-by-order in a systematic and hierarchic manner. The influence of the gradient index of material properties on the natural frequencies and their corresponding modal field variables of various FG piezoelectric and magneto-electro-elastic shells is presented.

Introduction

Recently, a new class of advanced material, namely the functionally graded (FG) material, has been developed and designed as the so-called smart (or intelligent) structures such as FG piezoelectric and FG magneto-electro-elastic plates/shells for sensing, actuating and control purposes. Unlike the conventional multilayered devices of which material properties suddenly change at the interfaces between adjacent layers, the material properties of these FG plates and shells are gradually varied through the thickness coordinate. That largely improves the working performance and lifetime of the devices composed of the FG material.

Since magneto-electro-elastic materials possess the coupled nature of constitutive equations, the coupling effects among elastic, electric and magnetic fields on the static and dynamic responses of FG magneto-electro-elastic plates and shells are significant. A comprehensive realization for the actual behaviors of FG magneto-electro-elastic plates and shells may be helpful for the design of these smart devices. However, three-dimensional (3D) analysis for the static and dynamic responses of FG piezoelectric and magneto-electro-elastic structures is scarce in the literature in comparison with that of multilayered piezoelectric and magneto-electro-elastic structures [1], [2], [3], [4], [5], [6], [7], [8], [9], [10].

Some literature relevant to the present subject has been presented in quiet recently. Based on the state space approach, Zhong and Shang [11], [12] presented an exact 3D analysis of FG piezoelectric and electro-thermo-elastic plates under mechanical, electric and thermal loads, respectively. The material properties of the FG plates have been assumed to obey the identical exponent-law distributions through the thickness coordinate. Using the method of propagator matrix (or transfer matrix), Zhong and Yu [13] proposed a state space formulation to study 3D free and forced vibration of FG piezoelectric plates. With an alternative state space formulation, Chen et al. [14] presented the free-vibration responses of FG magneto-electro-elastic plates. It has been pointed out that the coupling effect on higher-order natural frequencies has more significant than that on the lower-order ones.

Several numerical methodologies have also been developed for the accurate analysis of magneto-electro-elastic plates and shells. Ramirez et al. [15] presented the discrete layer solution to free vibrations of FG magneto-electro-elastic plates. Natural frequency solutions of FG plates with two different edge boundary conditions and various values of aspect ratios have been discussed. A semi-analytical finite element (FE) method has been used for the free-vibration analysis of FG and layered magneto-electro-elastic structures by Bhangale and Ganesan [16]. It has been demonstrated that the FE solutions are in excellent agreement with the 3D solutions available in the literature. Annigeri et al. [17] proposed a finite layer element method to investigate the free-vibration responses of layered and multiphase magneto-electro-elastic cylindrical shells.

After a close literature survey, we found that the state space approach has being popularly used for the 3D analysis of multilayered as well as FG piezoelectric and FG magneto-electro-elastic plates, seldom used for that of shells. Recently, an alternative approach, namely the asymptotic approach different from the conventional state space approach, has been proposed and successfully applied for the 3D static analysis of laminated piezoelectric hollow cylinders as well as of FG piezoelectric and magneto-electro-elastic shells by Wu and his colleagues [18], [19], [20]. The formulations are based on either 3D piezoelectricity or 3D magneto-electro-elasticity, and the method of perturbation has been used in the mathematical derivations. The accuracy and the rate of convergence of the asymptotic solutions have been validated by comparing their solutions with the state space solutions available in the literature.

To the best of authors’ knowledge, the 3D free-vibration analysis of FG magneto-electro-elastic shells cannot be found in the literature so far. The purpose of this paper is to extendedly apply the asymptotic approach to the present 3D dynamic problem. The major differences from its static version [20] are that the method of multiple time scales is used to eliminate the secular terms raised from the regular asymptotic expansions and both the orthonormality and solvability conditions for various order problems are derived to uniquely solve for the modal field variables of various orders. The detailed derivation for the present dynamic version of 3D asymptotic formulation is given as follows.

Section snippets

Basic equations of magneto-electro-elasticity

A doubly curved FG magneto-electro-elastic shell with heterogeneous material properties through the thickness coordinate is considered. A set of the orthogonal curvilinear coordinates (α, β, ζ) is located on the middle surface of the shell. The total thickness of the shell is 2h. aα and aβ denote the curvilinear dimensions in α- and β-directions; Rα and Rβ denote the curvature radii to the middle surface of the shell, respectively.

The linear constitutive equations of magneto-electro-elastic

Nondimensionalization

A set of dimensionless coordinates and elastic field variables is defined asx=α/Rh,y=β/Rh,z=ζ/h;Rx=Rα/R,Ry=Rβ/R;u=uα/Rh,v=uβ/Rh,w=uζ/R;σx=σα/Q,σy=σβ/Q,τxy=ταβ/Q;τxz=ταζ/Qϵ,τyz=τβζ/Qϵ,σz=σζ/Qϵ2;where ϵ2 = h/R; R and Q denote a characteristic length of the shell and a reference elastic modulus, respectively.

The dimensionless electric and magnetic field variables are defined asDx=Dαϵ/e,Dy=Dβϵ/e,Dz=Dζ/e,ϕ=Φeϵ2/hQ;Bx=Bαϵ/q,By=Bβϵ/q,Bz=Bζ/q,ψ=Ψqϵ2/hQ;where e and q stand for a reference piezoelectric

Asymptotic expansion

Since (3.4), (3.5), (3.6), (3.7), (3.8), (3.9), (3.10), (3.11), (3.12), (3.13), (3.14) contain terms involving only even powers of ϵ, we therefore asymptotically expand the primary variables in the powers ϵ2 as given byf(x,y,z,ϵ,τ0,τ1,)=f(0)(x,y,z,τ0,τ1,)+ϵ2f(1)(x,y,z,τ0,τ1,)+ϵ4f(2)(x,y,z,τ0,τ1,)+We substitute (4.1) into (3.4), (3.5), (3.8), (3.10), (3.11), (3.12), (3.13), (3.14) and collect coefficients of equal powers of ϵ, then obtain the following sets of recurrence equations.

Order ϵ0:w

The leading-order problem

Observing the previously recursive sets of asymptotic equations, we found that the analysis can be carried on by integrating those equations through the thickness direction. We therefore integrate (4.2), (4.3), (4.4), (4.5) to obtainw(0)=w0(x,y,τ0,τ1,),u(0)=u0-zDw0-Eeϕ0-Eqψ0,ϕ(0)=ϕ0(x,y,τ0,τ1,),ψ(0)=ψ0(x,y,τ0,τ1,),where w0, u0=u0(x,y,τ0,τ1,)ν0(x,y,τ0,τ1,)T,ϕ0 and ψ0 represent elastic displacements, electric potential and magnetic potential on the middle surface of the shell at the

Applications to benchmark problems

The benchmark problems for the free-vibration analysis of simply supported, functionally graded magneto-electro-elastic shells with open-circuit surface conditions are studied using the present asymptotic formulation. The boundary conditions on the edges are of a shear diaphragm type specified as aforementioned in (2.13a), (2.13b).

For this problem, the governing equations of the leading-order problem (5.9), (5.10), (5.11), (5.12), (5.13) can be solved by lettingu0=U0cosm˜xsinn˜ycos(ωτ0-ϑ),v0=V0

Functionally graded piezoelectric plates

The benchmark free-vibration problem of FG piezoelectric plates with open-circuit surface conditions in the literature [13] is used to validate the accuracy and rate of convergence of the present asymptotic formulation. The 3D free-vibration analysis of FG piezoelectric plates can be regarded as a special case of the present analysis by letting 1/Rα = 1/Rβ = 0 and dij = qij = 0. In this example, the material properties of the plate are considered to vary exponentially through the thickness coordinate

Conclusions

The 3D solution for the free-vibration response of simply supported, doubly curved FG magneto-electro-elastic shells has been presented using the method of multiple time scales. The derivation is based on 3D magneto-electro-elasticity with neither kinematical nor kinetic assumptions in advance. The method of multiple scales is used to eliminate the secular terms and make the present asymptotic expansion feasible. It is shown that the present asymptotic formulation can be reduced to that of FG

Acknowledgment

This work is supported by the National Science Council of Republic of China through Grant NSC 96-2221-E006-265.

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