Free vibration behaviour of multiphase and layered magneto-electro-elastic beam

https://doi.org/10.1016/j.jsv.2006.06.044Get rights and content

Abstract

Free vibration studies of multiphase and layered magneto-electro-elastic beam for BaTiO3–CoFe2O4 composite is carried out. In-plane plate finite-element analysis is used to obtain the behaviour of magneto-electro-elastic beam. Studies are carried out for beams for different boundary conditions, as may be applicable in design of smart sensors/actuators. The ANSYS package is used to validate the results obtained by the computer code developed for piezoelectric (PE) beam. Plane stress condition is used for the Euler–Bernouli magneto-electro-elastic beam.

Introduction

Research on magneto-electro-elastic materials has gained considerable importance since the last decade. These smart composite materials exhibit a desirable coupling effect between electric and magnetic fields, which are useful in smart or intelligent structure applications. These materials have the capacity to convert one form of energy, viz., magnetic, electric and mechanical energy to another form of energy. The composites made of piezoelectric (PE) and piezomagnetic (PM) materials exhibit a magnetoelectric effect which is absent in single phase PE or PM materials [1]. The smart materials seem to provide unique capabilities of sensing and reacting to external disturbances, thus satisfying performance, reliability and light weight requirements imposed in any modern structural applications [2]. For smart structures, the two basic elements are actuators and sensors. Many materials are available for this purpose, but PE materials are used commonly due to the factors like capability to at either as actuator or sensor and directly relate electrical signals to material strains and vice versa. Composites consisting of PE and PM components have found increasing applications in engineering structures, particularly in smart materials/intelligent structure systems. Magneto-electro-elastic materials are extensively used as magnetic field probes, electric packaging, acoustic, hydrophones, medical ultrasonic imaging, sensors, and actuators with the responsibility of magneto-electro-mechanical energy conversion [3]. Pan [4] has derived exact solutions for three-dimensional (3-D), anisotropic, linearly magneto-electro-elastic, simply supported and multilayered rectangular plates under static loadings. The behaviour of finitely long cylindrical shells under pressure loading has been studied by Wang and Zhong [5]. Free vibration study of simply supported and multilayered magneto-electro-elastic plates, has been carried out using propagator matrix approach, reported by Pan and Heyliger [6]. Buchanan [7] has studied the behaviour of infinitely long magneto-electro-elastic cylindrical shell using semi-analytical finite-element method. Annigeri et al. [8] have carried out a free vibration study of clamped–clamped magneto-electro-elastic cylindrical shell using semi-analytical finite-element method. Aboudi [9] has carried out micromechanical analysis of fully coupled electro-magneto-thermo-elastic composites. In his study, a homogenization micromechanical method is employed for the prediction of the effective moduli of magneto-electro-elastic composites. His study includes determination of effective elastic, PE, piezomagnetic, dielectric, magnetic permeability and electromagnetic coupling moduli, as well as effective thermal expansion coefficients and the associated pyroelectric and pyromagnetic constants for magneto-electro-elastic composite. Buchanan [10] has studied the vibration behaviour of an infinite plate consisting of layered versus multiphase magneto-electro-elastic composites. Recently, Jiang and Ding [11] have obtained analytical solutions to magneto-electro-elastic beams for different boundary conditions. Annigeri et al. [12] have carried out static study on magneto-electro-elastic beams for different boundary conditions using finite-element method. Annigeri et al. [13] have conducted free vibration studies on simply supported layered and multiphase magneto-electro-elastic cylindrical shells by semi-analytical finite-element method. Ramirez et al. [14] have analysed the free vibration response of two-dimensional (2-D) magneto-electro-elastic laminated plates. Ramirez et al. [15] have conducted a static analysis of functionally graded elastic anisotropic plates using a discrete layer approach Heyliger et al. [16] have studied the behaviour of 2-D static fields in magneto-electro-elastic laminates. From the literature survey, it is found that only few studies on free vibrations of magneto-electro-elastic beam structures have been reported.

Hence in the present paper, free vibration study of magneto-electro-elastic beam is carried out for three different boundary conditions, viz., clamped–clamped, clamped–free and simply supported ends. For the study, magneto-electro-elastic material BaTiO3–CoFe2O4 composite is considered. The multiphase material properties vary and are dependant on the ratio of fiber material to matrix material for BaTiO3–CoFe2O4 composite, the volume fraction of BaTiO3 is increased in steps of 20% to obtain different multiphase beam configurations.

Section snippets

Constitutive relation

For an anisotropic and linearly magneto-electro-elastic solid the coupled constitutive relation [4] for a general three-dimensional solid is as follows:{σ1σ2σ3σ4σ5σ6}=[C11C12C13000C12C11C13000C13C13C33000000C44000000C44000000C66]{S1S2S3S4S5S6}-[00e3100e3100e330e150e1500000]{E1E2E3}-[00q3100q3100q330q150q1500000]{H1H2H3},{D1D2D3}=[0000e150000e1500e31e31e33000]{S1S2S3S4S5S6}+[ε11000ε11000ε33]{E1E2E3}+[m11000m11000m33]{H1H2H3},{B1B2B3}=[0000q150000q1500q31q31q33000]{S1S2S3S4S5S6}+[m11000m11000m33]{

Finite-element formulations

For modeling of magneto-electro-elastic beam, four noded in-plane plate element with four degree of per node, i.e., displacement in axial direction (u), vertical direction (w), electric potential (φ) and magnetic potential (ψ) is used. Fig. 1 shows FE discretization using the in-plane plate elements.

The mechanical displacements, electrical potential and magnetic potential can be represented by suitable shape functions as follows:u=[Nu]{u},φ=[Nψ]{φ},ψ=[Nψ]{ψ}.

Here Nu, Nφ and Nψ are shape

Validation

The computer programme developed is validated with ANSYS results for eigenfrequencies, by treating the beam made of PE material (vf=100% of BaTiO3). The three boundary conditions, viz., clamped–free (C–F), simply supported (S–S) and clamped–clamped (C–C) beams with plane stress case are considered for the study. The Plane13 element with plane stress condition is used in conjunction with coupled field analysis in ANSYS® multiphysics. Here [C], [e] and [ε] material properties are used in the PE

Conclusions

  • 1.

    Magneto-electro-elastic beam is modelled using four noded in-plane plate element.

  • 2.

    The piezoelectric (PE) beam (vf=100% BaTiO3) is modelled in ANSYS for validation of frequencies with the code developed for three boundary conditions, i.e., C–C, S–S and C–F.

  • 3.

    The free vibration study is carried out for multiphase and layered magneto-electro-elastic beam for three boundary conditions, i.e., C–C, S–S and C–F.

  • 4.

    Three different magneto-electro-elastic materials are studied. It can be concluded that the

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