EFFICIENT HIGHER ORDER ZIG-ZAG THEORY FOR COUPLED - PDF document

18 TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS EFFICIENT HIGHER ORDER ZIG-ZAG THEORY FOR COUPLED MAGNETO-ELECTRO-ELASTIC COMPOSITE LAMINATES J. Lee 1 , J.-S. Kim 2 , M. Cho 3* 1 School of Mechanical and Aerospace Engineering, Seoul National University, Seoul, Korea, 2 Department of Intelligent Mechanical Engineering, Kumoh National Institute of Technology, Seoul, Korea 3 Division of WCU Multiscale Mechanical Design, School of Mechanical and Aerospace Engineering, Seoul National University, Seoul, Korea. * mhcho@snu.ac.kr Keywords : Zigzag plate theory, Piezoelectric, Piezomagnetic, Magneto-electro-elastic laminates. 1 Introduction been reported for engineering applications, one of the major drawbacks of them is that the number of Multilayered structures made of piezoelectric and unknowns is dependent upon the number of layers, piezomagnetic materials have been used widely due which means those models are not efficient and thus to their special properties of converting the they have limitations to be applied to the large-scale, mechanical energy into electrical or magnetic energy sensing and actuating problem (i.e., dynamic and vice versa [1-3]. For the piezoelectric materials, analysis of the multilayered, coupled MEE plates). various studies have been carried out to analyze and Thus, more efficient theory which also contains the to design such multilayered smart structures. accuracy are required. Among the various studies of Furthermore, for the accurate prediction of static and the multilayered plate structures, an efficient higher dynamic behaviors, various coupled thermo-electro- order plate theory (EHOPT) proposed by Cho and elastic analysis containing thermal effects that are Parmerter [9] is the best performer in displacement- significant in multiphysics problems have been based zigzag theories [10] and recommended for the carried out. Elasticity solution has been proposed [4] analysis of the MEE plates since numerous studies and higher-order zigzag models have been reported have been verified the accuracy and efficiency of the [5,6]. EHOPT by analyzing the fully coupled piezoelectric In the same context, the analysis of magneto-electro- composite plates [7] as well as the conventional elastic (MEE) materials has been increasingly composite laminates. This theory reduces the known demanded recently due to their unique variables using the top/bottom boundary conditions characteristics. For the analysis of the multilayered and the transverse shear stress continuity conditions. rectangular MEE plates, Pan [2] obtained the In this study, the multilayered MEE plates are analytical solution under the static sinusoidal load considered to carry out the fully coupled magneto- based on the quasi-Stroh formalism and the electro-elastic analysis built upon the EHOPT. propagator matrix method. Moreover, Pan and The displacement field, electric potential and Heyliger [3] solved the cylindrical bending problem magnetic potential are assumed as a third order of MEE plates with simply-supported edge condition. zigzag functions. The number of unknowns is The vibration analysis of MEE plates has also been reduced effectively by applying the top/bottom carried out using a layerwise-type approximation by conditions and transverse direction flux continuity Ramirez et al. [7]. , In addition, Annigeri et al. [8] conditions. For the practical usage of the present studied the free vibration behavior of MEE beam method, finite element discretization based on the based on the membrane-type finite element model. beam-type model is applied. To investigate the However, even though the previous analyses has different responses of the elastic, electric and

magnetic quantities, various layups and different In this study, a fully coupled higher-order zigzag loadings are considered by comparing them with the theory is proposed for efficient modeling. The results reported in the literatures. displacement field is assumed by superimposing zigzag linear field to the globally varying field as follows: 2 Formulation      u x z u 0 x x z x z 2 ( , ) ( ) ( ) ( ) A three-dimensional multilayered magneto-electro- 1 1 1 1 elastic plate is shown in Fig. 1. The constitutive  N 1  k      x z 3 S x z z  z z ( ) ( )( ) ( ) equations for a linear, anisotropic and magneto- k k 1 1  electro-elastic solid can be written as [1-3] k 1 (4)    2 u x z w x r x z r x z ( , ) ( ) ( ) ( )      3 1 2 C e E q H ij ijkl kl ijk k ijk k  N 1  sk    r x z z  z z      ( )( ) ( ) D e E d H (1) k k i ikl kl ik k ik k  k 1      B q d E H i ikl kl ik k ik k where  ( z-z k ) is a Heaviside unit step function. u 0 where σ ij , D i and B i are the stress, electric 1 and w represent the in-plane displacement and the displacement and magnetic induction, respectively. out-of-plane displacement from the reference plane, ε kl , E k and H k are the strain, electric field and respectively. χ 1 are the rotation of the normal about magnetic field, respectively. C ijkl , γ ik and μ ik are the x 1 axis. For the in-plane displacement, a cubic elastic, dielectric and magnetic permeability varying field is applied. Whereas, for the out-of- coefficients, respectively. e ijk , q ijk and d ik are the plane displacement, a quadratic field is assumed in piezoelectric, piezomagnetic and magnetoelectric order to express the physical behavior of the coefficients, respectively. The piezoelectric and displacement field accurately. In Eq. (4), the beam- magnetostrictive layers are made of BaTiO 3 and type model which contains x , z and no y axis is CoFe 2 O 4 , respectively. Corresponding equilibrium considered for the simplicity of the problem. equations can be written as Similar to the construction of the displacement field, the electric and magnetic potential are obtained by     D B 0, 0, 0 (2) ij j i i i i , , , superimposing linear zigzag field onto the global cubic smooth field as follows: The strain-displacement, electric potential and magnetic potential relationship can be expressed as        2 x z x x z x z ( , ) ( ) ( ) ( ) 0 1 2 follows:  N (5) 1    1 sk                x z 3 x z z  z z u u E H ( ) ( )( ) ( ) , , (3) k k ij i j j i i i i i 3 , , , , 2  k 1        x z x x z x z 2 ( , ) ( ) ( ) ( ) 0 1 2 N  (6) 1      sk   3 x z x z z  z z ( ) ( )( ) ( ) k k 3 k   1 k The unknown variables in Eqs. (4-6) can be reduced by applying boundary conditions of top/bottom surface and introducing the continuity conditions of transverse stresses, transverse electric and magnetic displacement between each layer. The top/bottom Fig.1. Geometry and coordinates of magneto- boundary conditions can be expressed as electro-elastic plate.