Wave propagation in ring shaped electro - magneto - elastic plate of polygonal cross - sections

Introduction The wave propagation in magneto-electro-elastic materials has gained considerable importance since last decade. The electro-magneto-elastic materials exhibit a desirable coupling effect between electric and magnetic fields, which are useful in smart structure applications. These materials have the capacity to convert one form of energy namely, magnetic, electric and mechanical energy to another form of energy. The composite consisting of piezoelectric and piezomagnetic components have found increasing application in engineering structures, particularly in smart/intelligent structure system. The magnetoelectro-elastic materials are used as magnetic field probes, electric packing, acoustic, hydrophones, medical, ultrasonic image processing, sensors and actuators with the responsibility of magnetic-electro-mechanical energy conversion. A method, for solving wave propagation in arbitrary and polygonal cross-sectional plates and to find out the phase velocities in different modes of vibrations namely longitudinal, torsional and flexural, by constructing frequency equations was devised by (Nagaya, 1981a, 1981b, 1981c, 1983a, 1983b). He formulated the Fourier expansion collocation method for this purpose and the same method is used in this problem.The threedimensional behavior of magnetoelectroelastic laminates under simple support has been studied by (Pan, 2001) and (Pan and Heyliger, 2002). An exact solution for magnetoelectroelastic laminates in cylindrical bending has also been obtained by (Pan and Heyliger, 2003). (Pan and Han, 2005) studied the exact solution for functionally graded and layered magneto-electroelastic plates. (Feng and Pan, 2008) discussed the dynamic fracture behavior of an internal interfacial crack between two dissimilar magneto-electro-elastic plates. (Buchanan, 2003) developed the free vibration of an infinite magneto-electroelastic cylinder. (Dai and Wang, 2005, 2006) have studied thermo-electro-elastic transient responses in piezoelectric hollow structures and hollow cylinder subjected to complex loadings. Later (Kong et al, 2009) presented the thermo-magneto-dynamic stresses and perturbation of magnetic field vector in a nonhomogeneous hollow cylinder. (Annigeri et al, 2006, 2007, 2006) studied respectively, the free vibration of clampedclamped magneto-electro-elastic cylindrical shells, free vibration behavior of multiphase and layered magneto-electro-elastic beam, free vibrations of simply supported layered and multiphase magneto-electro-elastic cylindrical shells. (Hon et al, 2008) analyzed a point heat source on the surface of a semiinfinite transversely isotropic electro-magneto-thermo-elastic materials. (Sharma and Mohinder Pal, 2004) developed the Rayleigh-Lamb waves in magneto-thermo-elastic homogeneous isotropic plate. Later (Sharma and Thakur, 2006) studied the effect of rotation on Rayleigh-Lamb waves in magneto-thermoelastic media. (Gao and Noda, 2004) presented the thermalinduced interfacial cracking of magnetoelectroelastic materials. (Bin et al, 2008) studied the wave propagation in nonhomogeneous magneto-electo-elastic plates. (Ponnusamy, 2007, 2011, 2012) have studied the wave propagation in generalized thermo-elastic cylinder of arbitrary cross section, thermoelastic and generalized thermo elastic plates of arbitrary and polygonal cross-sections respectively. (Ponnusamy and Rajagopal, 2010, 2011) have studied, the wave propagation in a generalized thermo elastic solid cylinder of arbitrary cross-section and in a homogeneous transversely isotropic thermo elastic solid cylinder of polygonal cross-sections respectively using the Fourier expansion collocation method. Formulation of the Problem We consider a homogeneous transversely isotropic magneto-electro-elastic ring shaped plate of polygonal crosssections with thickness h and occupying the space 0 Z L ≤ ≤ , L is the length of plate considered along the zaxis is shown in Figure 2. The system displacements and stresses are defined by