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Example text

Thus the wave vectors can be determined. Substitution of Eq. (42) into Eq. (35) gives the system equation (APU + D)d = Rd = s0 (46) where R = APU + D is the system matrix. e. s0 = 0 and the free wave propagation problem is considered, the vanishing of the system matrix determinant yields the following dispersion equation R( k x ; k y ; ω ) = 0 (47) which may be solved numerically by a proper root searching technique (Guo, 2008). Thus, the complete dispersion curves of various waves can be obtained, as illustrated in Section 4 for multilayered anisotropic elastic structures.

E. x 12 v1E 1 v12 a 12 d y 12 z 1 12 2 (a) top surface a( N + 1) N n h 12 12 z( N + 1) N d( N + 1) N N h N ( N + 1) v( N + 1) N x ( N + 1) N N +1 ( N + 1) E v y ( N + 1) N (b) bottom surface Fig. 2. The top and bottom surfaces of the multilayered anisotropic structure vˆ 12 (0) = TE vˆ 1E , Tv vˆ ( N + 1) N (0) = TE vˆ (EN + 1) (22) where TE =< TuE , Tσ E > is a transformation matrix with TuE = I nv /2 and Tσ E = −I nv /2 ; Tv =< Tu , Tσ > is also a transformation matrix, with Tu =< −1,1, −1 > and Tσ =< 1, −1,1 > for elastic layers and Tu =< −1,1, −1, −1 > and Tσ =< 1, −1,1,1 > for piezoelectric layers.

A. (2008). Dynamics of anisotropic multilayers, Wave Motion, 45(5): 629-640. Auld, B. A. (1990). Acoustic Fields and Waves in Solids, Second Edition, Vol. I & II, Robert E. Krieger Publishing Company, Malabar, Florida. Brekhovskikh, L. M. (1980). Waves in Layered Media, Second Edition. Academic Press, New York. Chakraborty, A. & Gopalakrishnan, S. (2006). A spectral finite element model for wave propagation analysis in laminated composite plate, ASME Journal of Vibration and Acoustics, 128: 477-488.

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