13th International Conference on Fracture June 16–21, 2013, Beijing, China -6- Based on mode superposition principle, the dynamic displacement responses can be expanded as Eq. (22): ( ) ( ) ( ) ( ) ( ) ( ) ( , , ) ( , ) ( ) ( , , ) ( , ) ( ) ( , , ) ( , ) ( ) p p mn mn m n p p p x xmn mn m n p p p y ymn mn m n p w x y t W x y T t x y t x y T t x y t x y T t β β =∑∑∑ =∑∑∑Ψ =∑∑∑Ψ (22) Assume the time-dependent factor to be i t e ω. Thus, ( ) ( ) i ( ) p p t mn mn T t T e ω = (23) Substituting Eq. (22-23) into Eq. (1), in view of the Eq. (11) and the orthogonality for the modes, one can acquire Eq. (24), ( ) ( ) ( ) , , p p kl q mn mn mn kl mn k l q T Z Z T F + = ∑ (24) in which: ( ) 2 2 ( ) ( )2 2 ( ) ( ) + ( + ) 4 p p p p mn mn mn mn ab Z h J b c ω ω ρ ρ ⎧ ⎫ ⎡ ⎤ = − × × ⎨ ⎬ ⎣ ⎦ ⎩ ⎭ i sin ( cos sin ) 1 2 0 0 2 2 cos( cos )e sin sin 2 a b k x y mn i h h m x n y F P k dxdy a b φ θ θ π π φ − + + ⎧ ⎫ = ⎨ ⎬ ⎩ ⎭ ∫ ∫ 2 2 0 0 i ( ) ( ) 0 0 2 0 0 0 2 2 0 0 0 0 0 0 sin sin sin sin ( ) ( ) k x x y y a b a b kl mn k x l y e m x n y a b Z dx dy dxdy a b x x y y π π ρω π π π − − + − = − + − ∫ ∫ ∫ ∫ Solving Eq. (24), one can obtain the displacement response. By taking derivative with respect to t , the velocity can be acquired. In accordance with the Rayleigh’s integral, the sound pressure at the observation point ( , , p p p x y z ) above the plate can be got using the velocity (Eq. 25): i 0 i i ( , ) e ( , , , ) e d 2π kR t p p p v x y p x y z t A R ω ωρ − Ω ⋅ = ∫ (25) 3. Validation A simply supported rectangular sandwich plate with dimensions of 400×300×10 mm is considered here for numerical studies, which are carried out to test the validity of the analytical solution. 0.5 mm and 9 mm are the thickness for the facings and the core respectively. The properties for them are listed in Table 1. Structural damping ratio is taken as 0.001. Use Nastran to test the natural frequencies and modes with FEM (finite element method). The comparisons of the analytical results with those got by numerical approach are shown in Table 2, from which one can see that the two sets of results match with each other very well. Table 1. Material properties Material Young modulus Poisson’ Density Coefficient of thermal expansion
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