13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- The velocity field is assumed as 0 w w z L =& & when 0 z L ≤ ≤ , where 0w& is the velocity at the midspan of the plastic-string, shown in Fig. 3(a) for half of the plastic-string. In the phase of dynamic structural response for the plastic-string model [15], the dynamic response of symmetric sandwich beam is dominated by axial (membrane) force alone. However, the asymmetric sandwich beam has axial (membrane) force pN′ and bending moment M, shown in Fig. 3(b). This is because that the plastic neutral surface of asymmetric sandwich structure is usually different from the geometric surface, and the geometric and plastic neutral surfaces of the symmetric sandwich beam are coincident. (a) (b) Figure 3. Sketches of the deformation process of a fully clamped plastic-string. (a) The velocity field and (b) a free body diagram of the half of the plastic-string. Considering the conservation of the moment of momentum for half of the sandwich plate shown in Fig. 3(b) with respect to the fixed end support Point A at time t, we have ( ) 0 0 L p s t s b c c d M N w M bh bh bH wzdz dt ρ ρ ρ + ′ − =− + + ∫ & (2) where 0w is the deflection at the midspan. The continuity conditions are 0( 0) 0 w t = = and ( ) 0 0 f w t V = = & , where the final common velocity of the core and two face sheets at the end of the core compression stage ( ) f s t s b c c V I h h H ρ ρ ρ = + + [4]. If ( ) 0 0 f w t T= = & , the motion of the plastic-string ends. Then, the maximum deflection ( ) 1 0 f w w t T = = and structural response time fT of the asymmetric sandwich plate can, respectively, be written as ( )( ) 1 3 Y t Y b lY c s t s b c c IL w h h H h h H σ σ σ ρ ρ ρ = + + + + (3)
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