propagation characteristics are identified for mode I conditions. In the next section, we examine the energetics of crack growth for a through thickness reinforced DCB specimen loaded by a flying wedge. The double cantilever beam (DCB) specimen loaded dynamically by a flying wedge offers a relatively simple experimental approach to analyzing the mode I dynamic delamination problem. Regions of stable crack growth as a function of the material properties of the through thickness reinforcement, the size of the DCB specimen and the velocity of the wedge have been identified. Beam Theory Formulation and Solution Characteristics: For a beam element, the equations of motion are: 2 2 t u Bh x N ¶ ¶ = ¶ ¶ r (1a) 2 2 t w p(w,t )B Bh x Q ¶ ¶ = - ¶ ¶ r (1b) 2 2 t I Q x M ¶ ¶ - = ¶ ¶ f r (1c) where u(x,t) and w(x,t) are the in-plane and transverse displacements of the neutral plane respectively, f(x,t) is the clockwise rotation of the cross-section, t is the time variable, N is the axial force, Q is the shear force, M is the bending moment, 2h is the total thickness of the DCB specimen, B is the width of the specimen, r is the density, I (= Bh3/12) is the moment of inertia and p(w,t) is the bridging traction corresponding to the opening mode. In this work, the time dependent bridging traction p corresponding to the opening mode is assumed to depend only on the transverse displacement w. In the absence of an axial force N, u = 0. For a Timoshenko beam, the equations for steady state motion can be reduced to [7]: 0 p Eh 12R (R c )(1 c ) 1 X p Eh 1 (R c ) 1 X w h 12R ( R c )(1 c ) c X w 3 2 l 2 l 2 2 2 l 2 2 2 2 l 2 l 2 l 4 4 = - - + ¶ ¶ - - ¶ ¶ - - + ¶ ¶ (2a) 2 2 2 l X w R (R c ) REh p X ¶ - ¶ - = ¶ ¶f (2b) where X = x – v t , c v E l / 2 2 r= , R G E/ k= , v is the (constant) steady state velocity, G and E are the shear modulus and the Young’s modulus of the laminate and the dimensionless shear coefficient k=5/6 for a beam with rectangular cross section. For steady state dynamic delamination, the velocity v is the delamination crack tip velocity. For an Euler-Bernoulli (E-B) beam, where both shear deformation and rotational inertia are ignored, the equation for steady state motion reduces to a simple form given by: p 0 Eh 12 X w h 12c X w 3 2 2 2 2 l 4 4 = + ¶ ¶ + ¶ ¶ (3)
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