13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- to bear out a classical beam theory). Therefore, the motion only exhibits the scalar beam deflection ( , ) w x t . The general differential motion equation, associated with the boundary conditions may be written as follows, 0 , a ∀ ≤ ≤ l l l l : [ ] 4 2 4 2 2 0 2 0, 0, , (0, ) ( ), (0, ) 0, ( , ) 0, ( , ) 0, w w EI A x t x t w w t W t t x w w t t x ρ ⎧ ∂ ∂ + = ∀ ∈ ∀ ⎪ ∂ ∂ ⎪ ∂ ⎪ = = ⎨ ∂ ⎪ ⎪ ∂ = = ⎪ ∂ ⎩ l l l (1) where ,,, E I A ρ are respectively the elastic modulus, the area moment of inertia of the beam cross section, the area of this section, and the mass density of the material. As the dynamic stage is the only one which matters, the initial conditions will refer to the crack running onset time when ( 0) c t = = l l , so that the quasi-static growth stage ( [ ] 0 0, c t x < ∈ l ) is disregarded here. 3. Approximate equation of the DCB motion by modal decomposition 3.1. Approximate equation of motion. It is supposed, as already suggested by Freund [5], that the deflection of the beam is given by the following approximation : ( ) ( ) ( ) ( ) ( ) ( ) ˆ ( ), , , 1, stat stat i i w X t t w X w X t w X a t X i N φ = + = + = (2) where the Einstein summation convention is henceforth adopted, from now on. The new variable [ ] / , 0,1 X x X = ∈ l is the normalized beam coordinate, useful in the case of a growing crack, such that the moving integration domain [ ] 0, ( )t l is replaced by the stationary reference domain [ ] 0,1 , ( )t ∀l . ( ) stat w X is the quasi-static deflection solution, ( ) ˆ , w X t the perturbation around the equilibrium solution, iφ the i th normal mode shape for free vibration of the beam, and ia the unknown deflection amplitude associated to iφ. The homogeneous boundary conditions associated to the free vibration of the beam are derived from (1) :
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