13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- ( ) ( ) ( ) ( ) 1 1 0, 0 0 0, 1, i i i i i N φ φ φ φ ⎧ = ′ = ⎪ ⎨ ′′ = = ∀ = ⎪⎩ (3) 3.2. Free vibration mode shape. An harmonic motion for the free vibration mode is assumed : ( ) ( ) ( ) ˆ , sin , 1, i i w X t X t i N φ ω θ = + = (4), where iω is the angular frequency associated to the mode iφ, θ the phase angle. The motion equation may be written as : [ ] 4 2 (4) 4 4 ( ) ( ), , 1, , 0,1 , i i i i i A X X i i N X EI ρ ω φ μ φ μ = ∀ = ∀ ∈ = l (5) The general solution of (10) is : ( ) ( ) ( ) ( ) ( ) i i i i i cos X sin X sh X sh μ φ μ μ μ ⎡ ⎤ = − ⎢ ⎥ ⎣ ⎦ (6) and each angular frequency is given by the following equation: ( ) ( ), , 1, i i tg th i i N μ μ = ∀ = (7) 4. Hamilton’s principle applied to a domain with a moving crack. Hamilton’s principle of stationary action is widely used in classical mechanics [6]. It postulates that the dynamics of a system is governed by a variational problem for a functional integrating the Lagrangian function, which contains some information about the energy of the system and the forces acting on it. It is the weak form of the differential equations of motion of the system. 4.1. General Hamilton’s principle applied to a system described by a discrete set of degrees of freedom. Let us recall the outlines of the principle for a system described by the generalized coordinates ( ), 1, iq i N = , in a formal way. Hamilton’s principle states that the true evolution of ( ) iq along a path γbetween two mechanical states at time 0t and 1t is a stationary point of the action functional : 1 0 ( ) ( , , ) t i i t q q t dt ϕ γ =∫ & L (8), where ( , , ) i i q q t & L is the Lagrangian function of the system. This principle involves the following system of N Euler-Lagrange equations [6] :
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