13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- 0, , 1, i i d i i N q dt q ∂ ∂ − = ∀ = ∂ ∂& L L (9) Let us introduce the Hamiltonian function as the Legendre transform of the Lagrangian function : ( , , ) ( , , ), 1 i i i i i i q p t pq q q t i N = − ≤ ≤ & & H L (10) with i i p q ∂ = ∂& L . The following relations : , i i i i p q q p ∂ ∂ = − = ∂ ∂ & & H H (11) are called Hamilton’s equations. The system of Euler-Lagrange’s equations (14) is equivalent to the Hamilton’s equations (16) (under some assumptions). 4.2. Application to the DCB specimen. These previous considerations are very formal and general. They can be applied to an elastodynamic system such our model, considering that , 1 1, , i i N q a i N q + = = =l . Therefore, the Lagrangian is : ( ) ( ) ( ) 0 , , , , , , , ( ) , 1, i i i i i a a K a a U a x dx i N = − − Γ = ∫ l & & & & l l l l l L (12) and the Euler-Lagrange equations become : ( ) ( ) 0, ,1 i i d K U K U i i N dt a a ∂ − ∂ − ⎛ ⎞ − = ∀ ≤ ≤ ⎜ ⎟ ∂ ∂ ⎝ ⎠ & (13) ( ) ( ) ( ) 0, d K U K U dt ∂ − ∂ − ⎛ ⎞ − +Γ = ⎜ ⎟ ∂ ∂ ⎝ ⎠ l l& l (14) which are respectively the local motion equations similar to equ. (1) for each eigenmode and the expression of the energy release rate. Using the Legendre transform of L , the following relation holds : ( ) ( ) ( ) 0 , , , , , , , ( ) , 1, i i i i i a a K a a U a x dx i N = + + Γ = ∫ l & & & & l l l l l H (15) Thus, the Hamiltonian is the total energy of the beam. Furthermore, since the Lagrangian equations are fulfilled, the energy balance is fulfilled. Starting from now, the complete solution of the dynamic crack growth problem in the DCB is restricted to the determination of the time dependent variables (),() ia t t l
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