13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- Then, the traction vector ( , ) it τ x at a boundary point s ∈∂Ω x is approximated in terms of the same nodal values ˆ ( ) a τ u and ˆ() a τ w as 1 1 垐 ( , ) ( ) ( ) ( ) ( ) ( ) ( ) n n h a a a a w a a τ τ τ = = = + ∑ ∑ t x N x C B x u N x R B x w , (16) where N(x) is related to the normal vector n(x) on s ∂Ω and the matrices aB and a wB are represented by the gradients of the shape functions with 1 2 2 1 0 ( ) 0 n n n n ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ N x , ,1 ,2 ,2 ,1 0 ( ) 0 a a a a a φ φ φ φ ⎡ ⎤ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ B x , ,1 ,2 ,2 ,1 0 ( ) 0 a a a w a a φ φ φ φ ⎡ ⎤ ⎢ ⎥ = ⎢ ⎥ ⎢ − ⎥ ⎣ ⎦ B x , and the material matrices 11 12 12 22 66 0 0 0 0 c c c c c ⎡ ⎤ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ C , 0 0 0 0 R R R R R ⎡ ⎤ ⎢ ⎥ = − − ⎢ ⎥ ⎢ − ⎥ ⎣ ⎦ R . Similarly the generalized traction vector ( , ) ih τ x can be approximated by 1 1 垐 ( , ) ( ) ( ) ( ) ( ) ( ) ( ) n n h a a a a h h h hw a a τ τ τ = = = + ∑ ∑ h x N x R B x u N x K B x w , (17) where 1 2 2 1 0 0 ( ) 0 0 h n n n n ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ N x , 1 2 2 1 1 2 2 1 0 0 0 0 0 0 0 0 K K K K K K K K ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ = ⎢ − ⎥ ⎢ ⎥ − ⎢ ⎥ ⎣ ⎦ K , ,1 ,2 ,2 ,1 0 0 ( ) 0 0 a a a hw a a φ φ φ φ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ B x , 0 0 0 0 0 0 h R R R R R R R ⎡ − ⎤ ⎢ ⎥ − ⎢ ⎥ = ⎢ − ⎥ ⎢ ⎥ ⎣ ⎦ . Satisfying the essential boundary conditions and making use of the approximation formulae (14) and (15) one obtains the discretized form of these boundary conditions as 1 ( )ˆ ( ) ( , ) n a a a φ τ τ = = ∑ ζ u u ζ% , (18) 1 ( ) ( ) ( , ) n a a a φ τ τ = = ∑ ζ w w ζ% , for u ∈Γ ζ . (19)
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