13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- ⎪ ⎪ ⎭ ⎪ ⎪ ⎬ ⎫ ⎪ ⎪ ⎩ ⎪ ⎪ ⎨ ⎧ Φ Φ Φ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ =− ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ∫ ∫ ∫ c c c s ds s p s ds s p s ds s p Y p Y p Y p Y p Y p Y p Y p Y p Y p A p A p A p 0 3 0 2 0 1 33 32 31 23 22 21 13 12 11 (1) 3 (1) 2 (1) 1 ) ( , )sin( ) ( , )sin( ) ( , )cos( ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) 2 ( , ) ( , ) ( , ) ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ πξ ξ ξ ξ (17) where ( , ) ( , 1 3) = − Y p i j ij ξ are known functions. Satisfaction of the mixed boundary conditions (5) and (6) on the crack face plane leads to the simultaneous singular integral equations ∫ ∑ ∫ ∑ − = − = = ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ Ψ ⎥ Ψ + ⎦ ⎤ ⎢ ⎣ ⎡ + − − = ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ⎥Ψ ⎦ ⎤ ⎢ ⎣ ⎡ + − Ψ + 1 1 3 2 3 1 31 0 31 1 1 3 2 0 1 1 ( , , ) ( , ) 0 ( , , ) ( , ) ( , ) ( , , ) ( , , ) ( , ) s p ds s x p s x p s p x s U p T s p ds s x p s x U s x p s p j j j m j j mj mj m κ κ π κ κ ( 1,2) = m (18) where 0 2 0 1 , T P T D= = , ( , ) ( , ) cs p s p i iΨ =Φ , (, , )( 1,2,3) = s x p i ijκ are known kernel functions, the constants 0 ij U are defined as ( , ) lim 0 U U p ij ij ξ ξ→∞ = and ( , ) U p ij ξ are known functions. The functions ( ) ( , ) 1 3 = − Ψ s p i i satisfy the single-valuedness condition: ( ) 1 3 ( , ) 0, 1 1 = − = Ψ ∫ − i s p ds i (19) and ( , ) s p iΨ may be expressed as 2 ( , ) ( , ) 1 s s p H s p i i − Ψ = (20) where ( ) ( , ) 1 3 = − H s p i i are unknowns to be solved. Use the Lobatto-Chebyshev method [11], singular integral equations (18) can be reduced to the following algebraic equations: ( ) [ ] ( ) ( ) ( ) p T x s p H s p s x U x s p H s p A m n i j i j i mj k k i mj i i i m k π κ κ − = ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + − + ∑ ∑ = = 1 3 2 0 1 1 , , , , , , ( 1,2) = m (21) ( ) ( ) ( ) [ ] ( ) 0 , , , , , , 1 3 2 3 1 31 0 31 = ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + − ∑ ∑ = = n i j i j i j k i i k i k i x s p H s p x s p H s p x s U A κ κ (22) ( ) , 0 1 = ∑ = n i i j iAH s p ( 1,2,3) = j (23) where,
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