2 2 2 0, 0 0, 0, 0 y y x x y x y y x w y x x y μ ϕ ϕ ϕ ϕ ψ ψ ϕ ∂ ∂ ∂ ∂ + = + = ∂ ∂ ∂ ∂ ∂ ∂ ∂ − = = = ∂ ⎧ ⎪ ∂ ∂ ⎨ ∂ ⎪ ⎪ ⎪⎩ (4) The crack tip stress field would be equipped with the same square root singularity as that of homogeneous materials when the material properties of different composite materials at the interfaces are continuous [7,8]. Therefore, the generalized displacements , , r w θ ϕ ϕ and the stress function ψ can be expressed as follows [9] / 2 / 2 / 2 1 / 2 =1 =1 =1 =1 = ( ) , = (),= (),= () i i i i x xi y yi i i i i i i r r w w r r ϕ ϕ θ ϕ ϕ θ θ ψ ψ θ ∞ ∞ ∞ ∞ + ∑ ∑ ∑ ∑ (5) where, ( ) xiϕ θ、 ( ) yiϕ θ 、 ( ) iw θ are the eigen-functions of the generalized displacement components, ( ) iψ θ are the eigen-functions of the stress function. Substituting Eq. (5) into Eq. (4) and considering the linear independence of 3/2 r− , 1 r− , 1/2 r− ,…, /2 2 ir − ,…, the boundary conditions are 2 2 2 2 cos () 2 sin () sin () 2cos () 0 sin () 2cos () cos () 2sin () 0 sin ( ) 2cos ( ) 2 ( ) 0, 1,2,3, sin ( ) 2cos ( ) 0, 1 xi xi yi yi xi xi yi yi i i yi i i i i i i w w i i w w i i μ θ θ μ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ θ θ θ θ θ + + + + − + + = + + − = ′ + − = = ′ + = ′ ′ ′ ′ = L 2 2 2 2 2 ,2. 4 ( )sin 2 ( )sin2 ( )( cos 4sin 2 ) 0 4 ( )sin2 4 ( )cos2 ( )sin2 ( 4) 0 i i i i i i i i i i i ψ θ θ ψ θ θ ψ θ θ θ ψ θ θ ψ θ θ ψ θ θ ⎧ ⎨ ⎩ ′′ ′ − + + ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ + = ′′ ′ − + + − = ⎪ ⎪ ⎪ ⎪⎩ (6) where, ( ) ( ) θ d d = ′ , ( ) ( ) 2 2 θ d d = ″ . 4. The higher order crack-tip field Substituting Eq. (5) into Eq. (2) and Eq. (6) and utilizing the linear independence of 3/2 r− , 1 r− , 1/2 r− ,…, /2 2 ir − ,…, the system of ordinary differential equations are obtained. Solving the system, we can obtain the results
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