13th International Conference on Fracture June 16–21, 2013, Beijing, China -7- ( ) ( ) 2 2 2 2 2 i 1 , 2 a m m m ζ ω ζ ρ ζ ρ ζ ⎡ ⎤ ′ − ⎢ ⎥ ⎡ ⎤ =− − − ⎣ ⎦ ⎢ ⎥ + ⎣ ⎦ (35) It is found from Eq. (33) that the stress fields can be obtained from the relations between the stress and stress function. The stress intensity factor at the crack tip is a very important physical quantity in fracture mechanics, which can reflect the stress intensity around the crack tip. The vector of the stress intensity factors can be defined as ( ) 1 1 ,1 , lim 2 , T h III III z z K K z z π ⊥ → ⎡ ⎤ = = − ∑ ⎣ ⎦ k (36) where h III K and ⊥ III K denote the stress intensity factors of the phonon field and the phason field, respectively. Substituting Eq. (17) into Eq. (36) gives ( ) ( ) 1 1 0 , 2lim 2 T h III III z z K K z z z π ⊥ → ⎡ ⎤ = = − ⎣ ⎦ BF k (37) in which the condition that ( ) 0 z BF is imaginary along the 2x axis is used. In the ζ plane, Eq. (37) becomes ( ) ( ) ( ) ( ) 0 i 2 2 lim i , ζ ζ π ω ζ ω ω ζ →− = − − ′ BF k (38) where i ζ=− is the corresponding point of the crack tip z a= . It is obvious from Eqs. (33) and (34) that one finds ( ) i lim ζ ω ζ →− ′ exists and ( ) 0 i lim 0 ζ ζ →− ≠ BF . Thus, by the L’Hospital rule, Eq.(38) results in ( ) ( ) 0 i 2 2 lim , ζ ζ π ω ζ →− = ′′ BF k (39) ( ) ω ζ ′′ can be obtained by Eq.(34) ( ) 2 3 3 3 2 2 2 2 6 ( ) i , 2 1 a m m m ζ ζ ω ζ ρ ζ ρ ζ ⎡ ⎤ − ⎢ ⎥ ′′ = − ⎢ ⎥ + ⎣ ⎦ (40) Inserting Eqs. (33) and (40) into Eq. (39), the analytic expressions of the stress intensity factors at the crack tip ( ) ,0,0 a for the anti-plane shear problem are derived as follows 32 32 , aK H σ π ∞ ∞ ⎡ ⎤ = ⎢ ⎥ ⎢⎣ ⎥⎦ k (41) where
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