13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- where ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎫ − = ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ − + = + + = = ( ) ( )] 2 3 ( ) [ ( ) ( ) ( )] [5 ( ) 2 1 ( ) ( ) ( ) ( )] 2 3 ( ) 2 ( ) [4 ( ) ( ) 2.0 3 ' 3 4 2 3 ' 2 3 2 ' 2 1 r a f r a F r a F r a r a r a f r a F r a F r a F r a r a r a f r a F r a f r a f r a r a r a r r r r r β β β β (8) where ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎫ − ⎥⎦ ⎤ ⎢⎣ ⎡ ⋅ − ⋅ − = − ⋅ ⎥ − ⎦ ⎤ ⎢⎣ ⎡ ⋅ − = = 15 ( ) 3 ( ) ( ) ( ) ( ) ( ) 3 2 ( ) ( ) 15 ( ) 3 ( ) ( ) ( ) ( ) ( ) 15 2 ( ) ( ) ( ) 4 ( ) 3 2 1 1 1 2 3 3 2 1 3 1 1 2 1 r a E r a E r a F r a E r a r a F r a E r a F r a E r a E r a F r a E r a F r a E r a r a F r a f r a F r πφ πφ (9) with s r s s f r a a r a ( ( ) )d 1 ( ) 2 0 2 ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⋅ = ∫ φ (10) , 1, 2, 3 (1 2 2 ) ! ( ) 2 ( ) 7 0 0 1 = + + =∑ ∏= = + j j k r a m S r a E m m k m m m j (11) where r is the radius of the semi-circular notch; rf and mS are defined in Eq. (5) and Eq. (6). 3. Crack surface displacements for a radial crack emanating from a semi-circular notch in a semi-infinite plate For a radial crack emanating from a semi-circular notch in a semi-infinite plate, uniform remote tension is chosen as the reference load case, and the corresponding non-dimensional stress intensity
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