13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- x2-direction or the crack mouth widening energy release rate for cracked solids [11-13]. The axial strain + − 2 2 2 2, , ~ , ~ u u have been found[11-13] EA N u = − − 2 2, ~ (19) and ( ) ( ) ( ) ( )⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − − − − − =− − − − =−∫ − l t l t l t l t l t EA N l t EA Nd u 21 2 1 1 1 2 1 1 1 2 1 0 2 2 2 π ξ ξ arctan ] [ ~ , . (20) Because the free action of the substrate surface and ab s and ef s is out of the K-dominant regions, the integral in left-hand side of Eq. (18) is a small quantity, which can be neglected. Substituting Eqs.(19) and (20) into Eq.(18), it will yield ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = − t l f t N KI ind π , (21) where the normalized stress intensity factor can be found as ( ) ( ) ( ) 1 2 2 2 1 21 2 1 1 1 2 1 1 2 / arctan ⎪⎭ ⎪ ⎬ ⎫ ⎪⎩ ⎪ ⎨ ⎧ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ − − − − − − − − ⎟= ⎠ ⎞ ⎜ ⎝ ⎛ l t l t l t l t l t t l f π μ π . (22) Eq.(22) varying with perimeter l/t is given in Fig.4. Fig.3. Contours of integration for the Cell 1 and Cell 2. Fig.4. the scheme of the normalized stress intensity factor. Cell 1 x2 x1 N=P/n b c d e f g h i j a o n Cell 2 x2 x1 b c d e f g h i j a o n N=P/n 0.0 0.2 0.4 0.6 0.8 1.0 0.0 2.0 4.0 6.0 8.0 10.0 12.0 Normalized indenter width l/t Eq.(23) Eq.(22) Normalized stress intensity factor KI(πt) 1/2/N
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