13th International Conference on Fracture June 16–21, 2013, Beijing, China -6- that this integral is path independent. If the integration path afe s is half of a circle and within the Kind-dominant region, it is not difficult to get ( ) 2 2 1 1 1 2 1 I s i i K E wn Tu ds J afe µ − = − = ∫ , (plane strain). (22) Then, Eq. (21) becomes ( ) ∫ − = − = bcd s i i I wn Tu ds K E J 1 1 2 2 1 2 1 , µ (23) This equation is a key formula to construct a method to calculate the ISIFs induced by the indentation. Additionally, this method can be applied to the contact problems with the finite and infinite boundaries. Figure 4. Integration path for Mode-I indentation. 4.2. Application of J1-integral in the indentation fracture for layered substrate For the layered substrate, two closed integration paths as shown in Fig. 5 will be considered in present work. One is the pant mn nedcbam s s s = + 1 and the other mfn nm s s s ʹ′ ʹ′ ʹ′ ʹ′ = + 2 . Additionally, as 0 1 = n , iT and ,1 iu are continuous on the paths mn s and nm s ʹ′ ʹ′ , from conservation law, we have the following contour integrals. ( ) 0 ,1 1 ,1 1 = − + = ∫ ∫ nedcbam mn s i i s i i wn Tu ds J Tu ds for the closed path 1s , (24) ( ) 0 ,1 1 ,1 1 = − + = ∫ ∫ ʹ′ ʹ′ ʹ′ ʹ′ mfn mn s i i s i i wn Tu ds J Tu ds for the closed 2s . (25) and ∫ ∫ −= = ʹ′ ʹ′ mn mn s i i s i i J Tu ds Tu ds ,1 ,1 1 (26) According to Eqs. (18), (19), (24)-(26), it can be found that ( ) ( ) ∫ ∫ − = − = afe bcd s i i s i i wn Tu ds wn Tu ds J ,1 1 ,1 1 1 , (27) which indicates that the integral is path independent for composite substrate similar to the Eq. (21). Iindenter P(x1) P x1 x2 2l Substrate b a f e d c n n
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