13th International Conference on Fracture June 16–21, 2013, Beijing, China -7- form a closed integral loop. The ( ) 0 1 →s J | denotes the driving force of boundary cracking in direction x1 when the limit taken exists, or the energy release rate with unit boundary movement s in direction x1; ( ) 0 2 →s J | denotes the driving force in direction x2, or the energy release rate with unit boundary movement s in direction x2. For a homogenous and isotropic substrate Griffith’s criterion [15] states that the crack will extend when the critical value max G is reached [15,16] c G G= max . (27) For a standard cracked specimen subjected to Mode-I loading, 1J G = max and cG can be calibrated as ( ) E G J KIC c IC / 2 2 1 μ = = − , (28) where KIC is the Mode-I fracture toughness. However, in the present case of surface crack initiation, a mixed-mode condition is expected as the singular stress field is generated by the sliding contact with the rigid flat-ended indenter. 7.2 Critical cracking angle and critical load. Firstly, ( ) 0 1 → ob s J can be found. Let ob s s = in Eq.(25) as shown in Figs.2 and 6 adjacent to the right corner of the indenter and s be within the Kind-dominant region. Because 0= iT and 0 1 = n on the integration path ob s , the energy-based driving force for boundary cracking in x1 direction in this case can be found as ( ) 0 lim ∫ = = → → ob ob ob s s s wn ds J 1 0 0 1 . (29) Fig.6. Integration path adjacent to punch corner within the Kind-dominant region for 2J . Next step is to calculate 0 2 → ob s J | . As shown in Figs.2 and 6, let ob s s = , 1 odcb in s S = in Eq.(26) and take the limit 0→ ob s , 0 →R/δ and 0 →R , which means that the 1 odcb in s S = is within the Kind-dominant region and 0 1 → odcb S . Then, the energy-based driving force for boundary cracking in x2 direction becomes [10,16] ( ) ( ) ( ) ∫ ∫ + − ′ − − =− = → → → → 2 1 2 0 / 0 2 2 0 0 2 lim Im 41 | lim odcb ob ob ob S R R s s s nds z E wn ds J ΨΦ ΦΦ ΦΦ Φ μ δ , (30) where ( ) z K K i z II I π Φ 2 2 − = , ( ) z K K i z II I π Ψ 4 2 3+ = , (31) in which I ind I K K − = − and II ind II K K − = for indentation singular stress fields. From Eqs.(30) and (31), it can be readily obtained that x1 o Substrate b d c n S1 δ R x'2 Indenter
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