13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- Nadai [4] gave also the asymptotic stress field due to the tangential load as ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ − 2 1 3sin 2 cos 2 cos 2 3sin 2 1 3sin 2 sin 2 2 2 2 θ θ θ θ θ θ π σ σ σ θ θθ r KII ind II r II II rr , (15) where a fP K fKI ind II ind π = = − − . (16) Actually, Eq. (15) is identical to the classical Mode II singular stress fields when KII-ind = KII. 3.3. Characters of the stress fields It is clear from the above discussion that the asymptotic stress field, , induced by the sliding contact is a typical mixed-mode I-II singular stress field for incompressible substrates and friction free. This singular stress field is responsible for surface crack initiation on the crack-free surface of the substrate at the contact edge. This finding is significant as it shows that singularity and distribution of the stress field induced by surface contact of a flat-ended indenter are identical to those of a mixed-mode crack. As a result, the concepts of stress intensity factor and fracture toughness can now be introduced unambiguously into contact mechanics and associated contact damage. Therefore, Eqs. (14-17) represent an important advance by defining the indentation stress intensity factors, and , and the Kind-dominant region at the contact edge. In other words, the fracture mechanics theory, such as the Griffith’s criterion, is applicable in the case of the boundary fracture induced by the sliding contact. 4. Calculation methods on indentation stress intensity factor 4.1. Application of J1-integral in the indentation fracture For a closed integration path abcdefa s as shown in Fig.4, following J1-integral can be gotten. ( ) 0 1 1 1 = − = ∫ abcdefa s i i wn Tu ds J , . (17) If the path abcdefa s is divided into afe de bcd ab abcdefa s s s s s = + + + , because of 0 1 = n on surface of the substrate, 0= iT on the ab s , 0 1 = T and 0 21 = , u on the de s , we have ( ) 0 1 1 1 = − = ∫ ab s i i wn Tu ds J , (18) and ( ) 0 1 1 1 = − = ∫ de s i i wn Tu ds J , . (19) Then substituting Eqs. (18) and (19) into Eq. (17), it follows that ( ) ( ) ( ) 0 1 1 1 1 1 1 1 = − + − − = − = ∫ ∫ ∫ bcd afe abcdefa s i i s i i s i i wn Tu ds wn Tu ds wn Tu ds J , , , . (20) It can then be rearranged to give ( ) ( ) ∫ ∫ − = − = bcd afe s i i s i i wn Tu ds wn Tu ds J 1 1 1 1 1 , , , (21) which means that along any two paths, afe s and bcd s , starting from the any point on the left free boundary to any one on the contact boundary, the J1-integrals are identical. It shows theoretically II ij I ij ij σ σ σ = + -ind IK II ind K −
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