13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- ar fN 2 0 21 π σ θ = = | (6) for the two cases of 05. =μ and 0= f . 3.2. Singular stress fields due to the normal and tangential loads The singular stress field at the sharp edge of the contact between a rigid flat-ended indenter and substrate is known from the asymptotic contact analyses of and Nadai [4]. Using the polar coordinates (r, θ), Fig.2, the stresses at the left corner can be found as follows due to the normal load: ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + = − ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ − 2 2 2 2 1 2 2 2 3 2 θ θ θ θ θ π σ σ σ θ θθ cos sin cos sin cos r KI ind I r I I rr . (7) This expression indicates that the stress state for indentation is a “negative” Mode-I singular stress field for cracked solids, where a N KI ind π = − , (8) which defines actually an indentation stress intensity factor. The familiar Mode-I singular stress field is obtained by removing the negative sign and changing KI-ind into KI for cracked solids with mode-I loads. Only difference between tensile mode-I stress field and indentation stress field is sign “-” in their equations. Nadai [4] gave also the asymptotic stress field due to the tangential load as ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ − 2 1 3 2 2 2 3 2 1 3 2 2 2 2 2 θ θ θ θ θ θ π σ σ σ θ θθ sin cos cos sin sin sin r KII ind II r II II rr , (9) where I ind II ind K fK − − = . (10) Actually, the Eq.(9) 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, II ij I ij ij σ σ σ = + , 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.(7)-(10) represent an important advance by defining the indentation stress intensity factors, I ind K − and II ind K − , 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. It should be pointed out that for finite boundary contact problems, Eqs.(7), (9) and (10) are still effective, for which case the I ind K − should be
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