13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- Figure 3. Indentation configuration, integral path and . Eq. (7) shows that the stress state near the indenter corner may vary in the form ( ) ( ) ⎪ ⎩ ⎪ ⎨ ⎧ = = = − − 0 , 0 2 , 0 1 1 1 x r x a r ij λ λ σ as 0→r . (10) For special cases either with 05. =µ or 0= f , Eq. (9) leads to 0.5 =λ , showing the same order of stress singularity as that for a sharp crack tip. For 05. =µ , the substrate becomes incompressible. The asymptotic stress boundary conditions of the substrate in the contact area next to the left and right corners then become ar P 2 | 22 0 π σ θ −= = (11) and ar fP 2 | 21 0 π σ θ = = (12) 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. 3, the stresses at the left corner can be found as follows due to the normal load: ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + −= ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ − 2 cos 2 sin 2 cos 2 1 sin 2 cos 2 2 3 2 θ θ θ θ θ π σ σ σ θ θθ r KI ind I r I I rr . (13) This expression indicates that the stress state for indentation is a “negative” Mode I singular stress field for cracked solids, where a P KI ind π = − , (14) 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. Only difference between tensile mode-I stress field and indentation stress field is sign “-” in their equations. 0→s 0→Δ p(x1) q(x1 ) Q P x1 x2 θ r α Δ s 2a θ r
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