13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- 2. Stress intensity factors for a V-notch emanated crack Let us consider a re-entrant corner in an infinite homogeneous elastic medium with a polar coordinate system (r,ϑ) centred at the V-notch tip (see Fig.1a). After Williams, the asymptotical stress field is given by: ( ) ( ) ( ) ( ) ϑω π ϑω + π σ = −λ −λ , f r K , f r K * * II rr 1 II I rr 1 I rr II I 2 2 (1a) ( ) ( ) ( ) ( ) ϑω π ϑω + π σ = −λ ϑϑ −λ ϑϑ ϑϑ , f r K , f r K * * II 1 II I 1 I II I 2 2 (1b) ( ) ( ) ( ) ( ) ϑω π ϑω + π τ = −λ ϑ −λ ϑ ϑ , f r K , f r K * * II r 1 II I r 1 I r II I 2 2 (1c) where KI * and KII * are the GSIFs in mode I (symmetrical) and mode II (anti-symmetrical) loading conditions respectively, λI and λII are the well-known Williams’ eigenvalues and the functions fij are the angular shape functions (i.e. the eigenvectors). Both eigenvalues and eigenvectors depend on the notch opening angle ω. Note that the definition of the GSIFs is somewhat arbitrary, depending on the choice of the normalization factor, here taken equal to (2π)1−λi as in [2] but equal to 1 or to √2π in other papers (e.g. [7] and [3], respectively). As we shall see later, the advantage of such a choice is that the critical value of the mode I GSIF continuously varies from the material tensile strength to the material fracture toughness as the re-entrant corner amplitude diminishes from 180° (flat edge) to 0° (cracked plate). In order to apply the FFM criterion, we need to evaluate the energy necessary for the abrupt appearance of a finite length crack at the notch tip. This quantity can be easily computed if the SIFs KI and KII of a crack at the notch vertex are known. To this aim, we begin noticing that, if the crack occurs within the GSIFs dominated stress field, the SIFs depend only on the GSIFs, crack direction ϑ, crack length a and notch opening angle ω (see Fig.1b). A straightforward application of the Π-theorem (as well as the principle of effect superposition) shows that this dependency must take the following form [9]: ( ) ( ) 1 2 II 12 1 2 I I 11 II I λ − λ − +µ ϑω =µ ϑω , K a , K a K * * (2a) ( ) ( ) 1 2 II 22 1 2 I 21 II II I λ − λ − +µ ϑω =µ ϑω , K a , K a K * * (2b) In case of pure mode I loaded V-notches, the emanated crack grows along the notch bisector (ϑ = 0); hence KII is zero and KI simplifies into [10]: ( ) 1 2 I I 11 Iλ − =µ ω K a K * (3) While the dimensionless µij parameters can be found tabulated with great accuracy for a crack, i.e. for ω = 0° [11], their values for a generic notch opening angle ω are not available. Nevertheless, they can be obtained by exploiting the results provided in [12], where the SIFs for a pair of forces per unit thickness (either normal or tangential) acting on the faces of a V-notch emanated crack are given. Beghini et al. [12] evaluated such SIFs for several ω and ϑ values by proper finite element computations and provided accurate analytical expressions of the SIFs. From such expressions and by the principle of effect superposition the coefficients µij can be obtained analytically [13].
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