13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- Figure 1. Polar reference system at the tip of a V-notch (a); V-notch emanated crack within the GSIFs dominated stress field region (b). 3. Coupled criterion It is well known that both strength criteria and LEFM fail in predicting the failure load causing fracture propagation from a V-notch. In fact, the stress field given by eqn (1) is singular and strength criteria provide a vanishing failure load. On the other hand, the SIFs provided by eqn (2) vanish as the crack length a tends to zero and, consequently, LEFM provides an infinite failure load. These shortcomings can be overcome by resorting to Finite Fracture Mechanics [4,14], which couples the stress and energy approaches. Following the FFM approach proposed by Cornetti et al. [14], a crack propagates by a finite crack extension ∆ if the following two inequalities are satisfied: ≥σ ∆ σ ≥ ∆ ∫ ∫ ∆ ϑϑ ∆ u 0 c 0 ( ) d ( ) d r r a a G G (4) where σu is the material tensile strength and Gc is the fracture energy, related to the material fracture toughness by the well-known relation Gc = KIc 2 / E′, where E′ = E / (1−ν2), E being the Young’s modulus and ν the Poisson’s coefficient. The FFM criterion (4) can be regarded as a coupled Griffith-Rankine non-local failure criterion: the former inequality is an energy balance, whereas the latter is an (average) stress requirement for crack to propagate. It means that fracture is energy driven, but a sufficiently high stress field must act at the crack tip to trigger crack propagation. It is worth observing that, in the present case (which is the usual one, i.e. a positive geometry), the strain energy release rate function G(a) is monotonically increasing since the SIFs increase along with the crack length (see eqns (2)) while the stress σϑϑ(r) is monotonically decreasing with the distance r (see eqns (1)) from the notch tip (as far as both the modes provide a stress singularity, i.e. for a notch opening angle less than about 102.6°). This means that the lowest failure load (i.e. the actual one) is attained when the two inequalities are substituted by the two corresponding equations. In fact the first inequality is satisfied for crack steps larger than a threshold value, thus providing a lower bound for the set of admissible ∆-values; on the contrary, the second inequality is satisfied for crack advancements smaller than a certain value, thus providing an upper bound. For low load values, the upper bound is smaller than the lower bound and, consequently, the set of admissible ∆-values is empty. As the ω r ϑ ϑ a KI *, KII * dominated zone ω (a) (b) KI , KII γ
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