13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- external load increases, the upper bound increases and the lower bound decreases till a load value is met (i.e. the failure load) for which both conditions are strictly fulfilled. Therefore, we conclude stating that the system (4) reverts to a system of two equations in two unknowns: the crack advancement ∆ and the corresponding (minimum) failure load, represented by the values KIf * and KIIf * of the GSIFs in critical conditions, implicitly embedded in the functions σϑϑ and G. Exploiting the well-known Irwin’s relationship in plane strain and mixed mode, the system (4) becomes: =σ ∆ σ = ∆ + ∫ ∫ ∆ ϑϑ ∆ u 0 2 Ic 0 2 II 2 I ( ) d ( ) ( ) d] [ r r K a K a a K (5) It is worth observing that the failure load estimate provided by the system (5) does depend on the crack propagation direction ϑ (see Fig.1b). Among all the possible directions, the actual one will be the direction ϑc providing the minimum failure load. Upon substitution of the SIFs eqns (2) into the first equation of the system (5) and integrating between 0 and ∆, we get: = ∆ +µ ∆ +µ ∆ µ ∆ λ λ +λ λ 2 Ic 2 II 2 I II 22 12 2 I 2 11 ( ) ( ) II I II I K K K K K * * * * , (6) where, for the sake of simplicity, we have introduced the angular functions: I 2 21 2 11 11 2λ µ +µ µ = , II I 21 22 11 12 12 2 λ +λ µ µ +µ µ µ = , II 2 22 2 12 22 2λ µ +µ µ = (7) Equation (6) highlights that the variation in the elastic energy is a quadratic function of the GSIFs. Equation (6) can be found also in Yosibash et al. [7], where it was derived in a different way, i.e. by directly computing coefficients of the quadratic form by suitable path independent integrals of the stress and displacement fields before and after the appearance of the finite crack advancement. Upon substitution of the stress field represented by eqn (1) into the second equation of the system (5) and integrating between 0 and ∆, we get: ∆ + ∆ =σ ∆ λ ϑϑ λ ϑϑ u II II I I II I * * f K f K (8) where, for the sake of simplicity, we have introduced the angular functions: ( ) I 1 I I I 2 −λ ϑϑ ϑϑ λ π = f f , ( ) II 1 II II II 2 −λ ϑϑ ϑϑ λ π = f f (9) 4. Failure load, crack deflection and mode mixity Under pure mode I loading condition, for symmetry reason the crack propagates along the notch bisector, i.e. ϑc = ϑIc = 0. Upon substitution of eqns (6) and (8), limited to the mode I contributions, into the system (5), we get:
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