13th International Conference on Fracture June 16–21, 2013, Beijing, China -8- On the other hand, in the case of mixed mode loadings, we have to substitute both eqns (15) into the stress condition (8) for crack propagation. Accordingly, we get: ( ) ( ) ( ) ( ) II II I I 21 II 1 II 2 1 I 1 I u f , , 1 −λ − −λ ϑϑ −λ − −λ ϑϑ δ + ω δ ω = σ σ s f a b f s f a b f (17) where we introduced the brittleness number s as [15]: b l b K s ch u Ic = σ = (18) The terms in square brackets in eqn (17) show a modest variation with the size, so that the terms in round brackets dominate. It means that, for large sizes and/or brittle materials, the first addend at the denominator (i.e. mode I) prevails; on the other hand, for small sizes and/or less brittle materials, the second addend at the denominator (i.e. mode II) does govern the problem. The presence of the brittleness number s in (17) highlights that the transition from mode I- to mode II-governed failure depends both on size and material brittleness. Thus we conclude that the size effect for a V-notched structure under mixed mode loading is represented by a curve with two slant asymptotes in the bilogarithmic plot (see Fig. 5c): the right one with slope (1−λI), the left one with slope (1−λII). This is a general trend, i.e. independent of the geometry and fracture criterion adopted. Figure 4. Self-similar specimens with a re-entrant corner of amplitude ω. It is worth observing that the analysis of the mode mixity leads to the same conclusion. In fact substitution of eqns (15) into eqn (13) provides: ( ) ( ) ( ) ω ω ψ= λ −λI II 2 I II , , arctan s f a b f a b (19) Equation (19) clearly shows that, except in the crack case (λI = λII = 1/2), the mode mixity does not depend only on the shape factors, but also on the brittleness number, i.e. on the structural size. In fact, whatever is the ratio between fII and fI (provided they are both different from zero), for sufficiently large sizes the mode mixity will always tend to zero (i.e. pure mode I), whereas it will tend to π/2 for vanishing sizes (i.e. pure mode II). ω b a b′ b′′ a′′ a′ σ σ σ
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