13th International Conference on Fracture June 16–21, 2013, Beijing, China -7- 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 Figure 3. Resistance domain in the GSIFs plane (ω = 90°): points lying beneath the thick curve correspond to admissible stress states and vice-versa. In Fig. 2b we plotted the safety domains for different notch opening angles ω. It is evident that all the curves are similar. Of course, this fact does not imply that the failure load does not vary with ω, since the physical dimensions as well as the shape functions defining the GSIFs vary along with ω. It simply shows that the transition from mode I to mode II fracture is approximately the same for all the notch amplitudes. 5. Size effect and mode mixity The introduction of a physical length against which to scale the notch tip GSIFs enables further aspects of the solution to be drawn out, such as the influence of a V-notch on the so-called size effect. Hence let us consider a set of self-similar geometries as the ones drawn in Fig. 4. Dimensional analysis allows us to write directly: ( ) I 1 I I , −λ = ω σ f a b b K * , ( ) II 1 II II , −λ = ω σ f a b b K * (15) where σ is the nominal stress, b is a characteristic size of the structure and fI, fII are shape factors depending on the geometry, here synthetically defined by the notch opening angle ω and relative notch depth a/b. Now let us focus our attention to the size effect on the nominal stress at failure σf. If only either the mode I or the mode II GSIF is different from zero, failure will occur whenever the corresponding GSIF reaches KIc * or KIIc *, respectively (the latter value being given by the intercept with the vertical axis in Fig. 3). Hence, according to eqn (15), the logarithm of the nominal stress at failure is given by [1]: ( ) ( ) b f a b K * 1 ln , ln ln I I Ic f − −λ ω σ = or ( ) ( ) b f a b K * 1 ln , ln ln II II IIc f − −λ ω σ = (16) It means that in a bi-logarithmic plot the strength vs. size curve is a straight line with (negative) slope equal to either (1−λI) or (1−λII), see Figs. 5a,b. Since λII > λI, the size effect is stronger under mode I than under mode II loadings. KIf * / KIc * KIIf * × l ch λI−λII / KIc * loading path ψ A O B C higher lch lower lch
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