13th International Conference on Fracture June 16–21, 2013, Beijing, China -6- 15 30 45 60 75 90 15 30 45 60 75 90 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 Figure 2. Crack deflection vs. mode mixity for different notch opening angle (a); safety domains in the GSIFs plane for different notch opening angles: continuous line, ω = 90°; dotted line, ω = 60°; dashed line, ω = 30°; dot-dashed line, ω = 0° (b). The values of the crack orientation angle are plotted in Fig. 2a vs. the mode mixity ψ for different notch opening angle ω. On the other hand, the critical values of the GSIFs can be plotted in the (KI *, KII *) plane for a given notch opening angle ω and varying the mode mixity ψ. In this way we obtain a curve delimiting a safety region, i.e. points lying in this domain correspond to admissible stress states, whereas points lying outside correspond to failure. It is convenient to plot the results in a dimensionless form: the mode I GSIF is normalized with respect to the generalized fracture toughness KIc *, whereas the mode II GSIF is normalized with respect to KIc * × l ch λI −λII . The safety domains are plotted in Fig. 2b for different ω values and in Fig. 3 for ω = 90°. If the external loads are increased proportionally, the ratio between the GSIFs keeps constant. It means that in the (KI *, KII *) plane, the loading curve is represented by a straight line starting from the origin. Furthermore, in the dimensionless plane, the angle between the loading path and the horizontal axis is exactly ψ. According to the brittleness assumption, failure is attained suddenly when the straight line crosses the curve delimiting the safety domain, point A (Fig. 3). Apart from substitution of the SIFs with the GSIFs, this behaviour strictly resembles what occurs in the classical crack branching problem. Indeed, the crack branching problem is a particular case of the present one. However there is a substantial difference with respect to the crack kinking problem: if ω > 0°, the mode mixity ψ depends also on the material brittleness through lch (see eqn (13)) and not only on the loading, i.e. on the GSIFs ratio. For a given KII * / KI * ratio, the slope of the loading line will diminish for brittle materials (low lch), while will increase for less brittle materials (high lch): in other words, whatever is the GSIF ratio, the failure point migrates towards point B (pure mode I failure) as material brittleness increases, whereas it moves towards point C, if the brittleness decreases (within a certain range, otherwise, as explained in the following section, the asymptotic approach does not hold any more). As clearly shown by eqn (13), the effect of the material on the mode mixity increases as the notch opening increase. In fact, for larger ω, the gap between the Williams eigenvalues λII and λI grows. On the other hand, for a vanishing notch opening angle, both λII and λI tend to 1/2 and the effect of the material vanishes. This material dependence is valid also for the orientation of the V-notch emanated crack, see Fig. 2a: as brittleness increases, lch diminishes, ψ diminishes and the crack deflection ϑc tends to zero, i.e. the V-notch emanated crack tends to propagate in mode I along the bisector. On the other hand, for less brittle material, the crack deflection is higher under the same GSIFs ratio. Once more, it is worth emphasizing the fundamental difference with respect to the crack case, where the angle of the crack kinking depends only on the (dimensionless) SIFs ratio, i.e. is the same independently of the material. ϑc [°] ψ [°] KIIf * × l ch λI−λII / KIc * KIf * / KIc * ω = 90° ω = 60° ω = 30° ω = 0° safe region unsafe region (a) (b)
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