ANALYTICAL MODEL In the FE model the effect of ligament diameter and stiffness and applied stress intensity were not considered. An analytical model is now developed to elucidate these effects upon crack defection for a bridged versus an unbridged crack. A phase angle is used to describe the extent of mode mixity: ÷÷ ø ö çç è æ y= - I II 1 K K tan (4) Considering Fig. 4, when a bridged crack is loaded in mixed mode then the bridge causes crack closure stresses both perpendicular, py, and parallel, px, to the crack plane. These lead to a resistance energy in both mode I and II directions according to: ò = u* 0 x II R 2 p du (5) and ò = v* 0 y I R 2 p dv (6) where u* and v* are crack face displacements at the position of the crack bridge furthest from the crack tip which was taken to be 20 times the fibre diameter in this example. This leads to crack-tip shielding, Ks, and Eq(3) becomes, for a bridged crack: ( ) ( ) 2 cos K K sin 2 3 2 K K cos 2r 1 II s II 2 I s I br q úû ù êë é q - - q - s = qq (7) where K E Rc s = separately for modes I and II. To obtain the crack closure function, p, it is assumed that the bridging fibre acts as an elastic beam. There is assumed to be fibre delamination of length, L, equivalent to twice the fibre diameter, d, prior to loading. In Mode I the fibre acts as a beam loaded in tension while in mode II it is assumed to be fixed at the crack faces leading to a bending moment. Bending moments due to py in mode II are assumed to be negligible for simplification of calculations. The beam is assumed to be rectangular with width d and unit depth. One then obtains: v L E d p f y × = (8) and u v d 10 E p 3 f x ÷ ø ö ç è æ × = (9) where v and u are defined in equations (1) & (2). Figure 6 shows the calculated extent to which crack bridging reduces the deflection angle of a crack under mixed-mode loading. It can be seen that increasing the fibre diameter, and consequently the bridging length, and fibre stiffness reduces the deflection angle. It was also found (not shown) that, as applied stress intensity factor ( 2 II 2 I *K K K = + ) is increased, the deflection angle decreases. It should be noted that this does not represent an equilibrium solution because for a bridged crack (1) and (2) are functions of (K-Ks) and not just K as taken here. Nevertheless, the relative effects of bridging upon br qq s should remain correct. DISCUSSION Crack deflection is a result of mixed-mode loading which is inherent to a crack propagating in a material containing a stiffness gradient. Numerical and analytical calculations show that decreasing crack deflection with crack extension, which was observed in the experiment, can be explained by the effects of the observed fibre bridging.
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