13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- { } 4 I 0 0i B K Pf Pr I/O 1 exp B K ⎧ ⎫ ⎛ ⎞ ⎪ ⎪ = = − − ⋅ ⎜ ⎟ ⎨ ⎬ ⎝ ⎠ ⎪ ⎪ ⎩ ⎭ , (4) B0 is a freely definable normalization crack front length and K0i corresponds to a cumulative initiation probability of 63.2%. 2.2. Conditional cleavage propagation Eq. (4) would imply that an infinitesimal KI value might lead to a finite failure probability. This is not true in reality. For very small KI values the stress gradient becomes so steep that even if cleavage fracture can initiate, it cannot propagate into the surrounding and other adjacent grains, thus only causing a zone of microcracks in front of the main crack. If propagation in relation to initiation is very difficult, a stable type of fracture may evolve. This is an effect often seen with ceramics. The need for propagation leads to a conditional crack propagation criterion, which causes a lower limiting Kmin value below which cleavage fracture is impossible. For structural steels in the lower shelf temperature range, the fracture toughness is likely to be the controlled by the inability of propagation (Fig. 3). The question regarding propagation alters the above pure weakest link type argument somewhat. It means that initiation is not the only requirement for cleavage fracture, but additionally a conditional propagation requirement must be fulfilled. Thus one must examine the probability of cleavage initiation during a very small load increment, assuming that no initiation has occurred before. Such a probability constitutes a conditional event and the resulting function is known as the hazard function. When the hazard function for initiation is multiplied by the conditional probability of propagation (Pr{P/I}), the cumulative failure probability including propagation becomes thus as Eq. (5) [1]. { } I min K 3 I f I 4 0 0i K B 4 K P 1 exp Pr P/I dK B K ⋅ = − − ⋅ ⋅ ⋅ ∫ , (5) There are two requirements for Pr{P/I}. It must be an increasing function starting from Kmin and for large KI values it must saturate towards a constant probability corresponding to a uniform stress denoted by an infinite stress intensity factor K∞. One possible form of Pr{P/I} is given by Eq. (6). Other possible forms have also been discussed in e.g. [1]. { } ( ) 3 min I K Pr P/ I P K 1 K ∞ ⎛ ⎞ = ⋅ ⎜ − ⎟ ⎝ ⎠ , (6) All the possible equations are functions growing from 0 to P(K∞), where P(K∞) is a number smaller than 1. The constant P(K∞) reflects the finite probability of crack propagation even in a uniform stress field, being due to a possible miss-orientation between the micro-crack and the possible cleavage crack planes and the need to cross a grain boundary. P(K∞) will increase with increasing stress and decrease with increasing temperature. Insertion of Eq. (6) into Eq. (5) leads to the equation for the cumulative cleavage fracture probability including the conditional propagation criteria, (Eq. 7).
RkJQdWJsaXNoZXIy MjM0NDE=