13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- Due to the complexity of the cleavage process, a statistical model is needed to understand the effect of the different steps on the temperature dependence. 2. Statistical modeling of cleavage fracture initiation The basis of a general statistical model is the following. It is assumed that the material in front of the crack contains a distribution of possible cleavage fracture initiation sites i.e. cleavage initiators. The cumulative probability distribution for a single initiator being critical can be expressed as Pr{I} and it is a complex function of the initiator size distribution, stress, strain, grain size, temperature, stress and strain rate etc. The shape and origin of the initiator distribution is not important in the case of a "sharp" crack. The only necessary assumption is that no global interaction between initiators exists. This means that interactions on a local scale are permitted. Thus a cluster of cleavage initiations may be required for macroscopic initiation. As long as the cluster is local in nature, it can be interpreted as being a single initiator. All the above factors can be implemented into the initiator distribution and they are not significant as long as no attempt is made to determine the shape and specific nature of the distribution. A quantitative description of the initiator distribution is also hindered by the statistical variation in stress and strain between grains and laths. Further, also the local orientations would need to be known. This is one reason why all present cleavage fracture models have had difficulties in connecting the models to real microstructural variables. If a particle (or grain boundary) fails, but the broken particle is not capable of initiating cleavage fracture in the matrix, the particle sized microcrack will blunt and a void will form. Such a void is not considered able to initiate cleavage fracture. Thus, the cleavage fracture initiator distribution is affected by the void formation, leading to a conditional probability for cleavage initiation (Pr{I/O}).The condition being that the cleavage initiator must not have become a void. The cleavage fracture process contains also another conditional event, i.e. that of propagation. An initiated cleavage crack must be able to propagate through the matrix in order to produce failure. Thus the conditional probability will be that of propagation after initiation (Pr{P/I}). The cleavage fracture initiation process can be expressed in the form of a probability tree (Fig. 4). STRESS APPLIED TO MATERIAL ELEMENT NO INITIATION VOID INITIATION CLEAVAGE INITIATION PROPAGATION ARREST Pr{O} Pr{V/O} Pr{I/O} Pr{A/I} Pr{P/I} FAILURE Figure 4. Probability tree for cleavage fracture [1]. 2.1. Probability of initiation For a "sharp" crack in small scale yielding the stresses and strains are described by the HRR field. One property of the HRR field is that the stress and strain distribution is self-similar and another that the stresses and strains have an angular dependence. The term “small scale yielding” is in this derivation used to describe the loading situation where the self-similarity of the stress field remains unaffected by loading. For such a situation, it has been shown, by weakest link statistics, that the probability of initiation alone can be expressed in the form of Eq. (4) [1].
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