PSI - Issue 13

Michihiro Kunigita et al. / Procedia Structural Integrity 13 (2018) 198–203 Kunigita / Structural Integrity Procedia 00 (2018) 000 – 000

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2. Theory 2.1. Weakest link mechanism

Figure 1 shows a schematic of the present model. Active zone near a notch-tip is defined as a region in which cleavage fracture is possibly initiated due to concentrated stress. The active zone is regarded as an aggregate of volume elements. The model is based on the weakest-link mechanism; cleavage fracture of the specimen is assumed to take place if even one of the volume elements initiates a cleavage crack (Beremin, 1983). Each volume element is assumed to possess a probability distribution of local fracture stress, ( ) . Let be maximum principal stress exerted on the i -th volume element at , probability of the volume element fracture during time increment, ∼ + Δ is expressed as, = ( ) . Using the weakest-link mechanism, probability of specimen fracture during the time increment is expressed as: ( ) = 1 − ∏ (1 − ( )) =1 (1) where is total number of the volume elements in the active zone. By considering the event that the specimen has not been fractured up to this time step, probability that the specimen is fractured at time is expressed as, ℎ ( ) = ( ) ∏ (1 − ( )) = −1 1 (2) where is number of time steps up to .

Fig.1 Schematic of the model.

Cleavage fracture initiation from bainite containing brittle microphase can be modelled as multiple steps (Martin Meizoso et al, 1994). Crack nucleation from MA particles, stage-I, and propagation of the initiated crack into ferrite matrix, stage-II, are considered. Stage-I: Strain is recognized as a controlling factor in the crack nucleation from MA particles (Kawata et al, 2016). we assume that the probability of crack nucleation from MA particles is expressed as, = (3) where is plastic strain exerted on a volume element and is a constant. If plastic strain increment is exerted during time step ∼ + Δ on the volume element which contains 0 particles of MA, number of the nucleated cracks in the volume element is expressed as, = 0 = 0 (4) Then, probability density of maximum size of the nucleated cracks can be expressed as, ( ) = ( ) −1 ( ) (5) where ( ) and ( ) are cumulative distribution function and probability density function of MA particle thickness, t , respectively, see Fig.2.