1.16.3.4.1 Background

It is now well accepted that fracture of ferritic steels in the temperature range where it propagates by cleav­age originates in microcracks (mostly in precipitates) ahead of macrocracks, which could be precracks in a test specimen or surface cracks in structures. To explain the extremely high cleavage fracture tough­ness (>20 MPa Vm) compared with that («1 MPa Vm) calculated from surface energy alone assuming pure cleavage of Fe matrix, it was originally postulated by Orowan41 that fracture in ferritic steels could occur due to cleavage originated in microcracks situated ahead of the main crack. Later, it was found by experiments that these microcracks originate in precipitates,42,43 and that the propagation of these microcracks into the matrix was assumed to be the controlling step in the fracture of ferritic steels. Another observation, though less well established, is that the cleavage stress at fracture on these microcracks is invariant with

44,45

temperature.

Ritchie, Knott, and Rice (RKR)46 used the HRR solutions (Hutchinson, Rice, and Rosengren) and finite element analysis (FEA) to simulate the plastic zone, and used a critical tensile stress achieved over a characteristic distance ahead of the crack as the fail­ure criterion. This distance is essentially a fitting parameter, and RKR46 used a value equal to or twice the average the grain diameter. The model successfully predicts the lower-shelf fracture tough­ness, but fails to predict the upturn near the transition temperature. Statistical models were introduced to predict the brittle-ductile transition (BDT) in steels starting with Curry and Knott,43 most notable among them were by Beremin47 and Wallin et a/.48 In both of these models, FEA solutions of crack-tip plasticity were used to obtain the stress fields ahead of the crack. In the Beremin model,47 the maximum princi­pal stress is calculated for each volume element in the plastic zone and a probability of failure is assigned. The total probability of failure is then obtained by summing over the entire plastic zone. Wallin et a/.48 extended the modeling to the transition region by considering variation of the effective surface energy (gs + gp) with temperature, where gs is the true surface energy and gp the plastic work done during propaga­tion. This eventually led to the master curve (MC) hypothesis, which predicts that the BDT of all ferritic steels follows a universal curve.48,49 Even though the MC is used to check the reliability of structures under irradiation,50 a clear understanding of the physical basis of this methodology is still lacking.51 Odette and He52 explained the MC using a micro­scopic fracture stress varying with temperature. Most experimental findings53,54 indicate that the fracture stress is not sensitive to temperature, and more care­ful experiments and simulations may be required to resolve this issue.

Discrete dislocation simulations of crack tips were successful in predicting the BDT of simple single crystalline materials.55,56 The advantage of this approach over the continuum methods is that funda­mental material properties such as dislocation veloc­ity and their mutual interactions can be treated dynamically. By these simulations, it has been found that the dislocation mobility plays a significant role in determining the transition temperature.56 However, the variation of dislocation mobility alone cannot explain the BDT behavior. An earlier attempt to model the BDT of complex materials like steels57 predicted the lower-shelf fracture toughness, substan­tiating Orowan’s postulate41 of high fracture tough­ness measured at low temperature. However, the model failed to predict the sharp increase of fracture toughness around the transition temperature region. Here, we present a discrete dislocation simulation in which crack-tip blunting is accounted for the first time. The effects of blunting are incorporated in the simulation using elastic stress fields of blunted cracks. As the crack tip is blunted due to dislocation emission, and the position of the ‘virtual sharp crack tip’ retreats from the blunted tip thereby reducing the field at the microcrack further in addition to the contribution from emitted dislocations. The critical particle is assumed to be at a fixed distance from the blunted macrocrack tip.