From numerous careful experimental studies con­ducted on the BDT behavior of steels, it is now established that precracks (macrocracks) blunt sub­stantially before the fracture of the specimen occurs at the transition region. However, the examination of the fracture surface reveals that cracks propagate predominantly by cleavage.44 Several cracked brit­tle particles are found to be present in the broken samples,45,53 and the measured microscopic fracture stress (at the microcracks) is found to be a few orders of magnitude higher than that of the pure Griffith value.53,54 All these observations are considered in our model as follows:

1. We implemented the blunting of macrocracks by using the elastic crack-tip stress field for blunted cracks. As dislocations are emitted, the crack blunts and the radius of curvature increases. The notional crack tip, which is taken as reference for calculation ofimage stresses, retreats away from the blunted tip.

2. A microcrack is placed in the field of a macrocrack and the failure criterion used in the calculation of the cleavage crack propagation from this microcrack.

3. We consider the emission of dislocations and subsequent shielding from the microcrack tip (a detailed study of this and the observed constancy for microscopic fracture stress is reported in an earlier study ).

The geometry of the model used for simulation is shown in Figure 13. A semi-infinite crack (macro­crack) with a finite microcrack situated ahead of it on its crack plane is loaded starting from K = 0. Disloca­tion sources are assumed to exist at a distance x0 from the tip, and are situated on the slip planes passing through the crack tip. During loading, dislocations are emitted from source positions (x0) when the resolved shear stress reaches a value of 2 т, The resolved shear stresses are obtained using expressions based on deri­vations for a semi-infinite crack58 and a finite crack59 for the respective cases. The emitted dislocations move along the slip plane away from the crack tip, and the stress at the source increases until another dislocation is emitted. The emitted dislocations move with a velocity based on the following expression:

Vx,. = jj-^ (Itx,.)mAe(-Ea’iT> [24]

V

У

у

The values for the parameters were obtained by fit­ting the data of screw dislocation velocities in iron.59 The value of m has a linear dependence on tempera­ture T: m = 400/T + 1.2, A = 3.14 x 10-4, and Ea = 0.316 eV. The first term restricts the motion of dis­locations below the friction stress value (t,), making sure that v = 0 for |tx, |< t,, and hence, most of the dislocations are in near-equilibrium positions at any given time. When the dislocations are in their equi­librium positions, the temperature and strain-rate dependence of measured fracture toughness (KF), plastic zone size (df), crack-tip opening displacement, etc. are determined only by the temperature and strain-rate dependence of the friction stress (t,). The friction stress used is chosen to be equal to the shear yield stress ay/2 when the Tresca yield criterion is assumed. Thus, the temperature dependence of frac­ture toughness is obtained by inputting the corresponding friction stress value for each tempera­ture. Simulations were done for temperatures range from -180 to -60 °C with corresponding yield stress values (ot,,) from 910 to 620 MPa.

The arrays of emitted dislocations form the plastic zones of the crack. The crack may also get blunted due to dislocation emission. In our case, the effects of blunting will be negligible for microcracks since the number of dislocations emitted is only up to 102. However, the effects of blunting will be significant in the case of macrocracks, because the number of dislocations emitted is of the order 105; here the blunting effects are accounted for by using the elastic crack-tip field for a blunted crack.60 The plastic zone developed at the macrocrack modifies the field ahead

of it so that it is the same as an elastic-plastic material with hardening.61 The microcrack placed in this field experiences a tensile stress and is assumed to propa­gate, leading to fracture when it reaches a critical value F (estimated on the basis of similar dislocation simulation of finite crack emitting dislocations along the slip planes). Computer simulations are performed in two stages. First, the microcrack is loaded to failure and the microscopic fracture stress is estimated for specific crystallographic orientations and crack sizes. The obtained microscopic fracture stress (sf) values are then used as the fracture criterion in the macro­crack simulation.