A microcrack in the configuration shown on the right side of Figure 13 is loaded monotonically. To mimic the triaxial stress field surrounding the microcrack, we modify the expressions for the stress fields given in Wang and Lee59 by subtracting the applied normal stress (oAj) and adding the uniaxial yield stress (ay) for cases when sAy > sy. The value of the slip plane angle is chosen such that predominantly brittle crack configurations are simulated. The applied load is incremented at a rate dK/dt = 0.01MPax/ms~1; a variable time step is used to make sure dislocations move as an array, and numerical instabilities are avoided. Dislocations are emitted from source posi­tions (here chosen as 4b, b = 2.54 A) along the slip planes. Since source positions are equidistant from the respective crack tips, simultaneous emission occurs. The emitted dislocations move away from the crack tip along the slip planes with a velocity calculated using eqn [24]. The stress fields of these dislocations then shield the crack tip from external load. Once moved to their equilibrium position, the amount of shielding from each dislocation is calcu­lated using expressions from Wang and Lee,59 and the total shielding at the crack tip is obtained by summa­tion. We ignore antishielding dislocations in our simulations, since any dislocations nucleated in the present configuration around the crack tip will be absorbed by the crack. Thus, the number of disloca­tion absorbed is small, and any blunting effects due to them are neglected here. Because in our case disloca­tions are emitted from crack-tip sources, dislocations emitted from either end contribute to shield both crack tips.62 The applied load is increased monotoni — cally and the crack-tip stress intensity is calculated at each time step. When the crack-tip stress inten­sity reaches a preassigned critical value, the crack is assumed to propagate and the corresponding

K ^2r

Microcrack

Macrocrack

applied load is the microscopic fracture stress (sf). The following two predominantly brittle microcrack systems are considered here: (1) cleavage plane (001), crack front [110], slip system 1[T11](112), hence a’ = 35° 16′, and (2) cleavage plane (110), crack front [110],^Ш^Ш), hence a’ = 54°44′.

A schematic illustration of the macrocrack is shown on the left-hand side of Figure 14. The macrocrack is assumed to be semi-infinite, with dis­location sources close to the crack tip. Dislocations are emitted simultaneously along the two slip planes, symmetrically oriented to the crack plane. The plas­tic zone formed by emitted dislocations produces a field equivalent to an elastic-plastic crack with small scale yielding.61 The slip plane angle in this case is chosen to be 70.5° (the direction of maximum shear stress of the elastic crack-tip field), compared to the present slip plane angles, which match best with continuum estimates.61 The initial source position (x0) along the slip plane is chosen to be 4b, where b is the magnitude of Burgers vector. (For each positive dislocation emitted, a negative one is assumed to move into the crack.) The dislocation sources on either side of the crack plane are at equivalent posi­tions x0 and operate simultaneously.

During the simulation, the applied stress intensity is increased in small increments and the positions of dislocations are determined. It is found that the dis­locations reach near-equilibrium positions. As the load is increased, more and more dislocations are emitted and the crack gets blunted. The radius of the blunted crack p is taken to be equal to Nb sin a, where N is the number of dislocations and a is the slip plane angle. The blunting of the macrocrack is illustrated in Figure 14. As blunting increases,
(1) the crack-tip fields are modified to be that of a blunted crack, (2) source position is chosen to be equal to the crack-tip radius, that is, x0 = p, and (3) the notional crack tip from which the image stress of dislocations is calculated is moved back to the center of curvature of the blunted crack. At each time step, the stress ahead of the crack at a distance Xp along the crack plane (the microcrack is assumed to be at this position) is calculated using the expres­sion from Trinkaus et at23 The fracture criterion is the tensile stress (o?) at Xp reaching oF calculated in the previous stage for the microcracks for the same temperature and the corresponding yield stress. Thus, when the microcrack tip stress intensity is K = Kjc, cleavage fracture of the matrix is assumed to occur. The applied load at the macrocrack then gives the fracture toughness (KF) at that given tem — perature/yield stress. Throughout the simulation, a microcrack of size 1 pm is used; this value is chosen as a typical (average) value in the range of sizes of microcracks found in experiments with which we compare our results.43 The detailed simulations reported here are obtained using Xp equal to 10 pm. However, the KF calculated using 20 pm is also shown in Figure 14. These values were chosen as typical of the order of grain sizes in this type of steel. It should be noted that changing these parameters will give different values for fracture toughness; however, the behavior of their variation with temperature will remain the same, as will be discussed next.