The (micro) fracture stress oF as a function of tem­perature was calculated for two different slip plane angles. The fracture criterion chosen was a crack-tip stress intensity K = Kjc = 1.0MP^/m (value esti­mated for Fe with surface energy of 2J m~2). The source position (x0) is chosen to be 4b, and the micro­scopic fracture stress oF is estimated for different temperatures. The results indicate that oF is practi­cally independent of temperature, consistent with many of the experiments (e. g., Kubin et a/.6). The value of oF thus obtained for each temperature is used to calculate the macroscopic fracture toughness (KF) in the next stage of simulation.

Figure 15 shows the typical behavior of the ten­sile stress at the microcrack (o^) as a function of the applied load (K). The fracture criterion in this case is o? reaching the critical value oF calculated in the previous stage. For the case shown in this figure, the microcrack size is 1 pm, and the rate of loading dK/dt = 0.01MPaVms-1. The distance of the

microcrack (particle) from the macrocrack (Xp) is 10 pm. As the temperature is increased and yield stress (friction stress) is decreased, the applied stress intensity K required for the tensile stress at Xp to reach the critical value oF increases exponentially. Two factors contributing to this exponential increase could be the decrease in the tensile stress at the microcrack due to crack-tip blunting and the increas­ing effects ofstress field (predominantly compressive) from the emitted dislocations. This can be seen in Figure 16, where the plastic zone size (d) and the radius of the blunted crack tip (p) for each tempera­ture measured at fracture K-applied = KF are shown.

The plastic zone size is the distance measured along the slip plane to the farthest dislocation from the crack tip. The dislocation source distance (x0) is chosen as p for a crack-tip radius >4b; else x0 = 4b. Figure 17 shows the macroscopic fracture toughness

Figure 19 The fracture toughness values from Figure 17 and for Xp = 20 mm compared with experimentally determined values. Reproduced from Amodeo, R. J.; Ghoniem, N. M. Phys. Rev. 1990, 41, 6958.

KF as a function of temperature for cases with and without considering the effects of crack-tip blunting. For the case without blunting, the increase in the fracture toughness (KF) with temperature is small. However, a sharper increase in the fracture toughness is observed when blunting is accounted for in the simulation. This striking observation emphasizes the significant effect of blunting in the increase of fracture toughness with temperature. This exponen­tial increase in the fracture toughness corresponds to the transition from brittle to ductile behavior. In Figure 18, J2F (J2 — integral value at fracture) calculated from the number ofdislocations emitted at the corresponding load KF is shown. J2 is defined as the sum of the glide forces on all dislocations around the crack tip.26 In this case, the disloca­tions are in equilibrium against the friction stress (t,) and we can compute J2F as the product of the
total number of dislocations (N) and the friction stress (г,,). Considering the fact that J2F is calculated from the number of dislocations emitted at an applied stress intensity factor KF, it is striking to note that the prefactor of the exponent of the J2F-temperature curve (0.0255) matches that of the KF-temperature curve (0.0123) to hold the known proportionality between K2 and J

In Figure 19, the calculated values of fracture toughness are compared with the fracture toughness measurements reported in Amodeo and Ghoniem3. The carbide found in these samples ranges in size from 0.44 to 1.32 mm, and in our calculations we have used microcracks of comparable size (1 mm). The results for the blunted case are shown in Figure 17, along with another set of values calculated for Xp = 20 mm, shown here for comparison. We can see that the model predicts the rapid increase in fracture toughness at the transition temperature region, and reasonably fits the experimental data. Considering the simplicity of the present model, the agreement suggests that a good step has been taken in predicting the BDT behavior.

The crack-tip behavior and the BDT predicted are in good agreement in the transition region where the fracture toughness increases rapidly with temper­ature. However, it should be noted that the model ceases to be valid at higher temperatures where duc­tile tearing effects will be significant. According to our model, the two factors that contribute to the sharp increase in the fracture toughness with temper­ature are (1) the increase in the mobility of the emitted dislocations and (2) the effect of macrocrack tip blunting. The mobility of emitted dislocations determines the equilibrium position of dislocations
and thus determines the tensile stress at the micro­crack. Also, the mobility of dislocations around the microcrack determines the crack-tip stress intensity at the microcrack and thus the microscopic fracture toughness (sf), which ultimately determines the frac­ture toughness of the material (KF). However, and as can be seen in Figure 17, this alone cannot explain the sharp upturn at the transition. The amount of crack-tip blunting is found to be a significant factor in capturing the rapid increase in the fracture toughness with temperature in the transition region.

1.16.4 Outlook

While continuum approaches to modeling the mechanical properties ofstructural materials are lim­ited to the underlying experimental database, DD methods offer new opportunities for modeling micro­structure evolution from fundamental principles. The limitation to the method presented here is mainly computational, and much effort is needed to over­come several difficulties. First, the length and time scales represented by the present method are still short of many experimental observations, and meth­ods of rigorous extensions are still needed. Second, the boundary conditions of real crystals are more complicated, especially when external and internal surfaces are to be accounted for. Thus, the present approach does not take into account large lattice rotations, and finite deformation of the underlying crystal, which may be important for explanation of plastic deformation at certain length scales. Finally, a much expanded effort is needed to bridge the gap between atomistic calculations of dislocation proper­ties (as discussed by Osetsky and Bacon in Chapter

1.12, Atomic-Level Level Dislocation Dynamics in Irradiated Metals) on the one hand, and continuum mechanics formulations on the other. Nevertheless, with all of these limitations, the DD approach is worth pursuing, because it opens up new possibilities for linking the fundamental nature of the microstruc­ture (especially of irradiated materials) with realistic deformation conditions. It can thus provide an addi­tional tool to both theoretical and experimental investigations of plasticity and failure of materials.