The small defects formed in irradiated steels and model alloys can act as barriers to dislocation move­ment and therefore result in an increase in yield strength and hardness. Particularly important is the hardening from the copper-enriched precipitates/ clusters formed during irradiation in the high copper steels which can be modeled using the Russell-Brown model.94 The Russell-Brown model of hardening due to copper precipitates is a modulus interaction the­ory, based on the reduction in energy of the segment of dislocation, which passes through a relatively soft copper particle in the iron matrix. As the energy of the dislocation is proportional to the modulus of the host material, an attractive force will act on the dislo­cation because the modulus of copper is less than that of iron. Russell and Brown estimated the attractive force as a function of copper volume fraction, and demonstrated that this could adequately describe hardening in Fe-Cu alloys. A key element in applying the Russell-Brown model is the estimation of the modulus. Three approaches have been employed, using the modulus for fcc Cu, or computing values for bcc Cu,95 or fitting to experimental data. The last one is the most common approach. The matrix hard­ening may be estimated from the response of low Cu steels (Cu < 0.1 wt%).

The individual hardening contributions from CECs and the MD must be combined with one another, as well as with the hardening from the pre­existing microstructure. The limiting rules for such superposition are a linear sum (LS) law and a square root of the sum of the squares (RSS) law.96 Computer models can be employed to determine the exact superposition law to be employed.97,98 Figure 14 shows a scatter plot, where the measured Asy is compared to the predicted values.99 It can be seen that excellent agreement can be achieved.

Bacon and Osetsky100 carried out molecular static (MS) and molecular dynamics (MD) simulations of the passage of a dislocation through a bcc Cu precip­itate. The MS simulations led to a dependence of hardening on precipitate size which differed from that predicted by the Russell-Brown model. How­ever, Odette (see Section 2 of Eason et at29) found that the Russell-Brown model gave slightly better agreement with the experimental data.

It should be added that further insight into the parameters controlling the hardening is obtained from CECs by combining microstructural data with

mechanical property data (particularly hardening or yield stress increase) where the MD has been sub­tracted from the total measured increase.