With the increase in size (10 nm or so depending on the particular alloy system) in coherent precipitate, the coherency is eventually lost. This happens during

overaging of precipitation hardenable alloys (such as Al-4.5 wt% Cu). The disloca­tion line is repelled due to the incoherency of the particles. Orowan (1947) pro­posed that the stress (At) required to bow a dislocation line between two particles separated by a distance 1 is

where G is the shear modulus of the matrix material and b is the Burgers vector of dislocation.

Every dislocation gliding over the slip plane adds one dislocation loop around the particle (Figure 4.36) and these loops exert back stress on the dislocation sources that must be overcome to cause further slip. A real example of Orowan looping is shown in Figure 4.36b. Generally, Orowan bypassing leads to fine and wavy slip compared to the coarse and planar slip due to the particle cutting mechanism.

Cu alloy (GP zones are the primitive particles formed during aging followed by transitional precipitates в" and в’ and finally equilibrium precipitate в.

If we now take the appropriate models of the particle shearing and bypassing and try to see how the strength increments due to these mechanisms contribute to the overall particle strengthening, an interesting observation can be made. Figure 4.37a shows such a prediction. Particle cutting (underaged material) increases strength­ening as the precipitate size increases, whereas Orowan bypassing increases with decreasing particle size (in the overaging regime). The crossover (the peak aging regime) gives the highest strength. Figure 4.37b shows why the aging experiments on aluminum alloys lead to such yield strength profiles as a function of aging time. With the aging time, the precipitate size increases and it follows first the particle cutting mechanism when the particles remain coherent, but when particles lose their coherency, the Orowan bypassing mechanism becomes important. Thus, the trend can be explained.