Both the Cottrell-Bilby theory of carbon-locking of dislocations for the yield point behavior in steel [18] and the Hall-Petch (H-P) dislocation pile-up model for a reciprocal square root of average grain diameter dependence of the yield and cleavage fracture stresses[19, 20] entered into Cottrell obtaining an implicit description of the dbtt in the relationship [9] ky(σ0yℓ 1/2 + k y) = CGγ (1) In Eq. (1), ky is the H-P microstructural stress intensity required for breaking-free Cottrell-locked dislocation sources so as to provide transmission of plastic flow across grain boundaries, ℓ is the polycrystal grain diameter, σ0y is the friction stress resistance for individual dislocation movements in slip band pile-ups, G is the shear modulus, γ is the cleavage crack surface energy, and C is a numerical constant. Hull called attention in [17] to his measurements made with Mogford [21] showing an increase in σ0y from neutron irradiation damage, thus raising the magnitude of the left-side of Eq. (1), and promoting brittleness. Petch obtained an explicit dependence of the dbtt on the temperature dependent component of the friction stress and on the ductile fracture stress dependent H-P determined value of kf, as [10] TC = (1/β)[lnB – ln{(CGγ/kf) – kf} - lnℓ-1/2] (2) In Eq. (2), B is the limiting value of the temperature dependent component of the friction stress at T = 0 and β is the exponential temperature coefficient. Fig. 1. Tensile ductile-brittle transition temperatures for two grain sizes, A of 100 μm and B of 5 μm; and, then of A altered by neutron irradiation to A* with raised value of Δσoy* [22]. Figure 1 provides an illustration produced at the U.S. Oak Ridge National Laboratory of the dbtt behavior as obtained on the basis of computed tensile test results applicable to grain size and neutron irradiation hardening [22]. In the figure, a comparison is made between the temperature dependences of the yield and cleavage fracture stresses of two annealed mild steel materials, “A” having a conventional grain size of ~100 μm and “B” having a grain size of ~5 μm. In addition, the yield stress is shown to be raised after neutron irradiation hardening of steel “A”, now labeled “A*”, but for which its cleavage stress is unchanged. As indicated in the figure, steel A has a lower dbtt because of the H-P inequality: kC > ky. And steel “A*” has an appreciably raised dbtt
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