Environmental Engineering Reference
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where d is the grain size and k y is the Petch unpinning coeffi cient. Thus,
the source hardening term (
s ) is equivalent to the grain size dependent
term which can be determined from the grain size dependence of the yield
stress:
σ
k
y
.
[1.20c ]
σ s
=
d
Alternatively, it can be evaluated using the Makin-Minter 19 method by
extrapolating the work-hardening portion of the stress-strain curve in
Fig. 1.13a to the elastic range (Fig. 1.13b). The intercept is interpreted as the
friction stress (
i ) and the difference between the yield stress and the inter-
cept is the source hardening (
σ
s ). Murty 20 demonstrated the equivalence of
Hall-Petch relation and Makin-Minter method from experimental results
on grain size dependence of the mechanical properties of pure iron.
Effects of neutron radiation exposure in austenitic stainless steel (FCC) 21
and mild steel (BCC) 22 are shown in Fig. 1.14a and 1.14b, respectively. It
can be seen that the smooth stress-strain curve in the unirradiated stainless
steel (Fig. 1.14a) developed a distinct yield point subsequent to radiation
exposure accompanied by a decrease in strain hardening and a decrease
in the uniform and total elongations. The fact that the yield point and the
Luders strain observed in unirradiated BCC mild steel (Fig. 1.14b) increases
initially with increase in neutron fl uence and eventually disappears after
the highest value (10 19 n/cm 2 ) clearly demonstrates the decrease in source
hardening with increased neutron radiation dose. On the contrary, in FCC
metals the yield point appeared following radiation exposure indicating an
increased source hardening in the irradiated material.
Friction hardening (
σ
i ) arises mainly from the long range elastic interac-
tions of moving dislocations with other (forest) dislocations as well as short
range interactions with faulted dislocation loops, precipitates, etc., so that
σ
￿ ￿ ￿ ￿ ￿ ￿
,
[1.21 ]
σαρβ
Gb
Gb
Nd
=
+
= α
Gb
ρ
σσ
σ
i
LR
SR
where the subscripts LR and SR represent the long-range and short-range
stresses, G is shear modulus, b is the Burgers vector,
is dislocation density,
N and d are the number density and diameter of the clusters (faulted loops,
precipitates, etc.) and
ρ
are constants representing the strengths of
long range and short range forces. In general the defect densities (
α
and
β
ρ
and N )
are proportional to the fl uence (or dpa) and thus
Gb
,
[1.22 ]
Φ
σα
i
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