Environmental Engineering Reference
In-Depth Information
Hill formulation was shown to be convenient wherein the generalized stress,
σ
g
, is defi ned as the uniaxial stress along the tube axial direction,
110
2
2
2
(
)
(
)
R
(
)
2
R
P
(
)
2
P
(
)
σ
+
σ
−
σ
+
σσ
σ
−
[3.64 ]
θ
σ
2
σ
σσ
σ
θ
θ
)
(
σ
σ
θ
σ
σ
σσ
=
P
(
R
1
+
The parameters,
R
and
P
, are the mechanical anisotropy parameters. Using
the Prandtl-Reuss energy balance in conjunction with the above yield crite-
rion, the strain increments or strain rates along the three orthogonal direc-
tions are related to the respective stresses
⎡
⎤
⎡
⎤
⎡
(
)
⎤
RP
R
P
P
−
ε
σ
r
r
⎡
⎢
⎤
⎥
⎡
⎢
⎤
⎥
ε
⎡
⎢
⎤
⎥
g
(
)
RR
P
RP
[3.65 ]
⎢
⎥
−
RR
P
⎢
⎥
⎢
ε
⎥
⎢
⎥
⎢
σ
⎥
⎥
=
⎢
⎥
θ
θ
,
P
(
R
)
+
σ
⎢
⎣
⎢
⎣
⎥
⎢
⎣
⎥
+
g
(
R
1)
P
RPP
⎣
⎦
⎦
⎣
⎦
⎦
⎣
⎦
⎦
ε
σ
−
P
z
z
where
ε
g
is the generalized strain rate corresponding to the generalized
stress
g
. These equations defi ne the yield and fl ow loci, and the corre-
sponding creep locus is referred to at a constant energy dissipation-rate,
W
σ
iij
111
It is clear from the above equations that the mechan-
ical anisotropy parameters,
R
and
P
, are the transverse contractile strain
(-rate) ratios in uniaxial tests. The contractile strain ratios (CSRs),
R
and
P
, defi ne the resistance to wall thinning of an anisotropic material and thus
control the formability which is of importance to the material manufac-
turers, in tube reduction processes, sheet drawing and forming, etc.
112
T h e
mechanical anisotropy parameters are directly related to the preferred ori-
entations of the grains. Thus obtained creep loci for cold-worked stress-
relieved annealed and following complete recrystallization are shown in
Fig. 3.28 .
113
Crystallite orientation distribution function (CODF) creep
model predictions
113
are shown as solid lines along with the experimental
results.
These types of creep studies are not commonly found albeit materials in
real engineering structures experience such complex multi-axial loadings.
Additional complexity is encountered in attempting to predict transients in
creep under such multi-axial loading.
(
)
ij
ij
gg
.
ij
ij
ij
g
ij
ij
gg
3.8 Creep of zirconium alloys used for LWR cladding
Materials used in the reactor undergo irradiation-assisted creep as well
as thermal creep (which predominates if stress and temperature are high
enough). The in-pile creep deformation of a material is the net contribu-
tion by both of these processes and it is diffi cult to distinguished between
them. Thermal creep rate of unirradiated material is different from that of
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