Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
α
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
α
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
Figure 2-1: Inverse pole figure characterization of texture in tubing (from [Kearns & Woods, 1966]). Numbers on contour lines
indicate orientation densities (times random) with respect to reference direction. Histograms show corresponding
(calculated) volume distributions of basal poles.
(1 1 2 ̅ 0) α− (1 0 1 ̅ 0) α−Zr
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
Figure 2-2: Correlation of hydride orientations with basal pole textures in samples cooled under 138 MPa uniaxial tensile stress.
The solid line, labelled as average curve from Fig. 5 of the paper by Kearns and Woods [Kearns & Woods, 1966] is the
correlation in unstressed samples (from [Kearns & Woods, 1966]).
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
Figure 2-3: One of the earliest micrographs [Parry, 1966] showing the composite structure of a hydride cluster (transverse section
of a Zr-2.5Nb pressure tube cooled under no external stress).
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
16 2
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
̅
{1 0 1 7} αZr
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
Fe N 2
16
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
∆ ∗
∗
= ∗ (− ) (− )
∗ =
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
= 10 −2 10 −1
∗ =
=
=
∆ ∗ =
=
=
=
∗ 2 −1
= (2 )
≫
(− ) → 1 ∗
∆ ∗
∗
= ∗ (− )
∗
∆ ∗
⁄
2
1 (∆ ) −1 2
= [ ∙ ( ) ]
2 2 ∗
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
=
∗ =
∆ =
∗
∗
∗
=
4
∗ =
=
=
=
∗
∗
∗
∗2
= 2
∗ =
=
=
=
=
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
{0 0 0 2} − ∥ (1 1 1) δ
[1 1 2 ̅ 0] α−Zr ∥ (1 1 ̅ 0)
2
ℎ ℎ = (4/3)
4
2
2
∆ = (∆ ℎ + ∆ ℎ + ∆ ℎ ) + 2 + 4
3 ℎ
ℎ
∆ ℎ =
∆ ℎ =
∆ ℎ =
=
=
∆ ℎ
ℎ
∗
∆ ∗
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
2 ̅ 3
∗
∆ =
3 [ (−∆ ′ℎ − Δ ℎ ( ) − Δ ℎ ( ) − ∆ ℎ + ∆ )] 2
∗
∗
ℎ
̅
∗
̅ = + 2
̅
∗
=
ℎ
ℎ
∗
∗
(−∆ ′ − Δ ( ) − Δ ℎ ( ) − ∆ + ∆ )
ℎ
∗
⁄
∗
∗
⁄
= =
∆ ′ℎ
ℎ
, ( )
′
∆ ℎ = V ̅ ( )
−
=
=
=
̅ −
V =
, () =
=
= / ℎ
ℎ
⁄
≪ 1
1
2
2
∆ ℎ = 1 − ∆ + 1 − 2 [Δ ∙ + 4(1 + ) ]
=
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
=
Δ =
=
Δ ℎ ( ) = (1 − ) Δ 2
1 1 (2 − )
2
2
Δ ℎ ( ) = (1 − ) 2 [Δ ∙ + 4(1 + ) + 8 (1 + ) ]
ℎ
Δ ( )
ℎ
ℎ
Δ ( ) = Δ ℎ ( ) + Δ ( )
∆ ℎ ∆
∆ ℎ = −
Δ = −
=
=
= (α + )/
⁄
(α + )
= lambda = ⁄
ℎ
ℎ
ℎ
′
|−∆ + ∆ | ≪ |−∆ ℎ − Δ ( ) − Δ ( )| ∆ ∗
∗
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
2 ̅ 3
∗
∆ =
3 ℎ ℎ 2
′
∗
∗
[ (−∆ ℎ − Δ ( ) − Δ ( ))]
∗
4 [−∆ ℎ + ∆ ] ̅ 3
− { }
3 3
ℎ
∗
∗
[ (−∆ ′ − Δ ( ) − Δ ℎ ( ))]
ℎ
4
= 3 ( )
∗
∗
∗ 3
∗ ∗
̅
∗
=
ℎ
′
∗
(−∆ ℎ − Δ ℎ ( ) − Δ ( ))
∗
∗
=
ℎ
∗
∆ = ∆ − [−∆ + ∆ ]
∗
∗
∆ ∗
2 ̅ 3
∗
∆ = 2
3
′
∗
∗
[ (−∆ ℎ − Δ ℎ ( ) − Δ ℎ ( ))]
∆ ∗
∗
[−∆ ℎ + ∆ ]
∗
, = ∗ , { }
∗
,
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
∆ ∗
∗ = ∗ (− )
,
∗2
∗
= 2
ℎ
⁄
−∆ = ( )
ℎ
|∆ | > | ∆ |
̅ −
− V ̅ − ℎ ∆
̅
̅
̅ −
−ℎ −ℎ
⁄
∗
∗
∆ = ∆ − [( ) − ]
∗
∗
⁄
[( ) − ]
∗
, = ∗ , { }
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
ℎ
(α + )/
⁄
1.66 ≡ = 1.5
∆
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
α
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
α
3 −1
≈ 1 × 10 17 m s
∗
=
∗
5.253 10 −27 m 3 =
−3 −1
8.602 × 10 16 m s
̇ = 1.66 5.253 × 10 −27 × 8.602 × 10 16 = 7.501 × 10 −10 s −1
̇ 8.223 × 10 −6 s −1
3.333 × 10 −2 s −1 min −1 ̇
2.467 × 10 −4
10 4
−3 −1
≈ 10 20 m s
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
1/ (K)
∆
∆ 2 ∆
∆ = −√∆ 2 ∙ ∆ ∆ =
−1
−1
−29 953 J K mol H
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
′
2
−∆ ℎ = 1.584 × 10 J m −3
∆ ∗
∆ ∗
̅
∗
′
−∆ ℎ
Table 3-1: Enthalpies of dissolution and formation for TSSD, TSSP1 and TSSP2 derived from a fit of the experimental
data of Pan and co-workers [Pan et al., 1996] assuming a common value for the pre-exponential term,
4
= 3.9153 × 10 wppm. The corresponding value for TSSE was derived assuming ∆ =
−√∆ 2 ∙ ∆ .
Hydride Solubility Expressions (wppm) Enthalpies in Solubility Expressions (J mol H)
1
⁄
( ) = ∙ exp(∆ ) ∆ = −31 000
( 1) = ∙ exp(∆ ⁄ ) ∆ 1 = −27 7040
1
( 2) = ∙ exp(∆ ⁄ ) ∆ 2 = −28 942
2
( ) = ∙ exp(∆ ⁄ ) ∆ = −29 953
© ANT International 2018
−3 −1
20
≈ 10 nuclei m s
−3
= 5.85 × 10 J m −2
1 × 10 −6
−3
3.8 × 10 J m −2
20
−3 −1
≈ 10 nuclei m s
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
= 0.0760 77.23 J m 3
−ℎ
̅
−1
J mol H −ℎ = ̅ = 11.354 ×
−1
10 m mol H
−6
3
876.9 J mol −1
−1
−29 306 + 876.9 = −28 429 J mol H
∆
2.32 ×
20
10 nuclei m s ∆ = 572.6 − 558.1 = 14.5 K
−3 −1
590.4 − 558.1 K
∗
2.008 × 10 −27 m 3
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
Table 3-2: Calculation of critical nucleation parameters and nucleation rate at zero internal and externally applied
stress. Case of finite energy barrier model of solvus hysteresis - equilibrium solvus assumed to be given
midway between TSSD and TSSP2 solvi [Pan et al., 1996].
Critical nucleus dimensions
Note: Axis of rotation of ellipsoid is about x1:
3
Values of some Tn(K) = 558.1 VZr(m /(mol Zr)) = e11 = 0.172 rH(H/Zr) = 1.48
parameters 1.40110
-5
4
= 0.3121 (MPa) = 3.031 10 e22 = 0 β = 1.5
δ
E(MPa) = 7.9510 ‘kB(J/(K atom)) = e33 = 0 cH (rH/β ) =
δ
4
δ
-23
1.380 10 0.98667
o
c(J/m ) = 0.0038 Ao(g/mol) = 6.025 Z = 0.1 cH (at. fn) =
2
10 0.008969331
23
2
p(J/m )= c+2i= R(J/(K∙mol)) = 8.3144 cH (wppm) = 100 d(m) = 4.5010
-10
o
0.0114
TSS equations- H(TSSD; J/mol H) Nnucl(sites/m ) =
3
Pan et al; constant A: = 31000 4.3010
22
H(TSSP1; J/mol H) Heter.nucl.red.factor =
= 27704 110
-6
H(TSSP2; J/mol H)
= 28942
A(wppm) = 3.9210
4
Solvus temperatures(at TD(K) = 624.5 TD(C) = 351.5 DH(m s ) = 1.1410
8
2 -1
given CH (wppm)):
0
TP1(K) = 558.1 TP1(C) = 285.1 d(m) = 4.5010
8
TP2(K) = 583.1 TP2(C) = 310.1
Estimated equilibrium Teq(K) = 603.4 H(Teq; J/mol H) =
solvus values: 29953
Teq(C) = 330.4 cH s, eq (wppm) = 49.2
Chemical energy: g , nucl (chem., J/m )
3
= -2.321 10
8
Strain energy reduction factor: 1
Critical nucleus dimensions when depends only on the ratio of surface energies:
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
Table 3-2 cont’d: Calculation of critical nucleation parameters and nucleation rate at zero internal and externally applied
stress. Case of finite energy barrier model of solvus hysteresis - equilibrium solvus assumed to be given
midway between TSSD and TSSP2 solvi [Pan et al., 1996].
gstrain( gstrain(re ao*(m) co*(m) Vo*(nu i(J/ =c/ Go* exp(- *(s - no*( nu Jo(nucl
3
3 -1
3
2
exact) duced) =∙a* cl,m ) m ) i Go*/kBT 1 ) cl/m ) /m s )
n)
1.94E+ 1.94E+07 4.317 8.203E 6.405E 0.2 0.01 4.451 8.11728E 1.14E 3.54E- 4.04E+
07 E-09 -11 -27 90 E-19 -26 +04 03 00
2.59E+ 2.59E+07 3.396 8.604E 4.157E 0.15 0.02 2.754 2.98249E 7.07E 1.30E 9.19E+
07 E-09 -11 -27 53 E-19 -16 +03 +07 09
3.88E+ 3.88E+07 2.509 9.533E 2.513E 0.1 0.03 1.503 3.38214E 3.86E 1.47E 5.69E+
07 E-09 -11 -27 80 E-19 -09 +03 +14 16
4.85E+ 4.85E+07 2.183 1.037E 2.070E 0.08 0.04 1.138 3.84287E 2.92E 1.68E 4.89E+
07 E-09 -10 -27 75 E-19 -07 +03 +16 18
5.54E+ 5.54E+07 2.038 1.106E 1.924E 0.07 0.05 9.914 2.57559E 2.55E 1.12E 2.86E+
07 E-09 -10 -27 43 E-20 -06 +03 +17 19
5.96E+ 5.96E+07 1.973 1.153E 1.881E 0.06 0.05 9.294 5.75676E 2.39E 2.51E 5.99E+
07 E-09 -10 -27 5 85 E-20 -06 +03 +17 19
6.45E+ 6.45E+07 1.917 1.214E 1.868E 0.06 0.06 8.771 1.13632E 2.25E 4.95E 1.12E+
07 E-09 -10 -27 33 E-20 -05 +03 +17 20
7.03E+ 7.03E+07 1.872 1.294E 1.900E 0.05 0.06 8.372 1.90763E 2.15E 8.32E 1.79E+
07 E-09 -10 -27 5 91 E-20 -05 +03 +17 20
7.72E+ 7.72E+07 1.848 1.404E 2.008E 0.05 0.07 8.152 2.53804E 2.09E 1.11E 2.32E+
07 E-09 -10 -27 60 E-20 -05 +03 +18 20
8.57E+ 8.57E+07 1.856 1.567E 2.262E 0.04 0.08 8.225 2.30658E 2.11E 1.01E 2.12E+
07 E-09 -10 -27 5 44 E-20 -05 +03 +18 20
1.27E+ 1.27E+08 2.891 3.663E 1.283E 0.03 0.12 1.996 5.57495E 5.13E 2.43E 1.25E+
08 E-09 -10 -26 67 E-19 -12 +03 +11 14
1.36E+ 1.36E+08 3.757 5.098E 3.014E 0.02 0.13 3.370 1.0074E- 8.65E 4.39E 3.80E+
08 E-09 -10 -26 8 57 E-19 19 +03 +03 06
1.41E+ 1.41E+08 4.625 6.509E 5.831E 0.02 0.14 5.107 1.62364E 1.31E 7.08E- 9.28E-
08 E-09 -10 -26 7 07 E-19 -29 +04 07 04
© ANT International 2018
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
,
̅ ( )
,
( ℎ ) = , [ ℎ ]
̅
∆ ( )
,
( ′ ) = ( ) [− ]
, =
̅
( ) =
ℎ
( )
̅
∆ ( ), ( ) =
=
and =
( )
̅
∆ ( )
̅
∆ ( )
̅
∆ ( )
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
̅
( )
ℎ
̅
̅
∆ ( ) + ( )
,
( ) = ( ) [− ℎ ]
̅
( )
ℎ
̅
( )
ℎ
̅
∆ ( )
′
∆ ℎ
, ( ) ≅ , ( ′ )
8
′
−∆ ℎ = 2.321 × 10 J m −3
−3 −1
20
≈ 2.84 × 10 nuclei m s
−1
−3
6.2 × 10 J m −2 1 291 J mol H
J/m 3
−1
J mol H 31 000 +
−1
1 291 = 29 709 J mol H
≈ 2.84 × 10 nuclei m s
20
−3 −1
598.4 − 558.1 = 40.3 K
∗
= 1.276 × 10 −27 m 3
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
Table 3-3: Calculation of critical nucleation parameters and nucleation rate at zero internal and externally applied
stress. Case of accommodation energy model of solvus hysteresis - equilibrium solvus assumed to be
given by TSSD solvus [Pan et al., 1996].
Critical nucleus dimensions
Note: Axis of rotation of ellipsoid is
about x1:
Values of some Tn(K) = 558.1 VZr(m /(mol Zr)) = e11 = 0.172 rH(H/Zr) = 1.48 Nnud(sites/m ) = 4.30
3
3
parameters 1.401 10 10
5
22
= 0.3121 (MPa) = 3.031 e22 = 0 β = 1.5 Heter.nucl.red.factor
δ
-6
10 = 1.00 10
4
E(MPa) = 7.95 kB(J/(K atom)) = e33 = 0 cH (rH/β ) =
δ
δ
10 1.380 10 0.98667
4
-23
O
c(J/m ) = 0.0062 Ao(g/mol) = 6.025 Z = 0.1 cH (at. fn) =
2
23
10 0.008969331
O
2
p(J/m ) = R(J/(K∙mol)) = cH (wppm) = d(m) = 4.50
-10
c+2i= 0.0186 8.3144 100 10
TSS equations- H(TSSD; J/mol A(wppm) = 3.92
Pan et al; H) = 31000 10
4
constant A:
H(TSSP1;
J/mol H) = 27704
H(TSSP2;
J/mol H) = 28942
Solvus TD(K) = 624.5 TD(C) = 351.5 DH(m s ) =
2 -1
temperatures 1.14 10
-10
(at given
CH (wppm)): TP1(K) = 558.1 TP1(C) = 285.1 d(m) = 4.50
o
10
-10
TP2(K) = 583.1 TP2(C) = 310.1
Estimated Teq(K) = 624.5 H(Teq;J/mol H) =
equilibrium 29953
solvus values:
Teq(C) = 351.5 cH s, eq (wppm) =
49.2
Chemical g , nucl (chem., J/m ) = -2.321 10
8
3
energy:
Strain energy reduction factor: 1
Critical nucleus dimensions when depends only on the ratio of surface energies:
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
Table 3-3 cont’d: Calculation of critical nucleation parameters and nucleation rate at zero internal and externally applied
stress. Case of accommodation energy model of solvus hysteresis - equilibrium solvus assumed to be
given by TSSD solvus [Pan et al., 1996].
gstrain( gstrain(re ao*(m) co*(m) Vo*(nu i(J/ =c/ Go* exp(- *(s ) no*(nu Jo(nucl/
-1
3 -1
2
3
exact) duced) =∙a* cl,m ) m ) i Go*/kBT cl/m ) m s )
3
n)
3.17E+ 3.17E+07 2.994 9.280E 3.483E 0.2 0.03 3.491 2.0821E- 5.49E 9.08E 4.98E+
07 E-09 -11 -27 10 E-19 20 +03 +02 05
4.22E+ 4.22E+07 2.370 9.794E 2.304E 0.15 0.04 2.187 4.6596E- 3.44E 2.03E 6.99E+
07 E-09 -11 -27 13 E-19 13 +03 +10 12
6.32E+ 6.32E+07 1.775 1.101E 1.453E 0.1 0.06 1.228 1.1938E- 1.93E 5.20E 1.01E+
07 E-09 -10 -27 20 E-19 07 +03 +15 18
7.87E+ 7.87E+07 1.565 1.213E 1.244E 0.08 0.07 9.537 4.1996E- 1.50E 1.83E 2.75E+
07 E-09 -10 -27 75 E-20 06 +03 +17 19
8.98E+ 8.98E+07 1.475 1.307E 1.192E 0.07 0.08 8.480 1.6555E- 1.33E 7.22E 9.63E+
07 E-09 -10 -27 86 E-20 05 +03 +17 19
9.66E+ 9.66E+07 1.438 1.372E 1.189E 0.06 0.09 8.061 2.854E- 1.27E 1.24E 1.58E+
07 E-09 -10 -27 5 54 E-20 05 +03 +18 20
1.04E+ 1.04E+08 1.410 1.457E 1.212E 0.06 0.10 7.741 4.3253E- 1.22E 1.89E 2.30E+
08 E-09 -10 -27 33 E-20 05 +03 +18 20
1.1367 1.14E+08 1.393 1.570E 1.276E 0.05 0.11 7.559 5.4734E- 1.19E 2.39E 2.84E+
E+08 E-09 -10 -27 5 27 E-20 05 +03 +18 20
1.25E+ 1.25E+08 1.396 1.731E 1.414E 0.05 0.12 7.595 5.2271E- 1.19E 2.28E 2.72E+
08 E-09 -10 -27 40 E-20 05 +03 +18 20
1.38E+ 1.38E+08 1.435 1.977E 1.705E 0.04 0.13 8.022 3.0023E- 1.26E 1.31E 1.65E+
08 E-09 -10 -27 5 78 E-20 05 +03 +18 20
1.55E+ 1.55E+08 1.547 2.398E 2.403E 0.04 0.15 9.321 5.5568E- 1.47E 2.42E 3.55E+
08 E-09 -10 -27 50 E-20 06 +03 +17 19
1.75E+ 1.75E+08 1.852 3.281E 4.716E 0.03 0.17 1.337 2.9145E- 2.10E 1.27E 2.67E+
08 E-09 -10 -27 5 71 E-19 08 +03 +15 17
2.03E+ 2.03E+08 3.058 6.320E 2.476E 0.03 0.20 3.643 2.8785E- 5.73E 1.25E 7.19E+
08 E-09 -10 -26 67 E-19 21 +03 +02 04
© ANT International 2018
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
<0 0 0 1> ≡ = 0.0542
<1 1 ̅ 0 0> = <1 1 2 ̅ 0> ≡ = 0.0329
<1 0 1 ̅ 7>
<1 1 2 ̅ 0> = <1 1 ̅ 0 0> ≡ ∥
≡ ⊥
<1 0 1 ̅ 7>
<1 0 1 ̅ 7> ≡ = 0.172
⊥
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
=
=
=
=
+
= 1 +
( = 0) ≡ (0)
∆ ℎ ∆
() (0)
∗ [−∆ ℎ + ∆ ]
() = (0)exp { }
() ( )
= 0°
()
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
( )
(0) (0)
∗
[−∆ ℎ , + ∆ , ] ∙
∗
() = (0) { } = (0) { }
∗
∗
[−∆ ℎ , + ∆ , ] ∙
( ) = (0) { } = (0) { }
ℎ ,
= −∆ + ∆ ,
,
= −∆ ℎ , + ∆
( )
() (0) ∆ ∆
( ) = ( ) = (0) [ ∗ ] ≡ (0) [ ∗ ]
⁄
(0) = (0) (0)
∆ = −
(0) [ ∗ ∆ ]
( , %) = 100
1 + (0) [ ∗ ∆ ]
( , %)
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
100
( , %) =
1 + 1 [− ∗ ∆ ]
(0)
(0) = 1
(0, %) = 50%
0.3/0.6 = ½ (0) = ½ ( , %) = 33%
í í
(0) (∆ ( ) + ∆ ( ))
( ) = (0) [ ∗ ]
∆ ( ) ∆ ( )
≡ (0) [ ∗ ] [ ∗ ]
∆ ( ) ∆ ( )
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
∙
( ) = ⊥ − ⋅ ∙
∙
( ) = ∥ − ⋅ ∙ = − ⋅ ∙
∥
= 0
∥
∆ ( )
∆ ( ) = ( ) − ( ) = ⋅ ( ⊥ − ⋅ [ − ])
≈ 1 − = 0.0213
= 0.172 = 0.9867
⊥
= 1.5
− = 0.0213 ∆ ( ) = 0.0936
∗
= 2.008 ×
10 −27 m 3 = 558.1 K
Table 3-4: Effect of external circumferential (hoop) tensile stress on % radial hydride fraction, . (Starting from 2%
radial hydride for the externally unstressed tube material.)
Circumferential (Hoop) Tensile Stress, (MPa) ( , %), % Radial Hydrides
0 2
40 5
80 13
120 27
160 50
200 73
240 88
280 95
320 98
360 99
400 100
© ANT International 2018
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
(0)
Figure 3-1: Plot of the theoretical predictions given in Table 3-4 showing the sigmoidal variation of the % radial hydrides ( ) as a
function of stress.
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
[ − ]
∆ ( ) = ⋅ ( ⊥ ∥ − ⋅ [ − ])
= 1.5 = 0.072
⊥
= 0.0458 ∆ ( ) = −0.00355
∥
( ) = − ⋅ ⋅
( ) = − ⋅ ⋅
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
= [ ∗ ∆ ( ) ]
∆ ( ) = ⋅ ⋅ [ − ]
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
Figure 3-2: Blister grown on a pressurized loop tube on the circumferential (horizontal)-radial (vertical) section of the tube wall
(from [Leger et al., 1989]). The pressure applied inside the tube resulted in 133 MPa uniaxial tensile stress in the
circumferential direction
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
( )
( ) = ( )
=
=
( , %)
( ) 100
( , %) = 100 1 + ( ) ≡ 1
1 +
(− )
( , %)
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
Figure 4-1: Plots of the % radial hydrides ( ( , %)) in a Zr-2.5Nb flattened tube specimen containing 100 wppm H for two
different maximum temperatures from which the specimens were cooled as a function of uniaxial tensile stress applied
2
in the tube circumferential direction (MN/m MPa) (Series C specimen). Maximum temperatures from which the
specimens were cooled are indicated in the figure (from [Hardie & Shanahan, 1975]).
Figure 4-2: Curve fits to data of % radial hydrides ( ( , %)) versus uniaxial externally applied tensile stress in the tube’s
circumferential direction for different maximum temperatures from which the specimens were cooled (from [Hardie &
Shanahan, 1975]). The data used in these fits were of specimens from the ID stringer zone of flattened Zr-2.5Nb
pressure tube material, all with hydrogen content of 100 wppm. The ID stringer zone is a region of maximum
macroscopic residual compressive stress in the material. The number beside each curve corresponds to the specimen
number in Table 2 of [Hardie & Shanahan, 1975] and to the number given in the legend showing the maximum
temperature from which each of the specimens was cooled.
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
Figure 4-3: Curve fits to data of % radial hydrides ( ( . %)) versus uniaxial externally applied tensile stress in the tube’s
circumferential direction for different maximum temperatures from which the specimens were cooled (from [Hardie &
Shanahan, 1975]). The data used in these fits were of specimens from the OD mixed zone of flattened Zr-2.5Nb
pressure tube material, all with hydrogen content of 100 wppm. The OD zone is a region of maximum macroscopic
residual tensile stress. The number beside each curve corresponds to the specimen number in Table 1 of [Hardie &
Shanahan, 1975] and to the number given in the legend showing the maximum temperature from which each of the
specimens was cooled.
∆
(%)
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.