(1 1 1) (0 0 2) δ (3 1 1) δ
δ
(1 1 1) δ
(1 1 1) δ
(1 1 1) δ
α
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
α
α
α
(α + )/
α
α
6
−1
30 (× 10 )K
α
− (Å) = 3.23118 + 1.6626 × 10 ( ( ) − 298)
−5
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
−5
− (Å) = 5.14634 + 4.7413 × 10 ( ( ) − 298)
H/Zr =
1.66
(α + )/
(α + )/
α
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
̅
(1 0 1 1) α−Zr
(1 1 1) δ
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
Figure 4-77: Variations in relative strain in the zirconium matrix derived from changes in d-spacing of the (1 0 1 ̅ 1) αZr peak in a
sample containing 245 wppm hydrogen heated under zero external load and then cooled under an external tensile load
of 160 MPa during which no hydride reorientation occurred: (a) in the TD; and (b) in the RD (from [Colas et al., 2014]).
Figure 4-78: Variations in relative strain derived from the changes in d-spacing of the (1 1 1) peak in a sample containing 245
δ
wppm hydrogen heated under no external load, then cooled under a 160 MPa external tensile stress applied in the TD
during which no hydride reorientation occurred: (a) in the TD; and (b) in the RD (from [Colas et al., 2014]).
̅
(1 0 1 1) α−Zr
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
α
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
Figure 4-79: Variations in relative strain derived from the changes in d-spacing of the (1 1 1) peak in a sample with 192 wppm of
δ
hydrogen heated under zero external load, then cooled under a 240 MPa external tensile stress applied in the TD
during which partial hydride reorientation occurred: (a) in the TD; and (b) in the RD (from [Colas et al., 2014]).
4.6.3.1
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
Figure 4-80: Geometry and locations of the specimens, Z1 (located 0.5 mm from the Fusion Zone (FZ), and Z3. An additional
specimen, Z2 (not shown in this schematic) machined from this plate was located 5 mm from the FZ. (b) Micrograph of
the Z1 specimen under polarized light. (c) Micrograph of the Z3 specimen showing the normal microstructure of the
plate material. In these micrographs differences in grain orientations are revealed by differences in their colour (from
[Santisteban et al., 2010]).
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
Figure 4-81: Schematic showing the orientation of the specimen with respect to the incident X-ray beam and the area detector
including an example of the types of diffraction rings obtained. The directions RD, TD and ND of the specimens with
respect to the original plate material are as shown in Figure 4-80 (from [Santisteban et al., 2010]).
– 2
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
Figure 4-82: (a) Examples of images recorded by the area detector for each of the three specimens used (for clarity, only a quarter
of each image is shown).The colour scheme is proportional to the logarithm of the counts. Debye rings corresponding
to the Zr and δhydride phases are shown indexed. (b) Diffractograms at two different temperatures over the
angular range from 42.5 < ϕ < 47.5 which is indicated by the wedge depicted by the solid black lines in the Z3 image
(from [Santisteban et al., 2010]).
̅
̅
[1 1 0] ‖ [1 2 1 0] − (1 1 1) ≈ ‖ (0 0 0 1) −
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
(2 2 0) ‖ ND (1 1 1) ‖ RD
̅
(0 0 0 2) α−Zr ∥ RD (1 1 2 0) α− ∥ ND
Figure 4-83: Pole figures for zirconium hydride precipitates in and around the HAZ of the weld (Z1 and Z2) and in the bulk plate
material (Z3). On the left are the experimentally determined results obtained from analysis of a single diffraction image
and on the right are the calculated pole figures determined from the experimentally determined (not shown) pole figure
of the Zr phase and δhydride/Zr orientation relationships assumed to be given by [1 1 ̅ 0] ‖ [1 2 ̅ 1 0] −
and (1 1 1) ≈ ‖ (0 0 0 1) − (from [Santisteban et al., 2010]).
4.6.3.2
(1 1 1) δ (2 2 0) δ
ℎ
ℎ )
2
= ℎ √ℎ + + 2
2
1 = 4.7577 Å
(≡ )
1
Å
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
(1 1 1) δ (2 2 0) δ
1 10 6
Figure 4-84: Variations in lattice parameter versus ϕ (angle from RD) for hydride phase rings (a) (1 1 1) and (b) (2 2 0)
δ
δ
for specimens Z1, Z2 and Z3 indicating the magnitude of strains (stresses) inside the hydrides.The solid lines are fits to
the data.The elastic strains as a function of angle, ϕ, around the ring are defined by (ϕ) = ( (ϕ) − )/
assuming that = 1 (from [Santisteban et al., 2010]).
= 4.7577 Å
∆ = ND − RD
= ∙ ∆ (1 + )⁄ = 0.3
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
= 90°C
(1 1 1) δ (2 2 0) δ
Figure 4-85: A higher magnification view of the microstructure of the Z3 specimen showing the orientations of the hydride
precipitates and their locations within the αZr grains. The hydrides appear as dark-blue lines going from side to side
of the grains (from [Santisteban et al., 2010]).
(1 1 1) δ (2 2 0) δ
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
Figure 4-86: Variations versus temperature during dissolution (red) and precipitation (blue) in specimen Z3 of (a) hydride volume
fraction, (b) lattice parameter along ND and RD (c) differences between the lattice parameters along the ND and RD
and (d) average peak width of the (1 1 1) Debye ring (from [Santisteban et al., 2010]).
δ
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
̅
(1 1 1) δ (1 0 1 1) α−Zr
(2 2 0) δ (1 1 1) δ
∆ = ND − RD
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
(1 1 1) δ (2 2 0)
= 0° = 90°
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
4.6.4.1
α
( ) ( )
> 25°C
( )
( ) = (25°C) × (25°C)
α [H]
( )
[H] − ( )[H]
[H] = 1 − ( )
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
[H] =
[H] =
( ) =
α
α
( ) ( )
[H] +
[H] = 1 +
([H] − [H] )
′
=
1 − [H]
( )
′
=
(1 − ( ))
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
=
=
=
=
≡ [ ] Å ≡
[ ] [ ]
[ ]
= 2 and = 4
2([H] ( ) [ ]
= − [H] )
(1 − ( ))(1 − [H] ) [ ]
( )
[H]
α
( + ) [H]
⁄
H/Zr = 1.61
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
Figure 4-87: Variation of hydrogen in solid solution as determined from the temperature variation of the (3 1 1) peak intensity
δ
along the TD and RD of sample S1 (containing 475 wppm H) following the heat treatment procedure HT1 (5C/min)
(from [Zanelatto et al., 2012]).
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
( )
[H] = [H] (1 − (30°C) )
[H] =
[H] =
( ) =
(30°C) =
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
Figure 4-88: (a) Plots of the log of hydrogen concentration in solution versus reciprocal temperature during precipitation and
dissolution for samples S1 and S2 compared to selected data obtained with the use of other methods; (b) similar plots
to those in (a) but at lower cooling rate (from [Zanellato et al., 2012]); note that the log[H] versus 1/T relationship is now
linear down to 10 wppm hydrogen.
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
Figure 4-89: Variation of log hydrogen concentration versus reciprocal temperature during hydride precipitation and dissolution at
different heating and cooling rates for sample S1 (from [Zanellato et al., 2012]).
4.6.4.2
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
Figure 4-90: Plots (a) and (d): thermal cycles performed on S2 the purple solid line is the reference continuous cycle (HT3) and
the markers represent the interrupted cycles (HT4/HT5). Evolution of hydrogen content: plots (b) and (e) in the solid
solution and, plots (c) and (f), of the supersaturation (as defined in the text) during the corresponding heat treatments
(the same marker coding is used for all subplots for identifying the different stages (from [Zanellato et al., 2012])).
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
4.6.4.3
̅
{10.0} α−Zr ({1 0 1 1} α−Zr ) {00.2} α−Zr {0 0 0 2} −
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
Figure 4-91: Lattice strains in the matrix during HT1 on S1 (475 wppm H content) derived from changes in positions of the
{10.0} α−Zr ({1 0 1 ̅ 1} αZr ) and {00.2} α−Zr ({0 0 0 2} − ) reflections: (a) comparison with non-hydrogenated
material (the solid lines are linear fits for non-hydrogenated material) and, (b) comparison with the cell distortion model.
For clarity the strains derived from the {00.2} α−Zr reflection is shifted by 10 and only one fifth of the data points are
3
plotted (from [Zanellato et al., 2012]).
α = 5.77 ×
{10.0}
6
-1
6
10 K α = 7.62 × 10 K -1 {10.0} α−Zr {00.2} α−Zr
{00.2}
−6
−1
5.5 10.8 × 10 K
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
α
α
α
∆ ̅
[H]
̅̅̅̅ 1 − [H] ∆
∆ = ∆ ≡
2
= (√3 4) = α
⁄
[H] =
=
∆ =
α
∆
∆
α
= ≡ = 0.0317
2
1
≡ = 0.0519 0.0564 = 0.0329 =
3
0.0542
[H]
∆
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
̅̅̅̅
∆
√3
̅̅̅̅ ( ̅ ̅ − ̅ ̅ )
2
∆ = ̅ − ̅ =
4
=
̅ = ̅ (1 + )
2
2
2
̅ = ̅ (1 + ) = ̅ (1 + )
̅̅̅̅ √3 { ̅ (1 + ) ̅ (1 + ) − ̅ ̅ }
2
2
2
∆ = ̅ − ̅ =
4
̅̅̅̅ 2
∆ = ̅ {(1 + ) (1 + ) − 1}
≪ 1
̅̅̅̅
∆ = ̅ (2 + )
̅̅̅̅ ∆
∆
[H]
= ̅ (2 + )(1 − [H] ) ∆
̅ = (√3 4) ̅ ̅⁄ α
∆
∆ = Ω − (2 + )
∆
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
[H]
[H]
∆
∆
Ω −
̅
α
∆
α
= +
{10.0} α−Zr {00.2} α−Zr
= 1.49
6
= 4.4 × 10 K −1 α = 5.8 × 10 −6 K −1
6
α = 5.77 × 10 K −1 α = 7.62 ×
{00.2}
{10.0}
10 K −1
6
/
{10.0} α−Zr {00.2} α−Zr
α
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
Table 4-8: Lattice properties of hydrogen in αZr and in hydride; the latter as a function of hydrogen composition
corresponding to the ( phase boundary.
Part I. Hydride transformation strains as a function of temperature and composition based on pure lattice strain
transformation mechanism.
̅
− T − − − , − , − ̅ −
(H/Zr) (C − − − , − − , −
) − −
(Å /atom (Å /atom Zr) (10 (10
3
3
6
6
Zr) m /mol H m /mol
3
3
) H)
1.61 25 23.266 27.245 1.488 0.0446 0.0281 0.0542 0.0329 0.00965 0.00477 1.747
1.50 225 23.357 27.575 1.694 0.0494 0.0323 0.0542 0.0329 0.00482 0.00062 1.754
1.47 300 23.391 27.699 1.765 0.0510 0.0337 0.0542 0.0329 0.00323 –0.00083 1.757
1.42 400 23.437 27.845 1.870 0.0534 0.0358 0.0542 0.0329 0.00080 –0.00294 1.760
1.37 454 23.461 27.891 1.948 0.0564 0.0374 0.0542 0.0329 –0.00119 –0.00446 1.762
1.33 504 23.484 27.937 2.016 0.0571 0.0387 0.0542 0.0329 –0.00293 –0.00580 1.764
© ANT International 2018
Part II. Hydride transformation strains as a function of temperature and composition based on invariant plane
strain transformation mechanism.
− T − − ̅ − , − , − ̅ −
(H/Zr) (C) − − , − , −
− − − −
(Å /atom Zr) (Å /atom Zr) (10 (10
3
6
3
6
m /mol H) m /mol H)
3
3
1.61 25 23.266 27.245 1.488 0.1062 0.0 0.0542 0.0329 –0.0520 0.0329 1.747
1.50 225 23.357 27.575 1.694 0.1204 0.0 0.0542 0.0329 –0.0662 0.0329 1.754
1.47 300 23.391 27.699 1.765 0.1253 0.0 0.0542 0.0329 –0.0711 0.0329 1.757
1.42 400 23.437 27.845 1.870 0.1325 0.0 0.0542 0.0329 –0.0783 0.0329 1.760
1.37 454 23.461 27.891 1.948 0.1378 0.0 0.0542 0.0329 –0.0836 0.0329 1.762
1.33 504 23.484 27.937 2.016 0.1426 0.0 0.0542 0.0329 –0.0884 0.0329 1.764
© ANT International 2018
Ω − = √3 α
2
Ω −ℎ = 4 3 −ℎ
̅
−ℎ (Ω −ℎ − Ω − )/
−ℎ −ℎ −ℎ
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
−ℎ
α
α
̅ −
2
2
= Ω − ( + 2 + 2 + + )
̅ −
Ω −
̅ −
−
α
3
m
̅ −
= 1.700 × 10 −6 mol H 14
̅
−ℎ
α
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
Figure 4-92: Shape used in finite element model of hydride plate embedded in αZr matrix. The axis of rotation of this plate is along
the 2 direction. Taking account of the symmetry properties of this plate, only a quarter of it is shown. The numbers
beside the arrows refer to the 1 and 2 coordinates. These are referred to as x and y coordinates in Figure 4-93 (from
Singh et al., 2013]).
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
Figure 4-93: Contour plots of the 22 stress (referred to as S22 in the figure) for, a) and c): hydride formed at 25 and 400C,
respectively, from a stress-free matrix and, b) and d): region of prior hydride and matrix in a) and c) after complete
dissolution of hydride (from [Singh et al., 2013]).
22
22
, = 0
22
22
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
(α + )/
α
(α + )/
α
(α +
)/
4.6.4.4
α
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
Figure 4-94: Lattice parameter values derived from the d-spacing of hydride reflections from three different sets of planes during
HT1 on S1: (a) during heating, and (b) during cooling. Only one fifth of the data points are shown. The evolutions of the
strains relative to the room temperature values are plotted in (c) and (d) for heating and cooling, respectively. For
reference, the unconstrained (free) lattice strain derived from the variation of the hydride lattice parameter is
represented as a solid reddish-brown line. (The line is based on combining the hydrogen composition variations with
temperature at the (α + )/ phase boundary as given in [Zuzek et al., 2000] with the lattice parameter variation with
temperature and hydride H/Zr ratio given by Singh and co-workers [Singh et al., 2007]). For comparison with the
evolution of the relative strains in the hydride phase, the changes in strains in the αZr matrix obtained via shifts in the
̅̅̅̅
{10.0} − ({1 0 1 1} − ) peak from Figure 4-91b is plotted as a solid blue line (from [Zanellato et al., 2012]).
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
{10.0} α−Zr
(α + )/
4.6.4.5
̅
̅
[1 1 0] ‖ [1 2 1 0] − (1 1 1) ≈ ‖ (0 0 0 1) −
{0 0 0 2} −
{0 0 0 2} −
̅
{1 0 17} −
̅
{1 0 17} −
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
(ℎ ) = ∙ (ℎ )
=
+ 2
=
=
=
=
(ℎ )
(ℎ ) = (1 + ) (1 − 2 ) ( − ) + ( − )}
{
= /2(1 + ) = [(1 + )⁄ (1 − 2 )]
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
{
(ℎ ) = (1 + ) (1 − 2 ) ( − ) + ( 11 − )}
11
11
{
(ℎ ) = (1 + ) (1 − 2 ) ( − ) + ( 22 − )}
22
22
(ℎ ) = (1 + ) (1 − 2 ) ( − ) + ( 33 − )}
{
33
33
− = ( 11 + 22 + ) − ( 11 + 22 + )
33
33
−5
= 0.436 − 4.8 × 10 ∙ ( (K) − 300)
−2
= 95.9 − 5.74 × 10 ( (K) − 273)
1
= /
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
Table 4-9: Stresses (elastic solution) inside a spheroidal hydride precipitate (subscript ⊥ ( subscripts 11 in
Equation 4-77 to Equation 4-79)) refers to the direction of axis of rotation of the spheroid; i.e., the direction
along the semi-minor axis of length, c, whilst subscript ‖ ( subscripts 22 or 33 in Equation 4-77 to
Equation 4-79) refers to the direction along the semi-major axis of length, a); = ( + 2 )/3 is the
‖
⊥
mean stress; ∆ ℎ is the hydride-matrix strain energy. Part A: Values of hydride transformation strains
and taken from Table 2.1 in [Puls, 2012], derived assuming hydride forms via a pure lattice strain
⊥
‖
transformation. Part B: Values of strains are given by the fractional volume changes listed in Table 2.1
⊥
T
of [Puls, 2012] with the values of the e strains equal to zero.
‖
IA. Hydride stresses as a function of temperature- and composition-dependent hydride transformation strains:
pure lattice strain transformation strains.
T(C) = / (MPa) (MPa) (MPa) ∆ (MJ/m )
3
‖
⊥
⊥
‖
25 1.61 0.0717 0.0453 0.1 1 297 7 260 5 273 375.4
225 1.50 0.0742 0.0484 0.1 1 037 5 806 4 216 319.6
300 1.47 0.0751 0.0496 0.1 953.2 5 339 3 887 300.6
400 1.42 0.0760 0.0509 0.1 847.4 4 744 3 445 273.7
454 1.37 0.0759 0.0512 0.1 788.8 4 411 3 204 255.9
504 1.33 0.0759 0.0515 0.1 738.0 4 123 2 995 240.3
© ANT International 2018
IIA. Hydride stresses as a function of temperature at constant hydrogen composition, elastic moduli* and
spheroidal aspect ratio: pure lattice strain transformation strains.
25 1.47 0.0704 0.0440 0.1 871.0 4 753 3 459 239.8
225 1.47 0.0738 0.0481 0.1 930.9 5 182 3 765 283.6
300 1.47 0.0751 0.0496 0.1 953.2 5 339 3 877 300.6
400 1.47 0.0767 0.0516 0.1 982.0 5 547 4 026 323.9
454 1.47 0.0782 0.0536 0.1 1 010 5 756 4 174 348.0
504 1.47 0.0783 0.0538 0.1 1 012 5 777 4 189 350.5
© ANT International 2018
IIIA. Hydride stresses as a function of temperature, hydrogen composition, elastic moduli and spheroidal aspect
ratio: pure lattice strain transformation strains.
25 1.61 0.0717 0.0453 0.1 –1 297 7 260 5 273 375.4
225 1.50 0.0742 0.0484 0.2 –1 887 5 560 4 335 339.3
300 1.47 0.0751 0.0496 0.3 –2 384 4 912 4 069 333.1
400 1.42 0.0760 0.0509 0.4 –2 603 4 202 3 669 312.7
454 1.37 0.0759 0.0512 0.5 –2 807 3 777 3 454 299.9
504 1.33 0.0759 0.0515 0.6 –2 933 3 423 3 260 287.6
© ANT International 2018
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
IB. Hydride stresses as a function of temperature- and composition-dependent hydride transformation strains:
invariant plane strain transformation strains.
25 1.61 0.171 0.0 0.1 1 549 1 222 1 331 132.5
225 1.50 0.181 0.0 0.1 1 317 925.8 1 056 119.1
300 1.47 0.184 0.0 0.1 1 241 832.2 968.5 114.4
400 1.42 0.188 0.0 0.1 1 141 716.0 857.6 107.4
454 1.37 0.189 0.0 0.1 1 081 653.4 796.0 102.1
504 1.33 0.190 0.0 0.1 1 029 599.8 743.0 97.58
© ANT International 2018
IIB. Hydride stresses a function of temperature, hydrogen composition, elastic moduli and spheroidal aspect
ratio: invariant plane strain transformation strains.
25 1.61 0.171 0.0 0.1 1 549 1 222 1 331 132.5
225 1.50 0.181 0.0 0.2 2 564 1 562 1 896 232.0
300 1.47 0.184 0.0 0.3 3 479 1 787 2 351 320.1
400 1.42 0.188 0.0 0.4 4 092 1 754 2 534 384.7
454 1.37 0.189 0.0 0.5 4 648 1 734 2 705 439.2
504 1.33 0.190 0.0 0.6 5 083 1 664 2 804 482.8
*Constant moduli values used are those corresponding to T = 300C: = 0.3050; E = 78.68 10 MPa
3
© ANT International 2018
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
Table 4-10: Strains (elastic solution) inside a spheroidal hydride precipitate (subscript ⊥( subscripts 11 in
Equation 4-77 to Equation 4-79)) refers to the direction of axis of rotation of the spheroid; i.e., the direction
along the semi-minor axis of length, c, whilst subscript ‖ ( subscripts 22 or 33 in Equation 4-77 to
Equation 4-79) refers to the direction along the semi-major axis of length, a. Part A: Values of hydride
transformation strains and taken from Table 2.1 in [Puls, 2012], derived assuming hydride forms via
‖
⊥
a pure lattice strain transformation. Part B: Values of strains are given by the fractional volume changes
⊥
T
listed in Table 2.1 of [Puls, 2012] with the values of the e . strains equal to zero. ( = 11 + 22 + ;
33
‖
= 11 + 22 + ; ≡ ; ≡ 22 = ).The reference states for the delta strains (e.g.,
33
11
33
‖
∆( − )) in each table part are those for which the delta strains are equal to zero.
IA. Hydride strains as a function of temperature- and composition-dependent hydride transformation strains:
pure lattice strain transformation strains.
T(C) − ∆( − ∆( ‖ ∆(
‖
‖
‖
⊥
3 3 = / − ) − ) − − )
3
(10 ) (10 ) (10 ) ‖
3
(10 ) (10 ) (10 ) (10 ) (10 )
3
3
3
3
25 1.6 71.7 45.3 0.1 53.43 0 37.27 0 21.11 0
1
225 1.5 74.2 48.4 0.1 35.22 18.22 41.85 4.58 48.48 27.37
0
300 1.4 75.1 49.6 0.1 29.27 24.16 43.47 6.20 57.67 36.55
7
400 1.4 76.0 50.9 0.1 21.81 31.62 45.34 8.07 68.88 47.77
2
454 1.3 75.9 51.2 0.1 17.89 35.54 45.96 8.69 74.03 52.92
7
504 1.3 75.9 51.5 0.1 14.47 38.96 46.54 9.27 78.61 57.49
3
© ANT International 2018
IIA. Hydride strains as a function of temperature at constant hydrogen composition, elastic moduli* and aspect
ratio: pure lattice strain transformation strains
25 1.4 0.1 25.78 0 38.61 0 51.45 0
7 70.4 44.0
225 1.4 0.1 28.34 2.56 42.17 3.55 56.00 4.55
7 73.8 48.1
300 1.4 0.1 29.27 3.50 43.47 4.86 57.67 6.22
7 75.1 49.6
400 1.4 0.1 30.53 4.75 45.20 6.59 59.88 8.43
7 76.7 51.6
454 1.4 0.1 31.78 6.01 6.93 8.32 62.09 10.64
7 78.2 53.6
504 1.4 0.1 31.91 6.13 47.11 8.49 62.30 10.85
7 78.3 53.8
© ANT International 2018
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
IIIA. Hydride strains as a function of temperature, hydrogen composition, elastic moduli and spheroidal aspect
ratio: pure lattice strain transformation strains
25 1.6 71.7 45.3 0.1 53.43 0 37.27 0 21.11 0
1
225 1.5 74.2 48.4 0.2 22.95 30.48 36.40 0.87 49.85 28.74
0
300 1.4 75.1 49.6 0.3 7.78 45.65 34.15 3.12 60.52 39.41
7
400 1.4 76.0 50.9 0.4 6.09 59.52 33.63 3.64 73.35 52.24
2
454 1.3 75.9 51.2 0.5 15.20 68.63 32.30 4.97 79.81 58.69
7
504 1.3 75.9 51.5 0.6 22.63 76.07 31.46 5.81 85.56 64.45
3
© ANT International 2018
IB. Hydride strains as a function of temperature- and composition-dependent hydride transformation strains:
invariant plane strain transformation strains
25 1.6 171.0 0.0 0.1 5.09 0 0.12 0 5.33 0
1
225 1.5 180.8 0.0 0.1 8.27 3.17 1.94 1.82 12.15 6.82
0
300 1.4 184.4 0.0 0.1 9.32 4.23 2.54 2.42 14.41 9.08
7
400 1.4 188.3 0.0 0.1 10.60 5.51 3.27 3.16 17.15 11.82
2
454 1.3 188.9 0.0 0.1 11.16 6.07 3.62 3.50 18.39 13.06
7
504 1.3 189.6 0.0 0.1 11.66 6.57 3.92 3.80 19.50 14.17
3
© ANT International 2018
IIB. Hydride strains at 300C as a function of temperature, hydrogen composition, elastic moduli and
spheroidal aspect ratio: invariant plane strain transformation strains
25 1.61 171.0 0.0 0.1 5.09 0 0.12 0 5.33 0
225 1.50 180.8 0.0 0.2 18.06 12.97 1.87 1.75 21.80 16.47
300 1.47 184.4 0.0 0.3 30.37 25.28 2.30 2.18 34.97 29.64
400 1.42 188.3 0.0 0.4 43.75 38.65 3.45 3.33 50.65 45.32
454 1.37 188.9 0.0 0.5 55.08 49.99 3.72 3.60 62.51 57.18
504 1.33 189.6 0.0 0.6 65.60 60.51 3.99 3.87 72.59 68.26
Constant moduli values are those corresponding to T = 300C: = 0.3050; E = 78.68 103 MPa
© ANT International 2018
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
+ 2
=
−
=
+ 2
−
−
≡ σ (ℎ ) = − = ( − ) + 2 ( − )
−
−
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
( − ) = 0
−
Δ( − )
.
1
2 3 ‖
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
−
⊥
̅
(1 1 1) ∥ (0 0 0 1) − [1 1 0] ∥
̅
[1 2 1 0] −
{ 0 0 0 1} −
̅
{1 0 1 7} α−Zr
≈ {0 0 0 1} −
‖
(1 1 1) δ
(1 1 1) δ
(1 1 1) δ
‖
(1 1 1) δ
(1 1 1) δ
(1 1 1) δ ‖
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
(α + )/
= 1.47. ⊥
‖
‖
‖
(α + )/
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
⊥
⊥
∥
⊥
⊥
⊥
⊥
⊥
⊥
⊥
⊥
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
⊥ ⊥
⊥
⊥
⊥ ⊥
( − ) Δ( − )
,
− , − ‖
‖
− ‖ −
‖
−
‖
( − )
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
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