The Effect of Hydrogen and Hydrides - ebook first test

 







   

















/






































Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.

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








































( )
′
(∇ ) 2






= ∫ [ ( ) + (∇ ) + ⋯ ]
2
′





1
2
′
2
2
]
= −[ ∇ ] + [ ( |∇ |)⁄ 2 ′
⁄
2
∇
2
∇ 





2
2
2
( , ∇ , ∇ , … ) = ( ) + ∇ + (∇ ) + ⋯
1

2

Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.

2
2
= ∫ [ ( ) + ∇ + (∇ ) + ⋯ ]
2

1

2
∇
1

2
∫ ( ∇ ) = − ∫ ( )(∇ ) + ∫ ( ∇ ∙ )
2
⁄
1
1
1

∇ ∙ = 0
2
( ∇ )
1
⁄
( ⁄ )(∇ ) 2 = − + 2
1
1













( ) , < 0 ( ) , = 0
2
⁄
2
2
⁄
2






























 



Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.

̅
(1 0 17) −
 (0 0 0 1) − (0 0 0 1) −
̅
〈1 1 2 0〉 α−










̅
( , ), ( , ), ( , ) 〈1 1 2 0〉 α−
2
3
1







( , ) ( , )




( , )

( , ) 19




( , )


= − ( , ) + ( , )


( , ) = ∇ 2 + ( , )

( , )

= ( , )
=


=



( , ) ( , ) =

( , ) =












⁄
( , )



Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.


2
2
3
= ∫ [ ( ( ( ))) + ∑ (∇ ( )) + (∇ ) ] + +




=1 2 2
=
 =
=













2
3
2
= ∫ [ ( , ) − ( ) + ∑ 2 (∇ ( )) + 2 (∇ ) ]






=1



( , ( )) − ( , = 0)







( ) = ( , ( )) + − ( − )










= = 
,


= = 
,


=



, ,


Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.

( , ) =



1 2 2 2 3 4 4 6





2 ( − , ) + 2 ( − , ) ∑( ) − 4 ∑( ) + 6 ∑( ) +
=1 =1 =1

4
2
2 2
∑ + 6 ∑ ( + ) + 7 ∑
2 2 2
2





5
≠ ≠ , ≠ ≠ ≠
1 7 1 4

, ( , ) = 0 , ( , ) =


1 or − 1 5
7














( )














( )



















Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.


1


= 2 ∑ ( ) ( )



=1
=
=
ℎ










( ) ( )



1

( ) =
− ( )
2

( )

 
( )




1

− ( ) ]
= ∫ [
2






̃


( ) = ∑ ( ) ( )



=1
̃














( ) = ( ) = − ( ) +




Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.

( )


( ) = 0




− ( ) +
( ) = 0


( ) ( )




̃


( ) = ∑ ( ) ( )



=1


( ) =
( )




1

= ∫ [ ̃
− ∑ ( ) ( ) +]
2 =1








̅

( )





( ) = ( ) − ̅








∫ ( ) = 0


̅














Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.

ℎ

= + ℎ



ℎ
= 2
̅ ̅ − ∑ ( ) ̅
=1



1
̃

ℎ
= ∫ [ − ∑ ( )Δ ( ) ]





2
=1
̃
Δ

̃
̃
Δ ( ) = ( ) −


̃
∫ ( ) =

̃
( )



̅ ( )







1
(
= 2 + )








∂
̅ = 0





∂ = 0
( )





Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.




̅ − ∑ ( ) = 0
=1

( )





̅ = ∑ ( )


=1

⁄













ℎ
= − 2 ∑ ∑
( ) ( )
=1 =1


 
( )






3
1 ∗
̃
̃
ℎ
= − ∑ ∑ ∫ (2 ) 3 ( ){Δ ( )} {Δ ( )}



2

=1 =1


( ) = ̃ ( )Ω ( ) ̃ ( )





̂
̂
( ) = ̂ ( ) ( ) ̂ ( )


−1
where Ω ( ) = Ω =




̃ ( ) = Fourier transform of ( ) (=

̃ ( )) with ̃ ( ) =




3
∫ ( ) (− )

Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.

⁄
= = )
̃
̃
{Δ ( )} = Δ ( )


̃
̃
{Δ ( )} ∗ = {Δ ( )}


̃
̃
= 0 {Δ ( )} { ( )}


{ ( )} ≠ 0
̃
̃
̃
{∆ ( )} ≡ ∆ ( ) = ∫ [ ( ) − ] − ∙ = { 0














= 2 ∑ ( ) ( ) − 2 ∑ ∑ ( ) ( )






=1 =1 =1
3
1 ∗
̃
̃
− ∑ ∑ ∫ ( ){ ( )} { ( )}
2 (2 ) 3
=1 =1
 = /


̅
 































Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.


















̃
( )

̃
( )



( )










( ) = 0



















Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.




( ) = ∑ ( ) ( )
2



=1
± 1

→ −

( )



( )


( )


( ) ( )


( ) 



( )




( ) = ( ) − ( ) ≡ ( ) − ( ) = ( ) − ∑ ( ) ( )


2










=1


( )) 




2

( ) =
= [ ( ) − ∑ ( ) ( )]




=1

( )
2


2
= ∑ ( ) ( ( ))




=1
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.

( )





2
( ) = − ( ) ∑ ( ) { ( )}





=1
ℎ
( ) = ∫ ( ) − ∙





2
2

{ ( )} = ∫ ( ) − ∙




G ( )

⁄
= Ω ( )
(G ( )) = 2 2 −1 =
−1
=

Ω
1



( ) ( )
= 2 ∫ 3




2 ̅̅̅̅̅̅̅̅̅̅̅̅̅̅



2
̅ ̅ − ∑ ̅ ( ) ( ) +
= 2 2 ̅̅̅̅̅̅̅ 2 ∑ ∑ ( ) ( ) ( ) ( )




=1 =1 =1

1

3
2
+ ∫ [ − ∑ ( ) ( )]



2
=1
(…̅ ) (… ) ( ) ( )
2



2 ̅̅̅̅̅̅̅̅̅̅̅̅̅̅




̅ ̅ −
= 2 2 ̅̅̅̅̅̅̅ 2 ∑ ∑ ( ) ( ) ( ) ( )
̅ ∑ ( ) ( ) +
2





=1 =1 =1

3
1 ∗
2
2
− ∑ ∑ ∫ ( ){ ( )} { ( )}
2 (2 ) 3
=1 =1


( ) = ( )Ω ( ) ( )




Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.

̂
̂


( ) = ̂ ( ) ( ) ̂ ( )
 































̅ ≡




2 ̅̅̅̅̅̅̅̅̅̅̅̅̅̅






∑ ( ) ( ) +
= 2 2 ̅̅̅̅̅̅̅ 2 ∑ ∑ ( ) ( ) ( ) ( )
−
2





=1 =1 =1

3
1 ∗
2
2
− ∑ ∑ ∫ ( ){ ( )} { ( )}
2 (2 ) 3
=1 =1
,

= −
,


,
( ) 2 ̅̅̅̅̅̅̅
̅ −



̅ = ∑ ( ) ( ) − = 0
=1
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.


2 ̅̅̅̅̅̅̅


̅ = ∑ ( ) ( ) +




=1




−

2
3

∗
,


= ∑ ∑ ( ) ( )⌈ 2 ̅̅̅̅̅̅̅ 2̅̅̅̅̅̅̅ 2 ̅̅̅̅̅̅̅ 2̅̅̅̅̅̅̅ 1 (2 ) 3 ( ){ 2 ( )} { 2 ( )}
( )⌉ − ∑ ∑ ∫
( )
( )
( ) −


2
2

=1 =1 =1 =1

( ) ( )


− − ∑ 2 ̅̅̅̅̅̅̅
2
=1
























α 





,



, ( ) 





Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.

,

, ( )






,
,
  ′


( )

( ) ′ , ( )





,
,
,
,
2
( ) = ∑ ( ) ( ) +  ( ) + ′ ( )


=1
 ( )

,
( )





,










 α
 


,
 ( )


α



α


α
⁄
 ℎ






Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.

2
, ( ) ( )



α   α


α α α 

,
′ ( )
1
′ ( ) = ( ) − ( )



3



1
,
,
′ ( ) = , ( ) − ( )
3









,
[′ ( , )]

,
= − ′ ( , )

=


,
,
= ′ ( , ) = ′ ( )

=



















ℎ = +



1 1

,
,
,
,
3
3

3
= ∫ ( ) ( ) − 2 ∫ ( −) ∫ ( ) ′
′
2

3
1




∗
− ∫ ̃ ( )Ω ( ) ̃ ( ) − ∫ , ( ) −
3
2 (2 ) 3 2
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.

1 1
3
′
3
3
= ∫ ′ , ( )′ , ( ) − ∫ ′ , ( ) ∫ ′ , ( ) ′

2 2
3
1


∗
3
− ∫ ′ ̃ , ( )Ω ( ) ′ ̃ , ( ) − ′ ∫ ′ , ( ) − ′ ′
2 (2 ) 3 2
′ ′ = ′

,
,
,
3
′ ̃ ( ) ′ ̃ ( ) = ∫ ′ ( ) (− ∙ )



3
∂


∗
( ) = − = ∫ [ ̃ ( ) − ( ) ] (− ∙ ) +
Ω ( ) ̃


,
( ) (2 ) 3


,
( )








1 7























̅
〈1 1 2 0〉 α−
(0 0 0 1) − 


Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.

̅
̅
[1 1 0] ‖ [1 2 1 0] −

(1 1 1) ‖ (0 0 0 1) −

̅
〈1 1 2 0〉 α− (0 0 0 1) −
̅
〈1 1 2 0〉 α− (0 0 0 1) −



































Figure 5-1: Shape change of hydride spherical particles grown in the Zr matrix as governed by their elastic energy (from
[Ma et al., 2002]). Each particle was identified with a different order parameter, , according to the three orientation
variants.




































Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.

∗
∗
Figure 5-2: Simulated formation process of hydride precipitates in Zr in a 512  512 grid: (a) = 1000, (b) = 2000,
∗
∗
(c) = 4000, (d) = 6000 (from [Ma et al., 2002]).





































Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.

Figure 5-3: hydride precipitation under an external uniaxial load applied vertically in the figures (from [Ma et al., 2002]. Load was
applied during both the nucleation and growth process. The 〈1 1 2 ̅ 0〉 α− direction of the matrix is along the
∗
horizontal axis: (a) to (c), strain = 0.14%, = 500, 1000, 5000, respectively; (d) to (f): strain = 0.42%,
∗
= 500, 1000, 5000, respectively














































Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.

Figure 5-4: Microstructural evolution under different constant strain conditions. As in Figure 5-3, the 〈1 1 2 ̅ 0〉 α− direction of the
matrix is along the horizontal axis (from [Ma et al., 2002]).The external load (actually constant strain) of 0.42% was
applied in the vertical direction: (a) to (c), externally constrained nucleation followed by externally non-constrained
∗
variant growth, = 500, 1000, 5000, respectively; (d) to (f), externally non-constrained nucleation followed by
∗
externally constrained variant growth, = 1000, 2000, 5000, respectively.













































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


























































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Figure 5-5: Hydride precipitation process showing the evolution of hydride precipitate growths in a continuum matrix: (a) to (d):
∗
= 1000, 2000, 3000, 5000 (from [Guo et al., 2008a]).



























Figure 5-6: Comparison of computer simulation (a) with experimental observation (b) of hydride formation under a uniform uniaxial
external tensile stress in the vertical direction (from [Guo et al., 2008a]).









Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.





























































Figure 5-7: Applied tensile stress as a function of evolution time at which hydride cracking was first observed versus average
hydride length (experimental data is from [Shi & Puls, 1999], theoretical results (PFM) (from [Guo et al., 2008a]).
























Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.




























Figure 5-8: The variation of the stress, (in the vertical direction of the figure) in and around a plate-shaped hydride calculated
assuming: (a) elastic matrix, elastic hydride, (b) elastoplastic matrix, elastic hydride (from [Guo et al., 2008a]).The edge
of the hydride plate in the horizontal direction is located at the point where abruptly changes from negative
(compressive) to positive (tensile) values. The platelet has its lowest magnitude of compressive stress in the plate
normal direction over the middle portion inside the hydride platelet.















 





̅
〈1 1 2 0〉 α− (0 0 0 1) −












Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.


ℎ = 0.327  = 0.324 ℎ = 1.083  = 0.520
 
̅
̅
[1 0 1 0], [1 2 1 0], [0 0 0 1]
1
2
3




= = 0.0094; 33 = 0.0413 = 0.

22
11


̅
〈1 1 2 0〉 α− (0 0 0 1) −






̅
〈1 1 2 0〉 α−
 15
14









 





























̅
〈1 1 2 0〉 α−




Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.

 







̅
〈1 1 2 0〉 α−





̅
〈1 1 2 0〉 α−
























































Figure 5-9: Temporal evolution of a population of hydride precipitates when an external tensile load of (a) 0, (b) 300 and (c)
600 MPa is applied along the 〈2 1 ̅ 1 ̅ 0〉 − direction; t̃ is the number of dimensionless time steps (from
[Thuinet et al., 2013]).







Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.



















 






































 Zr H
2


















Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.












 

















 






α  




































Figure 5-10: (a) Crystallographic structure of α Zr: Zr atoms (respectively tetrahedral interstitial sites) are empty circles (respectively
full spheres); (b) Tetrahedral interstitial sites of H atoms in Zr2H () and ZrH () hydrides (●, H; ○, unoccupied site)
(from [Thuinet & Besson, 2012]).






Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.

Table 5-1: Symmetry (H= hexagonal, T= trigonal), equilibrium cell parameters, volumes of zirconium and zirconium
hydrides (from ab initio calculations) and transformation strains relative to the Zr phase


Space Point group a (Å) c (Å) V (Å ) Hydride transformation
3
group strains with  phase
T
T
e (%) e (%)
33
11
Zr P63/mmc 6/mmm (H) 3.239 5.188 47.13
Zr2H () P3m1 3m1 (T) 3.266 10.825 100.00 0.8 4.3
̅
3m1 (T)
ZrH () P3m1 ̅ 3.269 5.677 52.53 0.9 9.4
© ANT International 2018


   












 





14










4
=

2
2

[ 3 12 +2 44 ] 14 ( ) 2
12 +2 44 44 11


11 = 22
12 44 14
















Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.

Table 5-2: Elastic constants (in GPa) determined from the ab initio calculations with cell internal relaxations for pure
Zr (), Zr2H () and ZrH () phases. Values of elastic constants in parentheses for  Zr were obtained
experimentally at T = 4 K.

Zr () Zr2H () ZrH ()
C11 142 (155) 155 177

C12 62 (67) 84 93
C13 64 (65) 66 66
C33 164 (172) 186 227
C44 29 (36) 22 32
C14 0 –23 12
© ANT International 2018





















 = /

  



 



























Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.

Figure 5-11: Bulk formation free energies of the ,  and zirconium hydride phases, obtained from point defect modelling at
various temperatures (from [Thuinet & Besson, 2012]).









   
 
and 35 mJ/m 2  
≈ 0 and 35 mJ/m 2






 


(%) ≫ (%)
33
11
̅ ≈ 0.

< 0.6%.
̅ = 0

′




(%)/ (%)( ) > (%)/ (%)(),
11
11
33
33














Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.

Figure 5-12: Evolution of the elastic strain energy density associated with the nucleation of and  precipitates in  Zr with shape
factor,  = / (from [Thuinet & Besson, 2012]).








a ∗ o ε ∗

 =
650 K
8
2.67 × 10 J/m 3


ZrH 0.3 ZrH 0.4 a ∗ o 

(< 5% < 0.1
 





Zr 2- H

Zr H to Zr H 
4
2


 












Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.

 ε = 0.02
∗
ε ∗












































































Figure 5-13: Influence of hydrogen content, , on critical nucleus radii of (ZrH0.3 to ZrH0.4)  and stoichiometric hydride

phases for (a) T = 650 K; (b) T = 300K (from [Thuinet & Besson, 2012]).

Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.

Figure 5-14: Comparison between (ZrH0.3 to ZrH0.4)  and stoichiometric hydride phases at 650 K of the critical shape factor
SS
(plot (b shown here of plots (a) and (b) in the paper), as a function of hydrogen content, X (from [Thuinet & Besson,
H
2012]).








 


 

 
   

 
















Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.































 






















 
 
 
̅
{1 0 1 7} αZr  














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.


≈ 4C









í




 













= ⁄

= 


=







= ⁄ ( + )





= ⁄ (1 + )










Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.












































( , %) ≈ 2%








(0) ≅ 2%


( , %) ≈ 98%














100
( , %) =

(1 + 49 ∙ [− ∙ ])
3
(∆ )
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.

( , %)


( , %)

















( , %)



( , %) = 98%

( , %) = 3% ( , %)


(%) = 2%

( , %) 


( , %)

























 

















Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.



































































 








Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.

  





























( , %)








( , %) = 98%


































Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.








⁄
> 0.83
1
2
1
⁄
> 0.57
1
2













⁄
⁄
 = 0  = 0 . 8
1
2
1
2






1





= 1 3 ( + )
⁄
1

2







Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.





























































(40) = 40°
(45)









Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.

 






(45)  
 
 
 
  










(40) (45)

(40) (45)
(40) (45)

(40)
 


(40) = ∙ ( )

(40)  (40)

(40)







Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.

⁄
⁄
log( (40)) 1 1

log( (40))




⁄
1


⁄
log( (40)) 1 1/




log( ( 40)) log( (40)) 1 1/
⁄










































Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.












 








































( )





Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.