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Excellence through CollaborationExcellence through Collaboration
L. GutkinD.A. Scarth
Kinectrics Inc.Toronto, Ontario, Canada
Third International Seminar on Probabilistic Methodologies for Nuclear Applications, Rockville MD, October 22 - 24, 2019
Estimation of Threshold Parameter and Its Uncertainty Using
Multi-Variable Modeling Framework for Response Variable with
Binary Experimental Outcomes
2ISPMNA 2019 – UC-001
Outline
Background
Approach
Modeling Framework
Analysis Results
Summary
Scal
ed F
requ
ency
0
0.2
0.4
0.6
0.8
1(X 10000)
Threshold Stress at Planar Surface
0
0.2
0.4
0.6
0.8
1
Prob
abili
ty o
f DH
C in
itiat
ion
Nominal applied stress (away from flaw tip)
Lower threshold stress Higher threshold stress
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Prob
abili
ty o
f DH
C in
itiat
ion
Nominal applied stress (away from flaw tip)
Reference relation
Updated relation
3ISPMNA 2019 – UC-001
BackgroundPressure tubes in CANDU reactors
® CANDU is registered trademark of Atomic Energy of Canada Ltd.
CANDU® reactor core: 380 to 480 fuel channels with nuclear fuel and pressurized heavy water coolant
4ISPMNA 2019 – UC-001
BackgroundDelayed hydride cracking in CANDU pressure tubes
Formation and growth of Zr hydrides at in-service flaws
Corrosion reaction of Zr-based material with heavy water
Zr-2.5Nb pressure tubes
Preferential diffusion of hydrogen to locations of stress concentration
Increasing content of hydrogen, in the form of deuterium
In-service flaws in Zr-2.5Nb pressure tubes are evaluated
for DHC initiation
Potential fracture of hydrides under applied loads
Potential crack initiation and growth by Delayed Hydride
Cracking (DHC)
5ISPMNA 2019 – UC-001
DHC initiation occurs when applied stress exceeds threshold stress
Model based on strip-yield process-zone approach is used to predict threshold stress for DHC initiation (CSA Standard N285.8)
BackgroundThreshold stress for DHC initiation
1 IH yieldth tip th t
thth
σK aσ σ Ψ k wσσ w
(ps), (ps)(ps)
, , ,
threshold stress intensity factor for DHC initiation from cracks
threshold stress for DHC initiation at planar surface
yield stress
flaw depth
pressure tube wall thicknessstress concentration factor
a tip th tip
a n th n
σ σ
σ σ, ,
, ,
threshold stress for DHC initiation applied stress
6ISPMNA 2019 – UC-001
ApproachParameter estimation from experimental data
Statistical characterization of A as model output or directly
Model parameter A: Best estimate and uncertainty
Auxiliary model with variable A
as output
Statistical characterization of A as model input
Model parameter A: Experimental data are available or feasible to obtain
Experimental data for variable A
Experimental data for variable B correlated with A
Auxiliary model with variable B as output, variable A as input
yes no
7ISPMNA 2019 – UC-001
ApproachOverview of approach to characterization
Obtaining reliable experimental data for DHC initiation at planar surfaces is extremely challenging
Specimens with flaws of known and reproducible geometry
DHC initiation experiments
Predictable and reproducible morphology of flaw-tip hydrides
Predictable and reproducible stress concentration at flaw tip
Statistical characterizationof parameter
Experimental data for DHC initiation at flaws of known and reproducible geometry
Auxiliary model with as parameter and probability of DHC initiation as response
thσ (ps)
thσ (ps)
thσ (ps)
8ISPMNA 2019 – UC-001
ApproachResults of DHC initiation experiments
If multiple results are available at a given test condition, proportion of failures can be used
Expe
rimen
t res
ultin
g in
failu
re
Nominal applied stress (away from flaw tip)
Less severe flaw More severe flaw
yes
no
9ISPMNA 2019 – UC-001
ApproachAuxiliary model for probability of DHC initiation
Position of zero-to-one transition in probability of DHC initiation along
applied stress axis
a) Flaw geometryb) Material resistance
to DHC initiation, including
Threshold stress for DHC initiation
thσ (ps)
0
0.2
0.4
0.6
0.8
1
Prob
abili
ty o
f DH
C in
itiat
ion
Nominal applied stress (away from flaw tip)
Lower threshold stress Higher threshold stress
10ISPMNA 2019 – UC-001
ApproachLogistic regression analysis
L(ΠCI) = L(Π̃CI) + Π
expected probability of DHC initiation
observed probability of DHC initiation
residual uncertainty
( ) ln 1CI
CICI
πL Π π
aCI σ
th
σL Π B σ( ) ln
πCI: proportion of failures
Bσ: empirical parameter 0
0.2
0.4
0.6
0.8
1
Prob
abili
ty o
f DH
C in
itiat
ion
Nominal applied stress (away from flaw tip)
Lower threshold stress Higher threshold stress
11ISPMNA 2019 – UC-001
Modeling FrameworkGeneralized predictor in auxiliary model
a tip a nCI σ σ
th tip th n
σ σL Π B Bσ σ
, ,
, ,( ) ln ln
, ,a tip a ntσ σk
stress concentration factor
flaw-tip applied stress nominal applied stress
IHC
th
tth tip
hIσ
λ
wσ
λ Zw
KaF
σ
(p
,
s)
(ps)1
1 IH yieldth
ht tip
tthhσ
σKσ
aλΨ wσσ w
(ps)(ps)(p, s)
, , ,
Small effect of plasticity outside of process zone:
simplified relation for threshold stress
corrective function
geometry factor
flaw severity factor 1
0 1t
t
kλ k
12ISPMNA 2019 – UC-001
Modeling FrameworkRelation for threshold stress at planar surface
IHa nCI λ σ
σL Π B B B σ w
aσ
λ λ Fw
K,0
0 0
( ) (1 ) ln ln
σ CI λ σ CIthB B Z B
σB Zσ
((0) (0)
00
ps)
ln( ); ln ln( )
normalizing constant
σth
λB Bσσ B
00
(ps) exp
(0)
0
, IHCI CIZ Z f w
Kaσ w
Empirical parameters B0, Bλ, Bσ: correlated random variables
13ISPMNA 2019 – UC-001
Modeling FrameworkThreshold stress at planar surface: varying ZCI
IHa nCI λ σ F K
σL Π B B B B Bσ w w
aλ λ F
σλ
K,0
0 0
( ) (1 ) ln ln ln
σ CI λ σ CIthB B Z B
σB Zσ
((0) (0)
00
ps)
ln( ); ln ln( )
σth
λB Bσσ B
00
(ps) exp
( )( )
(0)
0 0
,
KFCICI
IH IHCI CI
ZZ
ZKa a
FZ Zw wwKσ σ w
F KF σ CI K σ CIB B Z B B Z( ) ( )[1 ]; [1 ]
14ISPMNA 2019 – UC-001
Modeling FrameworkPotential correlation between and KIHthσ (ps)
IH
IH
a nCI λ σ
Q
σL Π B B B σ w
aλ λ
B
K
K
σF
λ
w
σ w
,0
0 0
0
( ) (1 ) ln ln
(1 ) ln
σ CI λ σ CI Q σB B Z B B QZ B BσQ0
1(0) (0)
00
ln( ); ln ln( ) ;
λ Q
σ σ
B B BσQ QB B
000 1exp ;
IHIHth
QK
σ Kσ w
Q1
(ps)0
0
( )
15ISPMNA 2019 – UC-001
Modeling FrameworkRelation for stress : smooth surface datathσ (ps)
10t
t
kλ k
t
a n a nss ss ssCI σ
hσ
σ σL Π B B B σσ(ps
,0
0)
,( ) ln ln
thss ssσB B σ
σ(ps
00
)
ln
s
th
s
ssσ
Bσ σ
B0
0(ps) exp
Empirical parameters B0ss, Bσ
ss : correlated random variables
16ISPMNA 2019 – UC-001
Analysis ResultsBest estimate of and its uncertainty
simulated triplets of B0, Bλ, Bσ
probability densityfunction of
σth
λB Bσσ B
00
(ps) exp
Scal
ed F
requ
ency
0
0.2
0.4
0.6
0.8
1(X 10000)
Distribution 1 Distribution 2
Threshold Stress at Planar Surface
thσ (ps)
thσ (ps)
17ISPMNA 2019 – UC-001
Reference relation
Updated relation
Analysis ResultsReference and updated probabilistic relations
Location parameter μL
Scale parameter βL
Lower threshold δL
Median value = L Lδ μexp( )
th L L Lσ μ β δ(ps)3L ( , , )
three-parameter log-normal distribution
th W Wσ μ β(ps)2W ( , )
Scale parameter μW
Shape parameter βW
Median value =two-parameter
Weibull distribution WβWμ
1/ln(2)
18ISPMNA 2019 – UC-001
Analysis ResultsReference and updated probabilistic relations
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Prob
abili
ty o
f DH
C in
itiat
ion
Nominal applied stress (away from flaw tip)
Reference relation
Updated relation
19ISPMNA 2019 – UC-001
Auxiliary multi-variable modeling framework has been developed to characterize threshold stress for DHC initiation at planar surfaces,
, as a random variable: Modeling framework is based on closed-form representation of
threshold stress for DHC initiation at flaws, as developed from strip-yield process-zone approach to modeling crack initiation
Higher probability of DHC initiation is inferred for • flaws of greater severity• lower material resistance to DHC initiation
is one of parameters in developed framework
Obtaining reliable experimental data for initiation of delayed hydride cracking (DHC) at planar surfaces
is extremely challenging
(ps)thσ
(ps)thσ
Summary
20ISPMNA 2019 – UC-001
Developed modeling framework can be applied to:
Statistically assessing binary outcomes of DHC initiation experiments performed on specimens containing flaws of varying severity
Characterizing threshold stress for DHC initiation at planar surfaces as a non-deterministic input parameter for relevant probabilistic evaluations
Investigating likely correlation between threshold stress for DHC initiation at planar surfaces and KIH (threshold stress intensity factor for DHC initiation from a crack)
Application of developed framework is under consideration
Summary
21ISPMNA 2019 – UC-001
Acknowledgements
Valuable discussions held with D.H.B. Mok, J. Cui and G.K. Shek of Kinectrics Inc.
Experimental data provided by CANDU Owners Group
Support provided by CANDU Owners Group
22ISPMNA 2019 – UC-001
Thank you
23ISPMNA 2019 – UC-001
From: Scarth, D.A. and Smith, E., “Developments in Flaw Evaluation for CANDU Reactor Zr-Nb Pressure Tubes”, Proceedings of ASME PVP Conference, Boston, MA, USA, 1999, PVP-Vol. 391, pp. 35-45.
DHC initiation does not occur until vT
reaches vC , a critical flaw-tip displacement
DHC initiation does not occur as long as restraining stress pH
remains below , a threshold stress for DHC initiation at planar surface
Threshold stress for DHC initiation is the lowest applied stress prior to hydride formation, at which DHC initiation may occur
BackgroundProcess-zone approach to DHC initiation
Proc
ess-
zone
rest
rain
ing
stre
ss, p
H
Process-zone displacement, vT
DHC initiation
pH = σth(ps)
Threshold stress
No DHC initiationNo DHC initiation
vT = vC
thσ (ps)
24ISPMNA 2019 – UC-001
Threshold nominal stress for DHC initiation
σth,n decreases monotonically as kt increases
kt : elastic stress concentration factorKIH : threshold stress intensity factor for DHC initiation from a crack
σth,n increases monotonically as KIH increases
kt = 1 for a planar surface (ρ→∞) and increases with decreasing ρ, approaching infinity for a crack (ρ→ 0)
Background
Thre
shol
d st
ress
for D
HC
initi
atio
n
Elastic stress concentration factor
Lower threshold SIF for DHC initiationHigher threshold SIF for DHC initiation
KIH
σth(ps)
kt = 1
(KIH)(KIH)
25ISPMNA 2019 – UC-001
From: Cui, J., Shek, G.K., Scarth, D.A. and Wang, Z., “Delayed Hydride Cracking Initiation at Notches in Zr-2.5Nb Alloys”, Journal of Pressure Vessel Technology, August 2009, Vol. 131.
BackgroundHydrided regions in Zr-2.5Nb
example hydrided
region at tip of V-notch in Zr-2.5Nb