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