CANDU Owners Group Inc. Strength Through Co-operation Probabilistic Modeling of Hydride-Assisted Cracking in CANDU Zr-2.5%Nb Pressure Tubes by Means of Linear and Non-Linear Multi-Variable Regression Analysis L. Gutkin Kinectrics Inc. D.A. Scarth Kinectrics Inc. ASME V&V 2012, Las Vegas NV, May 2-4, 2012
Observed V Temperature Pressure Presentation Outline Observed s'ol 1.E-06 1.E-07 1.E-08 1.E-09 400 200 100 Hydrides are completely dissolved 50 Hydrides exist Critical nominal stress (MPa) for crack initiation due to hydrided region overload 50 p NO R 2 = 0.929 R 2 = 0.882 T ZH Temperature of Terminal Solid Solubility of Hydrogen in Zr-2.5%Nb p ZH Axial crack growth rate (m/s) 1.E-09 1.E-08 1.E-07 1.E-06 100 p ZC T ZC Time Predicted V 200 Predicted s ' OL p ZH T ZH Hydrides are completely dissolved Hydrides exist 400 p NO Background Hydride-assisted cracking Hydrided region overload Crack growth: linear regression Response and predictor variables Multi-variable linear regression Quality of statistical fit Crack initiation: non-linear regression Response and predictor variables Multi-variable non-linear regression Quality of statistical fit Model application Probabilistic assessments of reactor core Presented models as probabilistic inputs V&V 2012-6058 2
Background Hydride-assisted cracking in Zr-2.5%Nb CANDU Zr-2.5%Nb pressure tubes contain hydrogen and flaws Hydrogen preferentially diffuses to flaws (stress concentrators) Hydrided regions form and grow at flaw tips Hydrided regions may fracture under applied load, potentially leading to crack initiation Repeated formation, growth and fracture of hydrided regions results in crack propagation Typical hydrided region at the tip of a V-notch in non-irradiated Zr-2.5Nb 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. V&V 2012-6058 3
Temperature Pressure Background Hydrided region overload Cool-down: As temperature decreases, hydrided regions form and grow at flaw tips, often at pressure p HF < p NO Hydrides are completely dissolved Hydrides exist p NO Temperature of Terminal Solid Solubility of Hydrogen in Zr-2.5%Nb T ZH T ZH Hydrides are completely dissolved Hydrides exist p NO p ZH p ZH Heat-up: As temperature increases, flaw-tip hydrides dissolve, but often they would still exist at p HF < p < p NO p ZC T ZC Time NO: normal operation ZH: zero-power hot ZC: zero-power cold If hydrided region formed at pressure p HF during the reactor cool-down still exists at p > p HF during the reactor heat-up, hydrided region overload is said to occur V&V 2012-6058 4
Symbol Crack Growth Response V Crack growth rate Predictors Response and predictor variables T q f t Name Absolute evaluation temperature Operating temperature (effective full-power) Operating flux (effective full-power) Operating time (effective full-power) Arrhenius-type effect of evaluation temperature, T, on crack growth rate, V, has been established in previous modeling work Effects of irradiation temperature, q, and irradiation fluence, F = f (t ) dt, on V have been previously reported Both axial growth rate and radial growth rate are used V&V 2012-6058 5
Crack Growth Multi-variable linear regression Multi-variable linear regression framework is considered. Transformations are used to ensure adequacy of error term and broad variety of functional trends for response and predictors: G (V) = B 0 + B 1 F 1 (T) + B 2 F 2 (q ) + B 3 F 3 (f) + B 4 F 4 (t) + E V Arrhenius functional dependence V V 0 exp(b 1 /T) is preserved. Representative forms F j *(X j ) are selected for other variables on the basis of both statistical and non-statistical considerations: ln V = B 0 + B 1 T -1 + B 2 F 2 *(q ) + B 3 F 3 *(f) + B 4 F 4 *(t) + E V Regression analysis is performed to determine coefficients B j so that error term E V is minimized. V&V 2012-6058 6
Observed V Crack Growth Quality of statistical fit 1.E-06 Axial crack growth rate (m/s) R 2 = 0.929 1.E-07 1.E-08 1.E-09 1.E-09 1.E-08 1.E-07 1.E-06 Predicted V V&V 2012-6058 7
Crack Growth Trends in predicted response Model equation: lnv B 0 B T 1 B 2 θ B 3 f B τ 4 0.5 E V Values of all B j are positive B 1 : Crack growth rate B 2 : Crack growth rate + B 3 : Crack growth rate B 4 : Crack growth rate as evaluation temperature as operating temperature as operating flux as operating time All observed trends are consistent with our fundamental understanding of crack growth mechanism V&V 2012-6058 8
Symbol Crack Initiation Response and predictor variables Name Response s OL Critical stress for crack initiation due to hydride region overload Predictors s TH s HF K t Threshold stress for initiation of delayed hydride cracking (DHC) at constant load Applied stress during hydride formation Stress concentration factor at flaw tip Predictor selection is based on experimental results as well as on dual process-zone approach introduced by E. Smith in 2004 Total flaw-tip displacement is a sum of two contributions of proportional magnitudes: Displacement due to plastic deformation of Zr-Nb matrix Displacement due to formation of hydrided region V&V 2012-6058 9
Crack Initiation Functional framework Overload resistance is given by: σ' σ OL HF W σ σ TH HF ω w : overload crack initiation exponent ( 0 < w < 1 ) W : overload crack initiation coefficient Both w and W depend on flaw geometry: W W 0 Φ W W 1 K t ω 1 Ψ ω K t ω 1 1 Resultant regression framework is intrinsically non-linear V&V 2012-6058 10
Crack Initiation Multi-variable non-linear regression Multi-variable non-linear regression framework is applied. Response is transformed to ensure adequacy of error term E s : ln σ σ ' OL HF ln W 1 TH W ln ΦW ln σ 0 E K t K t σ HF 1Ψ ω ω1 1 σ Threshold stress for DHC initiation, s TH, depends on: Inherent material resistance to DHC initiation (K IH ) Hydrided region morphology Flaw geometry Regression analysis is performed to determine positive coefficients W 0, W 1 and w 1, so that error term E s is minimized. V&V 2012-6058 11
Crack Initiation Quality of statistical fit Critical nominal stress (MPa) for crack initiation due to hydrided region overload 400 R 2 = 0.882 Observed s'ol 200 100 Preliminary results 50 50 100 200 400 Predicted s ' OL V&V 2012-6058 12
Model Application Probabilistic assessments of reactor core Failure event: break-before-leak (BBL) Actual crack length, L A, exceeds critical crack length, L C Prob[BBL] = Prob[L A > L C ] Crack initiation (CI) is required for crack growth Prob[BBL] = Prob[BBL CI] x Prob[CI] Crack growth models Crack initiation models Crack susceptibility (CS) is required for crack initiation Prob[CI] = Prob[CI CS] x Prob[CS] Crack initiation models Crack susceptibility models V&V 2012-6058 13
Model Application Presented models as probabilistic inputs V Vˆ T,θ, f,τ exp E V σ σ ' OL HF σˆ σ ' OL HF σ σ TH HF, K t exp E σ Random variable V is used to calculate Prob[BBL CI] in Prob[BBL] = Prob[BBL CI] x Prob[CI] OL Random variable is used to calculate Prob[CI CS] in σ HF Prob[CI] = Prob[CI CS] x Prob[CS] σ' Presented crack growth model has been incorporated into Canadian Nuclear Standard N285.8 as representative model V&V 2012-6058 14
Acknowledgements E. Smith developed the dual process-zone approach to hydrided region overload, which forms the theoretical basis of the crack initiation model Valuable discussions held with: G. Bickel, H. Chaput, D. Metzger and M. Resta Levi of Atomic Energy of Canada Ltd. J. Cui, G. Shek and S. Xu of Kinectrics Inc. Experimental data provided by CANDU Owners Group Funding provided by CANDU Owners Group V&V 2012-6058 15
Reserve Slides V&V 2012-6058 16
Flaw represented by V-notch with depth a and root radius r Applied tensile stress is uniform and equal to p H over entire process zone DHC initiation occurs only when: v T > v C p H > p C From: Background Process-zone approach to DHC initiation Threshold stress for DHC initiation, s TH, is the lowest applied stress prior to hydride formation, at which DHC initiation may occur 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. V&V 2012-6058 17
Background σ nth C ~ C 0 f ( K IH ) exp 1 kt KIH : threshold stress intensity factor for DHC initiation from a crack Kt : elastic stress concentration factor Threshold nominal stress for DHC initiation Threshold stress for DHC initiation kt = 1 Kt = 1 for planar surface (r ) Kt increases as r decreases Kt for a crack (r 0) From: Low er threshold SIF for DHC initiation pc Higher threshold SIF for DHC initiation sth snth kt 1 2 kt a r a K IH Elastic stress concentration factor snth increases as KIH increases snth decreases as Kt increases Gutkin, L., Scarth, D.A. and Bickel, G.A., Statistical Assessment of Effect of Hydride Non-Ratcheting Conditions on Delayed Hydride Cracking Initiation in CANDU Pressure Tubes, Proceedings of 2010 ASME PVP Conference, Bellevue, WA, USA, PVP2010-25707. V&V 2012-6058 18
Background Dual process-zone concept (E. Smith, 2004) Flaw-tip process-zone displacement, v T, is represented as sum of two terms of proportional magnitudes: v T = v TP + v TH = (1 + l) v TP v TP is due to plastic deformation of Zr-Nb matrix ligaments v TH is due to hydrided region formation E. Smith showed that for small overloads: s OL = w s TH + (1 w)s HF s HF < s OL < s TH w : dual process-zone parameter 0 < w = < 1 s OL : critical flaw-tip stress for overload crack initiation s TH : threshold flaw-tip stress for DHC initiation s HF : applied flaw-tip stress during hydride formation V&V 2012-6058 19 1 1 λ
Background Dual process-zone formulation (authors, 2010) Additive framework is replaced with multiplicative one: s OL = s HF 1 w s TH w s HF < s OL < s TH 1 1 λ w : overload crack initiation exponent 0 < w = < 1 Additional parameter is used to account for evolution of stress state in hydrided region as the overload progresses: s OL = W s HF 1 w s TH w W : overload crack initiation coefficient Using nominal stresses instead of peak flaw-tip stresses: s nol = W s nhf 1 w s nth w V&V 2012-6058 20
Temperature Background Hydride formation conditions Ratcheting: If normal operating temperature does not exceed terminal solid solubility (TSS) of H in Zr-2.5Nb, flaw-tip hydrides accumulate incrementally with each thermal cycle TSS (complete dissolution of hydrides, w hen exceeded) Non-ratcheting: If normal operating Time temperature exceeds TSS, flaw-tip hydrides completely dissolve and do not accumulate Ratcheting Non-ratcheting Hydrided regions may form under either ratcheting or non-ratcheting conditions, depending on hydrogen concentration at flaw location V&V 2012-6058 21
Standard error, ln(m/s) Crack Growth Step-wise regression: error term Axial crack growth rate 0.50 T / θ / ϕ / τ T / θ / ϕ T / θ T 0.46 0.42 0.38 0.34 0.30 4 3 2 1 Number of predictors in model V&V 2012-6058 22