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V.Tvergaard, A.Needleman, “Analysis of the cup-cone fracture in a round tensile bar”, Acta Metallurgica,. Volume 32, issue 1, pp.157-169, 1984. F. Yunchang ...
18th International Conference on Structural Mechanics in Reactor Technology (SMiRT 18) Beijing, China, August 7-12, 2005 SMiRT18-G03-3

SIMULATING HYDRIDE EMBRITTLEMENT ON COLD-WORKED STRESS-RELIEVED ZIRCALOY-4 WITH GURSON-TVERGAARD-NEEDLEMAN DAMAGE MODEL

J. Desquines, V. Busser, F. Perales Institut de Radioprotection et de Sûreté Nucléaire DPAM/SEMCA/LEC-Bâtiment 702 Saint-Paul-Lez-Durance 13108 France Phone: 33(0)442257575, Fax: 33(0)442256143 E-mail: [email protected] ABSTRACT The influence of hydride precipitates on the ductility of cold-worked stress-relieved Zircaloy-4 is analyzed using a Gurson-Tvergaard-Needleman (GTN) damage model. A GTN model previously determined on recristallized and hydrided Zircaloy-4 is adapted to the cold-worked stress-relieved alloy. In a following part, the adapted model is checked by calculating the fracture toughness values of cold-worked stress-relieved Zircaloy-4 and by comparing the obtained values to literature data. The obtained GTN model demonstrated its ability to determine fracture toughness of hydrided stress-relived Zircaloy-4 with a reasonable accuracy.

Keywords: RIA, zirconium, Zircaloy, hydride, fracture, toughness, damage. 1. INTRODUCTION A general trend in the nuclear industry is the increase of fuel burnup for economical purpose. In order to support the corresponding safety evaluation, the French “Institut de Radioprotection et de Sûreté Nucléaire” (IRSN) is developing a research program. Experiments on Reactivity Initiated Accidents (RIA) described by Papin (2003), demonstrated that one of the potentially limiting phenomenon to burnup increase is the hydride embrittlement of the fuel rod cladding. The oxidation of the cladding by the primary water creates a zirconia layer at the Outer Diameter (OD) of the cladding, and part of the hydrogen is dissolved in the Zircaloy-4 cladding. When the solubility limit of hydrogen is reached, hydride platelets precipitate in the cladding. The diffusion process attracts zirconium hydrides at the OD during in-pile irradiation because the OD remains colder than the Inner Diameter (ID). Thus there is usually zirconium-hydride accumulation at OD. The OD of the cladding is therefore a preferential site for crack nucleation during the Pellet Cladding Mechanical Interaction phase induced by RIA transients on high burnup fuel rods. Such incipient cracks can have a deleterious effect on the cladding ductility. Efforts are underway to apply Elastic Plastic Fracture Mechanics (EPFM) in order to determine a failure criterion for fuel rods under RIA transients. One of the difficulty is that the cladding is a very thin structure (570 micron thickness), and there is up to now no available 1508

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procedure for experimental determination of the fracture toughness of such a thin component. Thus the aim of the present paper is to evaluate the ability of GTN damage models to predict the fracture toughness of a fuel rod cladding. The failure mechanism under strain controlled loading of a hydrided cladding was consistently described by several authors: Yunchang (1985), Arsène (1997), Grange (1998). Under significant plastic strain, void nucleation within the hydride precipitates followed by void growth and coalescence was usually observed for sufficiently high hydride contents. Such a failure process is usually accurately described by coupled damage-plasticity models. One of these models was introduced by Gurson (1977) and improved by Tvergaard and Needleman (1984) giving rise to the GTN model. A GTN model was previously developed by Prat (1997) and Grange (1998, 2000A, 2000B) for hydrided sheet samples machined in recristallized Zircaloy-4. However, cold-worked stress-relieved Zircaloy-4 is also of interest because it is widely used for fuel cladding in commercial reactors. There are noticeable differences in the material ductile properties but also in hydride microstructure and hydride orientation between the two alloys. One of the main goals of the present study is to adapt Grange’s (1998) model to cold-worked stress-relieved Zircaloy-4.

2. GRANGE’S (1998) MODEL FOR RECRISTALLIZED ALLOY The GTN yield surface is given by: 2 Σ eq

σ

2 0

 3 Σm + 2 fq cosh  2 σ0

  − 1 − ( fq )2 = 0 

q: is a geometrical constant equal to 4/e~1.47, σ0: is the yield stress of the undamaged Zircaloy-4, Σ: is the macroscopic stress tensor with associated deviatoric part S,

(

Σeq: macroscopic equivalent tensile stress Σ eq =

)

3 2S : S ,

Σm: macroscopic hydrostatic stress (Σ m = 1 3 trΣ ) , f: is the void volume fraction. The void growth rate during a tensile test is governed by the following equation: •



f = f



nucleation + f

growth

Two main contributions are included, void growth due to the loading by increasing the diameter of the porosities and void nucleation within the hydrides. The void growth is governed by material incompressibility in the plastic range: •

f

= (1 − f growth ) tr ε p •

growth



ε p : is the macroscopic plastic strain tensor. Grange (1998) correlated the void nucleation to the applied plastic strain using the following relation:

f nucleation = A exp(λε p ) The original Grange’s model for void nucleation is not exactly the one suggested by Tvergaard and Needleman (1984) and also implemented in the CAST3M (2005) finite element code developed by CEA and 1509

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also used in the present study. The Grange’s model was consequently slightly modified in order to perform calculations in the CAST3M finite element code. In the following, the void nucleation rate will be correlated, consistently with Tvergaard (1984), to the applied plastic strain using the following equation: •

f

nucleation

=

 1ε p −εN exp −   2  sN 2π 

fN sN

  

2

• ε p  

fN, sN and εN are material parameters that were identified using Grange’s (1998) mechanical testing results. The original Grange’s model and the proposed one are based on the same experimental data and will provide similar results. For Zircaloy-4, Grange (1998) observed that failure happened simultaneously with void coalescence. Thus there is no special need of describing the void coalescence phase. For recristallized material (RX) the following set of parameters for GTN model at room temperature were found in order to reproduce Grange’s (1998) testing results: [H]: is the hydride content in ppm.

fN =

[H ] , hydrogen content is normalized by the maximum hydrogen content, 16000

ε NRX = 6.43.10 −1 + 1.19.10 −4 [H ] s NRX = 9.90.10 −2 + 2.30.10 −4 [H ]

(

)

f cRX = 8.00.10 −3. exp − 10 −3.[H ]

Since mesh size is also an important material parameter for GTN model, Grange (1998) recommended a mesh size for Zircaloy-4 being 50x50 microns (quadratic interpolation on 2D square meshes) for crack propagation studies. The Grange’s (1998) model is valid for hydride contents between 300 ppm and 1000 ppm approximately. The fN parameter corresponds to the ratio of hydride content normalized by the maximum possible value of hydride content in a Zirconium alloy. It thus corresponds to the volume fraction of hydrides. Since the voids are nucleated in the hydrides, the fN parameter corresponds to the maximum possible value of void volume fraction. This parameter doesn’t depend on material heat treatment, we will take the same normalizing value for recristallized alloy (RX) and cold-worked stress-relieved (CW) Zircaloy-4.

3. ADAPTATION OF THE GRANGE’S(1998) STRESS-RELIEVED ZIRCALOY-4

GTN

MODEL

TO

COLD-WORKED

The adaptation of the GTN model will be based on JAERI results, Fuketa (2003), on cold-worked stress-relieved Zircaloy-4. The JAERI tests were performed on fresh and uniformly hydrided Zircaloy-4 ring samples with hydride contents between 0 and 1600 ppm at test temperatures between room temperature and 300°C. Our analysis will be restricted here to room temperature tests. Fig.1 plots the Total Elongations (TE) derived from ring tensile tests at room temperature versus hydride content. The hydride embrittlement takes 1510

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place for hydride contents over 500 ppm with a decrease of Total Elongation versus hydride content. For hydride contents lower than 500 ppm the failure is not induced by void nucleation in hydrides but by ductile failure of the Zircaloy-4. The GTN model will thus be identified on TE values for hydride contents over 500 ppm. In order to perform finite element simulations using the GTN model it will be assumed that the JAERI ring specimen with machined gage sections behaves like a plate having the same gage section. Thus, a 2D simulation can be performed. Furthermore, plain strain stress-state will be assumed. 35 30

TE(%)

25 20 15 10 5 0 0

200

400

600 800 1000 1200 Hydride content [H] (ppm)

1400

1600

1800

Fig. 1. Total elongation of the JAERI hoop tensile tests at room temperature. The cold-worked Zircaloy-4 tensile stress-strain curve follows a power law:

Σ eq (MPa) = k (MPa)ε p

n

The k parameter is taken equal to 650 MPa and the n exponent is taken equal to 0.03. The Young modulus is: E=100 000 MPa. The Poisson ratio is: ν=0.325. The GTN parameters for the cold-worked stress-relieved material are assumed to be proportional to the one of the recristallized alloy:

ε NCW = αε NRX s NCW = αs NRX f cCW = βf cRX The goal of this analysis is to find the α and β values that match the JAERI experimental data in fig.1. The stress-strain curve can be derived from a GTN model calculation. Fig.2 plots an example of stress-strain curve obtained with CAST3M finite element code, using large strains and large displacements assumptions. Plots of the two-dimensional void density at different strain levels shows strain localization in the middle of the specimen and finally, a strong decrease of stress is calculated when TE value is achieved. 1511

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The GTN based TE value depends on the following parameters:

TE GTN (E , v, k , n, α ([ H ]), β ([ H ]), f N ([ H ])) A sensitivity study has been performed on this TEGTN function: -

no sensitivity to E, ν and k, slight sensitivity to the n exponent value, large sensitivity to α, β and fN,

Void density plots

700 600

Σeq (MPa)

500 400 300 200

Elasticity

100 0 0.00

0.05

0.10

0.15 0.25 0.20 Plastic strain: ε p

0.30

0.35

TE

0.40

Fig.2. Example of GTN calculation using CAST3M finite element code. Accurate values of E, ν and k are not needed. The fN function doesn’t depend on the material RX or CW. This way, in order to determine the link between CW material GTN model and RX material we only need to determine the α and β values. The identification of the α and β values is performed by minimizing the difference between TE values derived from the GTN model and the one of JAERI tests:

Min

[TEGTN (E , v, k , n, α ([ H ]), β ([ H ]), f N ([ H ])) − TE JAERI ([ H ])]

α ([ H ] ), β ([ H ] )

The following set of parameters was determined:



α = 0.445 + 0.467. exp −

[H ]* − 700 

 62    [H ]* − 700   β = 0.447 + 0.441. exp −  58  

[H ]*ppm = Max(554; [H ])

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The GTN model data and the JAERI test results are compared in fig.3. There is a good agreement between experimental data and cold-worked material translated model. 35 JAERI tests

30

GTN Model 25

TE (%)

20 15 10 5 0 0

500

1000

1500

2000

[H] (ppm)

Fig.3. Comparison between the GTN model and the JAERI test results. 4. COLD-WORKED STRESS-RELIEVED ZIRCALOY-4 PLANE-STRAIN FRACTURE TOUGHNESS CALCULATION The GTN model for cold-worked Zircaloy-4 was used in order to determine the material fracture toughness. A CT specimen (see fig.4) was modeled using CAST3M finite element code. The applied load has been increased up to the maximum converging value. The fracture toughness was determined based on the load P at which the critical void volume fraction is achieved.

Ø=0.25w

P 0.6w

0.275w a w 1.25w

0.6w

P B

w = 4,0 mm a = 1,6 mm B = 0.5 w Fig.4. CT specimen and finite element model for fracture toughness calculations. The fracture toughness is calculated at the critical load based on the usual stress-intensity-factor equation: 1513

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K Ic =

P . B.W 1 / 2

a  2 + W 

2 3 4 a a a  a     . 0 . 886 + 4 . 64 − 13 . 32 + 14 . 72 − 5 . 6         W  W  W  W   3

a 2  1 −   W

The GTN model derived fracture toughness is compared to literature data, from Walker (1974) and Kreyns (1996) in Fig.4 and shows a good agreement between the results. The hydride content influence on fracture toughness is well reproduced by the proposed GTN model for cold-worked stress-relieved Zircaloy-4.

35 GTN model at 23°C

30 KIC (MPa√ m)

litterature data (RT) literature 25 20 15 10 5 0 0

500

1000

1500 2000 [H] (ppm)

2500

3000

Fig.5. Comparison between GTN model predicted fracture toughness and experimental data. The GTN model can now be used in order to perform sensitivity studies on the failure conditions of Zircaloy-4. For example in order to better understand irradiated cladding failure it is possible to simulate small thickness samples, radial gradients of hydride content, crack path.

5. CONCLUSIONS A coupled damage-plasticity model of the literature for recristallized fresh Zircaloy-4 has been adapted to cold-worked stress relieved fresh Zircaloy-4. This alloy is commonly used as cladding material. The adaptation was done based on hoop tensile tests performed by JAERI on fresh uniformly hydrided ring samples. The model has not been based on microscopic analysis of mechanical testing thus its accuracy is probably not has good as a GTN model derived from direct experimental measurements like the Grange’s (1998) model. There are of course some limitations to such models, for example we cannot reproduce the hydride platelets size that has certainly an influence on the failure conditions of a thin specimen. The mesh size with 50 microns implies that a 570 microns thick cladding cannot be meshed accurately with a nucleated incipient crack. Efforts are underway in order to develop less mesh sensitive local approaches (see Perales (2005)). However, the derived GTN model is shown to simulate with a reasonable accuracy the plain-strain fracture toughness of fresh hydrided cold-worked Zircaloy-4. 1514

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Based on this model, sensitivity studies can be performed in order to better understand the hydride embrittlement mechanisms of very thin irradiated claddings with hydride content gradients in the radial direction.

ACKOWLEDGMENTS The authors would like to thank Yann Monerie from IRSN for his contribution to the analysis of the results of the present study. REFERENCES S.Arsène (1997), “Effet de la microstructure et de la température sur la transition ductile-fragile des Zircaloy hydrurés”, Thesis of the Ecole Centrale de Paris (France), 1997/53. CAST3M (2005) Internet web-site - http://www-cast3m.cea.fr/cast3m T.Fuketa, T.Sugiyama, T.Nakamura, H.Sasajima, F.Nagase (2003), “Effect of pellet expansion and cladding hydrides on PCMI failure of high burnup LWR fuel during reactivity transients”, Proceedings of the 2003 NSRC conference, NUREG/CP-0185. M.Grange (1998), “Fragilisation du Zircaloy-4 par l’hydrogène: comportement, mécanisme d’endommagement, interaction avec la couche d’oxyde, simulation numérique”, Thesis of the Ecole des Mines de Paris, France. M.Grange, J.Besson, E.Andrieu (2000A), “Anisotropic behavior and rupture of hydrided Zircaloy-4 sheets”, Metallurgical and Material Transactions A, Volume 31 A, March 2000, pp. 679-689. M.Grange, J.Besson, E.Andrieu (2000B), “An anisotropic Gurson type model to represent the ductile rupture of hydrided Zircaloy-4 sheets”, International Journal of Fracture, Volume 105, pp.273-293, 2000. A.L.Gurson (1977), “Continuum theory of ductile rupture by void nucleation and growth: part I yield criteria and flow rules for porous ductile media”, Journal of Engineering Materials and Technology, Volume 99, pp.2-15, 1977. P.H.Kreyns, W.F.Bourgeois, C.J.White, P.L.Charpentier, B.F.Kammenzind, D.G.Franklin (1996), “Embrittlement of core materials” , ASTM-STP 1295, pp.758-782. J.Papin, B.Cazalis, J.M. Frizonnet, E.Fédérici, F.Lemoine (2003), “Synthesis of CABRI RIA tests interpretation” , Eurosafe meeting, November 25-26, 2003, Paris, France. F.Perales, Y.Monerie, F.Dubois, L.Stainier (2005), “Numerical Simulation of dynamical fracture in heterogeneous materials”, Third MIT conference on Computational Fluid and Solid Mechanics, June 14-17, 2005, Cambridge, MA. F.Prat, M.Grange, J.Besson, E.Andrieu (1997), “Behavior and rupture of hydrided Zircaloy-4 tubes and sheets”, Metallurgical and Material Transactions A, Volume 29A, June 1998, pp. 1623-1651. V.Tvergaard, A.Needleman, “Analysis of the cup-cone fracture in a round tensile bar”, Acta Metallurgica, Volume 32, issue 1, pp.157-169, 1984. F. Yunchang, D.A.Koss (1985), “The influence of multiaxial states of stress on the hydrogen embrittlement of zirconium alloy sheets” , Metallurgical Transaction A, Volume 16A, April 1985, pp.675-681. T.J.Walker, N.Kass (1974), “Variation of Zircaloy fracture toughness in irradiation” , ASTM-STP 551, pp.328-354.

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