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Prediction of the effects of neutron irradiation on the Charpy ductile to brittle transition curve of an A508 pressure vessel steel
Computational Materials Science 32 (2005) 294–300 www.elsevier.com/locate/commatsci
Prediction of the effects of neutron irradiation on the Charpy ductile to brittle transition curve of an A508 pressure vessel steel C. Bouchet a, B. Tanguy a
a,*
, J. Besson a, S. Bugat
b
Centre des Mate´riaux, Ecole Nationale Supe´rieure des Mines de Paris, BP 87, 91003 Evry Cedex, France b EdF les Renardie`res Route de Sens - Ecuelles 77250 Moret-sur-Loing, France
Abstract Nuclear pressure vessel steels are subjected to irradiation embrittlement which is monitored using Charpy tests. Reference index temperatures, such as the temperature for which the mean Charpy rupture energy is equal to 56 J (T56 J), are used as embrittlement indicators. In this work a material model integrating a description of viscoplasticity, ductile damage and brittle fracture is used to simulate the Charpy test. The model is adjusted on an unirradiated material. It is then applied to irradiated materials assuming that irradiation affects hardening. It is shown that irradiation probably also affects brittle failure. The decrease of the Charpy upper shelf energy is also interpreted. 2004 Elsevier B.V. All rights reserved. Keywords: Ductile to brittle transition; Charpy test; Irradiation; Pressure vessel steel
1. Introduction To ensure the structural integrity of nuclear reactor pressure vessels (RPV) against embrittlement, the fracture-safe analysis must be performed for the plant life-time. This analysis is traditionally based on linear elastic fracture mechanics. For the materials used for RPVs, a
*
Corresponding author. E-mail address: [email protected] (B. Tanguy).
fracture toughness curve KIC(TRTNDT) was introduced as an absolute deterministic lower bound based on numerous static fracture mechanics tests [1]. The shape of this curve is assumed to be unique and its position with respect to the temperature axis is parametrized by the nil-ductility reference temperature (RTNDT) which is determined from drop weight (Pellini) and Charpy tests (ASME code requirements). Irradiation embrittlement is one of the most important issue for the integrity of nuclear RPVs. In spite of its importance in structural integrity
0927-0256/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2004.09.039
C. Bouchet et al. / Computational Materials Science 32 (2005) 294–300
assessment, the evaluation of the shift of the reference temperature, DRTNDT, due to irradiation embrittlement, is monitored by the respective shift of the Charpy curve (e.g. Charpy energy vs temperature curve), taken at a certain level of energy, e.g. 56 J in the French nuclear surveillance program. It is then assumed that the irradiation induced shift of the fracture toughness in the ductile–brittle transition region is the same as the Charpy transition temperature shift. However, in many cases, RTNDT or the shift of RTNDT due to irradiation relative to the Charpy resilience can hardly be transfered to toughness data.
2. Effect of irradiation on mechanical properties of nuclear pressure vessel steels 2.1. Ductile to brittle transition characterized by the Charpy test Ferritic steels show cleavage fracture with low Charpy energies (and low fracture toughness) at low temperatures and ductile tearing with high Charpy energies (and high fracture toughness) values at higher temperatures. In the transition region cleavage fracture is preceded by some amount of ductile tearing, and both fracture mechanisms coexist and compete. For about a century the Charpy test has been used as an acceptance criterion to determine the transition temperature of different materials. The transition temperature is determined from the plot of the dissipated energy vs temperature. In the frame of local approach to fracture, some of the authors have proposed a micromechanical analysis of the Charpy test in order to model the ductile to brittle transition curve. Assuming that the average cleavage stress is temperature dependent beyond a given temperature, it is then possible to predict the whole Charpy energy curves [2], including the large scatter observed in the ductile to brittle range. This strategy allows one to transfer the results of the tests to larger structures and further to study the effect of different damage mechanisms as irradiation or ageing on the ductile to brittle transition temperature.
295
2.2. Effect of irradiation on the transition curve and hardening properties For a given fluence and irradiation temperature, irradiation-induced embrittlement is strongly dependent of the material chemical composition [3,4]. Volume fraction of copper being an important factor in the hardening-induced embrittlement due to the defects. However the following general features are reported on most of the studies devoted to the irradiation embrittlement of ferritic steels. On the physical point of view, irradiation produces fine scale microstructures which obstruct dislocation motion [5]. The mechanisms that produce these obstacles can be separated into two groups at a macroscopic level [5–7]: hardening mechanisms and other mechanisms where embrittlement is induced by local strength changes, one of the most well-known and well-described being phosphorous segregation at grain boundaries [5]. Hardening mechanisms include matrix and age hardening. Matrix hardening is due to radiationproduced point defect clusters and dislocation loops, referred to as the matrix damage contribution [7]. Age hardening is an irradiation-enhanced formation of copper-rich precipitates. These two hardening mechanisms cause an increase of the yield strength whereas phosphorous segregation causes grain boundaries embrittlement without any increase of hardness and may be responsible for intergranular fracture [8,9]. It should be underlined that the micromechanical models developed in this study do not consider the embrittlement of the grain boundaries due to phosphorous segregation. Intragranular cleavage is assumed to be the prevailing mechanism. Considering the impact properties, two main effects due to irradiation embrittlement have been reported in the literature (see Fig. 1): (i) an increase of the ductile-to-brittle transition temperature (DBTT) [4,5,10–14], (ii) a trend to a decrease of the upper shelf energy (USE) even if for low irradiation fluence (1018 n/cm2), and low Cu (0.07%) content an increase of the USE has also been reported [3]. Two main cases for the irradiation effect on the material stress–strain curve could be considered and are schematically drawn on Fig. 2. The first
296
C. Bouchet et al. / Computational Materials Science 32 (2005) 294–300
180 160
unirradiated
140
USE
E (J)
120
irradiated
100 80 60
T56J
40 20 0 -150
-50
50 T ( º C)
150
Fig. 1. Evolution of the Charpy transition curve under irradiation (U = 4.65 · 1019 n/cm2) [15].
irr. 1
f
irr. 2
base
p Fig. 2. Two hypothesis concerning the evolution of the flow stress after irradiation.
case (dashed curve labeled irr.1) considers that the whole stress–strain curve of the unirradiated material is shifted to higher stress values by DrY. The second cases considers an increase of the yield stress of DrY but that the ultimate stress remains unaffected by irradiation (dotted curve labeled irr. 2). Based on experimental data on A508 Cl.3 steel obtained from the french surveillance program [15], it appears that the first case is the most satisfactory hypothesis (see Fig. 3) and will be the only case treated in the present study. The aim of this study is to predict the shift of index temperature using a micromechanical description of the damage processes. The material description is however based on continuum mechanics so that dislocations loops and copperrich precipitates cannot directly be accounted for. It is therefore assumed that the irradiation damage
Fig. 3. Evolution of the yield stress (rY) and the ultimate tensile stress (Rm) as a function of the irradiation flux [15].
is reflected by the increase of the yield stress, DrY. Many experimental evidences of the index temperature shift were found in the literature. Mostly, linear correlations between yield stress increase under irradiation and the DBTT shift on Charpy specimens, DTX = aXDrY, have been found. However index temperatures may be defined for different values of the rupture energy X = 41, 45 or 56 J. The following values for the correlation coefficient aX have been proposed in the literature: a41 = 0.45 in [3], a41 = 0.55 in [13]. In the French nuclear program, the shift of the index T56 J is considered [9] and will be used in the present work. From the experimental values given in [13,9,11,15] a linear correlation between DT56 J and DrY for various RPV steels was established: DT56 J = 0.60DrY (see Fig. 4). This correlation will be considered as the reference experimental database to which the predictions developed in this study will be compared.
3. Modeling of the ductile to brittle transition The behavior of the reference unirradiated A508 steel of this study is presented in the following. The model of the behavior consists of three parts: (i) a temperature and strain rate dependent viscoplastic model describing the behavior of the undamaged material, (ii) a model for ductile tearing, (iii) a model for brittle failure.
C. Bouchet et al. / Computational Materials Science 32 (2005) 294–300
Rousselier model [19], which is able to handle strain rate and temperature dependence, is used to model void nucleation and growth and final failure. Cavities are assumed to nucleate on MnS inclusions (volume fraction: 1.75 · 104) and on carbides (Fe3C, volume fraction: 2.3%) for plastic strains higher than 0.5. The model relies on the definition of a yield surface, / depending on the stress tensor invariants req and rkk (trace) and on a single damage variable representing the porosity, f. This yield surface is expressed as: / = r* rf where the effective stress is implicitly defined by the following equation: req 2 q rkk þ fDR exp R 1¼0 ð1 f Þr 3 2 ð1 f Þr
140 [10,12] [14] 120 [3]
( ˚C)
100 80 60 40 20 0
0
50
100 Y
150 (MPa)
200
250
ð3Þ
Fig. 4. Evolution of DT56 J as a function of DrY.
3.1. Viscoplastic behavior The viscoplastic behavior has been described in [16]. Plastic hardening is described by: rf ¼ r0 þ Q1 ð1 eb1 p Þ þ Q2 ð1 eb2 p Þ
297
ð1Þ
where rf is the flow stress and p the equivalent plastic strain. r0 (yield stress), Q1, b1, Q2 and b2 are temperature dependent coefficients. The plastic strain rate is given by: n 1 1 1 req rf 1;2 ð2Þ ¼ þ e_ 1;2 ¼ K 1;2 p_ e_ 1 e_ 2 where req is the von Mises stress. The strain rates e_ 1 and e_ 2 are each representative of a deformation mechanism: (1) Peierls friction, (2) phonon drag [17]. The mechanism with the smallest deformation rate controls deformation. In practice, the phonon drag mechanism only prevails at very high strain rates. K1, n1, K2 and n2 are temperature dependent coefficients. The model is identified for strain rates between 104 and 4000 s1, temperature between 196 and 300 C and plastic strains up to 1.0 using the Bridgman analysis. 3.2. Model for ductile tearing The model for ductile failure is presented in [18] and is only briefly recalled here. A modified
where qR and DR are material parameters to be adjusted. Plastic flow is then computed assuming normality. Details of the numerical implementation of the model can be found in [20]. The constitutive equations lead to softening up to crack initiation and propagation so that a material scale length, h, is required. In the following, as in many other studies (see e.g. [21]), this length is identified to the mesh size which must be adjusted. The model was fitted using notch tensile bars (NT) at room temperature [18]. The following model parameters were used: qR = 0.89, DR = 2.2 and h = 100 lm. 3.3. Model for brittle failure The description of brittle failure is derived from the Beremin model [22] which is adapted to account for: (i) the partial unloading of the specimen when the crack propagates, (ii) the temperature dependence of the model parameters which were considered as constant in the original model. The following definition of a local effective stress is used to define brittle failure rIp: rI expðp=kÞ; if p_ > 0 rIp ¼ ð4Þ 0; otherwise where rI is the maximum principal stress. The rupture probability of a volume element is represented by x = (rIp/ru)m where both ru and m may be temperature dependent. The load history integrating
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C. Bouchet et al. / Computational Materials Science 32 (2005) 294–300
stress variations but also temperature changes is represented by: ~ ¼ max xðt0 Þ xðtÞ 0
ð5Þ
t 2½0;t
where t is the time at which the failure probability is evaluated. Finally the failure probability of the whole specimen is given by: Z dV ~ P R ðtÞ ¼ 1 exp xðtÞ ð6Þ V0 V m, ru and k are material parameters which must be identified. V0 is a reference volume which can be arbitrarily fixed (V0 = 0.001 mm3 in this study). The parameters have been identified at low temperature (T<150 C): m = 17.8, k = 4 and ru = 2925 MPa. Above 80 C it is necessary to use a temperature dependent value for ru in order to represent the transition (Fig. 5). 3.4. Model for the transition The transition is modeled by simulating ductile tearing and by post-processing the results in order to obtain the brittle failure probability. There is therefore no specific model for the transition. However, as previously mentioned, it is necessary to use a temperature dependent value for ru above a threshold temperature ( 80 C) to model the sharp upturn of the Charpy transition curve. This
4000 m = 17.8 k =4
300
80
3600 3400 3200 3000 2800 -150
unirr adia ted
(MPa)
3800
60
40
-50 50 T (˚ C)
150 88 45
Cte 150
Fig. 5. Variation of ru as a function of temperature for the unirradiated material and for irradiation levels corresponding to an increase of the yield stress equal to 45, 88 and 150 MPa. Arrows indicate the temperature shift, DTU caused by irradiation.
implies that the mean cleavage stress increases with temperature. The simulation is based on the finite element method. Details can be found in [18].
4. Simulation of the ductile to brittle transition Using the previously described models, it becomes possible to model the effect of irradiation on the ductile to brittle transition. The flow stress (Eq. (1)) of the unirradiated material is modified by adding a contribution representing the irradiation effect which is expressed as: 300 is the increase of the yield Dr300 Y gðT Þ where DrY stress at 300 C and g(T) a function of the temperature [15]. The increase of the flow stress has two consequences: (i) stresses in the material increase causing earlier brittle failure, (ii) the macroscopic load on the Charpy specimen increases causing an increase of the dissipated energy in the ductile regime so that the USE also increases. The USE is computed by propagating the ductile crack through the whole Charpy specimen. The simulated Charpy transition curve is shown in Fig. 6 for Dr300 ¼ 88 MPa. A first Y analysis of brittle failure probability was performed assuming that ru is unaffected by irradiation. In that case the shift on the reference temperature, DT56 J is equal to 20 C whereas a value of 53 C is expected from the data shown on Fig. 4. The correct value of the temperature shift can be obtained assuming that ru is also affected by irradiation. The parameter for the irradiated material is expressed as: rirr u ðT Þ ¼ ru ðT DT U Þ. The shift DTU depends on the level of irradiation. The transition curve was also simulated for Dr300 Y ¼ 45 (low irradiation level) and 150 MPa (very high irradiation level). DTU was adjusted for both irradiation levels; results are shown on Fig. 5. The simulated values of DT56 J are compared on Fig. 7. It is shown that DT56 J is always underestimated assuming that ru is unaffected by irradiation. Assuming that ru is a constant equal to the low temperature value for the unirradiated material (2925 MPa) gives a conservative estimation of DT56 J.
C. Bouchet et al. / Computational Materials Science 32 (2005) 294–300
299
Fig. 6. Simulation of the Charpy transition curve (failure probability equal to 50%) for the unirradiated material and the irradiated material (DrY = 88 MPa) assuming that ru is unaffected by irradiation or using a shifted ru function to account for irradiation. The finite element mesh accounting for usual symmetries are also shown.
140 _
100
u
( ˚ C)
120 Cte
80 60 40 u(T)
20 0
0
50
100
300
150 200 (MPa)
250
Fig. 7. Prediction of DT56 J. Three hypothesis are used for the parameter ru: (i) constant value (low temperature value for the unirradiated material), (ii) temperature dependent ru (unirradiated material), (iii) temperature and irradiation dependent ru (Fig. 5). Thin lines indicate the experimental range.
5. Conclusions and discussion The ductile to brittle transition characterized by the Charpy test has been modeled in the case of a
RPV steel using constitutive equations for viscoplasticity, ductile tearing and brittle failure. The transition curve of the based unirradiated material can be modeled provided the parameter ru of the Beremin model is considered to be an increasing function of temperature. Irradiation hardens the material causing an increase of the stresses during the Charpy test thus causing earlier brittle failure. This effect cannot account for the whole experimental reference temperature shift DT56 J. It is therefore necessary to consider that irradiation also affects the mean cleavage stress and therefore the value of ru. It is therefore likely that the formation of copper-rich precipitates and dislocation loops under irradiation, which causes hardening, also favors cleavage crack nucleation. The micromechanical mechanism needs however to be identified. The increase of the flow stress induced by irradiation leads to an increase of the dissipated energy so that the simulated upper shelf energy at 20 C also increases. The experimentally reported decrease of the USE can however be explained based on the present simulation. From Fig. 6 it can be concluded that at 20 C brittle failure still
300
C. Bouchet et al. / Computational Materials Science 32 (2005) 294–300
occurs. This implies that if the USE is estimated from tests at 20 C a decrease will be predicted. An other way to estimate the USE is to increase the test temperate until pure ductile failure is obtained. As the flow stress decreases with temperature this will also decrease the measured USE.
[11]
References [1] ASME, ASME boiler and pressure vessel code, section 11. In service inspection of nuclear power plants. [2] B. Tanguy, J. Besson, R. Piques, A. Pineau, Ductile–brittle transition of a A508 steel characterized by the Charpy impact test. Part–II: modelling of the Charpy transition curve, Eng. Fract. Mech., in press. [3] F.M. Haggag, Effects of irradiation temperature on embrittlement of nuclear pressure vessel steels, in: A.S. Kumar, D.S. Gelles, R.K. Nanstad, E.A. Little (Eds.), Effects of Radiation on Materials: 16th International Symposium, ASTM STP 1175, American Society for Testing and Materials, Philadelphia, PA, 1993, pp. 172– 185. [4] K. Dohi, T. Onchi, F. Kano, K. Fukuya, M. Narui, H. Kayano, Effect of neutron flux on low temperature irradiation embrittlment of reactor pressure vessel steel, J. Nucl. Mater. 265 (1999) 78–90. [5] Y.A. Nikolaev, A.V. Nikolaeva, Y.I. Shtrombakh, Radiation embrittlement of low-alloy steels, Int. J. Pressure Vessels and Piping 79 (2002) 619–636. [6] M. Kirk, M.E. Natishan, Shift in toughness transition temperature due to irradiation: dt0 vs dt41 J, a comparison and rationalization of differences, in: IAEA Specialists Meeting on Master Curve Testing and Results Application, Prague, Czech, 2001. [7] M. Wagenhofer, H.P. Gunawardane, M.E. Natishan, Yield and toughness transition prediction for irradiated steels based on dislocation mechanics, in: S.T. Rosinski, M.L. Grossbeck, T.R. Allen, A.S. Kumar (Eds.), Effects of Radiation on Materials: 20th International Symposium, ASTM STP 1045, American Society for Testing and Materials, West Conshohocken, PA, 2001, pp. 97–108. [8] S. Raoul, B. Marini, A. Pineau, Effect of microstructure on the susceptibility of a 533 steel to temper embrittlement, J. Nucl. Mater. 257 (1998) 199–205. [9] C. Brillaud, F. Hedin, In-service evaluation of french pressurized water reactor vessel steel, in: R. Stoller, A. Kumar, D. Gelles (Eds.), Effects of radiation on materials: 15th international symposium, ASTM STP 1125, American Society for Testing and Materials, Nashville, TN, 1992, pp. 23–49. [10] H.-W. Viehrig, J. Boehmert, J. Dzugan, H. Richter, Master curve evaluation of irradiated Russian vver type reactor
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[17]
[18]
[19]
[20]
[21]
[22]
pressure vessel steels, in: S.T. Rosinski, M.L. Grossbeck, T.R. Allen, A.S. Kumar (Eds.), Effects of Radiation on Materials: 20th International Symposium, ASTM STP 1045, American Society for Testing and Materials, West Conshohocken, PA, 2001, pp. 109–124. C. Brillaud, Y. Grandjean, S. Saillet, Vessel investigation program of CHOOZ A PWR reactor after shutdown, in: S.T. Rosinski, M.L. Grossbeck, T.R. Allen, A.S. Kumar (Eds.), Effects of Radiation on Materials: 20th International Symposium, ASTM STP 1045, American Society for Testing and Materials, West Conshohocken, PA, 2001, pp. 28–40. E. Lucon, R. Chaouadi, S. Saillet, Radiation damage assessment by the use of dynamic toughness measurments on pre-cracked Charpy-v specimens, in: S.T. Rosinski, M.L. Grossbeck, T.R. Allen, A.S. Kumar (Eds.), Effects of Radiation on Materials: 20th International Symposium, ASTM STP 1045, American Society for Testing and Materials, West Conshohocken, PA, 2001, pp. 68–78. M. Sokolov, R. Nanstad, Comparison of irradiationinduced shifts of KJc and Charpy impact toughness for reactor pressure vessel steels, Tech. rep., Oak Ridge National Laboratory, NUREG/CR-6609, 2000. C. Brillaud, F. Hedin, In-service evaluation of french pressurized water reactor vessel steel, in: Effects of Radiation on Materials: 15th International SymposiumASTM STP 1125, American Society for Testing and Materials, Philadelphia, 1992, pp. 23–49. EDF Industry—Nuclear generation division corporate laboratories, private communication, 2001. B. Tanguy, R. Piques, L. Laiarinandrasana, A. Pineau, Mechanical behaviour of A508 steel based on double nonlinear viscoplactic constitutive equation, in: D. Miannay, P. Costa, D. Franc¸ois, A. Pineau (Eds.), EUROMAT 2000, Advances in Mechanical Behaviour. Plasticity and Damage, Elsevier, Tours, France, 2000, pp. 499–504. J. Campbell, W. Ferguson, The temperature and strainrate dependance of the shear strength of mild steel, Phil. Mag. 21 (1970) 63–82. B. Tanguy, J. Besson, An extension of the Rousselier model to viscoplastic temperature dependent materials, Int. J. Frac. 116 (1) (2002) 81–101. G. Rousselier, Ductile fracture models and their potential in local approach of fracture, Nucl. Eng. Design 105 (1987) 97–111. J. Besson, D. Steglich, W. Brocks, Modeling of crack growth in round bars and plane strain specimens, Int. J. Solids Struct. 38 (46–47) (2001) 8259–8284. C. Ruggieri, T. Panontin, R. Dodds Jr., Numerical modeling of ductile crack growth in 3-d using computational cell elements, Int. J. Frac. 82 (1996) 67–95. F.M. Beremin, A local criterion for cleavage fracture of a nuclear pressure vessel steel, Met. Trans. 14 A (1983) 2277– 2287.
Prediction of the effects of neutron irradiation on the Charpy ductile to brittle transition curve of an A508 pressure vessel steel C. Bouchet a, B. Tanguy a
a,*
, J. Besson a, S. Bugat
b
Centre des Mate´riaux, Ecole Nationale Supe´rieure des Mines de Paris, BP 87, 91003 Evry Cedex, France b EdF les Renardie`res Route de Sens - Ecuelles 77250 Moret-sur-Loing, France
Abstract Nuclear pressure vessel steels are subjected to irradiation embrittlement which is monitored using Charpy tests. Reference index temperatures, such as the temperature for which the mean Charpy rupture energy is equal to 56 J (T56 J), are used as embrittlement indicators. In this work a material model integrating a description of viscoplasticity, ductile damage and brittle fracture is used to simulate the Charpy test. The model is adjusted on an unirradiated material. It is then applied to irradiated materials assuming that irradiation affects hardening. It is shown that irradiation probably also affects brittle failure. The decrease of the Charpy upper shelf energy is also interpreted. 2004 Elsevier B.V. All rights reserved. Keywords: Ductile to brittle transition; Charpy test; Irradiation; Pressure vessel steel
1. Introduction To ensure the structural integrity of nuclear reactor pressure vessels (RPV) against embrittlement, the fracture-safe analysis must be performed for the plant life-time. This analysis is traditionally based on linear elastic fracture mechanics. For the materials used for RPVs, a
*
Corresponding author. E-mail address: [email protected] (B. Tanguy).
fracture toughness curve KIC(TRTNDT) was introduced as an absolute deterministic lower bound based on numerous static fracture mechanics tests [1]. The shape of this curve is assumed to be unique and its position with respect to the temperature axis is parametrized by the nil-ductility reference temperature (RTNDT) which is determined from drop weight (Pellini) and Charpy tests (ASME code requirements). Irradiation embrittlement is one of the most important issue for the integrity of nuclear RPVs. In spite of its importance in structural integrity
0927-0256/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2004.09.039
C. Bouchet et al. / Computational Materials Science 32 (2005) 294–300
assessment, the evaluation of the shift of the reference temperature, DRTNDT, due to irradiation embrittlement, is monitored by the respective shift of the Charpy curve (e.g. Charpy energy vs temperature curve), taken at a certain level of energy, e.g. 56 J in the French nuclear surveillance program. It is then assumed that the irradiation induced shift of the fracture toughness in the ductile–brittle transition region is the same as the Charpy transition temperature shift. However, in many cases, RTNDT or the shift of RTNDT due to irradiation relative to the Charpy resilience can hardly be transfered to toughness data.
2. Effect of irradiation on mechanical properties of nuclear pressure vessel steels 2.1. Ductile to brittle transition characterized by the Charpy test Ferritic steels show cleavage fracture with low Charpy energies (and low fracture toughness) at low temperatures and ductile tearing with high Charpy energies (and high fracture toughness) values at higher temperatures. In the transition region cleavage fracture is preceded by some amount of ductile tearing, and both fracture mechanisms coexist and compete. For about a century the Charpy test has been used as an acceptance criterion to determine the transition temperature of different materials. The transition temperature is determined from the plot of the dissipated energy vs temperature. In the frame of local approach to fracture, some of the authors have proposed a micromechanical analysis of the Charpy test in order to model the ductile to brittle transition curve. Assuming that the average cleavage stress is temperature dependent beyond a given temperature, it is then possible to predict the whole Charpy energy curves [2], including the large scatter observed in the ductile to brittle range. This strategy allows one to transfer the results of the tests to larger structures and further to study the effect of different damage mechanisms as irradiation or ageing on the ductile to brittle transition temperature.
295
2.2. Effect of irradiation on the transition curve and hardening properties For a given fluence and irradiation temperature, irradiation-induced embrittlement is strongly dependent of the material chemical composition [3,4]. Volume fraction of copper being an important factor in the hardening-induced embrittlement due to the defects. However the following general features are reported on most of the studies devoted to the irradiation embrittlement of ferritic steels. On the physical point of view, irradiation produces fine scale microstructures which obstruct dislocation motion [5]. The mechanisms that produce these obstacles can be separated into two groups at a macroscopic level [5–7]: hardening mechanisms and other mechanisms where embrittlement is induced by local strength changes, one of the most well-known and well-described being phosphorous segregation at grain boundaries [5]. Hardening mechanisms include matrix and age hardening. Matrix hardening is due to radiationproduced point defect clusters and dislocation loops, referred to as the matrix damage contribution [7]. Age hardening is an irradiation-enhanced formation of copper-rich precipitates. These two hardening mechanisms cause an increase of the yield strength whereas phosphorous segregation causes grain boundaries embrittlement without any increase of hardness and may be responsible for intergranular fracture [8,9]. It should be underlined that the micromechanical models developed in this study do not consider the embrittlement of the grain boundaries due to phosphorous segregation. Intragranular cleavage is assumed to be the prevailing mechanism. Considering the impact properties, two main effects due to irradiation embrittlement have been reported in the literature (see Fig. 1): (i) an increase of the ductile-to-brittle transition temperature (DBTT) [4,5,10–14], (ii) a trend to a decrease of the upper shelf energy (USE) even if for low irradiation fluence (1018 n/cm2), and low Cu (0.07%) content an increase of the USE has also been reported [3]. Two main cases for the irradiation effect on the material stress–strain curve could be considered and are schematically drawn on Fig. 2. The first
296
C. Bouchet et al. / Computational Materials Science 32 (2005) 294–300
180 160
unirradiated
140
USE
E (J)
120
irradiated
100 80 60
T56J
40 20 0 -150
-50
50 T ( º C)
150
Fig. 1. Evolution of the Charpy transition curve under irradiation (U = 4.65 · 1019 n/cm2) [15].
irr. 1
f
irr. 2
base
p Fig. 2. Two hypothesis concerning the evolution of the flow stress after irradiation.
case (dashed curve labeled irr.1) considers that the whole stress–strain curve of the unirradiated material is shifted to higher stress values by DrY. The second cases considers an increase of the yield stress of DrY but that the ultimate stress remains unaffected by irradiation (dotted curve labeled irr. 2). Based on experimental data on A508 Cl.3 steel obtained from the french surveillance program [15], it appears that the first case is the most satisfactory hypothesis (see Fig. 3) and will be the only case treated in the present study. The aim of this study is to predict the shift of index temperature using a micromechanical description of the damage processes. The material description is however based on continuum mechanics so that dislocations loops and copperrich precipitates cannot directly be accounted for. It is therefore assumed that the irradiation damage
Fig. 3. Evolution of the yield stress (rY) and the ultimate tensile stress (Rm) as a function of the irradiation flux [15].
is reflected by the increase of the yield stress, DrY. Many experimental evidences of the index temperature shift were found in the literature. Mostly, linear correlations between yield stress increase under irradiation and the DBTT shift on Charpy specimens, DTX = aXDrY, have been found. However index temperatures may be defined for different values of the rupture energy X = 41, 45 or 56 J. The following values for the correlation coefficient aX have been proposed in the literature: a41 = 0.45 in [3], a41 = 0.55 in [13]. In the French nuclear program, the shift of the index T56 J is considered [9] and will be used in the present work. From the experimental values given in [13,9,11,15] a linear correlation between DT56 J and DrY for various RPV steels was established: DT56 J = 0.60DrY (see Fig. 4). This correlation will be considered as the reference experimental database to which the predictions developed in this study will be compared.
3. Modeling of the ductile to brittle transition The behavior of the reference unirradiated A508 steel of this study is presented in the following. The model of the behavior consists of three parts: (i) a temperature and strain rate dependent viscoplastic model describing the behavior of the undamaged material, (ii) a model for ductile tearing, (iii) a model for brittle failure.
C. Bouchet et al. / Computational Materials Science 32 (2005) 294–300
Rousselier model [19], which is able to handle strain rate and temperature dependence, is used to model void nucleation and growth and final failure. Cavities are assumed to nucleate on MnS inclusions (volume fraction: 1.75 · 104) and on carbides (Fe3C, volume fraction: 2.3%) for plastic strains higher than 0.5. The model relies on the definition of a yield surface, / depending on the stress tensor invariants req and rkk (trace) and on a single damage variable representing the porosity, f. This yield surface is expressed as: / = r* rf where the effective stress is implicitly defined by the following equation: req 2 q rkk þ fDR exp R 1¼0 ð1 f Þr 3 2 ð1 f Þr
140 [10,12] [14] 120 [3]
( ˚C)
100 80 60 40 20 0
0
50
100 Y
150 (MPa)
200
250
ð3Þ
Fig. 4. Evolution of DT56 J as a function of DrY.
3.1. Viscoplastic behavior The viscoplastic behavior has been described in [16]. Plastic hardening is described by: rf ¼ r0 þ Q1 ð1 eb1 p Þ þ Q2 ð1 eb2 p Þ
297
ð1Þ
where rf is the flow stress and p the equivalent plastic strain. r0 (yield stress), Q1, b1, Q2 and b2 are temperature dependent coefficients. The plastic strain rate is given by: n 1 1 1 req rf 1;2 ð2Þ ¼ þ e_ 1;2 ¼ K 1;2 p_ e_ 1 e_ 2 where req is the von Mises stress. The strain rates e_ 1 and e_ 2 are each representative of a deformation mechanism: (1) Peierls friction, (2) phonon drag [17]. The mechanism with the smallest deformation rate controls deformation. In practice, the phonon drag mechanism only prevails at very high strain rates. K1, n1, K2 and n2 are temperature dependent coefficients. The model is identified for strain rates between 104 and 4000 s1, temperature between 196 and 300 C and plastic strains up to 1.0 using the Bridgman analysis. 3.2. Model for ductile tearing The model for ductile failure is presented in [18] and is only briefly recalled here. A modified
where qR and DR are material parameters to be adjusted. Plastic flow is then computed assuming normality. Details of the numerical implementation of the model can be found in [20]. The constitutive equations lead to softening up to crack initiation and propagation so that a material scale length, h, is required. In the following, as in many other studies (see e.g. [21]), this length is identified to the mesh size which must be adjusted. The model was fitted using notch tensile bars (NT) at room temperature [18]. The following model parameters were used: qR = 0.89, DR = 2.2 and h = 100 lm. 3.3. Model for brittle failure The description of brittle failure is derived from the Beremin model [22] which is adapted to account for: (i) the partial unloading of the specimen when the crack propagates, (ii) the temperature dependence of the model parameters which were considered as constant in the original model. The following definition of a local effective stress is used to define brittle failure rIp: rI expðp=kÞ; if p_ > 0 rIp ¼ ð4Þ 0; otherwise where rI is the maximum principal stress. The rupture probability of a volume element is represented by x = (rIp/ru)m where both ru and m may be temperature dependent. The load history integrating
298
C. Bouchet et al. / Computational Materials Science 32 (2005) 294–300
stress variations but also temperature changes is represented by: ~ ¼ max xðt0 Þ xðtÞ 0
ð5Þ
t 2½0;t
where t is the time at which the failure probability is evaluated. Finally the failure probability of the whole specimen is given by: Z dV ~ P R ðtÞ ¼ 1 exp xðtÞ ð6Þ V0 V m, ru and k are material parameters which must be identified. V0 is a reference volume which can be arbitrarily fixed (V0 = 0.001 mm3 in this study). The parameters have been identified at low temperature (T<150 C): m = 17.8, k = 4 and ru = 2925 MPa. Above 80 C it is necessary to use a temperature dependent value for ru in order to represent the transition (Fig. 5). 3.4. Model for the transition The transition is modeled by simulating ductile tearing and by post-processing the results in order to obtain the brittle failure probability. There is therefore no specific model for the transition. However, as previously mentioned, it is necessary to use a temperature dependent value for ru above a threshold temperature ( 80 C) to model the sharp upturn of the Charpy transition curve. This
4000 m = 17.8 k =4
300
80
3600 3400 3200 3000 2800 -150
unirr adia ted
(MPa)
3800
60
40
-50 50 T (˚ C)
150 88 45
Cte 150
Fig. 5. Variation of ru as a function of temperature for the unirradiated material and for irradiation levels corresponding to an increase of the yield stress equal to 45, 88 and 150 MPa. Arrows indicate the temperature shift, DTU caused by irradiation.
implies that the mean cleavage stress increases with temperature. The simulation is based on the finite element method. Details can be found in [18].
4. Simulation of the ductile to brittle transition Using the previously described models, it becomes possible to model the effect of irradiation on the ductile to brittle transition. The flow stress (Eq. (1)) of the unirradiated material is modified by adding a contribution representing the irradiation effect which is expressed as: 300 is the increase of the yield Dr300 Y gðT Þ where DrY stress at 300 C and g(T) a function of the temperature [15]. The increase of the flow stress has two consequences: (i) stresses in the material increase causing earlier brittle failure, (ii) the macroscopic load on the Charpy specimen increases causing an increase of the dissipated energy in the ductile regime so that the USE also increases. The USE is computed by propagating the ductile crack through the whole Charpy specimen. The simulated Charpy transition curve is shown in Fig. 6 for Dr300 ¼ 88 MPa. A first Y analysis of brittle failure probability was performed assuming that ru is unaffected by irradiation. In that case the shift on the reference temperature, DT56 J is equal to 20 C whereas a value of 53 C is expected from the data shown on Fig. 4. The correct value of the temperature shift can be obtained assuming that ru is also affected by irradiation. The parameter for the irradiated material is expressed as: rirr u ðT Þ ¼ ru ðT DT U Þ. The shift DTU depends on the level of irradiation. The transition curve was also simulated for Dr300 Y ¼ 45 (low irradiation level) and 150 MPa (very high irradiation level). DTU was adjusted for both irradiation levels; results are shown on Fig. 5. The simulated values of DT56 J are compared on Fig. 7. It is shown that DT56 J is always underestimated assuming that ru is unaffected by irradiation. Assuming that ru is a constant equal to the low temperature value for the unirradiated material (2925 MPa) gives a conservative estimation of DT56 J.
C. Bouchet et al. / Computational Materials Science 32 (2005) 294–300
299
Fig. 6. Simulation of the Charpy transition curve (failure probability equal to 50%) for the unirradiated material and the irradiated material (DrY = 88 MPa) assuming that ru is unaffected by irradiation or using a shifted ru function to account for irradiation. The finite element mesh accounting for usual symmetries are also shown.
140 _
100
u
( ˚ C)
120 Cte
80 60 40 u(T)
20 0
0
50
100
300
150 200 (MPa)
250
Fig. 7. Prediction of DT56 J. Three hypothesis are used for the parameter ru: (i) constant value (low temperature value for the unirradiated material), (ii) temperature dependent ru (unirradiated material), (iii) temperature and irradiation dependent ru (Fig. 5). Thin lines indicate the experimental range.
5. Conclusions and discussion The ductile to brittle transition characterized by the Charpy test has been modeled in the case of a
RPV steel using constitutive equations for viscoplasticity, ductile tearing and brittle failure. The transition curve of the based unirradiated material can be modeled provided the parameter ru of the Beremin model is considered to be an increasing function of temperature. Irradiation hardens the material causing an increase of the stresses during the Charpy test thus causing earlier brittle failure. This effect cannot account for the whole experimental reference temperature shift DT56 J. It is therefore necessary to consider that irradiation also affects the mean cleavage stress and therefore the value of ru. It is therefore likely that the formation of copper-rich precipitates and dislocation loops under irradiation, which causes hardening, also favors cleavage crack nucleation. The micromechanical mechanism needs however to be identified. The increase of the flow stress induced by irradiation leads to an increase of the dissipated energy so that the simulated upper shelf energy at 20 C also increases. The experimentally reported decrease of the USE can however be explained based on the present simulation. From Fig. 6 it can be concluded that at 20 C brittle failure still
300
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occurs. This implies that if the USE is estimated from tests at 20 C a decrease will be predicted. An other way to estimate the USE is to increase the test temperate until pure ductile failure is obtained. As the flow stress decreases with temperature this will also decrease the measured USE.
[11]
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