Int J Fract (2013) 179:87–108 DOI 10.1007/s10704-012-9775-2

ORIGINAL PAPER

Radiation embrittlement modelling in multi-scale approach to brittle fracture of RPV steels Boris Margolin · Victoria Shvetsova · Alexander Gulenko

Received: 2 May 2012 / Accepted: 8 October 2012 / Published online: 2 November 2012 © Springer Science+Business Media Dordrecht 2012

Abstract The purpose of the present article is to develop a multi-scale brittle fracture modelling for irradiated RPV materials. For this development, applicability of local brittle fracture criteria for radiation embrittlement modelling is analysed through comparison of the predicted and test results on radiation embrittlement of RPV steels in terms of ductile-to-brittle transition temperature and fracture toughness. The influence of radiation-induced defects on the processes of cleavage microcrack nucleation and propagation is clarified. The physical-and-mechanical models of the effect of irradiation-induced defects on cleavage microcrack nucleation are developed on the basis of dislocation and brittle fracture theories. Stress-andstrain controlled fracture criterion is developed that allows the adequate prediction of radiation embrittlement by various mechanisms. The differences and commonalities are revealed in the nature of material embrittlement due to cold work and neutron irradiation. The mechanism is explained of significant recovery of fracture resistance properties with simultaneous increase of fraction of intercrystalline fracture after post-irradiation annealing. Engineering approach for prediction of the temperature dependence of fracture toughness as a function of neutron fluence is justified. B. Margolin (B) · V. Shvetsova · A. Gulenko Central Research Institute of Structural Materials “Prometey”, Shpalernaya str., 49, 191015 Saint-Petersburg, Russia e-mail: [email protected]; [email protected]

Keywords Local fracture criterion · Cleavage microcrack nucleation · Radiation defects · Modelling · RPV steel

Nomenclature σ1 σeq  p æ = d εeq p d εeq σY SC σd σ˜ d , σd0 and η  SC and ξ

KJC Ttr  T F

The maximum principal stress The equivalent stress The accumulated plastic strain The equivalent plastic strain increment The yield stress The critical stress for microcrack propagation (the critical brittle fracture stress) The critical stress for microcrack nucleation The Weibull parameters for the probability of microcrack nucleation in unit cell The Weibull parameters for the probability of microcrack propagation in unit cell Fracture toughness The ductile-to-brittle transition temperature The parameter controlling KJC (T) in the Unified Curve method Temperature Neutron fluence

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1 Introduction At the present time, a multi-scale approach for prediction of brittle fracture of RPV steels is being intensively developed. Ideally this modelling is expected to unify the mechanical and physical aspects of material behaviour. It is clear that a key link in multi-scale fracture modelling is a local cleavage fracture criterion which includes cleavage microcrack nucleation and propagation conditions. Indeed, the local criterion allows the prediction of fracture characteristics on a macro-scale in terms of solid mechanics parameters and at the same time the critical parameters included in the local criterion are related to the physical processes of cleavage microcrack nucleation and propagation. When modelling the effect of irradiation on cleavage fracture, a multi-scale approach should be understood as a chain from radiation-induced defects through the physical mechanisms of nucleation and propagation of cleavage microcracks up to fracture on a macro-scale. At present the most unstudied link in this chain is the influence of radiation-induced defects on the processes of cleavage microcrack nucleation and propagation. Thus, in spite of significant progress both in mechanical and microstructural studies there is a gap in modelling of the transition from defects evolution on a micro-scale to fracture prediction on a macro-scale that has to be made through local cleavage fracture criterion. The main objective of the present study is to try to bridge this gap that may be achieved through the consideration of the processes of cleavage microcrack nucleation and propagation in material with irradiationinduced defects. It seems that there are several principal questions that have to be considered at this point: • what does embrittlement, in particular, radiation embrittlement mean from viewpoint of local cleavage fracture criteria, in other words, what parameters in local criterion are responsible for embrittlement; • whether new initiators of cleavage microcracks arise in irradiated material as compared with unirradiated or old initiators “work” only; • how irradiation-induced defects affect cleavage microcracks nucleation on carbides being the main initiators for ferritic steels; • why the effect of material hardening (the increase of yield stress) is quite different as caused by irradiation or by cold work;

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• how non-hardening mechanism of radiation embrittlement, in particular, the phosphorus effect may be predicted with local criterion; • how an appearance of intergranular fracture in irradiated RPV steels may be explained in terms of local criterion. Analysis of the above questions may be very helpful for the development of multi-scale brittle fracture modelling for irradiated RPV materials and for deeper understanding of material embrittlement phenomenon from viewpoint of local cleavage fracture criteria. It should be mentioned that the widely accepted approach (Lidbury et al. 2006; Diard 2005) can be criticized on the grounds that no attempt has been made to study cleavage microcrack behaviour in material with irradiation-induced defects. This approach is practically based on multi-scale modelling of plastic behaviour of a material that is constructed with several well-developed tools such as Molecular Dynamics and Rate Equation Theory codes, Discrete Dislocation Dynamics, and so on. However it is clear that any output from plastic deformation behaviour (yield stress, stress-strain curve etc.) cannot directly be converted into fracture behaviour. That’s why to perform a transition to fracture prediction for irradiated RPV steels the accepted approach uses an empirical correlation Ttr versus  σY (here Ttr is the ductile-to-brittle transition temperature (DBTT) shift and  σY is the yield stress increment). At the same time more direct way for the achievement of a multi-scale fracture modelling for irradiated RPV materials may be currently developed. Indeed, the physical nature of irradiation-induced defects in RPV steels are studied in detail, local cleavage fracture criteria are formulated and applied for fracture prediction and basic properties of the neutron irradiation effect on mechanical behaviour of a material are known. Thus, in the present paper an attempt is undertaken to develop a multi-scale brittle fracture modelling for irradiated RPV materials on the basis of local fracture criterion. This development has been achieved through (i) the analysis of applicability of available local cleavage fracture criteria for radiation embrittlement modelling; (ii) the analysis of the influence of irradiation-induced defects on the processes of cleavage microcrack nucleation and propagation; (iii) the development of physical-and-mechanical models of the effect of irradiation-induced defects on cleavage microcrack nucle-

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ation. The paper discusses also the different effect on brittle fracture caused by material hardening due to cold work and neutron irradiation, and explains an appearance of the mode of intergranular fracture for RPV steels after irradiation and post-irradiation annealing in terms of local criterion.

2 Local cleavage fracture criteria and their applicability for radiation embrittlement modelling 2.1 Formulations and physical background Currently two local cleavage fracture criteria are mainly used in brittle fracture modelling. Traditional formulation (Yoffe et al. 1924; Davidenkov 1936; Fridman 1952; Pisarenko and Krasowsky 1972; Knott 1973; Ritchie et al. 1973) is written as σeq ≥ σY

(1a)

σ1 ≥ S C ,

(1b)

where σeq is the equivalent stress, σY , the yield stress, σ1 , the maximum principal stress and SC , the critical brittle fracture stress, which is generally assumed to be independent of temperature, strain rate and stress triaxiality. From the physical point of view, the first condition is the condition for the nucleation of cleavage microcracks, and the second one—the condition of their propagation. Another local criterion of cleavage fracture was formulated and verified in the papers (Margolin and Shvetsova 1992, 1996; Margolin et al. 1997a, 1998b, 2006). This formulation is written in the form σnuc ≡ σ1 + mT ε · σeff ≥ σd ,

(2a)

σ1 ≥ SC (æ),

(2b)

where  p the effective stress is σeff = σeq − σpY , æ = d εeq is the accumulated plastic strain, d εeq is the equivalent plastic strain increment, σd is the strength of carbides or carbide-matrix interfaces or other particles on which cleavage microcracks are nucleated and mT ε is the concentration coefficient for the local stress near the microcrack-nucleating particles. Formulation (2b) for the cleavage microcrack propagation condition is taken in the same form as usually used and the critical brittle fracture stress, SC (æ), is interpreted as the stress for microcrack start and

propagation through various barriers such as grain boundaries, microstresses, slip bands and boundaries of dislocation substructure. The parameter SC is related to the length of microcrack which is equal to the distance between barriers, and the effective energy of these barriers. When the plastic strain increases, microstresses and the effective energy of barriers increase and the distance between barriers decreases due to dislocation substructure formation. As a result, the critical fracture stress SC increases. The performed theoretical and experimental studies (Margolin and Shvetsova 1992, 1996; Margolin et al. 1997a) allowed one to obtain the dependence of SC on plastic strain in the form SC (æ) = [C1 + C2 exp(−Ad æ)]−1/2 ,

(3)

where C1 , C2 , Ad are material constants. The proposed formulation (2a) for the cleavage microcrack nucleation condition is based on the analysis of the known dislocation models of microcrack nucleation. As a common case, the cleavage microcrack nucleation condition may be formulated when considering the stress concentration near the head of dislocation pile-up arrested by some obstacle (Margolin and Shvetsova 1992; Margolin et al. 1997a). Microcrack nucleation is assumed to happen when the sum of the maximum principal stress and local stress approaches some critical stress σ1 + σloc (r)|r=rc = σd , where σloc (r)—the maximum local normal stress in the head of dislocation pile-up; r—the polar coordinate, rc —a certain characteristic size. The parameter σd may be interpreted as strength of obstacle or matrix-obstacle interface and termed the critical stress for microcrack nucleation. To determine the distribution of local stresses near the dislocation pile-up head, dislocation pile-up is presented as sliding crack or mode II crack with some blunting. It allows the derivation of equation σloc (r)|r=rc = mT ε · σeff (Margolin et al. 1997a). As distinct from traditional dislocation models, the dislocation pile-up geometry is considered to depend on plastic strain and temperature (Margolin and Shvetsova 1992): the blunting of dislocation pileup increases as temperature increases due to cross-slip of dislocations and the length of dislocation pile-up decreases as plastic strain increases due to formation of dislocation substructure. This consideration provides the dependence of the coefficient mT ε on temperature T and plastic strain in the form mT ε = mT (T) · mε (æ).

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The functions mT (T) and mε (æ) are calculated as (Margolin et al. 1998b, 2006) mε (æ) = S0 /SC (æ),

(4)

mT (T) = m0 σYs (T),

(5)

where S0 ≡ SC (æ = 0) is the stress of start for the nucleus microcrack, m0 is a constant which may be experimentally determined and σYs is the temperaturedependent component of the yield stress. The dependence σYs (T) is taken in the form σYs (T) = b · exp(−hTa )

(6)

where b and h are the material constants independent of temperature, Ta is taken in Kelvin degrees. Equations (5) and (6) allow, in particular, the prediction of the temperature dependence of fracture toughness. To describe the effect of strain rate on fracture toughness, constant h in Eq. (6) may be written according to the Zerilli– Armstrong equation (Zerilli and Armstrong 1987) in the form h = β0 − β1 · ln æ˙

(7)

where β1 and β0 are material constants, and æ ˙ is the accumulated plastic strain rate. For further analysis it is useful to compare the above criteria from viewpoint of the underlying physical processes. It is seen from criterion (1) that the condition of cleavage microcrack nucleation is taken in the simplest form (1a) as the requirement to reach a minimum plastic strain corresponding to yield stress that is usually equal to 0.2 %. In other words, it is assumed that cleavage microcracks are always nucleated at the beginning of plastic flow. As distinct from condition (1a), cleavage microcrack nucleation according to condition (2a) depends on the maximum principal stress, plastic strain and temperature and is characterized by the critical stress σd for microcrack nucleation. It is important that the plastic strain when microcrack is nucleated may exceed 0.2 % and increases with temperature. It should be noted that condition (2a) describes not only cleavage microcrack nucleation caused by dislocation pile-up near some obstacle but also caused by twin near grain boundary. Both dislocation-pile-up and twin may result in the stress concentration due to shear deformation arrested by obstacle or grain boundary. For microcrack nucleated by twin mT (T) = const as σYs (T) = const (Qiao and Argon 2003).

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It may be noted that the connection of cleavage microcrack nucleation with plastic deformation seems to be quite clear from the physical point of view. Nevertheless, when formulating the local cleavage fracture criterion this connection was explicitly used only near twenty years ago (Margolin and Shvetsova 1992; Chen et al. 1994). Now this consideration is widely used in other models, for example, Bordet et al. (2005). The important consequences follow from this difference between conditions (1a) and (2a). It is known that as a common case, two processes have to occur for cleavage fracture: microcrack nucleation and microcrack propagation (the start and unstable growth of microcrack through various barriers). According to criterion (2) the brittle fracture on a macro-scale may be controlled by both microcrack nucleation (2a) and microcrack propagation conditions (2b) that depends on material properties and loading conditions, mainly, on the ratio SC / σY , stress triaxiality and temperature. For example, the brittle fracture of smooth specimens is controlled by condition (2b) and, by contrast, the brittle fracture of notched or cracked specimens from medium and high strength steels, in particular, from RPV steels is controlled by condition (2a) (Margolin et al. 1998a, 2006). For smooth tensile specimens for that stress triaxiality is low, microcrack nucleation condition (2a) is satisfied earlier than propagation condition (2b). Therefore condition (2b) controls the brittle fracture of smooth tensile specimens. When stress triaxiality is high that is typical for notched and cracked specimens, condition (2b) is satisfied for medium and high strength steels already at very small plastic strain when cleavage microcracks are still not nucleated (condition (2a) is not fulfilled). That’s why brittle fracture occurs just after satisfaction of condition (2a). By other worlds, condition (2a) controls brittle fracture for this case. According to criterion (1) the brittle fracture on a macro-scale is controlled practically by the only process—microcrack propagation, i.e. by condition (1b) as the microcrack nucleation condition (1a) is practically always satisfied earlier than condition (1b). As a consequence, a contradiction arises when interpreting test results from smooth and notched tensile specimens from RPV steels ruptured over the brittle fracture temperature range (Beremin 1983; Margolin et al. 1998a, 2006). Interpretation with criterion (1) shows that there is no transferability of the parameter SC for specimens of different geometries. This conclu-

Radiation embrittlement modelling

sion contradicts the known statement about invariance of SC to stress triaxiality (Davidenkov 1936; Fridman 1952; Ritchie et al. 1973). An attempt to explain these results by considering a stochastic nature for parameter SC (Beremin 1983) is not convincing (Margolin et al. 1998a). When formulating the above criteria (1) and (2) in a probabilistic manner (Beremin 1983; Margolin et al. 1998b) the situation varies only for criterion (2) so that both conditions (2a) and (2b) control brittle fracture of a specimen. 2.2 Applications for the prediction of fracture properties for RPV steels Prediction of brittle fracture on a macro-scale in a stochastic manner may be performed on the basis of criterion (1) with the Beremin model (Beremin 1983) and on the basis of criterion (2) with the Prometey model (Margolin et al. 1998b, 2002, 2008a). Both models use the Weibull statistics for stochastic parameters and the weakest link model to predict the brittle fracture on a macro-scale. The Beremin model takes the critical stress SC as stochastic parameter, and the Prometey model uses two stochastic parameters—σd and SC . It should be mentioned that criterion (1) was used earlier in the Pisarenko and Krasowsky (1972) model and in the Ritchie–Knott–Rice (RKR) (1973) model as well as criterion (2) was used in the model (Margolin et al. 1997b), for fracture toughness prediction in a deterministic manner. The Beremin and Prometey models were in detail represented in international literature so that there is no need to consider these models once more. In Sect. 2.4 the main equations of the Prometey model are given that are required for the subsequent analysis. Applicability of the RKR and Beremin models was widely studied for various materials. It was found that there are difficulties with the use of the RKR and Beremin models for medium and high strength steels, in particular, for RPV steel (Margolin and Shvetsova 1992, 1996; Margolin et al. 2006). It was shown (Merkle et al. 1999; Margolin et al. 1999) that the prediction of fracture toughness of irradiated RPV materials with the RKR and Beremin models is not correct. The reason consists in the fact that according to these models the temperature dependence of fracture toughness, KJC (T), is determined practically by the temperature dependence of the yield stress σY (T) as SC does not

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depend on temperature. According to criterion (1) the rate of growth of KJC with temperature T is controlled Y by the parameter σ1Y · dσ dT . For highly embrittled material the variation of KJC with temperature T occurs over the temperature range where this parameter → 0. As a result, the parameter KJC does not practically depend on temperature that is in contradiction with test results. Some attempts (for example Parrot et al. 2006; Tanguy et al. 2006) were undertaken to reformulate these models by introduction of the temperature dependence for the parameter SC (or the parameter σu in the terms of the Beremin model). Thus, the parameter SC becomes not invariant relative to stress triaxiality and temperature. It is clear that from physical viewpoint such “reformulation” cannot be considered as reasonable. It has been found in Margolin and Shvetsova (1992, 1996), Margolin et al. (1997a) that difficulties with the use of the RKR model and the Beremin model for medium and high strength steels are connected with the use of a microcrack nucleation condition in the form (1a). The Prometey model was verified by application to RPV steels in various conditions (initial, irradiated and highly embrittled) (Margolin et al. 1998b, 2002). It should be noted that there is no problem with prediction of KJC (T) curve with the Prometey model for embrittled RPV steels as the parameter σYs (T) is used in criterion (2) as temperature dependent parameter. As a result, Y over the temperature range where σ1Y · dσ dT → 0, the Ys ratio σ1Ys · dσ dT = 0. This is why criterion (2) allows one to describe KJC (T) adequately even for highly embrittled material. On the basis of the Prometey model an engineering method named the Unified Curve was proposed (Margolin et al. 2003). The Unified Curve method as well as the Prometey model predicts a possible variation in KJC (T) curve shape under irradiation. The Unified Curve method was verified by using more than 35 sets of experimental data for ferritic steels with various degrees of embrittlement, including irradiated RPV materials. 2.3 Applicability of local criteria for radiation embrittlement modelling From the viewpoint of multi-scale modelling of the effect of irradiation on cleavage fracture it is important, first of all, to analyse the applicability of

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criteria (1) and (2) in terms of having an adequate link with the physical mechanisms of radiation embrittlement. At present, for RPV steels three basic mechanisms involved in radiation embrittlement are generally recognized: matrix damage caused by irradiation-induced lattice defects, such as clusters of point defects and dislocation loops, precipitation of various elements, namely, copper, nickel, manganese and other, and segregation of impurities, mainly phosphorus (Hawthorne 1983; Nikolaev and Rybin 1996; Gorynin et al. 1996; Alekseenko et al. 1997; Gurovich et al. 1997, 2000; Williams et al. 2002; Debarberis et al. 2005). These radiation damages and mechanical properties of irradiated RPV materials are linked as follows. The matrix damage and element precipitation result in an increase of σY as the lattice defects and precipitates affect the dislocation mobility. This increase of the yield stress is caused by an increase of the temperature-independent (athermal) component σYG of the yield stress. Segregation of impurities, as a rule, is not associated with changes in σY due to irradiation, at the same time these segregations may result in increase of the DBTT (Alekseenko et al. 1997). Thus, segregation of impurities, in particular, phosphorus, results in so-called non-hardening mechanism of embrittlement. These properties are shown in Fig. 1. The role of non-hardening mechanism in embrittlement of RPV steels is also clearly seen from the data on post-irradiation annealing (Gurovich et al. 1997). It was shown that after post-irradiation annealing of 2Cr–Ni–Mo–V RPV steel at temperature 475 ◦ C the full recovery of yield stress occurs and at the same time DBTT recovers not fully. When analysing the applicability of the above criteria for radiation embrittlement modelling it should be taken into account that the critical stress SC does not practically depend on neutron fluence (at least, for transcrystalline brittle fracture (Alekseenko et al. 1997; Margolin et al. 1999). Then criterion (1) contains the only parameter—σY that depends on fluence. Hence, criterion (1) describes radiation embrittlement as a result of the material hardening only. Concerning another physical mechanism of radiation embrittlement, segregation of phosphorus and other impurities, it is clear that criterion (1) can not describe the trends shown in Fig. 1. Thus, criterion (1) describes radiation embrittlement through the mechanical factor only—increase of σ1 due to increase of σY .

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Criterion (2) contains two parameters that depend on fluence: the critical stress for microcrack nucleation σd and σY . It means that criterion (2) takes into account not only the material hardening but also a possible weakening of microcrack initiators. Prediction of fracture toughness for irradiated RPV materials with criterion (2) has shown that the parameter σd decreases when a degree of embrittlement increases. Decreasing the parameter σd may be interpreted as the weakening of microcrack initiators. This process may be considered as the physical factor of embrittlement. It is clear also that segregation of impurities (phosphorus) resulting in the material embrittlement without material hardening (see Fig. 1) may be explained with criterion (2). Indeed, as known the phosphorus segregation may occur on ferrite-carbide interfaces (Gurovich et al. 2000) and result in decreasing the matrix-carbide interface strength (i.e. in decreasing the critical stress σd ). This means that the nucleation of cleavage microcracks becomes easier compared with unirradiated steel. It seems that an application of criterion (1) may be justified from viewpoint of empirical correlations for irradiated materials. Indeed, the correlation Ttr versus σY is known to follow qualitatively from the Yoffe scheme (Yoffe et al. 1924) based on criterion (1). However, if we estimate on the basis of criterion (1) for steels with various σY the response in Ttr caused by the same value of σY we obtain the predicted curve 1 shown in Fig. 1 that differs essentially from curve 2 constructed according to empirical correlation Ttr (σY ). This empirical correlation is used in the form 0.857   σY /σY , (8) Ttr = 1070 4.53 +  σY / σY proposed in Lidbury et al. (2006) on the basis of treatment of large experimental data set for RPV materials. Equation (8) shows that the value of Ttr for σY = const decreases when the yield stress increases as shown by curve 2 in Fig. 2. At the same time the opposite trend follows from criterion (1): the higher σY the larger Ttr value (see curve 1 in Fig. 2). These quite different trends for curves 1 and 2 mean that criterion (1) cannot be used for radiation embrittlement modelling for RPV steels. Thus, the above examples show that radiation embrittlement of RPV materials cannot be considered as a result of the material hardening only. Criterion (2) in principle allows the description of radiation embrittlement caused by both material hardening and the

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200

250 Yield stress increment (MPa)

Transition temperature shift ( 0C)

Radiation embrittlement modelling

- Tirr=(50÷80) 0C; F=(3÷5)⋅1023 n/m2 - Tirr=350 0C; F=1.0⋅1024 n/m2

150

100

50

0

0

0.01

0.02

0.03

0.04

Phosphorus content (wt. %)

(a) Fig. 1 The effect of phosphorus content on the transition temperature shift a and the yield stress increment b for Cr–Ni–Mo–V steel (concentration of other impurities: Cu = 0.04 ÷ 0.07 %, Sb

ΔTtr

ΔσY=const 1

2 σY Fig. 2 Ttr versus σY for σY = const according to criterion (1) (curve 1) and the experimental correlation (8) (curve 2) (scheme)

initiator weakening. That’s why just this criterion is used hereinafter for modelling of the effect of irradiation on cleavage fracture. Next necessary step for further development of this modelling is consideration of the processes of cleavage microcrack nucleation and propagation in material with irradiation-induced defects that provides, in particular, the dependencies of local criterion parameters on neutron fluence.

2.4 The main considerations of the Prometey model 2.4.1 Local criterion (2) in probabilistic statement The following assumptions are used in local cleavage fracture criterion with two stochastic parameters (Margolin et al. 2008a). 1. The polycrystalline material is viewed as an aggregate of cubic unit cells. The size of the unit cell ρuc is never less than the average grain size. The stress

- Tirr=50 0C; F=2.0⋅1023 n/m2 - Tirr=350 0C; F=1.3⋅1024 n/m2

200

150

100

50

0

0.01

0.02

0.03

0.04

Phosphorus content (wt. %)

(b) + Sn + As = 0.001 ÷ 0.002 %; neutron fluence is given for E ≥ 0.5 MeV) (Alekseenko et al. 1997)

and strain fields in the unit cell are assumed to be homogeneous. 2. The criterion of brittle fracture of a unit cell is taken as criterion (2). 3. It is assumed that the parameters σd and SC are stochastic and the remainder of the parameters in criterion (2) is deterministic. 4. To describe the distribution function for the parameter σd , the Weibull law is used     σd − σd0 η p(σd ) = 1 − exp − , (9) σ˜ d where p(σd ) is the probability of finding in the considered unit cell a carbide with minimum strength less than σd ; σ˜ d , σd0 and η are Weibull parameters. 5. The distribution function for the parameter SC is described by the Weibull two-parameter function     SC ξ p(SC ) = 1 − exp − , (10) S˜ C where p(SC ) is the probability that the critical brittle fracture stress for the considered unit cell is less than SC ; S˜ C and ξ are Weibull parameters. 6. The weakest link model is used to describe the brittle fracture of the polycrystalline material.

2.4.2 The Prometey probabilistic model The basic considerations of the model are given according to Margolin et al. (2008a).

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1. The probability of microcrack nucleation in a unit cell is calculated as     σnuc −σd0 η , (11) Pnuc = 1 − exp − σ˜ d if σnuc (æ) = max σnuc (0, æ) and dσnuc > 0 (12) dPnuc = 0 if σnuc (æ)< max σnuc (0, æ) or dσnuc ≤ 0 (13) These conditions follow from the physical particularities of microcrack nucleation and Eq. (9). Equation (9) describes the distribution of the minimum strength of carbides over unit cells. In each unit cell, carbides with higher strength are present. Therefore for a unit cell the process of microcrack nucleation begins when the parameter σnuc reaches the minimum strength of carbide and goes on as long as the parameter σnuc increases. If this parameter begins to decrease then the process of microcrack nucleation stops and recommences again when the value of σnuc (æ) reaches the maximum value of σnuc over the whole previous deformation interval from æ = 0 to the current value of æ. This maximum value is designated max σnuc (0, æ). Thus, continuous nucleation of cleavage microcracks occurs when conditions (12) are satisfied. For σnuc (æ) < maxσnuc (0, æ) microcracks can not be nucleated even if dσnuc > 0 as all the microcrack-nucleating carbides for which condition (2a) have been satisfied were cracked during previous deformation. Hence for σnuc (æ) < maxσnuc (0, æ) or dσnuc ≤ 0 the probability of microcrack nucleation in a unit cell does not vary (Eq. (13)). 2. The probability of initiation and propagation of the nucleus microcrack in a unit cell may be calculated according to Eqs. (2b) and (10) as     σ1 ξ , (14) Pprop = 1 − exp − S˜ C (Hereafter this probability is briefly denoted the probability of propagation.) The parameter S˜ C is a function of the accumulated plastic strain æ  −1/2 , (15) S˜ C (æ) = σ˜ C · 1 + β exp(−Ad æ) where σ˜ C is some constant and β = C2 /C1 , C1 and C2 are coefficients in Eq. (3).

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Then Eq. (14) may be rewritten as     σprop ξ , Pprop = 1 − exp − σ˜ C

(16)

where

σprop = σ1 1 + β exp(−Ad æ)

(17)

The parameter σprop may be interpreted as some driving stress (instead of σ1 ) which controls microcrack propagation in the unit cell. 3. To calculate the probability of brittle fracture of a unit cell the monotonically increasing parameter æ is used to describe loading history. The whole loading history of a specimen is divided into intervals æ0 −æ1 ; æ1 −æ2 ; . . .; æn−1 −æn , . . ., æN−1 −æN , where æ0 = 0 and æN is the final considered value of æ. The probability of brittle fracture of the ith unit cell over the whole loading history of a specimen is calculated with n=N 

Pif =



Pinuc (æn )−Pinuc (æn−1 ) · max Piprop (æn , æN )

n=1

(18) where Pinuc (æn−1 ) and Pinuc (æn ) are the probabilities of microcrack nucleation over the loading ranges from æ0 = 0 to æn−1 and æ0 = 0 to æn respectively. The second term in Eq. (18) is the maximum value of the probability of microcrack propagation in the ith unit cell over the range from æn to æN . Equation (18) takes into account that microcracks propagating by cleavage mechanism are nucleated continuously and the process of microcrack propagation in each unit cell may happen only once. 4. The probability of brittle fracture of a specimen is calculated allowing for the weakest link model as Pf = 1 −

i=k  

 1 − Pif ,

(19)

i=1

where k is the number of unit cells in a specimen. 2.4.3 Stress and strain field calculation and the model parameters calibration Stress and strain fields for specimen are calculated by finite element method on the basis of incremental theory of plasticity with von Mises criterion in finite strain

Radiation embrittlement modelling

statement. For cracked specimen the finite elements sizes near the crack tip is considerably less (by the order) that the unit cell size. When calculating the probabilities of nucleation and propagation the stress and strain components are averaged over the unit cell volume. This treatment is carried out for undeformed finite elements mesh. To use the Prometey model it is necessary to find six parameters: σ˜ d , σd0 , η, σ˜ C , ξ and ρuc . It is clear that calibration of all these parameters is practically impossible from one series of tests. To overcome this difficulty a procedure was proposed (Margolin et al. 2007) that allows the calibration of two sets of the parameters (˜σC , ξ, σd0 ) and (˜σd , η) independently of one another from test results of specimens of two different types. The parameters σ˜ C , ξ and ηd0 are determined from the test results of standard smooth cylindrical tensile specimens over the brittle fracture temperature range. For these known parameters the values of σ˜ d and η are calibrated from test results of fracture toughness specimens or tensile notched cylindrical specimens at one temperature when brittle fracture occurs. Numerical values of the parameters for 2Cr–Ni–Mo–V RPV steel are presented in Margolin et al. (2007). The unit cell size ρuc may be determined from best agreement of the calculated and experimental values of KJC for a given temperature. The most suitable temperature is one corresponding to the low shelf of KJC (T). 3 Cleavage microcrack nucleation and propagation in material with irradiation-induced defects 3.1 Initiators of nucleus cleavage microcrack For ferritic steels in unirradiated condition, as a common case, microcrack nucleation may occur as a result of the rupture of elongated carbide or carbide-matrix interface. For RPV steels, globular carbides are typical (Gurovich et al. 2000; Ortner 2006), and therefore the second mechanism appears to be predominant. From viewpoint of the dislocation models for microcrack nucleation, both initiators provide some barriers for dislocations and, hence for the stress concentration. For analysis of cleavage microcrack nucleation in material with irradiation-induced defects it is necessary, first of all, to answer the question whether new initiators of cleavage microcracks arise in irradiated

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material as compared with unirradiated. For the answer it is important to define which nucleus cleavage microcrack is able to initiate cleavage fracture. Only new (fresh) cleavage microcrack may result in brittle fracture and cleavage nucleus microcrack has to be atomically sharp microcrack with the tip blunting equal to lattice parameter a0 . Such microcrack starts if its length l0 satisfies Griffith’s condition 

γE S 0 π l0 = 2 (20) 1 − μ2 where γ is the surface energy, E is Young’s modulus, μ is Poisson’s ratio, S0 is the stress of initiation for the nucleus microcrack. It should be noted that such requirement—to have atomically sharp nucleus microcrack—means that nucleus microcrack has not emitted dislocations from its tip before the start. In other words, the requirement for the microcrack tip blunting to be equal to the lattice parameter a0 is the physical condition of the stability to dislocation emission for microcrack initiation by brittle manner. Otherwise, for example, when the tip blunting equals to 2a0 , it shows that dislocation has been emitted from crack tip. If condition (20) is not met at microcrack nucleation moment this microcrack transforms in micro-void which can not result in brittle fracture under posterior loading. In other words, stationary microcrack (not fresh) can not result in brittle fracture. To exclude dislocation emission from microcrack tip before fulfilment of Griffith’s condition (20) the following conditions must be satisfied. 1. Nucleus microcrack has to have such orientation which satisfies Kelly–Tyson–Cottrell condition (Thomson 1983) τmax /σmax <τth /σth

(21)

where τmax is the maximum shear stress near crack tip, and τth , the theoretical shear strength. It follows from condition (21) that when σmax = σth , condition τmax < τth is satisfied, and hence, dislocation emission does not happen at elementary fracture event—rupture of atomic bond in the crack tip. dl has to be 2. The relative microcrack growth rate ld τ significantly larger than the dislocation motion rate, i.e. larger than the plastic strain rate of a material (Margolin et al. 2008a).

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Apparently, this condition is satisfied when a microcrack is nucleated near strong barrier. For this case, very powerful dislocation pile-up is formed, and when rupturing barrier, all dislocation fall down into the microcrack cavity practically instantly, so the nucleated microcrack grows very quickly. Basing on the above definition of nucleus cleavage microcrack it becomes clear that new initiators of cleavage microcracks do not arise in irradiated material as compared with unirradiated. Indeed, on the basis of microstructural examination results (Gorynin et al. 1996; Gurovich et al. 2000) it may be concluded that all irradiation-induced defects are very small and weak to nucleate a sharp cleavage microcrack. For example, sizes of dislocation loops are near 5 ÷ 20 nm and the nucleus microcrack size estimated from the Griffith’s condition (20) is 100 ÷ 400 nm. Thus, even if some discontinuities arise near radiation-induced defects they are not able to propagate by cleavage mechanism. Therefore, it is necessary to answer a question, how the irradiation-induced defects may affect cleavage microcrack nucleation on carbides or other initiators existing in unirradiated steel. 3.2 The effect of radiation defects on cleavage microcrack nucleation As a common case, it may be expected that possible mechanisms of the effect of irradiation-induced defects on cleavage microcrack nucleation may be divided into two groups. The first group includes the mechanisms connected with decreasing the strength of carbides or carbidematrix interface. For both cases the critical stress for microcrack nucleation σd decreases. (For RPV steels the decrease of σd due to decreasing the carbide-matrix interface strength is more proper (Margolin et al. 1999, 2008a) as globular carbides are cleavage microcrack initiators). The second group includes the mechanism connected with more easy formation of dislocation pileups near microcrack initiators due to radiation defects (Margolin et al. 2008a). This process may be described by increasing the probability of dislocation pile-up formation. Let us consider the possible mechanisms in detail.

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3.2.1 Mechanisms of decreasing the critical stress for microcrack nucleation The impurity segregation from viewpoint of decreasing the critical stress σd is most understandable mechanism. In 1972 J. Hawthorne assumed (Hawthorne 1983) that the phosphorus role in radiation embrittlement is similar to its role in temper embrittlement and connected with the diffusion of phosphorus to ferritecarbide interfaces. Later this idea was independently considered by Nikolaev and Rybin (1996) and confirmed by results of microstructural examinations (Gurovich et al. 2000). At the same time it should be noted that the mechanisms of the phosphorus diffusion appear to differ for temper and radiation embrittlement. This difference becomes apparent when analysing the data in Fig. 1. As seen from this figure, the strong effect of phosphorus on the DBTT shift is clearly revealed both at the irradiation temperatures Tirr = 50 and 350 ◦ C. It means that the phosphorus diffusion under irradiation is not classical thermo-activated diffusion. It is clear that from viewpoint of classical diffusion, time to have the P effect on DBTT shift at Tirr = 50 ◦ C must be larger by several orders than time at Tirr = 350 ◦ C. One of possible mechanisms of phosphorus diffusion under irradiation is so-called “solute drag” when phosphorus is involved in fluxes of point defects arising under irradiation (Okamoto and Rehn 1979). It is clear that segregation of impurities may result in the weakening of the cleavage microcrack initiators as it decreases the matrix-carbide interface strength. A simple physical model for phosphorus and other impurity segregation on carbide-matrix interface under irradiation provides the dependence of σd on neutron fluence F (Margolin et al. 2005)  m   F σd (F) = σ0d exp − α1 (22) F0 where σ0d = σd (F = 0); α1 and m are constants for a given condition of irradiation; F0 is the normalizing factor which is taken to be equal to 1022 n/m2 . Another mechanism for decreasing the critical stress σd is the arising of internal stresses caused by irradiation-induced dislocation loops on carbide-matrix interface. These internal stresses result in rupture of the interface at stress σnuc being less than σnuc for unirradiated material, i.e. condition (2a) is satisfied for

Radiation embrittlement modelling

97

Taking into account that the dislocation loop density ρloop = (lloop )−3 Eq. (23) is rewritten as dislocation loops

carbide

2/3 0 σeff d = σd − σloop · Sloop · ρloop .

As a characteristic parameter for the loop density ρloop the Orovan stress (Ashby 1970) may be used   ρi di , (25) τorov = α Gb

matrix

σ 0

-

-

Fig. 3 Scheme to calculate internal stresses near dislocation loops located at carbide-matrix interface

smaller value of σnuc . Thus, the influence of internal stress may be taken into account by introducing the effective stress σeff d decreasing due to radiation defects. The dependence σeff d on neutron fluence F may be deduced by the following way. Stress on the carbidematrix interface is absent when σnuc = 0 and dislocation loops are absent. Let the carbide-matrix atomic bond be linear elastic. Dislocation loop located on the interface results in tension of carbide-matrix atomic bonds; by this the internal stress near dislocation loop is compressive. By another words, dislocation loop “works” as a wedge between carbide and matrix as schematically shown in Fig. 3. Rupture of carbidematrix bonds may be represented as follows. On the first stage the mutual action of σnuc and internal stress caused by dislocation loop results in rupture of carbide-matrix bonds. After that internal stress caused by dislocation loop relaxes and tensile stress σnuc increases sharply as the bonds have ruptured. This results in rupture of other bonds generated by dislocation loops. To obtain quantitative estimation the assumption may be taken that if the first stage has occurred the second stage happens automatically. The following designations are taken: σloop is the average tensile stress created by dislocation loop, Sloop is an area of stress σloop action, lloop is average distance between loops, σeff d is the effective stress averaged on carbide size that is required to rupture the carbidematrix interface. This stress is calculated allowing for additional internal stress as 0 σeff d = σd −

(24)

σloop · Sloop l2loop

.

(23)

where α—a constant dependent on barrier type, G— shear modulus, b—Burgers vector, ρi and di —the density and size of defects of ith type (Zelensky et al. 1979). Assuming that for irradiated material the value τorov varies mainly due to dislocation loops the following equation may be written

√ (26) τorov = c1 c2 + ρloop dloop − c2 ,  ρi di are constants, the where c1 = α Gb and c2 = ρ parameters i and di are related to unirradiated condition. √ Basing on the Mises theory σY = 3 · τY (here τY is the yield stress under shear) it may be written √ √ σY = 3τY = 3τorov . (27) As a common case, the dependence σY on neutron fluence F is described as (Alekseenko et al. 1997)   n1 F , (28) σY = AσY · F0 where AσY and n1 are material constants, F0 is a normalizing coefficient. Let us assume that the increase of σY caused by irradiation-induced dislocation loops may be described by equation similar to Eq. (28) for which material constants are designated as AσY ≡ Aloop and n1 ≡ nloop . Then the dependence σeff d on neutron fluence F may be deduced from Eq. (24) allowing for Eqs. (26), (27) and (28) as    2/3 2 F nloop √ eff 0 σd = σd − c3 c4 + c2 − c2 , F0 (29) Sloop σloop A where c3 = and c4 = √ loop are material con3c1 d2/3 loop stants. It is seen from Eq. (29) that the value of σeff d decreases as F increases. Equation (29) may be approximated for nloop < 3/4 by equation of type (22). This estimation for nloop follows from the fact that for ferritic steels for which irradiation hardening is caused

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mainly by dislocation loops (for example, for St.3 steel and 2Cr–Mo–V steel with low copper content) it may be written nloop ≡ n1 ≤ 0.5. The above consideration may be also applied to the arising of internal stresses caused by irradiationinduced precipitates on carbide-matrix interface. Thus, for two considered mechanisms of decreasing σd the dependence σd (F) may be described by equation of the same type as Eq. (22). 3.2.2 Increasing the probability of dislocation pile-up formation in irradiated material As a common case, a necessary process for cleavage microcrack nucleation near carbides or other initiators is the formation of dislocation pile-ups. It should be noted previously that for most dislocation models of microcrack nucleation, the influence of concentration of barriers on this process is not taken into account. In particular, the proposed condition (2a) does not contain any parameter characterizing concentration or distribution of barriers. Then for material with barriers of the same strength and different concentration, nucleation of a microcrack happens for the same σnuc . It is clear that this result is valid for the plane barriers only. For compact barriers (for example, spherical carbides) microcrack nucleation is easier for material with high concentration of barriers as the probability of formation of dislocation pile-up increases. It may be assumed that the probability of formation of dislocation pile-ups depends on concentration of various defects being barriers for dislocations. Radiation defects such as dislocation loops and precipitates that affect dislocation motion may result in easier formation of dislocation pile-ups near microcrack initiators. Hence, these defects increase the probability of formation of dislocation pile-ups (Margolin et al. 2008a). According to the consideration in Margolin et al. (2008a)  σ˜ eff d

= σ˜ d ·

σY + C √ 3·τ˜ orov

− k η

.

(30)

˜ d that determines where σ˜ eff d is the effective value of σ the probability of microcrack nucleation on globular carbides with the carbide-matrix interfacial strength σ˜ d provided that radiation defects are present.

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The dependence of σ˜ eff d on F may be presented in the form   n      F eff σ˜ eff σ ˜ = · exp −A · . (31) d d irr unirr F0 where A and n are constants for a given condition of irradiation. When comparing Eqs. (22) and (31) it is seen that the type of the dependence σ˜ eff d (F) is the same for all the considered mechanisms. This conclusion is very important for engineering application when predicting the transformation of KJC (T) curve as a function of neutron fluence with the Unified Curve method. 3.2.3 The effect of radiation defects on the concentration coefficient for local stress near the microcrack-nucleating carbides We have considered the mechanisms of the effect of irradiation-induced damage on σd . Evidently, irradiation-induced defects result also in an increase of “driving force” σnuc ≡ σ1 + mT ε · σeff in condition (2a) that is caused by increasing σ1 as a result of increasing σY . In principal, it is possible that the coefficient mT ε = m0 σYs (T)mε (æ) also increases due to increase of the parameter m0 . (Other parameters, σYs and σeff , do not practically vary under irradiation.) The parameter m0 , being to some degree sensitive to material microstructure, may depend on the particularities1 of plastic deformation in steels with radiation-induced defects. It is clear that as a common case, irradiationinduced dislocation loops and precipitates may affect the geometry of dislocation pile-ups arrested by carbides and, hence, the coefficient m0 . However, at present, the experimental data are too few to allow the exact analysis of this effect. Possible trends of this effect are expected to be the following. 1. Width of dislocation pile-up (or slip band) decreases as the density of radiation defects increases. As a result, the blunting of dislocation pile-up near initiator decreases and the coefficient m0 increases. This phenomenon is observed when localization of plastic deformation occurs. (For irradiated austenitic steels the typical mechanism of localization is so-called channel deformation. Although for irradiated ferritic steels the localization mechanism is 1

One of the possible particularities of plastic deformation in steels with radiation defects was taken into account in the mechanism proposed in Sect. 3.2.2.

Radiation embrittlement modelling

The above speculative considerations do not allow the determination of the dependence m0 (F). At the same time calculation of KJC (T) curves with the Prometey model for increasing function of m0 (F) has shown (Fig. 4a) that for this case the transformation of KJC (T) curve may be practically exactly described as a lateral shift. The KJC (T) curves calculated with the Prometey model for decreasing σd are represented in Fig. 4b. (Here the decrease in σd models increasing neutron fluence.). Transformation of KJC (T) curves shown in Fig. 4b is typical for irradiated (embrittled) RPV materials and may be approximately described as a lateral shift to higher temperature range for small degree of embrittlement, and as a variation in the KJC (T) curve shape for high degree of embrittlement (Margolin et al. 1999, 2002, 2003). Thus, criterion (2) and the Prometey model provide a possibility to predict not only this transformation of the KJC (T) curve but also pure lateral shift of KJC (T) curve to higher temperature range. It is appropriate to mention here that any models based on the stress controlled criterion (1) predict a variation in the KJC (T) curve shape for any degree of embrittlement. This is because the KJC (T) dependence is determined according to these models by the σY (T) dependence. These models may predict lateral shift of KJC (T) only if the dependence of SC (or σu in terms of the Beremin model) on temperature and neutron fluence is a priori introduced, for example, as it is made in (Parrot et al. 2006; Tanguy et al. 2006). Thus, it should be concluded that at present the Prometey model based 2

Equation (8) takes into account the effect of cross sliding in a scale of several grains by means of the parameter σYs (T).

300

(a)

KJC, MPa√m

m0=0.1

0.4

0.2

0.6

200

100

0 -200

300

KJC, MPa√m

not usually revealed such a possibility cannot be rejected a priori). 2. The opposite tendency, namely, an increase of width of dislocation pile-up is also possible. In this case, the blunting of dislocation pile-up near initiator increases and the coefficient m0 decreases. Width of dislocation pile-up may increase as a result of some “diffuse sliding” caused by large number of strong barriers for dislocations. For irradiated RPV steels various precipitates are such strong barriers. 3. Radiation defects may also affect the dislocation pile-up blunting by limitation of local cross sliding in dislocation pile-up head (in a scale more less than a grain).2 Then the coefficient m0 increases as neutron fluence increases.

99

-100

T, 0C

0

100

(b)

200

~ = 18000 MPa 1− σ d ~ = 12000 MPa 2−σ d ~ = 8000 MPa 3−σ

2

1

3 4

d

~ = 6000 MPa 4−σ d ~ = 4000 MPa 5−σ d

5

100

0 -200

-100

0

100

T, 0C Fig. 4 The KJC (T) curves calculated with the Prometey model for increasing function m0 (F) and σ˜ d = const (a) and for decreasing parameter σ˜ d and m0 = const (b) (Specimen thickness is 25 mm and Pf = 0.5)

on criterion (2) is the only model that allows the prediction of lateral shift of KJC (T) curves. 3.3 The effect of radiation defects on cleavage microcrack propagation When analysing the applicability of local cleavage fracture criteria for radiation embrittlement modelling in Sect. 2.3 it has been mentioned that the critical brittle fracture stress SC does not practically depend on neutron fluence. This consideration is based on available experimental results. Test results show that irradiation does not decrease the critical brittle fracture stress SC , at least, for transcrystalline cleavage fracture (Alekseenko et al. 1997; Margolin et al. 1999). (It is appropriate to note there that the data are too few to be conclusive.) It may be also shown that from a physical point of view, irradiation-induced lattice defects (dislocation loops) and precipitates result in not decreasing the critical stress SC . It follows from the model (Margolin et al. 1997a) of propagation of Griffith’s crack on cleavage plane through the microstress fields which are

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considered as barriers for microcracks. Assuming these microstress fields are created by dislocation loops and precipitates it may be obtained from this model that the stress required for microcrack propagation increases due to increasing the microstresses. It is necessary to note that for high volume concentration of phosphorus, significant segregation of P on grain boundaries may be formed under irradiation that may result in decreasing the stress SC by two mechanisms. The first mechanism is the transition from transcrystalline to intercrystalline fracture. The second mechanism appears to be observed when grain boundaries with twist component dominate. In this case, phosphorus probably locates in screw misfit dislocations and makes easy splitting of cleavage microcrack on strips when microcrack crosses grain boundary (Qiao and Argon 2003). In other words, the energy for transition of cleavage microcrack from one grain to another reduces. The Prometey model takes into account these processes by reducing the parameter σ˜ C in Eq. (16). 4 Application and discussion In this section the mechanical and physical factors of radiation embrittlement are considered for various BCC metals. The effect of material hardening caused by irradiation and by cold work on brittle fracture is discussed in terms of local criteria. Appearance of the mode of intergranular fracture for RPV steels after irradiation and post-irradiation annealing is explained from viewpoint of local criterion (2) and the proposed mechanisms of the effect of radiation defects on cleavage microcrack nucleation. The last subsection returns to fracture prediction on a macro-scale: the prediction of the transformation of KJC (T) curve as a function of neutron fluence is considered. 4.1 Particularities of radiation embrittlement of various BCC metals The performed study have shown that radiation embrittlement is caused by two factors: (1) the mechanical factor—the increase of σ1 as a result of the σY increase and (2) the physical factor—the decrease of the critical stress σd for microcrack nucleation. The contributions of the mechanical and physical factors in irradiation embrittlement may depend on material properties, mainly, on the ratio SC /σY . This ratio deter-

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mines which physical process—microcrack nucleation or microcrack propagation—controls brittle fracture of cracked specimen. For steels with large value of SC /σY , for example, 20 for low-alloy, low-strength steel (SC /σ20 Y ≥ 4, σY ≡ ◦ σY (T = 20 C)), condition σ1 = SC is satisfied only when the value of σ1 increases significantly due to strain hardening, i.e. after significant plastic strain. By this the microcrack nucleation condition (2a) has been satisfied earlier. It means that brittle fracture of cracked specimens from these steels is mainly controlled by the microcrack propagation condition (2b) and hence, the mechanical factor predominates in radiation embrittlement for this steel. For medium-strength steel with the ratio SC /σ20 Y ≤ 2.5, in particular for RPV steel, condition σ1 = SC near the crack tip is met earlier than microcrack nucleation condition, therefore the last condition mainly controls brittle fracture of cracked specimens. When fracture is considered in the probabilistic statement both conditions control brittle fracture. Therefore both the mechanical and physical factors determine radiation embrittlement of RPV steel. The physical and mechanical factors of radiation embrittlement have been analyzed in detail in Margolin et al. (2008b). In Fig. 5 the calculation results obtained on the basis of the Prometey model are shown for medium-strength 2Cr–Ni–Mo–V RPV steel (SC /σ20 Y ≈ 2.3) and for low-alloy, low-strength M16C steel (SC /σ20 Y ≈ 4). Here the relative contribution of the mechanical factor is shown. The value (Ttr )σ is the temperature shift determined for the level KJC = √ 100 MPa m from the KJC (T) curves calculated for σY = var. As a quantitative measure the parameter (Ttr )σ /σY may be used. From Fig. 5 it is seen that for M16C steel the ratio (Ttr )σ /σY ≈ 0.4 ÷ 0.6 ◦ C/MPa that is in good agreement with test results for the irradiated steel (Margolin et al. 2008b). It means that for low strength steel the contribution of the material hardening in embrittlement is dominant. For this case the parameter σd decreasing does not change brittle fracture resistance controlled by microcrack propagation. At the same time for RPV steel the mechanical factor provides only 20 % of total temperature shift as (Ttr )σ /σY ≈ 0.1 ◦ C/MPa (see Fig. 5) and for RPV steels with low impurity content experimental values of Ttr /σY are from 0.4 to 0.5 (Alekseenko et al. 1997).

Radiation embrittlement modelling

101

80

2

(ΔTtr)σ, oC

60

40

1 20

0 0

40

80

120

160

200

ΔσY, MPa Fig. 5 The ductile-to-brittle transition temperature shift (Ttr )σ caused by the material hardening σY for medium-strength 2Cr–Ni–Mo–V RPV steel (curve 1) and for low-strength M16C steel (curve 2)

Another typical particularity of brittle fracture of RPV steels is connected with cleavage microcrack nucleation. For these steels globular carbides are the main initiators of cleavage microcracks. For such initiators all the considered mechanisms resulting in decrease of the critical stress σd may be realized. It seems that for other types of initiators, some mechanisms of the considered ones may not work. For example, for plane elongated barriers, the mechanism of decreasing the critical stress σd due to increasing the probability of dislocation pile-up formation near microcrack initiator cannot be realized. The reason is clear: for plane barrier the probability of dislocation pile-up formation is close to the unity.

4.2 The effect of prestrain on brittle fracture Analysis of this issue helps one to understand a link of local criterion with the embrittlement mechanisms as prestrain is known to result in the yield stress increase as well as neutron irradiation. However, the test results (Troschenko et al. 1989; Yasny 1998) show that material hardening caused by prestrain may result in both decreasing and increasing fracture toughness (Fig. 6a) as distinct from the hardening caused by irradiation always resulting in material embrittlement. Preliminary, it should be noted that as a common case, a sign of prestrain affects SC : pretension results in increasing SC and precompression results in decreasing SC (Allen 1959; Mudry 1987; Margolin and Shvetsova 1996). A model describing the different effect of

prestrain on SC was developed in Margolin and Shvetsova (1996), Margolin et al. (1996). In the present consideration the effect of pretension is only analysed provided that pretension and subsequent loading occur in the same direction. This condition corresponds to available experiments (Troschenko et al. 1989; Yasny 1998) on the prestrain effect on fracture toughness. Let us consider applicability of local criteria (1) and (2) for analysis of the prestrain effect on brittle fracture. As discussed in Sect. 2, according to criterion (1) the only factor affecting embrittlement is the increase of σ1 caused by the increase of σY . From viewpoint of criterion (1) (that may be expressed by the Yoffe scheme) the degree of material embrittlement is the same for the same increment of σY both for irradiated and cold worked3 materials. At the same time experimental data show that the ratios Ttr /σY are different for irradiated and cold worked materials. For example, for cold-worked A533B steel Ttr /σY = 0.1 ÷ 0.15 ◦ C/MPa for ε0 < 10 % (Fortner et al. 1976) and for irradiated steel Ttr /σY ∼ = 0.5 ◦ C/MPa (Alekseenko et al. 1997). Calculations have shown also that according to criterion (1) the dependence KJC (ε0 ) is always decreasing function (Fig. 6b) as the ratio SC /σY decreases when ε0 increases. According to test results obtained in Troschenko et al. (1989), Yasny (1998) the dependence KJC (ε0 ) for RPV steels is non-monotonic. Thus, criterion (1) does not allow one to describe a difference in the behaviour of irradiated and coldworked material. It is connected with that criterion (1) takes into account only one physical process of brittle fracture—cleavage microcrack propagation and does not describe adequately microcrack nucleation. At the same time, not only the σY increasing but also the variation of the critical stress σd may determine material behaviour on a macro-scale. Let us consider whether criterion (2) and the Prometey model can be used to describe the effect of cold work on brittle fracture. As a common case, cold work may result in the following. 1. When plastic pre-strain ε0 increases the ratio SC /σY decreases as σY increases more intensively than SC . 2. Driving stress for microcrack nucleation σnuc ≡ σ1 + mT ε · σeff decreases due to decreasing σeff This consideration is valid for plastic pre-strain ε0 < 10 % that does not result in SC growth.

3

123

102

(a)

(b)

80 1

KIC, MPa√m

Fig. 6 The effect of cold work on fracture toughness: a test results for RPV steel; b the relative dependence KJC (ε0 )/K0JC predicted with criterion (1); (c) the relative dependence KJC (ε0 )/K0JC predicted with the Prometey model based on criterion (2)

B. Margolin et al.

T=-150 0C

60 40 20

0

4

8

12

16

20

24

ε0, %

0

ε0,σY

(c) K JC (ε 0 ) K 0JC

2 1.5 1 0.5 0

0

4

8

that is caused by decreasing the strain hardening coefficient with ε0 growth. 3. The probability of microcrack nucleation decreases as a number of possible initiators decreases with ε0 growth. Indeed, under preliminary plastic deformation, cleavage microcracks are nucleated on weak initiators but not start and transform into void. As a result, under subsequent loading new cleavage sharp microcracks begin to be nucleated when σnuc > (σnuc )ε , where (σnuc )ε is value of σnuc corresponding to cold work condition (plastic strain ε0 and temperature T0 ). 4. When ε0 increases the value of σ˜ eff d decreases due to arising the internal stresses on carbide-matrix interface caused by the dislocation density increasing. This mechanism is similar to the mechanism of the irradiation-induced dislocation loops effect as considered in Sect. 3.1. 5. When ε0 increases, the probability of dislocation pile-up formation increases due to the dislocation density increasing. This mechanism has been represented in Sect. 3.2. The factors in items 1, 4 and 5 decrease brittle fracture resistance and the factors in items 2 and 3 increase the resistance. Thus, the dependence KJC (ε0 ) may be non-monotonic. It should be noted that all the above factors may be taken into account by the Prometey model. The issue has been considered in detail in Margolin et al. (2010) where the dependence KJC (ε0 ) was investigated with the Prometey model for 2Cr–Ni–Mo– V RPV steel and for low-strength M16C steel.

123

12

ε0, %

16

20

24

As an illustrative example, in Fig. 6c the relative dependence KJC (ε0 )/K0JC (where K0JC is fracture toughness for initial condition) is shown as resulting from the Prometey model for RPV steel. It is seen that the dependence KJC (ε0 ) may be both decreasing and increasing function depending on the plastic pre-strain value.

4.3 On the intergranular fracture mode The proposed mechanisms of the effect of irradiationinduced defects on cleavage microcrack nucleation and propagation allow one to explain an appearance of the mode of intergranular fracture for RPV steels after irradiation and post-irradiation annealing. In a number of studies (see eg Gurovich et al. 1996, 1997) the results of SEM investigations of fracture surfaces of Charpy specimens from unirradiated and irradiated RPV steels and also steels annealed after irradiation are presented. These results may be summarized as follows. Brittle fracture of unirradiated specimens occurs, as a rule, by transcrystalline cleavage and microcleavage. However, for irradiated specimens both transcrystalline fracture and mixed trans- and intercrystalline brittle fracture are revealed and a fraction of intercrystalline fracture regions is usually small (never larger than 20 % of fracture surface). After annealing, significant recovery of mechanical properties is observed: the DBTT shift decreases as compared with this shift for irradiated steels and the yield stress decreases too. Degree of recovery depends

Radiation embrittlement modelling

significantly on the chemical composition of a steel and annealing temperature. At the same time a fraction of intercrystalline regions in the fracture surface of annealed specimens increases as compared with irradiated specimens. These findings may be explained with criterion (2) and the proposed mechanisms of the effect of irradiation-induced defects on the critical stress σd . In deterministic statement, value of σd is determined by minimum value of the two values σtrd and σint d —the critical stresses for nucleation of transcrystalline and intercrystalline microcracks. The fracture mode may be transcrystalline or intercrystalline that depends on which value is less if the critical stress for propagation SC is the same for both types of microcracks. In probabilistic statement, mixed transcrystalline and intercrystalline fracture may be expected if the difference of σtrd and σint d is not large. The following interpretation for variation of the parameters σtrd and σint d may be proposed, as shown schematically in Fig. 7, for RPV steels in various conditions (unirradiated, irradiated and after post-irradiation annealing). Two different steels are considered: the upper scheme may be referred to 2.5Cr–Mo–V steel (base metal for WWER-440 RPV) and the lower— to 2Cr–Ni–Mo–V steel (base metal for WWER-1000 RPV). For unirradiated steels (Fig. 7a) if intercrystalline phosphorus segregation is absent, grain boundary is some strip with extremely damaged lattice that results in two consequences. Firstly, it is very difficult to nucleate atomically sharp intercrystalline microcrack as damaged lattice results in microcrack tip blunting. Secondly, propagation of intercrystalline microcrack is practically impossible without dislocation emission from its tip (in contrast to transcrystalline microcrack which may propagate on cleavage plane without dislocation emission) that requires large expenditure of energy. Therefore for unirradiated steels it may be tr int tr int taken σtrd < σint d and SC < SC , where SC and SC are the critical stresses for propagation of transcrystalline and intercrystalline microcracks. Intercrystalline phosphorus segregation caused by tempering or irradiint ation decreases both σint d and SC as sharp microcrack nucleation and propagation on phosphorus monolayer weakening atomic bonds, require less energy. The difference between σtrd and σint d is larger for 2.5Cr–Mo– V steel than for 2Cr–Ni–Mo–V steel (Fig. 7a). It is connected with different sensitivity of these steels to

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temper embrittlement: 2.5Cr–Mo–V steel is not sensitive to temper embrittlement caused by phosphorus segregation, and 2Cr–Ni–Mo–V steel shows a tendency to temper embrittlement which is caused by phosphorus diffusion intensified by nickel to phase and grain boundaries4 (Kurdyumov et al. 1983). That’s why brittle fracture of unirradiated 2.5Cr–Mo–V steel occurs mainly by the transcrystalline mechanism, and fracture of unirradiated 2Cr–Ni–Mo–V steel occurs with some fraction of intercrystalline mode. For irradiated steels (Fig. 7b) the stress σd decreases for carbides locating both in a grain and on grain boundaries. For carbides located in a grain, all three considered mechanisms resulting in decrease of σd may be realized. For carbides located on grain boundary, although only two mechanisms work—segregation of impurities and internal stress caused by dislocation loops and precipitates, these processes occur more intensively. This is confirmed by the results of theoretical (Tikhonchev and Svetukhin 2011) and experimental (Gurovich et al. 1996) studies. In Tikhonchev and Svetukhin (2011) it was shown that the density of irradiation-induced point defects (that result in formation of dislocation loops) is higher near interface boundaries than in a grain. In Gurovich et al. (1996) it was shown that phosphorus segregation and precipitation occurs very intensively near grain boundaries. As a result, for irradiated 2.5Cr–Mo–V steel the conditions tr int σtrd < σint d and SC ≈ SC are more possible, so that transcrystalline brittle fracture is mainly typical. For irradiated 2Cr–Ni–Mo–V steel the situation is possible tr int when σtrd ≈ σint d (Fig. 7b) and SC ≈ SC so that mixed trans- and intercrystalline brittle fracture are observed. The most interesting situation may be revealed after post-irradiation annealing at Tann = 475 ◦ C (Fig. 7c). At this temperature phosphorus segregations dissociate in a grain only, and do not dissociate on grain boundary (Nikolaev and Rybin 1996; Gurovich et al. 1997). Irradiation-induced dislocation loops and precipitates may be dissociated practically completely both in a grain and on grain boundary. Therefore the σtrd value increases up to the value for the unirradiated condiint tion but the σint d and SC values remain less than for the unirradiated condition. As a result, after annealing at Tann = 475 ◦ C (Fig. 7c) a fraction of intercrystalline fracture regions may increase as compared 4

Pressure vessel passes the temperature interval of temper embrittlement under cooling after tempering at 650 ◦ C.

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104 Fig. 7 Variation of the critical stress σd for nucleation of transcrystalline microcracks (shaded bar) and intercrystalline microcracks (open bar) for RPV steels in various conditions: unirradiated (a), irradiated (b), after post-irradiation annealing (c,d)

B. Margolin et al.

σd

2.5Cr-Mo-V steel

unirradiated

σd

annealed, 475 0C

annealed, 600 0C

2Cr-Ni-Mo-V steel

unirradiated

(a)

with irradiated specimens as the values of σtrd and σint d become close. For 2Cr–Ni–Mo–V steel the situation is possible when intergranular carbides become “weakest link” (Fig. 7c) and intercrystalline fracture predominates. It is interesting to note that for 2.5Cr–Mo–V steel this annealing results in full recovery of the mechanical properties (the yield stress and the transition temtr perature Ttr ) as (σint d )irr+ann ≈ (σd )unirr (Fig. 7c) int tr and (SC )irr+ann ≈ (SC )unirr . For 2Cr–Ni–Mo–V steel although the yield stress may recover fully, Ttr recovers not fully so that (Ttr )unirr < (Ttr )ann < (Ttr )irr . The reason is clear: for this case the brittle fracture resistance is controlled by σint d that does not recover fully. Annealing at temperature Tann ≈ 600◦ C (Fig. 7d) fully recovers σint d as dissociation of intercrystalline phosporus segregations occurs at this temperature (Rellick and McMahon 1974; Seah 1977; Tyson 1978; Gurovich et al. 1997) and the situation become close to the unirradiated condition (Fig. 7a). Full recovery of the mechanical properties and the fracture modes is observed. New SEM study results in Gurovich et al. (2010), Erak et al. (2010) confirm this conclusion. It may be noted also that for 2.5Cr–Mo–V steel annealing at Tann ≈ 600 ◦ C, in principle, does not vary the situation from viewpoint of the recovery of mechanical properties as σtrd remains the same. However, it may be expected that a fraction of intercrystalline fracture decreases up to its fraction for unirradiated condition.

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irradiated

irradiated

(b)

annealed, 475 0C

(c)

annealed, 600 0C

(d)

4.4 Application for prediction of KJC (T) curve transformation due to neutron irradiation Although this paper is not aimed at the prediction of KJC (T) curve, the most important consequence has to be considered. Prediction of KJC (T) curve for irradiated RPV steel may be performed with the Unified Curve method. The Unified Curve is engineering method obtained from the Prometey model (Margolin et al. 2003). This method allows one to take into account a possible variation in KJC (T) curve shape under irradiation. According to the Unified Curve concept the temperature dependence of fracture toughness at Pf = 0.5 for specimens with thickness B = 25 mm from RPV steel for any degree of embrittlement is described by    √ T−130 KJC(med) (T) = Kshelf , MPa m, JC + · 1+tanh 105

(32) √ where Kshelf JC = 26 MPa m;  is a constant for a given state of a material; T is temperature in ◦ C. For the embrittled materials the only parameter, , varies, the rest of the numerical parameters in Eq. (32) are fixed. The parameter  is calibrated from fracture toughness test results. For the Unified Curve the scatter in KJC values and the thickness effect are the same as for the Master Curve (Wallin 1984, 1985; Merkle et al. 1999).

Radiation embrittlement modelling

The transformation of KJC (T) curve as a function of neutron fluence F may be predicted if the dependence of the calibration parameter  on F is known. The obtained dependence of the critical stress of microcrack nucleation σ˜ eff d on fluence provides the determination of a type of function (F). It follows from the consideration that the main physical process controlling brittle fracture of cracked specimens from irradiated RPV steels is cleavage microcrack nucleation, and the main parameter controlling the dependence KJC (T) is the parameter σ˜ eff d (Margolin et al. 2003). From the exponential dependence for σ˜ eff d (F) (see Eqs. (22) and (31)) a type of function (F) was found in Margolin et al. (2005) as exponential  m   F (33)  = 0 exp −CF F0 where 0 is a value of  for unirradiated condition, CF and m, materials constants dependent on irradiation temperature and neutron spectrum, unit of F: n/cm2 , F0 = 1 × 1018 n/cm2 . It should be noted that exponential functions (22), (31) and (33) have some shortcoming that appears for highly irradiated condition: it follows from these functions that for extremely high degree of embrittlement shelf = (when F → ∞)  → 0 and KJC (T) ∼ = Klow JC const. The reasons are quite clear: firstly, sufficiently rough calculations in the above models and, secondly, investigation of a type of function (F) in Margolin et al. (2005) with no account taken of microcrack propagation. At the same time it is of interest to note that according to the Prometey model, for very high degree of embrittlement (when σ˜ d → 0) brittle fracture is controlled by conditions σnuc = σd0 or σ1 = SC . For both cases over temperature range of brittle fracture until σY varies with T even if very weakly the dependence KJC (T) is not constant but slightly increasing. Taking into account this m and the wide-used  fact F the corrected dependependence TKJC = AF · F 0 dence (F) may be derived on the basis of the Unified Curve in the following form  m   F + min . (34)  = (0 − min ) exp −CF F0 Here TKJC is a shift of a conditional temperature TKJC due to neutron irradiation, and this temperature TKJC corresponds to some given level of KJC designated as

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KJC for the fracture probability Pf = 0.5 and specimen thickness of 25 mm; AF and m are material con√ stants. Taking K JC = 100 MPa m the value of min ≡ √ 0.5(K JC − Kshelf JC ) is found as min = 37 MPa m.

5 Conclusions 1. Applicability of available cleavage fracture criteria has been analyzed in terms of having an adequate link with the physical and mechanical features of material embrittlement. Two types of local criteria that are currently widely used for fracture prediction have been considered: stresscontrolled criterion used in the RKR and Beremin models and stress-and-strain controlled criterion used in the Prometey model. It has been shown that the stress-and-strain controlled criterion provides adequate prediction for the known properties of embrittlement of a material. 2. It has been found that radiation embrittlement of steels is caused both by the material hardening (the mechanical factor) and the microcrack initiator weakening (the physical factor). The contributions of the mechanical and physical factors in radiation embrittlement depend on material properties, mainly, on the ratio SC /σY . This ratio determines which physical process—microcrack nucleation or microcrack propagation—controls brittle fracture of a cracked specimen. For steels with large value of SC /σY microcrack propagation controls brittle fracture of cracked specimens, hence, the mechanical factor predominates. For steels with small value of SC /σY , in particular for RPV steel, microcrack nucleation mainly controls brittle fracture of cracked specimens, and therefore the physical factor contributes significantly. 3. It has been explained with stress-and-strain controlled criterion why the material hardening caused by cold work may result in both decreasing and increasing fracture toughness as distinct from the hardening caused by irradiation always resulting in material embrittlement. Cold work always results in easier cleavage microcrack propagation due to decreasing the ratio SC /σY . The effect of cold work on cleavage microcrack nucleation is caused by several considered mechanisms: some of them result in easier cleavage microcrack nucleation and others make more difficult their nucleation.

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4. Irradiation-induced defects result in easier cleavage microcrack nucleation. Physical-and-mechanical models of the effect of radiation defects (dislocation loops, precipitates and segregation) on cleavage microcrack nucleation have been proposed. The dependence of critical stress for microcrack nucleation on neutron fluence σd (F) has been obtained allowing for various irradiation-induced defects. 5. Appearance of the mode of intergranular fracture for RPV steels after irradiation and post-irradiation annealing is analysed on the basis of the proposed mechanisms of the effect of radiation defects on cleavage microcrack nucleation. It has been explained why for RPV steels after post-irradiation annealing a fraction of intercrystalline fracture may increase as compared with irradiated specimens although the recovery of the yield stress and DBTT occurs. 6. It has been shown that the Prometey model provides not only a prediction of the KJC (T) curve allowing for possible variation in the KJC (T) curve shape but, at present, is the only model that allows the prediction of lateral shift of KJC (T) curves. Any models based on the stress controlled criterion may predict lateral shift of KJC (T) only if the dependence of SC (or σu in terms of the Beremin model) on temperature and neutron fluence is a priori introduced. 7. The Unified Curve method based on the Prometey model provides a prediction of the KJC (T) curve for irradiated RPV steels. The dependence (F) determined from the obtained dependence σd (F) allows the prediction of the transformation of KJC (T) curve as a function of neutron fluence. Acknowledgments This work has been carried out within the framework of contracts with Concern “ROSENERGOATOM” and the EC-sponsored ISTC projects 3072 and 3973 performed in collaboration with the Integrated Projects PERFECT and PERFORM 60. The authors would like to devote this publication to the memory of our friend and colleague Dr. Elena Nesterova.

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