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A new engineering method for prediction of the fracture toughness temperature dependence for RPV steels
International Journal of Pressure Vessels and Piping 80 (2003) 817–829 www.elsevier.com/locate/ijpvp
A new engineering method for prediction of the fracture toughness temperature dependence for RPV steels B.Z. Margolin*, A.G. Gulenko, V.A. Nikolaev, L.N. Ryadkov Central Research Institute of Structural Materials “Prometey”, 49 Shpalernaja Street, St Petersburg 193015, Russia Received 20 May 2003; revised 2 December 2003; accepted 2 December 2003
Abstract In order to predict the temperature dependence of fracture toughness for RPV steels with various degrees of embrittlement, up to extremely high levels, an engineering method is proposed which is similar to the Master Curve concept and named the Unified Curve concept. Wide verification of the Unified Curve concept is performed and comparison of the Master and Unified Curve concepts is carried out. It is shown that the Master Curve concept is a partial case of the Unified Curve concept. q 2004 Elsevier Ltd. All rights reserved. Keywords: Cleavage fracture toughness; Local criterion; Probabilistic model; Master Curve; Unified Curve; Reactor pressure vessel steel
1. Introduction At present, the Master Curve approach [1 –5] is a widely used method for prediction of the temperature dependence of fracture toughness for RPV steels. This approach allows the prediction of the KIC ðTÞ curve for any given fracture probability and any specimen thickness on the basis of small-sized specimen testing. Analysis of the applicability of the Master Curve approach for RPV steels in various states (initial (as-received), irradiated, thermal embrittled) has shown that this approach provides adequate predictions for materials in the initial state and for cases when the degree of embrittlement is not high. For high degrees of embrittlement, the Master Curve may provide inadequate and nonconservative predictions [6,7]. This circumstance is connected with using the lateral temperature shift condition in the Master Curve approach, i.e. it is assumed that the shape of the KIC ðTÞ curve does not change as the degree of embrittlement increases. The Prometey probabilistic model [6 –9] does not have such shortcomings. This model is based on the local approach and a new formulation of the cleavage fracture criterion [10,11]. The Prometey model allows predictions to be made sufficiently adequately of the KIC ðTÞ curves * Corresponding author. Tel.: þ 7-812-274-1101; fax: þ7-812-274-1707. E-mail address: [email protected] (B.Z. Margolin). 0308-0161/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijpvp.2003.12.005
for both RPV materials in the initial state and materials with various degrees of embrittlement up to a highly embrittled state [6,7]. The Prometey model as well as the Master Curve may use small-sized specimen testing and predict the KIC ðTÞ curve for any given fracture probability and any specimen thickness. The major disadvantage of the Prometey model appears to be the rather intensive calculations and the necessity to carry out additional tensile tests to have data on the stress – strain curves for different temperatures and also the dependence of critical brittle fracture stress on plastic strain. In other words, the Prometey model is not an appropriate engineering method to predict the KIC ðTÞ curve, compared with the Master Curve approach. The purpose of the present paper is to elaborate an engineering method based on the generalized results obtained by the Prometey model that allows the prediction of the KIC ðTÞ curve for RPV steels with various degrees of embrittlement, including extremely high levels.
2. Material embrittlement modelling with the Master Curve concept The main properties of the Master Curve concept are represented as [1 – 5].
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(1) The brittle fracture probability Pf is described by a three-parameter Weibull model in the form [1] " # KJC 2 Kmin 4 ; ð1Þ Pf ¼ 1 2 exp 2 K0 2 Kmin where Pf is the probability of fracture at KI # KJC for an arbitrary chosen specimen from a specimen set; K0 is a scale parameter dependent on the test temperature and specimen thickness; Kmin is the minimum possible fracture toughness. p In accordance with Ref. [1], Kmin ¼ 20 MPa m: (2) The effect of specimen thickness on fracture toughness is described by [2] X KJC 2 Kmin BY 1=4 ¼ ; ð2Þ Y 2K BX KJC min X Y where KJC and KJC are fracture toughness values for specimens with thickness BX and BY at the same probability of brittle fracture Pf : (3) The fracture toughness median value (at Pf ¼ 0:5) as a function of temperature for specimens 1T-CT ðB ¼ 25 mmÞ is given in [3] shelf KJCðmedÞ ¼ KJC þ b expðgðT 2 T0 ÞÞ; ð3Þ p p shelf where KJC ¼ 30 MPa m; b ¼ 70 MPa m; g ¼ 0:019; T the temperature in 8C and T0 is temperature at which p KJCðmedÞ ¼ 100 MPa m: (4) It is assumed that for embrittled materials only the parameter, T0 ; varies, whereas the rest of the numerical parameters in Eq. (3) are fixed. This assumption is usually called the lateral temperature shift condition. This condition connects an increase in the degree of material embrittlement with a lateral shift of the KJC ðTÞ curve to an elevated temperature range. Assumption 4 may also be interpreted in another way. Eq. (3) may be written as shelf KJCðmedÞ 2 KJC ¼ vcðTÞ;
ð4Þ
where v ¼ expð2gT0 Þ; cðTÞ ¼ b expðgTÞ: As seen from Eq. (4), the embrittled state of a material may be represented as a decrease of fracture toughness shelf (more exactly, the difference KJCðmedÞ 2 KJC ) at each temperature by a factor vin =vem ; where vin corresponds to the initial state with T0 ¼ T0in and vem corresponds to the embrittled state with T0 ¼ T0em : Thus, the temperature dependence of fracture toughness for any state of a material may be represented as the product of a coefficient v depending on material state and a function cðTÞ which does not depend on material state. Representation of the transformation of the KJC ðTÞ curve with degree of material embrittlement by Eq. (4) may also be useful when the dependence cðTÞ is not described by an exponential function. It may be noted that, as a common case, the function cðTÞ may be either an exponential or another type. The only restriction on the function cðTÞ follows from its physical
meaning: the function cðTÞ describes KJC ðTÞ for brittle fracture without regard for ductile tearing, i.e. this function does not take into account that the KJC ðTÞ curve varies in character when KJC exceeds the upper shelf level of KJC ðTÞ: In Section 3, on the basis of the Prometey probabilistic model, it is shown that the representation of the KJC ðTÞ curve by Eq. (4) is quite general, and the nature of the function cðTÞ is determined.
3. Modelling the temperature dependence KJC(T) by the Prometey model 3.1. Presentation of the Prometey probabilistic model 3.1.1. The local criterion for cleavage fracture The formulation of the local cleavage fracture criterion in a probabilistic manner is as follows [6 – 9]. 1. The polycrystalline material is viewed as an aggregate of cubic unit cells. The mechanical properties for each unit cell are taken as the average properties obtained by standard specimen testing. The size of the unit cell ruc is never less than the average grain size. The stress and strain fields in the unit cell are assumed to be homogeneous. 2. The local criterion for cleavage fracture of a unit cell is taken as
s1 þ mT1 seff $ sd
ð5aÞ
s1 $ SC ðæÞ;
ð5bÞ
where the critical brittle fracture stress, SC ðæÞ; is calculated by SC ðæÞ ¼ ½C1p þ C2p expð2Ad æÞ21=2 :
ð6Þ
Here, s1 is the maximum principal stress, the effective stress is seff ¼ seq 2 sYÐ; seq is the equivalent stress, sY is the yield stress, æ ¼ d1peq is Odqvist’s parameter, d1peq is the equivalent plastic strain increment, C1p ; C2p ; Ad are material constants, sd is the strength of carbides or ‘carbide –matrix’ interfaces or other particles on which cleavage microcracks are nucleated, mT1 is a parameter that depends on temperature and plastic strain and may be written as mT1 ¼ mT ðTÞm1 ðæÞ;
ð7Þ
m1 ðæÞ ¼ S0 =SC ðæÞ;
ð8Þ
mT ðTÞ ¼ m0 sYs ðTÞ;
ð9Þ
where S0 ; SC ðæ ¼ 0Þ; m0 is a constant which may be experimentally determined and sYs is the temperaturedependent component of the yield stress. Condition (5a) is the nucleation condition for cleavage microcracks. Condition (5b) is the propagation condition for cleavage microcracks. 3. To formulate criteria (5) in a probabilistic way, it is assumed that the parameter sd is stochastic and the remainder of the parameters controlling brittle fracture are deterministic. Such an assumption is based on analysis of
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the stochastic nature of various critical parameters controlling cleavage fracture of RPV steels [12]. 4. To describe the distribution function for the parameter sd ; the Weibull law is used: the minimum strength of carbides in the unit cell on which cleavage microcracks are nucleated, is assumed to obey sd 2 sd0 h pðsd Þ ¼ 1 2 exp 2 ; s~d
ð10Þ
where pðsd Þ is the probability of finding in each unit cell a carbide with minimum strength less than sd ; s~d ; sd0 and h are Weibull parameters. 5. The weakest link model is used to describe the brittle fracture of the polycrystalline material. 6. It is considered that brittle fracture may happen only in unit cells for which the conditions seq $ sY and s1 $ SC ðæÞ are satisfied. 3.1.2. The probabilistic model for the KIC ðTÞ curve prediction The probabilistic model for fracture toughness prediction is based on the local criterion described above. The stress and strain fields near the crack tip are calculated by FEM or on the crack extension line with an approximate analytical solution [8,9]. The brittle fracture probability of a cracked specimen, Pf ; may be presented in the form used in [13] s h Pf ¼ 1 2 exp 2 w ; s~d
ð11Þ
where the Weibull stress sw is "
sw ¼
k X
#1=h ðmaxðSinuc Þ
2 sd0 Þ
h
;
ð12Þ
i¼1
( Sinuc
;
sinuc ; if si1 $ SC ðæi Þ and sinuc . sd0 sd0 ;
if si1 , SC ðæi Þ or sinuc # sd0
:
Here snuc ; s1 þ mT1 seff ; k is the number of unit cells in the plastic zone, i is the number of a unit cell. For each unit cell, the parameter maxðSinuc Þ is the maximum value of Sinuc from the beginning of deformation up to the current loading. The above equations provide the calculation of the dependence of the brittle fracture probability on stress intensity factor, Pf ðKI Þ; as the parameter sw is a function of KI : To predict the KJC ðTÞ curve on the basis of the model proposed above, it is necessary to know the parameters SC ðæÞ; mT ðTÞ; s~d ; sd0 and h and also parameters describing plastic deformation to enable the stress and strain fields to be calculated.
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3.2. Modelling the KJC ðTÞ curve for a material with various degrees of embrittlement 3.2.1. Analysis of the main parameters controlling KJC ðTÞ It follows from Section 3.1, that for modelling the dependence KJC ðTÞ; the following parameters of a material have to be known: – –
parameters describing the stress – strain curve; parameters in the local criterion of cleavage fracture.
According to Margolin et al. [7] the stress – strain curve may be approximated by
seq ¼ sY ðTÞ þ A0 æn :
ð13Þ
Here A0 and n are material constants, as a common case, dependent on the state of the material. The temperature dependence of the yield stress sY ðTÞ is approximated by
sY ðTÞ ¼ sYG þ sYS ðTÞ ¼ a þ b expð2hTa Þ;
ð14Þ
where sYG and sYs are the temperature-independent component and the temperature-dependent component of the yield stress; a; b and h are material constants, in Eq. (14) Ta is taken in degrees Kelvin. As shown in Refs. [4,14 – 16], when material embrittlement is accompanied by a yield stress increase (for example, due to irradiation) and this increase is caused by an increase in sYG ; i.e. the value of the parameter a; the temperaturedependent component of sY does not vary practically. It should be noted that the temperature-dependent component of sY for various RPV steels may be described by the same dependence. Strain hardening of unirradiated material does not differ from strain hardening of irradiated material, i.e. the parameters A0 and n depend weakly on embrittlement degree. Now consider the parameters in the cleavage fracture local criterion (5). As shown in Ref. [16], the dependence SC ðæÞ does not vary practically for various degrees of embrittlement of a material, at least, for all cases when the dominant mechanism of fracture is cleavage or microcleavage. The parameter mT is calculated by Eq. (9) where m0 ¼ 0:1 [7 – 9]. Hence, the dependence mT ðTÞ is practically invariant to embrittlement degree. sd0 is calculated by equation sd0 ¼ a þ b [6,7], where a and b are coefficients in Eq. (14). It is clear that if for embrittled material the parameter sY increases then the parameter sd0 also increases. The main parameters which control the dependence KJC ðTÞ for various degrees of embrittlement [16] are s~d and h: Increasing material embrittlement degree may be modelled by decreasing the parameter s~d [6,16]. For 2Cr – Ni – Mo – V RPV steel in the initial state, h ¼ 6 ðsY ðT ¼ 20 8CÞ ¼ 580 MPaÞ; for irradiated weld of this steel, h ¼ 8 ðsY ðT ¼ 20 8CÞ ¼ 640 MPaÞ; and for this steel
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Table 1 Parameters for the stress–strain curves for 2Cr–Ni–Mo –V steel in the initial state approximated by seq ¼ sY þ A0 æn Parameter
A0 (MPa) n
Temperature (8C) 2196
2100
260
220
20
100
350
679 0.50
685 0.47
622 0.46
629 0.49
590 0.49
557 0.49
537 0.50
after thermal embrittlement, h ¼ 12 ðsY ðT ¼ 20 8CÞ ¼ 900 MPaÞ [6,7]. On the basis of these data the conclusion may be drawn that as embrittlement degree increases (that is accompanied with increase in yield stress) the parameter h increases. For the materials investigated, the minimum value of h ¼ 6, and the maximum value h ¼ 12: 3.2.2. Selection of the model parameter values for the KJC ðTÞ calculation Taking into account the analysis performed of the model parameters, the following numerical values of these parameters may be taken. The first variant: (first set of parameters) a ¼ 510 MPa; b ¼ 1083 MPa; h ¼ 9:31 £ 1023 K21 ; C1p ¼ 2:01 £ 1027 MPa22 ; C2p ¼ 3:90 £ 1027 MPa22 ; Ad ¼ 1:71; h ¼ 6; the parameter s~d is varied. Values of the coefficients A0 and n for various temperatures are given in Table 1. This set of data corresponds to 2Cr –Ni – Mo – V RPV steel in the initial state [6]. Variation of the parameter s~d when other parameters are fixed, allows the description of transformation of the KJC ðTÞ curve as a function of embrittlement degree for cases when this embrittlement is accompanied with insignificant increase in the yield stress. The second variant (second set of parameters) differs from the first variant by two parameters a ¼ 710 MPa and h ¼ 12; the parameter s~d is varied as well as for the first case. This variant is more typical for radiation-embrittled material when the yield stress increases. The typical increase in sY is equal to about 200 MPa [4,15,16], therefore the parameter a increases by 200 MPa as compared with the first variant. For both the first and second variants the size of unit cell is taken to be equal to 0.05 mm [6 –9]. As shown in Refs. [6,7] the descriptions of the scatter in KJC results and the effect of specimen thickness (the crack front length) on KJC as obtained on the basis of the Master Curve concept and the Prometey model coincide practically. Hence, in the present work, there is no need to study the scatter in KIC and the scale effect on the basis of the Prometey model. To describe these regularities, Eqs. (1) and (2) are sufficient. To provide the comparison with the Master Curve, the KJC ðTÞ curve is calculated for B ¼ 25 mm and Pf ¼ 0:5: 3.2.3. Calculation results In Fig. 1 the calculation results are presented for the data sets corresponding to the first and second variants.
Fig. 1. The dependence of KJC on temperature for various values of s~d : calculation for the first (a) and second (b) variants of the data set; the dotted curve is the upper shelf level.
Calculations were performed for plane strain assuming that ductile crack growth is absent for any high level of KJC : Actually, ductile crack growth begins when KI exceeds the ductile ductile upper shelf level KJC : For RPV steels, KJC is p approximately between 200 and 250 MPa m. It is seen from Fig. 1 that for each variant the KJC ðTÞ curves for various levels of s~d have similar shape. At the same time if parts of these curves, which locate below the upper shelf, are considered then the regularity of transformation of KJC ðTÞ may be revealed that was considered in Refs. [16,17]. For small degrees of embrittlement, transformation of KJC ðTÞ may be described by the lateral temperature shift. For large degrees of embrittlement, the slope of the KJC ðTÞ curve decreases (if curves restricted by the upper shelf are considered). Let us analyse whether the results obtained can be presented as Eq. (4). For this, each KJCðmedÞ ðTÞ curve is represented in the coordinates KJCðmedÞ ðTÞ 2 KJshelf versus T; 100 shelf KJCðmedÞ 2 KJC
ð15Þ
shelf 100 where KJCðmedÞ at T ¼ 2200 8C is taken as KJC ; KJCðmedÞ ¼ KJCðmedÞlT¼100 8C : It is clear that the temperature T ¼ 100 8C in Eq. (15) was taken in an arbitrary manner, because this value does not affect verification of the similarity of the above curves.
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data set, the KJCðmedÞ ðTÞ curves approaches the lower shelf at T , 2200 8C: For the second variant data set for all the cases shelf considered, the value of KJC varies over a very narrow p range: from 24 to 28 MPa m. Therefore, it may be taken shelf with a reasonable degree of accuracy that KJC does not depend on the embrittlement degree and is equal to p 26 MPa m.
4. Concept of a Unified Curve 4.1. The main considerations of the concept On the basis of the results obtained in the previous section, the following concept of the Unified Curve may be proposed. 1. The temperature dependence of fracture toughness at Pf ¼ 0:5 for specimens with thickness B ¼ 25 mm from RPV steel for any degree of embrittlement may be described by KJCðmedÞ ¼
shelf KJC
T 2 130 þ V 1 þ tanh 105
; MPa
p
m; ð17Þ
shelf 100 shelf Fig. 2. The ratio ðKJCðmedÞ 2 KJC Þ=ðKJCðmedÞ 2 KJC Þ as a function of temperature for the first (a) and second (b) variants of the data set: the solid curves are the calculated curves for various values of s~d ; the solid curve with points is a fitting curve calculated by formula (16).
In Fig. 2 the KJCðmedÞ ðTÞ curves are represented for various s~d in the coordinates of Eq. (15) for both the first and second variants. It is seen from this figure that all the curves corresponding to one variant differ from each other insignificantly. Moreover the analysis performed shows that the difference between the curves corresponding to the first and second variants is insignificant. This means that for materials with different strength levels, the KJCðmedÞ ðTÞ dependence for any degree of embrittlement may be approximately presented in the form of Eq. (4) where 100 shelf v ¼ KJCðmedÞ 2 KJC : For engineering estimations, the curve sets of the first and second variants may be considered as a unified set and this set may be described by a unified dependence cðTÞ: The cðTÞ curve corresponding to the data sets of the first and second variants is approximated by the function
p shelf ¼ 26 MPa m; V ¼ vM and T the temperawhere KJC ture in 8C. As seen from Eq. (17), the parameter V is equal to shelf KJCðmedÞlT¼1308C 2 KJC : As the degree of embrittlement increases, the parameter V decreases. 2. To describe the KJC ðTÞ dependence at Pf – 0:5; Eq. (1) is used. 3. To describe the KJC ðTÞ dependence for B – 25 mm; Eq. (2) is used. 4.2. Determination of parameters of the Unified Curve It follows from Section 4.1, that to predict the KJC ðTÞ curve for any given thickness of specimen and any fracture probability it is necessary to know the parameter V only. The parameter V as well as the parameter T0 in the Master Curve may be determined on the basis of test results at one temperature (single temperature method) or at several temperatures (multiple temperature method).
ð16Þ
4.2.1. Single temperature method This method is similar to the method proposed in ASTM E 1921-02 and consists of the following steps.
Results of treatment of the calculated data presented in Fig. 2 by the least squares method provide: M ¼ 1:23; TM ¼ 130 8 C; Z ¼ 105 8C: It is seen from Fig. 1, as the lower shelf, some value KJCðmedÞ at T ¼ 2200 8C may be taken which was obtained for the second variant of data set, as for the first variant of
1. Determination of KJCðmedÞ according to ASTM E 1921-02 (on the basis of the maximum likelihood method) for specimens with thickness B ¼ Btest tested at T ¼ Ttest : 2. Calculation of KJCðmedÞ for B ¼ 25 mm at T ¼ Ttest : 3. Determination of the parameter V with Eq. (17) on the basis of the known values of KJCðmedÞ and T ¼ Ttest :
T 2 TM cðTÞ ¼ M 1 þ tanh Z
:
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4.2.2. Multiple temperature method This method is also similar to that proposed in ASTM E 1921-02 and consists of the following steps.
where KJCðiÞ is the experimental value of KJC obtained at Ttest ¼ Ti : Derivation of this equation is given in Appendix A.
1. Recalculation of test results from specimens with thickness B to thickness B ¼ 25 mm by using Eq. (2). 2. Determination of V on the basis of solution of the nonlinear equation 82 N < X ðKJCðiÞ 2 Kmin Þ4 lnð2Þ 40 0 0 15 11 : i¼1 T 2 130 @V@1 þ tanh@ i AA 2 Kmin þ K shelf A JC 105 3 1 5 0 11 2 0 T 2 130 i AA 2 Kmin þ K shelf V@1 þ tanh@ JC 105 2 0 139 = T 2 130 A5 ¼ 0; ð18Þ £41 þ tanh@ i ; 105
5. Comparison of experimental data and results calculated with the Master Curve and Unified Curve concepts In the present section, results obtained at CRISM ‘Prometey’ and available experimental data on fracture toughness for RPV steels with various degrees of embrittlement are treated according to the Master Curve and Unified Curve concepts. These available data are taken from papers [6,18 –24] and the database summarized by Wallin and presented in Refs. [25 – 27]. In addition, unpublished experimental data obtained at CRISM ‘Prometey’ are also used. The base and weld metals of western and Russian RPV steels are considered in different conditions: as received, thermally embrittled state and after neutron irradiation. Moreover, other ferritic structural steels are considered.
Table 2 Values of the parameters dmc and duc for treatment of fracture toughness test results for various materials by the Master Curve and the Unified Curve Set number
Material
sy of T ¼ 20 8C (MPa)
T0 (8C)
Reference
dmc
duc
dmc =duc
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33
A533B steel (in) A508 steel (in) HY130L (in) ABS DS (in) A470 steel (in) 2CrNiMoV steel (in) NVA (in) 3CrNiMoV steel (in) WF-70 weld (in) HSST weld 73W (in) HSST weld 72W (in) A533 steel JRQ (in) WF-70 weld (in) A508 steel (TSE-5 and 6) KWO RPV A508 steel A508 steel (TSE-7) A533 steel A508 steel (TSE-5 and 6) NiCrMoV steel E36 WF-70 weld (irr) HSST weld 72W (irr) HSST weld 73W (irr) A533B steel 2.5CrMoV steel (embr) WF-70 weld (irr) 2CrNiMoV steel (embr) NP2 A533 steel JRQ (irr) Weld KS01 (irr) 2.5CrMoV steel PTS-1 (embr) Weld KS01 (irr)
567 650 955 270 – 565 218 550 740 513 496 480 790 605 – – 450 – 710 925 303 930 620 648 – 730 860 900 676 630 820 1037 950
2148 2140 2131 291.3 286.5 282.5 272.1 267.4 263.5 261.3 260.4 259.7 255.8 252.2 249.2 248 232.7 228.5 220.1 211.3 21.2 24.9 29.3 37.2 39.9 45.6 49 57.1 69.0 86.9 137 163 251
[25] [25] [25] [25] [27] [6] [25] [19] [24] [18] [18] [4] [24] [26] [22] [27] [26] [27] [26] [25] [25] [24] [18] [18] [23] Fig. 3 [24] [7] [25] [4] [20] [21] [26]
17.4 26.8 16.9 192 27.1 207 43.4 54.7 77.9 11.3 31.1 61.9 25.6 115.3 29.6 15 66 25.3 112 387 51.2 110 39 15.5 80.5 32.5 14.6 99.5 83.7 32 62.6 168 26.2
19.2 29.4 18.4 188 28.5 178 41.5 56.6 77 11.9 32 58.3 26.6 96.7 31.9 15.9 54.2 22.8 79.9 185 42.1 99.1 17.5 15.9 58.6 25.4 11.4 21.1 36.2 23.8 12.4 11.3 8
0.91 0.91 0.92 1.02 0.95 1.16 1.05 0.97 1.01 0.95 0.97 1.06 0.96 1.19 0.93 0.94 1.22 1.11 1.40 2.09 1.22 1.11 2.23 0.97 1.38 1.28 1.28 4.72 2.31 1.34 5.05 14.87 3.28
in, as received condition; embr, thermally embrittled condition; irr, after irradiation.
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The brittle fracture mode for these materials varies from cleavage to mixed mode (cleavage and intergranular fracture). For all cases, the parameters T0 and V are determined by the multiple temperature method. To determine T0 ; the equation from Refs. [4,5] was used. For impartial comparison of the Master Curve and Unified Curve approaches it is necessary to use some quantitative measure. It is clear that the main difference between the Master and Unified Curves is connected with the median value of KIC : From this the scatter in KIC is described by the same Eq. (1). Therefore, for comparison of the two approaches it is necessary to use a measure connected with the median values of KIC : For a given case, a suitable measure of deviation in describing the median values of KIC may be represented by vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u X u1 L exp ð19Þ d¼t ðK pr 2 KJCðmedÞj Þ2 ; L j¼1 JCðmedÞj where d is the root mean square deviation; L is the number pr of temperatures at which tests were carried out; KJCðmedÞj is the predicted median value of KJC calculated by the Master exp or Unified Curves at test temperature Tj ; KJCðmedÞj is the median value of KJC determined by treatment of experimental data at T ¼ Tj with the maximum likelihood method. According to ASTM E 1921-02 [5] " #1=4 ‘ X ðKJCðiÞ 2 Kmin Þ4 exp 1=4 KJCðmedÞj ¼ ½lnð2Þ þKmin ; ð20Þ i¼1
‘
where KJCðiÞ is the experimental value of fracture toughness at T ¼ Tj recalculated for B ¼ 25 mm; ‘ is the number of tests at T ¼ Tj : It follows from Eq. (19) that the lower the value of d the more adequate is the description of experimental data by the Master Curve or Unified Curve methods. For an ideal case, when we have representative sampling (sufficient number of tests) at each temperature and the predicted curve (3) or (17) describes the experimental data absolutely adequately, d ! 0: In Table 2 the parameters d calculated for treatment by the Master Curve, dmc ; and the Unified Curve, duc ; for various materials are presented. In Fig. 3 for some data sets, comparisons of the test results and KJC ðTÞ curves calculated by the Master Curve and the Unified Curve are presented. As seen from Table 2 and Fig. 3, for low T0 the Master Curve and Unified Curve provide very close predictions of KJC ðTÞ and the predicted KJC ðTÞ curves describe experimental data sufficiently adequately. As the embrittlement degree increases (T0 increases), the description with the Unified Curve becomes more adequate than predictions by the Master Curve. This tendency is well seen from Fig. 4, where the ratio dmc =duc versus T0 is presented. It is clear that if the ratio dmc =duc is close to unity, then prediction of KJC ðTÞ both
823
by the Master Curve and the Unified Curve is equivalent. The larger the ratio dmc =duc ; the less adequate the prediction of KJC ðTÞ by the Master Curve.
6. Discussion The approach proposed in the present paper allows an adequate description of KJC ðTÞ for ferritic steels which have very different properties: sY varies from about 300 up to 1000 MPa, T0 varies from about 2 100 up to about 250 8C. That is why we believe, that the title of the proposed method as the Unified Curve concept appears to be justified and this concept may be applied both to RPV steels and to other ferritic structural steels. As seen from Fig. 3 and Table 2, the curves predicted with the Master Curve and Unified Curve are very close for materials with high fracture toughness, i.e. for materials with low values of T0 : Such a result is connected with variation of KJC from the lower shelf up to the upper shelf over a range of temperatures lower than TM (see Eq. (17)). For TM ¼ 130 8C and Z ¼ 105 8C; when the temperature of reaching the upper shelf does not exceed 20 –50 8C the dependence (17) may be well approximated by an exponential with the constant parameter g ¼ 0:0186; i.e. in this case, the Unified Curve practically coincides with the Master Curve. When the embrittlement degree increases, the temperature range over which variation of KJC from the lower shelf up to the upper shelf shifts to a higher temperature range. As the temperature for reaching the upper shelf approaches TM ; the shape of the KJC ðTÞ curve predicted by the Unified Curve begins to differ from that predicted by the Master Curve. For this situation the lateral temperature shift condition begins to be invalid. For subsequent increase of embrittlement degree, the difference between the results predicted by the Master Curve and Unified Curve increases. This difference will be largest when the KJC ðTÞ curves predicted by the Master and Unified Curves will have fundamentally different tendencies. For example, for the case when over a temperature range, the dependence dKJC ðTÞ=dT calculated by the Unified Curve becomes a decreasing one, for the Master Curve, dKJC ðTÞ=dT is always increasing. Thus, it may be concluded that the Master Curve concept may be considered as a particular case of the Unified Curve concept. Applicability of the Master Curve is restricted to materials in the initial (as-received) state and to degrees of embrittlement which are not high. At the present time in order to extend the range of applicability of the Master Curve approach both for materials with low and high degrees of embrittlement, an additional condition is proposed to be used [5]. The dependence, KJC ðTÞ calculated by the Master Curve is
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valid over a temperature range restricted by the condition lT 2 T0 l , 50 8C:
ð21Þ
Introduction of condition (21) restricts application of the Master Curve concept for RPV structural integrity assessment. Consider this consideration in detail.
Assume, we consider RPV steel with a high degree of embrittlement. When analysing RPV structural integrity, a crack-like flaw is considered which has the crack front length Bflaw q 25 mm: The KIC ðTÞ curves for B ¼ 25 mm and B ¼ Bflaw are schematically shown in Fig. 5.
Fig. 3. Comparison of experimental data and KJC ðTÞ curves for B ¼ 25 mm calculated by the Master Curve (on the left) and Unified Curve (on the right) for steels with various degrees of embrittlement: all test results are recalculated for a specimen thickness B ¼ 25 mm:
B.Z. Margolin et al. / International Journal of Pressure Vessels and Piping 80 (2003) 817–829
825
Fig. 3 (continued )
As shown from Fig. 5, if for the loading condition p ; then application of the considered of the RPV, KI . KJC Master Curve for RPV structural integrity assessment is incorrect and nonconservative. Thus, the following conclusion may be drawn from the represented analysis. On the one hand, condition (21) allows expansion of applicability of the Master Curve to describe the KJC ðTÞ curves for highly embrittled material, on the other hand, the temperature range of the correct
prediction of KJC ðTÞ (condition (21)) may be very small to analyse RPV structural integrity. The Unified Curve concept allows adequate predictions to be provided for any degree of embrittlement of a material and therefore may be used to analyse RPV structural integrity without any restrictions. The last problem which we would like to discuss in this section may be formulated as follows. Is it possible to describe the variation of the shape of the KJC ðTÞ curve with
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Fig. 3 (continued )
Eq. (3) if the parameter g is taken to decrease as the KJC ðTÞ curve moves to higher temperatures? Let us show that the answer to this question is negative. According to the Master Curve a shift of the KJC ðTÞ curve occurs either for T0 increasing or B increasing. Hence, for a given T0 and B increasing, the parameter g must decrease (as B increases, temperature corresponding, for example, to p KJC ¼ 100 MPa m; increases). But on the other hand,
according to Eq. (2) the parameter g does not vary as B increases. Hence, either Eq. (2) is not correct (consideration ‘a’) or transformation of a shape of curve is not described by Eq. (3) with the varying parameter g (consideration ‘b’). As Eq. (2) was derived theoretically and confirmed experimentally the conclusion may be drawn that the consideration ‘b’ is incorrect.
B.Z. Margolin et al. / International Journal of Pressure Vessels and Piping 80 (2003) 817–829
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described by p T 2 130 shelf þ V 1 þ tanh ; MPa m; KJC ¼ KJC 105
Fig. 4. The dependence of the ratio dmc =duc on T0 :
-scatter band.
where for specimen thickness B ¼ 25 mm and Pf ¼ 0:5; p shelf KJC ¼ 26 MPa m; V is the only parameter which depends on embrittlement degree of a material and T temperature in 8C. This approach is here named the Unified Curve. 2. The parameter V may be determined on the basis of fracture toughness test results at one or several temperatures. Requirements for the number and size of specimens are the same as for determination of T0 : For determination of V on the basis of test results at several temperatures, an Eq. (18) is proposed. 3. For the known parameter V; the KJC ðTÞ dependence may be determined with Eqs. (1) and (2) for any given fracture probability and specimen thickness, i.e. with equations used in the Master Curve concept. 4. Comparison of the KJC ðTÞ curves predicted by the Master Curve and Unified Curve and experimental data is performed. More than 30 sets of experimental data are considered for materials with various degrees of embrittlement. It has been shown that for materials in the initial (asreceived) state and with small degrees of embrittlement, the curves predicted with both methods coincide practically. As the degree of embrittlement increases (T0 increases), the description of KJC ðTÞ with the Unified Curve becomes more adequate than the prediction by the Master Curve. The Master Curve concept appears to be a partial case of the Unified Curve concept.
Appendix A. Determination of the parameter V in the Unified Curve from fracture toughness test results at several test temperatures Fig. 5. The dependence of KJC on temperature for a crack front length B ¼ 25 mm (specimen thickness 25 mm) and for a flaw front length B ¼ Bflaw : the solid curves are predicted by the Master Curve; the dotted curve is a real curve.
Thus, the represented arguments show once more that the Unified Curve appears to be one from correct methods to describe transformation of KJC ðTÞ on embrittlement degree.
For determination of the parameter V in the Unified Curve, the maximum likelihood method is used [4]. An equation for determination of V is derived as well as an equation for determination of T0 by a multitemperature method [3,4]. According to the maximum likelihood method, the parameter V may be found from
› ln L ¼ 0; ›V
ðA1Þ
where 7. Conclusions L¼ 1. It has been shown on the basis of calculations performed with the Prometey model that the temperature dependence of fracture toughness for ferritic structural steels with various degrees of embrittlement may be
N Y
f ðxi ; VÞ:
ðA2Þ
i¼1
Here f ðxi Þ is the probability density function at x ¼ xi ; x is the random value; xi is experimental value; N is the number of experimental points.
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The probability density function f ðxi Þ may be defined from Eq. (1) of the present paper z dPf zxz21 x i ¼ exp 2 i f ðxi Þ ¼ ; dx az a
ðA3Þ
where z ¼ 4;
ðA4Þ
xi ¼ KJCðiÞ 2 Kmin ;
ðA5Þ
a ¼ K0 2 Kmin ;
ðA6Þ
where KJCðiÞ is the experimental value of KJC obtained at Ttest ¼ Ti ; K0 is the scale parameter, which may be calculated from Eq. (1) for Pf ¼ 0:5 K0 ¼
KJCðmedÞ 2 Kmin þ Kmin ; ðlnð2ÞÞ1=4
ðA7Þ
where KJCðmedÞ is fracture toughness for Pf ¼ 0:5: According to Eqs. (A2) and (A3) the likelihood function L is written as L¼
z x ðzxz21 Þa2z exp 2 i : i a i¼1
N Y
ðA8Þ
Eq. (A1) may be represented as
› ln L › ln L ›a ¼ ¼ 0: ›V ›a ›V
ðA9Þ
Here N X › ln L z ¼ 2 þ zxzi a2ðzþ1Þ : ›a a i¼1
ðA10Þ
For determination of the parameter a; Eqs. (A6), (A7) and (17) of the present paper are used T 2 TM shelf V 1 þ tanh 2 Kmin þ KJC Z a¼ : ðlnð2ÞÞ1=4
ðA11Þ
Let us differentiate a with respect to V
›a ¼ ›V
T 2 TM 1 þ tanh Z ðlnð2ÞÞ1=4
:
ðA12Þ
On making the substitution of Eqs. (A10) and (A12) in Eq. (A9) and taking into account Eqs. (A4), (A5) and (A11), we find the nonlinear equation for determination of
the parameter V 82 N < X ðKJCðiÞ 2 Kmin Þ4 lnð2Þ 40 0 0 15 11 : i¼1 T 2 130 i @V@1 þ tanh@ AA 2 Kmin þ K shelf A JC 105 3 1 5 11 0 2 0 T 2 130 AA shelf V@1 þ tanh@ i 2 Kmin þ KJC 105 2 0 139 = T 2 130 A5 ¼ 0: ðA13Þ 41 þ tanh@ i ; 105 References [1] Wallin K. The scatter in KIC results. Engng Fract Mech 1984;19: 1085–93. [2] Wallin K. The size effect in KIC results. Engng Fract Mech 1985;22: 149 –63. [3] Wallin K. Fracture toughness transition curve shape for ferritic structural steels. In: Teoh S, Lee K, editors. Fracture of engineering materials and structures. Amsterdam: Elsevier Applied Science; 1991. p. 83–8. [4] Merkle JG, Wallin K, McCabe DE. Technical basis for an ASTM standard on determining the reference temperature, T0 for ferritic steels in the transition range. NUREG/CR-5504, ORNL/TM-13631, 1999. [5] ASTM E 1921-02. Standard test method for determination of reference temperature, T0 ; for ferritic steels in the transition range. In: Annual Book of ASTM Standards, vol. 03.01, 2002. [6] Margolin BZ, Karzov GP, Shvetsova VA, Keim E, Chaouadi R. Application of local approach concept of cleavage fracture to VVER materials. Service experience and failure assessment applications. The 2002 ASME Pressure Vessels and Piping Conference, vol. 437.; 2002. Vancouver, BC, Canada, pp. 113 –20. [7] Margolin BZ, Shvetsova VA, Gulenko AG, Ilyin AV, Nikolaev VA, Smirnov VI. Fracture toughness predictions for a reactor pressure vessel steel in the initial and highly embrittled states with the Master Curve approach and a probabilistic model. Int J Pressure Vessel Piping 2002;79:219 –31. [8] Margolin BZ, Gulenko AG, Shvetsova VA. Improved probabilistic model for fracture toughness prediction based for nuclear pressure vessel steels. Int J Pressure Vessel Piping 1998; 75:843–55. [9] Margolin BZ, Kostylev VI, Minkin AI. The effect of ductile crack growth on the temperature dependence of cleavage fracture toughness for a RPV steel with various degrees of embrittlement. Int J Pressure Vessel Piping 2003; 80:285–296. [10] Margolin BZ, Shvetsova VA. Brittle fracture criterion: physical and mechanical approach. Problemy Prochnosti 1992;N2:3 – 16. in Russian. [11] Margolin BZ, Shvetsova VA, Karzov GP. Brittle fracture of nuclear pressure vessel steels. Part I. Local criterion for cleavage fracture. Int J Pressure Vessel Piping 1997;72:73– 87. [12] Margolin BZ, Gulenko AG, Shvetsova VA. Probabilistic model for fracture toughness prediction based on the new local fracture criteria. Int J Pressure Vessel Piping 1998;75:307–20. [13] Beremin FM. A local criterion for cleavage fracture of a nuclear pressure vessel steel. Met Trans 1983;14A:2277 –87. [14] Necludov IM. Radiation Hardening of Metals and Alloys. Radiation Damage of Pressure Water Reactor Steels. St Petersburg: Politechnica; 1997. pp. 173–220 (in Russian).
B.Z. Margolin et al. / International Journal of Pressure Vessels and Piping 80 (2003) 817–829 [15] Alekseenko NN, Amaev AD, Gorynin IV, Nikolaev VA. Radiation damage of nuclear power plant pressure vessel steels. La Grange Park, IL: American Nuclear Society; 1997. [16] Margolin BZ, Shvetsova VA, Gulenko AG. Radiation embrittlement modelling for reactor pressure vessel steels: I. Brittle fracture toughness prediction. Int J Pressure Vessel Piping 1999;76: 715–29. [17] Margolin BZ, Shvetsova VA, Gulenko AG. Comparison of the Master Curve and Russian Approaches as applied to WWER RPV steels. Use and applications of the Master Curve for determining fracture toughness (Work shop MASC 2002); 2002. Helsinki-Stockholm, pp. 12.1– 12.22. [18] Nanstad RK, McCabe DE, Menke BH, Iskander SK, Haggag FM. Effects of radiation on KIC curves for high-copper welds. In: Packan N, Stoller R, Kumar A, editors. Effects of radiation on materials. 14th International Symposium ASTM STP 1046, American Society for Testing and Materials; 1990. Philadelphia, p. 214– 33. [19] Margolin BZ, Shvetsova VA, Gulenko AG, Ilyin AV. Cleavage fracture toughness for 3Cr–Ni–Mo –V reactor pressure vessel steel. Theoretical prediction and experimental investigation. Int J Pressure Vessel Piping 2001;78:715–29. [20] Sokolov MA, Nanstad RK, Miller MK, Fracture toughness characterization of a highly embrittled RPV weld. To be published in: Effects of radiation on materials, ASTM STP 1447, Grossbeck M, editor, ASTM International, West Conshohocken, PA, 2003.
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[21] Planman T, Keinanen H, Wallin K, Rintamaa R. Master Curve analysis of highly embrittled pressure vessel steel. Irradiation embrittlement and mitigation. Proceed. of the IAEA Specialists Meeting in Gloucester, UK, 2001, Work. doc. IAEA TWG-LMNPP01/2; 2002. pp. 521 –35. [22] Keim E, Bartsch R, Nagel G. Application in licensing West European reactor. In: Use and applications of the Master Curve for determining fracture toughness (Work shop MASC 2002), Helsinki-Stockholm, 2002. [23] Ortner SR. Outstanding issues in Master Curve applications. In: Use and applications of the Master Curve for determining fracture toughness (Work shop MASC 2002), Helsinki-Stockholm, 2002, pp. 19.1– 19.14. [24] McCabe DE. Irradiation effect on engineering materials. Heavysection steel irradiation program. NUREG/CR-5591, vol. 8, no. 2, 2000. [25] Wallin K. Introduction to the Master Curve approach and ASTM E 1921. In: Use and applications of the Master Curve for determining fracture toughness (Work shop MASC 2002), Helsinki-Stockholm, 2002, p. 4.1–4.17. [26] Wallin K. Master Curve approach and SI assessment. In: Use and applications of the Master Curve for determining fracture toughness (Work shop MASC 2002), Helsinki-Stockholm, 2002, p. 8.1– 8.19. [27] Wallin K. Recommendations for the application of fracture toughness data for structural integrity assessments. NUREG/CR-0131, ORNL/ TM-12413, 1993.
A new engineering method for prediction of the fracture toughness temperature dependence for RPV steels B.Z. Margolin*, A.G. Gulenko, V.A. Nikolaev, L.N. Ryadkov Central Research Institute of Structural Materials “Prometey”, 49 Shpalernaja Street, St Petersburg 193015, Russia Received 20 May 2003; revised 2 December 2003; accepted 2 December 2003
Abstract In order to predict the temperature dependence of fracture toughness for RPV steels with various degrees of embrittlement, up to extremely high levels, an engineering method is proposed which is similar to the Master Curve concept and named the Unified Curve concept. Wide verification of the Unified Curve concept is performed and comparison of the Master and Unified Curve concepts is carried out. It is shown that the Master Curve concept is a partial case of the Unified Curve concept. q 2004 Elsevier Ltd. All rights reserved. Keywords: Cleavage fracture toughness; Local criterion; Probabilistic model; Master Curve; Unified Curve; Reactor pressure vessel steel
1. Introduction At present, the Master Curve approach [1 –5] is a widely used method for prediction of the temperature dependence of fracture toughness for RPV steels. This approach allows the prediction of the KIC ðTÞ curve for any given fracture probability and any specimen thickness on the basis of small-sized specimen testing. Analysis of the applicability of the Master Curve approach for RPV steels in various states (initial (as-received), irradiated, thermal embrittled) has shown that this approach provides adequate predictions for materials in the initial state and for cases when the degree of embrittlement is not high. For high degrees of embrittlement, the Master Curve may provide inadequate and nonconservative predictions [6,7]. This circumstance is connected with using the lateral temperature shift condition in the Master Curve approach, i.e. it is assumed that the shape of the KIC ðTÞ curve does not change as the degree of embrittlement increases. The Prometey probabilistic model [6 –9] does not have such shortcomings. This model is based on the local approach and a new formulation of the cleavage fracture criterion [10,11]. The Prometey model allows predictions to be made sufficiently adequately of the KIC ðTÞ curves * Corresponding author. Tel.: þ 7-812-274-1101; fax: þ7-812-274-1707. E-mail address: [email protected] (B.Z. Margolin). 0308-0161/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijpvp.2003.12.005
for both RPV materials in the initial state and materials with various degrees of embrittlement up to a highly embrittled state [6,7]. The Prometey model as well as the Master Curve may use small-sized specimen testing and predict the KIC ðTÞ curve for any given fracture probability and any specimen thickness. The major disadvantage of the Prometey model appears to be the rather intensive calculations and the necessity to carry out additional tensile tests to have data on the stress – strain curves for different temperatures and also the dependence of critical brittle fracture stress on plastic strain. In other words, the Prometey model is not an appropriate engineering method to predict the KIC ðTÞ curve, compared with the Master Curve approach. The purpose of the present paper is to elaborate an engineering method based on the generalized results obtained by the Prometey model that allows the prediction of the KIC ðTÞ curve for RPV steels with various degrees of embrittlement, including extremely high levels.
2. Material embrittlement modelling with the Master Curve concept The main properties of the Master Curve concept are represented as [1 – 5].
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(1) The brittle fracture probability Pf is described by a three-parameter Weibull model in the form [1] " # KJC 2 Kmin 4 ; ð1Þ Pf ¼ 1 2 exp 2 K0 2 Kmin where Pf is the probability of fracture at KI # KJC for an arbitrary chosen specimen from a specimen set; K0 is a scale parameter dependent on the test temperature and specimen thickness; Kmin is the minimum possible fracture toughness. p In accordance with Ref. [1], Kmin ¼ 20 MPa m: (2) The effect of specimen thickness on fracture toughness is described by [2] X KJC 2 Kmin BY 1=4 ¼ ; ð2Þ Y 2K BX KJC min X Y where KJC and KJC are fracture toughness values for specimens with thickness BX and BY at the same probability of brittle fracture Pf : (3) The fracture toughness median value (at Pf ¼ 0:5) as a function of temperature for specimens 1T-CT ðB ¼ 25 mmÞ is given in [3] shelf KJCðmedÞ ¼ KJC þ b expðgðT 2 T0 ÞÞ; ð3Þ p p shelf where KJC ¼ 30 MPa m; b ¼ 70 MPa m; g ¼ 0:019; T the temperature in 8C and T0 is temperature at which p KJCðmedÞ ¼ 100 MPa m: (4) It is assumed that for embrittled materials only the parameter, T0 ; varies, whereas the rest of the numerical parameters in Eq. (3) are fixed. This assumption is usually called the lateral temperature shift condition. This condition connects an increase in the degree of material embrittlement with a lateral shift of the KJC ðTÞ curve to an elevated temperature range. Assumption 4 may also be interpreted in another way. Eq. (3) may be written as shelf KJCðmedÞ 2 KJC ¼ vcðTÞ;
ð4Þ
where v ¼ expð2gT0 Þ; cðTÞ ¼ b expðgTÞ: As seen from Eq. (4), the embrittled state of a material may be represented as a decrease of fracture toughness shelf (more exactly, the difference KJCðmedÞ 2 KJC ) at each temperature by a factor vin =vem ; where vin corresponds to the initial state with T0 ¼ T0in and vem corresponds to the embrittled state with T0 ¼ T0em : Thus, the temperature dependence of fracture toughness for any state of a material may be represented as the product of a coefficient v depending on material state and a function cðTÞ which does not depend on material state. Representation of the transformation of the KJC ðTÞ curve with degree of material embrittlement by Eq. (4) may also be useful when the dependence cðTÞ is not described by an exponential function. It may be noted that, as a common case, the function cðTÞ may be either an exponential or another type. The only restriction on the function cðTÞ follows from its physical
meaning: the function cðTÞ describes KJC ðTÞ for brittle fracture without regard for ductile tearing, i.e. this function does not take into account that the KJC ðTÞ curve varies in character when KJC exceeds the upper shelf level of KJC ðTÞ: In Section 3, on the basis of the Prometey probabilistic model, it is shown that the representation of the KJC ðTÞ curve by Eq. (4) is quite general, and the nature of the function cðTÞ is determined.
3. Modelling the temperature dependence KJC(T) by the Prometey model 3.1. Presentation of the Prometey probabilistic model 3.1.1. The local criterion for cleavage fracture The formulation of the local cleavage fracture criterion in a probabilistic manner is as follows [6 – 9]. 1. The polycrystalline material is viewed as an aggregate of cubic unit cells. The mechanical properties for each unit cell are taken as the average properties obtained by standard specimen testing. The size of the unit cell ruc is never less than the average grain size. The stress and strain fields in the unit cell are assumed to be homogeneous. 2. The local criterion for cleavage fracture of a unit cell is taken as
s1 þ mT1 seff $ sd
ð5aÞ
s1 $ SC ðæÞ;
ð5bÞ
where the critical brittle fracture stress, SC ðæÞ; is calculated by SC ðæÞ ¼ ½C1p þ C2p expð2Ad æÞ21=2 :
ð6Þ
Here, s1 is the maximum principal stress, the effective stress is seff ¼ seq 2 sYÐ; seq is the equivalent stress, sY is the yield stress, æ ¼ d1peq is Odqvist’s parameter, d1peq is the equivalent plastic strain increment, C1p ; C2p ; Ad are material constants, sd is the strength of carbides or ‘carbide –matrix’ interfaces or other particles on which cleavage microcracks are nucleated, mT1 is a parameter that depends on temperature and plastic strain and may be written as mT1 ¼ mT ðTÞm1 ðæÞ;
ð7Þ
m1 ðæÞ ¼ S0 =SC ðæÞ;
ð8Þ
mT ðTÞ ¼ m0 sYs ðTÞ;
ð9Þ
where S0 ; SC ðæ ¼ 0Þ; m0 is a constant which may be experimentally determined and sYs is the temperaturedependent component of the yield stress. Condition (5a) is the nucleation condition for cleavage microcracks. Condition (5b) is the propagation condition for cleavage microcracks. 3. To formulate criteria (5) in a probabilistic way, it is assumed that the parameter sd is stochastic and the remainder of the parameters controlling brittle fracture are deterministic. Such an assumption is based on analysis of
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the stochastic nature of various critical parameters controlling cleavage fracture of RPV steels [12]. 4. To describe the distribution function for the parameter sd ; the Weibull law is used: the minimum strength of carbides in the unit cell on which cleavage microcracks are nucleated, is assumed to obey sd 2 sd0 h pðsd Þ ¼ 1 2 exp 2 ; s~d
ð10Þ
where pðsd Þ is the probability of finding in each unit cell a carbide with minimum strength less than sd ; s~d ; sd0 and h are Weibull parameters. 5. The weakest link model is used to describe the brittle fracture of the polycrystalline material. 6. It is considered that brittle fracture may happen only in unit cells for which the conditions seq $ sY and s1 $ SC ðæÞ are satisfied. 3.1.2. The probabilistic model for the KIC ðTÞ curve prediction The probabilistic model for fracture toughness prediction is based on the local criterion described above. The stress and strain fields near the crack tip are calculated by FEM or on the crack extension line with an approximate analytical solution [8,9]. The brittle fracture probability of a cracked specimen, Pf ; may be presented in the form used in [13] s h Pf ¼ 1 2 exp 2 w ; s~d
ð11Þ
where the Weibull stress sw is "
sw ¼
k X
#1=h ðmaxðSinuc Þ
2 sd0 Þ
h
;
ð12Þ
i¼1
( Sinuc
;
sinuc ; if si1 $ SC ðæi Þ and sinuc . sd0 sd0 ;
if si1 , SC ðæi Þ or sinuc # sd0
:
Here snuc ; s1 þ mT1 seff ; k is the number of unit cells in the plastic zone, i is the number of a unit cell. For each unit cell, the parameter maxðSinuc Þ is the maximum value of Sinuc from the beginning of deformation up to the current loading. The above equations provide the calculation of the dependence of the brittle fracture probability on stress intensity factor, Pf ðKI Þ; as the parameter sw is a function of KI : To predict the KJC ðTÞ curve on the basis of the model proposed above, it is necessary to know the parameters SC ðæÞ; mT ðTÞ; s~d ; sd0 and h and also parameters describing plastic deformation to enable the stress and strain fields to be calculated.
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3.2. Modelling the KJC ðTÞ curve for a material with various degrees of embrittlement 3.2.1. Analysis of the main parameters controlling KJC ðTÞ It follows from Section 3.1, that for modelling the dependence KJC ðTÞ; the following parameters of a material have to be known: – –
parameters describing the stress – strain curve; parameters in the local criterion of cleavage fracture.
According to Margolin et al. [7] the stress – strain curve may be approximated by
seq ¼ sY ðTÞ þ A0 æn :
ð13Þ
Here A0 and n are material constants, as a common case, dependent on the state of the material. The temperature dependence of the yield stress sY ðTÞ is approximated by
sY ðTÞ ¼ sYG þ sYS ðTÞ ¼ a þ b expð2hTa Þ;
ð14Þ
where sYG and sYs are the temperature-independent component and the temperature-dependent component of the yield stress; a; b and h are material constants, in Eq. (14) Ta is taken in degrees Kelvin. As shown in Refs. [4,14 – 16], when material embrittlement is accompanied by a yield stress increase (for example, due to irradiation) and this increase is caused by an increase in sYG ; i.e. the value of the parameter a; the temperaturedependent component of sY does not vary practically. It should be noted that the temperature-dependent component of sY for various RPV steels may be described by the same dependence. Strain hardening of unirradiated material does not differ from strain hardening of irradiated material, i.e. the parameters A0 and n depend weakly on embrittlement degree. Now consider the parameters in the cleavage fracture local criterion (5). As shown in Ref. [16], the dependence SC ðæÞ does not vary practically for various degrees of embrittlement of a material, at least, for all cases when the dominant mechanism of fracture is cleavage or microcleavage. The parameter mT is calculated by Eq. (9) where m0 ¼ 0:1 [7 – 9]. Hence, the dependence mT ðTÞ is practically invariant to embrittlement degree. sd0 is calculated by equation sd0 ¼ a þ b [6,7], where a and b are coefficients in Eq. (14). It is clear that if for embrittled material the parameter sY increases then the parameter sd0 also increases. The main parameters which control the dependence KJC ðTÞ for various degrees of embrittlement [16] are s~d and h: Increasing material embrittlement degree may be modelled by decreasing the parameter s~d [6,16]. For 2Cr – Ni – Mo – V RPV steel in the initial state, h ¼ 6 ðsY ðT ¼ 20 8CÞ ¼ 580 MPaÞ; for irradiated weld of this steel, h ¼ 8 ðsY ðT ¼ 20 8CÞ ¼ 640 MPaÞ; and for this steel
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Table 1 Parameters for the stress–strain curves for 2Cr–Ni–Mo –V steel in the initial state approximated by seq ¼ sY þ A0 æn Parameter
A0 (MPa) n
Temperature (8C) 2196
2100
260
220
20
100
350
679 0.50
685 0.47
622 0.46
629 0.49
590 0.49
557 0.49
537 0.50
after thermal embrittlement, h ¼ 12 ðsY ðT ¼ 20 8CÞ ¼ 900 MPaÞ [6,7]. On the basis of these data the conclusion may be drawn that as embrittlement degree increases (that is accompanied with increase in yield stress) the parameter h increases. For the materials investigated, the minimum value of h ¼ 6, and the maximum value h ¼ 12: 3.2.2. Selection of the model parameter values for the KJC ðTÞ calculation Taking into account the analysis performed of the model parameters, the following numerical values of these parameters may be taken. The first variant: (first set of parameters) a ¼ 510 MPa; b ¼ 1083 MPa; h ¼ 9:31 £ 1023 K21 ; C1p ¼ 2:01 £ 1027 MPa22 ; C2p ¼ 3:90 £ 1027 MPa22 ; Ad ¼ 1:71; h ¼ 6; the parameter s~d is varied. Values of the coefficients A0 and n for various temperatures are given in Table 1. This set of data corresponds to 2Cr –Ni – Mo – V RPV steel in the initial state [6]. Variation of the parameter s~d when other parameters are fixed, allows the description of transformation of the KJC ðTÞ curve as a function of embrittlement degree for cases when this embrittlement is accompanied with insignificant increase in the yield stress. The second variant (second set of parameters) differs from the first variant by two parameters a ¼ 710 MPa and h ¼ 12; the parameter s~d is varied as well as for the first case. This variant is more typical for radiation-embrittled material when the yield stress increases. The typical increase in sY is equal to about 200 MPa [4,15,16], therefore the parameter a increases by 200 MPa as compared with the first variant. For both the first and second variants the size of unit cell is taken to be equal to 0.05 mm [6 –9]. As shown in Refs. [6,7] the descriptions of the scatter in KJC results and the effect of specimen thickness (the crack front length) on KJC as obtained on the basis of the Master Curve concept and the Prometey model coincide practically. Hence, in the present work, there is no need to study the scatter in KIC and the scale effect on the basis of the Prometey model. To describe these regularities, Eqs. (1) and (2) are sufficient. To provide the comparison with the Master Curve, the KJC ðTÞ curve is calculated for B ¼ 25 mm and Pf ¼ 0:5: 3.2.3. Calculation results In Fig. 1 the calculation results are presented for the data sets corresponding to the first and second variants.
Fig. 1. The dependence of KJC on temperature for various values of s~d : calculation for the first (a) and second (b) variants of the data set; the dotted curve is the upper shelf level.
Calculations were performed for plane strain assuming that ductile crack growth is absent for any high level of KJC : Actually, ductile crack growth begins when KI exceeds the ductile ductile upper shelf level KJC : For RPV steels, KJC is p approximately between 200 and 250 MPa m. It is seen from Fig. 1 that for each variant the KJC ðTÞ curves for various levels of s~d have similar shape. At the same time if parts of these curves, which locate below the upper shelf, are considered then the regularity of transformation of KJC ðTÞ may be revealed that was considered in Refs. [16,17]. For small degrees of embrittlement, transformation of KJC ðTÞ may be described by the lateral temperature shift. For large degrees of embrittlement, the slope of the KJC ðTÞ curve decreases (if curves restricted by the upper shelf are considered). Let us analyse whether the results obtained can be presented as Eq. (4). For this, each KJCðmedÞ ðTÞ curve is represented in the coordinates KJCðmedÞ ðTÞ 2 KJshelf versus T; 100 shelf KJCðmedÞ 2 KJC
ð15Þ
shelf 100 where KJCðmedÞ at T ¼ 2200 8C is taken as KJC ; KJCðmedÞ ¼ KJCðmedÞlT¼100 8C : It is clear that the temperature T ¼ 100 8C in Eq. (15) was taken in an arbitrary manner, because this value does not affect verification of the similarity of the above curves.
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data set, the KJCðmedÞ ðTÞ curves approaches the lower shelf at T , 2200 8C: For the second variant data set for all the cases shelf considered, the value of KJC varies over a very narrow p range: from 24 to 28 MPa m. Therefore, it may be taken shelf with a reasonable degree of accuracy that KJC does not depend on the embrittlement degree and is equal to p 26 MPa m.
4. Concept of a Unified Curve 4.1. The main considerations of the concept On the basis of the results obtained in the previous section, the following concept of the Unified Curve may be proposed. 1. The temperature dependence of fracture toughness at Pf ¼ 0:5 for specimens with thickness B ¼ 25 mm from RPV steel for any degree of embrittlement may be described by KJCðmedÞ ¼
shelf KJC
T 2 130 þ V 1 þ tanh 105
; MPa
p
m; ð17Þ
shelf 100 shelf Fig. 2. The ratio ðKJCðmedÞ 2 KJC Þ=ðKJCðmedÞ 2 KJC Þ as a function of temperature for the first (a) and second (b) variants of the data set: the solid curves are the calculated curves for various values of s~d ; the solid curve with points is a fitting curve calculated by formula (16).
In Fig. 2 the KJCðmedÞ ðTÞ curves are represented for various s~d in the coordinates of Eq. (15) for both the first and second variants. It is seen from this figure that all the curves corresponding to one variant differ from each other insignificantly. Moreover the analysis performed shows that the difference between the curves corresponding to the first and second variants is insignificant. This means that for materials with different strength levels, the KJCðmedÞ ðTÞ dependence for any degree of embrittlement may be approximately presented in the form of Eq. (4) where 100 shelf v ¼ KJCðmedÞ 2 KJC : For engineering estimations, the curve sets of the first and second variants may be considered as a unified set and this set may be described by a unified dependence cðTÞ: The cðTÞ curve corresponding to the data sets of the first and second variants is approximated by the function
p shelf ¼ 26 MPa m; V ¼ vM and T the temperawhere KJC ture in 8C. As seen from Eq. (17), the parameter V is equal to shelf KJCðmedÞlT¼1308C 2 KJC : As the degree of embrittlement increases, the parameter V decreases. 2. To describe the KJC ðTÞ dependence at Pf – 0:5; Eq. (1) is used. 3. To describe the KJC ðTÞ dependence for B – 25 mm; Eq. (2) is used. 4.2. Determination of parameters of the Unified Curve It follows from Section 4.1, that to predict the KJC ðTÞ curve for any given thickness of specimen and any fracture probability it is necessary to know the parameter V only. The parameter V as well as the parameter T0 in the Master Curve may be determined on the basis of test results at one temperature (single temperature method) or at several temperatures (multiple temperature method).
ð16Þ
4.2.1. Single temperature method This method is similar to the method proposed in ASTM E 1921-02 and consists of the following steps.
Results of treatment of the calculated data presented in Fig. 2 by the least squares method provide: M ¼ 1:23; TM ¼ 130 8 C; Z ¼ 105 8C: It is seen from Fig. 1, as the lower shelf, some value KJCðmedÞ at T ¼ 2200 8C may be taken which was obtained for the second variant of data set, as for the first variant of
1. Determination of KJCðmedÞ according to ASTM E 1921-02 (on the basis of the maximum likelihood method) for specimens with thickness B ¼ Btest tested at T ¼ Ttest : 2. Calculation of KJCðmedÞ for B ¼ 25 mm at T ¼ Ttest : 3. Determination of the parameter V with Eq. (17) on the basis of the known values of KJCðmedÞ and T ¼ Ttest :
T 2 TM cðTÞ ¼ M 1 þ tanh Z
:
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4.2.2. Multiple temperature method This method is also similar to that proposed in ASTM E 1921-02 and consists of the following steps.
where KJCðiÞ is the experimental value of KJC obtained at Ttest ¼ Ti : Derivation of this equation is given in Appendix A.
1. Recalculation of test results from specimens with thickness B to thickness B ¼ 25 mm by using Eq. (2). 2. Determination of V on the basis of solution of the nonlinear equation 82 N < X ðKJCðiÞ 2 Kmin Þ4 lnð2Þ 40 0 0 15 11 : i¼1 T 2 130 @V@1 þ tanh@ i AA 2 Kmin þ K shelf A JC 105 3 1 5 0 11 2 0 T 2 130 i AA 2 Kmin þ K shelf V@1 þ tanh@ JC 105 2 0 139 = T 2 130 A5 ¼ 0; ð18Þ £41 þ tanh@ i ; 105
5. Comparison of experimental data and results calculated with the Master Curve and Unified Curve concepts In the present section, results obtained at CRISM ‘Prometey’ and available experimental data on fracture toughness for RPV steels with various degrees of embrittlement are treated according to the Master Curve and Unified Curve concepts. These available data are taken from papers [6,18 –24] and the database summarized by Wallin and presented in Refs. [25 – 27]. In addition, unpublished experimental data obtained at CRISM ‘Prometey’ are also used. The base and weld metals of western and Russian RPV steels are considered in different conditions: as received, thermally embrittled state and after neutron irradiation. Moreover, other ferritic structural steels are considered.
Table 2 Values of the parameters dmc and duc for treatment of fracture toughness test results for various materials by the Master Curve and the Unified Curve Set number
Material
sy of T ¼ 20 8C (MPa)
T0 (8C)
Reference
dmc
duc
dmc =duc
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33
A533B steel (in) A508 steel (in) HY130L (in) ABS DS (in) A470 steel (in) 2CrNiMoV steel (in) NVA (in) 3CrNiMoV steel (in) WF-70 weld (in) HSST weld 73W (in) HSST weld 72W (in) A533 steel JRQ (in) WF-70 weld (in) A508 steel (TSE-5 and 6) KWO RPV A508 steel A508 steel (TSE-7) A533 steel A508 steel (TSE-5 and 6) NiCrMoV steel E36 WF-70 weld (irr) HSST weld 72W (irr) HSST weld 73W (irr) A533B steel 2.5CrMoV steel (embr) WF-70 weld (irr) 2CrNiMoV steel (embr) NP2 A533 steel JRQ (irr) Weld KS01 (irr) 2.5CrMoV steel PTS-1 (embr) Weld KS01 (irr)
567 650 955 270 – 565 218 550 740 513 496 480 790 605 – – 450 – 710 925 303 930 620 648 – 730 860 900 676 630 820 1037 950
2148 2140 2131 291.3 286.5 282.5 272.1 267.4 263.5 261.3 260.4 259.7 255.8 252.2 249.2 248 232.7 228.5 220.1 211.3 21.2 24.9 29.3 37.2 39.9 45.6 49 57.1 69.0 86.9 137 163 251
[25] [25] [25] [25] [27] [6] [25] [19] [24] [18] [18] [4] [24] [26] [22] [27] [26] [27] [26] [25] [25] [24] [18] [18] [23] Fig. 3 [24] [7] [25] [4] [20] [21] [26]
17.4 26.8 16.9 192 27.1 207 43.4 54.7 77.9 11.3 31.1 61.9 25.6 115.3 29.6 15 66 25.3 112 387 51.2 110 39 15.5 80.5 32.5 14.6 99.5 83.7 32 62.6 168 26.2
19.2 29.4 18.4 188 28.5 178 41.5 56.6 77 11.9 32 58.3 26.6 96.7 31.9 15.9 54.2 22.8 79.9 185 42.1 99.1 17.5 15.9 58.6 25.4 11.4 21.1 36.2 23.8 12.4 11.3 8
0.91 0.91 0.92 1.02 0.95 1.16 1.05 0.97 1.01 0.95 0.97 1.06 0.96 1.19 0.93 0.94 1.22 1.11 1.40 2.09 1.22 1.11 2.23 0.97 1.38 1.28 1.28 4.72 2.31 1.34 5.05 14.87 3.28
in, as received condition; embr, thermally embrittled condition; irr, after irradiation.
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The brittle fracture mode for these materials varies from cleavage to mixed mode (cleavage and intergranular fracture). For all cases, the parameters T0 and V are determined by the multiple temperature method. To determine T0 ; the equation from Refs. [4,5] was used. For impartial comparison of the Master Curve and Unified Curve approaches it is necessary to use some quantitative measure. It is clear that the main difference between the Master and Unified Curves is connected with the median value of KIC : From this the scatter in KIC is described by the same Eq. (1). Therefore, for comparison of the two approaches it is necessary to use a measure connected with the median values of KIC : For a given case, a suitable measure of deviation in describing the median values of KIC may be represented by vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u X u1 L exp ð19Þ d¼t ðK pr 2 KJCðmedÞj Þ2 ; L j¼1 JCðmedÞj where d is the root mean square deviation; L is the number pr of temperatures at which tests were carried out; KJCðmedÞj is the predicted median value of KJC calculated by the Master exp or Unified Curves at test temperature Tj ; KJCðmedÞj is the median value of KJC determined by treatment of experimental data at T ¼ Tj with the maximum likelihood method. According to ASTM E 1921-02 [5] " #1=4 ‘ X ðKJCðiÞ 2 Kmin Þ4 exp 1=4 KJCðmedÞj ¼ ½lnð2Þ þKmin ; ð20Þ i¼1
‘
where KJCðiÞ is the experimental value of fracture toughness at T ¼ Tj recalculated for B ¼ 25 mm; ‘ is the number of tests at T ¼ Tj : It follows from Eq. (19) that the lower the value of d the more adequate is the description of experimental data by the Master Curve or Unified Curve methods. For an ideal case, when we have representative sampling (sufficient number of tests) at each temperature and the predicted curve (3) or (17) describes the experimental data absolutely adequately, d ! 0: In Table 2 the parameters d calculated for treatment by the Master Curve, dmc ; and the Unified Curve, duc ; for various materials are presented. In Fig. 3 for some data sets, comparisons of the test results and KJC ðTÞ curves calculated by the Master Curve and the Unified Curve are presented. As seen from Table 2 and Fig. 3, for low T0 the Master Curve and Unified Curve provide very close predictions of KJC ðTÞ and the predicted KJC ðTÞ curves describe experimental data sufficiently adequately. As the embrittlement degree increases (T0 increases), the description with the Unified Curve becomes more adequate than predictions by the Master Curve. This tendency is well seen from Fig. 4, where the ratio dmc =duc versus T0 is presented. It is clear that if the ratio dmc =duc is close to unity, then prediction of KJC ðTÞ both
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by the Master Curve and the Unified Curve is equivalent. The larger the ratio dmc =duc ; the less adequate the prediction of KJC ðTÞ by the Master Curve.
6. Discussion The approach proposed in the present paper allows an adequate description of KJC ðTÞ for ferritic steels which have very different properties: sY varies from about 300 up to 1000 MPa, T0 varies from about 2 100 up to about 250 8C. That is why we believe, that the title of the proposed method as the Unified Curve concept appears to be justified and this concept may be applied both to RPV steels and to other ferritic structural steels. As seen from Fig. 3 and Table 2, the curves predicted with the Master Curve and Unified Curve are very close for materials with high fracture toughness, i.e. for materials with low values of T0 : Such a result is connected with variation of KJC from the lower shelf up to the upper shelf over a range of temperatures lower than TM (see Eq. (17)). For TM ¼ 130 8C and Z ¼ 105 8C; when the temperature of reaching the upper shelf does not exceed 20 –50 8C the dependence (17) may be well approximated by an exponential with the constant parameter g ¼ 0:0186; i.e. in this case, the Unified Curve practically coincides with the Master Curve. When the embrittlement degree increases, the temperature range over which variation of KJC from the lower shelf up to the upper shelf shifts to a higher temperature range. As the temperature for reaching the upper shelf approaches TM ; the shape of the KJC ðTÞ curve predicted by the Unified Curve begins to differ from that predicted by the Master Curve. For this situation the lateral temperature shift condition begins to be invalid. For subsequent increase of embrittlement degree, the difference between the results predicted by the Master Curve and Unified Curve increases. This difference will be largest when the KJC ðTÞ curves predicted by the Master and Unified Curves will have fundamentally different tendencies. For example, for the case when over a temperature range, the dependence dKJC ðTÞ=dT calculated by the Unified Curve becomes a decreasing one, for the Master Curve, dKJC ðTÞ=dT is always increasing. Thus, it may be concluded that the Master Curve concept may be considered as a particular case of the Unified Curve concept. Applicability of the Master Curve is restricted to materials in the initial (as-received) state and to degrees of embrittlement which are not high. At the present time in order to extend the range of applicability of the Master Curve approach both for materials with low and high degrees of embrittlement, an additional condition is proposed to be used [5]. The dependence, KJC ðTÞ calculated by the Master Curve is
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valid over a temperature range restricted by the condition lT 2 T0 l , 50 8C:
ð21Þ
Introduction of condition (21) restricts application of the Master Curve concept for RPV structural integrity assessment. Consider this consideration in detail.
Assume, we consider RPV steel with a high degree of embrittlement. When analysing RPV structural integrity, a crack-like flaw is considered which has the crack front length Bflaw q 25 mm: The KIC ðTÞ curves for B ¼ 25 mm and B ¼ Bflaw are schematically shown in Fig. 5.
Fig. 3. Comparison of experimental data and KJC ðTÞ curves for B ¼ 25 mm calculated by the Master Curve (on the left) and Unified Curve (on the right) for steels with various degrees of embrittlement: all test results are recalculated for a specimen thickness B ¼ 25 mm:
B.Z. Margolin et al. / International Journal of Pressure Vessels and Piping 80 (2003) 817–829
825
Fig. 3 (continued )
As shown from Fig. 5, if for the loading condition p ; then application of the considered of the RPV, KI . KJC Master Curve for RPV structural integrity assessment is incorrect and nonconservative. Thus, the following conclusion may be drawn from the represented analysis. On the one hand, condition (21) allows expansion of applicability of the Master Curve to describe the KJC ðTÞ curves for highly embrittled material, on the other hand, the temperature range of the correct
prediction of KJC ðTÞ (condition (21)) may be very small to analyse RPV structural integrity. The Unified Curve concept allows adequate predictions to be provided for any degree of embrittlement of a material and therefore may be used to analyse RPV structural integrity without any restrictions. The last problem which we would like to discuss in this section may be formulated as follows. Is it possible to describe the variation of the shape of the KJC ðTÞ curve with
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Fig. 3 (continued )
Eq. (3) if the parameter g is taken to decrease as the KJC ðTÞ curve moves to higher temperatures? Let us show that the answer to this question is negative. According to the Master Curve a shift of the KJC ðTÞ curve occurs either for T0 increasing or B increasing. Hence, for a given T0 and B increasing, the parameter g must decrease (as B increases, temperature corresponding, for example, to p KJC ¼ 100 MPa m; increases). But on the other hand,
according to Eq. (2) the parameter g does not vary as B increases. Hence, either Eq. (2) is not correct (consideration ‘a’) or transformation of a shape of curve is not described by Eq. (3) with the varying parameter g (consideration ‘b’). As Eq. (2) was derived theoretically and confirmed experimentally the conclusion may be drawn that the consideration ‘b’ is incorrect.
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described by p T 2 130 shelf þ V 1 þ tanh ; MPa m; KJC ¼ KJC 105
Fig. 4. The dependence of the ratio dmc =duc on T0 :
-scatter band.
where for specimen thickness B ¼ 25 mm and Pf ¼ 0:5; p shelf KJC ¼ 26 MPa m; V is the only parameter which depends on embrittlement degree of a material and T temperature in 8C. This approach is here named the Unified Curve. 2. The parameter V may be determined on the basis of fracture toughness test results at one or several temperatures. Requirements for the number and size of specimens are the same as for determination of T0 : For determination of V on the basis of test results at several temperatures, an Eq. (18) is proposed. 3. For the known parameter V; the KJC ðTÞ dependence may be determined with Eqs. (1) and (2) for any given fracture probability and specimen thickness, i.e. with equations used in the Master Curve concept. 4. Comparison of the KJC ðTÞ curves predicted by the Master Curve and Unified Curve and experimental data is performed. More than 30 sets of experimental data are considered for materials with various degrees of embrittlement. It has been shown that for materials in the initial (asreceived) state and with small degrees of embrittlement, the curves predicted with both methods coincide practically. As the degree of embrittlement increases (T0 increases), the description of KJC ðTÞ with the Unified Curve becomes more adequate than the prediction by the Master Curve. The Master Curve concept appears to be a partial case of the Unified Curve concept.
Appendix A. Determination of the parameter V in the Unified Curve from fracture toughness test results at several test temperatures Fig. 5. The dependence of KJC on temperature for a crack front length B ¼ 25 mm (specimen thickness 25 mm) and for a flaw front length B ¼ Bflaw : the solid curves are predicted by the Master Curve; the dotted curve is a real curve.
Thus, the represented arguments show once more that the Unified Curve appears to be one from correct methods to describe transformation of KJC ðTÞ on embrittlement degree.
For determination of the parameter V in the Unified Curve, the maximum likelihood method is used [4]. An equation for determination of V is derived as well as an equation for determination of T0 by a multitemperature method [3,4]. According to the maximum likelihood method, the parameter V may be found from
› ln L ¼ 0; ›V
ðA1Þ
where 7. Conclusions L¼ 1. It has been shown on the basis of calculations performed with the Prometey model that the temperature dependence of fracture toughness for ferritic structural steels with various degrees of embrittlement may be
N Y
f ðxi ; VÞ:
ðA2Þ
i¼1
Here f ðxi Þ is the probability density function at x ¼ xi ; x is the random value; xi is experimental value; N is the number of experimental points.
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The probability density function f ðxi Þ may be defined from Eq. (1) of the present paper z dPf zxz21 x i ¼ exp 2 i f ðxi Þ ¼ ; dx az a
ðA3Þ
where z ¼ 4;
ðA4Þ
xi ¼ KJCðiÞ 2 Kmin ;
ðA5Þ
a ¼ K0 2 Kmin ;
ðA6Þ
where KJCðiÞ is the experimental value of KJC obtained at Ttest ¼ Ti ; K0 is the scale parameter, which may be calculated from Eq. (1) for Pf ¼ 0:5 K0 ¼
KJCðmedÞ 2 Kmin þ Kmin ; ðlnð2ÞÞ1=4
ðA7Þ
where KJCðmedÞ is fracture toughness for Pf ¼ 0:5: According to Eqs. (A2) and (A3) the likelihood function L is written as L¼
z x ðzxz21 Þa2z exp 2 i : i a i¼1
N Y
ðA8Þ
Eq. (A1) may be represented as
› ln L › ln L ›a ¼ ¼ 0: ›V ›a ›V
ðA9Þ
Here N X › ln L z ¼ 2 þ zxzi a2ðzþ1Þ : ›a a i¼1
ðA10Þ
For determination of the parameter a; Eqs. (A6), (A7) and (17) of the present paper are used T 2 TM shelf V 1 þ tanh 2 Kmin þ KJC Z a¼ : ðlnð2ÞÞ1=4
ðA11Þ
Let us differentiate a with respect to V
›a ¼ ›V
T 2 TM 1 þ tanh Z ðlnð2ÞÞ1=4
:
ðA12Þ
On making the substitution of Eqs. (A10) and (A12) in Eq. (A9) and taking into account Eqs. (A4), (A5) and (A11), we find the nonlinear equation for determination of
the parameter V 82 N < X ðKJCðiÞ 2 Kmin Þ4 lnð2Þ 40 0 0 15 11 : i¼1 T 2 130 i @V@1 þ tanh@ AA 2 Kmin þ K shelf A JC 105 3 1 5 11 0 2 0 T 2 130 AA shelf V@1 þ tanh@ i 2 Kmin þ KJC 105 2 0 139 = T 2 130 A5 ¼ 0: ðA13Þ 41 þ tanh@ i ; 105 References [1] Wallin K. The scatter in KIC results. Engng Fract Mech 1984;19: 1085–93. [2] Wallin K. The size effect in KIC results. Engng Fract Mech 1985;22: 149 –63. [3] Wallin K. Fracture toughness transition curve shape for ferritic structural steels. In: Teoh S, Lee K, editors. Fracture of engineering materials and structures. Amsterdam: Elsevier Applied Science; 1991. p. 83–8. [4] Merkle JG, Wallin K, McCabe DE. Technical basis for an ASTM standard on determining the reference temperature, T0 for ferritic steels in the transition range. NUREG/CR-5504, ORNL/TM-13631, 1999. [5] ASTM E 1921-02. Standard test method for determination of reference temperature, T0 ; for ferritic steels in the transition range. In: Annual Book of ASTM Standards, vol. 03.01, 2002. [6] Margolin BZ, Karzov GP, Shvetsova VA, Keim E, Chaouadi R. Application of local approach concept of cleavage fracture to VVER materials. Service experience and failure assessment applications. The 2002 ASME Pressure Vessels and Piping Conference, vol. 437.; 2002. Vancouver, BC, Canada, pp. 113 –20. [7] Margolin BZ, Shvetsova VA, Gulenko AG, Ilyin AV, Nikolaev VA, Smirnov VI. Fracture toughness predictions for a reactor pressure vessel steel in the initial and highly embrittled states with the Master Curve approach and a probabilistic model. Int J Pressure Vessel Piping 2002;79:219 –31. [8] Margolin BZ, Gulenko AG, Shvetsova VA. Improved probabilistic model for fracture toughness prediction based for nuclear pressure vessel steels. Int J Pressure Vessel Piping 1998; 75:843–55. [9] Margolin BZ, Kostylev VI, Minkin AI. The effect of ductile crack growth on the temperature dependence of cleavage fracture toughness for a RPV steel with various degrees of embrittlement. Int J Pressure Vessel Piping 2003; 80:285–296. [10] Margolin BZ, Shvetsova VA. Brittle fracture criterion: physical and mechanical approach. Problemy Prochnosti 1992;N2:3 – 16. in Russian. [11] Margolin BZ, Shvetsova VA, Karzov GP. Brittle fracture of nuclear pressure vessel steels. Part I. Local criterion for cleavage fracture. Int J Pressure Vessel Piping 1997;72:73– 87. [12] Margolin BZ, Gulenko AG, Shvetsova VA. Probabilistic model for fracture toughness prediction based on the new local fracture criteria. Int J Pressure Vessel Piping 1998;75:307–20. [13] Beremin FM. A local criterion for cleavage fracture of a nuclear pressure vessel steel. Met Trans 1983;14A:2277 –87. [14] Necludov IM. Radiation Hardening of Metals and Alloys. Radiation Damage of Pressure Water Reactor Steels. St Petersburg: Politechnica; 1997. pp. 173–220 (in Russian).
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