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Analysis of structure integrity of RPV on the basis of brittle fracture criterion: new approaches
International Journal of Pressure Vessels and Piping 81 (2004) 651–656 www.elsevier.com/locate/ijpvp
Analysis of structure integrity of RPV on the basis of brittle fracture criterion: new approaches George Karzova,*, Boris Margolina, Eugene Rivkinb a
Central Research Institute of Structural Materials Prometey, Shpalernaja, St Petersburg, Russian Federation b Research and Development Institute of Power Engineering, Moscow, Russian Federation
Abstract Based on the Master curve conception, the condition for fracture toughness is formulated for a heterogeneous distribution of stress intensity factor along a crack front and non-monotonic, non-isothermal loading. This formulation includes an elaborated procedure for taking into account the effects of shallow cracks and biaxial loading on fracture toughness. q 2004 Elsevier Ltd. All rights reserved. Keywords: Brittle fracture; Fracture probability; Fracture toughness; Brittle strength of reactor pressure vessels; Shallow crack effect; Biaxial loading effect; Weakest link model
1. Background
2. Introduction
The problems of substantiation and assuring nuclear power plant (NPP) equipment integrity were always the main attention in the scientific and organizing activities of Myrddin Davies. He believed that one of the main tasks in this field would be the consolidation of specialists’ efforts in various countries for the creation of a common understanding of approaches towards calculated assessments of safe operation time of the main components of NPPs. At the same time he always supported the creation and development of new progressive calculation procedures, within the framework of the general philosophy of assuring safety. He promoted the development and integration of European approaches for analysis of reactor pressure vessel (RPV) brittle fracture resistance in Russia. He assisted in the establishment of a mutual understanding between specialists of various countries in this area and gave not only scientific but also moral support to the authors of this work. He contributed much to the harmonization of the methods developed in Russia with the western approaches, and this provided (to a considerable degree) the formation of the general Russian-European position on the subject. The authors of the present article are very grateful to Myrddin Davies and devote this publication to his memory.
At the present time, when evaluating the resistance to brittle fracture for RPVs, the following considerations are used:
* Corresponding author. Tel.: þ 812-274-3796; fax: þ 812-274-1707. E-mail address: [email protected] (G. Karzov). 0308-0161/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijpvp.2004.03.001
1. Fracture toughness KIC which is dependent on temperature is used as a main characteristic of a material for evaluation of its resistance to brittle fracture. 2. It is taken that brittle fracture of the RPV does not happen if the condition: n·KI , KIC is fulfilled for each point on the crack front (where n— safety margin, KI —stress intensity factor (SIF)). Besides, it is assumed that parameter KIC is a deterministic parameter, and does not depend on flaw size. At the same time, it is well known that brittle fracture is a stochastic process, and may be described by the weakest link model [1 – 5]. As a result, the parameter KIC depends on flaw size and, in particular, on the crack front length. It should be also taken into account that fracture toughness may vary for various load biaxialities [6 – 11] and for various crack depths (a phenomena of shallow cracks) [12 –14]. In the present paper, main approaches are presented on the basis of which a new procedure for evaluation of brittle strength of RPV have been elaborated.
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3. Estimation of the shallow crack effect on fracture toughness For plate with a central crack under uniaxial tension, the fracture energy density 2g as a function of crack length a and the size of plastic zone near the crack tip is given by equation [15]: 2g ¼ sY dC ½m·expð1=mÞ· pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi arccosðexpð21=mÞÞ 1 2 expð22=mÞ 2 1;
ð1Þ
where sY ; the yield strength, dC ; the critical crack tip opening displacement, a ð2Þ m¼ ; c pE dC ; ð3Þ c¼ 8ð1 2 n2 ÞsY
n; Poisson’s ratio, E; elastic modulus, a; crack length. It follows from Eq. (1) that the fracture energy density is not a material constant and only for m ! 1 (small loads and big crack length) the fracture energy density is a constant of a material, which is equal to sY dC : Taking into account that sY dC ¼ ð1 2 n2 Þ
2 KIC E
and assuming 2g ¼ ð1 2 n2 ÞðKC2 =EÞ; we obtain [17]: KC ¼ vKIC ;
ð4Þ
where KC ; fracture toughness for specimen with shallow crack, v; coefficient which takes into account the shallow crack effect: v ¼ ½m expð1=mÞarc cosðexpð21=mÞÞ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 expð22=mÞ 2 11=2 : The dependence of v on parameter m is shown in Fig. 1. According to Eqs. (2) and (3) we have: m¼
8a pðKIC =sY Þ2
ð5Þ
Thus, fracture toughness for shallow cracks may be calculated by Eq. (4). Applicability of Eq. (4) for RPV materials of WWER was verified by using data of CRISM
Fig. 2. Comparison of the calculated and experimental data: 1,3–treatment of experimental data (W ¼100 mm), 2, calculation by Eq. (4); 1,2, KC ; 3, KIC :
‘Prometey’ published in Ref. [16]. In Ref. [16] 2.5Cr – Mo – V RPV base metal (for WWER-440) in various states was studied. Specimens with sizes W ¼100 mm, B ¼50 mm and the crack length a ¼ 10; 25 and 50 mm were tested by threepoint bending, where W; specimen width, B specimen thickness. In Fig. 2 the fracture toughness versus temperature curves are presented, which were obtained experimentally in Ref. [16] and calculated by Eq. (4). As seen from Fig. 2, the calculated curve predicts an increase of fracture toughness values for shallow cracks, but these values are less than experimental values. Thus, in spite of clear theoretical simplifications accepted for deducing Eq. (4), this equation may be used for engineering estimations of fracture toughness for shallow cracks as it gives conservative results. It should be also noted that, according to Dowd and Bass [12 –14], an increase in fracture toughness for specimens with shallow cracks is realized mainly at a=W ¼ 0:1 to 0.2. Taking into account that for a RPV with wall thickness S; W ¼ S; the following relation may be taken for estimation of the shallow crack effect on fracture toughness: ( KC ¼
KIC ;
at a . 0:15S
vKIC ; at a # 0:15S
ð6Þ
4. Estimation of the biaxial loading effect on fracture toughness
Fig. 1. The dependence of parameter v on m:
It is known that under normal operation of the RPV, and also under pressurised thermal shock (PTS) loading, the RPV experiences biaxial loading. In this connection, the problem arises how to estimate the biaxial loading effect on fracture toughness. It was shown in [6 – 11] that biaxial loading may result in decreasing fracture toughness. This
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653
decrease is revealed for specimens with shallow cracks and large-scale yielding [6]. For small-scale yielding or deep cracks, biaxial loading does not practically affect fracture toughness [6]. Thus, the biaxial loading effect on fracture toughness has to be taken into account for shallow cracks and large-scale yielding. Hereafter the procedure for determination of this parameter will be considered in detail. The following procedure may be proposed for an estimation of the biaxial loading effect on fracture toughness [17,18]. For a # 0:15 S and mv # 0:7: KCp ¼ vb vKIC ;
ð7Þ
where KCp ; fracture toughness for specimen with regard for the biaxial loading effect, vb ; coefficient which takes into account the biaxial loading effect on fracture toughness. Coefficient vb is calculated by equation:
vb ¼ 1 2 0:1b;
ð8Þ
where b; the load biaxiality ratio, i.e. ratio of maximum nominal stress parallel to the crack plane and acting along the crack front to maximum nominal stress perpendicular to the crack plane. Eq. (7) is valid when b varies from 0 up to 2. For a . 0:15S or for mv . 0:7; the biaxial loading effect on fracture toughness may be neglected, i.e. it may be taken vb ¼ 1: Taking into account Eqs. (6) – (8) fracture toughness for biaxial loading of RPV may be determined by equation: ( at a . 0:15S KIC ; p KC ¼ ; ð9Þ vb vKIC ; at a # 0:15S ( 1; at mv . 0:7 vb ¼ ; 1 2 0:1b; at mv # 0:7 where mv ¼ m=v2 : The proposed procedure may be justified by results obtained in Ref. [16]. These results allow one to draw the following considerations. The biaxial loading effect on fracture toughness is observed for cases for which parameter b affects the dependence CTODðJ=sY Þ (here CTOD, the crack tip opening displacement, J – J-integral). If the dependence CTODðJ=sY Þ is invariant to b; then the biaxial loading does not affect KJC : The dependences CTODðJ=sY Þ at b ¼ 0 and b ¼ 2 are presented in Fig. 3 for small and large scale yielding. As seen from Fig. 3a, the dependence CTODðJ=sY Þ at b ¼ 0 coincides with CTODðJ=sY Þ at b ¼ 2 for small scale yielding. It follows from Fig. 3a also, that for large scale yielding and deep cracks, these dependences begin to differ each from other at J=sY ¼ 0:5 mm and this difference is very small up to J=sY ¼ 1:0 mm (that corresponds to KI ¼ 360 MPa m at sY ¼ 600 MPa): values of CTOD for these two curves differ less than 4%.
Fig. 3. The dependence of CTOD on J=sY for various values of b and a=W : SSY, small-scale yielding; LSY, large-scale yielding [6]; sY ¼ 600 MPa.
Thus, it may be taken that the biaxial loading does not practically affect KJC for small scale yielding and for large scale yielding and deep cracks. As seen from Fig. 3b and c, for shallow cracks and large scale yielding, the difference between the dependences CTODðJ=sY Þ at b ¼ 0 and b ¼ 2 is very large and increases as the crack length decreases. Hence, for this case the biaxial loading affect KJC [6]. Estimate the value of the parameter mv presented in the form similar to Eq. (5): mv ¼
2:55asY E J 1 2 n 2 sY
ð10Þ
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The biaxial loading effect on fracture toughness is observed for cases for which the difference between the dependences CTODðJ=sY Þ at b ¼ 0 and b ¼ 2 is observed. It follows from Fig. 3b and c, that at a=W ¼ 0:1 (W ¼ 100 mm, a ¼ 10 mm), these dependences begin to differ each from other at J=sY ¼ 0:1 mm, and at a=W ¼ 0:05 (W ¼100 mm, a ¼ 5 mm)—at J=sY < 0:05 mm. Substituting a ¼10 mm, J=sY ¼ 0:1 mm, E ¼ 2·105 MPa, sY ¼ 600 MPa and n ¼ 0:3 in Eq. (10), we obtain mv ¼ 0:69: The same result is obtained at a ¼ 5 mm and J=sY < 0:05 mm. Thus, it may be taken that the biaxial loading effect on fracture toughness has to be taken into account at mv # 0:7: Taking into account that for shallow cracks value of J-integral is restricted by the critical value JC ¼ ð1 2 n2 Þ
ðvKIC Þ2 ; E
we substitute the value of J ¼ JC in Eq. (10), then the value of mv may be calculated by the following formula: 8a mv ¼ vKIC 2 p sY
ð11Þ
Parameter mv in Eq. (11) may be presented with regard for Eqs. (5) and (11) as: mv ¼ m=v2
ð12Þ
To estimate the b parameter effect on KCp ; we use the calculated dependence of vb on b obtained pffiffi in Ref. [6] (in Ref. [6] v is designated as a and b pffiffi a ¼ ððKJClb–0 Þ=ðKJClb¼0 ÞÞ). The linear Eq. (8) may be used to approximate data presented in Ref. [6]. As parameters KJC and KCp are identical in a physical sense, the biaxial loading effect on fracture toughness of component of RPV with shallow cracks may be presented as Eqs. (7) and (8).
Bi ðTÞlPf ¼P f is the fracture toughness of a specimen where KIC with thickness Bi for fracture probability P f : According to the Master curve conception [3 – 5] the condition (14) may be calculated by the formula: 1=4 B Bi KIClP ¼P ¼ ðK IC 2 Kmin Þ þ Kmin ; ð15Þ f f Bi
where K IC is some reference value of fracture toughness for a given specimen thickness B (for example B ¼ 25 mm) and a given fracture probability P f (for example P f ; 0:05), Kmin ; minimum fracture toughness. If the considered crack is shallow, i.e. its depth a # 0:15S; then fracture toughness K IC in Eq. (14) should be substituted by vb vK IC : 5.2. Formulation of condition of brittle strength for heterogeneous distribution of SIF along the crack front As an example consider a surface semi-elliptical crack. It will be taken that for each moment of a loading, KI ðLÞ; TðLÞ (see Fig. 4) are known (L; curvilinear co-ordinate). Consider a part of the crack front dL; on which SIF and temperature are taken to be constant and to be equal to KI ðLÞ and TðLÞ respectively. According to the Master curve concept [3 – 5]: " !4 # KI ðLÞ 2 Kmin dL ; ð16Þ Pf ¼ 1 2 exp 2 K0dL ðLÞ 2 Kmin where K0dL ¼ K0dL ðTðLÞÞ; scale parameter for the crack front length dL and temperature T ¼ TðLÞ: On the basis of [4] it may be written: 1=4 dL KIC ðLÞ 2 Kmin B ¼ ; ð17Þ dL K IC ðLÞ 2 Kmin dL where KIC ðLÞ; fracture toughness of a specimen with thickness dL at Pf ¼ P f and temperature T ¼ TðLÞ: On the basis of equation from Ref. [3] it may be deduced: dL 2 Kmin Þ4 ðK0dL ðLÞ 2 Kmin Þ4 ½2lnð1 2 P f Þ ¼ ðKIC
ð18Þ
5. Analysis of brittle strength of RPV component with a crack in a probabilistic statement 5.1. Formulation of condition of brittle strength for the homogeneous distribution of stress intensity factor (SIF) along the crack front As a condition of brittle strength it is taken that: Pf , P f ;
ð13Þ
where Pf ; fracture probability, P f ; a given level of fracture probability. For a homogeneous distribution of the SIF, condition (13) is equivalent to the following condition: Bi ; KI , KIClP ¼P f
f
ð14Þ
Fig. 4. For the calculation of brittle strength of a reactor pressure vessel for heterogeneous distribution of SIF along the crack front.
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Solving Eqs. (17) and (18) simultaneously we have: ðK0dL ðLÞ 2 Kmin Þ4 ½2lnð1 2 P f Þ B ¼ ðK IC ðLÞ 2 Kmin Þ4 dL
ð19Þ
Substituting Eq. (19) into Eq. (16) we have: PdL f
½lnð1 2 P f ÞðKI ðLÞ 2 Kmin Þ4 dL ¼ 1 2 exp K IC ðLÞ 2 Kmin Þ4 Bð
! ð20Þ
Consider various loading histories for a cracked component. As the loading parameter, time t is taken that is appropriate for the analysis of the PTS condition. As seen from Eq. (20), for values KI and T; which vary arbitrarily in time (and hence, KIC ), the dependence of the probability PdL on time may be non-monotonic. This f contradicts the physical meaning of cumulative probability as this parameter may only increase. Hence, Eq. (20) is not always valid and the brittle strength condition has to be reformulated allowing for this circumstance. First of all, consider isothermal monotonic loading. For this case, K IC ðLÞ ¼ const ¼ K IC and the parameter KI for each part of the crack front dL is a monotonically increasing function of time. Then, according to Eq. (20), the dependence of the probability PdL on time is a monof tonically increasing function too. Hence, Eq. (20) may be used for this case of loading. When using the weakest link theory, the probability of brittle fracture of a component with a crack of the front length B may be calculated by equation: ! lnð1 2 P f Þ ðB KI ðLÞ 2 Kmin 4 Pf ¼ 1 2 exp ð21Þ dL B 0 K IC 2 Kmin 4
Fig. 5. Possible dependence of KI on t under non-monotonic loading and regions in which fracture is possible and impossible.
calculated by Eq. (20) as [18]: lnð1 2 P f ÞZ dL dL max Pf ð0; tÞ ¼ 1 2 exp B
ð24Þ
where Z ¼ max að0; tÞ: When using Eq. (24) and making transformations similar as for the derivation of Eq. (22), we have the brittle strength condition for non-isothermal non-monotonic loading [18]: 1 ðB Z dL , 1 B 0
ð25Þ
Now, consider non-isothermal non-monotonic loading. As a common case, for this loading for each part of the crack front dL; the parameter
When KI varies non-monotonically, a phenomenon of preloading has to be considered (Fig. 5), which is similar to a so-called warm pre-stressing (WPS). After WPS, brittle fracture of a cracked component is known to happen at a KI value never less than the value of KI under pre-stressing, i.e. over range from t1 to t2 (see Fig. 5), brittle fracture does not happen. From a probabilistic point of view, over the above interval, PdL f ¼ const. In order to take into account this phenomenon, for nonisothermal non-monotonic loading, the parameter Z has to be calculated as follows: Z ¼ max að0; tÞ; where the maximum value of a is found over a time range where for each time moment t; the condition KI ðtÞ $ max KI ð0; tÞ is satisfied. (For illustration, in Fig. 5, these time ranges are hatched.) If it is necessary to take into account the shallow crack effect and biaxial loading effect, K IC in Eqs. (22) and (23) should be substituted by vb vK IC :
KI ðLÞ 2 Kmin 4 a¼ K IC ðLÞ 2 Kmin 4
6. Conclusion
Then, on making some transformation, the brittle strength condition Pf , P f may be written as [17,18]: 1 ðB KI ðLÞ 2 Kmin 4 dL , 1 B 0 K IC 2 Kmin 4
ð22Þ
ð23Þ
may vary arbitrarily. It follows from Eq. (20) that the parameter PdL f varies in the same manner. It is clear that the cumulative probability for the considered part of crack front dL cannot decrease and, at time t; it is determined by the maximum values of PdL f over a time range from 0 to t which is calculated by Eq. (20) and dL is denoted as max PdL f ð0; tÞ: The value maxPf ð0; tÞ is
1. A condition of brittle strength is formulated in a probabilistic statement for a RPV with a crack-like flaw. As the condition of brittle strength it is taken that Pf , P f ; where Pf ; fracture probability, P f ; a given level of fracture probability. Formulation of this condition is based on the weakest link model and takes into account the variation of KI and KIC along the crack front.
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2. Dependencies are proposed which take into account the shallow crack effect and the biaxial loading effect on fracture toughness of RPV steels. 3. Using approaches presented in the present paper allows one to decrease conservatism and to increase adequacy of evaluations of brittle strength of RPVs. Now these approaches have been included in the Russian Standard on evaluation of brittle fracture of WWER-type RPVs.
References [1] Weibull WA. A statistical theory of strength of materials. R Swed Inst Engng Res 1939;15:5–45. [2] Bolotin VV. Statistical methods in structural mechanics. M.: Stroyizdat; 1961 (in Russian). [3] Wallin K. The scatter in KIC result. Engng Fract Mech 1984;19: 1085–95. [4] Wallin K. The size effect in KIC results. Engng Fract Mech 1985;22: 149–63. [5] ASTM E 1921-97. ‘Test method for determination of reference temperature, T0 ; for ferritic steels in the transition range’ In: annual book of ASTM standards. [6] Margolin BZ, Kostylev VI. Analysis of biaxial loading effects on fracture toughness of reactor pressure vessel steels. Int J Press Vessel Piping 1998;75:589–601. [7] Aurich D, et al. The influence of multiaxial stress states on characteristic parameters for cleavage fracture in the elasticplastic range. Proc Int Conf Appl Fract Mech Mater Struct 1984; 345–56. [8] McAfee WJ, Bass BR, Pennell WE, Bryson JW. Analyses and evaluation of constraint models. USNRC Report NUREG/CR-4219 (ORNL/TM-9593/V12 and N1); 1996. p. 16 –24.
[9] McAfee WJ, Bryson JW. Test and analyses of unclad finite-length flaw cruciform specimens. USNRC Report NUREG/CR-4219 (ORNL/TM-9593/V12 and N1); 1996. p. 25–31. [10] Pennell WE, Corwin WR. Reactor pressure vessel structural integrity research. Nucl Engng Des 1995;157:159 –75. [11] Pennell WE, McAfee WJ, Bryson JW, Bass BR. Shallow-flaw fracture technology development. Transactions of the 14th International Conference on SMIRT, Lyon, France, vol. 4.; 1997. p. 1–11. [12] O’Dowd NP, Shih CF. Family of crack-tip fields characterized by a triaxiality parameter: Part I. Structure of fields. J Mech Phys Solids 1991;39:989–1105. [13] O’Dowd NP, Shih CF. Family of crack-tip fields characterized by a triaxiality parameter: Part II. Fracture applications. J Mech Phys Solids 1992;40:939 –63. [14] Bass BR, et al. Constraint effects in heavy-section steels, Proceedings Of the Joint IAEA/CSNI Specialists Meeting on Fracture Mechanics Verification by Large-scale Testing. USNRC Report NUREG/CP0131 (ORNL/TM-12413); October 1993 pp. 495–530. [15] Parton VZ, Morozov EM. Elastic-plastic fracture mechanics. Alma Alta: Nauka; 1974. in Russian. [16] Anikovskiy VV, Ignatov VA, Timofeev BT, Filatov VM, Chernaenko TA. Analysis of flaws sizes in weld reactor pressure vessels for energetic equipment and their influence on fracture resistance. Voprosy Sudostroenia, ser. Svarka. Vol. 34, TSNII RUMB, (in Russian). pp. 17 –32. [17] Margolin BZ, Rivkin EYu, Karzov GP, Kostylev VI, Gulenko AG. New method for calculation of brittle fracture resistance of RPV. Proceedings VI International Conference: Material Issues in Design, Manufacturing and Operation of Nuclear Power Plants Equipment, Saint-Petersburg, Russia, vol. 2.; 2000. pp.100-119. [18] Margolin BZ, Rivkin EYu, Karzov GP, Kostylev VI, Gulenko AG. New approaches for evaluation of brittle strength of reactor pressure vessels. Transactions of the 17th International Conference on Structural Mechanics in Reactor Technology (SMiRT 17), # G01-4, Prague, Czech Republic; 2003.
Analysis of structure integrity of RPV on the basis of brittle fracture criterion: new approaches George Karzova,*, Boris Margolina, Eugene Rivkinb a
Central Research Institute of Structural Materials Prometey, Shpalernaja, St Petersburg, Russian Federation b Research and Development Institute of Power Engineering, Moscow, Russian Federation
Abstract Based on the Master curve conception, the condition for fracture toughness is formulated for a heterogeneous distribution of stress intensity factor along a crack front and non-monotonic, non-isothermal loading. This formulation includes an elaborated procedure for taking into account the effects of shallow cracks and biaxial loading on fracture toughness. q 2004 Elsevier Ltd. All rights reserved. Keywords: Brittle fracture; Fracture probability; Fracture toughness; Brittle strength of reactor pressure vessels; Shallow crack effect; Biaxial loading effect; Weakest link model
1. Background
2. Introduction
The problems of substantiation and assuring nuclear power plant (NPP) equipment integrity were always the main attention in the scientific and organizing activities of Myrddin Davies. He believed that one of the main tasks in this field would be the consolidation of specialists’ efforts in various countries for the creation of a common understanding of approaches towards calculated assessments of safe operation time of the main components of NPPs. At the same time he always supported the creation and development of new progressive calculation procedures, within the framework of the general philosophy of assuring safety. He promoted the development and integration of European approaches for analysis of reactor pressure vessel (RPV) brittle fracture resistance in Russia. He assisted in the establishment of a mutual understanding between specialists of various countries in this area and gave not only scientific but also moral support to the authors of this work. He contributed much to the harmonization of the methods developed in Russia with the western approaches, and this provided (to a considerable degree) the formation of the general Russian-European position on the subject. The authors of the present article are very grateful to Myrddin Davies and devote this publication to his memory.
At the present time, when evaluating the resistance to brittle fracture for RPVs, the following considerations are used:
* Corresponding author. Tel.: þ 812-274-3796; fax: þ 812-274-1707. E-mail address: [email protected] (G. Karzov). 0308-0161/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijpvp.2004.03.001
1. Fracture toughness KIC which is dependent on temperature is used as a main characteristic of a material for evaluation of its resistance to brittle fracture. 2. It is taken that brittle fracture of the RPV does not happen if the condition: n·KI , KIC is fulfilled for each point on the crack front (where n— safety margin, KI —stress intensity factor (SIF)). Besides, it is assumed that parameter KIC is a deterministic parameter, and does not depend on flaw size. At the same time, it is well known that brittle fracture is a stochastic process, and may be described by the weakest link model [1 – 5]. As a result, the parameter KIC depends on flaw size and, in particular, on the crack front length. It should be also taken into account that fracture toughness may vary for various load biaxialities [6 – 11] and for various crack depths (a phenomena of shallow cracks) [12 –14]. In the present paper, main approaches are presented on the basis of which a new procedure for evaluation of brittle strength of RPV have been elaborated.
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3. Estimation of the shallow crack effect on fracture toughness For plate with a central crack under uniaxial tension, the fracture energy density 2g as a function of crack length a and the size of plastic zone near the crack tip is given by equation [15]: 2g ¼ sY dC ½m·expð1=mÞ· pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi arccosðexpð21=mÞÞ 1 2 expð22=mÞ 2 1;
ð1Þ
where sY ; the yield strength, dC ; the critical crack tip opening displacement, a ð2Þ m¼ ; c pE dC ; ð3Þ c¼ 8ð1 2 n2 ÞsY
n; Poisson’s ratio, E; elastic modulus, a; crack length. It follows from Eq. (1) that the fracture energy density is not a material constant and only for m ! 1 (small loads and big crack length) the fracture energy density is a constant of a material, which is equal to sY dC : Taking into account that sY dC ¼ ð1 2 n2 Þ
2 KIC E
and assuming 2g ¼ ð1 2 n2 ÞðKC2 =EÞ; we obtain [17]: KC ¼ vKIC ;
ð4Þ
where KC ; fracture toughness for specimen with shallow crack, v; coefficient which takes into account the shallow crack effect: v ¼ ½m expð1=mÞarc cosðexpð21=mÞÞ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 expð22=mÞ 2 11=2 : The dependence of v on parameter m is shown in Fig. 1. According to Eqs. (2) and (3) we have: m¼
8a pðKIC =sY Þ2
ð5Þ
Thus, fracture toughness for shallow cracks may be calculated by Eq. (4). Applicability of Eq. (4) for RPV materials of WWER was verified by using data of CRISM
Fig. 2. Comparison of the calculated and experimental data: 1,3–treatment of experimental data (W ¼100 mm), 2, calculation by Eq. (4); 1,2, KC ; 3, KIC :
‘Prometey’ published in Ref. [16]. In Ref. [16] 2.5Cr – Mo – V RPV base metal (for WWER-440) in various states was studied. Specimens with sizes W ¼100 mm, B ¼50 mm and the crack length a ¼ 10; 25 and 50 mm were tested by threepoint bending, where W; specimen width, B specimen thickness. In Fig. 2 the fracture toughness versus temperature curves are presented, which were obtained experimentally in Ref. [16] and calculated by Eq. (4). As seen from Fig. 2, the calculated curve predicts an increase of fracture toughness values for shallow cracks, but these values are less than experimental values. Thus, in spite of clear theoretical simplifications accepted for deducing Eq. (4), this equation may be used for engineering estimations of fracture toughness for shallow cracks as it gives conservative results. It should be also noted that, according to Dowd and Bass [12 –14], an increase in fracture toughness for specimens with shallow cracks is realized mainly at a=W ¼ 0:1 to 0.2. Taking into account that for a RPV with wall thickness S; W ¼ S; the following relation may be taken for estimation of the shallow crack effect on fracture toughness: ( KC ¼
KIC ;
at a . 0:15S
vKIC ; at a # 0:15S
ð6Þ
4. Estimation of the biaxial loading effect on fracture toughness
Fig. 1. The dependence of parameter v on m:
It is known that under normal operation of the RPV, and also under pressurised thermal shock (PTS) loading, the RPV experiences biaxial loading. In this connection, the problem arises how to estimate the biaxial loading effect on fracture toughness. It was shown in [6 – 11] that biaxial loading may result in decreasing fracture toughness. This
G. Karzov et al. / International Journal of Pressure Vessels and Piping 81 (2004) 651–656
653
decrease is revealed for specimens with shallow cracks and large-scale yielding [6]. For small-scale yielding or deep cracks, biaxial loading does not practically affect fracture toughness [6]. Thus, the biaxial loading effect on fracture toughness has to be taken into account for shallow cracks and large-scale yielding. Hereafter the procedure for determination of this parameter will be considered in detail. The following procedure may be proposed for an estimation of the biaxial loading effect on fracture toughness [17,18]. For a # 0:15 S and mv # 0:7: KCp ¼ vb vKIC ;
ð7Þ
where KCp ; fracture toughness for specimen with regard for the biaxial loading effect, vb ; coefficient which takes into account the biaxial loading effect on fracture toughness. Coefficient vb is calculated by equation:
vb ¼ 1 2 0:1b;
ð8Þ
where b; the load biaxiality ratio, i.e. ratio of maximum nominal stress parallel to the crack plane and acting along the crack front to maximum nominal stress perpendicular to the crack plane. Eq. (7) is valid when b varies from 0 up to 2. For a . 0:15S or for mv . 0:7; the biaxial loading effect on fracture toughness may be neglected, i.e. it may be taken vb ¼ 1: Taking into account Eqs. (6) – (8) fracture toughness for biaxial loading of RPV may be determined by equation: ( at a . 0:15S KIC ; p KC ¼ ; ð9Þ vb vKIC ; at a # 0:15S ( 1; at mv . 0:7 vb ¼ ; 1 2 0:1b; at mv # 0:7 where mv ¼ m=v2 : The proposed procedure may be justified by results obtained in Ref. [16]. These results allow one to draw the following considerations. The biaxial loading effect on fracture toughness is observed for cases for which parameter b affects the dependence CTODðJ=sY Þ (here CTOD, the crack tip opening displacement, J – J-integral). If the dependence CTODðJ=sY Þ is invariant to b; then the biaxial loading does not affect KJC : The dependences CTODðJ=sY Þ at b ¼ 0 and b ¼ 2 are presented in Fig. 3 for small and large scale yielding. As seen from Fig. 3a, the dependence CTODðJ=sY Þ at b ¼ 0 coincides with CTODðJ=sY Þ at b ¼ 2 for small scale yielding. It follows from Fig. 3a also, that for large scale yielding and deep cracks, these dependences begin to differ each from other at J=sY ¼ 0:5 mm and this difference is very small up to J=sY ¼ 1:0 mm (that corresponds to KI ¼ 360 MPa m at sY ¼ 600 MPa): values of CTOD for these two curves differ less than 4%.
Fig. 3. The dependence of CTOD on J=sY for various values of b and a=W : SSY, small-scale yielding; LSY, large-scale yielding [6]; sY ¼ 600 MPa.
Thus, it may be taken that the biaxial loading does not practically affect KJC for small scale yielding and for large scale yielding and deep cracks. As seen from Fig. 3b and c, for shallow cracks and large scale yielding, the difference between the dependences CTODðJ=sY Þ at b ¼ 0 and b ¼ 2 is very large and increases as the crack length decreases. Hence, for this case the biaxial loading affect KJC [6]. Estimate the value of the parameter mv presented in the form similar to Eq. (5): mv ¼
2:55asY E J 1 2 n 2 sY
ð10Þ
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The biaxial loading effect on fracture toughness is observed for cases for which the difference between the dependences CTODðJ=sY Þ at b ¼ 0 and b ¼ 2 is observed. It follows from Fig. 3b and c, that at a=W ¼ 0:1 (W ¼ 100 mm, a ¼ 10 mm), these dependences begin to differ each from other at J=sY ¼ 0:1 mm, and at a=W ¼ 0:05 (W ¼100 mm, a ¼ 5 mm)—at J=sY < 0:05 mm. Substituting a ¼10 mm, J=sY ¼ 0:1 mm, E ¼ 2·105 MPa, sY ¼ 600 MPa and n ¼ 0:3 in Eq. (10), we obtain mv ¼ 0:69: The same result is obtained at a ¼ 5 mm and J=sY < 0:05 mm. Thus, it may be taken that the biaxial loading effect on fracture toughness has to be taken into account at mv # 0:7: Taking into account that for shallow cracks value of J-integral is restricted by the critical value JC ¼ ð1 2 n2 Þ
ðvKIC Þ2 ; E
we substitute the value of J ¼ JC in Eq. (10), then the value of mv may be calculated by the following formula: 8a mv ¼ vKIC 2 p sY
ð11Þ
Parameter mv in Eq. (11) may be presented with regard for Eqs. (5) and (11) as: mv ¼ m=v2
ð12Þ
To estimate the b parameter effect on KCp ; we use the calculated dependence of vb on b obtained pffiffi in Ref. [6] (in Ref. [6] v is designated as a and b pffiffi a ¼ ððKJClb–0 Þ=ðKJClb¼0 ÞÞ). The linear Eq. (8) may be used to approximate data presented in Ref. [6]. As parameters KJC and KCp are identical in a physical sense, the biaxial loading effect on fracture toughness of component of RPV with shallow cracks may be presented as Eqs. (7) and (8).
Bi ðTÞlPf ¼P f is the fracture toughness of a specimen where KIC with thickness Bi for fracture probability P f : According to the Master curve conception [3 – 5] the condition (14) may be calculated by the formula: 1=4 B Bi KIClP ¼P ¼ ðK IC 2 Kmin Þ þ Kmin ; ð15Þ f f Bi
where K IC is some reference value of fracture toughness for a given specimen thickness B (for example B ¼ 25 mm) and a given fracture probability P f (for example P f ; 0:05), Kmin ; minimum fracture toughness. If the considered crack is shallow, i.e. its depth a # 0:15S; then fracture toughness K IC in Eq. (14) should be substituted by vb vK IC : 5.2. Formulation of condition of brittle strength for heterogeneous distribution of SIF along the crack front As an example consider a surface semi-elliptical crack. It will be taken that for each moment of a loading, KI ðLÞ; TðLÞ (see Fig. 4) are known (L; curvilinear co-ordinate). Consider a part of the crack front dL; on which SIF and temperature are taken to be constant and to be equal to KI ðLÞ and TðLÞ respectively. According to the Master curve concept [3 – 5]: " !4 # KI ðLÞ 2 Kmin dL ; ð16Þ Pf ¼ 1 2 exp 2 K0dL ðLÞ 2 Kmin where K0dL ¼ K0dL ðTðLÞÞ; scale parameter for the crack front length dL and temperature T ¼ TðLÞ: On the basis of [4] it may be written: 1=4 dL KIC ðLÞ 2 Kmin B ¼ ; ð17Þ dL K IC ðLÞ 2 Kmin dL where KIC ðLÞ; fracture toughness of a specimen with thickness dL at Pf ¼ P f and temperature T ¼ TðLÞ: On the basis of equation from Ref. [3] it may be deduced: dL 2 Kmin Þ4 ðK0dL ðLÞ 2 Kmin Þ4 ½2lnð1 2 P f Þ ¼ ðKIC
ð18Þ
5. Analysis of brittle strength of RPV component with a crack in a probabilistic statement 5.1. Formulation of condition of brittle strength for the homogeneous distribution of stress intensity factor (SIF) along the crack front As a condition of brittle strength it is taken that: Pf , P f ;
ð13Þ
where Pf ; fracture probability, P f ; a given level of fracture probability. For a homogeneous distribution of the SIF, condition (13) is equivalent to the following condition: Bi ; KI , KIClP ¼P f
f
ð14Þ
Fig. 4. For the calculation of brittle strength of a reactor pressure vessel for heterogeneous distribution of SIF along the crack front.
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Solving Eqs. (17) and (18) simultaneously we have: ðK0dL ðLÞ 2 Kmin Þ4 ½2lnð1 2 P f Þ B ¼ ðK IC ðLÞ 2 Kmin Þ4 dL
ð19Þ
Substituting Eq. (19) into Eq. (16) we have: PdL f
½lnð1 2 P f ÞðKI ðLÞ 2 Kmin Þ4 dL ¼ 1 2 exp K IC ðLÞ 2 Kmin Þ4 Bð
! ð20Þ
Consider various loading histories for a cracked component. As the loading parameter, time t is taken that is appropriate for the analysis of the PTS condition. As seen from Eq. (20), for values KI and T; which vary arbitrarily in time (and hence, KIC ), the dependence of the probability PdL on time may be non-monotonic. This f contradicts the physical meaning of cumulative probability as this parameter may only increase. Hence, Eq. (20) is not always valid and the brittle strength condition has to be reformulated allowing for this circumstance. First of all, consider isothermal monotonic loading. For this case, K IC ðLÞ ¼ const ¼ K IC and the parameter KI for each part of the crack front dL is a monotonically increasing function of time. Then, according to Eq. (20), the dependence of the probability PdL on time is a monof tonically increasing function too. Hence, Eq. (20) may be used for this case of loading. When using the weakest link theory, the probability of brittle fracture of a component with a crack of the front length B may be calculated by equation: ! lnð1 2 P f Þ ðB KI ðLÞ 2 Kmin 4 Pf ¼ 1 2 exp ð21Þ dL B 0 K IC 2 Kmin 4
Fig. 5. Possible dependence of KI on t under non-monotonic loading and regions in which fracture is possible and impossible.
calculated by Eq. (20) as [18]: lnð1 2 P f ÞZ dL dL max Pf ð0; tÞ ¼ 1 2 exp B
ð24Þ
where Z ¼ max að0; tÞ: When using Eq. (24) and making transformations similar as for the derivation of Eq. (22), we have the brittle strength condition for non-isothermal non-monotonic loading [18]: 1 ðB Z dL , 1 B 0
ð25Þ
Now, consider non-isothermal non-monotonic loading. As a common case, for this loading for each part of the crack front dL; the parameter
When KI varies non-monotonically, a phenomenon of preloading has to be considered (Fig. 5), which is similar to a so-called warm pre-stressing (WPS). After WPS, brittle fracture of a cracked component is known to happen at a KI value never less than the value of KI under pre-stressing, i.e. over range from t1 to t2 (see Fig. 5), brittle fracture does not happen. From a probabilistic point of view, over the above interval, PdL f ¼ const. In order to take into account this phenomenon, for nonisothermal non-monotonic loading, the parameter Z has to be calculated as follows: Z ¼ max að0; tÞ; where the maximum value of a is found over a time range where for each time moment t; the condition KI ðtÞ $ max KI ð0; tÞ is satisfied. (For illustration, in Fig. 5, these time ranges are hatched.) If it is necessary to take into account the shallow crack effect and biaxial loading effect, K IC in Eqs. (22) and (23) should be substituted by vb vK IC :
KI ðLÞ 2 Kmin 4 a¼ K IC ðLÞ 2 Kmin 4
6. Conclusion
Then, on making some transformation, the brittle strength condition Pf , P f may be written as [17,18]: 1 ðB KI ðLÞ 2 Kmin 4 dL , 1 B 0 K IC 2 Kmin 4
ð22Þ
ð23Þ
may vary arbitrarily. It follows from Eq. (20) that the parameter PdL f varies in the same manner. It is clear that the cumulative probability for the considered part of crack front dL cannot decrease and, at time t; it is determined by the maximum values of PdL f over a time range from 0 to t which is calculated by Eq. (20) and dL is denoted as max PdL f ð0; tÞ: The value maxPf ð0; tÞ is
1. A condition of brittle strength is formulated in a probabilistic statement for a RPV with a crack-like flaw. As the condition of brittle strength it is taken that Pf , P f ; where Pf ; fracture probability, P f ; a given level of fracture probability. Formulation of this condition is based on the weakest link model and takes into account the variation of KI and KIC along the crack front.
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2. Dependencies are proposed which take into account the shallow crack effect and the biaxial loading effect on fracture toughness of RPV steels. 3. Using approaches presented in the present paper allows one to decrease conservatism and to increase adequacy of evaluations of brittle strength of RPVs. Now these approaches have been included in the Russian Standard on evaluation of brittle fracture of WWER-type RPVs.
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[9] McAfee WJ, Bryson JW. Test and analyses of unclad finite-length flaw cruciform specimens. USNRC Report NUREG/CR-4219 (ORNL/TM-9593/V12 and N1); 1996. p. 25–31. [10] Pennell WE, Corwin WR. Reactor pressure vessel structural integrity research. Nucl Engng Des 1995;157:159 –75. [11] Pennell WE, McAfee WJ, Bryson JW, Bass BR. Shallow-flaw fracture technology development. Transactions of the 14th International Conference on SMIRT, Lyon, France, vol. 4.; 1997. p. 1–11. [12] O’Dowd NP, Shih CF. Family of crack-tip fields characterized by a triaxiality parameter: Part I. Structure of fields. J Mech Phys Solids 1991;39:989–1105. [13] O’Dowd NP, Shih CF. Family of crack-tip fields characterized by a triaxiality parameter: Part II. Fracture applications. J Mech Phys Solids 1992;40:939 –63. [14] Bass BR, et al. Constraint effects in heavy-section steels, Proceedings Of the Joint IAEA/CSNI Specialists Meeting on Fracture Mechanics Verification by Large-scale Testing. USNRC Report NUREG/CP0131 (ORNL/TM-12413); October 1993 pp. 495–530. [15] Parton VZ, Morozov EM. Elastic-plastic fracture mechanics. Alma Alta: Nauka; 1974. in Russian. [16] Anikovskiy VV, Ignatov VA, Timofeev BT, Filatov VM, Chernaenko TA. Analysis of flaws sizes in weld reactor pressure vessels for energetic equipment and their influence on fracture resistance. Voprosy Sudostroenia, ser. Svarka. Vol. 34, TSNII RUMB, (in Russian). pp. 17 –32. [17] Margolin BZ, Rivkin EYu, Karzov GP, Kostylev VI, Gulenko AG. New method for calculation of brittle fracture resistance of RPV. Proceedings VI International Conference: Material Issues in Design, Manufacturing and Operation of Nuclear Power Plants Equipment, Saint-Petersburg, Russia, vol. 2.; 2000. pp.100-119. [18] Margolin BZ, Rivkin EYu, Karzov GP, Kostylev VI, Gulenko AG. New approaches for evaluation of brittle strength of reactor pressure vessels. Transactions of the 17th International Conference on Structural Mechanics in Reactor Technology (SMiRT 17), # G01-4, Prague, Czech Republic; 2003.