Modeling for ductile-to-brittle transition under ductile crack growth for reactor pressure vessel...

Modeling for ductile-to-brittle transition under ductile crack growth forreactor pressure vessel steels

B.Z. Margolin* , V.I. Kostylev

Central Research Institute of Structural Materials ‘‘Prometey’’, 49 Shpalernaja Street, St. Petersburg 193015, Russia

Received 26 October 1998; accepted 28 November 1998

Abstract

The local stress–strain state (SSS) near the crack tip is investigated by the finite element method in the finite strain statement (with regardto a change of the crack tip blunting) for both stationary cracks and crack growing by a ductile mechanism. Using the revealed particularitiesof SSS near the stationary and growing crack tips and the local cleavage fracture criterion the phenomenon of the ductile-to-brittle transitionis explained for reactor pressure vessel steels. The model is proposed to predict the amount of ductile crack extension preceding the ductile-to-brittle transition as a function of the test temperature. The procedure for calculation of the cleavage fracture toughness is also elaboratedwith regard to ductile crack extensions. Analysis of the obtained calculated results and available experimental data is made. Alternativeapproaches for the interpretation of the ductile-to-brittle transition are discussed.q 1999 Elsevier Science Ltd. All rights reserved.

Keywords:Ductile-to-brittle transition; Ductile crack extension; Reactor pressure vessel steels

1. Introduction

At present the phenomenon of the ductile-to-brittle tran-sition for cleavage fractures, over a particular range oftemperatures is preceded by a certain amount of ductilecrack extension (a crack grows by the mechanism of nuclea-tion, growth and coalescence of voids) may be considered tobe well known [1–10]. All the investigations devoted to theinterpretation of the ductile-to-brittle transition may bedivided into three directions. The first direction explainsthe ductile-to-brittle transition by an increase in the strainrate which occurs during the ductile growth of a crack. Thisexplanation appears to be qualitative, for example [10].Probabilistic approaches [3–5,8,9] may be considered asthe second direction. The essence of the probabilisticapproaches consists of the following. The ductile growthof a crack is known to occur at increasing stress in thenet-section of a specimen. By this, some characteristicvolume of a material near the crack tip, in which a brittlefracture may be initiated, increases. Hence, the probabilityof finding a weak link in the vicinity of the crack tipincreases. Thus, as the ductile crack grows the brittle frac-ture probability increases. It should be noted that the prob-abilistic models describe mainly the size scale factor, which

accompanies the ductile crack growth, no attention beinggiven for investigation of the physical processes occurringunder the ductile-to-brittle transition. According to work ofthe third direction the ductile-to-brittle transition isconnected with a change of stress state near the crack tipduring ductile crack growth [1,2,6]. A stress state near thecrack tip in the process of the blunting of the stationarycrack tip and at the growth of the ductile crack was consid-ered in Ref. [2]. The gradient of the stress near the stationarycrack tip decreases as a result of the crack blunting with anincrease in load. Further, when in the beginning of theductile crack growth a ligament between the bluntingcrack tip and voids ruptures the tip radius of the formedcrack appears considerably less than the blunting radiusbefore the crack starts. As a result the stress gradient nearthe tip of the crack which grows by the ductile mechanismbecomes higher than the stress gradient near the bluntingstationary crack. By this the peak of triaxialitys1/seq

(wheres1 is the maximum principal stress is the crack open-ing stress andseq, the equivalent stress) is the same for thestationary and the growing cracks (Fig. 1). As the triaxialitypeaks1=seq does not change, the authors of Ref. [2] explainthe ductile-to-brittle transition by a sharp increase of thestress triaxiality (the movement from a point A in a pointB (Fig. 1)) in the previously deformed material. It willbecome clear by the present study that the cause of theductile-to-brittle transition in reactor pressure vessel steelsis indeed connected with increasing stress in a zone of the

International Journal of Pressure Vessels and Piping 76 (1999) 309–317

IPVP 1923

0308-0161/99/$ - see front matterq 1999 Elsevier Science Ltd. All rights reserved.PII: S0308-0161(98)00135-5

* Corresponding author. Tel.:17-812-274-1101; fax:17-812-274-1707.

E-mail address:[email protected] (B.Z. Margolin)

effect of the crack blunting, i.e. in a zone in whichs1=seq

does not reach its peak. However, the explanation of theductile-to-brittle transition given in [2] does not allow itsprediction, as the ductile-to-brittle transition criterion is notformulated.

In Ref. [6] a model has been proposed which explains theductile-to-brittle transition in low strength carbon steels.This model is based on the universally accepted formulationof the brittle fracture criterion:s1 � Sc, where Sc is thecritical brittle fracture stress. It has been shown in Ref. [6]by finite element method (FEM) that the peak ofs1

increases a little during the ductile crack extension. In this

case the following situation is possible. If at some tempera-ture the peak ofs1 is less thanSc a crack begins to grow bythe ductile mechanism and for a certain amount of theductile crack extension the peak ofs1 becomes equal toSc when the ductile-to-brittle transition is observed.

As it has been shown in Refs. [11–13] the brittle fracturecriterion s1 � Sc does not describe the brittle fracture inreactor pressure vessel steels. In particular, it has beenshown for cracked specimens that the cleavage initiationsites were located in a zone where strain increased, whereasthe stress peaks had passed that location and the stressdecreased with increasing applied load. Hence, the modelof the ductile-to-brittle transition proposed in Ref. [6]cannot be used to analyze the behavior of cracks in reactorpressure vessel steels.

The purpose of the present article is to propose the deter-ministic model of the ductile-to-brittle transition in the reac-tor pressure vessel steels. The model is based on the analysisof the stress–strain state (SSS) near the tip of a ductilegrowing crack and an application of the local cleavage frac-ture criterion [11,12]. Over a range of the ductile-to-brittletransition temperatures the cleavage origins are located in azone in which the crack tip blunting affects SSS, and there-fore, the SSS calculation by FEM is carried out on the finitestrain statement.

2. The local criterion for cleavage fracture

The local criterion for cleavage fracture has been formu-lated as [11,12]

snuc ; s1 1 mT1seff $ sd; �1a�

s1 $ Sc�æ�; �1b�wheres1 is the maximum principal stress;mT1 the para-meter which may be interpreted as the concentrator coeffi-cient for the stress in the dislocation pile-up tip;seff � seq 2 sY the effective stress;seq the von Misesequivalent stress;sY the yield stress;sd strength of carbidesor another particles, on which the cleavage microcracks arenucleated; æ� R

d1peq Odqvist’s parameter;

d1peq�

��2=3�d1p

ij 2 d1pmdij ��d1p

ij 2 d1pmdij �

qthe equivalent plastic strain increment;d ij the Kroneckersymbol; 1p

m � 1pii =3; Sc the critical brittle fracture stress.

DependenceSc(æ) the is a monotonic increasing functionof æ [12]

Sc � �C1* 1 C2* exp�2Adæ��21=2; �2�

whereC1* ; C2* ; Ad are material constants.ParametermT1 may be represented asmT1 � mTm1,

wheremT andm1 are determined by the widthd sl and thelength Lsl of the dislocation pile-up:mT , 1=

��dslp

andm1 ,

��Lslp

. ParametermT depends only on temperatureTand decreases as the temperature increases. Parameterm1

B.Z. Margolin, V.I. Kostylev / International Journal of Pressure Vessels and Piping 76 (1999) 309–317310

Fig. 1. Distribution of parameters1/seq on the crack extension line [2]:solid line – before crack start; dashed line – after ductile crack extension.

Fig. 2. Distribution ofs1 and æ on the crack extension line (a stationarycrack is considered) in the moment of brittle fracture of cracked specimen(a) and the loading history of the nearest to the crack tip unit cell (b).

depends only on æ and decreases as æ increases. Accordingto Ref. [12]

mT � ksY ; �3a�

m1 � �C1* 1 C2* exp�2Adæ��1=2; �3b�wherek is the material constant.

Condition (1a) in the local cleavage fracture criterion isthe nucleation condition for cleavage microcracks, condi-tion (1b) is the propagation condition for the cleavagemicrocracks. Condition (1b) describes both the cleavagemicrocrack starts1 � S0 ; Sc�æ� 0� and the overcomingof the microcracks through various barriers (such as grainboundaries, cell boundaries, microstresses) existing andformed in a material during plastic deformation.

It has been shown [11,12] that the brittle fracture of thecracked specimens from reactor pressure vessel steels iscontrolled only by condition (1a) over a temperature rangefrom low to the brittle–ductile transition temperatures.Condition (1b) is satisfied forKI , KIC. Over the brittle–ductile transition temperature range condition (1a) is satis-fied at stress being less thanspeak

1 , i.e. for points which arelocated in a zone adjoining the crack tip (r # 2CTOD,where r is the distance from the crack tip, CTOD thecrack tip opening displacement). According to Ref. [12] acrack starts when the unit cell nearest to the crack tip isbroken. The SSS near the crack tip in this moment�KI �KIC� may be represented as a scheme in Fig. 2(a) in whichdistributions ofs1 and æ on the crack extension line areshown when conditions (1a) and (1b) are satisfied in the unitcell nearest to the crack tip (r � ruc/2, whereruc is the unitcell size). The loading history for the unit cell nearest to thecrack tip is shown in Fig. 2(b). At some strain æ� ænuc

condition (1a) is satisfied (snuc� sd), by this we haves1 . Sc. If the temperature increases the yield stresssY

decreases and as a results1 decreases. This results in thefollowing: at brittle–ductile transition temperatureTtr

conditions (1a) and (1b) are satisfied simultaneously (seepoint A in Fig. 3); at the temperatureT . Ttr when condi-tion (1a) is satisfied (point B), condition (1b) has not beensatisfied (s1 , Sc) and fracture of the unit cell may beductile only. Hereafter it will be shown that for the ductile

crack growth under certain conditions the inequalitys1 , Sc

may change sign and as a result cleavage fracture occurs.In Sections 3 and 4 a feature of the ductile crack growth

will be considered.

3. The ductile fracture criterion

The ductile fracture of the structural steels is known tooccur on the mechanism of voids nucleation on inclusions,voids growth and coalescence [14,15]. The ductile fracturecriterion may be formulated as [15]

dFeq

dæ� 0; �4�

whereFeq� �1 2 SS�seq, (SS is the relative area of voids).If under loadingsm=seq (here,sm is the hydrostatic stress) isa constant, the ductile fracture criterion (4) may be writtenas [14,15]

æ� 1f ; �5�where1 f is the critical strain of the ductile fracture whichdepends on the parametersm=seq. The ductile fracturecriterion formulation as in Eq. (5) appears to be now widelyused. Ifsm=seq changes during loading the critical strain1 f

depends on the loading history of a material, namely

1f � wsm

seq�æ�

!: �6�

It will be shown in Section 4 that the loading history for unitcells located on the crack extension line varies as a crackgrows, but this variation is small. Therefore the criticalstrain may be taken the same for all the unit cells locatedon the crack extension line. Hence, the ductile crack growthmay be simulated as a consecutive fracture of the unit cells.The fracture of the nearest to the growing crack tip unit cellhappens if the plastic strain of this unit cell reaches thecritical value �æ� 1f �. Such a modeling for the ductilecrack growth appears to be similar to modeling of McClin-tock [16] and Rice [17].

4. SSS analysis near the growing crack tip

The purpose of this section is to analyze SSS near the tipof the ductile growing crack. In these investigations it istaken that the ductile fracture of a material near the cracktip occurs when condition (5) is satisfied. The SSS analysiswas performed by solving the elastic–plastic problem in thefinite strain statement by FEM. Material deformation hasbeen described by the isotropic hardening scheme and thevon Mises criterion. Calculation was carried out for a planestrain condition. The stress–strain curve of a material isdescribed by the following equation:

seq� sY 1 A0æn; �7�

wheresY � 540 MPa,A0 � 596 MPa andn � 0.464 (the

B.Z. Margolin, V.I. Kostylev / International Journal of Pressure Vessels and Piping 76 (1999) 309–317 311

Fig. 3. Brittle–ductile transition condition for specimen with stationarycrack.

values of these parameters correspond to the properties of2.5Cr–Mo–V reactor pressure vessel steel atT � 08C).Four-point bend specimen with a crack was considered(a � 50 mm, W � 100 mm,L � 600 mm,c � 100 mm(Fig. 4)). The size of the finite elements near the crack tipequals to 0.005 mm.

Simulation of the crack growth was performed by thefollowing procedure. In the first stage a specimen with astationary crack was loaded up toJ � Jc (Jc � 220 N/mm,this value corresponds to the upper shelf value for 2.5Cr–Mo–V steel [18]). Then forJ� Jc a distancerc�rc � ruc=2�from the blunting tip of a stationary crack was determinedfor which condition æur�rc

� 1f (Fig. 5(a)) was fulfilled. Inthe second stage a crack growth was simulated by a noderelease technique. An external load acting on a specimenwas fitted in order to the condition æur�rc

� 1f (Fig. 5(b))was fulfilled on the distancer � rc from the growing cracktip. A value of the crack jumpDr varied and has equalled torc/n (parametern varied from 1 to 10). For the performedcalculations a value of the critical strain1 f was varied.Three values of1 f were considered: 0.22, 0.46 and 0.72.

The following regularities of SSS near the crack tip wererevealed as a result of the performed calculations.

1. The points locating near the crack tip on the crack exten-sion line have the same loading historys1=seq�æ� up tothe moment of the crack start (see stationary crack curvein Fig. 6). This conclusion explains the self-similarity ofthe stress and strain fields near the stationary crack tipand is in agreement with Refs. [19,20].

2. When a crack grows the points locating on the crackextension line have the different loading historys1=seq�æ�. The dependencess1=seq�æ� for points

locating on the crack extension line during a process ofcrack growth are represented in Fig. 6. The values of thecrack extensionDa shown in Fig. 6 correspond to condi-tion æur�rc

� 1f (the calculation results are shown for1 f�0.72 andrc� 0.025 mm). It is seen from Fig. 6 the peakof stress triaxialitys1=seq does not practically change fordifferent curves corresponding to different values ofDa.Over a range æ$ 0:034 0:04 in which the crack blunt-ing affects SSS parameters1=seq increases monoto-nously asDa increases, the rate of stress triaxialitychange decreases. It should be noted that the calculationresults do not practically depend on the crack jump value


Fig. 4. Sketch of the cracked specimen and its loading taken for calculationby FEM.

Fig. 5. Simulation for crack start (a) and crack growth (b).

Fig. 6. Loading history of points located on the crack extension line duringthe crack growth.

Fig. 7. Shape of the crack tip for various values of the ductile crackextensionDa.

Dr. The increase of the stress triaxiality is caused byvarious shape of the tip for stationary and ductile grow-ing crack (Fig. 7). The decrease of the rate of stresstriaxiality change appears to be because some stationarylocal SSS near the growing crack tip is established.

In order to characterize a change of the stress state theparameterV is introduced as

V�Da� ;s1

seq�Da�2

s1

seq�Da� 0�

" #uæ�1f

� Ds1

sequæ�1f

:

�8�The calculated dependenciesV (Da) are represented in Fig.8 for various values of1 f. It is seen from Fig. 8 that depen-denceV (Da) appears to have an asymptote, i.e. parameterV reaches the maximum value forDa q CTODc (hereCTODc is the crack tip opening displacement atJ � Jc).The maximum value ofV depends on the critical strain1 f; the dependence of max(V ) on 1 f being not monoto-nously. This regularity may be explained consideringscheme shown in Fig. 9. Assume, that the critical strain1f ! ∞, then rc ! 0, i.e. the considered point is locatednear the crack tip surface where the condition of the planestress state is satisfied. Under this condition the maximumvalue of s1=seq is known to be equal to 2=

��3p

both forstationary and growing cracks. Hence, max�V� ! 0 for1f ! ∞. For small values of1 f the loading histories forpoints both for stationary and growing cracks differ insig-nificantly (Fig. 6). Evidently that max(V )! 0 for this casealso. Thus, it is clear that the dependence max(V ) on1 f has

the extreme (maximum) value for average values1 f. Inorder to keep the subsequent analysis as simple as possible,the calculated dependencesV (Da) were approximated byequation

V � a�1 2 exp�2bDa��; �9�wherea � 0.12,b � 17 for1 f � 0.22;a � 0.15,b � 11 for1 f � 0.46;a � 0.11,b � 15 for 1 f � 0.72 (see Fig. 8).

5. Prediction of the ductile-to-brittle transition

5.1. Scheme for the prediction of the ductile-to-brittletransition

The scheme for the prediction of the ductile-to-brittletransition (Fig. 10) is based on the analysis of loadinghistory for the nearest to the crack tip unit cell (seeFigs. 3 and 8). This scheme allows one to determine theductile crack extensionDa being necessary for arising


Fig. 8. DependencesV on Da for various1 f: point – FEM calculation;solid line – approximation by Eq. (9).

Fig. 9. Scheme for evaluation of max(V ).

Fig. 10. Scheme for prediction of ductile-to-brittle transition and evalua-tion of relationDa(T).

the ductile-to-brittle transition. In the interest of calcula-tion simplicity, we take that ænuc < 1f . For the ductile-to-brittle transition to realize it is then necessary that inthe nearest to the crack tip unit cell the conditions1 �Sc is satisfied for æ� 1f .

The dependenciess1(æ) are represented in Fig. 10(a) forthe unit cell being the nearest to the stationary crack tip,which were calculated for two temperaturesT1 andT2 beinghigher than the brittle–ductile transition temperatureTtr. Asshown earlier, atT . Ttr the crack start and growth occur bythe ductile mechanism. For the ductile-to-brittle transitionto be realized it is necessary that the stressess1 for æ� 1 f

increase onDs�1�1 at T� T1 and onDs�2�1 at T� T2. For thisthe crack length must increase onDa1 andDa2 respectively(Fig. 10(b)). Hence, when using the scheme represented inFig. 10(a) and (b) relationDa(T) may be constructed (seeFig. 10(c)). Thus, in order to evaluateDa(T) it is necessaryto know

• the dependenceSC(æ);• the dependencess1(æ) for stationary crack at various

temperatures;• the dependenceV (Da) for the value1 f.

The calculation of dependences1(æ) by FEM for varioustemperatures is very intensive. Therefore, it is expedient touse the approximated analytical solutions. The analyticalmethod for calculation of SSS near the stationary crack tiphas been proposed in Refs. [21,22]. This method takes intoaccount the crack tip blunting which changes under loading.In Section 5.2 the main relations of the proposed methodwill be briefly considered.

5.2. Method for SSS calculation near the crack tip

According to Refs. [12,21,22] the approximated solutionhas been obtained by using the following statements:

• Elastic–plastic material is considered.• SSS analysis is performed for a case of plane strain.• The stress–strain curve is approximated by Eq. (7).• The crack blunting change under loading is taken into

account.

According to Refs. [12,21,22] we consider a crack withthed blunting (the blunting radiusr � d /2) under loadingon mode I. For the equivalent stressseq and strain1eq thefollowing equation may be written according to Refs.[22,23]

seq1eq� seeq1

eeq �10�

where seeq and 1e

eq are the equivalent stress and strain,respectively, according to the elastic solution for problemabout SSS near the crack tip.

The distribution of the normal to the crack line stressse1

near the crack tip with the bluntingd is expressed by Crea-ger’s approximation [24]

se1 � KI��

2prp f �~r�; f �~r� � 1 1

11 1 2~r

� � ��1

1 1 �1=2~r�

s;

~r ;rr;

�11�whereKI is the stress intensity factor;r of the distance fromthe tip of blunted crack. It should be underlined that inregion adjoining to the crack tip, Creager’s approximation(11) gives values ofse

1 coinciding with those for notchhaving a curvature radiusr and in points remote from thecrack tip approximation (11) gives values ofse

1 for sharpcrack.

Assume, that the correlation ofseeq with se

1 for a crackwith bluntingd ± 0 may be written as

seeq�r� � se

1�r�qe�~r� ; �12�

whereqe�~r� is some function which is defined below.The average values�se

eq and �1eeq in the i-th unit cell of the

sizeruc are defined with regard to Eqs. (11) and (12) by thefollowing equations:

�seeq ;

1ruc

Ziruc

�i 2 1�ruc

seeq�r�dr

� 1ruc

KI��2pp

Ziruc

�i 2 1�ruc

f �~r�qe�~r�

��rp dr ; �13�


Fig. 11. Relationss1(æ) at various temperature calculated using methodrepresented in Section 5.2.

Fig. 12. DependencesDa on T for 2.5Cr–Mo–V steel: 1 – for yield stressequalsY; 2 – for yield stress equal 0.97sY; 3 – for yield stress equal1.03sY (sY is calculated by Eq. (22)).

�1eeq� 2�1 1 m�

3E�se

eq

� 2�1 1 m�3E

1ruc

KI��2pp

Ziruc

�i 2 1�ruc


��rp dr ; �14�

where E is the Young’s modulus andm the Poisson’sratio.

Below, the SSS parameters are considered as the valuesaveraged on the unit cell, and in the interest of simplifyingthe notation, the lines under corresponding values areomitted.

Using Eqs. (10), (13) and (14) we have forZ ; seq1eq

Z � b2JZiruc

�i 2 1�ruc


��rp dr

� �2

�15�

where b2 � 1=3p�1 2 m�r2uc; J � �1 2 m2�K2

I =E is the J-integral of Rice–Cherepanov.

Differentiating Eq. (15) with respect toJ atd � constant,we obtain

dZdJ� b2

Ziruc

�i 2 1�ruc


��rp dr

� �2

: �16�

Assume Eq. (16) to be correct at valued changing underloading and take into account the relationd � b3J=sY (forhardening materialsb3 � 0:41 [20]). Then integrating Eq.(16) we have

Z � b2

ZJ

0

Ziruc

�i 2 1�ruc


��rp dr

� �2

dJ; �17�

where ~r � 2rsY =b3J. According to Ref. [21] relationqe�~r�may be represented by a function

qe�~r� ��

2�1 2 m�2 1 m2 1 1

s

11

1 2 2m2

��2

�1 2 m�2 1 m2 1 1

s !�1

2 exp�2k~r�� 18�

in which k � 0.75.According to Ref. [22] loading of the material near the

crack tip is close to the radial one. Then æ< 1peq and

1eq� 1eeq 1 1p

eq, where according Hooke’s law:1e

eq� �2�1 1 m�=3E�seq. Hence, after determining thevalue Z according to Eq. (17) parametersseq; 1eq and æmay be determined by using Eq. (7).

For the knownseq and 1eq the principal stress compo-nents on the crack extension line may be found from thefollowing equations obtained on the base of theory for work-hardening plasticity, von Mises yield criterion and plane

strain condition [21]

s1 � seq

��2p ��1 2 �q�2 1 � �q 2 n�1 1 �q��2

1 �1 2 n�1 1 �q��2�21=2;

s2 � �qs1;

s3 � n�s1 1 s2�;

n � 0:5 2�1 2 2m�seq

3E1eq

" #= 1 1

�1 2 2m�seq

3E1eq

" #;

�19�

wheres1 is the stress perpendicular to the crack line,s2 thestress parallel to the crack line,s3 the stress out of the plane.In Eq. (19) the parameter�q is calculated as

�q� 1ruc

Ziruc

�i 2 1�ruc

q�r�dr : �20�

The dependenceq(r) has been obtained in [12] for work-hardening material. This dependence generalizes the knownformula derived by using the slip line theory [25]

q�r� � 1 20:85

0:851 ln�1 1 r=r� ; if r =r # 3:81

0:65; if r =r $ 3:81

:

8><>: �21�

5.3. Prediction of the ductile-to-brittle transition for reactorpressure vessel steels

In order to evaluate the conditions for which the ductile-to-brittle transition occurs in 2.5Cr–Mo–V steel we use theaforementioned proposed scheme (Section 5.1). Over thetransition temperature range parameters in Eq. (7) describ-ing the stress–strain curve for this steel may be takenaccording to Ref. [12] asA0 � 596 MPa,n � 0.464 andthese parameters do not practically depend on temperature.The relation of the yield stress on temperature may beapproximated as [22]

sY � �15262 400�exp�27:771× 1023�T 1 273��1 400;

�22�where units ofsY: MPa, andT: 8C. The relation of thecritical brittle fracture stressSc on æ is described by Eq.(2) with the following values of parameters: C1* � 2.26×1027 MPa22, C2* � 3.64 × 1027 MPa22, Ad � 1.32 [11].The unit cell sizeruc is taken to be equal to 0.05 mm. It wascalculated by the equations given in Section 5.2 that thecritical strain for ductile fracture1 f � 0.72 at the fracturetoughness valueJc � 220 N/mm. It was taken that thedependenceV�Da� is described by Eq. (9), wherea �0.11,b � 15 (see Fig. 8). For stationary crack the depen-dencess1(æ) calculated by equations represented in Section5.2 are shown in Fig. 11. The dependencesDa�T� calculatedby the procedure presented in Section 5.1 are shown in Fig.12. In Fig. 12 the curvesDa�T� are also given which areobtained for a material with the yield stress which differs on


^ 3% from the yield stress calculated by Eq. (22). As seenfrom Fig. 12 small variation of the yield stress results insignificant variation of value of the ductile crack extensionpreceding the cleavage fracture.

6. Discussion

The calculation results represented in Fig. 12 agree withexperimental data [7] in which it has been shown that:

• the value of ductile crack extension preceding the clea-vage fracture increases as temperature increases;

• the temperature range in which cleavage fracturehappens after the ductile crack extension is very narrowand is approximately equal to 508C;

• there exists a temperature above which cleavage fracturedoes not happen for any ductile extension of a crack.

According to the performed calculation the ductile-to-brittle transition for 2.5Cr–Mo–V steel is not observed attemperaturesT $ 508C. This conclusion agrees with theexperimental data of investigation [18] in which it hasbeen shown that atT $ 408C fracture of a cracked specimenoccurs by the ductile mechanism only.

As follows from Fig. 12, for some scatter of the yieldstress (this may take place under testing) a significant scatterof the ductile crack extensionDa may be observed at thesame temperature. This result agrees with the data in Refs.[4,7].

It should be noted that the calculated dependence in Fig.12 is correct only at temperatureT . Ttr. At T # Ttr theductile crack extension has to be absent as for the nearest tothe crack tip unit cell the criterion (1a) and (1b) is satisfied atDa� 0 (see Section 2). According to Ref. [12] the brittle–ductile transition temperatureTtr < 08C for 2.5Cr–Mo–Vsteel. As seen from Fig. 12, atT� 08C the calculation givessome valueDa ± 0. This connects with the assumption

ænuc� 1 f which was taken for simplicity in the calculationprocedure. Actually ænuc , 1 f and as temperature increases(T . Ttr) the value of ænuc begins to be comparable to1 f.Therefore the calculation ofDa becomes more exact as thetemperature increases (atT . Ttr). Besides, as seen fromFig. 12, the ductile crack extensionDa is sufficiently smallat T � Ttr (in order of the size ofruc, i.e. of the grain size)and such a error of calculation may be neglected.

The calculated results represented in Fig. 12 may be usedto evaluate cleavage fracture toughness with regard for theductile crack extension. The calculated curvesJR�Da�whichwere obtained by numerical simulation of the ductile crackgrowth using criterion æur�rc

� 1f are given in Fig. 13. Thecalculation was performed atT � 08C andT � 308C. J-integral was calculated for contour the size of which is ofthe order of 5CTOD, i.e. this size is larger than a size ofzone of the effect of the crack blunting with regard forincreasing this zone size during crack growth [20]. Asseen from Fig. 13, for various temperatureJR-curves coin-cide practically. Taking into account the relationDa�T�(curve 1 in Fig. 12) cleavage fracture toughnessJcl

c maybe calculated at any temperature. For example, at tempera-tureT� T* the ductile-to-brittle transition happens forJ �Jcl

c and as a result unstable (brittle) fracture occurs (see Fig.13). This instability is not to be connected with an instabilitywhich is observed under crack growth by the ductilemechanism and described as [17]

2J2a

� �P� dJR

da�23�

where P represents load. The instability of crack growthunder the ductile-to-brittle transition cannot be describedby Eq. (23). This may be explained by the following exam-ple. According to Fig. 13 atT � 208C the instability pointcorresponds toDa � 0.14 mm andJcl

c � 335 N/mm. Bythis 2J=2a � 56 N/mm2, dJR=da � 803 N/mm2, i.e.2J=2a p dJR=da. Hence, Eq. (23) predicts an absent of theinstability atT � 208C.

It should also be noted the following. As follows from Eq.(23), if JR-curve does not vary as temperature varies then theunstable crack growth occurs at the same value ofDa.However it is seen from Fig. 13 that for the ductile-to-brittletransition Da varies significantly with the variation oftemperature.

Consider once more explanation for the ductile-to-brittletransition given in Ref. [2] proceeding from the modifiedcriterion of brittle fracture (Section 2) and the scheme repre-sented in Section 5.1. As follows from Section 5.1 and Fig.10 the ductile-to-brittle transition happens when at plasticstrain æ� 1 f stress increases onDs1 owing to the ductilecrack extension and the conditions1� Sc is satisfied. Such asituation may be shown as the transition from point A topoint B in Fig. 1. Thus, the authors of Ref. [2] have correctlynoted that the stress fields near the crack tip are different upto and after the ductile crack growth and that the variation ofstress fields is the basic reason of the ductile-to-brittle


Fig. 13. The evaluation of cleavage fracture toughnessJclc after some

ductile crack extension.

transition. At the same time it should be noted that in Ref.[2] a qualitative model of the ductile-to-brittle transition hasbeen only proposed. In order to have a quantitative modelthe modified criteria (1a) and (1b) of cleavage fracture andthe presented above regularities of SSS should be used.

7. Conclusions

1. The cleavage fracture criterion proposed in Ref. [11]allows one to predict the ductile-to-brittle transitionafter some amount of the ductile crack extension.

2. Numerical investigation of SSS has shown that the peakof the stress triaxialitys1/seq is practically the same forboth stationary and growing crack. In a zone of the crackblunting effect the stress triaxiality is larger for a growingcrack than for stationary one and the stress triaxialityincreases up to some limit value as the ductile crackextension increases.

3. The procedure was elaborated for prediction of the valueof the ductile crack extension after which the ductile-to-brittle transition is observed.

4. The procedure for cleavage fracture toughness predictionwith regard for the ductile crack extension was elabo-rated.

5. Small variation of the yield stress results in significantvariation of value of ductile crack extension precedingthe cleavage fracture.

References

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[8] Amar E, Pineau A. Application of a local approach to ductile-brittletransition in a low-alloyed steel. Nuc. Eng. Des. 1987;105:89–96.

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