SUPER-DUPLEX STAINLESS STEEL: A CASE STUDY OF

INCORPORATING ANISOTROPIC MATERIAL PROPERTIES

INTO RELIABILITY ASSESSMENTS

N. C. Renton, W. F. Deans & M. J. Baker.

University of Aberdeen, U.K.

ABSTRACT

Tubular components used in high-pressure, high-temperature (HT-HP) oil wells require materialswith robust mechanical properties and high corrosion resistance to withstand the in-service conditions.A number of applications have made use of super-duplex stainless steel. The material has a highlyanisotropic microstructure and as a result its in-service performance depends on the orientation of theapplied loading. This study examines the failure mode of ductile fracture from a pre-existing crackand the application of probabilistic structural reliability techniques to the problem. The material’selastic-plastic fracture behaviour in two orientations is investigated using standard test specimens withdiffering crack depths. The results show that the measured Ji is geometery dependent and that SDSShas a higher crack resistance for radial flaws. The limit state function for the tangency condition ofthe R-curve for an axial flaw under hoop stress is developed, and the experimental results analysed toquantify the crack resistance term. The results can be used to carry out an assessment of the probabilityof tubular failure by ductile fracture.

1. INTRODUCTION

The development of high-temperature, highpressure (HT-HP) oil wells has seen the appli-cation of high strength corrosion resistant alloysin the manufacture of downhole casing andtubing components to cope with the combineddemands of highly corrosive production fluids,and large applied hoop and tensile stresses. Onesuch material is super-duplex stainless steel(SDSS).

The duplex grades of stainless steel are two-phase materials, containing nominally equalvolumes of austenite and ferrite achieved viatheir chemical composition and thermomechan-ical processing during manufacture. With 0.2%proof stresses of around 900MPa and ultimatetensile strength in excess of 1050MPa, combinedwith excellent corrosion resistance, the materialis ideally placed to cope with the demands ofHT-HP service [1][2].

However, recent in-service tubing failuresat stresses well within design limits havehighlighted gaps in the understanding of theinitiation and propagation of cracks in the ma-terial in chloride containing environments [3][4].The extreme in-service conditions, coupled withthe relative small population of HT-HP wellsintroduce a number of uncertainties into theengineering problem which can be examinedsuccessfully using a probabilistic structuralreliability approach.

Defects can be introduced into the material viain-service problems such as fatigue or stress cor-rosion cracking. While these time-dependentphenomena are of interest, this study inves-tigates the catastrophic failure mode of duc-tile fracture from a pre-existing crack. Elastic-plastic fracture mechanics problems have beencharacterised using the J-integral. Unlike morebrittle materials, unstable and catastrophic frac-ture of elastic-plastic materials does not occurat some critical value of the J-integral. Instead,

0a

a

ca

( , )A cJ F a

2( , )

AJ F a

1( , )AJ F a

Crack Length, a

iJ

cJ

J-In

tegra

l

0

0( )RJ a a

Figure 1: The tangency condition for failure

unstable tearing only occurs if the following con-dition is met:

dJR

d∆a≤

dJA

d∆a(1)

This is known as the tangency condition of theJ-∆a curve and is sometimes referred to as theR-curve approach.

The l.h.s of equation (1) is a material propertythat represents the material’s crack resistanceto an incremental amount of crack growth, andthe r.h.s is the applied energy or crack drivingforce. Carrying out a probabilistic failureassessment requires an understanding of bothsides of the above equation, and their possiblevariation. The R-curve approach is summarisedin Figure 1, which shows the material curveJR(a − a0) and three different J-integral curvesfor three different applied forces F1 < F2 < Fc

as a function of crack length. Stable tearinginitiates at F1 and unstable tearing at Fc.

There is large body of work aimed at modellingthe applied J-integral, JA for different tube andcrack geometeries using finite element codes[5][6][7]. Characterising the crack resistance ofSDSS has received less attention and containsa number of challenges. These challengesare centred on the anisotropic nature of themicrostructure of the duplex family of stainless

steels.

Seamless casing and tubing components man-ufactured from the duplex grades are forgedand typically have hot and cold mechanicaltreatments applied creating elongation of theaustenite islands in the ferrite matrix. Theresult is a highly anisotropic microstructure thatcan lead to the ratio between crack resistancesof different orientations being as great as 2.5:1[8]. Any attempt to model failure from aninitial flaw using the tangency condition musttherefore reflect the properties of the materialin that orientation.

A second challenge arises in the experimen-tal measurement of the J-∆a curve. Thetraditional approach to measuring fracturemechanics parameters such as the J-integralis to use small standard geometry specimensin the belief that their fracture behaviour istransferable to full scale structural components[9][10]. The elastic-plastic nature of SDSS,and the comparatively small size of the wallcross-section of tubular components means atransferable measurement of crack resistance isnot guaranteed using the standard proceduresset out in the ASTM and BSI codes [11][12].This geometery dependence of J-∆a curveshas been noted in the literature [8][13][14], thepractical consequence of which is the possibleoverestimation of the stress required to causeunstable tearing from environmental or fatigueinduced flaws.

This paper describes an investigation into the ef-fect of microstructural anisotropy on the fracturemechanics behaviour of an SDSS used for tubu-lar manufacture and the transferability of exper-imentally measured SDSS crack resistances. Thestudy reviews previous work in the area of SDSSmaterial properties and the limitations of the J-integral to characterise crack growth in elastic-plastic materials. An experimental procedurefor single specimen measurement of the J-∆a

curve is set out, involving the use of nominallyidentical single-edge notched bend (SENB) spec-imens with different initial starting crack lengthsmachined from two different orientations of thetubular wall. The results of the tests are pre-sented, together with a discussion and conclu-sions with a view to improving the understand-ing of the failure of the material in HT-HP con-ditions and its modelling for the purposes of re-liability assessment.

2. FRACTURE MECHANICS

CHARACTERISATION

OF SDSS

2.1 DUPLEX STAINLESS STEEL

The modern duplex grades of stainless steelare a high strength, low nickel alternativeto traditional austenitic grades, and highernickel alloys. The lower nickel content achievessignificant cost and weight savings over alloyswith comparable corrosion resistance. The highstrength is achieved through the material’s two-phase microstructure which contains nominallyequal volumes of austenite and ferrite, normallyin the form of multi-grained austenite islandssurrounded by a ferrite matrix.

The difference between the duplex and super-duplex grades is one of chemical composi-tion based on the use of an empirical pittingresistance equivalent number (PREN) whichcombines the percentage weight of chromium,molybdneum, tungsten and nitrogen to give:

PREN = Cr + 3.3(Mo + 0.5W) + 16N (2)

Duplex grades with a PREN greater than40 are known as super-duplex. The typicalchemical composition of various duplex grades iscontained in table 1. The chemical compositionof the two phases varies as a result of chemicalpartitioning. Chromium, molybdneum andtungsten exist in higher quantities in the ferrite,

and nitrogen and nickel in higher quantities inthe austenite. However, the differences in com-position are relatively small with no more than16% between the composition of the two phasesfor any one individual element [1]. The additionof small amounts of nitrogen significantlystrengthens the austenitic phase producingsuperior mechanical properties [15][16][17] andcan lead to the austenite being the harder phase.

In addition to the chemical composition, mi-crostructure morphology and crystallographictexturing as a result of thermomechanical pro-cessing play an important role in determiningthe material’s mechanical properties. In com-mon with many duplex grades, the SDSS ofinterest in this study has anisotropic and in-homogeneous microstructure. This is demon-strated in Figure 2 which contains two micro-graphs taken using a scanning electron micro-scope (SEM) from the radial (Y-Z) and axial(X-Z) planes as defined in BS7448-4 [12]1

A detailed review of all aspects of the duplexgrades including super-duplex is contained in thepreviously mentioned references [1][2].

2.2 EFFECT OF ANISOTROPYON TENSILE AND FRACTUREBEHAVIOUR

The effect of anisotropic microstructure onglobal material properties of duplex stainlesssteel has been studied in the literature. Theanisotropy in the tensile properties of cold-rolled duplex sheet with elongated austeniteislands in the direction parallel to rolling wasinvestigated separately by Makinde and Ferron[18] and Hutchinson et al [19]. Both studiesfound an increase of approximately 10 percentin the 0.2% proof stress and ultimate tensilestrength in the direction perpendicular torolling direction. The physical reason behindthe result was the presence of crystallographictexturing: the preferential alignment of active

1See figure B.2 (b) pg 40 of [12]

STANDARD TRADE Cr Ni Mo N Others PREN DUPLEXMARKS %b.w. %b.w. %b.w. %b.w. %b.w. GRADE

UNS S 32304 SAF 2304 23.0 4.0 0.2 0.1 - 25 Low alloyUNS S 31803 SAF2205 22.0 5.3 3.0 0.16 - 35 Med alloyUNS S32760 Zeron100 25 7.0 3.6 0.25 0.7W 41 SuperUNS S32750 SAF 2507 25 7.0 3.8 0.27 - 42 SuperUNS S32740 SMDP3W 25 6.7 3.1 0.26 2.0W 43 Super

Table 1: Comparison of chemical compositions of different grades of duplex stainless steel.

(a) Radial Plane (b) Axial Plane

Figure 2: The anisotropic microstructure of SDSS

slip-planes of the two-phases in a particulardirection, introduced by the cold rolling of thematerial. This effect outweighed the expectedmechanical fibering in the direction of theelongated microstructure. The tubular materialof interest to this study is also cold-worked atthe last stage of the manufacturing process,and there is the potential for some degree ofcrystallographic texturing in the material.

The crack resistance of a 25Cr 4Ni 0.34Nduplex stainless steel was investigated byKolednik et al [8]. Three separate planes of asolution-annealed forged rod were tested usingcompact tension specimens denoted A, B, andC which correspond to BS7448-4 fracture planes(Z-Y),(X-Z) and (Y-Z) respectively2. Figure

2The first letter in the BS7448-4 [12] fracture planesdenotes the direction perpendicular to the crack plane,and the second to the direction of crack propagation.

3 demonstrates the relation of these planes tothe tubular component of interest to this study.The figure shows the reference axes, two viewsof the same pipe component containing an axialflaw and radial flaw, and the orientation of themicrostructure. Note that the Z direction is notabsolute, but perpendicular to the tube surface.

Kolednik et al used the direct-current potentialdrop technique to measure crack growth and avalue of the J-integral, Jqi the quasi-initiationpoint, to characterise the initiation of stabletearing in three specimens machined from therod. The reported value of Jqi measured foreach plane was 200kJ/m2 for A, 450kJ/m2 forB, and 200kJ/m2 for C3. The energy dissapa-tion rate D was also examined in [8]. D is thetotal crack resistance of the material to an incre-mental amount of crack growth, and is closely

3table 3, pg 3311 of [8]

X

ZAxial flaw

X

Y

Z

Axial flaw

Radial flaw

Radial flaw

Figure 3: Flaw geometery and microstructure orientation in relation to tubing component

related to the gradient of the J-∆a R-curve viathe specimen depth W , the initial crack lengtha0, and the geometery factor η:

dJR

d∆a≈

ηD

W − ao

(3)

where η is equal to 2 for SENB. D was foundto peak at crack initiation, then decrease to alimiting value after a short amount of crackgrowth to a value approximately equal to 2000,5000, and 2000kJ/m2 for planes A, B and Crespectively4.

These differences in crack resistance are at-tributed to the anisotropic microstructure in[8]. It is reported that the crack propagatedpreferentially in the more brittle ferrite or alongferrite/austenite boundaries thus avoiding themore ductile austenite in specimens A and C.The mechanism suggested for the increase in Jqi

in specimen B is that the crack front is forcedto cross the more ductile austenite islands inorder to propagate.

4Kolednik et al [8] were unable to confirm if steadystate values of D were achieved in their experiments asthey only measured relatively short amounts of crackpropagation.

While the study contained in [8] is of usein understanding the role of microstructuralanisotropy, it is of limited use to a reliabil-ity analysis as only single specimen results arequoted. In addition, the results of Jqi and D ap-pear to have been rounded without quoting theexact results gained. The heterogeneous natureof the microstructure suggests there will be vari-ation in the measured value of Ji for nominallyidentical specimens which needs to be quantified.

The results presented in [8] suggest that thecrack resistance of duplex stainless steels for aradial flaw in a tubular component would be2.25-2.5 times greater than that for an axialflaw orientated longitudinally in the pipe. Axialflaws are then of more concern than radialones, and indeed this seems to be reflected bythe in-service experience noted in [3][4] whereaxial flaws have propagated by tearing followinginitial environmental attack.

2.3 FRACTURE MECHANICSPARAMETERS

The crack resistance of duplex stainless steelcan be characterised using a number of dif-ferent parameters. The main requirement is

that the parameter used to characterise thematerial’s behaviour is transferable: smalllaboratory specimen results should reflect thebehaviour of full-scale tubular components.The two traditional parameters used are the J-integral and the crack tip opening displacement(CTOD). This study will focus on the J-integral.

The problem for a single parameter approachis the known dependence of the measuredinitiation value Ji on the degree of constraintpresent at the crack-tip [13][14]. In practice,this means that laboratory tests using standardsingle-edge notched bend (SENB) or compacttension (CT) specimens may not reflect in-service performance.

The phenomena behind this effect are of interestand have been studied in detail by Stampfl andKolednik [20]. First, an energy balance acrossa single specimen during an increment of crackextention is considered:

D ≡ Rtot ≡1

B

d(Wpl + Γ)

d(∆a)

=1

B

d(U − Wel)

d(∆a)≡ C (4)

The terms Rtot and C are the total crack growthresistance and crack driving force, respectively.On the l.h.s, Wpl is the total non-reversiblestrain energy required to cause an incrementalamount of crack growth, Γ is the surface energy,and B is the specimen thickness. On the r.h.s,U is the total elastic and plastic strain energyapplied to the specimen and Wel the reversibleelastic energy.

The r.h.s is used to measure experimentally thel.h.s. since tearing occurs when Rtot = C whichvia equation (1) is equivalent to the tangencycondition of the J − ∆a curve. In the standardASTM and BSI single specimen measurementtechniques [11][12], U is the area under theforce vs load-line displacement curve, and

Wel = Uel can be found from the complianceof the machine + specimen. This approach isdemonstrated graphically in Figure 4.

It is this technique that leads to the geometrydependance of the measured value of the crack-driving force at initiation Ci.

plU

elU

A

O

Lo

ad, F

(k

N)

Load Line Displacement, q (mm)q'

F’

pl elU U U

Figure 4: Plastic and Elastic Energies from theLoad vs Load Line Displacement Curve.

The part of the external energy being used bynon-reversible processes is denoted Upl, and theJ-integral is estimated using an equation of theform5:

JR = Jel + Jpl (5)

= G +ηUpl

BN(W − a0)

∴ JR =K2

E ′+

ηUpl

BN(W − a0)(6)

where G is the elastic energy release rate, BN

is the effective width of the specimen, K isthe stress intensity factor, and E ′ is the elasticmodulus for either plane stress or plane strainconditions.

In using this method to determine a transfer-

5Both the stress intensity K and the ligament length(W − a) can be corrected for crack growth.

2. Plastic strain zone

(fracture)

3. Plastic strains from

bending

4. Plastic strains

from roller

Roller

Process Zone

Close-UpSENB Specimen

1. Fracture

process zone

Void nucleation

and growth

Cup and Cone

Ductile Surfaces

Crack-tip

blunting

Figure 5: Non-reversible processes occuring in an SENB specimen during ductile fracture.

able material property, the unstated assumptionbeing made is that the plastic component of theexternal energy applied is being used entirelyby the fracture process so that Up = Wpl. Ananalysis of the different physical phenomenathat contribute to Up demonstates that this isnot the case.

Elastic-plastic materials such as SDSS experi-ence significant amounts of plastic deformationdistant from the crack tip. The followingnon-reversible processes contibute to Up:

• Crack-tip blunting prior to initiation of frac-ture, Wpl,cbl

• Creation of new fracture surfaces in thefracture process zone, Wpl,sf

• The energy consumed in the fracture pro-cess zone by void nucleation and growthahead of the crack tip,Wpl,v

• Plastic deformation local to the processzone, Wpl,l

• Plastic deformation in the ligament aheadof the plastic zone of the fracture process,Wpl,lig.

• Lateral contraction of the specimen at theedge, Wpl,lat, and

• The formation of shear lips at the specimenedges Wpl,slant

This list is not exhaustive. Some of these pro-cesses are highlighted in the diagram shown inFigure 5. Up is given by:

Upl = Wpl,cbl + Wpl,sf + Wpl,v + Wpl,l

+ Wpl,lig + Wpl,lat + Wpl,slant (7)

It is possible to separate the energy componentsof Up into those that are geometery independent,and those that are not. To formalise this, a newexpression is proposed:

Up = Wpl + Φpl (8)

where Wpl represents the sum of plastic strainenergies that are independent of specimen geom-etry, and Φpl is the sum of plastic strain energiesthat do vary as the geometry of the specimen,and the remaining ligament, change. Using thephysical phenomena identified in equation (7)gives:

Wpl = Wpl,cbl + Wpl,sf + Wpl,v (9)

Φpl = Wpl,lig + Wpl,lat + Wpl,slant + Wpl,l

(10)

This analysis has an important practical conse-quence: any experimental measurement of the

J-integral using the load-line displacement ap-proach may overestimate the amount of energyrequired to achieve the tangency condition. Thiscan occur if the specimen geometery does notmatch either the real flaw’s crack-tip constraintor ligament size, resulting in a increase in Φpl

over what would be experienced in the real com-ponent.

L

W

Z

Y

Specimen B

Specimen A

Z

X

Figure 6: Orientation of H and V specimens withrespect to the tube, showing fatigue pre-cracks.

This section has highlighted the need for a care-ful investigation of the fracture behaviour ofthe cold-worked SDSS used in seamless tubingmanufacture. Of particular interest is the ef-fect of microstructure anisotropy on crack re-sistance, the transferability of standard exper-imental measurements of the J-integral, and thephysical variation of such measurements for usein a structural reliability assessments. An ex-perimental procedure was developed to addressthese issues.

3. EXPERIMENTAL

PROCEDURE

In the present research, the experimentalmeasurement of J-∆a curves was carried outusing BS7448-4 [12] and ASTM 1820-05 [11]as guidance. The single specimen procedure

was used to allow the direct measurement ofcrack resistance during crack growth, and tostudy the variation of crack resistance betweenspecimens.

SENB specimens were machined from a 7-inchdiameter tubular manufactured from cold-worked SDSS for use in the three-point bendtest procedure using a 100kN Instron 1185.The single specimen technique requires themeasurement of crack growth ∆a during thetest. This was achieved using a direct currentpotential drop technique (DCPD).

To investigate the issues raised in the previoussection, two variables relating to the specimenswere examined:

• Orientation: Two nominally identicalgroups of 18 side-grooved SENB specimens(36 in total) were machined from two differ-ent orientations in the pipe.

• Notch Depth: Within each orientationgroup, three different notch depths wereused.

The two specimen orientations are shown in Fig-ure 6 relative to the tubular. The two groups ofspecimens are denoted ‘H’ and ‘V’. The H spec-imens simulate the behaviour of an axial flaw inthe tube, the V specimens simulate a radial flaw.

The test matrix is shown in Table 2, with thenumber of specimens of each orientation andinitial crack size a0.

Specimen Initial Crack Depth (a0)Orientation ≈2mm ≈4mm ≈6mm(Y-Z) H 6 6 6(X-Z) V 6 6 6

Table 2: Test Matrix of SENB specimens.

The DCPD system used in this study was

developed using published studies on the use ofthe DCPD technique in R-curve determinationcontained in [21][22]. The background noiseof the measured voltage across the crack was≈ 40µV which corresponded to a crack depthresolution of 100µm. Load-line displacementq was determined using the indirect methodoutlined in Annex C, Figure C.3 of [12].

The three-point bend tests were carried out onthe Instron in displacement control mode at1.0mm/min. All relevant data from the Instron,the DCPD equipment, the LVDT and the straingauge meter were recorded using an 8-channelcomputerized data-logger.

4. EXPERIMENTAL

RESULTS

The raw data from the tests was processed ac-cording to the single specimen J-∆a R-curveprocedure set out in the ASTM standard [11].

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Crack Growth, a (mm)

J (

MJ

/m2)

J 0.2BL

J i

Figure 7: J-∆a experimental R-curve for H speci-men #6 showing Ji and J0.2BL.

4.1 J-∆a CURVES

The determination of Ji was done by a com-bination of strain gauge measurement andmaximum load fractography, which confirmedthat ≈ 0.1mm of crack growth occured prior

to maximum load. A polynominal fit of theR-curve was then used to identify the pointof initiation. An example J-∆a curve fromhorizonal specimen #6 with initial crack size of2mm is shown in Figure 7.

Two example curves from H and V specimenswith the same initial starting crack of 2mm areshown in Figure 8. The full set of results for Ji

for both orientations is presented in Figure 9.

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Crack Growth, a (mm)

J (

MJ

/m2

)H Specimen V Specimen

Figure 8: J-∆a experimental R-curves for an H andV specimen with a0 = 2mm.

4.2 VARIATION

The results presented in the previous sectionwere analysed using a statistical analysis givingsample mean x(Ji), sample standard deviations(Ji), and sample coefficient of variation (COV).The results of the statistical analysis are pre-sented in Table 3. Using Student’s t-test onthe data, the results show significant differences

J i = 1.5483 (a 0)-1.2994

J i = 1.0745 (a 0)-1.2382

0

0.1

0.2

0.3

0.4

0.5

0.6

2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7

Initial Crack Length, a o (mm)

Ji (

MJ

/m2

)

V Specimens

H Specimens

Power (V Specimens)

Power (H Specimens)

Figure 9: Experimental Ji values describing the initiation of stable tearing for H and V specimens

Specimen a0 = 2mm a0 = 4mm a0 = 6mmType x(Ji) s(Ji) COV x(Ji) s(Ji) COV x(Ji) s(Ji) COV

MJ/m2 MJ/m2 MJ/m2 MJ/m2 MJ/m2 MJ/m2

H 0.346 0.038 0.111 0.172 0.032 0.189 0.118 0.012 0.103V 0.442 0.056 0.127 0.234 0.027 0.115 0.133 0.008 0.063

Ratio of V/H 1.279 - - 1.361 - - 1.127 - -

Table 3: Statistical Analysis of Ji results for H and V specimens

between the H and V specimens at the 99%confidence level.

5. FORMULATION OF

LIMIT STATE FUNCTION

The results of the experimental programme shednew light on the structural integrity of SDSStubulars. The first comment on the overall re-sults is the obvious need for a probabilistic ap-proach. The onset of stable tearing Ji has a COVof between 0.063 and 0.189, with the higher val-ues observed at smaller initial crack sizes.

A probabilistic structural reliability analysisrequires the engineering problem to be written

as a limit state function. This section describesthe development of such a function for thetubular failure mode of ductile fracture. Theexperimental results presented in the previoussection are then discussed with reference totheir role in modelling the various parts of thelimit state function.

The tube shown previously in Figure 3 hasa wall thickness t defined by the outer andinner radius dimensions R0 and Ri so thatt = R0 − Ri. This paper is concerned with theapplication of an internal pressure P creating ahoop stress σh. Tensile and bending stresses arenot considered. As a result, only an axial flawwill be examined.

It is assumed that a crack-like flaw is presenton the external surface of the tube, with anaxial orientation as shown in Figure 3. Theflaw geometry is defined by its depth a and itslength on the surface 2c. The ratio a/2c is, forthe moment, assumed fixed for all crack depths.

For a given crack geometry and initial size a0,the tangency condition described in the intro-duction can be written as a limit state functionas follows:

M =dJR

d∆a−

dJA

d∆a(11)

where M takes its usual meaning as a scalarrandom variable such that M ≤ 0 correpondsto failure and M = 0 describes the boundarybetween a safe operating region and the failureregion. This expression can be developed byexamining the crack resistance and crack drivingforce terms individually.

Begining with the crack resistance term, the R-curves generated in the experiments show that,for any given value of a0, JR is a function of crackgrowth ∆a and that there is random variation ofJR for a given ∆a between nominally identicalspecimens. JR is then defined as a random vari-able described by its distribution function whosemoments are themselves a function of two othervariables:

• Initial crack size a0

• Crack growth ∆a = a − a0

The random behaviour of JR is demonstratedin Figure 10 which shows two separate JR

curves for two different initial crack sizesa1 < a2. The R-curves are denoted JR1(a − a0)and JR2(a − a0). The value of the J-integralat initiation Ji is shown for both curves as arandom variable denoted Ji1 and Ji2 respectively.

The types of distributions and the relationshipsbetween the moments E[Ji]

k, E[JR(a − a0)]k

(where k = 1, 2, 3....), and the initial crack

0 2a a

Crack Length, a

0 2[ | ]iE J a a

J-In

teg

ral

0

2 0( )RJ a a

0 1[ | ]iE J a a

0 1a a

1 0( )RJ a a

1( )

iJf J

2( )

iJf J

Figure 10: The random nature of JR curves.

length a0 and crack growth a − a0 are requiredto evaluate the probability that the limit statefunction M is less than or equal to zero. The ex-perimental results in this study go some way toachieving this, and will be discussed in the nextsection.

For a given crack length a(i) = a0 + ∆a corre-sponding to a given load F(i) and point on theLLD curve q(i):

JR(i) =K2

I(i)

E ′+

ηUpl(i)

BN(W − a(i))(12)

For side-grooved SENB specimens:

KI(i) =F(i)S

W 1.5 (BBN)0.5 × g1(a(i)/W ) (13)

E ′ =E

(1 − ν2)(14)

where, S is the span length between rollers, Eis the elastic modulus, and ν is Poisson’s ratio.Equations (14) and (13) can be substitutedinto (12) which can then be incorporated backinto the limit state function expression M .These equations demonstrate that dJR/d∆acan be measured experimentally, subject to thetransferability problem discussed earlier in thepaper.

The crack driving force term in M is also re-quired to complete a probabilistic structural re-liability analysis. For a given crack geometery,JA can be split into its elastic and plastic com-ponents:

JA = Jel + Jpl (15)

=K2

I

E ′+ Jpl (16)

For the axial flaw problem in the SDSS tubularKI is expressed by:

KI = g1

(

a(i)/W)

σh√

πa(i) (17)

∴ KI = g1 (a(i)/W )PR0

(R0 − Ri)

√πa(i) (18)

where g1(a(i)/W ) is the geometry functionquoted in [12] evaluated at the current crackdepth a(i), P is the internal pressure, and R0

and Ri the internal and external radii of thetube. The applied Jpl requires finite elementsolution, either directly, or using a polynomial ofthe type published in the literature6. Whicheverapproach is used, some model error shouldbe expected since FE codes rely on the fullstress-strain curve for the material to estimatethe value of the J-integral. SDSS containstwo phases with their own separate tensileproperties. The effect of this on FE estimatesof JA is not clear, but some residual modellingerror will be incorporated.

This study has been focussed on the material’scrack resistance term. It is one thing to defineJR as a random variable, but another to quan-tify the distribution that describes its randomnature. The experimental results of the previ-ous section give direction on how to model twoeffects that have a large impact on the variationof JR - microstructural anisotropy and geometrydependent plastic strains on JR. The experimen-tal results will now be examined in this context.

6See [23] as an example

5.1 EFFECT of MICROSTRUCTURALANISOTROPY

The SEM analysis of the microstructure shownin Figure 2 in combination with the work ref-erenced in [8][14][19] suggested crack resistancewould be dependent on orientation. The experi-mental results confirm that an orientation effectis present in the SDSS investigated in this study.Figure 8 shows the two main effects observed:

• The initiation value for stable tearing Ji isgreater in the V specimens.

• The gradient of the J-∆a curve is alsogreater in the V specimen demonstrating ahigher incremental crack growth resistance.

The V-specimens correspond to the (Y-Z) crackplane which replicates a radial flaw in the tube.The results for all specimens, summarised inFigure 9, confirm these effects at all three initialcrack lengths.

The statistical analysis of the results given inTable 3 shows that the ratio of the V and Hspecimen sample means x(Ji) was between 1.12-1.36 depending on initial crack size. Providedthe specimen results are transferable in thefracture mechanics sense, then the tube willdemonstrate a 12 − 36% increase in Ji for aradial flaw compared with an axial flaw. Thisincrease is caused by the crack propagatingpreferentially in the ferrite regions and alongferrite/austenite phase boundaries in the Hspecimens.

It is interesting to note that while there is aconsiderable difference between the two planes,it is far smaller than the 250% measured byKolednik et al [8]. The main reason for thisis likely to be the fact that the tubular SDSSmaterial is heavily cold-worked whereas thestudy in [8] used solution annealed material.This reinforces the need to treat SDSS gradesindividually: Thermo-mechanical processing is

as important as chemical composition.

The variation in the test results was also af-fected by the anisotropy. A comparisson of thesample standard deviation s(Ji) for both V andH specimens shows a smaller amount variationin the V specimens at both 4 and 6mm nominalinitial crack lengths. The COV for V specimenswas 0.127 for nominal 2mm a0 reducing to 0.063at 6mm a0. The COV of the H specimens showsno relationship with a0, although the lowestCOV is for the nominal 6mm a0.

Microstructural anisotropy is also responsible forthe difference in s(Ji) between the two orien-tations. The crack propagates by a number ofdifferent mechanisms in the (X-Z) plane beingtested by the H specimens. Note the varia-tion in the incremental data points between thetwo specimen types in Figure 8. Localised areasof austenite can cause higher incremental crackgrowth resistances leading to the higher varia-tion in Ji for the H specimens.

5.2 EFFECT OF CRACK DEPTH

The analysis of experimental measurement ofthe J-integral via Up means that some formof geometry dependence in the results wasexpected. Figure 9 confirms that the measuredJi is strongly dependent on a0. As the ligamentdecreases, so too do the geometry dependentterms in Φpl. This leads to a decrease in themeasured Ji, with close to a 50% reduction froma0 = 2mm to a0 = 6mm in the H specimens anda 70% reduction in the V specimens over thesame range.

This powerful result highlights a practical pointof primary importance to modelling the failureof tubulars manufactured with an elastic-plasticmaterial such as SDSS. Standard laboratoryspecimen test results are not immediately trans-ferable to real cracks. The laboratory specimen

geometry must be aligned to the real crack inthe component, otherwise the amount of Φpl

measured in the test will be different to thatexperienced in-service. If the real crack beinganalysed is greater in depth than the initialcrack size in the laboratory specimen then thecritical force required Fc to cause failure maybe overestimated which is undesirable.

Understanding the variation within the groupsis difficult due to the small sample sizes. How-ever, the analysis in Section 2 suggests a possibleeffect of crack depth on the variation in Ji. Re-call that Upl = Wpl + Φpl. The ligament reducesas the initial crack size increases, with a corre-sponding reduction in Φpl as a result. Since Wpl

cannot change with crack size (it is a materialproperty), then the proportion of Upl made upfrom Wpl increases with increasing a0. As a re-sult deeper cracks should have a higher sensitiv-ity to microstructural heterogeniety. However,shallower cracks Upl will be susceptible to geom-etry variation which results in varying amountsof Φpl. It is difficult to tell from the experimentalresults which of these effects is dominant.

5.3 JR MODELLING SUGGESTIONS

The results suggest a number of steps that mustbe taken to model SDSS tubular failure usingprobabilistic reliability techniques. Recognisingthe effect of microstructural anisotropy, radialand axial flaws must be modelled using sepa-rate limit state functions. Section 5 containsa development of the limit state of interest tothis project, that of an axial flaw subjected to athrough wall hoop stress.

Evaluating the limit state function describedin this paper requires the distributions of therandom variables identified. The experimentalwork presented can be used to derive a relation-ship between the expectation of Ji and initialcrack length a0.

A least squares best-fit line of a power law formhas been plotted through all of the Ji data pointsfor each specimen type V and H. This gives thefollowing empirical relations between J

(V )i , J

(H)i

and 2(mm) ≤ a0 ≤ 6(mm):

E[

J(V )i

]

= 1.5483(

a0

)

−1.2994

for radial flaws (19)

E[

J(H)i

]

= 1.0745(

a0

)

−1.2382

for axial flaws (20)

The relationship between E[

Ji

]2and a0 is less

clear from the current results. Further tests arein progress to develop a suitable expression forvariation.

These results will form part of an attempt toevaluate the probability of failure using the limitstate function for an axial flaw developed in thissection. Future work will see the experimentalresults used to develop distributions for the vari-ables contained in dJR/d∆a. The crack drivingforce dJA/d∆a will also be studied using exist-ing FE studies of the form contained in the pre-viously mentioned references [5][6][7].

6. CONCLUSIONS

This paper has examined how to incorporate mi-crostructural anisotropy of a cold-worked super-duplex stainless steel tube into a probabilisticstructural reliability assessment. The main con-clusions reached during the theoretical and ex-perimental sections are:

• The crack resistance of SDSS is random innature and requires a probabilistic approachto modelling failure.

• The microstructural anisotropy meansE

[

Ji

]

is 12 − 36% higher for a V speci-men (radial flaw) over an H specimen (axialflaw).

• The incremental crack growth resistance ishigher in the (X-Z) plane for a radial flaw.

• There was a higher level of variation in the(Y-Z) plane (axial flaw) due to the abilityof the crack to move over longer distancesin ferrite before arresting at localised areasof high austenite content.

• To carry out a structural reliability anal-ysis two separate limit state functions arerequired - one for each orientation.

• Non-reversible energy measured during astandard J-∆a test contains two separablecomponents - the geometry independentWpl and geometry depedent Φpl.

• Φpl varies with starting ligament size in side-grooved SENB specimens. There is a 50%reduction in Ji in the (Y-Z) plane froma0 = 2mm to a0 = 6mm and a 70% re-duction in the (X-Z) plane over the samerange.

• Standard tests must be performed at thecrack depth of concern to avoid over orunder-estimation of the behaviour of realcomponents.

The use of probabilistic structural reliabilitytechniques require detailed understanding of thephysical nature of materials such as SDSS, andhave an important role to play in the successfuldevelopment of challenging HT-HP oil wells.

ACKNOWLEGEMENTS

The authors would like to thank Shell U.K. (Ltd)for their financial and technical support duringthis project.

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