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Engineering Structures 21 (1999) 924–936www.elsevier.com/locate/engstructAnalysis of ductile fracture in damaged pipelines by a geometricparameter methodAlberto Corigliano * , Giulio Maier, Stefano MarianiDepartment of Structural Engineering, Politecnico of Milano, piazza L. da Vinci 32, 20133 Milan, ItalyReceived 9 October 1997; received in revised form 16 February 1998; accepted 24 February 1998AbstractNumerical simulations by a geometric parameter approach of ductile fracture processes in damaged pipelines are here presentedand discussed. The geometric parameters are the crack-tip-opening-displacement (CTOD) and the crack-tip-opening-angle (CTOA):following recent literature, they are assumed to govern the onset of crack propagation and the subsequent propagation phase,respectively. A simplified procedure is proposed for the identification of critical CTOA c as a function of the tip advancement.Numerical simulations of crack propagation in Three-Point-Bending Tests and in pressurised damaged pipelines are criticallypresented and compared with experiments and with numerical simulations based on the use of Gurson’s model. © 1999 ElsevierScience Ltd. All rights reserved.Keywords: Ductile fracture; Geometric parameters; Pipelines1. IntroductionDuctile fracture characterised by large-scale yielding(LSY) is a phenomenon of major importance for thedesign and integrity assessment of steel structures, suchas pressure vessels and pressurised pipelines, in a varietyof industrial situations. In the presence of LSY, andespecially of local unstressing of plastically deformedmaterial, the traditional J-integral approach to elasticplasticfracture mechanics is known to become inaccurateor even inapplicable to engineering purposes [1–3].In the recent literature alternative approaches havebeen proposed. A noteworthy one rests on an extensionof the J-integral theory and is characterised by the use,besides the J-integral, of a second fracture parameter,which is related to higher order approximations of theasymptotic (near tip) solution [4–8]. Another approach,which is at present increasingly studied, is rooted inmicromechanics and rests on constitutive laws whichinclude a phenomenological description of local damageprocesses leading to crack initiation and propagation.Representative of this kind of damage laws is the Gur-* Corresponding author. Tel.: 39-2-2399-4244; Fax: 39-2-2399-4220.son’s model, in which crack generation and advancementare linked to the evolution of porosity as a manifestationof void initiation and growth up to coalescence [9,10].The geometric approach adopted herein for numericalsimulations is motivated by its computational simplicityand by empirical and phenomenological arguments [11–18]. The basic hypotheses are that the shape of the cracknear the tip can be characterised by the crack tip openingdisplacement (CTOD) and the crack tip opening angle(CTOA) and that these geometric parameters, and theirexperimentally determined critical values, govern thefracture process, at least when it occurs in mode I. Thisunderlying assumption, not endowed with micromechanicalor physical motivation, has been corroborated byexperiments on pipeline steels [19,20].In this paper the geometric method is investigatedthrough its usage in simulating ductile crack propagationpreliminarily in three-point-bending (TPB) specimens toparameter identification purposes and, subsequently, inpressurised pipes (for industrial purposes). The relevanceand timeliness of fracture mechanics in the design andintegrity assessment of important infrastructures, such aspressurised pipelines, have been elucidated, e.g. by Kanninenand O’Donoghue [15].The objectives of the paper are: to assess the feasibility,in an industrial environment, of ductile fracture0141-0296/99/$ - see front matter © 1999 Elsevier Science Ltd. All rights reserved.PII: S0141-0296(98)00040-6

A. Corigliano et al. /Engineering Structures 21 (1999) 924–936925simulations based on the geometric approach andspecifically concerning pressurised pipelines; to examinethe aptitude of these simulations to attain a satisfactorycompromise between the conflicting requirements ofsimplicity and realism; to perform a calibration and validationof the geometric analysis procedure, keeping inmind the high costs of fracture experiments on gas-ducts.In the simulations large strains and displacements areallowed for and, as suggested by symmetry in the geometriesand loading conditions, rectilinear mode I crackpropagation is assumed. The software employed is basicallythe commercial finite element code ABAQUS [21]coupled with user-defined procedures which govern thecrack advancement.The final objective of the present critical rather thanmethodologically innovative study is to provide anorientation in planning and interpreting computerisedanalyses by industrial analysts in similar engineeringsituations.2. Identification of critical values of geometricparameters by means of simplified models of threepoint-bendingtestsExperimental data on the mode I advancement of thecrack tip in a ductile material (see e.g. Han et al. [20];Guan et al. [22]) suggest that the following empiricalcriteria can be regarded as satisfactory to engineeringanalysis purposes: (a) in mode I fracture processes, crackpropagation starts when the crack tip opening displacement(CTOD, Fig. 1(a)) reaches a critical level, sayCTOD c ; (b) the subsequent crack growth is controlled bythe parameter CTOA (Fig. 1(b)) which, after a transitionphase, reaches a constant critical value CTOA c duringthe steady-state propagation phase.It is worth noting that the geometrical definitions ofCTOD and CTOA given in Fig. 1(a,b) have a clear,unambiguous meaning oriented to the use in the contextof customary finite element modelling. It is also worthnoting that the CTOA parameter is assumed to characterisesituations of growing crack, in which the crack surfaceshave already lost their initial round blunted shapenear the tip.As for the assumption (a), the critical value of CTODat the onset of crack propagation can be computedaccording to the British Standards [23] which make useof a formula proposed by Dawes [24]. More recently, adiscussion on a procedure for computing CTOD in bentspecimens has been presented by Kolednik [25,26]. Thesuggested new procedure is based on experimental plots(obtained from TPB tests) of load versus crack-mouthopening-displacement(CMOD). This empirical procedureis used herein only in order to identify the criticalCTOD c at the onset of crack propagation with referenceto situations of large-scale yielding in single-edgenotched(SEN) bent specimens endowed with deepnotches and made of steels usually employed in pressurevessels.The variation of the parameter CTOA CTOA c upto its constant value CTOA c is identified in Section 2.2on the basis of the response of a TPB test interpretedby means of the simplified model discussed below inSection 2.1.2.1. A simplified (kinematic) description of crackgrowth during three-point-bending testsFig. 1.Definition of the geometric parameters referred to herein.As anticipated in Section 1, the geometric parameterCTOA can be assumed to govern mode I crack propagationin the sense that the crack tip advances only ifCTOA CTOA c . Rice and Sorensen [27] pointed outthat, in elastic-perfectly plastic materials, the deformedprofile of a growing crack asymptotically for r→0 (rbeing the distance from the crack tip) is proportional to(logr)r, which implies a vertical tangent at the tip. Inview of this the definition of CTOA has to be suitablychosen. Herein CTOA is interpreted as the anglebetween two straight lines emanating backward from thetip and reaching two points located on the crack flanks ata predetermined, constant distance from the tip measuredalong the propagation direction (Fig. 1(b)). The conceptof cohesive stresses across the crack near the tip, notexplicitly envisaged in the geometric approach to ductilefracture simulations, might interpret the experimentalevidence of increasing sharpening of the crack at the tipduring the propagation phase.In the available literature CTOA c is usually interpretedas a material constant (see e.g. Kanninen and O’Don-

926 A. Corigliano et al. /Engineering Structures 21 (1999) 924–936oghue [15]; Wang [28]; Martinelli and Venzi [29]). Inthis paper, CTOA c is instead assumed to vary during afirst stage of transient crack propagation, and to reach astationary value in a subsequent phase of steady-statepropagation. Specifically, in the present interpretation,the crack starts propagating with CTOA at the highestvalue (say CTOA 0 c); after some amount a t of tipadvancement with decreasing CTOA c CTOA c (a), asteady-state value CTOA c is reached, which is generallymuch lower than CTOA 0 c. In such a model CTOA 0 c,CTOA c and the transition advancement a t are consideredas material parameters which define the dependenceCTOA c (a) through interpolations to be suitablychosen later (see Eq. (6)).In this and in the subsequent section a simple procedureis developed for the identification of the abovethree parameters through simulations of TPB tests. Tothis aim, use is made of a simplified interpretation ofthe TPB test, mainly based on kinematic assumptions,proposed by Martinelli and Venzi [29], on the behaviourof the specimen during the test. This simplified modelallows one to obtain in analytical form a theoretical loadversus-displacementrelationship P P(u) during crackpropagation and this relationship can be compared to itsexperimental counterpart in order to identify the parametersgoverning the function CTOA c (a).With reference to a SEN bent specimen, for the crackpropagation phase in a TPB test the following hypothesesare assumed: (a) the material behaviour is rigid perfectly-plasticwith a yield limit f 1 2 ( y u ), y and u being the elastic limit and the ultimate stress, respectively;(b) the two halves (or arms) of the specimen rigidlyrotate one respect to the other around a plastic hingecentred on the mid-section; (c) the centre of relativerotation is located in the mid-section at a distance (W a) from the crack tip, and the plastic rotational factor is independent from the crack depth a/W (see e.g. Wuet al. [30]).By virtue of the hypothesis (c), some geometrical considerations[29] lead to the following relation whichlinks the infinitesimal crack advancement da to theinfinitesimal rotation d of one arm of the specimen withrespect to a fixed reference (see Fig. 2):da1 tan tan(CTOA/2) (W a) (1)d tan(CTOA/2)By separating the variables a and and integratingfrom the initial to the current situation, Eq. (1) yields arelation between the current ligament length (W a)and the current rotation angle :W aW a 0exp 1 tantan(CTOA/2)tan(CTOA/2) 0d (2)Fig. 2.Three Point Bending (TPB) test: definition of symbols.where denotes the integration variable and 0 therotation angle at the onset of the crack propagation, i.e.when a a 0 ; the ligament (W a) is measured in areference frame rigidly rotating with one arm of thespecimen, Fig. 2.By virtue of the above hypotheses (a) and (b), the loadcarrying capacity (in terms of the load P applied on themid-section) of the specimen during crack propagationcan be expressed as follows (see Fig. 2):P th fBS (W a) 2 (3)where S (S 2R 2 sin); (W a) (W a)/cosThis expression merely represents the plastic collapse(ultimate) load of a simply supported notched beamunder central concentrated loading, with thickness B,current span S and current mid-section height (W a) .The constraint factor which shows up as a correctivefactor in Eq. (3) is intended to take into account somefeatures of the actual two-dimensional collapse mechanismnot reflected by the drastic idealisation of thespecimen as a rigid-perfectly plastic beam [31]. Factor can be determined in two different ways, the latter ofwhich is adopted herein: (i) from the slip-line field solutionfor the TPB test (see e.g. Wu et al. [30]); (ii) byimposing that the ultimate load for a a 0 in Eq. (3)equals the maximum load value recorded in the experimentalload-versus-displacement plot.Let now the current span S (Eq. (3)) and mid-sectionheight (W a) (Eq. (3)) be expressed in terms of theirinitial values and of the rotation angle (see Fig. 2);thereafter, making use of Eq. (2), we obtain:P th () fBS (W a) 2

f B(S 2R 2 sin) W a 20cos1 tantan(CTOA/2)exp 2tan(CTOA/2) 0for 0 eA. Corigliano et al. /Engineering Structures 21 (1999) 924–936d(4)where e denotes the value of at the end of the fractureprocess, i.e. for a a e W.The load-point displacement u can be computed as afunction of the rotation angle on the basis of thegeometry of the TPB specimen in its deformed configuration(see Fig. 2):u th () S 2 tan (W R 1 R 2 ) 1 cos , (5)cosfor 0 eEq. (4) and (5) can be interpreted as a parametric representationof the load-versus-displacement relationshipfor the TPB specimen during crack propagation, the variableparameter being the rotation angle . This approximaterepresentation of a TPB test will be used in Section2.2 in order to identify, for some steels, the functionCTOA c () and, hence, CTOA c (a). It is worth stressingthat the identification procedure will rest only on theexperimental data after crack initiation, so that its resultsare practically not influenced by the very crude approximationof the above simplified model for the TPB testbefore crack initiation.2.2. Identification of CTOA c as a function of the cracktip advancementLet the variation of CTOA CTOA c be modelled bythe following function of the rotation angle of one armof the specimen with respect to a ground reference(Fig. 2):CTOA c CTOA c (CTOA 0 c CTOA c ) t 2 t 0for 0 t (6)CTOA c CTOA c for t eHere CTOA 0 c and CTOA c represent the values ofCTOA c at the onset of crack propagation and during steady-statepropagation, respectively. Symbols 0 , t , e inEq. (6) denote the values at the following situations:the start of transient crack propagation with varyingCTOA c ; the beginning of steady-state crack propagation;the end of the propagation phase, respectively.927The procedure here proposed for the identification ofthe above five parameters is based on the knowledge ofthe following data from a TPB test under displacementcontrol: the experimental load-versus-displacementrelationship P ex P ex (u); the experimental valuesu 0 ex, P 0 ex of the point in the P ex (u) graph correspondingto the onset of crack propagation, as revealed by a potential-dropdevice; the initial a 0 and final a e crack lengths,measured (as averages along the thickness) on the specimenafter the test.In SEN bent specimens made of low-strength hightoughnesssteels (currently employed in transmissionpipelines), such as those considered herein, the externalwork during a test turns out to be dissipated almost completelyin the mid-section fracture and plastic collapse(plastic hinge activation). Hence, the hypotheses (b) and(c) in Section 2.1 appear to be acceptable and the modelproposed there reasonably accurate for interpretingTPB tests.Among the five parameters which govern the functionCTOA c () according to Eq. (6), parameters 0 and e canbe directly obtained from Eq. (5) evaluated at the onsetand at the end of crack propagation. The optimal choiceof the other three is obtained by minimising, with respectto the parameters CTOA 0 c, CTOA c , and t , a measure ofthe discrepancy between the experimental load P ex (u)and the theoretical one P th (u) i.e.:u emin (PCTOA 0 c , CTOA c t th (u) P ex (u)) 2 (7), u 0duu 0 and u e being the values of the load-point displacementat the onset and at the end of the propagation phase,respectively. The minimisation (7) has been carried outhere by the simplest direct search, namely by selectingthe optimal choice of parameters within a set of predefinedtrial points.Once the parameters governing the relationshipCTOA c (), Eq. (6), have been identified, the functionCTOA c (a) can be numerically computed by means ofEq. (2) and (6) combined.The above procedure for the identification of the functionCTOA c (a) has been applied to three SEN bentspecimens labelled 1, 2 and 3 of steels with differentfracture toughness and equal elastic moduli (E 207 000 MPa; 0.3). Geometrical and material dataare collected in Table 1 together with the identifiedvalues of parameters CTOD c , CTOA 0 c, CTOA c and a t .As far as specimen (2) is concerned, the extremelyhigh value of CTOA 0 c is partially due to the assumedquadratic variation of CTOA c (a) during the first stageof crack propagation. More accurate results can be achievedby increasing the order of the simplified polynomialinterpolation used during this transient propagation

928 A. Corigliano et al. /Engineering Structures 21 (1999) 924–936Table 1Geometrical and material data for TPB specimens. E 207 000 Mpa, 0.3Specimen Steel S (mm) B W a 0 (mm) y u K V Charpy CTOD c CTOA 0 c CTOA c a t (mm)number (mm) (N/mm 2 ) (N/mm 2 ) (J/mm 2 ) (micron) (degrees) (degrees)(1) API 5L-X60 48 12 6.26 482. 565. 1.9 33.52 25.00 5.25 1.99(2) API 5L-X65 56 14 5.33 553. 605. 2.6 56.00 180.00 40.00 1.92(3) API 5L-X60 40 10 4.11 545. 629. 1.9 38.57 59.00 7.50 1.33phase, as shown by recent experimental results [20]. Thevalue of CTOA c is, by contrast, close to the one observedduring the steady-state propagation phase if crackgrowth is governed by means of the Gurson’s model (seeSection 3.2).In Table 1 the Charpy toughness K V of the specimensteels are also reported for the sake of completeness.These values have been measured during a separate seriesof tests conducted in order to determine thematerial properties.The experimental load-versus-displacement diagramsfor the three specimens are compared in Fig. 3 to thetheoretical ones achieved by means of the simplified(rigid-plastic) model of the specimen described in Section2.1 after its calibration by the above procedure. Fig.3 shows also the theoretical load-versus-displacementplots which result from the hypothesis (adopted in theliterature earlier cited) that CTOA c is constant throughoutthe whole crack propagation process. The proposedtheoretical model turns out to accurately interpret thesoftening branch of the load-versus-displacement plotsobtained experimentally. The change of curvature afterthe peak load is seen to be correctly represented by thevariable CTOA c approach. The comparable accuracyachieved by the constant CTOA c approach is mainly dueto the incorrect localisation of the starting point of crackgrowth. In fact, Fig. 3 shows that the propagation onsetis always estimated, by the constant CTOA c approach,to occur after the peak load on the load-versus-displacementplots, while the experimental beginning of crackgrowth happens before it, as assumed in the simulationprocedure discussed in what precedes.For the three TPB specimens considered, Fig. 4 visu-Fig. 4. Functions CTOA c (a) for the TPB specimens of Table 1.Fig. 3. Experimental and theoretical load-versus-displacement for three TPB specimens (R 1 R 2 5 mm; other data in Table 1). (a) TPB 1;(b) TPB 2; (c) TPB 3.

A. Corigliano et al. /Engineering Structures 21 (1999) 924–936929alises the relationship between CTOA c and crackadvancement, resulting from the adopted identificationprocedure. These results show a considerable differencebetween the CTOA c (a) for TPB1 or TPB3 and TPB2,which well represents the significant scatter in the steeltoughness measured by the values K V from Charpy tests(see Table 1).Clearly more experimental results are needed in orderto statistically assess the intrinsic uncertainties for eachof the identified parameters.3. Comparative simulations of fracture testsThe analyses of fracture processes briefly describedbelow (Section 3.1) rest on the assumption that the fractureprocesses are governed by the critical values CTOD cand CTOA c (a) according to the geometric criteriaassumed in Section 2. Analyses based on Gurson’smodel are briefly discussed in Section 3.2. In Section3.3 simulations of TPB tests by both approaches arecomparatively presented and compared to experimentaldata, in order to achieve more insight into the performanceof the kinematic approach adopted in what precedes.In all simulations performed in this study thefinite element code ABAQUS [21] has been employedfor analyses at fixed boundary conditions along the ligament,i.e. at constant crack length, while ad hoc proceduresexternal to that code have been implemented inorder to govern the crack advancement. For bothemployed procedures the same finite element mesh hasbeen used.3.1. Finite element simulations based on the geometricparametersSince rectilinear mode I crack propagation can a prioribe conjectured in TPB fracture tests, the adopted finiteelement discretisation of the specimen by displacementmodelling in space rests on structured meshes, withcharacteristic element size along the ligament of about1/50 of the specimen width W. It is important to noticethat a further reduction of the element size along theligament does not change significantly the global structuralresponse [32]. In all computations the materialsconsidered are typical pressure-vessel structural steels(see Table 1), the plastic behaviour of which is describedby means of Mises J 2 yield criterion and slight isotropichardening with saturation.During each simulation, the geometric parametersCTOD and CTOA are computed on the basis of displacementsof nodes lying on the crack flanks near thetip according to the proposal of Rice and Sorensen [27].Specifically (cf. Fig. 1(a)), the value of CTOD isobtained as the opening of the crack where two linesemanating from the tip with a slope of 45° with respectto the propagation direction intersect the lines which bestfit the deformed crack faces along the initial crack(which has been generated in the specimen by fatigueloading). When CTOD reaches its critical value CTOD cand, hence, the crack may start propagating, the node atthe tip is released (i.e. is replaced by two separatenodes).Along the subsequent propagation phase controlled bythe function CTOA c (a), the value of CTOA is computedas the angle between the tangents to the modelledcrack surfaces at the tip; these tangent lines are determinedby means of the displacements of the last releasednode. At each step of the analysis, the CTOA is computedin the current configuration and compared to thecurrent critical value given by the function CTOA c (a),identified as described in Section 2.2. When CTOA CTOA c (a) the tip node is released.3.2. Finite element simulations based on Gurson’smodelThe material model originally developed by Gurson[9] can be regarded as an elastic-plastic softening constitutivelaw in which the softening behaviour is governedby an evolution law for the porosity (or void volumefraction) f, i.e. by an internal variable susceptible to beinterpreted as a measure, in an average sense, of thevoid growth.The present finite element simulations are centred on afinite-strain generalisation of the Gurson’s yield criterionwith associated flow rule, in the following modified formproposed by Tvergaard [33,34]: 2 e 2q 2 1 f cosh q 2 k kM2 M 1 (q 1 f) 2 0 (8)where e (3s ij s ij /2) 1/2 is the effective Mises stress,s ij ij G ij k k/3 is the stress deviator, G ij are thecomponents of the metric tensor in the current configuration, M is the equivalent tensile flow stress in thematrix material which surrounds the void. Besides theelastic moduli (E, ) and yield and ultimate stresses ( y , u ) the set of parameters in this model includes thosewhich govern the influence of porosity on the materialresponse, namely the initial value f 0 of the void volumefraction and the two Tvergaard parameters q 1 , q 2 . Alsothe void nucleation, induced and controlled by plasticstrains, is taken into account and reflected at thephenomenological macroscale level by the model (seee.g. Chu and Needleman [35]; Tvergaard [10] for a comprehensivethorough description of this model).In the simulations presented in Section 3.3 f 0 isassumed to be zero; q 1 , q 2 are identified by a directsearchminimisation of discrepancy between numericaland experimental results concerning each TPB test.

930 A. Corigliano et al. /Engineering Structures 21 (1999) 924–936For computational economy, material which behavesaccording to Gurson’s model has been confined to a stripof elements (or computational cells) along the propagationdirection, similarly to what done in Xia and Shih[36] and Xia et al. [37]; all other elements in the meshobey a standard J 2 plasticity constitution, like in thesimulations based on the geometric approach. The thicknessof the strip of Gurson elements has been chosen inthe range 200–300 m. This range has recently beensuggested [36,37] on the basis of micromechanical considerations(average distance among inclusions) and, ina sense, can be regarded as a simple regularisation provisionin the presence of constitutive instability due tosoftening.Using Gurson’s model, a complete simulation of theductile fracture process from void initiation to coalescenceup to rupture (material separation) is in principlepossible [10,36]. However, this would require regularisationprovisions and ad hoc finite element modellinghardly compatible with the use of a commercial generalpurposecode. Therefore, the node release techniquealready employed in the geometric parameter method(Section 2) is adopted herein. Specifically, the currenttip node is released when the porosity f reaches theextinction value f E 0.2, which can be interpreted as acritical threshold at which the material is no more ableto sustain tractions. The value f E 0.2 has been assumedwith satisfactory results in Xia and Shih [36], also onthe basis of an earlier study by Tvergaard and Needleman[38].identified on the basis of the experimental load-versusdisplacementplot for each specimen).In order to check the accuracy of the simplified procedureof Section 2.1 for the identification of therelationship CTOA c CTOA c (a), in Fig. 6 the curvestheoretically obtained for SEN bent specimens 2 and 3by the geometric method are compared with the valuesof CTOA computed in the simulation based on Gurson’smodel at the instant when the current tip node isreleased. A fairly good agreement can be observedbetween results attained by the two so diverse approaches.For the TPB 2 only, Fig. 7 shows the final deformedmesh as arrived at by means of the geometric and Gurson’ssimulations, both carried out up to almost completecrack propagation across the ligament and, hence, withina phase of large displacements and rotations.The results shown in Figs. 5–7 can be considered asa positive test for the applicability, at least to TPB testsimulations, of the geometric parameter approachdescribed in Section 3.1 and for the validity of the identificationprocedures discussed in Section 2. It can be concludedthat the simple geometric approach leads toresults satisfactorily close to those based on Gurson’smodel and to experimental data. It is worth noting, inparticular, that allowing for the (identified) variability ofthe function CTOA c (a) always leads to better resultsthan the assumption CTOA c const., both in terms ofload-versus-displacement plots and in terms of ligamentversus-displacementdiagrams.3.3. Simulation of three-point-bending tests andcomparisons with experimental resultsReference is made here again to the three TPB testsconsidered in Section 2.2 (for parameter identification).The relevant geometrical and material data are gatheredin Table 1. In the finite element analyses performed inthis study, plane-strain hypothesis has been adopted (inview of the square section of the specimen) and fournodeisoparametric elements have been used for the spatialdiscretisation.In Fig. 5 experimental results concerning the threeTPB tests of Table 1 are compared to results of simulationscarried out by the geometric parameter approach, interms of load-versus-displacement graphs and ligamentversus-displacementplots. For the crack propagationphase both the hypothesis of CTOA c constant and ofCTOA c CTOA c (a) have been assumed, in the lattercase employing the values of the geometric parametersidentified in Section 2 and shown in Table 1.For the specimens 2 and 3, Fig. 5 also presents in (b)and (c) results provided by simulations based on Gurson’smodel (Section 3.2) with the parameters q 1 , q 2given in Table 2 (parameters q 1 , q 2 have been directly4. Simulation of ductile fracture in a pipelineThe proposed geometric parameter approach to ductilecrack propagation analysis has been conceived primarilyfor studying fracture phenomena in pressurised pipes.This section concerns the simulation of crack propagationalong the thickness of an indented-pressurisedpipe. The numerical results are compared to experimentaldata relevant to a series of experiments whichhas not been completed yet. More details on the experimentsand a more deep discussion of numerical simulationswill be presented in a forthcoming paper.The phenomenon here described is a simplification ofreal-life situations in which indentation and notch (dentand-gaugeaccident) can be caused by, for example, anagricultural tool hitting the pipeline; subsequent crackpropagation can be observed due to the internal pressurein the pipe. In the study, only crack propagation alongthe thickness has been considered. This phenomenonusually precedes de-pressurisation and dynamic crackpropagation in the axial direction of the pipe driven bythe internal gas (see e.g. Alpa et al. [39]; Kanninen andO’Donoghue [15]).Full scale tests on segments of pipelines are conducted

A. Corigliano et al. /Engineering Structures 21 (1999) 924–936931Fig. 5. Experimental and numerical results for TPB 1 (a), TPB 2 (b) and TPB 3 (c): load P versus displacement u and residual ligament (W a) versus displacement u.Table 2Gurson’s model parameters for TPB 2 and 3Specimen number q 1 q 2(2) 1.50 1.00(3) 1.60 1.52by Snam S.p.a. in order to assess the safety of pressurisedpipelines against dent-and-gauge accidents. Theexperiments concern segments of pipes and the dentand-gaugephase precedes the crack propagation phaseduring pressurisation. Fig. 8 represents a segment ofpipeline with a longitudinal notch or defect. First, thestructure is subjected to a radial load which causes anindentation around the defect. This loading phase ()leads to a residual deformed configuration in which thediameter of the pipe below the defect is reduced (by 3%of its original length). A second phase () consists of aninternal pressurisation which leads to crack propagationalong the thickness.The above two phases () and () of the loading pro-Fig. 6. CTOA c (a) dependence on tip advancement a for TPBspecimens 2 and 3.

932 A. Corigliano et al. /Engineering Structures 21 (1999) 924–936Fig. 7. Final deformed meshes for the TPB specimen 2. (a) Geometricapproach; (b) Gurson model.Fig. 8. Simulated dent-and-gauge accident in a pipeline. In mm: D 660, t 11.13, a 0 2.23, L 2200, L i 200.cess have been simulated by means of both plane strain(2D) and three-dimensional (3D) finite element discretisationsin space. The spatial discretisation, in the former(2D) case has been performed by four-node isoparametricelements, in the latter (3D) case by shell elementsfar from the defect zone and brick elements around it.In the 2D case the element size along the ligament hasbeen chosen equal to about 1/50 of the pipe thickness,as done in TPB simulations described in the previousSection 3. In the 3D case the brick element size to thicknessratio has been increased to about 1/20 in order toreduce the computational burden.In the 2D interpretation of the system, the analysesfor the phase () have been carried out in two ways:with a traditional Mises elastic-plastic model over thewhole domain; with Gurson’s model in the strip ofelements along the ligament on the radial directionbelow the notch. The final states of the above analyseshave been used as starting states for the numerical simulationsof crack propagation phase (): the former bymeans of the geometric method described in Section 3.1;the latter by means of Gurson’s model (Section 3.2).For the 3D simulation of phase () aJ 2 (von Mises)plasticity model was adopted over the whole mesh.Fig. 9(b) provides load (applied on the indentor) versusdisplacement (of the point below the indentor, asdepicted in Fig. 9(a)) plots concerning the pre-deformation(indentation) phase (). As it can be observed,the 3D analysis result is not far from the experimentalone, while the (by far more economical) 2D simulationdoes not reproduce the actual response (as expectedsince L i ≪L in Fig. 8 and, hence, the indentation is localisedin both the circumferential and longitudinaldirections).In order to simulate the response of the pipe along theloading phase (), with crack propagation through thethickness, the numerical approaches of Section 3.1 andSection 3.2 have been applied, with parameters identifiedon the basis of TPB 3 (see Tables 1 and 2). 3D simulationshave been carried out by the geometric parameterapproach only. When CTOA, in the transversal symmetryplane of the pipe, reaches a critical value the crackpropagates along the thickness maintaining a rectilinearfront in the longitudinal direction. This simplified strategyfor crack advancement disregards the existence of acurved crack front in the longitudinal direction and givesrise to a final critical situation which is a lower boundof the real one.The computed response during this phase (), drivenby increasing internal pressure, is represented in Fig. 10.Namely, in Fig. 10(b) through plots which depict therelationship between the pressure and the displacementss of the point lying on the internal surface of the pipejust below the defect (as shown in Fig. 10(a)); in Fig.10(c) through diagrams of residual ligament versus thesame displacement s (positive upward and referred to thefinal configuration after phase () of loading). In thesefigures, 2D simulations by the geometric and Gursonapproaches can be compared to the 3D simulation carriedout by the geometric approach only. A comparisonin Fig. 10(b) between these graphs and those of the computed2D and 3D responses without crack propagationacross the thickness of the pipe quantifies the loss ofstiffness due to the crack growth.In Fig. 10(b) also the experimental response is shown.The 2D simulations show a response which appearsmore compliant than the 3D one. This difference can bedue to the fact that in 2D the contribution to resistance

A. Corigliano et al. /Engineering Structures 21 (1999) 924–936933Fig. 9.For the pipe in Fig. 8 during the indentation phase: (a) Visualisation of displacement s and load P; (b) load-versus-displacement.Fig. 10. For the pipe in Fig. 8 during the pressurisation phase: (a) visualisation of displacement s and pressure p; (b) pressure-versus-displacementsduring crack propagation up to failure; (c) residual ligament-versus-displacement.given by the non indented sections around the damagedzone cannot be taken into account. The differencebetween the experimental plot and the 3D simulated oneis partially justified by the way in which crack propagatesin 3D simulations, maintaining a straight front anddisregarding the effect of curved crack fronts.All the analyses have been made under load-control,the internal pressure being the controlled variable. Thevalue of failure pressure has been considered to becoincident with the value of the control variable in thelast converged step of the analyses.At the end of the simulations without crack propagationin the thickness the following values of failurepressure have been obtained: p cr 18.12 MPa for the2D case; p cr 16.84 MPa for the 3D case.At the end of the analyses with crack propagation in

934 A. Corigliano et al. /Engineering Structures 21 (1999) 924–936the thickness, governed by the geometric parameters, theobtained values of failure pressures and relevantreductions with respect to the non-propagation case are:p cr 5.49 MPa for the 2D case ( 70%) and p cr 6.17 MPa ( 63%) for the 3D case. Such strongdecreases quantify the carrying capacity loss due to thecrack propagation across the thickness. The ultimatevalues of pressure at the end of the propagation phaseare not very different in 2D and 3D interpretations ( 3.4%), mainly because after the crack propagation alongits thickness, the pipe recovers almost completely theoriginal cylindrical shape.Fig. 11 shows the deformed configuration of the pipeat the onset of crack tip advancement in the 3D simulation.In Fig. 12 the deformed shapes of the mesh aroundthe process zone (discretised by means of brickelements) at the onset and at almost complete crackgrowth are depicted: in this figure the darkest zones representthe residual ligaments in the longitudinal planeof symmetry.Generally, the fracture analyses by means of the geometricparameter method, compared to the correspondingones using Gurson’s model, turned out to be easierto interpret and to implement in computer codes and lessdemanding in terms of computing time.5. Closing remarksA computational procedure based on geometric parametersfor the analysis of ductile fracture processes hasbeen presented with reference to three-point-bendingtests and pressurised steel pipelines affected by dentand-gaugeaccidents. The governing geometric parametersare the crack-tip opening displacement (CTOD)and the crack tip opening angle (CTOA); in an initial(transient) propagation phase the latter depends on thetip advancement. By comparisons with experiments andalternative approaches (specifically, fracture analysisprocedures by Gurson’s porosity model), it has beenshown that a noticeable advantage of the geometric parametermethod proposed herein is provided by its costeffectiveness,namely by the combination of simplicityand technically acceptable accuracy with which crackinitiation and propagation up to failure can be numericallysimulated by means of commercial state-of-the-artfinite element codes with marginal user’s interventions.The practical applicability of the present methodologyrests on the actual possibility of routinely identifying,through standard laboratory tests and their simplifiedsimulations, the critical values of geometric parametersFig. 11.3D deformed mesh for the pipe at the end of the indentation phase.

A. Corigliano et al. /Engineering Structures 21 (1999) 924–936935of crack processes governed by softening material models,like the Gurson’s one.AcknowledgementsThis work has been carried out in the framework ofa research project sponsored by Snam Spa (San Donato,Milan, Italy), which provided the experimental data. Thepermission to publish some of the results, fruitful interactionswith G. Bolzoni, S. Venzi and their co-workers,and the contribution of S. Testolina, before and after hisgraduation (laurea), are also gratefully acknowledged.ReferencesFig. 12. Particular of the 3D deformed mesh at the end of the indentationphase and at failure.which describe the crack shape near its tip during thefracture process.The limitations of the above conclusions should bestressed: they hold in the peculiar kind of engineeringsituations considered in this paper, namely situations inwhich ductile fracture can be a priori conjectured to substantiallyoccur in mode I, as it is frequently the case inpipelines and pressure vessels. Lack of physical foundationsand of micromechanical interpretations does notpermit to warrant a general use of the elastic-plastic fractureanalysis method based on geometric parameters asinvestigated in this paper. The alternative analysismethod resting on the use of Gurson’s porosity model,has been employed in this study for comparisons andrecognised as more versatile and physically sound, butalso significantly more laborious as for routine computationsto be carried out in industrial environments. Furthermore,simulations of crack propagation driven bycritical values of the geometric parameters do not showmesh dependent results, which occur during simulation[1] Rice JR. Mathematical analysis in the mechanics of fracture. In:Liebowitz H, editor. Fracture: an advanced treatise. New York:Academic Press, 1968;II:191–311.[2] Broek D. Elementary engineering fracture mechanics. TheNetherlands: Martinus Nijhoff, 1982.[3] Schmitt W, Kienzler R. The J-integral concept for elastic-plasticmaterial behavior. Engng Fracture Mech 1989;32:409–18.[4] Betegon C, Hancock JW. Two-parameter characterization of elastic-plasticcrack tip fields. J Appl Mech 1991;58:104–10.[5] O’Dowd NP, Shih CF. Family of crack-tip fields characterizedby a triaxiality parameter—I. Structure of fields. J Mech PhysSolids 1991;39:989–1015.[6] O’Dowd NP, Shih CF. Family of crack-tip fields characterizedby a triaxiality parameter—II. Fracture applications. J Mech PhysSolids 1992;40:939–63.[7] Dodds RH, Shih CF, Anderson TL. Continuum and micromechanicstreatment of constraint in fracture. Int J Fract1993;64:101–33.[8] Nevalainen M, Dodds RH. Numerical investigation of 3-D constrainteffects on brittle fracture in SE(B) and C(T) specimens.Int J Fracture 1995;74:131–61.[9] Gurson AL. Continuum theory of ductile rupture by voidnucleation and growth. I. Yield criteria and flow rules for porousductile media. J Engng Mat Techn 1977;99:2–15.[10] Tvergaard V. Material failure by void growth to coalescence. AdvAppl Mech 1990;27:83–151.[11] Venzi S, Martinelli A, Re G. Measurement of fracture initiationand propagation parameters from fracture kinematics. In: Sih GC,Mirabile M, editors. Analytical and experimental fracture mechanics.The Netherlands: Sijthoff and Noordhoff, 1981:737–56.[12] Battacharya A. Crack tip opening displacement in relatively hightoughness materials. Engng Fracture Mechanics 1991;40:187–200.[13] Kanninen MF, Grant TS, Demofonti G, Venzi S. The developmentand validation of a theoretical ductile fracture model. In:Proceedings of the Eight Symposium on Line Pipe Research,Houston, Texas. Arlington, Virginia: American Gas Association,1993.[14] Brocks W, Eberle A, Fricke S, Veith H. Large stable crackgrowth in fracture mechanics specimens. Nuclear Engng andDesign 1994;151:387–400.[15] Kanninen MF, O’Donoghue PE. Research challenges arisingfrom current and potential applications of dynamic fracture mechanicsto the integrity of engineering structures. Int J Solids Structures1995;32:2423–45.[16] McClintock FA, Kim Y-J, Parks DM. A criterion for plane strain,

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