Implementation and validation of a triaxiality dependent cohesive model: experiments and simulations

Int J Fract (2013) 181:227–239DOI 10.1007/s10704-013-9837-0

ORIGINAL PAPER

Implementation and validation of a triaxiality dependentcohesive model: experiments and simulations

Faizan Md. Rashid · Anuradha Banerjee

Received: 13 January 2013 / Accepted: 12 March 2013 / Published online: 26 March 2013© Springer Science+Business Media Dordrecht 2013

Abstract A recently formulated triaxiality dependentcohesive model for plane strain is implemented andits versatility is tested in simulation of ductile frac-ture of mild steel at different states of stress. The tri-axiality dependent model was implemented as lineardisplacement formulation based elements whose con-stitutive behaviour was dependent on the stress-stateof the neighbouring continuum element. By comparingthe experimental data and predictions of correspondingplane strain simulations, the model parameters are esti-mated. The model is shown to be effective in reproduc-ing characteristic features of the macroscopic responseof both pre-cracked as well as geometries without apreexisting nominal defect. Since the model parame-ters are held constant for simulations at different stress-states, they are effectively material constants.

Keywords Triaxiality · Cohesive zone model ·Cohesive zone parameter determination ·Failure locus · Stress-state

1 Introduction

Development of fracture models that incorporate thestrong effect of state of stress on ductile fracture mech-anisms is vital for life assessment of real structures. By

F. Md. Rashid · A. Banerjee (B)Department of Applied Mechanics, IIT Madras,Chennai 600 036, Indiae-mail:[email protected]

using such models, the significant problem of transfer-ability can be resolved, as fracture characterisation evenunder ideal laboratory conditions would be able to pro-vide accurate predictions of ductile fracture initiationand growth for a much wider range of applications.

In simulation of fracture in ductile solids, among theseveral different approaches that have been adopted,cohesive zone model finds wide popularity in predic-tion of crack growth initiation and crack growth resis-tance curves. Ever since the model was introduced byNeedleman (1987) as a numerical technique to sim-ulate the process of debonding of inclusions from ametal-matrix, it has been used for various studies ongrowth analysis of a macroscopic crack in homogenousductile metals, interfaces between dissimilar metals,concrete structures as well as to describe crazing inpolymers (Bazant 2002; Tijssens et al. 2000; Tvergaardand Hutchinson 1992, 1993). Departing from the tradi-tional numerical implementation of the cohesive modelas special interface elements in pre-defined plane ofcrack propagation, more recent implementations basedon the partition-of-unity property of the shape functionsof finite elements are able to simulate crack growthindependent of the structure of the mesh (de Borst 2003;Remmers et al. 2003).

Based on void growth mechanics and controlledexperiments, it has been well established that fracturemechanisms in ductile metals are strongly influencedby the state of stress (Hancock and Mackenzie 1976;McClintock 1968; Rice and Tracey 1969; Ghahrema-ninezhad and Ravi-Chandar 2012). To quantify this

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228 F. Md. Rashid, A. Banerjee

effect, a measure of the triaxiality of the stress-stateis defined as the ratio between the mean stress andthe effective stress: H = σmean/σe f f . The equivalentplastic strain at failure, a measure of the fracture duc-tility, has been shown to decrease exponentially withincrease in the triaxiality parameter, H . More recentstudies have shown that the mechanism of ductilefracture is different depending on the level of triax-iality. Under conditions of low triaxiality, the domi-nant mechanism is internal shear localisation of plasticstrain in the ligament between voids. The quantifica-tion of the effect of the stress-state on this mechanismalso requires the Lode parameter apart from triaxial-ity parameter (Bao and Wierzbicki 2004; Barsoum andFaleskog 2007; Pardoen and Brechet 2004; Nahsonand Hutchinson 2008). However, for plane strain orhigher triaxiality, the shear localisation is minimal andit is sufficient to use triaxiality parameter as a mea-sure of the stress-state as described in the previousstudies.

In the development cohesive zone model, however,the importance of triaxiality was first discussed onlya decade ago by Siegmund and Brocks (1999). Theyargued that the model parameters, cohesive strengthand cohesive energy, cannot be assumed to be mater-ial constants, as most studies till then did, as they aredependent on the stress-state. For wider applicabilityof the cohesive model, Siegmund and Brocks (2000)developed a cohesive model which accounts for theeffects of triaxiality in the traction-separation law. Intheir study, a plane strain unit cell model with degra-dation of material properties represented through GTNmodel was subjected to a state of stress with a constantin-plane stress ratio, α = σ11/σ22. The triaxiality para-meter under the assumption of incompressible plasticdeformation was calculated to be:

Hpl = 1√3

(1 + α

1 − α

). (1)

Using the response of the unit cell, the dependenceof the cohesive parameters on the triaxiality was estab-lished and the cohesive law was then based on cohesiveparameters as a function of triaxiality. The proposedmodel extended the applicability of the cohesive zoneframework for a wider range of fracture problems, how-ever, its basis being the GTN model for porous plas-ticity required the determination of the GTN modelparameters first. Later Anvari et al. (2006) includedthe effect of strain rates in the triaxiality dependent

model of Siegmund and Brocks (2000). A differentapproach was taken by Wnuk and Legat (2002) byusing the description of the distribution of cohesivetraction ahead of the crack tip, however, the presenceof a crack like nominal defect was essential, renderingthe model unsuitable for simulation of ductile fracturein geometries such as notched bars which do not havea pre-existing defect.

Accounting for the dominant effect of the stress-state on the ductile fracture process, Banerjee andManivasagam (2009) proposed a cohesive zone modelfor prediction of ductile fracture in different stress-states. The formulation developed for mode-I, planestrain includes the effect of stress-state through aneffective triaxiality parameter explicitly. The onset aswell as the progressive degradation of material degra-dation due to extensive void growth and coalescence,is characterised by the model parameters: C and S. Themodel has been used in its modified form for charac-terising fracture of thin-walled structures (Rajendranet al. 2010; Scheider et al. 2011), also to developthe formulation of a stress-state dependent cohesivemodel for fatigue (Jha and Banerjee 2012), and morerecently, for estimating stiffness losses in a rotor witha transverse breathing crack Toni Liong and Proppe(2012). While the study of Banerjee and Manivasagam(2009) estimated the model parameters for plane strainby comparing its results with that of cohesive mod-els based on GTN damage models (Anvari et al. 2006;Siegmund and Brocks 1999, 2000), a direct determina-tion of the parameters from experiments was left as afuture task.

In the present work, a combined computational andexperimental study is undertaken to validate the tri-axiality dependent cohesive zone (TCZ) model for-mulated by Banerjee and Manivasagam (2009) andarrive at a procedure to determine the model parame-ters. To achieve conditions of different stress-states,notched mild steel (IS 2062) bar specimens as well aspre-cracked specimens were manufactured and testedfor monotonic ductile fracture. For simulation, theTCZ model was implemented in FEM as linear dis-placement formulation based elements whose consti-tutive behaviour was dependent on the stress-state ofthe neighbouring continuum element. Comparison ofpredictions from simulations with the experimentaldata of notched bars and a CT specimen, the modelparameters, C and S/C , are estimated. For valida-tion, the estimated values are then used in prediction

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Implementation and validation of a triaxiality dependent 229

of crack initiation and growth in other CT speci-mens. A discussion is developed on the physical sig-nificance of the parameters and appropriateness ofdifferent test specimens for determination of modelparameters.

2 Experiments

With the dual objective of experimental validationof the TCZ model as well as establishing a pro-cedure for determination of the model parametersfrom experiments, geometries developing differentstress-states were tested under monotonic load untilfinal failure using a 100 kN servo-hydraulic testingmachine.

2.1 Moderate triaxiality specimens: notched bars

From a 20 mm thick mild-steel (IS 2602) plate, a seriesof notched flat-bar specimens were machined as shownin Fig. 1. The different notch radii manufactured were10, 6 and 3 mm. Also, a v-notch geometry correspond-ing to vanishingly small radius was manufactured. Tomake direct comparison of the macroscopic responsemeaningful, the ligament length was kept uniformlyat 3 mm and thickness at 16 mm for all the speci-mens. Standard monotonic tensile tests were carriedout for at least 5 specimen for each geometry. From theSEM micrographs of upper and lower fracture surfacesas shown in Fig. 2a–d for v-notched bar and 10 mmnotched bar, it can be inferred that plane strain con-ditions prevail during the fracture process as the sur-face profile appears similar across the thickness foreach micrograph. The upper and lower fracture sur-faces of the v-notched specimen appear largely flat inmost of the middle portion of the ligament thickness,also the entire fracture surface shows flat ruptured dim-ples implying ductile fracture. The edges of the liga-ment corresponding to the notch-tip is receded in boththe upper and lower surfaces confirming them to bethe location of the damage or crack initiation. In con-trast, on the fracture surfaces of the 10 mm notched bar,the flat dimpled surface appears only at the center of theligament where the damage and fracture initiates, theremaining ligament has shear failure on both the sidesas indicated by the cup and cone shape of the fracturesurface.

Fig. 1 Flat mild steel bars with different notch geometries.

2.2 High triaxiality specimens: fatigue pre-crackedCT specimens

From the same plate as the notched bars, several CTspecimens with a width, W = 60 mm and thickness,B = 16 mm were manufactured. Fatigue pre-crackingwas done till the crack length ahead of the notch tipwas greater than at least 2 mm. From the fracture test-ing under monotonic load, the force-deflection datawas recorded. The fracture surface showed flat ruptureddimples indicative of void growth under high constraintin the middle thickness and shear lip formation near thefree surfaces.

3 Implementation of TCZ model and FEMsimulations

For simulation of the experimental data, finite elementmeshes of the geometries were modeled. Two dimen-sional, 4-noded, plane strain elements were used for allthe analyses. The finite strain elastic-plastic analysisof these meshes were performed using ABAQUS. Theplasticity being based on J2 flow theory, for each geom-etry two types of analysis were performed: without anydamage model and with damage accounted througha stress-state dependent cohesive model as proposedby Banerjee and Manivasagam (2009). The elastic-plastic properties of the mild steel used for the studywere determined from the uniaxial tension test of roundtensile bars, the experimental data was approximatedwith a strain hardening material such that the uniaxialresponse was represented by:

ε = σ

Eσ ≤ σy, (2)

= σy

E

(σ

σy

) 1n

σ > σy, (3)

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Fig. 2 SEM micrographs of fracture surfaces of a v-notch top surface b v-notch bottom surface c 10 mm notch top surface d 10 mmnotch bottom surface

0

100

200

300

400

500

600

700

800

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

σ

ε

Exp Power law

Fig. 3 True stress-true strain response curve for mild steel (IS2602) and its power law approximation

where E is the elastic modulus, σy is the propor-tional limit and n is the strain hardening exponent.For E = 210 GPa, σy = 197 MPa and n = 0.228,the true stress-true strain curve from experiments iswell approximated by the power law curve as shown inFig. 3.

Using the elastic-plastic data, plane strain analysisof the notched bars shows good agreement with cor-responding 3-dimensional analysis as well as experi-mental data as shown in Fig. 4 for a typical notchedbar. The simulation results shown from the planestrain analyses as well as throughout the study arecorrected by a factor (1 − ν2) to account for thefact that except where localised deformation takesplace most of the specimen has plane stress conditionsprevailing. Such a correction significantly improvesthe correlation between the experiments and two-dimensional approximations and makes direct compar-isons possible.

To simulate ductile fracture process under condi-tions of widely different stress states, the dominantrole of stress-state has been incorporated by Banerjeeand Manivasagam (2009). In the model, the constitu-tive behaviour of the fracture process zone or cohesivelayer of thickness that of average void spacing, D, isrepresented by a traction-separation law with three dis-tinct regions:

Tn = (1+√3Hef f )

2E

3δ̂n 0 < δ̂n ≤ δ̂n1, (4)

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0

5

10

15

20

25

30

35

40

0 0.5 1 1.5 2 2.5 3

Forc

e, k

N

Deflection, mm

Experiment2D corrected

3D sim

Fig. 4 The force-deflection curve from experiment and simula-tion for notch radius = 3 mm

= (1+√3Hef f )

σy√3

(2E√3σy

δ̂n

)n

δ̂n1 ≤ δ̂n ≤ δ̂n2

(5)

= σmax exp

⎡⎣−0.01

(δ̂n − δ̂n2

δ̂n2

)4⎤⎦ δ̂n2 ≤ δ̂n,

(6)

where δ̂n is the opening displacement normalised withD (taken to be 100 microns for a typical mild steel).The triaxiality dependent shape parameters, δ̂n1 =√

3σy/(2E) and δ̂n2, define the bounds of the linear andstrain hardening behavior of the process zone respec-tively.

δ̂n2 =√

3

2

(Ce−1.5Hef f + σy

E

)(7)

represents the onset of rapid material degradation due toextensive void nucleation and growth leading to duc-tile failure.The triaxiality of the stress-state is incor-porated through an effective triaxiality parameter, Thenon-dimensional constant C is a model parameter thatdefines the effect of triaxiality on the onset of the soft-ening process as described in Eq. 7. It is in effect ameasure of an upper bound on equivalent plastic strainrequired at very low triaxiality, assuming Eq. 7 holds,for failure due to void growth and coalescence mech-anism. The second model parameter, S, on the otherhand defines a lower bound on theequivalent plastic

strain that is sufficient for rapid damage growth at hightriaxiality (H > Hsat ). Such that,

Hef f = 1

2

(H �

pl +Hsat

)− 1

2

(H �

pl −Hsat

)tanh

(α−αc

0.1

), (8)

that follows H �pl = ( 1 + 2α + 2α2 + 2α3)/

√3 at

low triaxiality and smoothly increases to the saturationlimit, Hsat .

Sσy

E= Ce1.5Hsat . (9)

Banerjee and Manivasagam (2009) reasoned that sat-uration of the effective triaxiality starts at the criticalvalue of biaxiality ratio,

αc =√

3Hsat − 1√3Hsat + 1

. (10)

Based on their parametric study the authors concludedthat initiation of ductile fracture at moderate biaxialityratios, α < αc, is entirely controlled by parameter Cand thus, the parameter can be determined from a lowconstraint test. The second parameter, S, was inferredto have a significant role on fracture process at higherconstraint such as ahead of a crack like defect.

The traction, σmax , at separation δn = δ̂n2 is the peakstrength of the process zone that is yet to start softeningdue to damage. As observed in Siegmund and Brocks(1999) as well as Banerjee and Manivasagam (2009),the peak strength cannot be assumed to be a materialparameter as it is strongly influenced by the state ofstress. In the present work by modifying the basic rou-tine by Gao and Bower (2004), the stress-state depen-dent model described in Eqs. 4-6 is implemented asa 4-noded element with linear displacement formula-tion. The stiffness matrix of each cohesive element isderived on the basis of its contribution to the internalvirtual work, δ�, for a virtual increment in the separa-tion vector, δ [u], defined as

δ� =∫

T. δ [u] , (11)

where, T is the cohesive stress vector acting in the ele-ment. The finite element analyses is performed usingcommercially available software package ABAQUS asit provides the option of writing sub-routines for defin-ing elements which are not available in their elementlibrary. The average triaxiality of the stress-state of theadjacent element at the previous time step is extractedusing anothersubroutine (UVARM) and passed on to

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Fig. 5 Refined mesh of theCT specimen withuniformly sized continuumelements adjacent to thecohesive elements

Cohesive elements

Continuum elements

Fig. 6 Contours ofequivalent plastic strain at(i) 90

(iii)

v−notch

(i) (ii) (iii) (i) (ii) (iii)

3 mm−notch

(ii)

6 mm−notch 10 mm−notch

(i) (ii) (iii) (i)

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the user-element subroutine (UEL) for calculation ofstiffness matrix and load vector for each cohesive ele-ment at every time increment. A layer of cohesive ele-ments was introduced at the central plane of all thespecimens, as shown in Fig. 5 the mesh was signifi-cantly refined near the process zone with the elementsize uniform at 0.1 mm for all specimens.

4 Results and discussion

4.1 Notched bar results

For developing a better understanding of the failureprocess in the different notched bars initially 2-D planestrain, elastic-plastic analysis of the bar geometrieswithout any damage is performed. Contours of theequivalent plastic strain are presented at three instantscorresponding to 90 at 10 For clarity and comparisonthe undeformed shape is used for the presentation. Thefigure shows that at near peak conditions, the 10 mm bardevelops the maximum plastic strain at the center of theligament. In comparison, for bars having smaller notchradii at near peak conditions the plastic deformation ismaximum near the notch tip before it spreads all alongthe ligament in post-peak conditions. The extreme caseof vanishing radius, v-notch geometry, shows severelocalisation of plastic deformation at the notch tip andnegligible deformation at the center of the ligament innear peak conditions.

Besides the extent of plastic deformation, the ductilefailure process is greatly influenced by the triaxialityof the stress-state as well. Before onset of full fieldplasticity, the triaxiality parameter variation across theligament for all the bars is shown in Fig. 7a. The barswith circular notches develop the lowest possible triaxi-ality of 0.577 at the notch surface which is traction free.Among the notched bars, v-notched geometry developsthe highest triaxiality parameter all across the ligament.By virtue of its sharpness, even the notch tip developsnon-uniaxial conditions and therefore higher triaxial-ity compared to other notched geometries. For all thebars, the maximum triaxiality develops at the centerof the ligament, the smaller the radius, the higher thetriaxiality it develops. Post significant plastic deforma-tion, the triaxiality across the ligament increases andthe maximum continues to be higher than at the notchsurface. However, post necking only the 10 mm barshows further increase in the triaxiality at the center of

0.6

0.8

1

1.2

1.4

1.6

0 0.2 0.4 0.6 0.8 1 1.2 1.4

H, T

riax

ialit

y

Distance, mm

notch radius = 0 mmnotch radius = 3 mmnotch radius = 6 mm

notch radius = 10 mm

0.6

0.8

1

1.2

1.4

1.6

0 0.2 0.4 0.6 0.8 1 1.2 1.4

H, T

riax

ialit

y

Distance, mm

notch radius = 0 mmnotch radius = 3 mmnotch radius = 6 mm

notch radius = 10 mm

(a)

(b)

Fig. 7 Triaxiality variation across the ligament a before onsetof full field plasticity b after 10 drop

the ligament. In the other circular notches, the triaxi-ality saturates across the ligament and for v-notch, thetriaxiality shows an overall decrease. Since the failureprocess begins when there is a combination of plas-tic deformation and triaxiality satisfying critical con-ditions, the likelihood of crack initiation at the centerseems most likely for the 10 mm circular notch whereboth the conditions of triaxiality as well as extent ofplastic deformation are favourable at all instants con-sidered. Even though v-notched bars also experiencehighest triaxiality inside the ligament, the triaxialityat the notch surface is not insignificant. As discussedpreviously, the plastic deformation is entirely localisedat the notched tip, and therefore it becomes the likelysite of crack initiation and growth as observed in theexperiments.

With the ultimate objective being characterisationof the ductile fracture process using the TCZ model, aparametric study is undertaken to establish the effectof the model parameters, C and S/C on the macro-scopic response of each notched bar geometry. Keeping

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0

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30

35

40

45

0 0.5 1 1.5 2 2.5 3 3.5 4

Forc

e, k

N

Deflection, mm

C=2.5C=3.0C=3.5

No damageExpt

0

5

10

15

20

25

30

35

40

45

0 0.5 1 1.5 2 2.5 3 3.5 4

Forc

e, k

N

Deflection, mm

C=2.5C=3.0C=3.5

No damageExpt

0

5

10

15

20

25

30

35

40

45

0 0.5 1 1.5 2 2.5 3 3.5 4

Forc

e, k

N

Deflection, mm

C=2.5C=3.0C=3.5

No damageExpt

0

5

10

15

20

25

30

35

40

45

0 0.5 1 1.5 2 2.5 3 3.5 4

Forc

e, k

N

Deflection, mm

C=2.5C=3.0C=3.5

No damageExpt

(a) (b)

(c) (d)

Fig. 8 Effect of parameter C on the force-deflection curve of a v-notched bar b 3 mm notched bar c 6 mm notched bar d 10 mm notchedbar

S/C at 20, for a v-notched geometry, the macroscopicbehaviour for a range of parameter C is comparedwith the experimental data in Fig. 8a. The figure alsoshows the response when there is no damage accountedfor, showing a typical elastic-plastic response followedby a peak-load and load drop associated with local-isation of deformation. In presence of cohesive ele-ments with a high value of C , the analysis replicatesmost of this undamaged macroscopic response and thedamage initiation occurs only after post-peak defor-mation. However, as the value of C decreases, theinitiation of significant damage occurs closer to theplastic deformation localisation. Damage increases andpropagates as the load drops rapidly before final fail-ure.

The circular notches as shown in Fig. 8b–d have avery similar effect of parameter C on the macroscopicbehaviour. The summary of the effect of parameterC on the characteristic features of the macroscopicresponse, the peak load and the fracture load, is

presented in Fig. 9a–b. Both, the average peak load andthe average fracture load’ from the experiments show adecrease with increasing notch radii. Since the experi-mental force-deflection curves are fairly smooth priorto final failure, from simulations the fracture load istaken to be the load at which there is a sharp transitionin the slope of the curve. From the figures, the effectof the parameter C is most evident on the fracture loadof the circular notches. A close approximation to theexperimental data from all notched bars which repre-sent conditions of different stress-states is obtained forparameter C between 3.0 and 3.5.

For a given material, to relate ductile fracture underconditions of different triaxiality, typically the locus ofthe equivalent plastic strain at failure is used. For thepresent study, the locus is generated by first identify-ing the fracture loads from the experiments on differentnotched bars as shown in Fig. 10. For a particular frac-ture load the corresponding equivalent plastic strain aswell as the triaxiality of the stress-state is obtained at

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the location of failure initiation from 2-D plane strainfinite element analysis without cohesive elements. Asexpected the equivalent plastic strain at failure decaysexponentially and the fit to the experimental data isgiven by:

ε p = 4.2e−1.7H (12)

It is interesting to note that if the exponential fit isforced to have the argument as −1.5H then the mul-tiplicative factor comes out to be 3.45 which is in therange of parameter C that reproduces the macroscopicbehaviour of the notched bars closely.

To determine its effect, the parameter S/C is nextvaried keeping C at 3.0, Fig. 11a–b shows the force-deflection curves for bar with notch radii of 10 mm andv-notch. For the range of S/C considered, the effect ofparameter S/C on the macroscopic behaviour of 10 mmbar as well as on v-notch bar is negligible. This trend islargely due to the fact that for S/C ranging between10 and 30 the corresponding αc, calculated as perEqs. 9–10, ranges between 0.68 to 0.61. The saturationlimit on the biaxiality ratio is therefore, well above themaximum biaxiality that develops in the notched barsof approximately α = 0.5.

4.2 CT specimen results

Ductile fracture in fatigue pre-cracked CT specimenstend to develop significantly higher triaxiality. Initially,to understand the spatial variation of equivalent plas-tic strain and state of stress ahead of the crack tip, a2-dimensional plane strain analysis without any cohe-sive element of a CT specimen with initial crack depthratio, a/w = 0.54 is performed. At the fracture loadof crack growth initiation that is obtained from theexperiments, the corresponding analysis shows a verysharp variation in both α as well as the equivalentplastic strain right ahead of the crack tip as shown inFig. 12. Including the equivalent plastic strain at failureand the corresponding triaxiality from the CT spec-imen in the failure locus changes the exponential fitto ε p = 5.0e−1.9H . The important implication is thatincluding the CT specimen data in the failure locuscan lead to significant changes in the locus constantswhich may not be accurate representative of the mate-rial’s fracture behaviour, and therefore, not suitable indetermination of model parameter C .

25

30

35

40

45

0 2 4 6 8 10

Forc

e, k

N

Notch radius, mm

C=2.5C=3.0C=3.5

Expt

25

30

35

40

45

0 2 4 6 8 10

Forc

e, k

N

Notch radius, mm

C=2.5C=3.0C=3.5

Expt

(a)

(b)

Fig. 9 Effect of parameter C on a the peak load and b fractureload of the different bar geometries

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

0.6 0.7 0.8 0.9 1 1.1 1.2

ε p a

t fai

lure

Triaxiality, H

Experimental dataExponential fit

3.45 exp(-1.5 H)

Fig. 10 Locus of equivalent plastic strain at failure at differentstress-states

Next the stress-state dependent cohesive elementsare applied for simulation of ductile fracture in fatiguepre-cracked CT specimens. To understand the effectof model parameters, a parametric study is under-taken. The energy dissipated during ductile fracture

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0 0.5 1 1.5 2 2.5 3 3.5 4

Forc

e, k

N

Deflection, mm

S/C=10S/C=20S/C=30

No damageExpt

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0 0.5 1 1.5 2 2.5 3 3.5 4

Forc

e, k

N

Deflection, mm

S/C=10S/C=20S/C=30

No damageExpt

(a)

(b)

Fig. 11 Effect of parameter S/C on the force-deflection curveof a 10 mm-notch geometry and b v-notch geometry

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.2 0.4 0.6 0.8 1 1.2

Distance from crack tip, mm

εpα

Fig. 12 The variation of the equivalent plastic strain and thebiaxiality ratio ahead of the crack tip

is known to be due to the work done for creation ofnew surfaces as well as dissipation due to plasticityremote from the crack tip (Tvergaard and Hutchinson1992). The combined effect is manifested as the signif-icant non-linearity in the macroscopic force-deflectionresponse. The evaluation of fracture resistance curve

0

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60

70

0 5 10 15 200

1

2

3

4

5

6

7

8

Forc

e, k

N

Γ/(σ

y D

)

Displacement, mm

Experimentwithout TCZM

C=3.5C=3.0C=2.5

0

10

20

30

40

50

60

70

0 5 10 15 200

1

2

3

4

5

6

7

8

Forc

e, k

N

Γ/(σ

y D

)

Displacement, mm

ExperimentS/C=30S/C=20S/C=10

C=3.45, S/C=10

(a)

(b)

Fig. 13 Effect of a parameter C b parameter S/C , on the force-deflection curve of a CT specimen

requires both the macroscopic response as well ascrack-extension data. In the present study, however, thefocus is primarily on comparison between the macro-scopic force-deflection behaviour from experimentsand simulations.

Figure 13a shows the effect of parameter C onthe force-deflection curve of the CT specimen with acrack-depth ratio of 0.54 keeping parameter S/C at 20.The figure compares the simulation predictions withthe experimental data of the CT specimen. The 2-Dplane strain analysis without cohesive elements showshigh load bearing capacity which does not get compro-mised as crack extension is not allowed in the analysis.However, with triaxiality dependent cohesive elementsincluded in the analysis, the crack extension decreasesthe load bearing capacity. For higher values of parame-ter C , as expected, the behaviour is that of a materialwith higher resistance to both initiation of crack exten-sion as well as continued growth and therefore a higherpeak load is achieved.

For C = 3.0, the effect of parameter S/C is estab-lished as shown in Fig. 13b. For the parameter rangingbetween 10 and 30, higher S/C shows higher resistance

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Table 1 Characteristics ofthe force-deflection curvefrom experiments andsimulations for a fatiguepre-cracked CT specimengeometry

C S Initiation load (kN) Peak load (kN)

2.5 20 32.8 44.93.0 20 34.5 47.63.5 20 36.0 50.13.0 10 30.4 41.53.0 20 34.5 47.63.0 30 37.6 51.6

Fig. 14 The contours of normal stress that acts along direction perpendicular to crack plane at a 90 load b peak load c after 10 drop

to both initiation of crack extension as well as continuedcrack growth. This is primarily because the parameterS/C controls the onset of saturation of the effective tri-axiality. Higher S/C implies saturation at a lower valueof biaxiality ratio as illustrated in Fig. 3 of Banerjee andManivasagam (2009) which in terms of TSL impliesthat higher than a limiting value of biaxiality ratio (αc

as in Eq. 10), the TSL doesn’t change its shape. Thesummary of the effect of both the parameters on theload of initiation of crack extension and the peak loadthat develops is presented in Table 1 illustrating thestrong effect the parameters have on the initiation ofcrack extension and continued crack growth.

To establish the model parameters for the represen-tative mild steel, C = 3.45 from the failure locus ofnotched bar is taken and S/C is adjusted to find a goodcomparison with experimental force-deflection curve.Figure 13b also includes the macroscopic responsewith parameter C = 3.45 in combination with S/Cadjusted to 10 to closely reproduce the experimentalcurve for the CT specimen, such that the peak loadfrom simulation is within 2 value of S/C is thereforethe second model parameter for the given material.

From the analysis, the contour of the normal stressacting along the direction perpendicular to the crackplane at different stages of crack growth is presented inFig. 14a–c. The contour limits are held constant for thethree instances shown: at 90 load and after 10 indicatesthe location of the initial crack tip. Even though theapplied load after 10 same magnitude as at 90 highlystressed zone ahead of crack tip is much bigger in com-parison. This is as a result of much larger crack sizewhich increases the stress intensification ahead of thecrack tip.

With both the model parameters established, C =3.45 and S/C = 10, next the model’s effectivenessis tested in prediction of ductile fracture of specimenswith a/w = 0.51 and 0.65. The experimental force-deflection curves, including that for a/w = 0.54 , areshown in Fig. 15a. All the specimens show initial lin-ear response followed by non-linearity due to onsetof full field plasticity, later initiation of crack growthand continued stable ductile crack propagation resultin load drop. The onset of the non-linearity and thepeak load capacity is directly dependent on the initialcrack size: the shorter the initial crack the higher is the

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load bearing capacity. The corresponding simulations,as shown in Fig. 15b, using the established values forthe model parameters, C = 3.45 and S/C = 10, areable to closely reproduce the peak loads and the overallload reduction due to continued stable crack growth.Interestingly, the simulations are also able to capturethe ’popping’ phenomenon associated with short spurtsof rapid crack growth as observed in the experiments.Since the cohesive parameters of the triaxiality depen-dent cohesive model are independent of stress-state andare able to well approximate the characteristic featuresof the ductile fracture at different triaxialities, they canbe considered to be non-dimensional material constantsof a material.

5 Conclusions

The computational implementation and the experi-mental validation of the triaxiality dependent cohe-sive zone model recently proposed by Banerjee andManivasagam (2009) is presented. To achieve planestrain ductile fracture at different triaxialities, a seriesof flat notched bars of mild steel (IS 2602) with dif-ferent notch radii were tested. From the 2-D planestrain elastic-plastic analysis of the bar geometries, it isestablished that the location of the initiation of failuredepends on both the extent of plastic deformation aswell as the triaxiality of the stress-state. While the crit-ical combination is achieved at the center for notchedbar with large radii, for v-notched bar it is achieved atthe notched tip, as was observed in the fracture surfacemicrographs.

The TCZ model is implemented as cohesive elementsthat use the triaxiality from the neighbouring contin-uum elements in calculation of its traction-separationlaw and thus, its constitutive behaviour. It is shown thatthe model parameter C characterises the toughness ofthe material for fracture under conditions of moderatetriaxiality and it can be determined from experimentsas it is directly related to the equivalent failure locus ofthe notched bars. Apart from giving repeatable failuredata, notched bars of large radii such as 10 mm havebeen found to have no effect of the second parameterand therefore suitable for determination of C .

A parametric study of C and the second model para-meter, S/C , for CT specimen which develops high tri-axiality prior to fracture, shows that both the parametersplay an important role in the initiation of growth as well

0

10

20

30

40

50

60

70

0 5 10 15 20

Forc

e, k

N

Displacement, mm

a/w=0.51a/w=0.54a/w=0.65

0

10

20

30

40

50

60

70

0 5 10 15 20

Forc

e, k

N

Displacement, mm

a/w=0.51a/w=0.54a/w=0.65

(a)

(b)

Fig. 15 Force-deflection curves of CT specimens a from exper-iments b from corrected plane strain analysis

as propagation of a pre-existing nominal defect. The Cparameter obtained from notched bar experiments withan adjusted value of S/C is able to closely reproducethe experimental data obtained from CT specimens. Itis also argued that including the CT specimen data inthe failure locus should be done cautiously as it maylead to inaccurate representation of the material’s frac-ture behaviour.

The present study does show the effectiveness ofthe model in approximating the characteristic featuresof the macroscopic response: peak load, fracture load,location of failure initiation and the equivalent failurelocus at different triaxiality ranging between moderateto high. However, the validation presented is limited inits scope as the focus has been on comparison of themacroscopic response between experiments and sim-ulations. While the macroscopic response does havemanifestations of the fracture process as well as plastic

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dissipation, a detailed study including crack extensiondata for characterisation of crack growth resistancecurves (J -R curves) for different initial crack sizes andin different ductile metals is required to further substan-tiate the versatility of the model. Such a study is part ofan ongoing project by the authors. Also, being a planestrain model its applicability is limited to stress-stateswith triaxiality 1/

√3 or higher.

Acknowledgments Authors gratefully acknowledge the finan-cial support received from ISRO-IITM cell through project titled“Stress-state dependent ductile fracture model”

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Implementation and validation of a triaxiality dependent cohesive model: experiments and simulations

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