Ductile crack growth based on damage criterion ...nptel.ac.in/courses/105108072/mod08/application-12.pdf · Ductile crack growth based on damage criterion: Experimental and numerical

Mechanics of Materials 39 (2007) 623–636

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Ductile crack growth based on damage criterion:Experimental and numerical studies

M. Mashayekhi a, S. Ziaei-Rad a, J. Parvizian a, J. Niklewicz b, H. Hadavinia b,*

a Mechanical Engineering Department, Isfahan University of Technology, Isfahan, Iranb Faculty of Engineering, Kingston University, SW15 3DW, UK

Received 23 December 2005

Abstract

The continuum mechanical simulation of the microstructural damage process is important in the study of ductile frac-ture mechanics. In this paper, the continuum damage mechanics framework for ductile materials developed by Lemaitrehas been validated experimentally and numerically for A533-B1 alloy steel under triaxial stress conditions. An experimen-tal procedure to identify the damage parameters was established and the experimental calibrated damage parameters werethen used in a finite element model. A fully coupled constitutive elastic–plastic-damage model has been developed andimplemented inside the ABAQUS implicit FEA code. The model is based on a simplified Lemaitre ductile damage modelwhose return mapping stage requires the solution of only one scalar non-linear equation. A local crack growth criterionbased on the critical damage parameter was proposed; the validity of this criterion was examined by comparing the sim-ulation with the experimental results on standard three point bending (3PB) test. The critical load at crack growth initi-ation and the fracture toughness, JIc, has also been predicted from the simulation. These numerically predicted valuescompared favourably with those obtained from experiments.� 2006 Elsevier Ltd. All rights reserved.

Keywords: Damage model; Ductile fracture; Crack growth; Triaxiality

1. Introduction

Fracture of engineering components is often pre-ceded by considerable changes in the microstructureof its material. It is now well known that the appli-cability of the J-based single parameter fracturemechanics is restricted to high constraint crackgeometries and materials of low ductility. Forhighly ductile materials, the fracture process zone

0167-6636/$ - see front matter � 2006 Elsevier Ltd. All rights reserved

doi:10.1016/j.mechmat.2006.10.004

* Corresponding author. Tel.: +44 208 547 8864.E-mail address: [email protected] (H. Hadavinia).

is large and the crack tip fields are no longer ade-quately characterized by the J-integral alone.Micro-mechanically based damage models, whichsimulate the physical process of void nucleation,growth and coalescence using continuum mechanicsequations, are among the most promising methodsto investigate fracture behaviour in ductile materials(Lemaitre and Chaboche, 1990). The advantage of amicromechanical damage model, compared withconventional fracture mechanics, is that, in general,the model parameters are only material dependent,and not geometry dependent. The damage model

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624 M. Mashayekhi et al. / Mechanics of Materials 39 (2007) 623–636

allows damage assessments at every point of a struc-ture for any geometry or loading, as long as thedamage mechanisms and stress/strain fields areknown (Lemaitre, 1992).

The development of microstructural damage inengineering materials can be effectively modelledusing continuum damage mechanics (CDM) (Lemai-tre, 1992). CDM introduces a field variable to repre-sent the damage in a continuum sense. In this paper,this concept has later been used to model the initia-tion and growth of cracks.

Accurate failure predictions can only be obtainedif microstructural damage is taken into account inthe fracture modelling. This requirement has led tothe development of the so-called local or continuumapproaches to fracture, in which fracture is regardedas the ultimate consequence of the material degrada-tion process (Lemaitre and Chaboche, 1990; Lemai-tre, 1985). In these methods, the degradation isoften modelled using continuum damage mechanics(Lemaitre, 1985; Rice and Tracey, 1969; Needlemanand Tvergaard, 1984). Continuum damage theoryintroduces a set of field variables (damage variables)which explicitly describe the local loss of materialintegrity. A crack at macro scale size is representedby that part of the material domain in which thedamage has become critical, i.e., where the materialcannot sustain stress anymore. Redistribution ofstresses results in the concentration of deformationand damage growth in a relatively small region infront of the crack tip. It is the growth of damagein this process zone that determines in which direc-tion and at what rate the crack will propagate. Crackinitiation and growth thus follow naturally from thestandard continuum mechanics theory, instead offrom separate fracture criteria.

In the CDM model, the effect of void growth onmaterial behaviour is incorporated by introducingan (internal) damage variable in the constitutiverelation. The effect of void nucleation can also beincluded by modifying the damage growth lawappropriately. But as far as void coalescence is con-cerned, it has to be incorporated as an additionalcondition (in terms of continuum parameters) basedon a suitable micro model. Dhar et al. (1996) com-bined Lemaitre’s (1985) CDM model and Thoma-son’s (1990) void coalescence condition in a largedeformation finite element analysis of different casestudies to show that the critical value of damagevariable is a geometry independent material param-eter and can be used for predicting micro-crackinitiation. The literature contains quite a few local

criteria for crack growth initiation. Ritchie et al.(1973) proposed a critical stress criterion for cleav-age fracture. For mild steel at low temperature, theyfound that predictions from this criterion agreedwell with experimental results. For ductile fracture,Rice and Johnson (1970) expressed the critical cracktip opening displacement in terms of an inter-parti-cle distance. For structural steels, the predictionsfrom the RJ model match well with experimentalresults. Ritchie et al. (1979) used a strain based cri-terion to predict the fracture toughness of A533-Band A302-B alloy steels.

In this paper, an experimental procedure to iden-tify the damage parameters has been established forA533-B1 alloy steel under triaxial stress conditions.The CDM model has been used to simulate the duc-tile damage behaviour of flat rectangular notchedbar specimen tests. The model is validated by model-ling the three point bending fracture specimen (3PB).The comparisons indicate good agreement betweenthe simulated and experimental results. The criticalload for crack growth initiation and the fracturetoughness are predicted by using the proposed crite-rion for crack growth initiation. Comparison ofnumerical and experimental results shows that theproposed criterion reasonably predicts the crackgrowth initiation in Mode-I ductile fracture.

The present paper has been organized as follows.In Section 2, the CDM model formulation is brieflyreviewed and a local criterion for crack growth ini-tiation and propagation is proposed. In Section 3,the experimental procedure used to calibrate thedamage parameters in A533-B1 steel is explained.The CDM model has been used to simulate the duc-tile damage behaviour of flat rectangular notchedbar specimens. Comparisons between the simulatedand experimental results are also presented in thissection. In Section 4, to validate our present crite-rion, experiments have been conducted on standard3PB specimens. By using the damage parametersidentified in Section 2, the CDM model with dam-age propagation criterion are applied to crackgrowth in 3PB tests. The FEA simulation comparedwith the experimental results. In Section 5, the mainhighlights of the work are summarized.

2. Ductile fracture model

2.1. Constitutive law

In the Lemaitre damage model (Lemaitre, 1985),the damage variable is defined as the net area of a

M. Mashayekhi et al. / Mechanics of Materials 39 (2007) 623–636 625

unit surface cut by a given plane corrected for thepresence of existing cracks and cavities. By assum-ing homogeneous distribution of microvoids andthe hypothesis of strain equivalence, which statesthat the strain behaviour of a damaged material isrepresented by constitutive equations of the virginmaterial (without damage) in the potential of whichthe stress is simply replaced by the effective stress(Lemaitre, 1992), the effective stress tensor, ~r, canbe represented as

~r ¼ r

1� Dð1Þ

where r is the stress tensor for the undamaged mate-rial. The corresponding effective stress deviator, ~s, isrelated to the stress deviator, s, by an analogousexpression:

~s ¼ s

1� Dð2Þ

Therefore, the evolution equation for internal vari-ables can be derived by assuming the existence ofa potential of dissipation, W, as a scalar convexfunction of the state variables, which is decomposedinto plastic, Wp, and damage, Wd, components as

W ¼ Wp þWd ¼ Uþ rð1� DÞðsþ 1Þ

�Yr

� �sþ1

ð3Þ

for a process accounting for isotropic hardening andisotropic damage, in which r and s are material andtemperature-dependent properties and U and Y are,respectively, the yield function and the damagestrain energy release rate, given by

Uðr; epeq;DÞ ¼

req

ð1� DÞ � ½r0Y þ Rðep

eqÞ� ð4Þ

and

�Y ¼r2

eq

2Eð1� DÞ22

3ð1þ mÞ þ 3ð1� 2mÞ p

req

� �2" #

ð5Þ

where r0Y is the initial yield stress, R represents the

radial growth of the yield surface, epeq is the equiva-

lent plastic strain, epeq ¼

ffiffiffiffiffiffiffiffi2=3

pkepk, ep is the plastic

strain tensor, req is the equivalent stress, req ¼ffiffiffiffiffiffiffiffi3=2

pksk, p is the hydrostatic stress, p = (1/3)tr(r),

E is the Young’s modulus, and m is Poisson’s ratio.By the hypothesis of generalized normality, the

plastic flow equation is

_ep ¼ _c

ffiffiffi3

2

rs

ksk ¼ _c3

2

s

req

ð6Þ

and the evolution law of the internal variables are

_epeq ¼ � _c

oWoR¼ _c

_D ¼ � _coWoY¼ _c

1

1� D�Y

r

� �s ð7Þ

where _c is the plastic consistency parameter, whichis subject to the so-called Kuhn-Tucker conditions(Simo and Hughes, 1998) for loading and unloadingas

_c P 0; U 6 0; _cU ¼ 0 ð8Þ

The evolution problem is highly non-linear therebyrequiring an efficient integration algorithm, as dis-cussed in Section 2.2.

2.2. Numerical integration for damaged solids

An algorithm for the numerical integration of theelastic–plastic-damage constitutive equations will bepresented in this section. Algorithms based on theoperator split concept, resulting in the standardelastic predictor/plastic corrector format, are widelyused in computational plasticity (Simo and Hughes,1998). Let us consider what happens to a typicalGauss point of the finite element mesh within a(pseudo-) time interval [tn, tn+1].

Given the incremental strain:

De ¼ enþ1 � en ð9Þ

and the values rn, epn, ep

eq;n and Dn at tn, the numericalintegration algorithm should obtain the updatedvalues at the end of the interval, rn+1, e

pnþ1, ep

eq;nþ1

and Dn+1 such that they become consistent withthe constitutive equations of the model.

The elastic predictor/return mapping algorithmfor the elastic–plastic-damage model can be summa-rised as following:

Step 1. Elastic predictor: Given the incrementalstrain, De, and the state variables at tn, computeelastic trial stresses:

eetrialnþ1 ¼ ee

n þ De; eptrialeq;nþ1 ¼ ep

eq;n;

strial ¼ sn þ 2GDe;

ptrial ¼ pn þ KDv Dnþ1 ¼ Dn:

ð10Þ


In the above equations G and K are, respectively,the shear and bulk moduli, e and v denote, respec-tively, the strain deviator and the volumetric strain.

Step 2. Plastic consistency check: First compute

Utrial ¼ffiffiffi3

2

rkstrialkð1� DnÞ

� ½r0Y þ Rðep

eq;nÞ� ð11Þ

and then check:

IF Utrial6 etol THEN (elastic state)

Update (Æ)n+1 = (Æ)trial RETURN

ELSE (plastic state)Step 3. Return mapping: Find Dc such that

DðDcÞ � Dn �Dc

1� DðDcÞ�Y ðDcÞ

r

� �s

¼ 0 ð12Þ

Step 4. Update the variable:

snþ1 ¼ 1�ffiffiffi3

2

r2GDckstrialk

!strial; pnþ1 ¼ ptrial;

rnþ1 ¼ snþ1 þ pnþ1I ; epeq;nþ1 ¼ ep

eq;n þ Dc;

eenþ1 ¼

1

2Gsnþ1 þ

1

3Kpnþ1I ; Dnþ1 ¼ DðDcÞ

ð13Þ

ENDIF

RETURN

The full details of the above algorithm can befound elsewhere Mashayekhi et al. (2005).

Table 1Chemical composition of A533-B1 steel

%C %Mn %P %S %Si %Cr %Ni %Mo

0.19 1.41 0.011 0.011 0.20 0.13 0.69 0.48

The balance is Fe.

2.3. Damage growth model

The fracture of ductile materials is mainly due togrowth and coalescence of microscopic voids exist-ing within the material. In the numerical simulation,when the fracture threshold within an element isreached, that element fractures and a crack occurs.The direction of crack propagation and the cracktip position are then determined by the damageparameter value at each element of the model.

Crack propagation in the structure is often simu-lated with finite element (FE) analysis based on con-tinuum mechanics. There are four common methodsto simulate crack propagation in a finite elementmodel. These are element splitting, nodes separa-tion, decreasing elements stiffness and deleting dam-aged elements. In this paper, the last technique hasbeen used in order to simulate crack propagation.The displacement control method was adopted forloading the structure. After each increment in FE

analysis, the damage parameter for all the elementsin the model was calculated. The initiation of acrack in the structure is assumed to occur at anypoint when the damage parameter reaches its criti-cal value, Dc, a value at which ductile failure willoccur. In our numerical analysis this value at anyGauss point was chosen 0.9. At this stage the dam-aged elements are removed and the crack willadvance.

3. Measurement of damage parameters and numerical

validation

3.1. Material specification

Calibration and validation of the continuumdamage model was conducted on sample of A533-B1 steel, extracted from a 110 mm width rolled steelblock. The chemical composition of the material isgiven in Table 1. Engineering stress–strain proper-ties of the steel block were obtained using standardsmooth round-bar tensile specimens with 25 mmgauge length and 6.2 mm diameter, tested accordingto ASTM Standard E8 in a servo-hydraulic 100 kNcapacity testing machine. The measured yield andultimate stresses of the material are 430 MPa and600 MPa, respectively. Other mechanical propertiesof the material extracted from tests are tabulated inTable 2.

3.2. Identification of the damage parameters

Selection of the damage parameter is one of themost important and most contentious aspects ofdamage mechanics. Many experimental techniquesexist to measure damage or damage related events(Lemaitre, 1992). It is often difficult to performa quantitative measurement with the majority ofthese techniques. The simplest method to do this isby the use of one-dimensional damage measurementthrough tension tests.

The first natural way is that damage is measuredfrom the microstructure, such as measuring of thefraction area of voids at each different plastic strainlevel (Lemaitre, 1992):

Table 2Identified mechanical and damage parameters for A533-B1 steel

Yield stressr0

Y ðMPaÞUltimatestress rTS

(MPa)

Ultimatestrain eu (%)

Young’sModulus E

(GPa)

Poisson’sratio, m

Hardeningcoefficienta k

(MPa)

Hardeningpowera n

Damage parameters

r (MPa) s

430 600 9.3 200 0.3 998 0.18 2.8 1

a Power-law work hardening is: rY ¼ kenp.


D ¼ 1� Aeff

Að14Þ

where Aeff is the effective resisting area and A is theoverall cross-sectional area.

The second method is that damage is measuredthrough physical parameters, such as electricalpotential difference measured at each different plas-tic strain level:

D ¼ 1� V

Vð15Þ

In which V and V are electrical potential differ-ence for the virgin and the damaged material,respectively.

A straightforward manner to quantify the isotro-pic elastic damage variable is to measure thedecrease of the stiffness of the material. Successiveloading and unloading of the material permits tomeasure different stages of damage which is thencomputed through:

D ¼ 1� ED

E0

ð16Þ

where ED is the effective elastic modulus of the dam-aged material, derived from measurements, and E0

is the Young’s modulus of the virgin material.Lemaitre (1985) was the first to measure D

through the degradation of the elastic modulus. Aspecimen similar to the one depicted in Fig. 1, calledthe damage specimen, was loaded in tension. Theplastic strain was recorded by local strain gaugesfixed at the minimum cross-section. The specimen

Fig. 1. The usual specimen geometry for measuring the damageparameter, D, [Lemaitre, 1992]. All dimensions in mm.

is then unloaded and the elastic modulus is mea-sured from the slope of the unloading stress–straincurve. After some level of deformation, typicallyproducing strains less than 0.10, the strain gaugesfail and they are replaced by a new set. The speci-men is further loaded to increase plastic strain andthen unloaded to obtain the current elastic modulus.This process is repeated until a visual crack isdetected. For ductile metals with failure strains ofaround unity, at least ten pairs of strain gauges wereused.

In this work, in order to evaluate the efficiency ofthe CDM model in describing damage evolutionunder stress triaxiality, flat rectangular notchedbar specimens were tested. The experimental testingprocedure is subdivided into the following steps:

(i) Two flat rectangular notched bar specimenswere prepared. The block material is machinedinto rectangular specimens of length 78 mmand minimum cross-sectional area of 8 ·8 mm. The opposite faces of the specimenwere polished parallel to each other. The spec-imen geometry and dimensions are shown inFig. 2a.

Fig. 2. Flat rectangular notched bar specimen. (a) Geometry anddimension (all in mm), (b) FE model of 1/8 of the specimen usingsymmetry condition.

Table 3Measured effective Young’s modulus, plastic strain and damageparameter in flat rectangular notched bar specimen duringdeformation

ED (GPa) ep (mm) D

187.4 0.0445 0.1309168.4 0.0723 0.1720158.3 0.0956 0.2042153.6 0.1095 0.2225149.4 0.1244 0.2413145.6 0.1396 0.2596145.0 0.1425 0.2630140.9 0.1623 0.2853136.3 0.1885 0.3126131.3 0.2231 0.3447122.4 0.3064 0.4038118.1 0.3613 0.4286


(ii) Strain gauges with size of 10 · 8 mm wereattached to the minimum cross-section of thespecimen using epoxy glue; the strain gaugeresistance was 120 X and its nominal deforma-tion limit is ±20.0%. A sample flat rectangularnotched bar specimen with the attached straingauge is shown in Fig. 3.

(iii) A monotonic load was applied to the specimenunder displacement control. The rate of dis-placement has been fixed at 0.5 mm/min on aservo-hydraulic Instron testing machine, witha load capacity of 100 kN.

(iv) The tests were carried out with a series ofpartial unloading–reloading to measure thechange in the elastic slope while the strainincreased.

(v) The tests were continued until the deformationlimit for the strain gauge is reached. Then thespecimen was unloaded and removed from thetesting machine and a new strain gauge wasattached, as required.

The test results are presented in Table 3. Thecoefficients s and r are calibrated from these results.According to experimental results reported byLemaitre (1992), the s parameter for this materialwas assumed to be 1. For the one-dimensional case,Eq. (5) changes to:

�Y ¼ r2 � 1

2Eð1� DÞ2ð17Þ

Then, we can substitute Y into Eq. (7) and deter-mine dD/dep:

Fig. 3. Specimen used for the calibration of damage parameters (a) sp

dDdep¼ r2

2Erð1� DÞ2ð18Þ

dD/dep can be calculated from the slope of the linefitted to Di and ep

i data. The parameter r, can thenbe obtained from Eq. (18):

r ¼ r2

2Eð1� DÞ2 dDdep

ð19Þ

Up to 12 data points were considered in order tocalculate an accurate r.

A sample load versus displacement diagram mea-sured on a flat rectangular notched bar specimen isshown in Fig. 4. For this specimen, three sets ofstrain gauges were used to measure the deformationof the specimen until the sudden failure of the last

ecimen before loading (b) specimen with attached strain gauge.

0

10000

20000

30000

40000

50000

0 0.5 1 1.5 2 2.5

Displacement (mm)

Loa

d (N

)

First gauge Second gauge Third gauge

Fig. 4. Experimental load–displacement at the strain gauge position in flat rectangular notched bar specimen.


strain gauge. The loading–unloading stages used fordamage measurement are clearly visible. Part of anelastic unloading–reloading ramp is enlarged inFig. 5 showing a limited hysteretic loop. Accordingto the suggestions made by Lemaitre (1992), theYoung’s modulus was measured during the unload-ing ramp. However, the uncertainty in the estima-tion of the Young’s modulus during either theunloading or the elastic reloading is less than 2%.The calibrated damage parameters extracted fromthe tests results are summarised in Table 2.

3.3. Finite element model of flat rectangularnotched bar

In reality, the state of stress in a flat rectangularnotched bar specimen geometry does not completelyfollow fully plane stress or plane strain conditions.An accurate study of the evolution of stress triaxial-ity with plastic strain across the minimum sectionrequires a three-dimensional finite element simula-tion. Simulations have been carried out withABAQUS Standard code and a user’s subroutine

17000

22000

27000

32000

37000

0.134 0.144 0.1

Displacem

Loa

d (N

)

Fig. 5. A stage of elastic unloading–reloadi

UMAT. Due to symmetry, only one fourth of thespecimen has been modelled by using eight-noded,isoparametric, hexahedral elements (Fig. 2b). Inorder to have a detailed map of stress and damage,the minimum section was meshed with evenlyshaped bricks with a side length of 0.2 mm. The pre-dicted load–displacement results from FEA arecompared with the experimental values in Fig. 6.The FEA results follow very closely the experimen-tal values. The maximum difference was found to beless than ±10%.

In the next step, the damage evolution across theminimum section was analysed in order to determinethe location of first ductile failure. The contour plotsof damage evolution are shown in Figs. 7a–c. Theseresults shows that ductile failure does not initiate atthe centre of the minimum section as it would beexpected. Here, the competition of stress triaxialityand plastic strain accumulation, controlled by thenotch effect, determines the failure initiation site tobecome near the notch root. From there, ductilefracture spreads across the minimum section. Thefailure of the elements along the minimum section

54 0.164 0.174

ent (mm)

ng for measurement of elastic moduli.

0

10000

20000

30000

40000

50000

0 0.5 1 1.5 2.0 2.5

Exp.data

FEM+CDM

Loa

d (N

)

Displacement (mm)

Fig. 6. Comparison of experimental and FE simulation of load-displacement at the strain gauge position in flat rectangular notched barspecimen.

Fig. 7. Damage evolution across the minimum section (notch). (a) At applied displacement of 0.04 mm, (b) at applied displacement of1 mm, (c) at applied displacement of 1.2 mm.


occurs in a few load increments. Once a ductile crackinitiated by the failure of few elements, catastrophicfailure immediately follows it.

4. Continuum approach to fracture

After a certain amount of loading which resultsin some damage growth, three regions can generallybe distinguished in the material domain S0 (Fig. 8).

In region S0, the damage variable has its initialvalue (D = 0) and the material properties are thoseof the virgin material. In the second region, Sd,some development of damage has occurred, butthe damage is not yet critical, i.e., 0 < D < 1. Thelimiting value D = 1 has been reached in the thirdregion, i.e., Sc. The mechanical integrity andstrength of the material have been completely lostin this region. The completely damaged region, Sc,

SC: D=1

S0: D=0

Sd: 0<D<1

Fig. 8. Damage distribution in a solid.


is the continuum damage representation of a crack.It is important to realize that the complete loss ofstrength in Sc, implies that in this region stressesare identically zero for any arbitrary deformationfields.

During the initiation stage, the damage variablesatisfies D < 1 everywhere in the domain S and nocracks are therefore present. Cracks initiate at posi-tions when the damage parameter, D, approaches itscritical value, i.e., D = 1. As soon as the crack hasbeen initiated, the deformation field contains a dis-continuity. This means that the most critical pointin front of the crack tip will fail instantaneouslyand the crack starts to grow.

In this section, the prediction of crack initiation,crack propagation and resistance curve of J � Da

for a three point bending (3PB) specimen of theA533-B1 steel has been carried out using the dam-age growth model suggested in Section 2.3.

4.1. Criterion for setting the element size in FEA

The FE analysis will be valid in the continuummechanics sense if an appropriate element size waschosen in the model. The upper and lower boundssize of the elements in the model can be set accord-ing to the following rules.

Fig. 9. Element size limits: (a) low

In order to satisfy the requirements of a contin-uum media, the element size should be many timesgreater than the size of the material grain size

d� q ð20Þwhere q is the mean grain size and d is the elementsize around the crack tip (see Fig. 9).

On the other hand, since local damage processesoccur essentially within the plastic zone, the size ofthe elements must be smaller than the size of thesmallest plastic zone possible around the stressconcentrator:

d� rpl ð21ÞThe mean material grain size of most steels used

in engineering application is about q 6 0.01 mm.Assuming that the requirements of continuummechanics are satisfied if the element size exceedsfive times the grain size, according to Eq. (20) theminimum size for the element will be

dmin ¼ 0:05 mm ð22ÞAn estimate for the size of the plastic zone at a

crack tip can be obtained using Dugdale-Barenblattcohesive zone model. Based on their analysis, theplastic zone size for plane strain condition is(Anderson, 1995):

rp ¼1

3pK2

c

r2Y

ð23Þ

where Kc is the crack tip stress intensity factor.The ratio of K2

c=r2Y in Eq. (23) can be varied be-

tween 1 mm and 200 mm for commonly used steels.Thus the smallest plastic zone size can be estimatedas:

dmax <1

3p� 12 ¼ 0:106 mm ð24Þ

From the above discussion, one can conclude thatfor commonly used steels, the element size shouldvary between 0.05 mm 6 d < 0.106 mm.

er bound, (b) upper bound.


In this study, the element size around the cracktip was set at 0.1 mm and remained the same forall the simulations to minimize the effect of meshdependency.

4.2. Experiment on three point bending specimen

Fracture toughness testing was performedaccording to the ASTM E1820-99 standard 3PBspecimen. The geometry and dimension of thespecimen is shown in Fig. 10. The notch was‘‘sharpened’’ by ‘‘precracking’’ using cyclic fatigueloading. The final precrack length (notch plus pre-

Fig. 10. Experimental setup for three point bending test (a) exper

0

20000

40000

60000

80000

0 1 2

Loa

d (N

)

Load line dis

Fig. 11. Load versus load-line-displaceme

crack) was 19 mm, with a presumed atomicallysharp crack tip. The specimen was then loaded tofailure under displacement control with a UniversalTesting Machine Model 1195 at ambient tempera-ture with a cross-head displacement rate of0.5 mm/min. The magnitude of the loads and thecorresponding displacements were simultaneouslyrecorded during the test. Fig. 11 presents the loadversus load-line-displacement for the 3PB specimen.In order to calculate JI value at a certain load level,the following formulae were used (ASTM, 1999):

J I ¼2ABb

ð25Þ

imental setup (b) specimen geometry. All dimensions in mm.

3 4 5placement (mm)

nt in three point bending specimen.


where A is the energy absorbed during loading,measured by the area under the tensile loadingcurve, B is the thickness of the specimen and b isthe ligament length.

A J � Da resistance curve is produced accordingto ASTM E1820, from which the critical fracturetoughness JI c, corresponding to ductile crack initi-ation, is calculated. The resistance curve (J � Da)illustrated in Fig. 12. Exclusion lines are drawnat crack extension (Da) values of 0.15 mm and1.5 mm. The slope of the exclusion lines corre-sponds approximately to the component of crackextension that is due to crack blunting, as opposedto ductile tearing. All data that fall within the exclu-sion limits are fit to a power-law expression:

J ¼ C1ðDaÞC2 ð26Þ

where C1 and C2 are constants and for our accepteddata C1 = 350 and C2 = 0.4632.

The JQ is defined as the intersection betweenpower-law expression and 0.2 mm offset line. If all

0

100

200

300

400

500

600

0 0.5 1

CRACK EXTENS

J (k

Pa.m

)

350(QJ Δ= a

Fig. 12. The resistance curv

Fig. 13. Finite element model of the three point bending test. (a) Globathe position O.

other validity criteria are met, JIc = JQ as long asthe following size requirements are satisfied:

B; b P25J Q

rY

ð27Þ

The value of JIc is found to be 240 kN/m.

4.3. Simulation of crack growth in 3PB test

A finite element model was constructed to evalu-ate how well the CDM predicts the measuredfracture initiation from the test. Also, the transfer-ability of damage parameters to fracture mechanicstest specimens ought to be validated. The A533-B1steel was characterised by fracture test of three pointbending specimen in previous section. Since for thegiven tests B,b P 25JQ/rY, a plane strain state couldbe expected, allowing a two-dimensional FE analy-sis. The analysis was performed by using ABAQUS/standard FEA software. A representative finiteelement mesh for one half of the 3PB model with

1.5 2 2.5

ION Δa (mm)

Power Law Regression Line

Points used for regression

Points out of domain

Blunting line

0.2 mm offset line

0.15 mm Exclusion line

1.5 mm Exclusion line

Exclusion line

0.4632)

es for A533-B1 steel.

l mesh, (b) local mesh near the crack tip. The initial crack tip is at


a/W = 0.48 is shown in Fig. 13. The size of the ele-ments surrounding the crack tip is about 0.1 mm.The stable ductile crack growth is simulated auto-

Fig. 14. Simulation of crack growth in three point bending test. (a) Dethe end of loading.

matically by the damage model and leads to a chainof failed elements along the ligament. The details inFig. 14 show how the crack propagates to form a

tails of local mesh near the crack tip, (b) details of global mesh at

0

100

200

300

400

500

600

0 0.2 0.4 0.6 0.8 1

CRACK EXTENSION Δa (mm)

J (k

Pa*m

)

FEM+CDM

exp. data

0.2 mm offset line

Fig. 15. Experimental and FE simulation of crack resistance curves for A533-B1 steel.


blunted crack tip. In Fig. 14a the evolution of dam-age growth around the crack tip and advance of thecrack front has been consequently shown. Byincreasing the loading, the damage around the cracktip will grow until eventually the first element willreach to the critical damage parameter, Dc. At thisstage this element will be removed from the modeland the crack will advance as much as the deletedelement length. The loading will continue toincrease and the above process will be checked atthe end of each loading increment. This process con-tinues until final failure of the specimen. In Fig. 14bthe final step of the crack growth in 3PB test hasbeen shown. A few cases of FE simulation withcrack length less than 0.1 mm were also performedand no significant difference in the crack path wasobserved.

The far field J-integral was also computed at allincrements. The crack growth resistance curveJ � Da was obtained from these analyses. Theseresults were compared with the experimental J � Da

in Fig. 15 with very good agreement. This confirmsthat the experimental damage parameters calibra-tion is very accurate and the measured J � Da curveis very well predicted by the simulations. The resultsof JIc value from FE analysis of 3PB specimen is260 kN/m, while the experimental value of JIc for3PB specimen is 240 kN/m. The simulated value is8.3% higher than the experimental value. Thisdiscrepancy partly attributed to the different condi-tions in two-dimensional plane strain FE model andthe condition in the real three-dimensional test.

5. Conclusions

In this paper, the effect of stress triaxiality onductile damage evolution in metals was investigatedfrom both experimental and theoretical points ofview. Ductile fracture process is influenced by boththe plastic strain and triaxiality. Stress triaxialityplays a major role on the damage evolution, whichis demonstrated by the progressive reduction ofmaterial ductility under increasing triaxial states ofstress. These effects have been studied by examiningdamage evolution in notched flat rectangular barspecimen. The CDM model predictions are in verygood agreement with the experimental damage mea-surements. The validity and transferability of dam-age parameters were checked by applying them to3PB specimen commonly used for materials testingin damage mechanics.

The elastic–plastic-damage analysis was success-fully applied to study ductile fracture. Furthermore,by numerical simulation of ductile crack growth,fracture toughness values were successfully deduced.Comparison of the computer simulated results withthe experimental data showed that the overall agree-ment is satisfactory. The proposed crack growth ini-tiation criterion is reasonably good in explainingMode-I fracture.

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