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Comput MechDOI 10.1007/s00466-012-0739-1
ORIGINAL PAPER
Numerical study of electron beam welded butt joints with the GTNmodel
Haoyun Tu · Siegfried Schmauder · Ulrich Weber
Received: 29 March 2012 / Accepted: 25 May 2012© Springer-Verlag 2012
Abstract The fracture behavior of S355NL electron beamwelded steel joints is investigated experimentally and numer-ically. The simulation of crack propagation in an electronbeam welded steel joint was performed with the Gurson–Tvergaard–Needleman (GTN) damage model. A parameterstudy of the GTN model was adopted which reveals the influ-ence of parameters on the material behavior of notched roundand compact tension specimens. Based on the combinedmethod of metallographic investigations and numerical cal-ibration, the GTN parameters were fixed. The same param-eters were used to predict the ductile fracture of compacttension specimens with the initial crack located at differentlocations. Good match can be found between the numericaland experimental results in the form of force versus CrackOpening Displacement as well as fracture resistance curves.
Keywords Parameter study · Gurson–Tvergaard–Needle-man model · Crack propagation
1 Introduction
Since the invention, the welding technique has found wideapplications in industry fields as its weight saving effectcompared to riveting is attractive. Weldments are frequentlyapplied in critical components of the heat transport piping ofnuclear reactors [1], integral parts of power plant equipments[2], pressurized water reactors [3], boiling water reactors [4]
H. Tu · S. Schmauder (B) · U. WeberInstitute for Materials Testing, Materials Science and Strengthof Materials (IMWF), University of Stuttgart, Pfaffenwaldring 32,70569 Stuttgart, Germanye-mail: [email protected]
S. Schmaudere-mail: [email protected]
and offshore pipelines [5]. Compared to the traditional arcwelding technique, electron beam welding (EBW) has lowerheat input per unit length of the weld, which results in anarrow heat affected zone and little distortion and residualstress. For thick workpieces, joints obtained from multi-passarc welding can be replaced by a single pass EBW, which ledto less energy consumption during welding process. RecentlyEBW are used widely in transportation and energy indus-tries. Their failure always draws attention as the fracturebehaviour of the welded joints influences the crack growthof structures, which affects the lifetime and safety of compo-nents. The structural steel S355 and its weldments are usedwidely building the wind-powered tower tube [6] or the float-ing support structures for offshore wind turbines [7]. Thesewelded structures are critical components for the wind toweror wind turbines, they influence the stability of wind powerplant components during the creation of the power and theycan help to prevent from fracture of these structures influ-encing the service lives of the wind power facilities. Withthe development of the finite element method, attention hasbeen focused on the fracture behaviour of welded joints in anumerical way. In this paper, a damage model was adopted tostudy the fracture behaviour of S355 EBW joints, as appliedby wind energy industries.
Various models exist for the description of void growthin ductile materials. McClintock [8] in 1968 developed amodel which approximates a single void by a cylindricalhole in an unbounded domain subjected to a radial tensilestress. Similarly, Rice and Tracey [9] in 1969 investigatedthe growth of a single void in an infinite elastic–plastic solid.Based on the fundamental work of Bishop and Hill [10], thewidely known porous ductile material model is that devel-oped by Gurson [11], and later modified by Tvergaard andNeedleman [12,13]. Due to the development of computertechnology and the desire to use damage models for studying
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Table 1 Chemical composition of the steel S355NL, mass contents in %
Steel C Si Mn P S Cr Mo Ni Al Co
S355NL 0.198 0.260 1.386 0.026 0.013 0.020 <0.005 <0.005 0.013 0.006
the fracture behaviour of materials, the Gurson–Tvergaard–Needleman (GTN) model [11–13] was used to predict crackpropagation in homogeneous and inhomogeneous structures[14–16]. After initial analyses of the fracture behaviour of alaser welded joint [17,18] the GTN model was used to studythe ductile failure of a laser welded joint [19–22]. Like thesuccessful application of the Rousselier model studying thefracture behaviour of an EBW joint [23,24], the GTN modelwill be used here to study the fracture behaviour of EBWjoints.
The paper is divided into three parts. In the first part of thispaper, experimental investigations on the base material andfor welded joints are presented, which provide the dimen-sions of the finite element model and the model input data.The second part of this paper is devoted to the parameterstudy of the GTN model. The influences of the GTN param-eters on the material behaviour are shown; this can help tofind appropriate numerical GTN parameters. The last part isdevoted to the application of the same parameters obtainedfrom the numerical calibration to study the fracture behaviourof compact tension (C(T)) specimens.
2 Experimental investigations
Low-alloyed structural steel S355NL was chosen in theexperiments as the base material (BM) to produce weldedjoints. After the electron beam welding process, a butt jointis obtained from two S355NL plates with the thickness of60 mm. The chemical components of S355NL are shown inTable 1, which is obtained from spectrometric analysis.
In order to identify the different weld regions, especiallythe fusion zone (FZ) and the heat affected zone (HAZ), thehardness is measured across the weld, as shown in Fig. 1.From the hardness test results, see Fig. 2, the dimensionsof FZ and HAZ are found to be 2.8 and 3.1 mm, respec-tively. The mechanical properties of different weld regions ofS355 EBW joints are obtained from flat specimens along theweld line of which the gauge length is 50 mm. These stressstrain curves are used as finite element model input data.Table 2 shows the mechanical properties of base material(BM) and fusion zone (FZ) of the welded joints containingyield strength Re, tensile strength Rm, uniform elongationAg and strain at rupture A.
After surface etching, the microstructures of differentweld regions are shown in Fig. 3a–c. As shown in Fig. 3a, the
Middle
Root
Top
weld centre
FZ HAZ BM
Fig. 1 Hardness test across the weld joint
-15 -10 -5 0 5 10 150
50
100
150
200
250
300
HAZ BMFZHAZ
Vic
kers
Har
dnes
s (H
V5)
Distance from the weld centre (mm)
Top Middle Root
BM
Fig. 2 Profile of hardness results cross the weld
BM shows typical dual phase structures, which is comprisedof Ferrite and Perlite. The microstructure of the HAZ consist-ing of Ferrite, Perlite and Martensite around the fusion lineis shown in Fig. 3b, which is a transitional region betweenthe BM and the FZ. In Fig. 3c, acicular Martensite structuresobtained after the welding process can be found in the FZ.This can explain the brittle behaviour of the fusion zone.
In order to identify the volume fraction of non-metallicinclusions and the average distance between two neighbour-ing inclusions, metallographic investigations on polished
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Table 2 Mechanical properties of different weld regions of S355NL EBW joint
RB Me (MPa) RF Z
e (MPa) RB Mm (MPa) RF Z
m (MPa) AB Mg AF Z
g AB M AF Z
348 513 533 687 0.151 0.037 0.246 0.052
(a)
Ferrite Martensite
(c)(b)
Perlite
weld seam
Fig. 3 Microstructures of different weld regions of an S355 EBW joint: a base material (BM), b heat affected zone (HAZ) and c fusion zone (FZ)
surfaces of the specimens were performed, both for BM andHAZ. In Fig. 4, it can be seen that the inclusions are notequally distributed in the material but are rather localized orclustered in some regions. Quantitative analyses were madein order to quantify these experimental observations. Afterquantitative analyses of the microscope results, the volumefraction of inclusions for the BM and HAZ is found to be0.0009 and 0.0016, respectively. The mean distances betweenneighbouring inclusions for BM and HAZ are 0.10 and 0.12mm, respectively. The fracture surfaces of unnotched roundspecimens obtained from the BM are shown in Fig. 5. Typ-ical dimple ductile fracture characteristics are found, wherelarge voids are next to smaller voids.
In this work, fracture toughness tests were performed onC(T) specimens with different configurations, i.e. the initialcrack is created in the BM, in the middle of the FZ and at theinterface between the FZ and the HAZ, separately. Standardcompact tension specimens (C(T)25) with 20 % side grooveswere chosen for the test. During testing, the load was con-trolled by quasi-static displacements. After the C(T) test, theexperimental results are shown in terms of force versus crackopening displacement (COD) as well as fracture resistance
JR curves. As the FZ shows higher tensile strength valuescompared to that of the BM, see Fig. 6, a C(T) specimenwith a crack in the FZ shows higher fracture toughness com-pared to the BM as shown in Fig. 7. For a C(T) specimen witha crack in the FZ, the specimen suddenly ruptures, showinga more brittle fracture behaviour.
The fracture surfaces of C(T) specimens extracted fromthe BM and HAZ are displayed in Fig. 8a–b. The stable crackgrowth regions show typical dimple fracture characteristics,large voids are next to smaller voids. In Fig. 8a, broken parti-cles are visible at the bottom of large dimples. For the HAZ,a similar structure can be found at the same magnification(Fig. 8b). Both these pictures indicate that the fracture of theBM and the HAZ is controlled by void nucleation, growth andvoid coalescence during deformation. For the FZ–C(T) sit-uation, similar dimple fracture structures are also observed,however smaller ductile regions compared to the BM–C(T)and HAZ–C(T) are obtained in the fracture surface pointingat a reduced ductility. Furthermore, besides the dimple areas,some flat regions (marked regions in Fig. 8c can be found onthe fracture surface, which are due to the unstable brokenway of the FZ. These evidences can explain the less ductile
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Fig. 4 Typical micrographs of different weld regions of an S355 EBW joint: a base material and b heat affected zone (dark regions: inclusions)
Fig. 5 SEM pictures of the fracture surface for unnotched round spec-imen extracted from S355 BM under different magnifications: a over-view of fracture surface, b enlarged image of marked region of picture
(a), c enlarged image of marked regions of picture (b), and d enlargedimage of marked region of picture (c)
behaviour of the FZ. In this paper, the GTN model has beenapplied to study the ductile crack growth of the BM and theHAZ structures.
3 Damage model
In the frame of continuum damage mechanics, a model forporous metal plasticity was presented by Gurson [11], laterthe original yield condition was modified by Tvergaard and
Needleman [12,13]. The equation of the so called Gurson–Tvergaard–Needleman model (GTN) model is as follows:
φ =(σeq
σ
)2 + 2q1 f ∗ cosh(q2σm
2σ
)− 1 − q3 f ∗2 = 0, (1)
where σeq is the von Mises equivalent stresss, σm is the hydro-static stress and σ is the flow stress for the matrix material ofthe cell, while f represents the current void volume fraction.The parameters q1, q2 and q3 are parameters introduced byTvergaard [25]. The f ∗ is a function of f which represents
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Fig. 6 Stress–strain curves obtained from from tensile test results offlat specimens extracted from different weld regions of S355NL elec-tron beam welded joints
the accelerated damage originated from the void coalescencewhen a critical volume fraction fc is reached:
f ∗ ={
f for f ≤ fC
fC + k( f − fC ) for f> fC, k = f ∗
U − fC
fF − fC,
(2)
According to this model, a crack appears when the currentf value reaches the final void volume fraction, ff . The mate-rial looses its stress carrying ability totally when the ultimatevoid volume fraction f ∗
u is reached, where f ∗u = 1/q1.
The rate of the void volume fraction consists of thegrowth of existing voids and the nucleation of new voids,as explained by Rice, and Needleman [26,27]:
0
10
20
30
40
50
F (
KN
)
CT--BM CT--FZ CT--HAZ
3.02.52.01.51.00.50
COD (mm)
Fig. 7 Force versus crack opening displacement (COD) curves of com-pact tension (C(T)) specimens with the initial crack located in the basematerial (BM), the fusion zone (FZ) and the heat affected zone (HAZ),respectively
f = fgrowth + fnucleation . (3)
The void growth is given according to the following equation:
fgrowth = (1 − f )εplkk , (4)
where εplkk is the plastic volume dilatation rate.
When the matrix behaves plastically incompressible, Chuand Needleman [27] suggest the void nucleation rate is givenas follows:
fnuclation = fn
sn√
2πexp
[−1
2
(ε
pleq − εn
sn
)]ε pl , (5)
(a) (b) (c)
Prenotch from Electron Discharge Machining (EDM)
fatigue pre-crack
Stable fracture
EDMfatigue pre-crack
Stable fracture
EDMfatigue pre-crack
Stable fracture
Fig. 8 Fracture surfaces for S355NL: a BM, b HAZ and c FZ
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Table 3 Parameters used in the parameter study with the GTN model
f0 fc fn ff εn sn q1 q2 lc (mm) f ∗U
0.0001 0.05 0.01 0.2 0.2 0.1 1.5 1 0.05 0.65
where fn is the void volume fraction of the void-nucleatingparticles, εn is the mean strain when void nucleation occursand sn is the corresponding standard deviation of the nucle-ation strain.
In this paper, all simulations are performed on ABAQUSplatform with the GTN model as a user subroutine (UMAT)[28].
4 Parameter study using the GTN model
From the previous explanation it can be derived that thereexist nine GTN parameters to be defined before beingapplied to the simulation. The parameters (εn, fn, sn) are usedto describe the void nucleation, the parameters (f0, fc, ff )describe the procedure of damage from void initiation to voidcoalescence until final failure and the parameters (q1, q2, q3)
are model parameters. As described by Steglich [29], the ini-tial void volume fraction f0 interacts with the volume fractionwhen void coalescence happens fc and the volume fraction ofvoid nucleating particles fn. It is very difficult to judge whichparameter plays the major role with respect to the simulationresults. Thus, the combined influence of these parametersis investigated in this work. Moreover, additional parameterstudies have been undertaken and are described below. Inthis paper, we consider the following values for q1 = 1.5,q2 = 1, q3 = 2.25, which are suggested by Tvergaard andNeedleman [30,31].
The material parameter set used for the parameter studyis shown in Table 3. Our method is to change one param-eter at one time while keeping the rest parameters constantduring the simulation. In this section, the parameter studywill show how the GTN parameters influence the simulationresults. The influence of parameters on the fracture positionand the final slope of force versus diameter reduction curvesof notched round specimens and the slope of JR curves ofC(T) specimens are shown in the following.
4.1 Influence of f0
The initial void volume fraction f0 is taken as the volumefraction of inclusions. As can be found in Fig. 9, the f0 valueinfluences the fracture position of the F–�D curve of notchedround specimens. Higher f0 values let the final fracture hap-pen earlier while the slope of the curves after the final fracturepoint is not affected by the f0 value. In Fig. 10, the value f0
0
5
10
15
20
25
0.0001
F (
KN
) f0=0.0001
f0=0.0004
f0=0.0006
f0=0.0008
f0=0.001
1.81.61.41.21.00.80.60.4 2.00.20.0
ΔD (mm)
Notch radius R=4mm
0.001
Fig. 9 Force versus diameter reduction curves for different f0 values
0
100
200
300
400
500
f0=0.0001
f0=0.0004
f0=0.0006
J-In
tegr
al (
N/m
m)
3.02.52.01.51.00.50.0
Δa (mm)
Fig. 10 Fracture resistance curves for different f0 values
results in a significant influence of the slope of the fractureresistance (JR) curve. Higher f0 values result in a more flatJR curve because crack coalescence can be reached earlierwith higher f0 values, which means less energy is required forcrack propagation. The initial void volume fraction is foundto be the parameter of highest impact on the J-energy resultsand thus most relevant for crack propagation.
4.2 Influence of fc
The critical volume fraction fc is the volume fraction whenvoid coalescence happens. As can be seen in Fig. 11, the fc
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0
5
10
15
20
25
fc=0.01
fc=0.02
fc=0.03
fc=0.04
fc=0.05
F(K
N)
1.81.61.41.21.00.80.60.4 2.00.20.0
ΔD (mm)
Notch radius R=4mm
Fig. 11 Force versus diameter reduction curves for different fc values
0
100
200
300
400
500
fc=0.01
fc=0.02
fc=0.03
fc=0.04
fc=0.05
J-In
tegr
al (
N/m
m)
3.02.52.01.51.00.50.0
Δa (mm)
Fig. 12 Fracture resistance curves for different fc values
value influences the fracture position of the F–�D curves ofnotched round specimens. Higher fc values let the final frac-ture happen later while the slopes of the curves after the finalfracture point are not affected by the fc values. In Fig. 12,the value fc creates also a significant influence on the slopeof the fracture resistance (JR) curve. Higher fc values meanthe threshold for void coalescence is higher; more energy isrequired to drive the crack propagation, resulting in steeperJR curves.
4.3 Influence of ff
The final void volume fraction is the volume fraction whenthe material looses its stress carrying ability completely.Lower ff values let the final fracture happen earlier. As canbe seen from Fig. 13, for notched round specimens, the ff
value does not influence the fracture position, however, itinfluences the slope of the curve after fracture; a steeper
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.00
5
10
15
20
25
0.35
Notch radius R=4mm
ΔD (mm)
F (
KN
)
ff=0.15
ff=0.20
ff=0.25
ff=0.30
ff=0.35
0.15
Fig. 13 Force versus diameter reduction curves for different ff values
0
100
200
300
400
500
ff=0.15
ff=0.20
ff=0.25
ff=0.30
J-In
tegr
al (
N/m
m)
3.02.52.01.51.00.50.0
Δa (mm)
Fig. 14 Fracture resistance curves for different ff values
slope can be obtained with a lower ff value. The ff valuedoes not influence the fracture resistance (JR) curve dramat-ically; moreover, all the JR curves are parallel, as shown inFig. 14.
4.4 Influence of fn
The fn value is the volume fraction when new void initiationhappens during deformation. More new voids let the materialfail earlier. As can be found from Fig. 15, the fn value influ-ences the fracture position of the F–�D curve of notchedround specimens. Higher fn values can lead to earlier failureof the specimen while the slope of all the curves after frac-ture initiation is constant. The fn value does not influence thefracture resistance (JR) curve too much; all the JR curves areparallel, as shown in Fig. 16.
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0
5
10
15
20
25F
(K
N)
fn=0.01
fn=0.011
fn=0.012
fn=0.013
fn=0.014
1.81.61.41.21.00.80.60.4 2.00.20.0
ΔD (mm)
Notch radius R=4mm
Fig. 15 Force versus diameter reduction curves for different fn values
0
100
200
300
400
500
fn=0.01
fn=0.011
fn=0.012
fn=0.013
fn=0.014
J-In
tegr
al (
N/m
m)
3.02.52.01.51.00.50.0
Δa (mm)
Fig. 16 Fracture resistance curves for different fn values
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.20
5
10
15
20
25
εn=0.2
εn=0.25
εn=0.3
εn=0.35
εn=0.4
Notch radius R=4mm
ΔD (mm)
F (
KN
)
Fig. 17 Force versus diameter reduction curves for different εn values
0
100
200
300
400
500
εn=0.20
εn=0.25
εn=0.30
J-In
tegr
al (
N/m
m)
3.02.52.01.51.00.50.0
Δa (mm)
Fig. 18 Fracture resistance curves for different εn values
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.00
5
10
15
20
25
ΔD (mm)
Notch radius R=4mm, lc=0.05mm
F (
KN
)
experiment--BM f
c=0.03, f
0=0.0001, ε
n=0,2, f
n=0.01
fc=0.05, f
0=0.0001, ε
n=0,2, f
n=0.01
Fig. 19 Comparison of experimental and numerical force versus crosssection reduction curves for notched round specimens extracted fromthe BM when fc = 0.05 and fc = 0.03 are used
4.5 Influence of εn
The εn value is the mean strain when void nucleation happens.The influence of the εn value is interrelated to the influenceof fn. The εn values has the same influence as that of fn, how-ever, with opposite in direction. As can be seen in Fig. 17,higher εn values result in later broken position of the F–�Dcurves. The influence of εn on the JR curves can be foundin Fig. 18, higher εn values result in a higher the fractureresistance (JR) curve.
5 Numerical application of the GTN model
The parameter set shown in Table 3 is used to study thefracture behaviour of the base material and of welded struc-tures. The distances between voids lc (lc = 0.05mm for
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(a)
0
10
20
30
40
50lc=0.05 mm
f0=0.0001, f
c=0.05, f
n=0.01, ε
n=0.2
3.02.52.01.51.00.50
F (
KN
)
COD (mm)
experiment--BM GTN
(b)
0
100
200
300
400
500lc=0.05 mm
experiment--BM GTN
f0=0.0001, f
c=0.05, f
n=0.01, ε
n=0.2
J-In
tegr
al (
N/m
m)
2.52.01.51.00.50.0
Δa (mm)
Fig. 20 Comparison of experimental and numerical a force versus crack opening displacement (COD) curves, and b fracture resistance curves forC(T) specimens with initial crack located in the BM (lc = 0.05mm, fc = 0.05)
(a)
0
10
20
30
40
50lc=0.05 mm
f0=0.0001, f
c=0.03, f
n=0.01, ε
n=0.2
3.02.52.01.51.00.50
F (
KN
)
COD (mm)
experiment--BM elastic-plastic GTN
(b)
0.0 0.5 1.0 1.5 2.0 2.50
100
200
300
400
500
J-In
tegr
al (
N/m
m)
Δa (mm)
experiment--BM GTN
f0=0.0001, f
c=0.03, f
n=0.01, ε
n=0.2
Fig. 21 Comparison of experimental and numerical a force versus crack opening displacement (COD) curves, and b fracture resistance curves forC(T) specimens with initial crack located in the BM (lc = 0.05mm, fc = 0.03)
BM and lc = 0.1mm for HAZ) are chosen according to theprevious results in the Rousselier model [24]. As shown inFig. 19, the simulated F–�D curves of notched round spec-imens extracted from the BM are very close to the experi-mental ones when fc = 0.05 or fc = 0.03. The same GTNparameter set as in Table 3 except different fc values is usedto study the fracture behaviour of C(T) specimens obtainedfrom the BM. For the homogeneous BM, good agreementcan be achieved between the numerical and experimentalF–COD curves, as shown in Fig. 20a. However, the GTNmodel predicts much higher J values at the crack initiationstage, as shown in Fig. 20b. In order to obtain improved JR
curves with lower values especially for small � a values, thefc values should be reduced. Good agreement can be obtainedbetween the numerical and experimental results in the formof force versus Crack Opening Displacement as well as frac-ture resistance curves when fc is reduced to 0.03, as shown inFig. 21. For C(T) specimens with the initial crack located atthe interface between the FZ and the HAZ, good agreementcan also be obtained between the numerical and experimentalF–COD as well as fracture resistance JR curves when anotherGTN parameter set was chosen, as depicted in Fig. 22. TheGTN parameters applied for the C(T) specimens with the
initial crack located in the BM and at the interface betweenthe FZ and the HAZ are summarized in Table 4.
6 Discussion and conclusions
Experimental and numerical investigations on the fracturebehaviour of S355NL electron beam welded joints have beenpresented. Fracture toughness tests were performed on dif-ferent C(T) specimens, i.e., the initial crack is located in theBM, in the middle of the FZ and at the interface between theFZ and the HAZ. Microstructures of different weld regionsare shown. For the C(T) specimens obtained from the BMand from the interface situation, the SEM pictures of frac-ture surfaces show large voids close by to with smaller voids.This confirms that void initiation, growth and coalescenceare the main reasons for ductile fracture. For C(T) spec-imens with an initial crack located in the FZ, less stablefracture regions compared to that of the BM and the HAZare found. Moreover, some flat regions can be found in thestable fracture region, showing the C(T) specimens werebroken in an unstable way. This can explain why the C(T)
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(a)
0
10
20
30
40
50
f0=0.0013, f
c=0.03, f
n=0.01
εn=0.2
3.02.52.01.51.00.50
F (
KN
)
COD (mm)
experiment--HAZ elastic-plastic GTN
(b)
0
100
200
300
400
500 experiment--HAZ GTN
f0=0.0013, f
c=0.03, f
n=0.01, ε
n=0.2
J-In
tegr
al (
N/m
m)
3.53.02.52.01.51.00.50.0
Δa (mm)
Fig. 22 Comparison of experimental and numerical a force versus crack opening displacement (COD) curves, and b fracture resistance curves forC(T) specimens with the initial crack located at the interface between the FZ and the HAZ (lc = 0.1mm, fc = 0.03)
Table 4 The GTN parameters used for the C(T) structures
C(T) extracted from different weld regions f0 fc fn ff εn sn q1 q2 lc (mm)
BM 0.0001 0.03 0.01 0.2 0.2 0.1 1.5 1 0.05
Interface between FZ and HAZ 0.0013 0.03 0.01 0.2 0.2 0.1 1.5 1 0.1
specimens from FZ broke suddenly and show more brittlebehaviour.
Parameter studies of the GTN model were performed onnotched round specimens and C(T) specimens. For notchedround specimens, the f0, fc, fn, εn values influence the finalfracture positions of F–�D curves. Higher f0 and fn valuesresult in earlier fracture, however lower fc and εn values pro-duce earlier fracture positions of notched round specimens.During the deformation of all materials, higher f0 and fn val-ues result in that the current void volume fraction reaches thefc threshold earlier. The stage when the material decreasesits stress carrying ability happens earlier, leading to an ear-lier fracture position of the F–�D curves. Lower fc valuesmean that void coalescence arises earlier. Lower εn valuesmeans the void nucleation occurs earlier, and results in thatthe current void volume fraction reaches the fc limit ear-lier. The ff value does not influence the position of frac-ture point on notched round specimens but influences theslope of F–�D curve after the fracture point. Lower ff val-ues produce a smaller slope after the fracture point. Aftervoid coalescence, the material stress carrying ability becomesweaker. When the current void volume fraction f reaches thefinal void volume fraction at failure ff , the material loosesits stiffness totally. A lower ff means its stress carrying abil-ity will decrease faster after void coalescence, leading to asmaller slope of the F–�D curve after facture initiation ofnotched round specimens. For the C(T) specimens, f0 andfc do have strong influences on the JR curves. Lower f0 val-ues or higher fc values can produce steeper JR curves andhigher J values. The ff , fn and εn values do not influencethe JR curve dramatically but shift the JR curve in a parallel
manner. Higher ff and εn values or lower fn produce higherJR curves.
Based on the metallographic investigations and numericalcalibrations on the notched round specimen, the GTN param-eters could be fixed. The same parameters are used to studythe fracture behaviour of C(T) specimens. Good agreementcan be obtained between the numerical and experimentalF–COD as well as fracture resistance JR curves of C(T) spec-imens when the initial crack is located in the BM and at theinterface between the FZ and the HAZ. This shows the GTNmodel can be used to study the fracture behaviour of electronbeam welded joints and can predict the crack propagation ofwelded joints very well.
Acknowledgements The research has been financially supported byChina Scholarship Council (CSC). This financial support is gratefullyacknowledged. The authors would like to thank Prof. V. Ploshikhin andDr. Y. Rudnik from Neue Materialien Bayreuth GmbH, for providingsome of the experimental results.
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