3D CZM Ductile Crack Growth Triaxiality

Three-dimensional modeling of ductile crack growth:

Cohesive zone parameters and crack tip triaxiality

C.R. Chen a,b,c, O. Kolednik d,*, J. Heerens e, F.D. Fischer f

a Materials Center Leoben, A-8700 Leoben, Austriab Shenyang National Laboratory for Materials Science, Shenyang 110016, China

c Centre for Advanced Materials Technology, School of Aerospace, Mechanical and Mechatronic Engineering,

University of Sydney, Sydney 2006, Australiad Erich Schmid Institute of Materials Science, Austrian Academy of Sciences, Jahnstrasse 12, A-8700 Leoben, Austria

e GKSS Forschungszentrum Geesthacht GmbH, D-21502 Geesthacht, Germanyf Institute of Mechanics, Montanuniversitat Leoben and Erich Schmid Institute of Materials Science,

Austrian Academy of Sciences, A-8700 Leoben, Austria

Received 31 March 2004; received in revised form 6 December 2004; accepted 19 January 2005

Available online 9 April 2005

Abstract

For 10 mm thick smooth-sided compact tension specimens made of a pressure vessel steel 20MnMoNi55, the inter-

relations between the cohesive zone parameters (the cohesive strength, Tmax, and the separation energy, C) and the crack

tip triaxiality are investigated. The slant shear-lip fracture near the side-surfaces is modeled as a normal fracture along the

symmetry plane of the specimen. The cohesive zone parameters are determined by fitting the simulated crack extensions

to the experimental data of a multi-specimen test. It is found that for constant cohesive zone parameters, the simulated

crack extension curves show a strong tunneling effect. For a good fit between simulated and experimental crack growth,

both the cohesive strength and the separation energy near the side-surface should be considerably lower than near the

midsection. When the same cohesive zone parameters are applied to the 3D model and a plane strain model, the stress

triaxiality in the midsection of the 3D model is much lower, the von-Mises equivalent stress is distinctly higher, and the

crack growth rate is significantly lower than in the plane strain model. Therefore, the specimen must be considered as a

thin specimen. The stress triaxiality varies dramatically during the initial stages of crack growth, but varies only smoothly

during the subsequent stable crack growth. In the midsection region, the decrease of the cohesive strength results in a

decrease of the stress triaxiality, while the decrease of the separation energy results in an increase of the triaxiality.

Ó 2005 Elsevier Ltd. All rights reserved.

Keywords: Cohesive zone model; Ductile fracture; Crack growth resistance; Stress triaxiality; Fracture process zone

0013-7944/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.engfracmech.2005.01.008

* Corresponding author. Tel.: +43 3842 804114; fax: +43 3842 804116.

E-mail address: [email protected] (O. Kolednik).

Engineering Fracture Mechanics 72 (2005) 2072–2094

www.elsevier.com/locate/engfracmech

1. Introduction

This paper deals with the 3D finite element (FE) modeling of ductile crack growth in smooth-sided com-

pact tension (CT) specimens using the cohesive zone model. The main problem is that the cohesive zone

parameters, i.e., the separation energy, C, and the cohesive strength, Tmax, may vary along the crack front

and also with the crack extension. The question is, therefore, how to find the relationship between the cohe-

sive zone parameters and the crack tip constraint.

In the cohesive zone model, the role of the cohesive zone is to reflect the mechanical effects of the fracture

process zone. The fracture process zone is the local region ahead of the crack tip where the micro-voids or

micro-cracks initiate, grow, and coalesce with the main crack. This small and highly inhomogeneous region

is poorly defined and poorly known [1]. In the cohesive zone model, the fracture process zone is simplified

as being an initially zero-thickness zone, which is composed of two coinciding cohesive surfaces. Under

loading, the two cohesive surfaces separate, and the traction between them varies with the separation dis-

tance in accordance with a specified traction–separation function. Different traction–separation functions

have been applied to model different types of fracture [2–12]. The primary parameters characterizing the

traction–separation function are the cohesive strength, Tmax, and the separation energy, C. The cohesive

strength is the peak value of the traction in the traction–separation curve; the separation energy is the area

under the curve, giving the work spent in the cohesive zone for the creation of a unit crack area. In the

following, Tmax and C are referred to as the cohesive zone parameters. The exact shape of the traction–sep-

aration function has been claimed to be of secondary importance for the fracture in homogeneous materials

[4], but for the interface fracture some doubts have grown recently whether this is really true [5].

In the cohesive zone modeling of crack growth, two groups of parameters affect the crack growth resis-

tance curve: The conventional material parameters, such as the YoungÕs modulus, E, the yield strength, ry,

and the strain hardening exponent, n, and the cohesive zone parameters, Tmax and C. The conventional

material parameters can be easily determined, but the determination of the cohesive zone parameters is

not easy, as they are not only influenced by the continuum properties of the material but also by the local

constraint conditions at the crack tip (and maybe by some micro-structural parameters [13]). For elastic–

plastic materials, one appropriate parameter to quantify the local constraint at a growing crack tip is the

stress triaxiality, i.e., the ratio of the hydrostatic stress to the von-Mises equivalent stress, t = rh/req.

Siegmund and Brocks [14–16] analyzed the dependence of the cohesive zone parameters on the triaxiality

by using a unit cell model containing a void under plane strain conditions. They change the ratio of the

stresses in x- and y-directions to create different stress triaxiality. Their analyses show that with a decrease

of the stress triaxiality, the cohesive strength decreases, while the separation energy increases. A decrease of

the cohesive strength results in an increase of the crack growth rate, while an increase of the separation

energy results in a decrease of the crack growth rate. This means that, when the stress triaxiality varies,

the resulting variations of the separation energy and the cohesive strength have counteractive effects on

the crack growth rate. This might be the reason why, for some cases, 3D simulations of crack growth in

different specimens, which were made with constant cohesive zone parameters, fit to the experimental re-

sults [17–20]. It should be noted that the variations of triaxiality are comparatively small in side-grooved

specimens, whereas in smooth-sided specimens, the triaxiality decreases drastically when approaching the

side-surface. For the general application of the cohesive zone model for simulating ductile crack growth

it is, therefore, necessary to investigate more thoroughly the interaction between the local constraint con-

ditions and the cohesive zone parameters.

The difficulties in characterizing the fracture toughness of thin ductile plates and in transferring the frac-

ture parameters to real structures lie in the three-dimensional (3D) character of the crack tip stress and

strain fields [23,24]. In a thin plate containing a through-crack, the stress conditions resemble that of plane

stress, except in the regions near the crack front where the stress state is three-dimensional [25]. The con-

straint tends to be relaxed near the free surfaces, which often leads to shear lip formation [26]. Therefore,

C.R. Chen et al. / Engineering Fracture Mechanics 72 (2005) 2072–2094 2073

full 3D calculations are necessary to describe the fracture behavior of thin specimens, as well as to under-

stand the competition between the flat-fracture and shear-lip fracture modes [27].

In this paper, a 3D finite element analysis is performed to test the application of the cohesive zone model

for simulating crack growth in thin compact tension (CT) specimens with smooth-sided surfaces. The mate-

rial is a pressure vessel steel, 20MnMoNi55, with a medium strength and high fracture toughness. The cohe-

sive zone parameters are determined by fitting the simulated values of the local crack extension to the

experimental data. The following questions shall be investigated: How does the crack tip triaxiality vary

with the crack extension? How does the crack tip triaxiality vary along the crack front? What is the influ-

ence of the cohesive zone parameters on the crack tip triaxiality? The purpose of this paper is to gain some

understanding about the relationship between the cohesive zone parameters and the crack tip triaxiality.

2. Material properties and experimental data

The material investigated is a pressure vessel steel 20MnMoNi55 with a YoungÕs modulus, E = 210 GPa,

a yield strength, ry = 465 MPa, and an ultimate tensile strength, ru = 600 MPa. The average strain harden-

ing exponent is n = 0.13. The true stress vs. true plastic strain curve is shown in Fig. 1.

A conventional multi-specimen fracture mechanics test was conducted on CT-specimens according to

the ESIS Standard (ESIS P2-92, 1992). The geometry of the specimens was: Thickness B = 10 mm, width

W = 50 mm, initial crack length a0 � 30.4 mm. In total, five specimens were tested. Table 1 collects the data

of the load, F, the load line displacement, vLL, the J-integral, J, and the average crack extension, Da, for

each specimen. The test resulted a value of J0.2/Bl = 400 kJ/m2. It should be noted that, due to the high frac-

ture toughness and the relatively small specimen thickness, J0.2/Bl and the J vs. Da curve cannot be regarded

as being independent of the specimen size, as the J-validity limit Jmax = Brf/20 = 267 kJ/m2 < J0.2/Bl. Here-

by, the flow strength rf is given by rf = (ry + ru)/2.

To determine the average crack extension, Da, the local crack extension, Da, was measured with an opti-

cal microscope (and in the scanning electron microscope for very small Da). The measurements were taken

0.1 mm inside the side-surfaces and on seven equidistant positions along the crack front. In Table 1, the

maximum local crack extension, Damax, and the average value of the crack extension at the side-surfaces,

Dass are also listed. As the F–vLL curve of Specimen 2 lies distinctly above the curves of all other specimens,

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

0

100

200

300

400

500

600

700

800

900

Tru

e s

tress [M

Pa]

True plastic strain

Fig. 1. True stress vs. true plastic strain curve of the 20MnMoNi55 steel.

2074 C.R. Chen et al. / Engineering Fracture Mechanics 72 (2005) 2072–2094

we have to exclude the data of Specimen 2 from the cohesive zone modeling. Fig. 2 shows the distribution

of the local crack extension along the crack front for the remaining four specimens. The specimens exhibit

shear lips and an appreciable amount of lateral contraction; this is the reason why the side-surface data

points are increasingly shifted towards the midsection with increasing vLL. Due to the micro-structural scat-

ter, the crack front is wavy and asymmetric.

3. Traction–separation function and finite element model

3.1. The traction–separation function for the cohesive elements

Different traction–separation functions have been frequently applied in the literature for ductile fracture:

A polynomial function [2], an exponential function [3], and a trapezoidal function [4]. In this paper, we use

a polynomial function [2] where the relationship between the cohesive normal traction, T, and the separa-

tion distance normal to the crack plane, d, is defined as

T ¼

27

4Tmax

d

df

� �

1ÿ 2d

df

� �

þd

df

� �2" #

for d 6 df

0 for d > df

8

>

<

>

:

ð1Þ

-5 -4 -3 -2 -1 0 1 2 3 4 50

1

2

3

4

5

6 vLL

=8.556mm

vLL

=5.767mm

vLL

=4.203mm

vLL

=1.985mm

z [mm]

∆a [

mm

]

Fig. 2. Experimental Da–z curves of four specimens, where Da is the local crack extension and z the distance from the midsection of the

specimen.

Table 1

Data of the multi-specimen fracture mechanics test

Specimen F (kN) vLL (mm) J (k/m2) Da (mm) Damax (mm) Dass (mm)

1 16.15 1.985 290 0.216 0.443 0.127

2 18.56 2.502 440 0.503 1.316 0.139

3 16.84 4.203 720 1.152 2.301 0.558

4 14.82 5.767 925 2.616 3.777 1.769

5 12.63 8.556 1294 4.508 5.787 3.608

The data of Specimen 2 were excluded from the cohesive zone analysis.


The function is plotted in Fig. 3. Tmax is the cohesive strength, df the characteristic normal separation dis-

tance at which the traction drops to zero. The normal separation energy, C, is defined as

C ¼

Z

df

0

T dd ð2Þ

Inserting Eq. (1) into Eq. (2), the three parameters C, Tmax, and df are related by

C ¼9

16Tmaxdf ð3Þ

A function similar to Eq. (1) can be introduced for the tangential separations, and coupling rules are needed

for a mixed-mode fracture. Such a mixed-mode fracture appears along the slant shear-lip fracture regions.

The realistic modeling of the crack growth in specimens exhibiting both flat-fracture and shear-lip fracture

regions is, however, beyond the current investigation. A crucial task would be how to pre-define the com-

plicated 3D shape of the shear lip faces. In the current investigation, the crack plane is assumed to be along

the symmetry plane, and only the normal separation is considered. As only a quarter of the specimen is

analyzed in the finite element computation, only half of the normal separation between the top and bottom

faces of the cohesive zone is taken into account.

An important point in treating the simulation results of cohesive zone model is the definition of the crack

tip location. The crack tip can be defined at three different locations:

I. At the position d = df where T just decreases to zero or to a very small value, e.g., 0.05Tmax for the

exponential traction–separation function [4,14,20,21] (Point D in Fig. 4),

II. at the position d = dmax where T just reaches Tmax [5] (Point B in Fig. 4),

III. at a position d = dc in the softening region, with dmax < dc < df (Point C in Fig. 4).

In this paper, we define the crack tip as the position where the separation distance reaches df. This is a

relatively reasonable choice for a steel exhibiting micro-ductile fracture.

3.2. Finite element model

The simulation of crack growth is performed with the finite element code ABAQUS. A quarter of the

CT-specimen is modeled. In the 3D FE model, the plane z = 0 represents the midsection of the specimen,

and the plane z = 5 mm represent the side-surface. There are 11 layers in the thickness direction. The layer

Fig. 3. Traction–separation function, Eq. (1), for the cohesive elements.


thickness is 0.5 mm for the Layers 1–9 in the interior, 0.3 mm for the Layer 10, and 0.2 mm for the Layer 11

(the side-surface layer). Eight-node solid elements are used for the bulk elements. The local mesh in the x–y

plane around the crack tip is shown in Fig. 5. The initial crack front is simplified as being a straight line.

The x-coordinate of the initial crack front is set to be x = 0. The size of the elements in x-direction is 0.1 mm

for 0 6 x 6 9.1 mm.

The cohesive elements are placed in the crack plane, y = 0, between the location of the initial crack front,

the point O at x = 0, and the point at x = 9.1 mm. Each cohesive element has a top and a bottom face with

four nodes on each face. The top face is connected with the adjacent solid element, while the bottom face is

connected with the symmetry plane, y = 0.

The constraint conditions are defined as follows: the displacement uy = 0 for the nodes at the plane y = 0

from x = 9.1 mm to x = 19.6 mm, and uy = 0 for the nodes at the bottom faces of the cohesive elements

from x = 0 to x = 9.1 mm; uz = 0 for the nodes at the midsection z = 0; ux = 0 for the nodes at the loading

position. The displacement load, uy = 4.5 mm, is applied in 1500 equal sub-steps to the nodes at the loading

B

CD

LSoft

a

Tmax

x

TC

C

T

δmaxδ

Cδ

f

BD

Fig. 4. On the definition of the location of the crack tip. The points D, C, and B correspond to the points D, C, and B in Fig. 3. Lsoft is

the softening zone size.

O

Fig. 5. Local mesh near the crack tip on the x–y plane. The initial crack tip is at the position O.


position. The load line displacement, vLL, is two times the y-direction displacement at the point (ÿ30.4,

0, 0).

The FE calculations are performed using the finite-deformation formulations unless it is specified

otherwise.

4. Characteristics of cohesive zone parameters and crack tip triaxiality

4.1. Characteristics of local crack extension and stress triaxiality in the 3D model for constant cohesive

zone parameters

4.1.1. Variation of the local crack extension along the thickness

When the cohesive zone parameters are set to be constant, the computed crack extension varies along the

thickness due to the variation of the crack tip constraint. First, we assume the cohesive zone parameters to

be C = 120 kJ/m2 and Tmax = 1460 MPa (i.e., Tmax/ry = 3.14). This set of C- and Tmax-values was deter-

mined by Cornec et al. [10] for the identical 20MnMoNi55 steel and was applied with good success to model

the behavior of side-grooved CT and center cracked tension specimens of the same gross thickness. The

cohesive strength, Tmax, was evaluated as the local maximum stress in the center of a notched cylindrical

tensile specimen (8 mm diameter, 2.5 mm notch radius) in the moment of fracture. The separation energy

was taken as being equal to the value of the J-integral at the physical point of fracture initiation, Ji. Hereby,

Ji was determined from measurements of the stretched zone width [10]. For our smooth sided specimens,

however, this set of parameters is not suitable, as the computed crack extensions near the midsection are

distinctly larger than the experimental values. To make the simulated crack extensions near the midsection

fit to the experimental results, C (or Tmax) must be increased.

When the parameters are set to C = 180 kJ/m2 and Tmax = 1460 MPa, the computed crack extensions

near the midsection are similar to the experimental data, see Fig. 6. The simulated crack extension curves

show a strong tunneling effect. However, the initiation of crack growth near the side-surface occurs in the

simulation significantly later than in the experiments. The main reason is that the value of

Tmax = 1460 MPa is too high for the cohesive elements near the side-surface. It is also seen in Fig. 6 that

in the crack plane a strong contraction along the z-direction occurs which increases with the external dis-

placement load.

-5 -4 -3 -2 -1 0 1 2 3 4 5

0

1

2

3

4

5

6

Simulated

A : vLL

= 8.55

B : vLL

= 5.78

C : vLL

= 4.21

D : vLL

= 2.00

D

C

B

A

Γ = 180 kJ/m2

Tmax = 1460 MPa

∆a [m

m]

z [mm]

Experimental

vLL

= 8.556

vLL

= 5.767

vLL

= 4.203

vLL

= 1.985

Fig. 6. Simulated crack extension curves when C and Tmax are set to be constant.


It is well known that for the plane strain case under small scale yielding conditions, the separation energy

comes close to the value of the J-integral at the point of physical facture initiation, Ji [29]. This is strictly

true, if no energy is dissipated outside the process zone. Therefore, it is not surprising that in our case (thin

specimens and large-scale yielding conditions) the separation energy near the midsection (about 180 kJ/m2)

is much smaller than the J-value at 0.2 mm tearing (J0.2/Bl = 400 kJ/m2).

4.1.2. Variation of the stress triaxiality along the thickness

The stress triaxiality is defined as t = rh/req, where rh and req are the hydrostatic stress and the von-

Mises equivalent stress, respectively. For any cohesive element, the values of rh and req are obtained at

the center of the solid element adjacent to the cohesive element.

The stress triaxiality changes with the thickness position, z, and the distance from the current crack tip,

Dx, and varies also with the crack extension, Da. Fig. 7a shows the stress triaxiality distribution ahead of

the current crack tip in different layers at a load line displacement of vLL = 6.0 mm. It is seen that with

increasing distance from the propagating crack tip, the triaxiality first increases drastically, reaches a peak

value, and than decreases. Although the highest triaxiality peak (tmax � 1.6) does not appear in the midsec-

tion layer (Layer 1), it is seen that Layer 1 shows the highest triaxiality values after the triaxiality peak, and

the triaxiality decreases from Layer 1 to the side-surface layer (Layer 11). The location of the triaxiality

peak in the layers near the midsection is about 1.0 mm ahead of the crack tip. This means that near the

midsection the micro-fracture processes are initiated about 1 mm ahead of the crack tip. At the side-surface

layer no distinct triaxiality peak (tmax � 0.7) is observed.

Fig. 7b and c show the distributions of the hydrostatic stress rh and the von-Mises equivalent stress reqahead of the local crack tip in different layers at vLL = 6.0 mm. The shapes of the rh-distributions resemble

strongly those of the triaxiality distributions and also the locations where the peak values appear are sim-

ilar. The req distributions look totally different. Near the current crack front (i.e., at small Dx), req increases

significantly from the midsection to the side-surface layer. At the far region (Dx > 1 mm), the req values in

the Layers 1–9 become similar, only in the side-surface layer req is distinctly lower than in the other layers.

Fig. 8a shows the variation of the triaxiality peak values along the thickness at vLL = 2.0, 3.0 and

6.0 mm. The triaxiality peak varies smoothly near the midsection, but decreases drastically when approach-

ing the side-surface. Comparing the three curves, it is seen that the triaxiality near the midsection decreases

with increasing vLL. For vLL = 2.0 mm, the highest triaxiality peak occurs in Layer 1, however, for vLL = 3.0

and 6.0 mm the highest triaxiality peaks are observed in the Layers 5 and 6, around z � 2.5–3.0 mm.

Fig. 8b shows the variations of rh and req (both taken at the location of triaxiality peak) along the thick-

ness. It is seen that the variation of the rh peak along the thickness is similar to the variation of the triax-

iality peak, while the variation of req along the thickness is much smoother. Only near the side-surface, an

increase of req is observed. Remarkable is, however, the increase of req with increasing vLL. As near the

midsection the req-increase is much more significant than the variation of the rh peak, the stress triaxiality

decreases near the midsection with increasing loading.

From the results shown in Figs. 7 and 8, it can be concluded that the distribution of the stress triaxiality

ahead of the crack tip and the variation of the stress triaxiality along the thickness are mainly decided by

the variation of the hydrostatic stress rh, while the variation of the stress triaxiality near the midsection with

the crack extension is mainly determined by the variation of the equilibrium stress req.

4.1.3. Variation of the stress triaxiality with crack extension

To help us understand whether the cohesive zone parameters for the fracture initiation can be the same

as those for stable crack growth, we investigate the variation of triaxiality with the crack extension. Fig. 9a

shows the variation of the stress triaxiality peak in the Layers 1, 3, 5, and 7 with the local crack extension in

each layer. For the Layers 1 and 3, the triaxiality peak at fracture initiation (Da = 0) is distinctly higher than

during the stable crack growth. Compared at the points of the local fracture initiation (Da = 0), the stress


0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

0

200

400

600

800

1000

1200

Γ = 180 kJ/m2

Tmax

= 1460 MPa

∆x [mm]

vLL

=6.0 mm

Layer 1

Layer 3

Layer 5

Layer 7

Layer 9

Layer 11

σh

[M

Pa]

(b)

σe

q [M

Pa]

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

500

550

600

650

700

750

800

850

Γ = 180 kJ/m2

Tmax

= 1460 MPa

vLL

=6.0 mm

Layer 1

Layer 3

Layer 5

Layer 7

Layer 9

Layer 11

∆x [mm](c)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6 Layer 1

Layer 3

Layer 5

Layer 7

Layer 9

Layer 11

vLL

= 6.0 mm

Γ = 180 kJ/m2

Tmax = 1460 MPa

σh/ σ

eq

∆x [mm](a)

Fig. 7. Stress state ahead of the local crack tip in different layers at a load line displacement of vLL = 6.0 mm: (a) Stress triaxiality

rh/req; (b) hydrostatic stress rh; (c) von-Mises equivalent stress req. The symbol Dx denotes the distance from the local crack tip.


triaxiality peaks decrease monotonically from the Layer 1 to the Layer 7. During stable crack growth, how-

ever, the triaxiality peak in Layer 5 becomes higher than in the Layer 1. Contrary to the behavior in other

layers, the triaxiality in Layer 7 increases during the first stages of local crack extension, reaches a maxi-

mum value and then decreases. If the cohesive zone parameters have a close relation with the stress triax-

iality peak, then the cohesive zone parameters for the fracture initiation might be different from those for

stable crack growth.

Fig. 9b shows the corresponding variations of rh and req (taken at the position Dx where the triaxiality

reaches its peak value). The rh-peak keeps nearly constant during the local crack growth in the Layers 1

and 3, but increases in Layer 5. It is seen that req increases due to strain hardening in all layers during

the first stages of local crack extension. As from the midsection region to the side surface the point of frac-

ture initiation is shifted to higher vLL-values, req increases from the Layer 1 to the Layer 7. Near the mid-

section the stress req is distinctly lower at the crack growth initiation than during stable crack growth. This

is the main reason why near the midsection the stress triaxiality peak is higher at the point of fracture ini-

tiation than that during stable crack growth.

It is usually thought that the highest stress triaxiality should occur in the specimen midsection and that

the triaxiality should decrease monotonically from the midsection to the side-surface. This is observed at

the fracture initiation stage, but not during the crack growth. With the increase of crack extension, the

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

Γ = 180 kJ/m2, T

max= 1460 MPa

A: vLL

= 2.0 mm

B: vLL

= 3.0 mm

C: vLL

= 6.0 mm

C

B

A

z0 [mm]

(σh/σ

eq) m

ax

500

600

700

800

900

1000

1100

1200 Γ = 180 kJ/m2, T

max= 1460 MPa

σhm

ax [M

Pa

]σ

eq [M

Pa

]

vLL

= 2.0 mm

vLL

= 3.0 mm

vLL

= 6.0 mm

z0

[mm]

(a)

(b)

Fig. 8. Variations of triaxiality peak along the thickness at vLL = 2.0, 3.0 and 6.0 mm: (a) Triaxiality peak; (b) stresses rh and req

corresponding to the triaxiality peak. z0 denotes the distance from the midsection in the initial geometry.


triaxiality near the midsection decreases more significantly than at the locations far away from the midsec-

tion. As a result, the highest triaxiality peak may move during crack growth to the boundary between the

midsection region and the influence zone of the side-surface.

4.1.4. Influence of cohesive zone parameters on local crack extension and stress triaxiality

Fig. 10 shows the Da–vLL curves at the midsection for three different sets of cohesive zone parameters:

(A) C = 180 kJ/m2, Tmax = 1460 MPa; (B) C = 180 kJ/m2, Tmax = 1360 MPa; (C) C = 120 kJ/m2, Tmax =

1460 MPa. By comparing the curves A and B, it is seen that a variation of Tmax affects the slope of Da–

vLL curve, but has little or no influence on the initiation point of the Da–vLL curve. A decrease of Tmax re-

sults in a higher crack growth rate. By comparing the curves A and C, it is seen that a variation of C affects

both the slope and the initiation point of the Da–vLL curve. A decrease of C results in an earlier crack

growth initiation and a higher crack growth rate. It is noticed that the crack growth rate is more sensitive

to the cohesive strength than to the separation energy.

In order to understand the relationship between the cohesive zone parameters and the crack tip triaxi-

ality, it is necessary to investigate also the influence of cohesive zone parameters on the crack tip triaxiality.

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

1.4

1.5

1.6

1.7

1.8

1.9

(σh/σ

eq) m

ax

layer 1

layer 3

layer 5

layer 7

Γ = 180 kJ/m2, T

max= 1460 MPa

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

600

700

800

900

1000

1100

σh

max [M

Pa

]σ

eq [M

Pa

]

∆a [mm]

∆a [mm]

layer 1

layer 3

layer 5

layer 7

(a)

(b)

Γ = 180 kJ/m2

Tmax

= 1460 MPa

Fig. 9. Variation of the triaxiality peak with the crack extension: (a) Triaxiality peak; (b) stresses rh and req corresponding to the

triaxiality peak. Da is the local crack extension in each layer.


0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

1.4

1.5

1.6

1.7

1.8

1.9

2.0

2.1

C

B

A

∆a [mm]

(σh/σ

eq) m

ax

Γ = 180 kJ/m2, T

max = 1460 MPa

Γ = 180 kJ/m2, T

max = 1360 MPa

Γ = 120 kJ/m2, T

max = 1460 MPa

Γ = 180 kJ/m2, T

max = 1460 MPa

Γ = 180 kJ/m2, T

max = 1360 MPa

Γ = 120 kJ/m2, T

max = 1460 MPa

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

500

600

700

800

900

1000

1100

σh

ma

x [

MP

a]

σe

q [

MP

a]

∆a [mm]

(a)

(b)

Fig. 11. Effects of variations of Tmax and C on the triaxiality peak in the midsection: (a) Triaxiality peak; (b) stresses rh and req

corresponding to the triaxiality peak.

1 3 4 5 6 7

0

1

2

3

4

5

6

B

C

A

A: Γ = 180 kJ/m2, T

max = 1460 MPa

Γ = 180 kJ/m2, T

max = 1360 MPa

Γ = 120 kJ/m2, T

max = 1460 MPa

B:

C:

∆a [

mm

]

vLL

[mm]

2

Fig. 10. Effects of variations of the cohesive strength, Tmax, and the separation energy, C, on the crack extension in the midsection.


Fig. 11a shows the variations of the stress triaxiality peaks in the midsection layer with the local crack

extension for the three different sets of C and Tmax. The three curves are nearly parallel. By comparing

the curves A and B, it is seen that a decrease of Tmax results in a decrease of the triaxiality peak. By com-

paring the curves A and C, it is seen that a decrease of C results in an increase of the triaxiality peak.

Fig. 11b shows the corresponding rh and req vs. Da curves. It is seen that the decrease of Tmax results in a

decrease of both rh and req. The decrease of rh is, however, more significant than the decrease of req; as a

result, the stress triaxiality decreases. It is also seen that a decrease of C has very little effect on rh, but re-

sults in a decrease of req. Thus, a decrease of C results in an increase of the stress triaxiality.

4.1.5. Effects of small-deformation conditions on the simulated crack growth

Our finite element simulations are performed using the finite-deformation formulations. Most cohesive

zone studies in the literature, however, have been modeled using the small-strain assumptions. To check

whether the use of the finite-deformation formulations is necessary, we compare the simulated crack exten-

sion curves for both the finite-deformation and the small-deformation conditions. When applying the

small-deformation conditions, all computations refer to the initial (undeformed) geometry; for the

finite-deformation conditions, the computations in the nth load-increment refer to the deformed geometry

obtained in the (n ÿ 1)th load-increment. For the 3D model, when finite-deformation conditions are ap-

plied, the x- and z-displacements of the bottom face (located in the symmetry plane of the specimen) of

the cohesive element are set equal to the displacements of the upper face (which is connected to the adja-

cent solid element). In this way, the reduction of area of the crack plane by the plastic contraction in thick-

ness direction can be taken into account. It is already known from the literature that, when determining

the cohesive zone parameters by fitting the simulated crack growth data to the experimental data, the

small-deformation condition yields a slightly lower cohesive strength than the finite-deformation condition

[20,32].

Fig. 12 compares the simulated crack extension curves obtained under small- and finite-deformation

conditions. It is seen that there is not much difference for small load line displacements, at vLL = 2.0 and

4.21 mm. For larger loading, however, at vLL = 5.78 and 8.56 mm, the crack extensions are distinctly

smaller under the small-deformation condition than under the finite-deformation condition. Therefore,

for the crack growth modeling of 20MnMoNi55, it is necessary to use the finite-deformation

formulations.

-5 -4 -3 -2 -1 0 1 2 3 4

0

1

2

3

4

5

6A : v

LL= 8.55

B : vLL

= 5.78

C : vLL

= 4.21

D : vLL

= 2.00

D

C

B

A

Γ = 180 kJ/m2, T

max= 1460 MPa

∆a [m

m]

z [mm]

Finite-deformation

Small-deformation

5

Fig. 12. Comparison of the simulated crack extension curves between the small-deformation and the finite-deformation conditions.


4.2. Comparison with a plane strain model

It is usually thought that, at least for thick specimens, the stress state near the crack front in the specimen

midsection is similar to that of plane strain conditions. To check whether a plane strain model can be used

for our specimens to simulate the crack growth at the specimen midsection, a comparative computation

under plane strain conditions is performed. To do this, the midsection layer (0.5 mm thick) of our 3D model

is taken. The boundary condition uz = 0 is applied to all nodes of this layer of 3D elements. The behavior of

the layer is identical to that of a 2D plane strain model. The cohesive zone parameters are set to be

C = 180 kJ/m2 and Tmax = 1460 MPa.

4.2.1. Comparison of the crack extensions

Fig. 13 compares the local Da–vLL curve for the midsection of 3D model to the Da–0vLL curve of the

plane strain model. It is seen in the diagram that fracture initiation occurs at the same vLL and that the

curves coincide for the very first stages of crack growth, but then the curves deviate and the crack growth

rate in the plane strain model is by approximately a factor 2 higher than in the midsection of the 3D model.

4.2.2. Comparison of the stress triaxialities

Fig. 14a compares the triaxiality distributions ahead of the current crack tip in the plane strain model

and in the midsection layer of the 3D model. The curves are presented for the point of fracture initiation,

as well as for Da = 1 and 4 mm. It is seen that, for the same cohesive zone parameters, the triaxiality in the

plane strain model (t = 2.8 at fracture initiation, t = 2.5 for Da = 4 mm) is distinctly higher than in the mid-

section layer of 3D model (t = 1.95 at fracture initiation, t = 1.5 for Da = 4 mm). Higher stress triaxiality

means that less energy will be consumed by plastic deformation during crack growth. This explains why

for the same cohesive zone parameters, the crack growth rate is much higher in the plane strain model than

at the midsection of 3D model.

Fig. 14b shows the corresponding rh- and req-distributions. In the plane strain model, rh is higher

and req is significantly lower than in the midsection layer of the 3D model. This can explain why for

the same cohesive zone parameters, the triaxiality ahead of the crack tip is significantly higher in the

1 2 3 4 5 6 7

0

1

2

3

4

5

6

B

A

A : Midsection of 3D model

B : Plane strain model

Experimental data at Midsection

∆a [m

m]

vLL

[mm]

Γ = 180 kJ/m2, T

max= 1460 MPa

8 9

Fig. 13. Comparison of the Da–vLL curves between 3D and plane strain models for the same cohesive zone parameters.


plane strain model than in the midsection layer of the 3D model. It is also seen that during the crack

growth the req-increase in the midsection layer of 3D model is much more significant than in the plane

strain model.

From Fig. 14a, it is further seen that the distance between the crack tip and the position of maximum

triaxiality varies during the crack extension. This distance is significantly smaller at fracture initiation than

during the stable crack growth. According to McMeeking [28] (who did not consider a fracture process

zone), the maximum stress triaxiality lies at a distance of about two times the crack tip opening displace-

ment (CTOD) in front of the crack tip; the distance depends on the amount of hardening. From Fig. 14a, it

is seen that in the cohesive zone model this estimate is not bad for the point of fracture initiation, but with

the increase of crack extension, the distance between the maximum stress triaxiality and the crack tip be-

comes distinctly larger than 2 CTOD (the crack tip opening is about 0.22 mm). It is also seen that the crack

tip triaxiality in the midsection layer of the 3D model decreases with the crack extension. A decreasing

stress triaxiality during the crack extension causes more plastic deformation around the crack tip during

the crack propagation. In terms of the analysis by Cotterell and Atkins [30], this means that a true R-curve

effect does appear.

0 4

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Γ = 180 kJ/m2, T

max= 1460 MPa

Plane strain

∆a= 4∆a= 1∆a= 0

3D

σh/σ

eq

x [mm]

x [mm]

5 6 7

0 1 2 3 6

0

200

400

600

800

1000

1200Γ = 180 kJ/m

2, T

max= 1460 MPa

D

C

B

A

D

C

B

A

C

A

B

A: h, plane strain

B: σ

σ

h, 3D

C: σeq

, 3D

D: σeq

, plane strain∆a= 4∆a= 1

D

=0

σeq a

nd σ

h [M

Pa]

7

(a)

(b)

1 2 3

∆a

4 5

Fig. 14. Comparison of the triaxiality distributions ahead of the crack tip between 3D and plane strain models: (a) Triaxiality; (b)

stresses rh and req. The results of the 3D model are taken from the midsection layer.


4.2.3. Comparison of the cohesive tractions

Fig. 15 shows the cohesive tractions ahead of the crack tip for the plane strain model and for the mid-

section of 3D model. The curves are shown for the point of fracture initiation and for Da = 2 mm. With

increasing distance from the current crack tip, the cohesive tractions increase drastically to reach the cohe-

sive strength, and then the tractions decreases gradually. For the same set of cohesive zone parameters, the

variation of the cohesive traction is smoother in the plane strain model than in the midsection of 3D model.

If we compare the softening zone sizes (i.e., the x-distance between the positions of T = 0 to the position of

T = Tmax), it is seen that the softening zone sizes are equal in two models at the point of fracture initiation.

During the crack growth, however, the softening zone size is distinctly larger in the plane strain model than

in the midsection of 3D model.

By comparing the local Da–vLL curves, triaxiality distributions, and cohesive traction distributions

between the plane strain model and the midsection layer of the 3D model for the same cohesive zone para-

meters, it can be concluded that the stress state at the midsection of our CT specimens with 10 mm thick-

ness is significantly different from that of a plane strain state. Thus, the plane strain model cannot correctly

simulate the crack growth at the midsection; instead, 3D models have to be applied.

4.3. Characteristics of the crack tip triaxiality when Tmax varies in thickness direction

Experimental results show that fracture mechanisms in the specimen midsection and near the side-sur-

faces are different. In the midsection region, the fracture surface is flat; near the side-surfaces, a slant shear-

lip fracture surface appears, tilted 45° to the symmetry plane. The proportion of the shear-lip fracture to the

flat fracture increases with the crack extension. Thus, when applying a 3D cohesive zone model to simulate

crack growth in a specimen with shear lips, one difficulty appears: how to pre-define the shape of the crack

plane. In this paper, for simplification, the slant fracture near the side-surface is taken as being equivalent to

the normal separation along the symmetry plane y = 0; and the cohesive zone parameters are kept constant

during the crack extension.

It is seen from Fig. 6 that, when the cohesive zone parameters are set to be C = 180 kJ/m2 and

Tmax = 1460 MPa, the initiation of crack growth near the side-surface occurs in the simulation much later

than in the experiment. To let the simulated fracture initiation near the side-surface occur earlier, the cohe-

sive zone parameters near the side-surface need to be reduced. We use a simple guideline to find a more

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

A : Midsection of 3D model

B : Plane strain model

B

A

B

A

Γ = 180 kJ/m2, T

max= 1460 MPa

∆a= 2∆a= 0

T / T

ma

x

x [mm]

Fig. 15. Comparison of the cohesive traction distributions ahead of the crack tip between 3D and plane strain models.


appropriate distribution of the cohesive strength along the thickness: At first, we apply a set of constant

cohesive zone parameters to make the simulated crack extensions near the midsection fit to the experimen-

tal results. As the influence of the cohesive strength is much stronger than that of the separation energy

(Figs. 10 and 11), we keep the separation energy constant along the thickness. In Fig. 8a, the stress triax-

iality remains roughly constant from Layer 1 to Layer 6 (from z = 0 to z = 3 mm) and decreases sharply

from Layer 7 to Layer 11 (from z = 3 to z = 5 mm). Referencing to this variation of the stress triaxiality,

we assume that the cohesive strength remains constant, Tmax = 1480 MPa, between z = 0 and z = 3 mm,

and decreases then linearly from z = 3 to z = 5 mm. Directly at the side-surface, the cohesive strength is

set to be two times the yield strength, Tmax(at z = 5) = 930 MPa, because Li and Siegmund [31] have found

that under plane stress conditions a suitable value for the cohesive strength is about two times the yield

strength. The so adopted cohesive zone parameters are depicted in Fig. 16a.

Fig. 16b shows a comparison between the simulated and experimental crack extension curves. Compared

to Fig. 6, the local crack extension remains unchanged in the midsection region. Near the side surfaces, the

local Da-values are increased, but the simulated crack extensions near the side-surface are still significantly

smaller than the experimental crack extension data. Although Tmax has been reduced considerably near the

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

z [mm]

180

930

1480

Tmax

[MPa]

Γ [kJ/m2]

-5 -4 -3 -2 -1 0 1 2

0

1

2

3

4

5

6

Simulated

A: vLL

=8.55

B: vLL

=5.78

C: vLL

=4.21

D: vLL

=2.00

D

C

B

A

∆a

[m

m]

z [mm]

Experimental

vLL

=8.556

vLL

=5.767

vLL

=4.203

vLL

=1.985

3 4 5

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

A: vLL

= 2.0 mm

B: vLL

= 3.0 mm

C : vLL

= 4.0 mm

D : vLL

= 6.0 mm

D

C

B

A

z0 [mm]

(σh/σ

eq) m

ax

0

0 1 2 3 4 5 6 7 8

3

6

9

12

15

18

Experimental:

Specimen 1

Specimen 3

Specimen 4

Specimen 5

Simulated

F [kN

]

vLL [mm]

(a) (b)

(c) (d)

Fig. 16. Assumed variation of the cohesive strength in thickness direction and the corresponding simulation results: (a) Variation of

Tmax along the thickness; (b) comparison of the simulated crack extension curves with the experimental data; (c) variation of the

triaxiality peak along the thickness; (d) comparison of the simulated load vs. load line displacement curve with the experimental data.


side surfaces, the local crack growth rate has not changed significantly enough. Fig. 16c shows the variation

of the stress triaxiality peak along the thickness for four different vLL-values. Whereas the triaxiality peak

decreases distinctly with the increasing vLL, its change within the region between z = 3 and z = 5 mm is

small. The curves resemble strongly those of Fig. 8a. This means that the decrease of the cohesive

strength within the region influenced by the side-surface has only marginally changed the triaxiality

distribution.

Fig. 16d shows a comparison between simulated and experimental load vs. load line displacement

(F–vLL) curves. It is seen that the simulated F–vLL curve lies somewhat below the experimental F–vLL data.

The possible reason might be that the initial (average) crack length in the FE model may be somewhat too

large. The initial crack front is a tunneling curve in experiment, while it is simplified as a straight line in the

FE model. Another reason might be that the hardening of the material is not correctly modeled for high

strains.

4.4. Additional possible measures for a better fit to the experimental data

From the findings in [31], it does not make much sense to reduce further the cohesive strength within the

region influenced by the side-surface in order to make the simulated crack extensions near the side-surface

fit to the experimental data. An alternative way is to reduce the separation energy near the side-surface.

Such a decrease of separation energy could be argued from a comparison of the micro-structure of the frac-

ture surfaces in the flat-fracture and shear-lip fracture regions. The latter consists of distorted dimples

which are much smaller and shallower than the dimples in the flat-fracture region. The separation energy

consists of two parts, the specific energy for void initiation and the specific energy for the formation of the

dimple structure during void growth and coalescence [22]. The latter term is, according to StuweÕs model

[33], proportional to the average height of the dimples. Therefore, it is conceivable that the separation en-

ergy in the shear-lip fracture regions can be considerably smaller than in the flat-fracture region. It should

be noted, however, that the specific energy for void initiation which represents only a small fraction of the

separation energy for the flat-fracture region with large dimples, might play a bigger role for the shear-lip

fracture region, as due to the lower hydrostatic stress component void initiation occurs at considerably

higher plastic strains.

As the extension of the shear-lip fracture region increases with the crack extension and the fact that we

work with constant cohesive zone parameters during the crack extension, a gradual decrease of the sep-

aration energy could be assumed. Fig. 17a presents a possible variation of the cohesive zone parameters in

the thickness direction. The separation energy is constant in the midsection region and decreases linearly

within the influence zone of the side-surface, from C = 180 kJ/m2 at z = 3 mm to C = 113.1 kJ/m2 at

z = 5 mm. Hereby, the decrease of C is made so that df keeps constant (0.2162 mm) through the thickness.

Fig. 17b shows the comparison between the simulated and experimental crack extension curves. Com-

pared to Fig. 16b, the simulated local crack extensions increase very little near the midsection, but quite

significantly near the side-surfaces. Nevertheless, the simulated local Da-values near the side surfaces are

still too small, especially for large vLL-values. The task of finding such a variation of the cohesive zone

parameters so that the simulated crack extensions fit better to the experimental data is left for a future

study.

4.5. Comparison of cohesive zone parameters between the steels 20MnMoNi55 and St37

Chen et al. [22] studied the cohesive zone parameters and stress triaxiality distributions in a thick

(25 mm-thickness) CT specimen made of the low strength steel St37 (with ry = 270 MPa, ru = 426 MPa,

n = 0.20, JIC = 120 kJ/m2). It is meaningful to compare the cohesive zone parameters between the two

steels.


It should be noted that, contrary to the current experiment, for the St37 the thickness was large enough

to have a wide region around the midsection with high stress triaxiality. In this region, the triaxiality was

comparable in magnitude to the triaxiality for plane strain conditions (2.8 compared to 3.1 for plane strain).

It is remarkable that also in this case, the highest triaxiality peak did not appear in the midsection but at the

boundary between the constant-triaxiality region around the midsection and the influence zone of the side-

surface where the triaxiality exhibits a steep decrease.

For the St37, the separation energy is low (C � 16 kJ/m2 near the midsection). This means that the resis-

tance against fracture initiation is low for this material. For the 20MnMoNi55, the separation energy is

high (C � 180 kJ/m2 near the midsection). Thus the resistance against fracture initiation is very high for this

material.

For the St37, the ratio of the cohesive strength to the yield strength is high (about 4:1 near the midsec-

tion). As a result, the ratio of the energy dissipation rate to the separation energy is very high for this mate-

rial. Thus, although the separation energy of the St37 is low, the resistance against crack growth is not low.

For the 20MnMoNi55, the ratio of the cohesive strength to the yield strength is not high (about 3.2:1 near

the midsection). As a result, although the plastic dissipation rate of the 20MnMoNi55 is much higher than

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

δf = 0.216 mm

113.1

180

930

1480

Γ [kJ/m2]

Tmax

[MPa]

z [mm](a)

z [mm]

-5 -4 -3 -2 -1 0

0

1

2

3

4

5

6

Simulated

A : vLL

= 8.55

vLL

= 5.78

vLL

= 4.21

vLL

= 2.00

B :

C :

D :

D

C

B

A

∆a [

mm

]

Experimental

vLL

= 5.767

vLL

= 4.203

vLL

= 1.985

vLL

= 8.556

1 2 3 4 5

(b)

Fig. 17. Assumed variations of the cohesive strength and the separation energy in thickness direction and the corresponding simulation

results: (a) Variations of Tmax and C along the thickness; (b) comparison of the simulated crack extension curves with the experimental

data.


that of the St37, the ratio of the energy dissipation rate to the separation energy for the 20MnMoNi55 is

not as high as that for the St37. The high separation energy and high energy dissipation rate make the

20MnMoNi55 to have a very high resistance against crack growth.

For the St37, the resistance against shear fracture is higher than the resistance against normal fracture, as

no shear lip appears near the side-surfaces; for the 20MnMoNi55, the resistance against shear fracture is

lower than the resistance to normal fracture, as wide shear fracture regions occur near the side-surfaces.

The reason might be that the strain hardening exponent of the 20MnMoNi55 is lower than that of the St37.

5. Effects of mesh size and load increment on the results of the simulation

In 2D cohesive zone modeling, very fine mesh and very small load increment are usually applied, thus it

is unnecessary to especially consider the effects of element size and load increment on the simulation results.

In 3D cohesive zone modeling, however, due to the limitation of the computer capability, we cannot use

very fine meshes and very small load increments. Therefore, it is necessary to discuss the effects of element

size and load increment on the results of the simulations.

5.1. Influence of element size in crack growth direction

To resolve the influences of the element size along the direction of the crack growth x, we compare the

simulated Da–vLL curves of two plane strain models with a coarse and a fine mesh. The coarse model has

the same mesh as our 3D model in the x–y plane, and the size of the cohesive elements in x-direction is

0.1 mm. The number of elements in the fine model is about six times larger, and the size of the cohesive

elements in x-direction is 0.05 mm. The cohesive zone parameters are C = 180 kJ/m2 and Tmax = 1460 MPa

for both models. The results show that the crack extension in the coarse model is a little bit larger than in

the fine model. For example, at a load line displacement of vLL = 5.0 mm, the crack extensions in the fine

and coarse model are 5.3 mm and 5.4 mm, respectively. Because the differences of the Da–vLL curves are so

small, it is acceptable to use the coarse model for the simulation.

5.2. Influence of element size in thickness direction

In 3D modeling of crack growth, one numerical issue is the effect of the aspect ratio of the elements near

the crack plane. Here, the aspect ratio means the ratio of the element sizes in z-direction and x-direction. To

study the influence of the element size in z-direction, we compare the simulated crack extension curves for

two 3D models with high and medium aspect ratio of the elements. The mesh in the x–y plane is the same

for the two models, and the only difference is the element size in the thickness direction. The high-aspect

ratio model has 6 layers, and the layer thickness is 1 mm for the Layers 1–4, 0.5 mm for the Layers 5

and 6. The medium-aspect ratio model is the 3D model applied in Section 4. The cohesive zone parameters

are as Fig. 16a, i.e., C is 180 kJ/m2 through the thickness; Tmax is 1480 MPa from z = 0 to z = 3 mm, and

then decreases linearly to 930 MPa at z = 5 mm.

The results show that the crack extension in the model with high aspect ratio is somewhat larger than

in the model with medium aspect ratio. For example, at a load line displacement of vLL = 8.0 mm, the

model with medium aspect ratio yields a local crack extension of Da = 5.5 mm at z = 0, and Da = 1.9 mm

at z = 5 mm; the model with the high aspect ratio yields Da = 5.64 mm at z = 0, and Da = 2.2 mm at

z = 5 mm. Therefore, a larger element size in thickness direction tends to make the crack extension lar-

ger, especially near the side-surface, but again the differences are not very large. Therefore, and due to

the limitation of our computer capabilities, it is acceptable to use the model with the medium aspect

ratio.


5.3. Influence of load increments

To study the influence of the size of the load increments, we compare the simulated crack extension

curves of our 3D model for two different increments of the displacement at the load application point

Duy: large increments (Duy = 0.06 mm) and small increments (Duy = 0.03 mm). The results show that the

two load increment sizes create nearly the same local crack extension values. For example, at a load line

displacement of vLL = 8.0 mm, the largest difference of crack extension is only about 20% of one element

size in the x-direction, i.e., the difference is only 0.02 mm. All our FE calculations in Section 4 have been

performed with the small load increments (Duy = 0.03 mm). Therefore, the load increment in our FE cal-

culations is small enough.

6. Summary

Crack growth in 10 mm thick CT-specimens made of a pressure vessel steel 20MnMoNi55 has been sim-

ulated using the cohesive zone model. The cohesive zone parameters have been determined by fitting the

simulated crack extension values to the experimental data of a multi-specimen fracture mechanics

test. The interrelation between cohesive zone parameters and crack tip triaxiality can be summarized as

follows:

(1) The crack tip triaxiality at the midsection is significantly lower than in a plane strain model; the von-

Mises equivalent stress at the midsection is distinctly higher than in the plane strain model. For the

same cohesive zone parameters, the crack tip softening zone during the crack growth is larger and the

simulated crack growth rate is significantly higher in the plane strain model than that in the midsec-

tion of 3D model.

(2) In thickness direction, the stress triaxiality varies smoothly near the midsection, but decreases dras-

tically when approaching the side-surfaces. When the cohesive zone parameters are taken to be con-

stant, at the point of fracture initiation the highest triaxiality occurs at the midsection. With

increasing crack extension, the triaxiality near the midsection decreases more significantly than at

other locations; thus, after some crack extension, the highest triaxiality peak appears at the boundary

between the influence zone of the side-surface and the midsection region.

(3) The stress triaxiality varies dramatically during the initial stages of crack growth, but varies smoothly

during the subsequent stable crack growth. Near the midsection, the triaxiality at the point of fracture

initiation is distinctly higher than during the stable crack growth, caused by the increasing von-Mises

equivalent stress due to the hardening.

(4) Near the midsection, a decrease of the cohesive strength results in a decrease of the stress triaxiality; a

decrease of the separation energy results in an increase of the stress triaxiality. At the side-surfaces,

the influence of cohesive zone parameters on the stress triaxiality is weak.

(5) If the slant shear-lip fracture near the side-surfaces is modeled as a normal fracture along the symme-

try plane of the specimen, both the cohesive strength and the separation energy near the side-surface

should be considerably lower than near the midsection.

Acknowledgments

The authors acknowledge the financial support of this work by the Materials Center Leoben under

the project numbers SP7 and SP14. C.R. Chen thanks the support from the G19980615 project of

China. The finite element calculations were finished at the Center for Advanced Materials Technology,


University of Sydney, Australia, and C.R. Chen gratefully acknowledges the financial support from Prof.

Y.W. Mai.

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Documents

3D CZM Ductile Crack Growth Triaxiality