A Finite Element Ductile Failure Simulation Method Usig Stress Modified Fracture Strain Model

7/29/2019 A Finite Element Ductile Failure Simulation Method Usig Stress Modified Fracture Strain Model

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A finite element ductile failure simulation method using

stress-modified fracture strain model

Chang-Sik Oh a, Nak-Hyun Kim a, Yun-Jae Kim a,, Jong-Hyun Baek b, Young-Pyo Kim b,Woo-Sik Kim b

a Korea University, Department of Mechanical Engineering, Anam-Dong, Sungbuk-Ku, Seoul 136-701, Republic of Koreab Korea Gas Corporation Research and Development Center, 638-1 Ansan, Kyonggi-do 425-150, Republic of Korea

a r t i c l e i n f o

Article history:

Received 14 January 2010

Received in revised form 4 September 2010

Accepted 13 October 2010

Available online 20 October 2010

Keywords:

Ductile fracture simulation

Finite element analysis

Stress-modified fracture strain

a b s t r a c t

This paper proposes a new method to simulate ductile failure using finite element analysis

based on the stress-modified fracture strain model. A procedure is given to determine the

stress-modified fracture strain as a function of the stress triaxiality from smooth and

notched bar tensile tests with FE analyses. For validation, simulated results using the pro-

posed method are compared with experimental data for cracked bar (tensile and bend)

tests, extracted from API X65 pipes, and for full-scale burst test of gouged pipes, showing

overall good agreements. Advantages in the use of the proposed method for practical struc-

tural integrity assessment are discussed.

2010 Elsevier Ltd. All rights reserved.

1. Introduction

For structural integrity analyses, performing full-scale tests is important, but is in general quite expensive and time-

consuming. Furthermore, it is often very difficult to perform full-scale tests reflecting complex geometries and loading

conditions in practical assessment. For these reasons, an efficient tool may be needed not only to design complex full-scale

tests but also possibly to minimize expensive and time-consuming full-scale tests. Finite element ductile failure simulations

based on the local approach are quite useful in this respect and thus increasingly important.

A number of works have been reported in the literature up to present on finite element ductile failure simulations.

Depending on the model employed for simulating damage, existing works can be broadly classified into two categories.

The first one is using a micro-mechanical model for ductile fracture, incorporating void nucleation, growth and coalescence,

for instance, the GursonTvergaardNeedleman model [116] and the Rousellier model [1720]. The second category is

using a phenomenological model for ductile fracture. One popular model is, for instance, the cohesive zone model [21

27]. As many researchers have already published a number of papers using these methods, applicability and validity of these

methods have been well discussed in the literature. From the authors point of view, it is felt that a few issues need to be

resolved in practical application of these methods. The first one is how to find parameters embedded in these models. For

instance, the GTN model has eight parameters related to micro-mechanism of ductile fracture. Determination of these

parameters are not an easy task, and often not robust. Although other models tend to have less parameters, robust parameter

determination is a common problem in finite element ductile failure simulations. The second issue is that, when these

methods are to be implemented into commercial finite element programs, special subroutines or elements often need to

be developed.

0013-7944/$ - see front matter 2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.engfracmech.2010.10.004

Corresponding author. Tel.: +82 2 3290 3372; fax: +82 2 926 9290.

E-mail address: [email protected] (Y.-J. Kim).

Engineering Fracture Mechanics 78 (2011) 124137

Contents lists available at ScienceDirect

Engineering Fracture Mechanics

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / e n g f r a c m e c h
http://dx.doi.org/10.1016/j.engfracmech.2010.10.004mailto:[email protected]://dx.doi.org/10.1016/j.engfracmech.2010.10.004http://www.sciencedirect.com/science/journal/00137944http://www.elsevier.com/locate/engfracmechhttp://www.elsevier.com/locate/engfracmechhttp://www.sciencedirect.com/science/journal/00137944http://dx.doi.org/10.1016/j.engfracmech.2010.10.004mailto:[email protected]://dx.doi.org/10.1016/j.engfracmech.2010.10.004


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This paper proposes a new method to simulate ductile failure using finite element method, based on a phenomenological

stress-modified fracture strain model for ductile fracture. Section 2 briefly summarizes experimental results. In Section 3, the

proposed method is introduced, together with explanations on how to simulate progressive failure. Section 4 compares sim-

ulated results with experimental ones. The present work is discussed in Section 5 and concluded in Section 6.

2. Summary of experimental programs

This paper attempts to propose a finite element based ductile failure simulation method. Tests were performed for two

objectives; firstly to show how to apply the proposed method to simulate ductile failure, and secondly to validate the pro-

posed method by comparing with experimental data. These tests include smooth and notched bar tensile tests, cracked bartests and full-scale burst tests of gouged pipes. The material is the API X65 steel [28] popularly used for gas transportation

within Korea. Chemical composition of the API X65 steel is given in Table 1. This section briefly explains these tests and

results.

2.1. Smooth and notched round bar tensile tests

Standard round bar specimens for tensile tests were extracted (in the longitudinal direction) from a API X65 pipe having

outer diameter Do = 762 mm and wall thickness t= 17.5 mm. To investigate the effect of triaxial stress states on tensile prop-

erties, notched round bar specimens with three different notch radii, 6.0 mm, 3.0 mm and 1.5 mm, were also made. For all

specimens, the minimum section has a diameter of 6.0 mm. Schematic diagrams for smooth and notched round tensile spec-

imens, employed in the present work, are depicted in Fig. 1.

Tensile tests were performed for smooth and round notched bars. In testing, axial displacement was monitored using

extensometer with the gauge length of 25 mm. For a given specimen geometry, tests were repeated three times. Engineeringstressstrain data from all tests are shown in Fig. 2a. Yield and tensile strengths determined from smooth bar tensile tests

were about 465 MPa and 563 MPa, respectively, as tabulated in Table 1. True stressstrain data, obtained from smooth round

bar tests using the Bridgman correction [29], are shown in Fig. 2b. The true fracture strain is $1.2, indicating that the mate-

rial is sufficiently ductile. Notched bar tests show that, as the notch radius decreases, the yield and tensile strengths increase,

but the strain to fracture decreases due to the higher stress triaxialties introduced by the notch.

Nomenclature

a crack lengthDo outer diameterd gouge depthE Youngs modulus gouge length

P, Pb pressure and burst pressureR radius of the minimum section of tensile barsr radial distance from the center of the smooth and notched barst thicknessw widthd displacement at the gouge mouth, see Fig. 5ee equivalent strain, Eq. (5)ep (equivalent) plastic strainef fracture straine1, e2, e3 principal strain componentsm Poissons ratiore, rm effective stress and mean normal stress, respectively, see Eqs. (1) and (2)r1, r2, r3 principal stress componentsr

y, r

uyield (0.2% proof) strength and tensile strength, respectively

x, Dx damage and incremental damage

Table 1

Chemical compositions and mechanical tensile properties at room temperature of the API X65 steel, used in the present work.

Youngs modulus Poissons ratio Yield strength Tensile strength Element (wt.%)

E (GPa) m ry (MPa) ru (MPa) C P Mn S Si Fe Ceq

210.7 0.3 464.5 563.8 0.08 0.019 1.45 0.03 0.31 Balance 0.32

C.-S. Oh et al. / Engineering Fracture Mechanics 78 (2011) 124137 125
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2.2. Cracked bar tests

Single-edge-cracked bar tests were performed. To investigate constraint effects due to the loading mode on failure behav-

iours, two types of cracked specimens were extracted from API X65 pipes. One is for three-point bending loading (single-

edge cracked bend specimen, SE(B), Fig. 3a) and the other is for tensile loading (single-edge cracked tension specimen,

SE(T), Fig. 3b). Relevant dimensions are also shown in Fig. 3. The width w and thickness t of SE(B) specimens were

Fig. 1. Schematic illustrations of (a) smooth bar and (b) notched bar tensile specimens (units: mm).

Fig. 2. (a) Engineering stressstrain curves from smooth and notched round bar tests, and (b) true stressstrain data.

Fig. 3. Schematic illustrations of (a) SE(B) and (b) SE(T) specimens (units: mm).

126 C.-S. Oh et al. / Engineering Fracture Mechanics 78 (2011) 124137


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10 mm. The SE(T) specimens had the width ofw = 12 mm and the thickness oft= 25 mm. Note that sizes of SE(T) and SE(B)

bars were small, as they were extracted directly from a pipe piece. An initial notch was machined and the subsequent pre-

cracking was introduced by fatigue. No side-groove was made for all specimens. For SE(B) specimens, the final crack length,

a, was about 50% of the width. Specimens were loaded using three-point bend, and the crack mouth opening displacement

(CMOD) was monitored as a function of load. Load-CMOD curves resulting from two identical specimens are shown in Fig. 4a.

For SE(T) specimens, the final crack lengths were about 25%, 50% and 75% of the width, to see the effect of the crack length on

failure behaviours. A total of seven specimens were tested, two for a/w = 0.25, three for a/w = 0.5 and two for a/w = 0.75. Ten-

sile loading was applied using the grip, and the displacement was monitored using the extensometer with the 10 mm gauge

length. Resulting loaddisplacement curves are summarized in Fig. 4b. It shows that overall behaviours for SE(T) specimens

are quite different from those for SE(B) ones.

2.3. Burst tests of full-scale pipes with gouge defect

Test pipes having the outer diameter ofDo = 762 mm, the thickness oft= 17.5 mm, and the total length of 2300 mm were

prepared. To simulate gouge defect, a 45 V-notch with a circular notch radius of 2 mm was made on the outer surface of the

pipe, as shown in Fig. 5. The depth of the gouge, d, was fixed as d = 8.75 mm which is 50% of the pipe thickness (d/t= 0.5). The

axial length of the gouge, on the other hand, was systematically varied from = 100 mm to = 600 mm (Table 2). Both ends

were capped by circumferential welding. The pipes were pressurized by water and burst pressures were experimentally

determined at the point when the ligament failed. Fig. 6 depicts photos of the gouge before and after the test. The measured

maximum (burst) pressures, Pb, from full-scale pipe tests are summarized in Table 2, which shows that the burst pressure

decreases with increasing gouge length.

Fig. 4. (a) Experimental load-CMOD curves for SE(B) specimens, and (b) experimental loaddisplacement curves for SE(T) specimens.

Table 2

Summary of full-scale tests of pipes with gouge defect.

Pipe no. MNA MNB MNC MND MNE

l (mm) 100 200 300 400 600

Pb (MPa) 24.68 22.48 17.7 18.14 16.57

Fig. 5. Schematic illustrations of a full-scale pipe with gouge defect (units: mm).



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3. Proposed ductile failure simulation method

3.1. Damage model and failure simulation

The damage model proposed in this paper is based on the concept of the stress-modified fracture strain model. It has been

well-known that (true) fracture straine

f for dimple fracture strongly depends on the stress triaxiality (defined by the ratio ofthe mean normal stress rm and equivalent stress re) [3036]:

rm

re

r1 r2 r33re

1

re ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi1

2fr1 r22 r1 r32 r3 r22g

r2

where ri (i = 13) denotes the principal stress. Although detailed expressions differ slightly, the dependence of ef on the

stress triaxiality can be modelled using an exponential function

ef / exp armre

3

where a is a material constant. A detailed form of the fracture strain can be found from notched bar tensile tests, as will bedescribed in Section 3.3. Once the form of ef is available as a function of the stress triaxiality, incremental damage due to

plastic deformation, Dx, is calculated (at each gauss point) using

Dx Depef

4

where Dep is the equivalent plastic strain increment, calculated from FE analysis.

When the accumulated damage becomes unity, x =RDx = 1, ductile failure is assumed and incremental crack growth is

simulated by reducing all stress components at the gauss point sharply to a small plateau value, as schematically depicted in

Fig. 7. Decreasing stresses to zero can cause numerical problems, and thus stresses are assumed to remain a finite (cut-off)

value. The results from sensitivity analysis for the effects of the decreasing slope and cut-off stress value will be given later.

3.2. Implementation to FE program ABAQUS

Suppose the fracture strain ef is determined for a given material. Then damage simulation using the proposed method

requires information on the stress triaxiality and incremental plastic strain values, which can be determined from elas-

ticplastic FE analysis. Thus the proposed method can be easily implemented in commercial FE programs. To implement

the proposed damage model to ABAQUS [37], two user subroutines were developed. The first one is the USDFLD subroutine

to re-define field variables. By defining as state variables, information on the mean normal (hydrostatic) stresses and equiv-

alent stresses/strains at gauss points is passed into the UHARD subroutine. In the UHARD subroutine, accumulated damage is

calculated according to Eq. (4) using information passed from the USDFLD subroutine. When the accumulated damage be-

comes critical (unity), stresses are relaxed simply by changing the yield surface.

3.3. Determination of fracture strain ef

To apply the present damage model, the first step is to determine the fracture strain ef

as a function of the stress triaxiality

for a given material. By combining notched bar test results and elasticplastic FE analysis, the explicit form ofef can be found

Fig. 6. Gouge defect (a) before the test, and (b) after the test.

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as follows. Elasticplastic, axi-symmetric FE analyses simulating tensile tests of smooth and notched round bar specimens

were performed to determine variations of the triaxial stresses and strains within the specimens. Symmetric conditions were

fully utilized and the second order, reduced integration elements (CAX8R within ABAQUS) were used for efficient computa-

tion. Typical FE meshes with the different notch radii are shown in Fig. 8. The number of elements and nodes in typical FE

meshes ranged from 484 elements/1557 nodes to 658 elements/2089 nodes. To incorporate the large geometry change effect

in tensile testing, the large geometry change option was chosen.

Fig. 9 compares experimental engineering stressstrain data from smooth and notched (notch radius = 3 mm) tensile

tests with the FE results. Although the FE analysis cannot reproduce failure of tensile test specimens, it can well simulate

deformation behaviour even after necking up to failure initiation points. Agreements between the test results and FE ones

are quite good up to failure initiation points which are indicated in Fig. 9 using the cross symbols.

Fig. 8. FE meshes for notched tensile bars: (a) notch = 1.5 mm and (b) notch = 3 mm.

Fig. 7. Schematic illustration of simulating ductile failure using stress relaxation.

Fig. 9. Comparison of experimental engineering stressstrain data for (a) smooth and (b) notched (radius = 3 mm) tensile bars with FE results.



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Local stress and strain fields in the minimum section of the tensile bars were extracted from the FE results as a function of

applied load. Fig. 10 show radial variations of the stress triaxiality rm/re and equivalent strain ee in the minimum (necked)

section of the smooth and notched bars at the point of failure initiation (see Fig. 9). The equivalent strain, ee, is defined by

ee ffiffiffi2

p

3

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffife1 e22 e1 e32 e3 e22g

q5

where ei (i = 13) denotes the principal strain. In the figure, the distance (r) is normalized with respect to the radius of the

minimum section (R), and the values ofr/R = 0 and r/R = 1 mean the center and the free surface of the specimen, respectively

(Fig. 8). The stress triaxiality decreases with increasing notch radius, but always attains its maximum value in the center of

the specimen, regardless of the notch radius. Furthermore, equivalent strain attains its maximum value in the center of the

bar, except for the 1.5 mm case where the maximum value of equivalent strain occurs at the notch tip. Thus for all cases

except for the 1.5 mm case, failure is expected to initiate in the center of the minimum section in the specimen. Even for

the 1.5 mm case, as the stress triaxiality in the center is much higher than that in the notch tip, failure is also expected

to initiate in the center of the minimum section in the specimen.

Fig. 11a shows the evolution of the stress triaxiality in terms of the equivalent strain for smooth and notched round ten-

sile bars in solid lines. Both the stress triaxiality and the equivalent strain are extracted in the center of the minimum section

of test specimens, where failure initiation is expected to occur. The last points in Fig. 11a correspond to the failure initiation

points. It shows that the stress triaxiality in the center of the specimen depend on the equivalent strain. As a ductile failure

criterion should include the history of stress and strain, average stress triaxiality is introduced, defined by

rm

re

ave 1

eefZeef0

rm

re

dee

6

Fig. 10. (a) Stress triaxiality distributions and (b) equivalent strain distributions for smooth and notched tensile bars at the failure initiation point,

determined from the FE analyses.

Fig. 11. (a) Variations of the stress triaxiality with the equivalent strain for smooth and notched bar tensile tests, and (b) fracture strain as a function of thestress triaxiality.



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where eef denotes the equivalent strain to failure initiation. Such definition can incorporate the history effect on stresses and

strains on ductile fracture. For a given notch radius, the calculated average stress triaxiality is constant, and is shown in

Fig. 11a with dotted lines. Resulting equivalent strains to failure initiation (called the fracture strain) are shown in

Fig. 11b, as a function of the (average) stress triaxiality. Note that one point in Fig. 11b corresponds to the result for one notch

radius. It shows that the fracture strain decreases sharply with increasing the stress triaxiality. Noting that the fracture strain

is found to be exponentially dependent on the stress triaxiality [36], the following regression is proposed for the fracture

strain ef:

ef 3:29exp 1:54rmre

0:10 7

which is shown in Fig. 11b with a solid line. It shows that Eq. (7) agrees with the data and captures dependence of the stress

triaxiality on equivalent strain to fracture. Note that the history dependent ductile failure strain, similar to the proposed one,

was originally discussed by Marini et al. [38].

3.4. GursonTvergaardNeedleman (GTN) Model

For comparison, results from the proposed model are also compared with those using the GTN (GursonNeedleman

Tvergaard) model. The GTN model to simulate ductile damage and failure is given by [1,5,6].

U

r2e

r2

2q1f

cosh q23rm

2r

1

q21f

2

0

8

rij _epij 1 f _repf

f f6fc

fc f

u fcfffc f fc f > fc

(

_f _fgrowth _fnucleation_fgrowth 1 f _epkk; _fnucleation A _ep;A

fN

sNffiffiffiffiffiffiffi2p

p exp 12

epe eN

sN

2( ) 9

In Eqs. (8) and (9), r is the current flow stress of the matrix material;f defines the effective void volume fraction; _epij is the

macroscopic plastic strain rate tensor; _ep is the average effective plastic strain rate of the material; fdefines the current void

volume fraction;ff andfc denote the void volume fraction at failure and at a critical situation at whichfstarts to deviate from

f, respectively; fu is the value f at fracture (i.e., fu = f

u (ff)); _epkk is the trace of the macroscopic plastic strain rate tensor; fN is

the volume fraction of particles available for void nucleation, eN is the mean void nucleation strain; and sN is the standarddeviation of the distribution. Note that the GTN model has eight micro-mechanical parameters (eN, sN, fN; fo, fc, ff; q1, q2) that

must be calibrated for a given material.

For the API X65 steel, the authors have already calibrated parameters for the GTN model in Ref. [39]. Five parameters re-

lated to void nucleation and the refined yield locus were fixed to typical values suggested in literature [26]: eN = 0.3, sN = 0.1,

fN = 0.0008; q1 = 1.5 and q2 = 1.0. The parameterfo, related to initial void volume fraction, was determined from the Franklins

formula [40] (based on the assumption of a spherical inclusion):

fo 0:054 S% 0:001Mn%

10

which leads to fo = 0.000125 for the present material. Two parameters (fc and ff) were calibrated by comparing smooth and

notched tensile bar test results with simulated results using the GTN model. They were chosen so that the simulated results

give best fits to test results. Resulting values offc andff were found to befc = 0.015 andff = 0.25. In applying the GTN model to

simulate ductile failure, a proper choice of the parameters in the GTN model is crucial. It should be noted that the calibrated

parameters are purely empirical, and we believe that no unique parameter set can be found. The calibrated values resulted

from more than hundred trials, when led to a conclusion that the above sets gave best results, compared to experimental

notched bar tests.

Finally it should be noted that the calibrated parameters may depend on the finite element size. The above parameters

were calibrated using the element size of 0.15 mm (150lm) which will be used throughout this paper.

4. Results

4.1. Sensitivity analysis

To apply the proposed model to simulate ductile failure, several points related to the proposed FE damage analysis need

to be resolved. They include, for instance, the element type and size. For the element type, first-order elements (the

four-node quadrilateral element for two dimensional and axi-symmetric problems and the eight-node brick element for



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Fig. 13. Sensitivity analysis results from 2-D SE(B) and SE(T) test simulations: (a) and (b) the effect of the decreasing slope; (c) and (d) the effect of thetolerance.

Fig. 12. FE meshes for the SE(B) specimen with a/w = 0.46: (a) 2-D mesh and (b) 3-D mesh.



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three-dimensional problems) with full integrations are used in the present work. Although second-order elements can be

used in principle, they often exhibit fluctuating hydrostatic stress values and thus can cause numerical problems when

the accumulated damage is calculated. For the element size, it is rather obvious that the element size can affect simulated

results from FE damage analyses, and thus cannot be chosen arbitrarily. It should be chosen to reflect the materials length

scale (such as void spacing for ductile fracture), and this is true not only for the proposed method but also for all FE damage

analyses. In this paper, the mesh size effect on simulated results was not investigated, and the size was fixed to 0.15 mm

(150 lm).1

As the proposed method decreases stresses at the gauss point to the cut-off value when the accumulated damage be-

comes unity (as schematically depicted in Fig. 7), appropriate decreasing slope and cut-off value should be chosen. To inves-

tigate the effects of the decreasing slope and the cut-off value on simulated results, sensitivity analysis was performed for the

cracked (SE(B) and SE(T)) bars, as effects would be the most pronounced for crack problems. For efficient investigations, two-

dimensional (2-D) simulations were performed rather than three-dimensional (3-D) ones. A typical FE mesh is shown in

Fig. 12a. Four-node plane strain solid elements (element type CPE4) of the size 0.15 mm 0.15 mm were uniformly spaced

in the cracked section. Simulated loaddisplacement records for SE(B) and SE(T) specimens are compared with experimental

data in Fig. 13. It shows that simulated results do not depend on the decreasing slope and the cut-off value, as long the

decreasing slope was taken to be smaller than 1/5000 (when the strain increases by 0.1, the stress decreases more than

500 MPa) and the cut-off value is less than 50 MPa. Although the decreasing slope and cut-off value should be ideally zero,

such values can cause numerical problems. Thus, for numerical efficiency, the cut-off value and decreasing slope in this work

were taken to be 10% of the yield strength (46 MPa) and 1/5000, respectively.

4.2. Smooth and notched bar tensile test results

Fig. 14 compares smooth and notched bar tensile results with simulated FE results resulting from three different analyses.

One is conventional elasticplastic analysis without any damage model (shown in dotted lines). The other two are using

Fig. 14. Comparison of smooth and notched bar tensile test results with simulated ones: (a) notch radius = 1.5 mm, (b) notch radius = 3 mm, (c) notch

radius = 6 mm, and (d) smooth bar.

1

One reason to fix the element size as 0.15 mm is to compare the present results with those using the GTN model. As noted in Section 3.4, the GTNparameters were calibrated using the element size of 0.15 mm.



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damage models, one using the GTN model (shown in open symbols), described in Section 3.4, and the other using the pro-

posed method (shown in closed symbols). For all cases, the element size was fixed to 0.15 mm. To incorporate the large

geometry change effect in tensile testing, the large geometry change option was chosen.

The conventional elasticplastic FE analysis can predict plastic behaviours of smooth and notched tensile tests up to the

failure initiations, indicated in the figures. However, after failure initiation, predictions deviate from experimental data, as

the conventional analysis cannot simulate failure. The FE results using the GTN model agree well with the experimental re-

sults for all cases. This is not surprising as the parameters associated with the GTN model are determined so as to give good

fits to smooth and notched bar tensile data, as explained in Section 3.4. For the FE analysis using the proposed method, agree-

ment is very good for smooth bar tensile test data. However, for notched bar tensile test data, the results are not as good as

those using the GTN model. Although the failure initiation can be well predicted, subsequent crack growth seems to be un-

der-predicted. Considering the fact that the present method does not involve any fitting process, predictions for notched bar

tests can be regarded as good.

4.3. Cracked bar test results

For comparison with cracked bar test results, 3-D FE analyses were performed, rather than 2-D plane strain analyses. This

is because the tested cracked bars do not have any side-groove, and thus the plane strain assumption may not appropriate. A

typical FE mesh for 3-D analyses is shown in Fig. 12b. Eight-node brick elements with full integrations (element type C3D8)

of the size 0.15 mm 0.15 mm 15 mm were uniformly spaced in the cracked section. The number of elements and nodes

in FE meshes were from 8202 elements/9791 nodes to 15,390 elements/17,654 nodes. True stressstrain data were directly

given in the FE analysis, and incremental plasticity with the large geometry change option was invoked.Fig. 15 compares cracked bar test results with simulated FE results. Again for the FE results, those from three different

analyses are compared with experimental data; the conventional analysis without any damage model (in dotted lines),

the analysis using the GTN model (in open symbols), and that using the proposed method (in closed symbols). Deviations

of the conventional elasticplastic FE results from experimental data are due to crack growth. The proposed method gives

overall good predictions for all cases considered, and agreements with experimental data are overall slightly better than

those for the GTN model.

Fig. 15. Comparison of cracked bar test results with simulated ones: (a) SE(B)with a/w = 0.46; (b), (c) and (d) SE(T) with a/w = 0.25, a/w = 0.5 and a/w = 0.75,respectively.

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4.4. Burst tests of full-scale pipes with gouge defect

Burst tests of full-scale pipes with gouge defects are simulated using FE analyses. A typical finite element mesh is shown

in Fig. 16, with a detailed view of the defective region. Eight-node brick elements with full integrations (element type C3D8)

of the size 0.15 mm 0.15 mm 15 mm were uniformly spaced in the defective section. The number of elements and nodes

in FE meshes were from 43,082 elements/51,088 nodes to 74,028 elements/89,569 nodes, depending on the gouge length.

Internal pressure was applied to the inner surface of the pipe, together with end forces to simulate the closed end condition.

True stressstrain data were directly given in the FE analysis, and incremental plasticity with the large geometry change op-

tion was invoked.

Fig. 17a shows simulated pressuredisplacement curves of gouged pipes. The displacement d is measured at the gouge

mouth, as schematically shown in Fig. 5. For FE results, the results from two different analyses are compared; one from con-

ventional elasticplastic analysis and the other from the proposed method. Failure points predicted from the proposed meth-

od are indicated using the cross symbols. Estimated burst pressures from the proposed method are compared with

experimentally measured data in Fig. 17b. The estimated burst pressures are in good agreement with experimentally mea-

sured ones with differences less than $3%, except the MNC case where the difference is as much as $12%. Although such a

difference is still not significant, it can be seen from Fig. 17b that the experimental data for the MNC case are somewhat low-

er than expected from the trend curve. Estimated burst pressures using the GTN model are also compared in Fig. 17b, show-

ing that predictions are overall close to those using from the proposed method. Both experimental and estimated burst

pressures decrease with increasing gouge length, as shown in Fig. 17b. Such a trend can be readily explained by idealizing

gouge as an axial crack. Empirical-based burst pressures for axial cracked pipes are given by [41].

Pf ry ru2

2tDo

1 dt

1 dt 1

M

!;M

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi1 1:61

2

2Do tts

11

Fig. 16. Typical FE mesh for pipes with gouge.

Fig. 17. (a) FE pressuredisplacement curves including predicted failure points, and (b) comparison of predicted burst pressures with experimental ones.



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where d and denote the crack depth and length, respectively. This equation predicts that burst pressures should decrease

with increasing crack length. Burst pressures estimated from Eq. (11) are compared with experimental and estimated ones in

Fig. 17b, showing that Eq. (11) gives conservative (lower) failure loads, as expected.

5. Concluding remarks

In this paper, a new method to simulate ductile failure based on FE analysis is proposed. The method is based on the

stress-modified fracture strain model which has been well-known phenomenological model for ductile fracture. Incrementaldamage is defined by the ratio of incremental plastic strain and stress-modified fracture strain. When the accumulated dam-

age becomes unity, then stresses at the gauss point are decreased to a small value to simulate progressive failure. To validate

the proposed method, tests are performed using specimens extracted from API X65 grade pipes. The stress-modified fracture

strain as a function of the stress triaxiality is firstly obtained by combining FE analyses and experimental data of smooth and

notched bar tensile tests. Then simulated results using the proposed method are then compared with experimental data

from cracked bar (tensile and bend) tests and from full-scale burst test of gouged pipes. Good comparisons with experimen-

tal results support the confidence of the proposed method.

The proposed method adopts the stress-modified fracture strain model as a failure criterion for ductile fracture, which has

been known for many years. Using notched bar tensile tests, determination of the stress-modified fracture strain model for a

given material is straightforward and robust, as described in this paper. Once the stress-modified fracture strain model is

determined, the proposed damage model does not include any parameter and thus does not require any calibration proce-

dure, which offers significant advantage in practical application.

Acknowledgements

This research is supported by Basic Atomic Energy Research Institute and Nuclear R&D Program

(M2080608000208M060800210), funded by the Korea Science & Engineering Foundation.

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A Finite Element Ductile Failure Simulation Method Usig Stress Modified Fracture Strain Model