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CHAPTER FIVE
GURSON MODEL FOR DUCTILE FRACTURE
5.1. IntroductionCrack initiation and propagation is one of the common failure modes for tubular joints
subjected to tensile loads. However, simulation using finite elements based on continuum
mechanics formulation does not represent the effect of crack initiation and growth, which
violates the continuity and integrity of material and geometry.
Four numerical methods conventionally used to simulate crack initiation and propagation
include: discrete crack model, fracture mechanics, smeared crack model and continuum
damage mechanics. In the discrete crack model, crack develops only along existing
element boundaries and the crack growth depends on the mesh size and orientation
(Cofer and Will, 1992). For the fracture mechanics approach, the crack size and
orientation has to be assumed or known a-prior. In the smeared crack model, the material
stiffness decreases to zero when fracture is detected. The continuum damage mechanics
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approach introduces a damage variable to reflect the amount of damage in the structure
throughout the loading history. The constitutive relationship is modified using this
damage variable (Lemaitre 1985).
The present study adopts the void growth and nucleation approach which was established
by Gurson (1975). The Gurson model simulates the plastic yield behavior of a porous, or
void containing, material. Under plastic deformation, the material strain hardens, and
voids nucleate and grow, and subsequently lead to deformation localization and fracture.
Thomason (1990) reports that all engineering metals and alloys contain inclusions and
second-phase particles, to a greater or lesser extent, and this leads to void nucleation
and growth .
This chapter begins with a description of the Gurson model formulation. The next section
discusses the benchmark study on the classical bar-necking problem. The subsequent
section presents the Gurson model simulation in tubular joints, followed by a sensitivity
study on the Gurson model parameters. The findings in this chapter are summarized by
Qian et al. (2005a).
5.2. Gurson Model FormulationThe yield condition in the Gurson model is modified by Tvergaard (1981) by introducing
three qi (i = 1, 2 and 3) factors as shown in Eq. 5.1.
2
22
1 322 cosh (1 ) 0
2
e m
yy
qfq q f
= + + =
where q3 = q1
2(5.1)
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in which, e refers to the effective stress, m denotes the hydrostatic pressure, fdefines
the void volume fraction and y
is the material yield stress. The yield criterion in Eq. 5.1
becomes von Mises yield criterion whenf= 0. Forq1 = q2= q3 = 1.0, the yield function
bears the same form as the original Gurson model. Tvergaard (1981) reports that q1 = 1.5,
q2 = 1.0 and q3 = q12
= 2.25 better represents materials subjected to plain-strain condition.
Instead offin the original Gurson model, q1frepresents the void volume fraction in Eq.
5.1, and magnifies the void volume fraction by a factor ofq1 (Thomason, 1990).
The change of void volume arises from two sources: growth of existing voids and
nucleation of new voids (Tvergaard, 1981):
growth nucleationdf df df = + (5.2)
nucleation v pdf A d = (5.3)
where
2
1exp
22
p NN
v
NN
fA
ss
=
(5.4)
The plastic nucleation strain follows a normal distribution. The nucleated void volume
depends on the mean plastic nucleation strain, N, standard deviation, sN, and the void
volume fraction of the nucleating particles,fN (Tvergaard, 1981). These three parameters
depend on material properties and vary with different materials. Void nucleation initiates
once yielding occurs for high strength steel FeE 690 (Arndt and Dahl 1997). fN refers to
the ratio of the void volume in the nucleating particles over the entire volume of the
material and is thus less than the total void volume fraction of the material, which is
normally less than 10% (Thomason, 1990). Otsuka et al. (1987) reports that ductile crack
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initiation occurs when the void volume fraction reaches about 4-6% for bending mode.
The Gurson model is an upper-bound theory (Gurson, 1977) and does not include the
plastic limit load failure of the inter-void matrix (Thomason, 1990).
In this chapter, the load-deformation curves for tubular joints follow the notations below.
Curve name (G) Analysis using Gurson model
Curve name (CT) FE model with crack-front simulation
Curve name_f fine mesh is employed
Curve name_m medium mesh is employed
Curve name_c coarse mesh is employed
5.3. Benchmark Study5.3.1.Bar-Necking ProblemThe conventional bar-necking problem has been studied to verify different void growth
and nucleation models by researchers (Mahnken 1999, Needleman 1972, Tvergaard and
Needleman 1984). The FE simulation in the current study employs 2D plane-strain finite
elements. The applicability of the Gurson model in 3D continuum elements (C3D20R in
ABAQUS element library) is verified by comparing the behavior of the tensile bar
simulated using 3D elements with that of 2D axi-symmetric elements (CAX8R in
ABAQUS element library).
Figure 5.1 describes the details of the FE mesh and geometry, with the material and
geometric property obtained from ABAQUS benchmarks manual (2001). Figure 5.1(c)
indicates the Gurson model parameters used in the analysis, with qi (i =1, 2 or 3)
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parameters modified by Tvergaard (1981). Figure 5.1(c) indicates little difference
between 2D and 3D models.
(a) 3D solid model (b) 2D axi-symmetric model (c) Load-deformation curves
Fig. 5.1 3D solid model; (b) 2D axi-symmetric model; and (c) Load-deformation curves; for the bar
necking problem.
5.3.2.Model Parameters qi (i= 1, 2 and 3)The effect of void volume fraction in the Gursons yield function depends partly on the
definition of three qi (i = 1, 2 or 3) parameters as introduced by Tvergaard (1981). The
effect of these parameters is thus investigated using 3D tensile bar model.
Tvergaard (1981) reports that the qi parameters depend on the strain-hardening properties.
Faleskog et al. (1998) propose a relationship between strain-hardening and qi values.
According to Bessen et al. (2001), q2 = 1.15 provides the best fit for round bar and plane
strain specimens. Therefore, three values of q2 are selected: q2 = 1.0, 1.15 and 2.0. In
0.0 0.2 0.4 0.6 0.80.0
1.5
3.0
4.5
Tvergaard model
q1
= 1.5, q2
= 1.0, q3
= 2.25
P
(unit)
(unit)
3D model N
=0.30
2D model sN
=0.10 fN
=0.04
0.5l0 = 4
R=1
Rotating
axis
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addition, comparison is carried out between the Tvergaard model and the original Gurson
model. A relatively large value ofq1 = 2.0 is also included for comparison.
(a) Effect of q1 (b) Effect of q2
Fig. 5.2 (a) Effect of q1; and (b) Effect of q2; on the tensile bar.
Figure 5.2 compares the effect of qi on the 3D tensile bar model. The effect of qi is
noticeable only after the strength reduction becomes significant. Increasing qi either
magnifies the effect of void volume fraction f or hydrostatic pressure p, which causes
more severe reductions in the tensile strength. In addition, variation of qi has no
observable effect on the initiation of strength reduction in the tensile bar.
5.4. Tubular Joints5.4.1.Joints with Initial CrackTwo joint test specimens with initial crack: a T-joint with a surface crack (Zerbst et al.,
2002a) and an X-joint with a through-thickness crack (HSE, 1999), are simulated and
0.0 0.2 0.4 0.6 0.80.0
1.5
3.0
4.5
P(
unit)
(unit)
q2=1.0
N=0.30 s
N=0.10
q2=1.15 f
N=0.04
q2=2.0 q
1=1.5 q
3=2.25
0.0 0.2 0.4 0.6 0.80.0
1.5
3.0
4.5
q1=1.0
N=0.30 s
N=0.10
q1=1.5 f
N=0.04
q1=2.0 q
2=1.0 q
3=q
12
P(
unit)
(unit)
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compared using the Gurson model. Both joint specimens experience brace axial loads.
Table 5.1 lists the geometric parameters of these two joints. For both joints, the initial
crack locates at the chord saddle.
Table 5.1 Geometry of tubular joints with and without initial crack
Joint Reference Loading d0(mm)
T-joint Zerbst et al. (2002a) Axial 298.5 0.51 7.46 1.0 10.2Cracked
X_HSE HSE (1999) Axial 572.0 0.48 15.1 0.5 9.1
X1 Axial 407.4 1.0 25.5 1.0 17.5
X2
Sander & Yura
(1986)Axial 407.4 0.35 25.5 0.83 4.5
X3 Kang et al. (1998c) OPB 169.0 0.61 11.4 0.85 16.6Intact
K-joint Wang et al. (2000) Axial 217.4 0.65 24.8 0.75 13.9
5.4.1.1.T-JointZerbst et al. (2002a) report four T-joint tests with the same joint parameters (= 0.51, =
7.46, = 1.0, = 10.2) and different pre-crack geometry. The crack is introduced as a
surface notch at the saddle point. The crack length, 2c, is 46.5 mm for all four joints. The
T-joint with the crack depth, a = 9.94 mm, is selected. Figure 5.3(a) shows the schematic
configuration of a T-joint, with the crack front geometry. The numerical analysis includes
four different models. Figure 5.3(b) shows crack-front modeling of the surface notch,
while Fig. 5.3(c) (e) shows three continuous FE models which ignore the notch (i.e. an
intact model), with different mesh density, since the Gurson model depends on values
of plastic strains. The fine mesh utilizes forty elements around the quarter brace-chord
intersection curve in order to ensure the element aspect ratio to be close to 1:1:1.
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The boundary-loading condition of the T-joint assumes pin supports to the chord ends
and applies tensile loads along the brace axis. Figure 5.4 compares the load-deformation
curves of the numerical analyses and test results. In Fig. 5.4(a), the numerical analysis
includes the void growth without the void nucleation property. The initial porosity f0
imposes a critical effect on the joint strength. However, the initial porosity can be
improved during material production process. In spite of the large initial porosity (0.05)
for the crack-front model, no load reduction occurs in the Gurson model with only the
void growth process. Figure 5.4(b) compares joints with and without incorporation of the
void nucleation process. Material softening becomes significant with void nucleation
under large plastic deformations, and consequently causes a slight load reduction.
(a) T-joint configuration
(b) Crack-front modeling (c) Fine mesh (d) Medium mesh (e) Coarse mesh
Fig. 5.3 (a) T-joint configuration; (b) Crack-front modeling; (c) Fine mesh; (d) Medium mesh; and (e)
Coarse mesh; for pre-cracked T-joint.
d0
t0
Surface notch
Quarter
brace-to-chordintersection
t1
d1
2c
a
Crack tip
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0.00 0.05 0.10 0.15
0
4
8
12
f0
= 0.05
f0
= 0.01
q1=1.5, q
2=1.15
P/f
yt02
/d0
Test =0.51=7.46FE(CT) =1.0=10.18FE(CT)(G) d
0=298.5mm
FE(CT)(G)
0.00 0.05 0.10 0.15
0
4
8
12
No nucleation
fN
= 0.10
q1=1.5, q
2=1.15
P/fy
t02
/d0
Test =0.51=7.46FE(CT) =1.0=10.18FE(CT)(G) d
0=298.5mm
FE(CT)(G)
(a) f0 (b) Void nucleation
Fig. 5.4 Comparison of numerical results for the effect of: (a) f0; and (b) Void nucleation.
Figure 5.5(a) compares three fN values on the crack-front model. The Gurson model
depends significantly on the fN value. A large value offN corresponds to a less ductile
material, and the strength reduction initiates at a very small deformation level. The
comparison between different fN values shows that the difference between fN = 0.10 and
0.20 is much more dramatic than the difference between fN= 0.04 and 0.10. Figure 5.5(a)
incorporates the load-deformation curves for the three meshes of the intact model. For all
three meshes, the Gurson parameters assume the following values: q1 = 1.5, q2 = 1.15 N
= 0.10,sN = 0.05 andfN = 0.10. The crack-front model develops from the medium mesh
model. This implies that the mesh size further away from the crack-front is similar to the
medium mesh. Similar load-deformation characteristics are observed between the crack-
front model and the medium mesh for the same material parameterfN = 0.10. This
indicates that the mesh refinement in the crack-front may affect the local stress evaluation
around the crack front. It does not, however, impose a significant influence on the global
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joint strength computation since the very refined elements in the crack front do not form
a collapse mechanism along the joint load path. Variation of the material parameters, fN,
may exert a comparatively stronger effect on the joint strength. On the other hand, with
the same material parameters, the effect of void growth and nucleation becomes most
significant in the fine mesh which causes the largest plastic strain and most pronounced
strength reduction. Figure 5.5(b) compares the load-CTOD (Crack-Tip Opening
Displacement) curves. The difference between the numerical analysis and the test
increases as the joint deformation increase, since the current FE model does not simulate
crack propagations.
0.00 0.05 0.10 0.150
4
8
12
fN
= 0.04
fN
= 0.20
fN
= 0.10
fN
= 0.10
fN
= 0.10
fN
= 0.10
P/f
yt02
/d0
Test FE_f(G)
FE(CT)(G) FE_m(G)
FE(CT)(G) FE_c(G)
FE(CT)(G)
0.0 0.4 0.8 1.20
1000
2000
3000
P(kN)
CTOD (mm)
Test =0.51=7.46FE =1.0=10.18
d0=298.5mm
(a) Effect of fN (b) CTOD
Fig. 5.5 Comparison of numerical results for: (a) Effect of fN; and (b) CTOD.
5.4.1.2.X-JointThe HSE report (1999) describes a series of X-joint tests with and without cracks. One X-
joint with through-thickness crack at the saddle position is selected in this study. The
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crack length extends about 15% of the brace-chord intersection perimeter. The through-
thickness crack is simulated in two ways. In the normal fine FE mesh as indicated in Fig.
5.6(a), eighty elements are utilized to simulate half of the brace-chord intersection, with
the elements corresponding to the crack removed. The element aspect ratio in the brace-
chord intersection area is around 1:1:1. In the other mesh scheme, the crack-front is
simulated in the refined model.
Figure 5.6(b) compares the numerical crack-front model and test results. The tested joint
experiences strength reductions as crack propagates. Without the Gursons algorithm, the
FE crack-front model sustains increasing strength within the prescribed deformation.
DifferentfN values with q1 = 1.57, q2 = 0.97 (as suggested by Faleskog, et al., 1998) and
initial porosity equal to 0.01 are investigated. The Gurson model analysis with a smaller
fN (0.04) value terminates due to the numerical convergence at a deformation level
corresponding to the load reduction in the test. With a larger value offN (0.20), the
analysis terminates at an even earlier deformation level due to very large deformation of
the crack-tip elements. This proves the inappropriateness of a large fN value. Table 5.2
lists the joint strength for the two joint specimens with crack-front simulations. The
strength definition follows the plastic limit load approach presented in Chapter Four. In
cases where the FE analysis terminates earlier than the plastic limit load approach, the
joint strength refers to the load level at the end of the analysis.
Figure 5.6(c) presents the results of the normal fine FE models. With the absence of the
Gurson model, the crack propagation is not captured, and the joint strength prediction can
be un-conservative if a large deformation limit defines the joint strength. Both the crack-
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front model and normal fine mesh (with the same material properties) show similar
ultimate strength levels compared to the test results. This implies that the simulation of
the crack-front singularity is not necessary in the strength analysis with the Gurson model.
(a) Crack-front and fine mesh
0.00 0.02 0.04 0.06 0.08 0.100
5
10
15
20
termination of
analysis
=0.48=15.1=0.5=9.1d
0=572mm
15% crack
P/f
yt02
/d0
Test FE(CT)
FE(CT)(G) fN
=0.04
FE(CT)(G) fN
=0.20
0.00 0.02 0.04 0.06 0.08 0.100
5
10
15
20
termination of
analysis
=0.48=15.1=0.5=9.1d
0=572mm
15% crack
P/fy
t02
/d0
Test FE_f
FE_f(G) fN
=0.10
FE(CT)(G) fN
=0.10
(b) Crack-front FE results (c) Fine mesh results
Fig. 5.6 (a) Crack-front mesh and fine mesh; (b) Crack-front FE results; and (c) Fine mesh FE results; for
X_HSE joints.
Both cracked joint models demonstrate the importance of the Gurson model parameters.
Although the crack in the T-joint does not cause strength reductions, the inclusion of a
large fN (0.20) magnifies the effect of crack and leads to under-estimations of the joint
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strength. For the X-joint, a large fN (0.20) value terminates the numerical analysis at a
small deformation level with the joint strength lower than the test results, as shown in
Table 5.2. For both joints, the stress evaluation within the crack-front region differs
highly from the normal FE mesh without the crack-front simulation. However, the small
crack-front region does not form a critical failure in the joint load-path. As a result, only
marginal strength difference exists between the crack-front model and normal FE model.
Table 5.2 T- and X_HSE joint strength for varyingfNwith crack-front simulationFE results ISO Others FE
Joint fN
(kN) FE/Test (kN) ISO/test (kN) Others/test
Test
(kN)
0.04 2246 0.94
0.10 2204 0.92T-joint
0.20 2077 0.87
2617a
2790b
1.09
1.162170 0.91 2397
0.04 1885* 0.97
0.10 1908* 0.98X_HSE
0.20 1602* 0.82
1559a
2469b
0.80
1.27 2589* 1.33 1945
* Strength taken at the end of analysis
aStrength corresponding to first crack (with chord bending effect incorporated if present)
bUltimate strength (with chord bending effect incorporated if present)
Table 5.2 compares the current FE predictions against ISO (2001) mean strength
formulation and FE computation from the reported study (Zerbst et al., 2002b & HSE,
1999). The ISO joint strength corresponding to crack initiation under-estimates the
X_HSE joint strength by about 20%. Over-conservatism is observed for X_HSE joint
using ISO ultimate strength formulation. On the other hand, both ISO strength predictions
over-estimate the T-joint strength. The FE prediction presented by Zerbst et al. (2002b)
shows a lower joint strength when plasticity occurs, although the joint strength does not
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indicate any sign of strength reduction. For X_HSE joint, the FE results reported by the
researchers (HSE, 1999) shows increasing joint strength with increasing joint
deformation, since crack effect is not incorporated into the FE modeling.
5.4.2.Intact JointsTwo types of intact joints (X- and K-joints) in reported studies (Sanders and Yura 1986,
Kang et al. 1998c, Wang et al. 2000) are used to verify the Gurson model. Two types of
loading conditions are studied for X-joints: the brace tensile axial loading and OPB
moment. The K-joint is loaded compressively in one brace and supported on the other
member end. Table 5.1 shows the geometrical properties for these joints.
In the selected tests (except for X2), cracks initiate in the loading history, which leads to
reductions in the joint capacity. Since the crack locates in the chord wall, the Gurson
model property is assigned to the welds and chord material. The default Gurson model
parameters are: q1 = 1.5, q2 = 1.15, q3 = 2.25; and the material parameters are: N= 0.10,
sN = 0.05, fN = 0.10. The values forN , sN and fN are selected within the nominal range
empirically, since they are not reported in the respective tests. Section 5.5 investigates the
effect of these parameters, through a detailed sensitivity study.
5.4.2.1.X-Joints under Axial TensionThe current study selects two of the X-joints (X1 and X2) reported by Sanders and Yura
(1986), each with two duplicated tests. Similar to the cracked T-joint, three meshes are
generated for the X-joints with different levels of refinement. In the fine mesh, element
aspect ratio is maintained around 1:1:1.
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0.00 0.02 0.04 0.06 0.08 0.10
0
40
80
120 =1.0=25.5=17.5d
0=407.4mm
N
=0.10, fN
=0.10, sN
=0.05
First crack in test B
First crack in test A
P/fy
t02
/d0
X1 (test A) FE_f
X1 (test B) FE_f(G)
0.00 0.04 0.08 0.12
0
5
10
15
20
25
N
=0.10, fN
=0.10, sN
=0.05
First crack in test D
First crack in test C
P/f
yt02
/d0
X2 (test C) =0.35 =25.5X2 (test D) =0.834=4.49FE_f d
0=407.4mm
FE_f(G)
(a) X1 (b) X2
Fig. 5.7 Comparison of test and numerical results for: (a) X1; and (b) X2.
Figure 5.7 compares the X-joint behavior with and without the Gursons algorithm. The
two duplicated tests for X1 fail by fracture failure. The tests stop once the crack
penetrates through the chord wall, although no sign of load reduction exists. Similar trend
among different meshes is observed as that of the T-joint. Hence, Fig. 5.7 shows only
results from fine mesh, where substantial strength reduction occurs. For both X1 and X2,
the predicted peak loads remain similar to that in the tests, although the corresponding
deformation levels are larger than that in the tests. In X1 tests, the fracture failure is
defined as the through-thickness penetration of the crack in the chord wall, which has not
mobilized the full joint capacity. In X2 tests, no sign of crack failure exists at the end of
the tests. The first crack in all four tests does not limit the joint capacity as shown in Fig.
5.7. This implies that the joints offer a significant amount of reserve strength beyond the
crack initiation. It is the propagation of crack that leads to the loss of joint capacity.
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5.4.2.2.X-Joint under OPB
For X-joint subjected to OPB moment, one side of the chord wall is under compression
and the other under tension. The possible failure modes include the instability associated
with the compression chord wall, and the fracture failure of the tensile chord wall.
Kang et al. (1998c) reported the effect of chord pre-load on the moment capacity of X-
joints. The OPB test without the presence of chord load is selected as the reference in the
current study. In Kangs test, pure moment is applied through a four-point load
mechanism on the brace, as illustrated in Fig. 5.8(a). In the FE simulation, pure moment
is applied. The measurement of brace rotation is taken consistently within the pure
moment loaded brace corresponding to the test. The element size is refined in the brace-
to-chord intersection such that the element aspect ratio is around 1:1:1.
0.0 0.1 0.2 0.30
2
4
6
8
N
=0.10,
fN
=0.10,
sN=0.05Yura's
deformation limit
Mopb
/fy
d1
t02
(radian)
Test =0.61=11.4FE_f =0.85=16.6FE_f(G) d
0=169mm
(a) Load application in Kangs test (b) Comparison of FE and test
Fig. 5.8 (a) Load application in Kangs test; and (b) Comparison of FE and test; for OPB loaded X-joint.
F/2 F/2
F/2 F/2
x
Fx/2
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Figure 5.8(b) compares the test and numerical results. Without the Gurson model, the X-
joint sustains increasing load with further deformation. Although the load reduction is not
captured by the Gursons algorithm, the presence of voids softens the material and
reduces the joint strength when plasticity occurs and propagates in the joint. This
weakens the joint strength without introducing an abrupt load reduction. Fracture failure
occurs far beyond Yuras deformation limit, which is a deformation level that practical
joint may not be able to achieve (Zettlemoyer, 1988).
5.4.2.3.K-JointThe K-joint test reported by Wang, et al. (2000) is selected as the reference. No initial
crack is introduced in the joint test. Similar to the study of X- and T-joints, three FE
schemes are adopted to verify the effect of mesh density on the joint response. In the fine
mesh, four layers of elements are employed in the gap region of the chord. Sixteen
elements are utilized along the 30 mm gap length. The element aspect ratio in the gap and
the brace-to-chord intersection region is maintained around 1:1:1. Figure 5.9(a) shows the
boundary conditions of the K-joint.
Figure 5.9(b) shows the load-deformation curves for K-joints. The under-prediction of
the ultimate joint strength may be caused by the un-reported material strain-hardening. In
spite of the initiation and propagation of the crack in the K-joint, no load reduction exists
in the compression brace behavior. With the Gurson model, a slight load reduction is
observed at a relatively large deformation level, due to the reduction in the tensile brace
resistance initiated by void growth and nucleation processes. Conventional FE analysis
without the Gurson model does not indicate a reduction in the strength. The first crack
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reported in the test occurs much earlier than the ultimate strength. However, the joint
strength continues to increase with the presence of crack.
0.00 0.04 0.08 0.120
10
20
30
40
N
=0.10, fN
=0.10, sN
=0.05
First crack
Psin/f
yt02
/d0
Test =0.65=24.8FE_f =0.75=13.9FE_f(G) d
0=217.4mm
(a) Boundary condition (b) Comparison of FE and test results
Fig. 5.9 (a) Boundary condition; and (b) Comparison of FE and test results; for the K-joint
5.4.3.DiscussionTable 5.3 lists the joint strength obtained from the FE analyses (using the Gurson model
with N= 0.10,sN= 0.05 andfN= 0.10) and the test results for the intact joints. There are
three strength definition adopted for the FE results (obtained from the fine mesh): the
plastic limit load approach; Lus deformation limit (Lu et al., 1994) and 15% plastic
strain (Dexter and Lee, 1999a). For the T-joint, the joint strength at the end of the test is
recorded. Among the X- and K-joints, the peak loads in the tests correspond to fracture
failure of the joint except for X2, for which, no crack develops. Table 5.3 compares the
current FE results against the ISO mean strength, which shows very conservative
At all member ends, x = z = 0
x
zy
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predictions for X-joints under brace tension. For X-joints under OPB and K-joint under
brace compression, over-estimation of the ISO formulation is observed.
Table 5.3 Comparison of ultimate strength for intact joints
Plastic load Lus deformation 15% strain ISOJoint Test
FE FE/Test FE FE/Test FE FE/Test (kN) ISO/test
X1 (kN) 2248 2154 0.96 2055 0.91 2016 0.901054
a
2194b
0.47
0.98
X2 (kN) 397 392 0.99 388 0.98 334 0.84200a
302b
0.50
0.76
X3 (kNm) 12.8 10.4 0.81 - - 10.9 0.85 18.2 1.42
K- (kN) 225 210 0.93 208 0.92 206 0.92 277 1.27
aStrength corresponding to first crack
bUltimate strength
Table 5.4 Displacement at 15% plastic strain for three joint types
Joint Fine (/d0) Medium (/d0) Coarse (/d0) Coarse/Fine
X1 0.05 0.05 0.07 1.3
T-joint 0.04 0.04 0.12 3.3
K-joint 0.02 0.03 0.40 2.5
Lus deformation limit shows similar strength predictions as the plastic limit load
approach. Both approaches correspond to a state earlier than the fracture failure in the
tests. The plastic strain limit approach defines the joint strength corresponding to the first
attainment of 15% plastic strain at the element integration point around the brace-chord
intersection. The non-dimensional joint displacement /d0 corresponding to the 15%
plastic strain for different mesh schemes is tabulated in Table 5.4. The displacement in
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the coarse mesh increases up to 3.3 times that of the fine mesh, which implies significant
size dependence of the plastic-strain criterion.
Load path remains crucial in the void growth and nucleation process. Gursons approach
to simulate the effect of ductile fracture is based on the assumption of continuum
mechanics, with the yield condition and plastic flow rule modified by the void volume
fractionf. Stress redistribution is mobilized once material softens due to the propagation
of voids, with the adjacent material mobilized, contributing to the joint strength. It
therefore, delays the formation of the complete collapse of the tensile load path.
Consequently, load reductions predicted by the Gurson model correspond to larger
displacements as compared to those obtained from the tests for most of the joints studied.
For X-joints with high ratio (X1), a very small portion of the chord wall between the
two braces forms the critical load path. For small joints (X2), the chord material
between the two braces constitutes a relatively larger volume. Stress redistribution from
the highly stressed area to other regions help to release the high stress in the critical load
path, which in turn delays the effect of the void growth and nucleation. As shown by the
FE results in Fig. 5.7, there is an obvious strength reduction in X1 at around the
deformation level of/d0 = 0.08. However, there is only slight strength reduction for X2,
when the deformation level exceeds /d0 = 0.10. This shows that the effect of crack is
more significant for X-joints with high than lowratios, which is consistent with the
observation reported in the Health and Safety Executive (HSE) report (HSE, 1999).
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The brace-chord intersection of a T-joint, as compared to an X-joint, is under additional
tension due to bending generated by chord end reactions. This complicates the loading on
the crack front. The assumed Gurson properties for the intact joint may not be a good
representation of the real material property and this may be a possible cause for the
significant void growth and nucleation effect in the intact fine mesh with N = 1.0, sN =
0.05 andfN= 0.10.
The gap region between the braces forms the critical load path for K-joint. The gap near
the weld toe of the tension brace experiences tensile and shear stresses. However, the
brace tension is proportioned by the brace angle , and the tensile stress induced by the
local bending may not be of a significant magnitude. In Fig. 5.9, there is a marginal
difference between the analysis with and without the Gurson model around the first
plateau of the load-deformation curve. However, as plasticity spreads in the joint, the
difference becomes more pronounced.
5.5. Sensitivity StudyThe void nucleation material parameters: N,sNandfN, play a significant part in ensuring
the accuracy of the analysis as observed in the T-joint study. These material properties
vary with different types of steel properties and may be affected by the manufacturing
process (Thomason, 1990). Up to now, there is no sufficient material data to quantify the
parameters such as N, sN and fN for each type of material. In order to investigate the
effect of these material properties, a sensitivity study is conducted on three types of joints:
X- T- and K-joints, using the fine mesh for the intact models. In the study on the effect of
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each material parameter, the rest of the parameters remain as constants. As the strain
hardening parameter varies, N, sN and fN take the values of 0.10, 0.05 and 0.10
respectively. As N changes, sN and fN equal 0.05 and 0.10, with the reported material
strain hardening property included. For the variation ofsN, N andfNboth take the value
of 0.10. WithfNvaried, NandsNremain at 0.10 and 0.05.
5.5.1.Strain-HardeningDifferent strain-hardening relationship causes different levels of plastic strain under the
same loading condition. The current study adopts the power hardening law described in
Eq. 5.5.
,
,
y
N
y
yy
forE
forE
= >
(5.5)
The hardening relationships vary the value ofN: 5, 10, 20 and (corresponding to
elastic-perfectly-plastic property). When the hardening relationship changes, the Youngs
modulus remains at 205 GPa. The joint deformation at the load reduction, as tabulated in
Table 5.5, reflects the effect of void growth and nucleation.
Similar trends in the three types of joints are observed. Strength reduction occurs at a
smaller deformation level with a largerN(lower strain-hardening). The strain-hardening
relationship for the reported X-joint test resembles that represented by N= 10. For the T-
joint,N= 10 also approximates closely the reported strain-hardening. For all three joint
types, N = 5 corresponds to an un-realistically high strain-hardening, which produces
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very high joint strength. Higher strain hardening delays the development of plastic strain,
which postpones the void growth and nucleation process.
The joints with elastic perfectly plastic property, which is normally assumed in the design,
require special consideration in the design procedure. With zero strain-hardening, the
plastic strain attains easily the nucleation strain, which causes dramatic loss of the joint
strength as demonstrated in Table 5.5. This does not reflect the real material behavior, for
which strain-hardening exists to some extent.
Table 5.5 Peak joint strength and corresponding deformation for various material properties
X1 T-joint K-jointParameters
/d0 Ppeak/fyt02 /d0 Ppeak/fyt0
2 /d0 Ppeak/fyt0
2
5 0.095 191 0.117 17.2 0.064 38.0
10 0.051 109 0.067 12.6 0.024 30.5
20 0.029 82.2 0.043 10.8 0.018 28.7
N
(N=0.10,sN=0.05,
fN=0.10) 0.011 65.0 0.032 9.53 0.015 26.1
0.0 0.097 111 0.060 11.1 0.018 26.7
0.10 0.075 103 0.049 10.9 0.017 26.8N
(sN=0.05,
fN=0.10) 0.30 0.105 123 0.119 12.4 0.018 26.9
0.04 0.105 118 0.091 11.8 0.018 26.8
0.10 0.075 103 0.049 10.9 0.017 26.8
fN
(N=0.10,sN=0.05) 0.20 0.031 79.6 0.033 10.0 0.018 26.7
0.05 0.075 103 0.049 10.9 0.017 26.8sN(N=0.10,fN=0.10) 0.10
0.084 108 0.060 11.3 0.018 26.8
Test results 0.049 105 0.103 12.5 - 28.7
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5.5.2.Effect ofNThe value ofNtakes 0.0, 0.1 and 0.3 to assess the effect ofNon the Gurson model. The
case of N = 0.0 implies immediate void nucleation once material yields. A relatively
large N(0.3) corresponds to a very ductile material. The values ofsN and fN takes 0.05
and 0.10, respectively, as Nis varied. Figure 5.10 illustrates the effect ofNon X1 and T-
joint. The ultimate load level does not vary much with the N value, with strength
reductions first observed in both joints with N= 0.10.
0.00 0.04 0.08 0.120
50
100
150s
N=0.05
fN
=0.10
P/fyt0
2
/d0
N
=0.0 (G) =1.0=25.5
N
=0.10 (G) =1.0=17.5
N
=0.30 (G) d0=407.4mm
X1(test A) X1(test B)
0.00 0.04 0.08 0.120
5
10
15
sN
=0.05
fN
=0.10
P/f
yt02
/d0
N
=0.0 (G) =0.51=7.46
N
=0.10 (G) =1.0=10.18
N
=0.30 (G) d0=298.5mm
Test
(a) X1 (b) T-joint
Fig. 5.10 Effect ofN on Gurson model for: (a) X1; and (b) T-joint.
Table 5.5 reflects the effects ofN on three joint types via the joint peak load and the
corresponding deformation. The same trend exists for all three joints. The ultimate
strength level does not vary significantly with N. However, the deformation level
corresponding to the peak strength becomes relatively larger for X- and T-joints with N=
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0.30 than N = 0.10 and 0.0. A small value of N (0.0) does slightly reduce the joint
strength once plasticity occurs. This early initiation of void nucleation does not cause a
consequently earlier load reduction for all three joints as shown in Table 5.5 and Fig. 5.10.
For all the three joint types, the effect of void nucleation becomes more pronounced in N
= 0.10, followed by N = 0.0 and N = 0.30. For X-joints, fracture failure occurs at the
deformation level around /d0 = 0.05 in the test. All three values of N cause a strength
reduction at a larger deformation level. For T-joint, load reduction does not occur in the
test. The value of = 0.30 delays both the initiation of void nucleation and the
consequent load reduction, and thus shows a closer correlation with the tested joint. The
effect of the N on the tension brace of K-joint is least prominent, due to the fact that the
primary failure mode of the K-joint is not fracture failure.
5.5.3.Effect offNand sNThree values offN are selected to investigate the effect of volume fraction of the void
nucleating particles: 0.04, 0.10 and 0.20. The value of 0.04 is the nominal value
recommended by ABAQUS (2001). A very large valuefN= 0.20 is included to compare
the amplified effect of void volume of the nucleating particles. The values ofsNtake 0.05
and 0.10. Figure 5.11 shows the effect offN and sN on X1-joint. The joint response
exhibits a strong dependence onfNand a much lower dependence onsN.
Both Table 5.5 and Fig. 5.11 demonstrate the significant effect offN. From Eqs. 5.3 and
5.4, the void nucleation rate is directly proportional to fN. Small fN (0.04) delays the
development of void volumes and a relatively large fN (0.2) accelerates the void
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nucleation process, causing premature strength reductions. The joint strength differs more
significantly between large fN values (0.1 and 0.2) than between smallerfN values (0.04
and 0.1). As mentioned, however,fN= 0.2 does not describe a realistic material.
0.00 0.04 0.08 0.120
40
80
120N
=0.10
sN
=0.05
P/f
yt02
/d0
fN
=0.04 (G) =1.0=25.5
fN
=0.10 (G) =1.0=17.5
fN
=0.20 (G) d0=407.4mm
X1(test A) X1(test B)
0.00 0.04 0.08 0.120
40
80
120
N
=0.10
fN
=0.10
P/fy
t02
/d0
sN
=0.05 (G) =1.0=25.5
sN
=0.10 (G) =1.0=17.5
test A d0=407.4mm
test B
(a) fN (b) sN
Fig. 5.11 Effect of (a) fN; and (b) sN; on X1 joint.
The value ofsN does not affect the joint behavior significantly. Similar joint strength
among different values ofsN is observed for all joints, with slight variation in the peak
strength and the corresponding displacement. The effect offNandsN is more pronounced
for X- and T-joints, with very little effect on the K-joint behavior.
5.6. SummaryThe Gurson model reflects the fracture failure by load reductions in the load-deformation
curves of tubular joints. The accuracy of the Gurson model depends primarily on two
factors: the mesh size and the material properties. The detailed mesh refinement in the
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crack front does not form a critical load path, and thus is not necessary in joints with
initial cracks. The mesh size should remain sufficiently small near the critical load path to
compute accurately the strain value. The current study recommends more than three
layers of elements across the chord wall thickness with the element aspect ratio
maintained around 1:1:1 near the brace-chord intersection. The Gurson model parameters
qi (i = 1, 2 or 3) does not introduce an effect on the tensile bar until the post-peak state.
The void nucleation parameters (N, sN and fN) determine the accuracy of the Gurson
model, with the effect offN being most significant. A large value offN (> 0.10) does not
represent the realistic material. Without material data, fN shall vary between 0.0 to 0.10.
The value ofNhardly affects the ultimate joint strength, while a large value ofN (0.30)
postpones the void nucleation process. Hence, 0.0 N 0.10 is suggested. sN imposes
marginal effect on the joint response. A nominal range of 0.05 to 0.10 is recommended.
The effect of void growth and nucleation on the tubular joint depends also on the load
path. Among the three types of joints investigated, K-joint is the least sensitive to
variation of the nucleation parameters compared to X- and T-joints.