Experimental and ductile fracture model study of single-groove welded joints under monotonic loading

Engineering Structures 85 (2015) 36–51

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Experimental and ductile fracture model study of single-groove welded joints under monotonic loading Lan Kang a, Hanbin Ge b,⇑, Tomoya Kato b a b

School of Civil Engineering and Transportation, South China University of Technology, Guangzhou 510640, China Department of Civil Engineering, Meijo University, Nagoya 468-8502, Japan

a r t i c l e

i n f o

Article history: Received 27 May 2014 Revised 3 December 2014 Accepted 4 December 2014

Keywords: Ductile crack initiation Steel welded joint Monotonic loading Ductile fracture model Base metal Weld metal HAZ

a b s t r a c t Tests and finite element (FE) analyses of smooth flat bar, U-notch and V-notch specimens are presented to demonstrate the application and validation of proposed three-stage and two-parameter ductile fracture model for evaluating the ductile crack initiation, propagation and final failure in steel welded joints under monotonic loading. Modeling concepts and procedures for characterizing the material parameters of ductile fracture model using smooth flat bar and U-notch tests are described. Accuracy of the model is validated through a series of tensile tests of U-notch, V-notch and welded smooth flat bar specimens. Three types of materials used in welded structures including base metal, weld metal and HAZ are investigated. Furthermore, the effect of notch position on ductile fracture behavior of HAZ specimens and the effect of mesh size on ductile fracture behavior of U-notch and V-notch specimens are studied. Detailed finite element analyses that employ the ductile fracture model are shown to predict ductile fracture behavior with good accuracy across the specimen geometries and material types in terms of ductile crack initiation point, ultimate load point and load–displacement curve. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction As one of the features of steel bridge structures, the steel member is of comparatively thin-walled section, and local buckling occurs in the steel member. However, for thick-walled welded steel members or concrete-filled steel piers, ductile cracks may occur in the welding or the base metal due to extreme load before occurrence of local buckling [1–6]. Despite the importance of welded structures in civil engineering construction, rigorous computational methods to evaluate their fracture resistance are not well developed, due to the challenges of simulating fracture in these weld details [7]. Accordingly, ductile fracture becomes one of important failure modes in steel structures for developing fracture-resistant design provisions and for evaluating structural performance under extreme loads such as strong earthquakes, especially for the thick-walled welded steel structures, and needs to be investigated deeply [8]. Fracture governs the ultimate strength of steel structure in a variety of situations where discontinuities lead to the concentration of inelastic strain and triaxial stresses, such as in net-sections of bolted connections, welded connections, regions of localized

⇑ Corresponding author. Tel./fax: +81 52 838 2342. E-mail address: [email protected] (H. Ge). http://dx.doi.org/10.1016/j.engstruct.2014.12.006 0141-0296/Ó 2014 Elsevier Ltd. All rights reserved.

yielding in steel members [9]. Assuming the void growth to be the defining step for ductile crack initiation, models that aim to predict ductile fracture need to capture the combined effects of the triaxiality and plastic strain [7]. In McClintock’s model [10,11], ductile failure initiation strain was first related with stress triaxiality. Rice’s and Tracey’s model [12] and Chi’s model [13] showed that the extension rate of void growth size is dependent on the triaxiality ratio and plastic strain rate of the material. Kanvinde and Deierlein [9] employed two models (void growth model indicated as VGM and stress modified critical strain model indicated as SMCS) based on this theory. The VGM involves an explicit integration of the stress and strain histories, whereas the SMCS is a simpler approach that is based only on the instantaneous values of the stress–strain quantities at fracture initiation [7,14–16]. Lemaitre’s model [17], which is based on continuum damage mechanics (CDM), showed that triaxial stresses contribute to damage leading to ductile fracture. Some researchers employed CDM to evaluate damage initiation and evolution in structural steels [18,19]. The above mentioned models can be classified into micromechanicsbased fracture model because they predict fracture based on combinations of local stresses and strains (at the crack tip or in a continuum) determined through finite element (FE) analysis [7]. Some researchers employed similar micromechanics-based fracture model to evaluate structural steels or steel members subjected to static tension [20–26]. However, those studies concentrated on

L. Kang et al. / Engineering Structures 85 (2015) 36–51

37

Nomenclature DI d

eeq rdi edi eq eeqf g deeq e_ eq DIc

a F ðgÞ Dcr M

eR eD p

m

damage initiation parameter damage evolution parameter equivalent plastic strain yield stress at the onset of damage equivalent plastic strain at the onset of damage equivalent plastic strain at element failure stress triaxiality incremental equivalent plastic strain equivalent plastic strain rate critical value of damage initiation parameter material toughness parameter stress modification function material constant material constant material constant material constant material constant Poisson’s ratio

structural steels, not weld metal and heat-affected zone (HAZ), where ductile crack easily initiates during earthquake because of their material discontinuousness. The ductile fracture performance of welded structures under extreme load still is a puzzle, yet not to be effectively resolved. Modeling of the crack distribution and extension of cracking in three dimensions can be difficult and requires an enormous amount of computational effort and time [27] because a real structure often has welded connection components with a complicated geometry. Due to the complexity of simulating fracture in structural weld details, there has been limited research in the application and modern fracture predictive techniques to weldments [7,27]. Some previous studies focused on the ductile fracture strength of welded members [5,28,29]. These studies focused on the global fracture behavior of welded members, the local fracture behavior (such as the ductile crack initiation and propagation) cannot be simulated effectively. Besides, detailed FE analyses are employed to study fracture toughness requirements in welded beam–column connections by Chi et al. [30]. The ductile plastic damage behavior of weld HAZ was studied based on CDM theory by Wang [31]. A ductile failure model proposed by Peñuelas et al. [32] based on the Gurson model for integrating the constitutive equations that describe the ductile failure process of a metallic material in a welded joint, and Gurson model is a basic damage model recommended for use in the analysis of emergency condition for building structures according to the current European standards [33]. Kanvinde et al. [7] employed SMCS model to evaluate its effectiveness in predicting fracture deformation capacity of structural fillet welds. Fracture behavior of beam-to-column welded joints was predicted by Huang et al. [34] using micromechanics damage model. Qian et al. [35] present a new fracture formulation to describe the ductile tearing and unstable fracture failure for circular hollow section joints under monotonically increasing brace tension. Although these approaches have been applied to welded steels in steel coupons or mechanical components, and the toughness and ductility of steel material are prescribed in European standard [33], they are relatively new to civil engineering applications because few studies (including experimental and analytical research) have been done in both qualitative and quantitative senses to structural engineering details. In severe earthquake, the structural steel members usually have to resist enormous extreme load which show a short load duration with large plastic deformation. It mainly depends on the

r E0 E r ueq D(ueq) Le ufeq

abs awm ahaz COV Ke Ks

ry ey Est

damaged stress tensor damaged elastic modulus undamaged elastic modulus undamaged stress tensor equivalent plastic displacement damage evolution function of equivalent plastic displacement characteristic length of the element plastic displacement at element failure material toughness parameter of base metal material toughness parameter of weld metal material toughness parameter of HAZ coefficient of variation elastic stiffness degradation stiffness yield stress yield strain initial strain-hardening modulus

mechanical properties of materials themselves to resist such loads, which is described as the reaction of large strain in material level [36] including ductile crack initiation, propagation and final failure. A multiscale approach (as shown in Fig. 1), which has been employed in other research areas [37], should be employed to the complete analysis and design of a steel structure that involves the material property characterization (material level), verification (member level) and its applications on structure analysis subjected to extreme load (structure level). Therefore, ductile fracture model of welded steel structures under extreme load plays a quite important role in structural seismic design and analysis. This study applies micromechanics-based fracture model based on Rice’s and Tracey’s theory [12] to the compact tension tests for fracture characterization of welded steels described in this paper. The model is selected due to its accuracy and simplicity for prediction of fracture in base metals during previous research [7,9,12,13,15,16]. The investigation involves a range of compact tension experiments and complementary FE analyses, including U-notch fracture test, V-notch fracture test, and smooth flat-bar fracture test. The ductile fracture parameters of different materials (including base metal, weld metal and HAZ), which are widely applied to civil engineering, are obtained. The paper then presents results of FE simulations conducted to examine the fracture modeling technique. The effect of notch position on ductile fracture behavior of HAZ specimens and mesh size sensitivity are investigated. Finally, commentary is provided on the results and limitations of the approach.

2. Three-stage and two-parameter ductile fracture model The ductile fracture model of structural steel in this paper is described using three stages and two parameters, as shown in Fig. 2(a). The damage initiation parameter DI is the variable which judges whether the corresponding damage initiation criterion has been reached. The damage evolution parameter d is the accumulated damage once the damage initiation criterion is met. The 1st stage – elastic stage (O ? A): the material is in elastic state, at the end of this stage (point A), a metal or other material ceases to behave elastically. The 2nd stage – plastic stage (A ? B): the material is in plastic state. At point A, the parameter DI = 0 because the equivalent plastic strain eAeq ¼ 0, and point A is the onset of

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L. Kang et al. / Engineering Structures 85 (2015) 36–51

Structure level

Time Member level

-to

a

Material level

a h a a 6

a

Ducle fracture Ducle fracture Ducle fracture

Length Fig. 1. Multi-scale approach for the ductile fracture of steel structure [35].

σ

Plasc stage

σ di

A: the onset of accumulated damage iniaon parameter

σy

(DI=1, d=0) σ B k

(DI=0) A

B: the material reaches the damage iniaon criterion, the onset of damage(d)

dσ

Soen stage E

Elasc stage

(1-d)E

E

C (DI=1, d=1) ε ε eqf

di ε eq

O

C: element final failure, element erosion

(a) Three-stage and two-parameter ductile fracture model d 1

ueqf

ueq

(b) Damage evolution Fig. 2. Ductile fracture model considering damage evolution.

accumulated damage initiation parameter. The DI increases with increasing of eeq. At the end of this stage (point B), the material reaches the damage initiation criterion (i.e. DI = DIc) and eBeq ¼ edi eq . rdi and edi eq are yield stress and equivalent plastic strain at the onset of damage, respectively. The 3rd stage – soften stage (B ? C): softening material behavior, which results macroscopically in a loss of material stiffness with adjacent failure, is preceded by the initiation and accumulation of microscopical defects such as cracks, micro-pores, shear-bands or crazes. At the beginning of this stage

(point B), DI = 1 and d = 0. Point B is the onset of damage. At the end of this stage (point C), DI = 1 and d = 1, which means that the element reaches final failure, which leads to element erosion at this position. efeq is equivalent plastic strain at element failure. 2.1. Damage initiation criteria Most of the phenomenological ductile fracture models are based on a damage initiation diagram which gives the equivalent

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L. Kang et al. / Engineering Structures 85 (2015) 36–51

plastic strain at damage initiation ðedi eq Þ as a function of the stress state (i.e. stress triaxiality g) can be used directly as a damage initiation criterion in the case of a linear strain path [38]. For the more general case of a nonlinear strain path, an integral damage initiation criterion is necessary. Kolmogorov [39] has presented an integral criterion according to the following equation:

Z

edi eq

0

deeq ¼1 edieq ðgÞ

ð1Þ

in which, g = rm/req, rm, req is the mean and effective stress, respectively; deeq is the incremental equivalent plastic strain. Integral criteria can account for nonlinear strain paths. However, in more severe cases of loading path changes (i.e. compression-tension reversal) even the integral criteria are no longer valid. edi eq ðgÞ is regarded as the equivalent initiation strain when the corresponding damage initiation criterion has been reached. The Eq. (1) can be written in the form of stress-modified critical plastic strain:

DI ¼

Z

e_ eq dt F ðgÞ

ð2aÞ

DI ¼ DIc

ð2bÞ

where FðgÞ is the stress modification function, e_ eq is the equivalent plastic strain rate, and DIc is the critical value of damage initiation parameter when the corresponding damage initiation criterion has been reached. For the models based on Rice’s and Tracey’s theory [12], the stress modification function can be written as:

FðgÞ ¼ a expð1:5gÞ

F ðgÞ ¼

Dcr

2 3

 ð1 þ mÞ þ 3ð1  2mÞg2 p2=M

r ¼ ð1  dÞr

ð5aÞ

E0 ¼ ð1  dÞE

ð5bÞ

 is the effective, i.e. where d is the overall damage variable and r undamaged, stress tensor computed in the current increment. If the failure surface is achieved at any stress state, the yield surface is forced to remain constant by setting the hardening modulus to zero and stiffness degradation, controlled by the scalar damage variable d, starts until the material has lost its load-carrying capacity (d = 1). This expression introduces the definition of the equivalent plastic displacement ueq as the fracture work conjugate of the yield stress after the onset of damage:

ueq ¼ Le eeq

ð6Þ

where Le is the characteristic length of the element. The evolution of the damage variable with the relative plastic displacement can be specified in linear form, as shown in Fig. 2(b). Once the plastic displacement of the element reaches the plastic displacement at element failure ufeq , the local damage d of the element reaches to 1.0. The damage variable can be directly assumed as a function of equivalent plastic displacement:

  d ¼ d ueq

ð7Þ

ð3Þ

where a is material toughness parameter that is determined through fracture test. For the Lemaitre’s model [17] based on CDM, the stress modification function can be written as:

eR  eD

criterion has been reached. A scalar damage variable d is introduced to control the stiffness degradation. At any given time during the analysis, the damaged stress tensor r and damaged elastic modulus E0 in the material are given by the scalar damage equation:

ð4Þ

In which eR, eD, Dcr, M, p are material constants, m is Poisson’s ratio. The two typical models predict very similar trends in the case of proportional loading, as seen in Fig. 3, where the stress modification function FðgÞ versus stress triaxiality g is plotted for the two models. In this paper, the employed ductile fracture model (i.e. micromechanics-based fracture model) is based on Rice’s and Tracey’s model because of its simplification and availability [7,9,12,13,15,16]. 2.2. Damage evolution and stiffness degradation The damage evolution law describes the rate of degradation of the material stiffness once the corresponding damage initiation

3. Ductile fracture tests for calibration of micromechanicsbased fracture model parameter 3.1. Experimental program The experimental program in this study provides important results regarding the characterization of damage in different structural welded materials (including base metal, weld metal and HAZ) loaded to extreme load in details. 19 small-scale tensile specimens were used for investigating the ductile fracture parameter and behavior of single-groove welds with designed thickness t = 12 mm. The steel used is SM490Y steel (equivalent to ASTM A242), and all of welded specimens were extracted from assemblies of two plates single-bevel groove welded together, as shown in Fig. 4. All specimens were subjected to 2000 kN universal tension machine under displacement control (as illustrated in Fig. 5). During the testing, load P and observed point displacement were measured and recorded by a data logger (TDS-530). Moreover, surface crack initiation behaviors for all types of specimens were observed and recorded continuously by two high speed cameras. One of the most important factors in the present experiment is to define ductile crack initiation. Though there are solutions such as damage inspection by magnetic dye penetrant, or fractographic measurement, ductile crack initiation is defined as the point when crack length extends to 1–2 mm according to visual or video camera observation [8]. 3.2. Tested specimens

Fig. 3. The stress modification function F(g) versus stress triaxiality g.

The specimens include two kinds of notched specimens (as shown in Fig. 6(a)) with U-notch and V-notch, which would produce ductile cracking from the surface of the notch root. Other specimens include welded and non-welded smooth flat-bar for characterizing material constitutive response. In which, 2 nonwelded smooth flat-bar specimens (indicated as NWSFB) and 3

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Weld metal Base metal 280

268

22 5

WM-U and WM-V HAZ-U and HAZ-V BS-U and BS-V 20

Specimen Weld metal

5

WSFB

Specimen 45°

12

22 280

10

280

Base metal

(a) Specimens extraction

(b) Notch’s position

Fig. 4. Schematic diagram illustrating the extraction of various specimens from welded plate assembly.

Ω extensometer

V-notch

geometry and size of all tensile specimens with U- and V-notch were measured and recorded before test. In order to investigate the effect of notch position on the ductile fracture behavior of welded steel joint, four HAZ specimens, which’s notch was located at different positions, were tested. Because there might be machining precision error during manufacturing, total ten specimens were fabricated, in which four specimens were selected to be tested, including two HAZ-U specimens and two HAZ-V specimens. 3.3. Welding condition

Specimen

The welded joint with thickness of 12 mm plate were fabricated with YM-55C welding. The welding condition of the welded specimens is as follows: current = 330 A, voltage = 38 V, travel speed = 50 cm/min, heat input = 15 kJ/cm, total pass = 3, interpass temperature = 350 °C, shielding gas was 100% CO2, and welding process was semi-automatic.

Loading direcon

3.4. Constitutive relationship obtained from welded and non-welded smooth flat-bar tension tests

Fig. 5. Photograph of specimen being tested in tension.

The microstructure of the single-bevel groove weld consisted of 3 regions; i.e. weld metal, HAZ and base metal [40]. In order to investigate the effect of these regions on plastic and fracture behavior of specimens, the constitutive relationships of different regions were obtained from welded and non-welded smooth flatbar tension tests. Their mechanics properties are listed in Table 1 and their stress–strain curves obtained from tests are demonstrated in Fig. 7.

welded smooth flat-bar specimens (indicated as WSFB) are employed to obtain stress–strain relationship and material properties for base metal, weld metal and HAZ, as illustrated in Table 1. For weld metal material, the fracture specimens include 5 U-notched specimens (indicated as WM-U) and 3 V-notched specimens (indicated as WM-V). For HAZ material, the fracture specimens contain 2 U-notched specimens (indicated as HAZ-U) and 2 V-notched specimens (indicated as HAZ-V). For base metal material, the fracture specimens consist of 1 U-notched specimen (indicated as BS-U) and 1 V-notched specimen (indicated as BS-V). These ductile fracture tests were used to determine the toughness parameter, a, of each material by combining the test data with complementary FE simulations. Designed geometry and size of tensile specimens with U- and V-notch are illustrated in Fig. 6(b). Because the ductile crack initiation and fracture behavior of specimen are greatly affected by true detailed geometry of specimens and actual size of weldment, the specimens were polished for clarifying the actual geometry of weld in test (see Fig. 6(c)), the actual

3.5. Determination of toughness parameter a The calibration process for the toughness parameter a involves determining a critical combination of the stress and strain state at crack initiation and fracture points. The FE analysis without fracture model is carried out to obtain equivalent plastic strain and stress triaxiality histories. The actual ductile crack initiation and final fracture points are determined through tests. Then substitute the equivalent plastic strain and stress triaxiality histories determined by FE analysis into the following equation to back-calculate a, in which end point of integral equation is determined by test results:

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L. Kang et al. / Engineering Structures 85 (2015) 36–51

Smooth flat-bar specimen

V-notched specimen

U-notched specimen

(a) Samples of tested specimens

(b) Designed geometry and size of tensile specimens with V- and U-notch

(c) Side view of single-bevel groove weld in tested specimen Fig. 6. Tested specimens.

Table 1 Mechanical properties of various materials. Material type

E (GPa)

m

ry (MPa)

ey (%)

Est (GPa)

a

ufeq

Base metal

214.5

0.3

390.2

0.182

5.36

1.965

0.02 (U-notch) 0.05 (V-notch)

Weld metal

169

0.3

370.0

0.219

4.23

2.178

0.05 (U-notch) 0.125 (V-notch)

HAZ

171

0.3

386.2

0.226

4.28

1.641

0.078 (U and V-notch)

Notes: E = Young’s modulus, t = Poisson’s ratio, ry = yield stress, ey = yield strain, Est = initial strain hardening modulus, a = toughness parameter, ufeq = plastic displacement at element failure.

a¼

Z 0

edi eq

deeq expð1:5gÞ

ð8Þ

First of all, the parameter of base metal (abs) is obtained from two NWSFB specimens, and accuracy of this parameter is validated through fracture tests of BS-U and BS-V specimens. Secondly, the parameter of weld metal (awm) is obtained from five WM-U specimens, and accuracy of this parameter is validated through fracture tests of three WM-V specimens. Furthermore, the parameter of HAZ (ahaz) is obtained from HAZ-U2 and HAZ-V1, and accuracy of

ahaz is verified through fracture tests of HAZ-U1 and HAZ-V2. Finally, the accuracy of all parameters (including abs, awm and ahaz) is validated through three fracture tests of WSFB specimens.

3.6. FE models of all specimens This section describes the development of FE models to simulate the fracture experiments. Because the detailed geometries of specimens have great effect on the ductile crack initiation,

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L. Kang et al. / Engineering Structures 85 (2015) 36–51

Fig. 7. Stress–strain relationship obtained from smooth flat-bar tension tests.

the necking and bulging, and the associated nonlinear stress and strains at the notch tip. At the location of ductile crack initiation, the refined mesh size is about 0.1 mm for smooth bar and U-notch specimens, and 0.2–0.3 mm for V-notch specimens. Different material relationships (such as base metal, HAZ and weld) are introduced into different regions as shown in Fig. 8. The stress triaxiality and the equivalent plastic strain histories are obtained and recorded ahead of the notch tip. The effects of residual stress and initial deformation are not considered. All the FE models incorporate large-deformations and isotropic von Mises plasticity. Hardening properties are based on uniaxial stress and strain data obtained from material tension coupons of WSFB and NWSFB. Besides, for HAZ specimens with different notch positions, FE models and mesh details are illustrated in Fig. 9. 4. Fracture behavior of tested specimens

Weld metal

4.1. Fracture behavior of weld metal specimens (WM-V and WM-U)

HAZ WM-U

WM-V

Base metal

WSFB Fig. 8. FE models for WM-U, WM-V and WSFB specimens.

propagation, and final fracture behavior of coupons, detailed measurements were conducted on the fracture specimens before testing and all of specimens were respectively simulated based on actual geometry and size measured by using the general FE software ABAQUS [41]. The FE models of the WM-U, WM-V and WSFB specimens are shown in Fig. 8. The three-dimensional model (C3D8R in ABAQUS) is necessary for these specimens to capture

It is observed in test that, both in WM-V and WM-U specimens, ductile cracking was found to result from the growth and coalescence of numerous nucleated micro-voids from the surface of notch root region (tensile zone), leading to final shear mode failure. The tensile zone is a coarse surface. The final fracture surface (shear zone) is very smooth and along a local shear band oriented at an angle of about 45° in relation to the tensile axis, as shown in Fig. 10(c) and (d). In contrast, in WSFB and NWSFB specimens, ductile cracking occurred from the central region (tensile zone), resulting in final ductile mode failure. As illustrated in Fig. 10(a) and (b), the tensile zone of WSFB and NWSFB is obviously larger than that of WM-U and WM-V. Qualitatively, surface cracking in the WM-U and WM-V specimens implies shear mode ductile cracking, which is in contrast with the ductile mode of ductile cracking from the specimen center in the smooth flat-bar specimens. From the load–displacement curves of the WSFB and NWSFB, a significant delay was observed between the ultimate load point and final failure point. In the WM-U and WM-V specimens, the delay was

Weld HAZ

Mesh detail

1.5mm

Mesh detail

3.2mm

: Notch root p line

(a) HAZ-V1

(b) HAZ-V2

Base metal

Mesh detail

Mesh detail

1.0mm

1.5mm

: Weld fusion line

(c) HAZ-U1

Fig. 9. FE model and mesh detail for HAZ specimens.

(d) HAZ-U2

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L. Kang et al. / Engineering Structures 85 (2015) 36–51

Tensile zone Shear zone

Shear zone

Tensile zone Shear zone

Shear zone

Ducle crack iniaon

Ducle crack iniaon

(a) NWSFB

(b) WSFB

Ducle crack iniaon

Ducle crack iniaon

Shear zone

Shear zone

Tensile zone

Tensile zone

(c) WM-U

(d) WM-V

Fig. 10. Final failure surfaces of NWSFB, WSFB, WM-U and WM-V specimens.

Tensile zone Shear zone Ducle crack iniaon

Tensile zone

Tensile zone

Ducle crack iniaon

(a) HAZ-U1

(b) HAZ-U2 Ducle crack iniaon Shear zone

Tensile zone Ducle crack iniaon

Tensile zone

(a) HAZ-V1

(b) HAZ-V2

Fig. 11. Final failure surfaces of HAZ specimens.

Table 2 Experimental and FE simulation results of NWSFB specimens. Specimen

Displacement of extensometer (mm)

Load (kN)

Toughness parameter

abs

Ductile crack initiation point

Ultimate load point

Final failure point

Ductile crack initiation point

Ultimate load point

Final failure point

NWSFB1

⁄ (47.053)

29.204 (30.146)

46.561 (47.053)

⁄ (230.514)

269.333 (270.701)

228.603 (230.514)

1.885

NWSFB2

⁄ (47.028)

30.004 (31.055)

47.021 (47.028)

⁄ (224.066)

267.178 (263.905)

226.839 (224.066)

2.044

Average (COV)

1.965 (0.0405)

Note: ⁄ means that the ductile crack initiated from the center of specimens, and so it could not be observed by cameras. The value in bracket is FE prediction result using the toughness parameter of 1.965 for base metal.

Table 3 Experimental and FE simulation results of BS-U specimens. Specimen

Displacement of extensometer (mm)

Load (kN)

Ductile crack initiation point

Ultimate load point

Final failure point

Ductile crack initiation point

Ultimate load point

Final failure point

BS-U1

11.320 (11.585)

11.188 (11.215)

12.152 (12.662)

205.649 (201.912)

206.658 (202.236)

145.774 (145.163)

BS-V1

9.906 (9.958)

13.732 (13.046)

16.030 (16.118)

185.696 (184.078)

192.160 (190.169)

100.200 (105.417)

Note: The value in bracket is FE prediction result using the toughness parameter of 1.965 for base metal.

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Table 4 Experimental and FE simulation results of WM-U specimens. Specimen

Displacement of extensometer (mm)

Load (kN)

Toughness parameter

awm

Ductile crack initiation point

Ultimate load point

Final failure point

Ductile crack initiation point

Ultimate load point

Final failure point

WM-U1

12.506 (12.052)

10.515 (10.930)

13.520 (13.389)

195.640 (209.009)

202.776 (214.822)

125.664 (108.266)

2.190

WM-U2

12.509 (11.973)

11.663 (11.274)

13.011 (13.067)

206.352 (221.435)

214.464 (223.123)

179.832 (179.508)

2.268

WM-U3

13.048 (12.071)

11.735 (11.439)

14.212 (13.870)

203.816 (225.922)

215.344 (227.174)

104.096 (118.503)

2.236

WM-U4

12.309 (11.993)

11.263 (11.235)

13.133 (13.610)

190.592 (200.427)

201.096 (202.483)

90.136 (104.022)

2.019

WM-U5

10.337 (10.623)

9.801 (9.985)

11.979 (12.828)

195.348 (191.543)

196.621 (194.511)

107.030 (114.032)

2.176

Average (COV)

2.178 (0.0859)

Note: The value in bracket is FE prediction result using the toughness parameter of 2.178 for weld metal.

Table 5 Experimental and FE simulation results of WM-V specimens. Specimen

Displacement of extensometer (mm)

Load (kN)

Ductile crack initiation point

Ultimate load point

Final failure point

Ductile crack initiation point

Ultimate load point

Final failure point

WM-V1

11.513 (10.593)

13.719 (13.525)

17.197 (16.479)

230.840 (233.934)

234.748 (241.423)

151.893 (147.268)

WM-V2

8.517 (8.400)

10.022 (10.634)

13.018 (12.897)

216.336 (225.880)

219.680 (234.424)

141.312 (140.324)

WM-V3

10.020 (9.514)

12.061 (12.735)

14.849 (15.327)

211.080 (208.674)

216.006 (216.255)

137.780 (131.360)

Note: The value in bracket is FE prediction result using the toughness parameter of 2.178 for weld metal.

Table 6 Experimental and FE simulation results of HAZ specimens. Specimen

Displacement of extensometer (mm)

Load (kN)

Toughness parameter

Ductile crack initiation point

Ultimate load point

Final failure point

Ductile crack initiation point

Ultimate load point

Final failure point

HAZ-V1

6.647 (6.517)

8.322 (8.615)

10.835 (10.315)

187.072 (186.418)

194.096 (194.403)

90.512 (90.877)

HAZ-V2

10.593 (10.720)

13.452 (14.505)

16.503 (16.175)

199.456 (197.148)

203.944 (203.489)

99.624 (95.077)

HAZ-U1

12.093 (11.969)

11.535 (11.687)

14.702 (13.398)

204.408 (201.690)

204.894 (204.408)

60.504 (60.969)

HAZ-U2

11.007 (11.124)

10.619 (10.946)

12.380 (12.433)

201.080 (203.435)

202.912 (203.798)

114.096 (110.166)

Average (COV)

ahaz 1.647

1.635 1.641 (0.00376)

Note: The value in bracket is FE prediction result using the toughness parameter of 1.641 for HAZ.

Table 7 Experimental and FE simulation results of WSFB specimens. Specimen

Displacement of extensometer (mm)

Load (kN)

Ductile crack initiation point

Ultimate load point

Final failure point

Ductile crack initiation point

Ultimate load point

Final failure point

WSFB1

⁄ (32.026)

22.214 (22.104)

31.120 (32.026)

⁄ (203.216)

247.848 (249.720)

205.512 (203.216)

WSFB2

⁄ (32.021)

22.260 (22.103)

32.495 (32.021)

⁄ (201.529)

243.472 (247.912)

199.448 (201.529)

WSFB3

⁄ (31.327)

19.524 (21.773)

27.553 (31.327)

⁄ (199.117)

240.800 (239.846)

202.992 (199.117)

4.63%

6.02%

0.99%

1.36%

Average error

Note: ⁄ means that the ductile crack initiated from the center of specimens, and so it could not be observed by cameras. The value in bracket is FE prediction result using the toughness parameters of 1.965, 2.178 and 1.641 for base metal, weld metal and HAZ, respectively.

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L. Kang et al. / Engineering Structures 85 (2015) 36–51

250

relatively smaller because the U- and V-notch lead to the earlier initiation of ductile crack. Generally, a discontinuity in either the material properties or the geometry may lead to ductile crack initiation, propagation and final failure in a steel structure. The geometry discontinuity in V-notch and U-notch specimens results in the final shear mode failure due to ductile crack initiation.

200

4.2. Fracture behavior of HAZ specimens (HAZ-U and HAZ-V)

Delay between ulmate load point and ducle crack iniaon point

Load (kN)

300

150

100

Ultimate load point Crack initiation point

NWSFB1-E NWSFB2-E NWSFB1-A NWSFB2-A

50

NWSFB1-E NWSFB2-E NWSFB1-A NWSFB2-A

NWSFB1-A NWSFB2-A

NWSFB

0 0

10

20

40

30

50

Displacement (mm)

(-E: experimental result, -A: analytical result)

250

250

200

200

Load (kN)

Load (kN)

Fig. 12. Comparison of experimental and simulated load–displacement curves of NWSFB specimens.

The final failure surface of HAZ specimens are shown in Fig. 11. The ductile crack of HAZ-U2 and HAZ-V1 (notch root tip in the middle of HAZ zone and about 1.5 mm from weld fusion line) initiated from the surface of notch root tip region, but leading to final ductile failure. The total final failure surface of them is coarse tensile zone and perpendicular to the tensile axis, without smooth shear zone. In the HAZ-U1 specimen (notch root tip in the weld metal and about 1 mm from fusion line), the ductile crack initiated from the surface of notch root tip region and near the boundary of weld and HAZ, but final shear failure occurred. The tensile zone of HAZ-U1 was obviously larger than that of WM-U and WM-V. The failure mode of HAZ-U1 is weld-to-HAZ transition. Similarly, the ductile crack of HAZ-V2 (notch root tip in the base metal and about 3.2 mm from weld fusion line) initiated from the surface of notch

150 BS-U1-E BS-U1-A Ultimate load point

100

150

Ultimate load point Crack initiation point

WM-U1-E WM-U2-E WM-U3-E WM-U4-E WM-U5-E WM-U1-A WM-U2-A WM-U3-A WM-U4-A WM-U5-A

100

BS-U1-E BS-U1-A Crack initiation point

50

50

BS-U1-E BS-U1-A

BS-U1

0 0

2

4

6

8

10

12

14

0 0

16

2

WM-U1-E WM-U2-E WM-U3-E WM-U4-E WM-U5-E WM-U1-A WM-U2-A WM-U3-A WM-U4-A WM-U5-A

WM-U1-E WM-U2-E WM-U3-E WM-U4-E WM-U5-E WM-U1-A WM-U2-A WM-U3-A WM-U4-A WM-U5-A

4

6

8

10

WM-U

12

14

16

Displacement (mm)

Displacement (mm)

(a) WM-U

(a) BS-U1 250 250

200

Load (kN)

Load (kN)

200

150 BS-V1-E BS-V1-A 100

150

Ultimate load point Crack initiation point

100 WM-V1-E WM-V2-E WM-V3-E WM-V1-A WM-V2-A WM-V3-A

Ultimate load point

BS-V1-E BS-V1-A

50

Crack initiation point

50

BS-V1-E BS-V1-A 0

2

4

6

8

10

12

14

16

WM-V

0

BS-V1

0

WM-V1-E WM-V2-E WM-V3-E WM-V1-A WM-V2-A WM-V3-A

WM-V1-E WM-V2-E WM-V3-E WM-V1-A WM-V2-A WM-V3-A

0 18

2

4

6

8

10

12

14

16

Displacement (mm)

(b) WM-V

(b) BS-V1

(-E: experimental result, -A: analytical result)

Fig. 13. Comparison of experimental and simulated load–displacement curves of BS specimens.

18

20

Displacement (mm)

Fig. 14. Comparison of experimental and simulated load–displacement curves of WM-U and WM-V.

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L. Kang et al. / Engineering Structures 85 (2015) 36–51

root tip region and near the boundary of base metal and HAZ, and final shear failure occurred. The failure mode of HAZ-V1 is HAZ-tobase metal transition.

5. FE simulation results and discussion 5.1. Calibration of the ductile fracture parameters

250

250

200

200

150

Load (kN)

Load (kN)

Application of the ductile fracture model for a given material requires the characterization of the two material parameters: (i) the toughness parameter a, (ii) the equivalent plastic displacement at failure ufeq . The two parameters are determined from monotonic tests of smooth and notched flat bar specimens. Values of a and ufeq for the base metal, weld metal and HAZ considered in this study, are summarized in Table 1 with other relevant material properties. The average value of the material constant a is 1.965 for base metal, 2.178 for weld metal and 1.641 for HAZ with a coefficient of variation (COV) of 0.0405, 0.0859 and 0.00376, respectively. These parameters can be considered as material constants, and the ductile crack initiation, ultimate load

and final failure points predicted using these parameters are presented in Tables 2–7. In which, the toughness parameter of base metal abs comes from the average value of two NWSFB specimens (as illustrated in Fig. 12 and Table 2), and is validated through BS-U1 and BS-V1 specimens (as shown in Fig. 13 and Table 3). The toughness parameter of weld metal awm is obtained from the average value of five WM-U specimens (as illustrated in Fig. 14(a) and Table 4), and is validated through three WM-V specimens (as shown in Fig. 14(b) and Table 5). The toughness parameter of HAZ aHAZ is obtained from the average value of two HAZ specimens (HAZ-V1 and HAZ-U2 as illustrated in Fig. 15 and Table 6), and is verified through another two HAZ specimens (HAZ-V2 and HAZ-U1 as shown in Fig. 15 and Table 6). Finally, these parameters are verified through three WSFB specimens (as illustrated in Fig. 16 and Table 7). The toughness parameter a is to determine the onset of damage parameter. Besides, the values of ufeq for various metals are calibrated through trial and error process until the finite element load displacement behavior matched fairly well with the experimental results and the predictions of ductile crack initiation, ultimate load and final failure points are within 10% of the experimental results.

HAZ-V1-E HAZ-V1-A

Ultimate load point

100

HAZ-V1-E HAZ-V1-A

HAZ-V2-E HAZ-V2-A

Ultimate load point

100

HAZ-V2-E HAZ-V2-A

Crack initiation point

50

Crack initiation point

50

HAZ-V1-E HAZ-V1-A

HAZ-V2-E HAZ-V2-A

HAZ-V1

0 0

2

4

6

8

10

12

14

16

0

4

8

6

10

12

Displacement (mm)

(a) HAZ-V1

(b) HAZ-V2 250

200

200

150 HAZ-U1-E HAZ-U1-A

Ultimate load point HAZ-U1-E HAZ-U1-A

14

HAZ-U2-E HAZ-U2-A

Ultimate load point

100

Crack initiation point

50

HAZ-U2-E HAZ-U2-A

HAZ-U1

0

HAZ-U2

0 2

4

18

HAZ-U2-E HAZ-U2-A

HAZ-U1-E HAZ-U1-A

0

16

150

Crack initiation point

50

2

Displacement (mm)

250

100

HAZ-V2

0

Load (kN)

Load (kN)

150

6

8

10

12

14

16

0

2

4

6

8

10

Displacement (mm)

Displacement (mm)

(c) HAZ-U1

(d) HAZ-U2 (-E: experimental result, -A: analytical result)

Fig. 15. Comparison of experimental and simulated load–displacement curves of HAZ specimens.

12

14

16

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L. Kang et al. / Engineering Structures 85 (2015) 36–51 300

Ducle crack iniaon

250

Shear zone

Test

Load (kN)

200

Tensile zone 150 Ultimate load point Crack initiation point

100

WSFB1-E WSFB2-E WSFB3-E WSFB1-A WSFB2-A WSFB3-A

50

0 0

4

8

WSFB1-A WSFB2-A WSFB3-A

WSFB1-E WSFB2-E WSFB3-E WSFB1-A WSFB2-A WSFB3-A 12

16

Analysis

Ducle crack iniaon

WSFB

20

24

28

32

36

40

Displacement (mm)

Fig. 17. Comparison of tensile zone of failure surface for WM-U2 specimen obtained from test and analysis.

(-E: experimental result, -A: analytical result) Fig. 16. Comparison of experimental and simulated load–displacement curves of WSFB specimens.

Using this procedure the ufeq for base metal, weld metal and HAZ is obtained and listed in Table 1. It is noted that the value of this parameter is related to notch type of specimens (such as U-notch and V-notch specimens). ufeq is to define degradation rate of material, and further work is also in progress to overcome the notch type dependency. 5.2. Comparison of experimental and FE simulation results of NWSFB, BS and WM specimens The load versus displacement curves of NWSFB, BS and WM specimens obtained from FE analysis as well as the tests are shown in Figs. 12–14, respectively, where E represents test results and A represents predicted results using the proposed three-stage and two-parameter ductile fracture model. The ductile crack initiation point, ultimate load point, final failure point and toughness parameters of NWSFB, BS, WM-U and WM-V specimens are listed in Tables 2–5, respectively, in which final failure point of analysis is regarded as the point which has the equivalent load value compared to experimental results. Because the ductile crack of NWSFB specimens initiates from center of specimens, we could not observed their ductile crack initiation points during tests. It is then proved that the prediction results are in a good agreement with experimental results in terms of ductile crack initiation point, ultimate load point, final failure point and load–displacement curve predictions. The tensile zone of final failure surface of WM-U2 specimen obtained from experiment and analysis is compared in Fig. 17. It is observed that the similar

P

triangular tensile zone could be simulated during FE analysis although the limitation of mesh size leads to non-smooth edge of analytical tensile zone. Fig. 18 illustrates the P–D curve of smooth bar, U-notch and Vnotch specimens obtained from tests, in which important points, such as ductile crack initiation point, ultimate load point and fracture load point, are indicated as P1, P2 and P3. The similar fracture mechanism is also observed from FE analysis. The ductile crack initiation point of NWSFB (P1, as shown in Fig. 18(a)) occurs after ultimate load point (P2), and the delay between them is very large. For U-notch specimens such as BS-U1 and WM-U, the ductile crack initiation point of such specimens (P1, as shown in Fig. 18(b)) appears after but close to ultimate load point (P2). However, for the V-notch specimens such as BS-V1 and WM-V, the ductile crack initiation point of them (P1, as shown in Fig. 18(c)) appears before ultimate load point (P2), and in addition, fracture predictions show that after crack initiation the load capacity of welded joints does not immediately start dropping but increases slowly to maximum value (ultimate load point, P2), then slightly decreases to fracture load (P3), and finally drops rapidly until failure, which is consistent with the test results. It is concluded that the notch of specimens leads to earlier occurrence of ductile crack initiation, and the sharper the notch is, the earlier the ductile crack initiates. Besides, based on experimental and analytical results, the relationship of absolute value of Ks/Ke can be obtained: |Ks/Ke|V-notch < | Ks/Ke|U-notch < |Ks/Ke|smooth bar. This phenomenon can be explained as following. Fig. 19 illustrates the equivalent plastic strain contours of NWFB1, BS-U1 and BS-V1 just before ductile crack initiation. We can find out that equivalent plastic strain distribution of smooth flat bar specimen is relatively uniform, however, equivalent plastic strain of V-notch specimen unevenly distributes. Uniform distribution of equivalent plastic strain leads to rapid

P

P P2 Ke

P1 and P3

Ke

P2

P1

Smooth bar

P3

V-notch

U-notch D

P2

Ks

Ks

Ks

(a) Smooth flat bar specimen

Ke

P1 and P3

D

D

(b) U-notch specimen

(c) V-notch specimen

P1: ductile crack initiation point; P2: ultimate load point; P3: fracture load point. Fig. 18. P–D curves (load–displacement curves) of smooth flat bar, U-notch and V-notch specimens.

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L. Kang et al. / Engineering Structures 85 (2015) 36–51

In brief, referring to smooth specimen, ductile crack initiation occurs later but crack propagation is very rapid, and in V-notch specimen, ductile crack initiation appears earlier but crack propagation is slow. For U-notch specimen, occurrence point of ductile crack initiation and propagation speed of ductile crack are in the intermediate state between smooth flat bar and V-notch specimen.

NWSFB1

Maximum equivalent plasc strain: 1.035 Notch direcon BS-U1

5.3. Effect of notch position in HAZ fracture tests Generally, the grains on the HAZ adjacent to the weld fusion line are more coarsening compared with those of other areas, and therefore they have lower toughness values [42,43]. However, toughness of the HAZ might be dependent of welding condition and chemical components of material. Therefore, the mechanical properties should be measured directly through performing a series of tensile test. It is noted that the width of HAZ is so narrow (about 3 mm) that one cannot easily make specimen for tensile test. For this reason, total ten specimens were fabricated, in which four specimens were tested. First of all, the toughness parameter of HAZ is 1.641, which is obviously lower than that of base metal and weld metal. For these four HAZ specimens, the ductile crack initiated from the surface of notch root tip region, and the ductile crack of HAZ-V1 and HAZ-U2 specimens initiated in the HAZ region, that of HAZ-V2 and HAZ-U1 initiated in the base metal region and weld region, respectively. It is concluded that geometry discontinuous-

Maximum equivalent plasc strain: 0.957

BS-V1

Maximum equivalent plasc strain: 0.561 Fig. 19. Equivalent plastic strain contours of NWSFB1, BS-U1 and BS-V1 just before ductile crack initiation point.

propagation of ductile crack, and on the contrary uneven distribution of equivalent plastic strain results in slow propagation of ductile crack. 250

A Load (kN)

200

B

150

Ultimate load point

100

C

A B C

HAZ-U2-E HAZ-U2-A HAZ-U2-E HAZ-U2-A Crack initiation point

50

HAZ-U2-E HAZ-U2-A

HAZ-U2

0 0

2

4

6

8

10

12

14

16

Displacement (mm)

Test

Analysis

A: Crack iniaon

B: Crack propagaon Fig. 20. Ductile crack initiation and propagation.

C: Further propagaon

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L. Kang et al. / Engineering Structures 85 (2015) 36–51

ness of specimens plays more important role in ductile fracture of steel welded specimens. In this study, HAZ material has rather lower toughness value, when the notch tip position just on HAZ region, the ductile fracture behavior of such specimen is worse. Fig. 20 shows ductile fracture failure process of HAZ-U2 obtained from test and analysis. Crack initiates at the middle of surface of notch root region, growing along the thickness direction and the width direction from the surface to the center quickly. The predicted fracture process shown in Fig. 20 also well agrees with the observed experimental behavior.

5.4. Comparison of experimental and FE simulation results of WSFB specimens In order to validate the reliability of three-stage and twoparameter ductile fracture model, the tests of three WSFB specimens performed were used for analysis. The predicted load–displacement curve, ductile crack initiation point, ultimate load point and final failure point as well as test results are compared in Fig. 16 and Table 7. It can be seen that the ductile fracture predictions of the proposed ductile fracture model match closely with

Width direcon Notch direcon

Thickness direcon

250

250

200

200

Load (kN)

Load (kN)

(a) Three directions of meshing

150

100

100 Experimental result Mesh size: 0.483mm Mesh size: 0.261mm Mesh size: 0.147mm Mesh size: 0.103mm Mesh size: 0.064mm

50

Experimental result Mesh size: 0.483mm Mesh size: 0.261mm Mesh size: 0.147mm Mesh size: 0.103mm Mesh size: 0.064mm

50

0

0 0

2

4

6

8

10

12

0

14

2

4

6

8

10

12

Displacement (mm)

Displacement (mm)

(b) Notch direction

(c) Notch direction without damage evolution

(mesh size of width direction=0.333mm)

(mesh size of width direction=0.333mm)

250

250

200

200

Load (kN)

Load (kN)

150

150

14

150

100

100 Experimental result Mesh size: 0.5mm Mesh size: 0.333mm Mesh size: 0.2mm Mesh size: 0.1mm

50

Experimental result Mesh size: 0.5mm Mesh size: 0.333mm Mesh size: 0.2mm Mesh size: 0.1mm

50

0

0 0

2

4

6

8

10

Displacement (mm)

12

14

0

2

4

6

8

10

12

14

Displacement (mm)

(d) Width direction

(e) Width direction without damage evolution

(mesh size of notch direction=0.1mm)

(mesh size of notch direction=0.1mm)

Fig. 21. Load–displacement curves of FE simulations for HAZ-U2 specimen with different mesh sizes.

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L. Kang et al. / Engineering Structures 85 (2015) 36–51

the experimental results. The average error of ultimate load and final failure points in terms of displacement is 4.63% and 6.02%, respectively, and that in terms of load is 0.99% and 1.36%, respectively. These show excellent agreement (average error < 7%) between the experimental and analytical quantities. In summary, an examination of Fig. 16 and Table 7 indicates that the ductile fracture model offers a promising approach to predicting fracture in stress/strain situations that may be found in common structural details. Referring to the above observation, the predictions for the smooth flat bar and U-notch tests (where the geometry discontinuousness is not an important factor) are significantly more accurate as compared to those for the V-notch tests. Extrapolating this trend, one may imagine that the ductile fracture model is likely less accurate for sharp cracked specimens where geometry discontinuousness is a major factor. 5.5. Effect of mesh size To ensure the opportunity to initiate ductile fracture during numerical simulation, it is necessary to apply a proper mesh size. It is especially important in the region with high stress–strain gradients, for instance near the crack tip [44]. The HAZ-U2 and HAZV1 specimens are used for evaluation purpose to estimate mesh size dependency. Various simulations have been carried out using different mesh sizes (as illustrated in Fig. 21) near the notch region to investigate the mesh dependency nature of the local evaluations of ELCF performance for the welded specimens tested. Three directions including thickness, width and notch directions are indicated in Fig. 21(a). Because of mesh size non-dependency in thickness direction, the thickness is divided into 8 elements (about 1.3 mm). The ductile fracture behavior may be sensitive to mesh size in notch and width directions, so HAZ-U2 and HAZ-V1 are meshed with different mesh sizes. Referring to HAZ-U2 in Fig. 21(b) and (c), the mesh size of width direction is constant of 0.333 mm, and the mesh size of notch direction is 0.064, 0.103, 0.147, 0.261, and 0.483 mm, respectively. In Fig. 21(d) and (e), the mesh size of notch direction is constant of 0.1 mm, and the mesh size of width direction is 0.1, 0.2, 0.333, and 0.5 mm, respectively. In which the cases of Fig. 21(b) and (d) consider damage evolution, and the cases of Fig. 21(c) and (e) do not incorporate damage evolution. It is observed that without considering damage evolution, the prediction results are not sensitive to mesh size to a certain degree. However, when considering damage evolution, the prediction results of mesh size from 0.064 mm

250

Load (kN)

200

150

100 Experimental result Mesh size: 0.303mm Mesh size: 0.282mm Mesh size: 0.235mm Mesh size: 0.193mm Mesh size: 0.0923mm

50

HAZ-V1

0 0

2

4

6

8

10

12

14

16

Displacement (mm)

(mesh size of width direction = 0.333mm) Fig. 22. Load–displacement curves of FE simulations for HAZ-V1 specimen with different mesh sizes in notch direction.

to 0.261 mm in notch direction are consistent, and the prediction result of 0.333 mm in width direction is in good agreement with that of 0.5 mm. The mesh size dependency is not significant for smooth flat bar and U-notch specimens discussed herein because the stress and strain gradients are flat as compared to the V-notch specimens. In this study, for NWSFB, WSFB and U-notch specimens, 0.1 mm  0.333 mm  1.3 mm (notch direction  width direction  thickness direction) mesh density is employed because of computational efficiency and analytical accuracy. Fig. 22 illustrates the results of mesh sensitivity analysis of HAZ-V1 with evolution damage. The mesh size of 0.333 mm in width direction is constant, and the mesh size in notch direction varies from 0.0923 mm to 0.303 mm. It is noted from Fig. 22 that the prediction results are quite sensitive to mesh size. In this study, mesh size of 0.2–0.3 mm in notch direction is employed. Some effective methods can be employed to overcome the shortcoming (mesh size dependence), such as non-local model [45] and weakly coupled model [22]. When a geometrical discontinuity exists, such as sharp corners and edges, the mesh size sensitivity at the discontinuity can be observed, and the more obvious the geometrical discontinuity is, the stronger mesh size sensitivity is.

6. Conclusions The application of the three-stage and two-parameter ductile fracture model to predict ductile fracture behavior in welded joints is examined through a series of tests and analyses of 5 smooth flat bar specimens, 8 U-notch specimens and 6 V-notch specimens. The ductile fracture model simulates the micro-mechanisms of void growth, collapse, and degradation to predict ductile crack initiation, propagation and final failure at the continuum level. The model is based on two parameters, which are calibrated using monotonic tests of smooth and notched flat bar specimens. All the specimens qualitatively exhibit similar behavior, stable load–displacement response up to the point of ductile crack initiation during tensile loading. The key difference between the three types of specimens is related to the gradient of stresses and strains in the critical region. The V-notch specimens have relatively steeper gradients, which lead to the earlier appearance of ductile crack initiation and slower crack propagation, whereas the smooth flat bar specimens exhibit flatter gradients, such that ductile crack initiates later and crack propagates faster. By means of finite element method, the three-stage and twoparameter ductile fracture model is applied to simulate the welded joints under monotonic loading. Fracture mechanism and determination process of two parameters in proposed ductile fracture model are carefully investigated in this study. The average value of the material constant is 1.965 for base metal, 2.178 for weld metal and 1.641 for HAZ with a coefficient of variation (COV) of 0.0405, 0.0859 and 0.00376, respectively. The average error between the experimental observations and analytical predictions, as measured by ductile crack initiation point and ultimate load point, is within 7% for welded smooth flat bar specimens. It is concluded that the methodology adopting the three-stage and two-parameter ductile fracture model can well predict the ductile fracture behavior of welded steel joint subjected to monotonic loading, including ductile crack initiation, ultimate load and final failure points. Besides, HAZ material has lower toughness parameter, its ductile fracture performance is worst, and this conclusion is proved by final failure surface study for all tested specimens. The specimens with notch in HAZ (such as HAZ-V1 and HAZ-U2) only have tensile zone, and other specimens have tensile and shear zones in the final failure surface. It is proved from test and simulation that

L. Kang et al. / Engineering Structures 85 (2015) 36–51

notch in the HAZ region may result in reducing the ductile fracture performance of welded specimens. Furthermore, mesh size sensitivity analyses demonstrate that the prediction results of smooth flat bar and U-notch specimens using proposed ductile fracture model in this study are not obviously sensitive to mesh size, on the contrary, those of V-notch specimens are very sensitive to mesh size because of steep stress and strain gradients occurring in sharp notch root tip region. This study demonstrates the capability and feasibility of the proposed ductile fracture model to predict ductile fracture under monotonic loading. While the model simulates small welded tests with reasonable success, the use of this model for real-scale structural components will require accurate simulation of other phenomena, such as local buckling and extremely low cycle fatigue. On the other hand, the prediction of proposed ductile fracture model driven fracture in sharp crack situations that may exhibit sharper strain gradients, increasing the sensitivity to the mesh size, in this study whose determination is somewhat subjective. For these situations, it may be necessary to either characterize and simulate these detailed phenomena, or alternatively use an appropriate degree of caution in interpreting simulation results. Ongoing work by the writers examines these issues. Acknowledgements The study is supported in part by grants from the JSPS Grants in-Aid for Scientific Research (C) (No. 24560588), and the Advanced Research Center for Natural Disaster Risk Reduction, Meijo University, which supported by Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan. As the former JSPS foreign researcher, the first author is thankful for financial support from the JSPS postdoctoral fellowship program for foreign researchers (Grant No. 12067). References [1] Bruneau M, Wilson JC, Tremblay R. Performance of steel bridges during the 1995 Hyogo-ken Nanbu (Kobe, Japan) earthquake. Can J Civil Eng 1996;23(3):678–713. [2] Nakashima M, Inoue K, Tada M. Classification of damage to steel buildings observed in the 1995 Hyogoken-Nanbu earthquake. Eng Struct 1998;20(4):271–81. [3] Mahin SA. Lessons from damage to steel buildings during the Northridge earthquake. Eng Struct 1998;20(4):261–70. [4] Kuwamura H. Fracture of steel during an earthquake-state-of-the-art in Japan. Eng Struct 1998;20(4–6):310–22. [5] Kuwamura H. Classification of material and welding in fracture consideration of seismic steel frames. Eng Struct 2003;25(5):547–63. [6] Usami T, Ge HB. A performance-based seismic design methodology for steel bridge systems. J Earth Tsunami 2009;3(3):175–93. [7] Kanvinde AM, Fell BV, Gomez IR, Roberts M. Predicting fracture in structural fillet welds using traditional and micromechanical fracture models. Eng Struct 2008;30(11):3325–35. [8] Ge HB, Kang L. Ductile crack initiation and propagation in steel bridge piers subjected to random cyclic loading. Eng Struct 2014;59:809–20. [9] Kanvinde AM, Deierlein GG. Void growth model and stress modified critical strain model to predict ductile fracture in structural steels. J Struct Eng – ASCE 2006;132(12):1907–18. [10] McClintock FA. A criterion for ductile fracture by the growth of holes. J Appl Mech 1968;35:363. [11] McClintock FA. Local criteria for ductile fracture. Int J Fract 1968;4(2):101–30. [12] Rice J, Tracey DM. On the ductile enlargement of voids in triaxial stress fields. J Mech Phys Solids 1969;17(3):201–17. [13] Chi WM, Kanvinde AM, Deierlein GG. Prediction of ductile fracture in steel connections using SMCS criterion. J Struct Eng – ASCE 2006;132(2):171–81. [14] Kanvinde AM, Deierlein GG. Cyclic void growth model to assess ductile fracture initiation in structural steels due to ultra low cycle fatigue. J Eng Mech – ASCE 2007;133(6):701–12.

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