Micromechanical assessment of mismatch effects on fracture of high-strength low alloyed steel welded joints

Engineering Fracture Mechanics 109 (2013) 221–235

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Micromechanical assessment of mismatch effects on fracture of high-strength low alloyed steel welded joints M. Rakin a, B. Medjo a,⇑, N. Gubeljak b, A. Sedmak c a

University of Belgrade, Faculty of Technology and Metallurgy, Serbia University of Maribor, Faculty of Mechanical Engineering, Slovenia c University of Belgrade, Faculty of Mechanical Engineering, Serbia b

a r t i c l e

i n f o

Article history: Received 9 October 2012 Received in revised form 18 April 2013 Accepted 28 June 2013 Available online 5 July 2013 Keywords: Welded joints Micromechanics Strength mismatch Constraint effects

a b s t r a c t Fracture of steel welded joints produced with one or two weld metals is analysed; the latter (double mismatched) consist of overmatched and undermatched portion. Specimens with a pre-crack in the symmetry plane of each joint are examined. In double mismatched joints, the pre-crack front passes through both weld metals. Micromechanical approach (complete Gurson model) is used for ductile damage development modelling. Influence of joint width, crack length, as well as size and formulation of finite elements, is examined. The constraint effect caused by the different joint geometry is predicted and the effect of material inhomogeneity along the crack front of the double mismatched joint is assessed by transferring the parameters from the joints with one weld metal. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction Understanding the fracture process in welded structures is very important for integrity assessment, having in mind that welded joints often contain some initial defects. For structural steels, overmatched (OM) welded joints are often fabricated. In the case of high-strength low alloyed (HSLA) steels, undermatched (UM) joints are sometimes used [1,2], which leads to stress and strain concentration in the weld metal (WM). The constraint effects play an important role even in macroscopically homogeneous geometries. In welded joints, these effects are more pronounced, since the influence of material heterogeneity (mismatch) exists in addition to the geometry constraints. Mechanical heterogeneity is crucial for predicting the failure of such structures under different loading types and operating conditions [2–4]. The significance of the constraint and mismatch effects in fracture analysis can be illustrated, besides by a large number of journal papers, by many publications which consider exclusively these topics, e.g. [3,5–7]. One of the ways to take into account both constraints caused by material mismatching and those caused by geometry is J–Q–M formulation [8,9], derived by extending the J–Q theory [10] of the two-parameter fracture mechanics. In this formulation (utilizing three parameters), J is related with the load, Q with the geometrical constraint and M with the material constraint. Betegon and Penuelas [11] defined a procedure similar to the J–T one [12], also by establishing an additional parameter that quantifies the material mismatching. Also, one of the methodologies used for fracture assessment of welded joints is cohesive zone modelling, applied in [13]. Strength mismatch was analysed in terms of the two parameters of the cohesive zone – cohesive stress and critical fracture energy. Local approach, i.e. micromechanical modelling, was applied in analysis of steel welded joints in [14–22]. Burstow and Howard [14] applied the Rousselier ductile fracture model [23] to simulate crack growth in steel welded joints with different ⇑ Corresponding author. Tel.: +381 11 3303 653; fax: +381 11 3370 387. E-mail address: [email protected] (B. Medjo). 0013-7944/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.engfracmech.2013.06.010

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Nomenclature a0 Da A CTODi CTODf E f f⁄ f0 fc fv f_ growth f_ nucleation F K n q1, q2 r Rp0.2 Rm Sij

vLL W

initial crack length (mm) crack length increment (mm) strain controlled nucleation rate (–) crack tip opening displacement at crack initiation (mm) crack tip opening displacement at final fracture (mm) Young’s modulus (GPa) porosity or void volume fraction (–) damage function (–) initial porosity (–) critical porosity (–) volume fraction of non-metallic inclusions (–) porosity growth rate (s1) porosity nucleation rate (s1) force (kN) accelerating factor in the GTN model (–) hardening exponent (–) constitutive parameters in the GTN model and CGM (–) void space ratio (–) yield strength (MPa) ultimate tensile strength (MPa) stress deviator (MPa) load line displacement specimen width (mm)

Greek symbols a, b constants (in the CGM) introduced by Thomason (–) e_ pij component of the plastic strain rate tensor (s1)

epeq equivalent plastic strain (–) e_ peq equivalent plastic strain rate (s1) e1, e2, e3 principal strains (–) / k

r1 rm r

yield function of the Gurson–Tvergaard–Needleman model (–) mean free path between non-metallic inclusions (lm) maximum principal stress (MPa) mean stress (MPa) current flow stress of the material matrix (MPa)

levels of mismatching. The authors considered both overmatched and undermatched joints, reporting good results for the former. Li et al. [15] assessed the fracture of welded joints using the Gurson–Tvergaard–Needleman (GTN) model [24–26]. Unlike most of the other studies, they examined specimens without an initial crack in order to predict the fracture initiation. It was found that damage of undermatched joints mainly occurred in the weld metal and the damage resistance decreases with decrease of the joint width. The damage of overmatched joints mainly occurred in the base metal and the damage resistance increased with decrease of the width. GTN model was also applied in [16], to predict ductile fracture resistance of laser-hybrid steel weldments; a special attention was given to the deviation of the crack path obtained for the initial cracks located in the vicinity of the fusion line. Penuelas et al. [17] analysed ductile fracture of steel welded joints with different crack lengths and widths of the weld metal using the complete Gurson model (CGM) [27]. The level of material mismatching was variable; yield strength of the weld metal was kept constant while the yield strength of the base metal was varied. Betegon et al. [18] extended the description of the material failure by analysing both ductile and cleavage fracture of welded joints at different temperatures; Beremin model was used for cleavage, while the CGM was applied for ductile fracture. The resistance to cleavage was shown to depend on both temperature and joint configuration, while the resistance to ductile fracture turned out to be almost temperature independent. In a recent work, Chhibber et al. [19] presented a detailed micromechanical analysis of welded joints using the GTN model; a parametric study was performed in order to assess the effects of different parameters on material fracture resistance. In this work, the joints with two different weld metals are examined in addition to the overmatched and undermatched ones. These joints are denoted as double mismatched (DM), and are composed of overmatched and undermatched portion

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[28–30]. DM joints are often used for repair welding, but their application also includes welding of HSLA steels, because it enables the joint fabrication without preheating [31]. The fatigue pre-cracks are located in the middle of the weld metal for all examined cases. In DM joints, the pre-crack is in ‘through-thickness’ position, i.e. the crack front runs through both weld metals. This fact enabled modelling of the joints as bimaterials (tri-materials in the case of DM joints); it was shown in several studies [2,17,19,20,32] that if the crack is located in the middle of the weld metal, the joint can be analysed without taking the heat affected zone (HAZ) into account. However, in some cases it is crucial to model the fracture behaviour of HAZ [21,22,31,33], if the crack initiates in HAZ (e.g. around some initial defect) or the fracture path of a crack initiated in WM or base metal (BM) runs through HAZ. Micromechanical analysis is chosen as appropriate for fracture assessment in this work, since it correlates the local stresses and strains with resistance to crack initiation and growth. The aim is to decrease or eliminate the dependence of fracture parameters on structure geometry and material heterogeneity. The Gurson yield criterion is applied, through the complete Gurson model. This model includes the void coalescence criterion proposed by Thomason [34], and critical damage parameter value (used as the failure criterion) is not a material constant, but depends on the stress/strain state, constraint level, etc. The crack growth simulation is performed by tracking the deterioration of elements in front of the crack tip. 2. Micromechanical modelling The GTN model takes into account the influence of voids on the material deformation and yielding, by extending von Mises plasticity theory. The porosity f (void volume fraction – damage parameter) is introduced into the expression for plastic potential:

/¼

  3Sij Sij 3q2 rm 2   ½1 þ ðq1 f  Þ  ¼ 0 þ 2q f cos h 1 2r2 2r

ð1Þ

where r is the current flow stress of the material matrix, rm is the mean stress and Sij is the stress deviator. Constitutive parameters q1 and q2 were introduced by Tvergaard in [25], in order to improve the ductile fracture prediction of the original Gurson model. In Eq. (1), f is the damage function, or modified void volume fraction [26]:

f ¼



f

for f 6 fc

ð2Þ

fc þ Kðf  fc Þ for f > fc

where fc is the critical value of f, at the onset of void coalescence. The parameter K (sometimes referred to as the ‘‘accelerating factor’’) defines the final stage of ductile fracture – void coalescence, which leads to complete loss of load carrying capacity of the material. During the ductile fracture process, the voids nucleate due to the fracture of particles or their separation from the material matrix. The increase of porosity value occurs in two ways: growth of the existing voids and nucleation of the new ones under the external loading:

f_ ¼ f_ nucleation þ f_ growth

ð3Þ

f_ nucleation ¼ Ae_ peq

ð4Þ

f_ growth ¼ ð1  f Þe_ pii

ð5Þ _ peq

_ pii

where A is the void nucleation rate, e is equivalent plastic strain rate and e is plastic part of the strain rate tensor. The GTN model has undergone many modifications, some of which include combinations with other methods (e.g. combination with cohesive zone modelling presented in [35]). The modification used in this work is proposed by Zhang et al. [27], who applied the Thomason’s void coalescence criterion [34] to the GTN model, obtaining the complete Gurson model or CGM. This model introduces a criterion for the onset of void coalescence:

 





r1 1 b > a  1 þ pffiffiffi ð1  pr 2 Þ; r r r

ð6Þ

where r1 is the maximum principal stress, a and b are constants fitted by Thomason [34] (a = 0.1 and b = 1.2). Instead of a constant value, Zhang et al. [27] proposed a linear dependence of a on hardening exponent n, which is applied in the CGM. The void space ratio r from Eq. (6) is given by [27]:

r¼

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi,pffiffiffiffiffiffiffiffiffiffiffiffi 3 3f e e2 þ e3 ; ee1 þe2 þe3 4p 2

ð7Þ

e1, e2 and e3 being principal strains. Therefore, the critical void volume fraction fc can be calculated during the finite element (FE) analysis, i.e. it is not a material constant in the CGM. This critical value depends on the strain field (see Eq. (7)); it does not necessarily have the same value in all finite elements and even in all integration points within one element. This fact is

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especially important in joints, where the material heterogeneity causes severe gradients in the stress and strain fields around the crack tip before and after the crack growth initiation [9,11,36–38]. If the GTN model is used, it is necessary to determine this value (e.g. by transferring its value from a tensile specimen or from unit cell analyses) and use it as material parameter for fracture assessment.

3. Materials The base metal examined in this study is high-strength low-alloyed steel NIOMOL 490. The chemical compositions of the BM and fillers used for obtaining overmatched (OM) and undermatched (UM) joints are given in Table 1. Basic mechanical properties of the materials, determined on round tensile (RT) specimens, as well as true stress – true strain curves are given in Fig. 1. Microstructural observation of the materials revealed the presence of sulphides, oxides, silicates and complex inclusions. The highest fraction of sulphides and silicates is found in UM, while BM and OM contain a significant fraction of oxides. Microstructural parameters for all materials (see Table 2) are determined by quantitative microstructural analysis. Measurement fields were taken on surfaces of several prepared samples. Volume fraction of non-metallic inclusions fv is obtained as the mean value of their volume fractions for all fields. In order to determine the mean free path between non-metallic inclusions k, five measuring lines are drawn in each measurement field. The number of interceptions of inclusions per line unit is determined, and k is obtained as the mean edge-to-edge distance between inclusions. The average mean free path between non-metallic inclusions is determined based on the values of k in all fields. In the initial stage of ductile fracture of steel, the voids nucleate mainly around non-metallic inclusions. Hence, the initial porosity f0 is here assumed to be equal to the volume fraction of non-metallic inclusions fv, Table 2. This is equivalent to the assumption that all voids are initiated at a low loading level. Such an approach, i.e. setting the value of f0 as equal to the volume fraction of non-metallic inclusions in steel, was applied previously in [16,17,39–41]. Volume fraction of larger void-nucleating particles was also used as the initial porosity in fracture analysis of other materials, e.g. nodular cast iron [36,42] and aluminium alloys [43–45]. Microstructural observations on low-alloyed steel with 0.22 mass% of carbon given in [46] have shown that the effect of the secondary voids formed around Fe3C particles is very low during the process of ductile fracture. Since the percentage of carbon in both weld metals is even lower (Table 1) in comparison with the steel examined in [46], it is expected that they also contain less Fe3C. Therefore, nucleation of the secondary voids is neglected in this work.

4. Experimental testing As the first part of the analysis, single-edge notched bend (SENB) specimens are used for estimation of fracture behaviour of OM and UM welded joints with three different widths (2H = 6, 12 and 18 mm, 2H being the WM width in the root area). Crack tip opening displacement (CTOD) values are directly measured using a d5 clip gauge, developed by GKSS [47], as a very convenient procedure for both experimental and numerical application. Details about the welding process and preparation of the plates are given in [48]. Local approach to ductile fracture (the GTN model) was also used in [20,49] for assessment of ductile fracture initiation in these joints. The measures of the specimens are 25  25  130 mm, while the distance between the supports is 100 mm. Initial (fatigue) crack lengths on the specimens are mostly around 8 mm, which means that the initial crack length to specimen width ratio is a0/W  0.32. Detailed data for all specimens are given in Table 3. The crack length in some specimens is larger than 8 mm; the influence of this dimension on the fracture behaviour is assessed on OM joint (2H = 6 mm). In addition to the mismatched welded joints, fracture of double mismatched joints is analysed, also using SENB specimens, Fig. 2. The main idea of the presented work is to use the micromechanical model for prediction of the difference in behaviour of the two weld metals in the conditions of ductile fracture. The crack in the DM welded specimen is located in the symmetry plane of the joint, in the through-thickness direction (through both weld metals), Fig. 2. Dimensions of the specimens are 25  25.2  125 mm, distance between the supports is 100 mm, and the initial crack length is a0 = 7 mm. Unlike the configuration in Fig. 2, the crack can be located in one of the weld metals in DM joint. The influence of the interface between the two weld metals on ductile fracture initiation was previously examined by the authors in [29], while plastic collapse of such joints was studied by Kozak et al. [30].

Table 1 Chemical composition of the base metal and consumables (mass%). Material

C

Si

Mn

P

S

Cr

Mo

Ni

Filler OM Base metal Filler UM

0.04 0.123 0.096

0.16 0.33 0.58

0.95 0.56 1.24

0.011 0.003 0.013

0.021 0.002 0.16

0.49 0.57 0.07

0.42 0.34 0.02

2.06 0.13 0.03

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Fig. 1. True stress – true strain curves and properties of the base metal and weld metals.

Table 2 Microstructural parameters. Material

fv

k (lm)

BM UM OM

0.0122 0.0071 0.0063

103 126 157

Table 3 Pre-crack lengths for the tested SENB specimens.

a0 a0/W

OM (2H = 6 mm)

OM (2H = 6 mm)

OM (2H = 12 mm)

OM (2H = 18 mm)

UM (2H = 6 mm)

UM (2H = 12 mm)

UM (2H = 18 mm)

7.694 0.307

10.334 0.413

7.826 0.313

9.414 0.376

7.544 0.302

7.87 0.32

7.735 0.31

Fig. 2. Schematic view of the examined joints with pre-cracks.

5. Numerical analysis Finite element software package Abaqus (www.simulia.com) is used for numerical analysis, and the CGM is applied through user material subroutine (created by Zhang, based on [27]). Specimens with OM or UM joints are analysed under plane strain conditions, using 4-noded full integration elements. The finite element (FE) mesh is given in Fig. 3, with the

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Fig. 3. FE mesh of the specimen and detail around the crack tip.

magnification near the crack tip. The crack tip is modelled using refined FE mesh without singular elements. External loading is defined by prescribing vertical displacement of the rigid body, loading pin, which is in contact with the model; all other motions of the pin are restrained. Contact is also used for defining the boundary conditions for support. The supports and loading pin were modelled as rigid bodies (analytical surfaces). Surface-to-surface contact with a finite-sliding formulation is defined between the contact surfaces in all cases. The heat affected zone is not taken into account in the numerical analysis, since the crack is located in the weld metal, along the axis of symmetry of each joint (see Section 1).

Fig. 4. One half of double mismatched SENB specimen, FE mesh and locations for tracking the porosity change during the increase of loading.

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As explained previously, the initial porosity f0 is set as equal to the volume fraction of non-metallic inclusions fv. FE calculations are carried out with the values of Tvergaard constitutive parameters q1 = 1.5 and q2 = 1. One half of the SENB DM specimen is given in Fig. 4; as shown previously, the initial crack runs through both weld metals in the butt welded joint. The advantages of the d5 concept for determination of CTOD are especially pronounced for this geometry, because the difference between two specimen sides can be measured. Measurement location on the UM side of the specimen is shown in Fig. 4, while the location in OM weld metal is on the opposite side of the specimen. Three-dimensional FE model is created, since this geometry cannot be represented in 2D due to the changes in material properties and geometry along the crack front. Linear 8-noded elements with full integration are used. Locations for damage parameter tracking along the crack front are shown in Fig. 4.

6. Results and discussion 6.1. Overmatched joints The void volume fraction change during the loading increase is strongly affected by the geometry of the welded joint, as shown in Fig. 5 for OM joints and the three analysed widths. This diagram represents the growth of f at the integration point nearest to the crack tip (in the element in front of the crack tip). The detail of the diagram (Fig. 5b) shows the fc values for three joints obtained using the CGM. These values are different due to the differences in stress and strain state caused by the geometry (i.e. they do not represent a material property, as mentioned in Section 2). The FE meshes used for obtaining these results are created using elements with dimensions in the ligament 0.15  0.15 mm. The influence of the FE size and formulation on the crack growth initiation is analysed in Section 6.4. Equivalent plastic strain fields for all three OM joints (2H = 6, 12 and 18 mm) are shown in Fig. 6. It can be seen that high values of these variables are limited to the weld metal in OM (2H = 18 mm) joint; hence the material heterogeneity does not

Fig. 5. Dependence of porosity f on CTOD–OM joints (a) and detail of the same diagram (b).

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Fig. 6. Equivalent plastic strain fields – OM joints.

have a pronounced influence on this specimen. However, in OM (2H = 12 mm) joint, and even more in OM (2H = 6 mm) joint, plastic strains are also located in the base material. Therefore, the plastic deformation of the base metal with lower strength is more pronounced with the joint width decrease, which increases the fracture resistance. Stable crack growth is modelled by tracking the development of damage in the ligament ahead of the crack tip, with criterion for the loss of load-carrying capacity defined using the CGM. The influence of the joint width on the crack growth can be seen in the same figure, where the ligament meshes for the three analysed widths are given. Dark colours represent high values of porosity f, in elements that have already lost load-carrying capacity. The influence of geometry is similar as for the crack growth initiation (Fig. 5) – increase of the joint width reduces the fracture resistance. This dependence can be seen as increase of crack length, since the porosity fields given in Fig. 7 correspond to the same value of the load line displacement vLL = 1.6 mm. Experimentally obtained F–CTOD and CTOD–Da diagrams are compared with the results obtained using von Mises plastic yield criterion and the CGM, Fig. 8 (F stands for force, while Da is crack length increment). It can be seen that the load-carrying capacity of the material is overestimated if damage due to the voids is not taken into account (von Mises criterion). It is shown that the CGM can predict the weakening of the material caused by the existence of voids during the process of ductile fracture. Very good agreement with experimental data is achieved for 6 and 12 mm wide joints, while certain differences exist for the widest joint (18 mm), where the estimated values of force are larger, but with correct trend that reflects the loss of load-carrying capacity. The influence of an important dimension of the specimen, initial crack length a0, is analysed for the joint width 6 mm (a0 = 8 mm and a0 = 10.4 mm), Fig. 8a. The difference in behaviour of these two specimens can be assessed using the CGM, i.e. damage modelling is in good agreement with actual material behaviour. From CTOD–Da diagrams (Fig. 8d) it can be seen that the increase of joint width causes lower crack growth resistance. Similar conclusion can be drawn from the measured values of CTODf (at final fracture), also given in Fig. 8d. It can be noticed that difference between the crack growth resistance curves for joints OM (2H = 6 mm) and OM (2H = 12 mm) is larger than the difference between OM (2H = 12 mm) and OM (2H = 18 mm). This can be attributed to the fact that the pre-crack in OM

Fig. 7. Crack growth – OM joints,

vLL = 1.6 mm.

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Fig. 8. F–CTOD curves for OM joints obtained experimentally, using von Mises criterion (without crack growth) and CGM (with crack growth) (a–c) and CTOD–Da curves (d) for the three joint widths.

(2H = 18 mm) joint is somewhat longer in comparison with the other two (9.4 mm/8 mm). Therefore, the influence of the joint width in this case is coupled with the influence of the initial crack length. 6.2. Undermatched joints When the damage development in UM joints is considered (Fig. 9), it can be noticed that the joints UM (2H = 18 mm) and UM (2H = 12 mm) exhibit very similar behaviour. In this case, there are no significant differences in initial crack lengths like those measured on OM specimens. However, these results can be related to the strain fields, Fig. 10. Plastic deformation in

Fig. 9. Dependence of porosity f on CTOD–UM joints.

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Fig. 10. Equivalent plastic strain fields – UM joints.

UM (2H = 12 mm) and UM (2H = 18 mm) specimens is restricted mainly to the weld metal, hence the material heterogeneity does not have a pronounced effect for the joints wider than 12 mm. On the other hand, large strain values in the 6 mm wide joint are also located in the base metal. Base metal prevents the deformation of lower-strength weld metal, enabling the faster damage growth in comparison with the two other specimens. The influence of the joint width on the crack growth can be seen in Fig. 11, where the ligament meshes for three analysed widths are given. The influence of geometry is opposite to that obtained for OM joints – increase of the joint width enhances the fracture resistance. This dependence can be seen as decrease of crack length, since the porosity fields given in Fig. 11 correspond to the same value vLL = 3.5 mm. Force–CTOD diagrams for UM joints are given in Fig. 12a–c. The FE size used for OM joints (0.15  0.15 mm) results in earlier loss of load-carrying capacity in comparison with experimental data. Similar can be said for the elements sized 0.3  0.3 mm, while the size 0.45  0.45 mm gives a more appropriate trend. Therefore, appropriate finite element size in the undermatched joints is three times larger in comparison with the overmatched ones, despite the fact that the mean free paths between the inclusions in these two weld metals are rather similar, Table 2. A more precise determination of the strain and stress field would be possible with a finer mesh around the crack tip; however, elements smaller than 0.45  0.45 mm result in too rapid damage development in the material. This mesh dependence is considered to be one of the main drawbacks of the local approach to ductile fracture in general. Diagrams F–CTOD for UM joints obtained using the micromechanical model exhibit a deviation from experimental data for CTOD values up to 0.3 mm. One of the possible explanations is that plane strain state in the entire specimen is not adequate for modelling in the case of significant plastic yielding of UM welded joints. A proposed approach for better stress field description is subdivision of the region into plane stress and plane strain sub-regions [50,51]; plane strain is than used in the

Fig. 11. Crack growth – UM joints,

vLL = 3.5 mm.

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Fig. 12. F–CTOD curves for UM joints obtained experimentally and using CGM (a–c) and CTOD–Da curves (d) for the three joint widths.

region around the crack tip and the ligament, while plane stress is used in the rest of the model. Possibilities for application of this approach will be elaborated in the future work. The crack growth curves for the element size 0.45  0.45 mm are given in Fig. 12d; the difference between UM (2H = 12 mm) and UM (2H = 18 mm) is very small, which is consistent with the similar trend of damage (porosity) development in these two specimens, Fig. 9. Small differences between these two joints are also obtained for measured values CTODf (at final fracture), given in the same diagram. 6.3. Double mismatched joints The FE size in the ligament ahead of the crack front in the DM joints is adopted based on the examination of OM and UM joints (0.15  0.15 mm for OM and 0.45  0.45 mm for UM portion), and the mesh consists of 8-noded finite elements with full integration, Fig. 4. During the experimental and numerical analysis of DM joints, CTOD is measured using the d5 concept on both sides of the specimen, due to the different material properties of the weld metals. Curves F-CTOD obtained experimentally and using the micromechanical model are shown in Fig. 13. There are two sets of curves (designated as OM and UM), because CTOD is determined at both sides of the specimen. The difference caused by the material inhomogeneity can be quantified using the CGM. The increase of the damage parameter (porosity f) with the increase of loading is given in Fig. 14; locations correspond to those in Fig. 4. The values of f increase more prominently in the middle of the specimen, as a consequence of high stress triaxiality. Regardless of the location, the porosity increases faster in the OM portion of the joint, in comparison with the UM. It is important to notice that such behaviour can be captured only if appropriate finite element sizes are transferred from the OM and UM joints. Otherwise, the damage development would be wrongly predicted to be faster in the UM portion in comparison with the OM one. Distribution of void volume fraction f on the symmetry plane of DM joint is shown in Fig. 15. Red1 colour represents the areas with high void volume fraction, with elements that have already lost their load-carrying capacity due to the void coalescence. The differences between the two materials can be predicted by the micromechanical criterion (CGM); the crack growth 1

For interpretation of color in Fig. 15, the reader is referred to the web version of this article.

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Fig. 13. F–CTOD curves for DM joints obtained experimentally and using CGM.

Fig. 14. Dependence of porosity f on CTOD – DM joints; middle of the specimen and surface of the UM and OM weld metal.

Fig. 15. Distribution of porosity f in the symmetry plane and photo of the fracture surface.

resistance is much lower in the OM part of the joint. This assessment is obtained by transferring the micromechanical parameters, initial void volume fraction and the finite element size, from the single mismatched (OM and UM) to the double mismatched (OM + UM weld metal) joints, as mentioned previously.

6.4. Influence of the FE mesh on the prediction of crack growth initiation The influence of the FE mesh on the prediction of crack growth initiation is analysed on the example of OM welded joints. The size of the elements near the crack tip is varied: 0.15  0.15 mm, 0.3  0.3 mm and 0.075  0.075 mm. The effect of FE

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Fig. 16. Influence of the FE size (a) and formulation (b) on CTODi values for OM joints.

size on the crack tip opening displacement at crack growth initiation (CTODi) is shown in Fig. 16a. Crack growth initiation is determined by reaching the critical porosity fc in the FE in front of the crack tip. In accordance with the analysis given in [52], the value of f is tracked in the integration point nearest to the crack tip. Calculation with FE size of 0.3  0.3 mm gives higher material resistance in comparison with experimental data, while the opposite is obtained with FE size of 0.075  0.075 mm. FE size 0.15  0.15 mm is the most suitable for this type of joints (OM). The influence of the geometry (joint width) on CTODi values can also be seen in Fig. 16a. The results obtained using various finite elements (Fig. 16b) imply that the integration order influences the predicted values of CTODi. The use of elements with dimensions 0.3  0.3 mm and 3  3 integration gives similar results as those with dimensions 0.15  0.15 mm and 2  2 integration. These two elements have similar distance between integration points, which means that distance between integration points (instead of FE size itself) is adequate for ductile fracture description. On the other hand, for the same number of integration points and distance between them, prediction of ductile fracture initiation is not much affected by the change of the interpolation order. The results obtained using 8-noded and 4-noded elements with 2  2 integration do not differ significantly, Fig. 16b. Of course, this number of Gauss (integration) points corresponds to reduced integration for the former and to full integration for the latter. 7. Conclusions Ductile fracture of welded joints made of high-strength low-alloyed steel has been analysed using local approach, i.e. micromechanical modelling. It is concluded that the complete Gurson model can quantify the constraint effect in examined joints with one or two weld metals:  Geometry (width of the joint) affects the obtained results. Narrow weld metal in overmatched welded joint has higher resistance to ductile fracture than the wide one, while opposite trend is obtained for undermatched joints.

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 It is shown that damage and deformation of the undermatched joints with width 12 and 18 mm is very similar; this means that the material heterogeneity does not play a significant role with further increase of the joint width over 12 mm. The appropriate FE size for UM turns out to be three times larger than for OM weld metal.  The influence of the initial crack length on ductile fracture, analysed on overmatched joints, is predicted correctly.  The influence of material inhomogeneity along the front (for the through-thickness position of the crack) is obtained; ductile fracture parameters are transferred from OM and UM joints to DM joint containing both OM and UM portion.  Besides the size of the finite elements near the crack tip, integration order influences ductile fracture initiation; hence the distance between integration points is more relevant than element size for ductile fracture modelling. On the other hand, interpolation order (quadratic or linear FEs) does not influence the results significantly if elements with same number of integration points are used.

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