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Numerical modelling of damage behaviour of laser-hybrid welds
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Engineering Fracture Mechanics 75 (2008) 3251–3263 www.elsevier.com/locate/engfracmech
Numerical modelling of damage behaviour of laser-hybrid welds A. Nonn *, W. Dahl, W. Bleck Department of Ferrous Metallurgy, Aachen University, Intzestr. 1, 52072 Aachen, Germany Received 9 October 2006; received in revised form 1 October 2007; accepted 20 October 2007 Available online 4 November 2007
Abstract The effect of laser-hybrid welds on deformation and failure behaviour of fracture mechanics specimens is investigated in order to provide quantitative prediction of damage tolerance and residual strength. The simulation of crack initiation and crack extension in hybrid welds is performed by applying GTN damage model. The identification of damage parameters requires combined numerical and experimental analyses. The tendency to crack path deviation during crack growth depends strongly on the constraint development at the interface between base and weld metal. In order to quantify the influence of local stress state on the crack path deviation, the initial crack location is varied. Finally, the results from fracture mechanics tests are compared to real component, beam-column-connection, with respect to fracture resistance. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: GTN model; Laser-hybrid welds; Damage; Crack path deviation
1. Introduction Laser-hybrid welding process has been developed in 1970s, but the growing interest in different fields of manufacturing industry has been observed recently due to the widespread application of laser welding methods. The laser-hybrid welding benefits from advantages of both conventional arc beam and laser welding processes. Some of the advantages resulting from the laser welding are narrow weld seam and heat affected zone (HAZ), low thermal input and thus residual stresses, high welding speed, minimal work-piece distortion, etc. The advantages associated with conventional arc welding include high process stability, great tolerance to join gaps, deep welding penetration, ability to include filler wire, etc. Porosity, solidification cracks, gaps resulting from the lack of fusion and insufficient penetration are considered to be possible defects in the hybrid-laser welds. Porosity originates from dissolved gases in material, gas entrapment or gases from contaminated surfaces. On the other hand, the formation of solidification crack depends on the nature of precipitation as well as on weld geometry. With growing application of laser-hybrid welding, the demand for characterisation of mechanical and fracture behaviour under consideration of these defects increases constantly. The accurate fracture prediction is *
Corresponding author. Tel.: +49 241 8095841; fax: +49 241 8092253. E-mail address: [email protected] (A. Nonn).
0013-7944/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2007.10.015
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required for the utilisation of possible safety reserves thus providing economical and safe design of components. The objective of this paper is hence to investigate the mechanism responsible for crack initiation and crack propagation in order to predict crack resistance of hybrid welded components under quasi-static loading. For this purpose, the Gurson–Tvergaard–Needleman (GTN) damage model is applied in numerical analyses. This model, implemented in the finite element program ABAQUS, establishes the connection between stress state and the microstructural damage development and requires the identification of the adequate parameters for the analysed material. Since these parameters show no dependency on the specimen geometry and load type, they are applicable in the specimens and components with different geometrical configurations for the calculation of the ductile crack growth. However, combined experimental and numerical studies have to be performed for the identification of the damage parameter set. The results from the investigations on the laser weld with respect to fracture behaviour have been demonstrated in [1–3]. The quantification of the constraint effect before the crack initiation occurs is presented in [1]. Thus, the recommendations have been derived with respect to possible constraint corrections. The damage analyses performed in [2,3] on laser welded Al sheets by applying GTN model have shown the influence of the crack location on the stress state and fracture behaviour during stable crack growth. In the first part of the paper, the identification of damage parameters for a structural steel is presented. The fracture mechanics tests on the compact and bending specimens are used for the verification and calibration of the parameter set. The second part is concerned with the influence of the initial crack location on the crack growth. Depending on the crack configuration, the crack path deviation (CPD) is observed, which is characterised by the path change of the growing initial crack originally located in the fusion zone (FZ) towards base metal (BM). Regarding component safety, the CPD can be evaluated positively due to the higher component toughness resulting from higher energy consumption necessary for the crack propagation in the base metal. On the other hand, there is a risk of sudden load drop occurrence (pop-in’s) or even brittle fracture, since the crack could meet the brittle zones in the heterogeneous HAZ before it can reach the BM. This risk is strongly reduced once the crack reaches BM and continues to propagate there. The aim of numerical investigations is to quantitatively determine conditions under which the CPD occurs. At the end of the paper, the crack resistance of real components (beam-column-connection) is determined numerically and compared to the crack resistance from fracture mechanics testing. 2. The GTN damage model Based on the von Mises criterion extended with material damage parameters, the GTN damage model describes the formation, development and the coalescence of the voids during the stable crack growth by following equation [4]: r2eq 3q2 rm Uðrm ; req ; f Þ ¼ 2 þ 2q1 f cosh ð1Þ ð1 þ q3 f 2 Þ ¼ 0; r 2r where req is the von Mises equivalent stress, rm the hydrostatic stress and r the flow stress of the ‘‘voidless” matrix material, which is function of the accumulated plastic strain. The coefficients q1, q2, and q3 are phenomenologically based parameters, introduced by Tvergaard [5] to consider the interaction between adjacent voids. The original equation developed by Gurson [6] contains either coefficients q1, q2, and q3 nor the modification of void volume fraction. Tvergaard and Needleman [4] substituted the void volume fraction f in the original equation by modified void volume fraction f* in order to take into account the loss of the load bearing capacity due to the void coalescence. The function f*(f) (see Fig. 1) is given by 8 f 6 fc > : fu f P ff
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f* f u*
κ fc 1 fc
ff
f
Fig. 1. The modified void volume fraction f*(f).
The modified void volume fraction f* corresponds to the void volume fraction f, unless the critical porosity fc is exceeded. Subsequently, the accelerated void growth, quantified by the factor j, occurs until f and f* reach their final values ff and fu , respectively. For the unloaded case f* is equal to the initial void volume fraction f0, which can be derived from the microstructural analysis. The parameter at the onset of the void coalescence is the critical void volume fraction fc. One of the possibilities to determine the parameter fc is given by the unit cell model calculations, which can be performed using FE-analysis. The GTN model requires also the specification of the mesh size, viz. element height, which is normal to the direction of crack propagation. Once this parameter is chosen, it should be held constant regardless of the model dimensions. The element height corresponds to the dissipated deformation energy per crack extension and affects the level of the crack resistance curve. This energy depends on the metallographically determined ‘‘microstructural” length lc, defined as the average distance between void forming inclusions, and on the hardening behaviour of the material matrix. Therefore, the relation between the element height and lc is not directly proportional but related to dissipated energy [7]. The recommendation for the applied element height values results from the numerical simulations with GTN model for different materials. Accordingly, the element height should be estimated as a multiple of lc (factor 6–12) in order to obtain the correct Ji values [8]. Attention should be paid to the meshing of half of the specimen due to the symmetry condition. In this case, the actual damaged zone consists of two element rows and the element height is then 3– 6 times lc. The damage value fu is reached at the occurrence of the macroscopic fracture characterised by the loss of stress bearing capacity and it can be calculated as follows: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q1 þ q21 q3 fu ¼ : ð3Þ q3 With the chosen parameters q1 = 1.5, q2 = 1.0 and q3 ¼ q21 based on the recommendation by Tvergaard and Needleman [9,10], the corresponding damage value fu equals 1/q1. For the application of the GTN model, all parameters must be known, both for the BM and for FZ. The micromechanical processes in the microstructure can be described quantitatively by implementing the GTN model into the numerical analysis. The rate of the void volume fraction consists of terms for growth of existing voids and nucleation of new voids, proposed by Needleman et al. [11,12] f_ ¼ f_ growth þ f_ nucleation :
ð4Þ
The growth term is derived from the assumption of incompressible matrix material f_ growth ¼ ð1 f Þ e_ pl kk ;
with plastic volume dilatation rate e_ pl kk :
ð5Þ
The strain-dependent nucleation term, suggested by Tvergaard [13], describes the formation of secondary voids between primary voids during the damage development f_ nucleation ¼ Ae_ pl ;
ð6Þ
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where e_ pl is equivalent plastic strain rate of the matrix material and parameter A the intensity of the void nucleation: " 2 # fN 1 epl eN pffiffiffiffiffiffi exp A¼ : ð7Þ 2 SN S N 2p According to this equation, the new voids with volume fraction fN are nucleated at a mean equivalent plastic strain eN with the standard deviation SN following the Gaussian distribution. The numerical calculations are carried out using the finite element code ABAQUS standard [14] with GTN model as a user material subroutine (UMAT) [15] and large deformation theory. 3. Mechanical properties and microstructure A structural steel S355 with its widespread application in the field of building constructions is chosen for investigations. Hybrid welded butt joints are produced by using CO2 lasers for 12 mm (S355) thick steel plates. The first information about the weld heterogeneity along the cross section can be obtained from the hardness mapping, resulting from ultrasonic contact impedance (UCI) procedure. Fig. 2 shows macro views of a weld with three material zones for analysed steel with corresponding hardness measurements. These three zones are base metal (BM), HAZ and nail shaped fusion zone (FZ). The hardness values for BM range between 160 and 220HV. The fusion zone possesses hardness values between 200–260HV with increasing values towards ends of weld cap and weld root. The analysed weld is overmatched configurations with mismatch factor MF defined BM as a ratio RFZ el =Rel , which is equal to 1.68 for this steel. Tables 1 and 2 contain the results obtained from chemical analysis and mechanical properties of the analysed steel including yield strength Rel, mismatch factor MF, tensile strength Rm, uniform strain Ag, strain at rupture A. The true stress–strain curves, used as input for numerical calculations, are obtained from tensile tests on 8 40 and 3 15 round bar specimens for BM and FZ, respectively. The analysis of microstructure provides an indication of fracture toughness properties. Due to the low carbon content, the microstructure of BM is ferritic with a few intermediate pearlitic grains. The microstructure of weld metal consists of ferrite phases, such as Widmansta¨tten and acicular ferrite with good toughness properties. By using the MAG (Metal Active Gas) filler wire, which delivers inclusions for intragranular nucleation, the ratio of acicular ferrite increases at
Fig. 2. Macro view with hardness measurement for steel grades S355.
Table 1 Chemical composition of the investigated steels, mass contents in % Steel
C
Si
Mn
P
S
Cr
Mo
Ni
Al
Co
S355
0.054
0.018
1.085
0.011
<0.001
0.015
<0.005
0.019
0.021
<0.005
Table 2 Mechanical properties of the different zones of the welded specimen Steel
RBM (MPa) el
RFZ el (MPa)
MF
RBM m (MPa)
RFZ m (MPa)
ABM g
AFZ g
ABM
AFZ
S355
349
586
1.68
439
690
0.180
0.051
0.349
0.158
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the cap of the weld [16]. At the root, however, where weld looks more like laser weld, the ratio of acicular ferrite decreases and microstructure is characterised for the most part by Widmansta¨tten ferrite. 4. Parameter identification-metallographic analyses The identification of the damage parameters for the application of GTN model requires additional metallographic analyses on the polished specimens with analysed area of 100 mm2. The initial porosity f0 depends on the volume fraction of non-metallic inclusions, like sulphides and oxides, both for BM and FZ. In addition, shape, size and distribution of inclusions are quantified for BM, HAZ and FZ. The maximal, minimal and average distances (kmax, kmin and kmid) between neighbouring particles together with their size (maximal, minimal and average diameter, dmax, dmin and dmid), void aspect ratio (vmax and vmin), number of particles (Np) and volume fraction (Vp) are evaluated at five randomly selected independent locations at a magnification of 200. The metallographic data given in Table 3 are the average values resulting for these five locations. Void aspect ratio is defined as ratio between maximal and minimal diameter of inclusions, which returns a value of 1 for a perfectly spherical inclusions and values below 1 for varying degrees of circularity. Except for minimal neighbour distance kmin, the results for BM and FZ of steel S355 differ little from each other. The shape factor with maximal value of 1.0 indicates that most particles are oxides with almost spherical form. The initial porosity f0 for S355 is set equal to volume fraction of large particles both for BM and FZ. A further crucial parameter for damage calculation is ‘‘length-scale” parameter, which is considered by element size in FE modelling. This parameter is chosen on the basis of combined FE-analyses and microstructural measurements of the average distance between adjacent inclusions. 5. Parameter identification-tests on the notched round bar specimens Additional numerical and experimental analyses are required to determine further parameters, such as, critical porosity fcand element size. One way to determine the critical porosity fc is to perform different numerical calculations for the notched tensile specimens by varying parameter fc until the best agreement between numerical and experimental results is achieved with respect to the load drop occurrence [17]. However, this calibration of fc is only possible with already known element size. The second way involves the calculations of cylindrical unit cell with spherical cavity for different stress triaxialities [18]. The triaxiality is quantified by the parameter h defined as a ratio between hydrostatic and von Mises equivalent stresses. In these unit cell calculations, the parameter fc corresponds to the void volume fraction at which the cell collapse occurs, characterised by rapid stress reduction. For investigated steel S355, the critical porosity fc is obtained by using unit cell model calculations, which are performed in dependence of stress triaxiality. The resulting critical void volume fraction fc differs slightly, when evaluated for three h values (h = 1, h = 2 and h = 3). This result coincides with the conclusion obtained from cell model calculation in [19] that no strong dependence of the parameter fc is evident when the initial porosity is small enough and no secondary voids are taken into account. On the other hand, non-metallic inclusions of S355 possess high void aspect ratio, hence void shape effects on the critical porosity can be neglected. The fc values determined from these calculations are 0.027 and 0.030 for BM and FZ, respectively. With the fixed fc values, the element size can be estimated numerically by using the results from the tests on the notched round bar specimens. Fig. 3 shows the force vs. diameter reduction curves for S355, which are provided by the tests on the 8 40 notched round bar specimens of homogeneous BM and with laser-hybrid weld (HW).
Table 3 Results from metallographic investigations of the inclusion distributions at a magnification of 200 Steel
Zone
kmax (lm)
kmin (lm)
kmid (lm)
dmax (lm)
dmin (lm)
dmax (lm)
vmid
Np
Vp
S355-12
BM FZ
104 134
34 19
69 66
3.04 3.33
2.33 2.62
2.77 3.05
0.93 0.94
89 87
0.0006 0.0006
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14
Load F [kN]
12 10
8x40, DN=4.08 mm, ρ=1.0 mm
8 6 4
BM-Exp. BM-GTN HW-Exp. HW-GTN
2 0 0.0
0.2
0.4
0.6
0.8
1.0
1.2
Diameter reduction ΔD [mm] Fig. 3. Load vs. diameter reduction for S355.
The damage analyses are carried out on the 2D axisymmetric models of notched round bar specimens. The models are simplified by neglecting the effects of the weld geometry and weld material heterogeneity. The weld is presented as a rectangular strip of 3.2 mm width containing only FZ with homogeneous properties. The notch is located in the centre of the specimen as well as in the centre of the hybrid weld. Since the symmetry conditions to the ligament and length axis are considered, only fourth of model is meshed. The same notch geometry defined by notch radius q = 1.0 mm and net diameter DN = 4.0 mm is used for BM and HW. The first calculations with identified parameters f0 and fc (‘‘BM-GTN” and ‘‘HW-GTN” in Fig. 3) correspond well to the experimental results, if the size of the element in the centre of the symmetric specimen is set to 0.3 and 0.125 mm for BM and FZ respectively. It is important to point out, that the element size can only be estimated according to the results from the tensile tests. The final calibration of this parameter is achieved by fracture mechanics test results. Although the average distance between the particles is almost the same (see Table 3) the difference in the element size is based on the different hardening behaviour of BM and HW. The load drop observed in simulation corresponds to the loss of carrying capacity of the central element, when it reaches critical porosity. The relative diameter reduction DD/D0 in [%] at the fracture is with 13.4% for BM higher than for HW with 9.75%. The parameter sets resulting from the analyses on the notched round bar specimens are given in Table 4. The element size (lx ly lz) is expressed by the element length lx in ligament direction, height ly and thickness lz. 6. Fracture mechanics investigations Fracture mechanics tests are performed on 0.4CT specimens with an objective to verify determined parameter sets and, if required, to adjust element size. Furthermore, the influence of laser-hybrid welds on the local stress and strain fields during stable crack extension is quantified. The fracture surfaces with their characteristic honeycomb structure for ductile failure are presented in Fig. 4. Base metal of S355 (see Fig. 4a) shows large primary voids connected with distinctly smaller secondary dimples. It is assumed that large voids originate from the coarse particles, like oxides and sulphides. Compared to BM, the fracture surface of FZ (see Fig. 4b) contains higher fraction of fine secondary dimples especially at the cap. The fraction of large voids increases with decreasing width of FZ in direction of root.
Table 4 GTN parameter set used for the damage calculations Steel
Zone
f0
fn
en
sn
fc (h = 2)
j
q1
q2
lx ly lz (mm)
S355-12
BM FZ
0.0006 0.0006
0.000 0.000
0.0 0.0
0.0 0.0
0.027 0.030
6 6
1.5 1.5
1.0 1.0
0.30 0.60 0.60 0.25 0.25 0.60
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Fig. 4. Fracture surfaces for S355 BM (a) and FZ (b).
The results from the fracture mechanics tests with respect to load vs. displacement and fracture resistance curves are given in Fig. 5. Very good agreement is achieved between all experimental and numerical results. In contrast to specimens of homogenous BM, the uneven crack extension is evident across the thickness on the fracture surfaces of hybrid welded specimens with decreasing crack growth towards weld root. With numerous small inclusions, which act as nucleating sites for secondary microvoids, the crack extension at the FZ cap is distinctly faster compared to the root. Other asymmetric crack extension is observed, when the initial crack is located in the FZ near the fusion line. During crack growth, the stress triaxiality and plastic strains increase in the softer BM at the fusion line. Besides the increasing plastic strains, it is the high triaxiality level at the material interface, which triggers the second crack initiation in the BM. With increasing plastic zones between these two cracks, the first crack leaves the original path and starts to propagate in direction of BM. Due to the high mismatch and narrow weld seams, this phenomenon is specially pronounced for laser and electron beam welds [20]. The mechanisms behind crack path deviation (CPD) are examined in the following paragraph. The crack resistance curves for S355 are given in Fig. 5b both for BM and HW. Up to Da = 0.4 mm the level and the slope of the curves are almost identical for specimens with BM and HW. There is also no significant difference regarding level of crack initiation values with Ji = 202 N/mm for BM and Ji = 190.9 N/ mm for HW. However, BM shows better fracture behaviour with increasing crack growth compared to HW. In order to verify the transferability of GTN parameters and to investigate the geometry and the loading effect on the crack growth, the SENB specimens with the HW are also tested for S355, see Fig. 6. For FE-analysis the initial crack is located in the centre of the weld seam. The crack initiation and the level of crack resistance curve for BM and HW are very similar to those obtained for the CT samples. However, the crack initiation values are overestimated by numerical analysis due to the lower triaxiality compared to the CT
10
1400
9
1200
J-Integral [N/mm]
Load F [kN]
8 7 6 5 4 3
BM-Exp. BM-GTN HW-Exp. HW-GTN
2 1 2.0
4.0
1000 800 600 400 200
0 0.0
BM-Exp. BM-GTN HW-Exp. HW-GTN
6.0
Opening displacement VLL [mm]
0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
Crack growth Δa [mm]
Fig. 5. Load vs. opening displacement (a) and J-Integral vs. crack growth Da (b).
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1400 SENB 20x10
1200
4
J-Integral [N/mm]
Load F [kN]
5 SENB 10x10
3 2 1 0 0.0
BM-Exp. BM-GTN HW-Exp. HW-GTN
1000
BM-Exp. BM-GTN HW-Exp. HW-GTN
SENB 20x10
800 600 400 200
SENB 10x10
0 2.0
4.0
6.0
Deflection [mm]
8.0
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
Crack growth Δa [mm]
Fig. 6. Comparison between experiment and simulation with respect to load vs. deformation (a) and fracture resistance curves (b) for SENB specimens with hybrid weld and of homogenous base metal, S355.
specimens. In contrast, the experimentally determined initiation shows no strong dependency on the constraint effect.
7. Crack path deviation (CPD) The influence of the initial crack location at the crack growth is studied for the steel S355. Therefore, 3D models of 0.4CT specimens are generated and meshed with 8-noded brick elements (C3D8). The initial crack length is a0 = 10 mm, thus a0/W = 0.5. If the initial crack is located 0.5 mm away from the fusion line in the CT specimen, the crack path deviation (CPD) is observed. On the other hand, the crack growth is straight with initial crack in the centre of the weld. The minimal distance between initial crack and fusion line required for CPD occurrence is about 0.9 mm. Due to the different stress states, the crack extension varies along the specimen thickness, see Fig. 7. With the plane strain conditions in the mid-thickness plane, the high constraint at fusion line contributes to the initiation and propagation of second crack. The increasing plastic deformation between two cracks leads to the deviation of the first crack into the softer BM. At the surface no crack path deviation (CPD) occurs, since the triaxiality is low because of the plane stress conditions. The uneven fracture surface from numerical results coincides with the experimental findings. In order to understand the mechanisms, which control the crack growth and CPD, the 2D 0.4CT specimens are modelled with the initial crack located 1.6 and 0.5 mm away from fusion line. The local stress and strain state are quantified by the stress triaxiality h and equivalent plastic strain epl v for both configurations. The results are evaluated and compared for Da = 2 mm along the ligament in the FZ (path 1) and along the fusion line (path 2) in the BM (path 2-BM) and FZ (path 2-FZ). In the case of the straight crack growth for
Fig. 7. The generated 3D FE model of the 0.4CT specimen used for the numerical analysis (a) with crack propagation at the surface (b) and in the mid-thickness plane (c).
A. Nonn et al. / Engineering Fracture Mechanics 75 (2008) 3251–3263 3.5
3.5 y
Stress triaxiality h
3.0
FL-BM
Initial crack
2.0
Path 1-FZ Path 2-BM Path 2-FZ
1.5
Fusion line y
3.0
Path 1
2.5
Path 2
y=1.6 mm
FZ
FL-FZ
1.0
Stress triaxiality h
Path 2 Fusion line
0.5
Initial crack
2.5
1.0
2.0
3.0
1.5
Path 1
FL-FZ
1.0
FZ y=0.5 mm
0.0 0.0
4.0
Distance from initial crack tip [mm]
FL-BM
Path 1-FZ Path 2-BM Path 2-FZ
2.0
0.5
0.0 0.0
3259
1.0
2.0
3.0
4.0
Distance from initial crack tip [mm]
Fig. 8. Distribution of stress triaxiality h for the cases when crack grows straight (a) and deviates (b) for Da = 2 mm.
0.80
0.80
y
0.70 0.60
y=0.5 mm
Fusion line pl
Path 2
y=1.6 mm
Equivalent plastic strain εv
Equivalent plastic strain εv
pl
0.90
Path 1 Initial crack
Path 1-FZ Path 2-BM Path 2-FZ
0.50 0.40
FZ
0.30 0.20 0.10
FL-BM FL-FZ
0.00 0.0
1.0
2.0
3.0
4.0
Distance from initial crack tip [mm]
Path 2
Fusion line
0.70
y FL-FZ
0.60
Initial crack
Path 1
0.50 Path 1-FZ Path 2-BM Path 2-FZ
0.40 0.30
FL-BM
0.20
FZ
0.10 0.00 0.0
1.0
2.0
3.0
4.0
Distance from initial crack tip [mm]
Fig. 9. Distribution of equivalent plastic strain epl v for the cases when crack grows straight (a) and deviates (b), Da = 2 mm.
y = 1.6 mm (see Fig. 8a and 9a), the maximal h value in the BM is achieved 3 mm away from initial and 1 mm away from the current crack tip. It is almost 50% higher than the maximal h value in the FZ located directly behind the current crack tip. In contrast, the maximal plastic strain level achieved between initial and current crack tip is ca. 30% higher in the FZ than in the BM. On the other hand, the decrease of epl v values is stronger in FZ, thus leading to lower plastic strains ahead of the current tip. Neither h nor epl v values in the BM are sufficient to trigger the initiation of the second crack. The results for second configuration with y = 0.5 mm and CPD occurrence are depicted in Fig. 8b and 9b. Due to increasing stress triaxiality at the fusion line with increasing loading, the second crack initiates in BM after 1 mm extension of the first crack in the fusion zone. In 3D model, however, crack initiates already after 0.5 mm of first crack extension in mid-thickness plane. On the one hand, the stress triaxility at the interface between BM and FZ in 3D model is slightly reduced compared to the 2D model with maximal out-of-plane constraint. On the other hand, the plastic strain development is more pronounced in 3D than in the 2D model leading to the second crack appearance at lower global loading for 3D model. The local maxima of h and epl v values responsible for CPD are reached at the fusion line on the FZ side (path 2-FZ) ca. 1 mm away from the initial crack. These maximal values are almost three times higher than in the FZ (path 1). With epl v values of almost 0.7, it can be concluded that the CPD is controlled by the preceding high plastic strains at the fusion line. For the better understanding of mechanism behind the CPD, it is important to present the damage development which finally leads to CPD, besides the evaluation of the local stress and strains at the moment of
A. Nonn et al. / Engineering Fracture Mechanics 75 (2008) 3251–3263 0.030
0.030 0.025 EL3
0.020 EL4
0.015 0.010 0.005
EL3-FZ-1mm EL4-FZ-2mm
0.000 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 pl
Equivalent plastic strains εv
Modified void volume fraction f*
Modified void volume fraction f*
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EL1
0.025 EL 2
0.020 0.015 0.010 0.005
EL1-FL-BM
EL2-FL-FZ 0.000 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 pl
Equivalent plastic strains εv
Fig. 10. Damage development against equivalent plastic strain epl v for the cases when crack grows straight (a) and deviates (b).
CPD occurrence. Fig. 10a shows the damage development (modified void volume fraction against the equivalent plastic strain) for straight crack growth in the element containing current crack tip after 1mm (EL3-FZ1 mm) and 2 mm (EL4-FZ-2 mm) crack growth. It is obvious that the damage development is intensified with increasing crack growth. In case of CPD, the similar damage distribution in dependence of plastic strains can be found in the element (EL1-FL-BM) on the fusion line at the BM side, see Fig. 10b. Low triaxiality level and consequent strain localisation in the fusion zone element at the fusion line (EL2-FL-FZ) cause the shift of the damage development curve towards higher plastic strains. Thus, when the critical porosity is reached, the strains in this element (EL2-FL-BM) are 2.5 times higher than in the base metal element, see Fig. 10b. 8. Damage modelling of steel frame connections Besides ship building industry, a further application for the use of laser-hybrid welds is within the field of steel constructions. As regards beam-column-connections subjected to seismic loading, one advantage of the hybrid welding compared to conventional welding is the lower residual stresses and thus lower probability for cleavage fracture to occur. The 3D non-linear FE-analysis of the beam-column-connection (BCC) is conducted by using GTN damage model for S355, see Fig. 11. The element size given in Table 4 is used to mesh the weld and base metal area
Fig. 11. FE model of beam-column-connection (BCC) with semi-elliptical crack located in the centre of the flange.
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0.25
longitudinal strain load
0.20
buckling
800
1400
700
1200
500
crack initiation
0.15
gross section (beam) yielding
0.10
400 300
Load F [kN]
600
200 0.05
J-Integral [N/mm]
average longitudinal strain [-]
around semi-elliptical crack. As expected, the pre-calculation with elastic–plastic material properties shows the highest stress concentrations in area of geometric and material discontinuities, which can be found in the welded connection. Due to the possible metallographic defects during welding, e.g. solidification defects, a pre-existing crack is assumed and additionally included in the FE model as a surface semi-elliptical crack. The resulting curves showing load and strains vs. vertical deflection are presented in Fig. 12a. The gross section yielding in the beam is observed for small strains of ca. 2%. The crack initiation and thus failure occurs shortly before the load maximum is reached at the strain level of 10%. The results also show that the crack initiates before the buckling of the beam flange under pressure sets in. The crack initiation level with Ji = 273 N/mm is significantly higher compared to CT specimen with different crack configurations, see Fig. 12b. With Da > 0.3 mm the resistance level converges to R-curve for hybrid welded CT specimens. The crack initiation is observed under angle / = 60°, at the location of the highest stress triaxiality, see Fig. 12 Thereafter the crack extension continues in direction of the surface / = 0° and towards / = 90°. The quantified local stress and strain fields are given in Fig. 13 along the initial crack path for J = 273 N/ mm, which corresponds to J value at the crack initiation in BCC. The stress triaxiality and local plastic strains are evaluated at / = 60° for BCC. For the chosen crack tip loading, the crack extension in the FZ is about 0.2 mm and no second crack or CPD are evident in the CT specimen with y = 0.5 mm. Hence, the level and course of triaxiality and plastic strains for y = 0.5 and 1.6 mm are similar along the initial crack path. Stress triaxiality ahead of the crack for y = 0 mm is intensified by high mismatch at the interface between base
1000
0.1
0.2
0.3
0.4
0.5
0° φ
800 60°
600 90°
400
0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
0 0
CT
200
100 0.00
BM HW y=0 HW y=0.5 HW y=1.6 BCC
0.6
Crack growth Δa [mm]
vertical deflection uv [m]
Fig. 12. Load and strains vs. vertical deflection curve (a), crack resistance curve (b) and crack propagation of semi-elliptical crack (c).
3.0
Equivalent plastic strain εv
pl
0.8
Stress triaxiality h
2.5 2.0 1.5 1.0 CT
0.5 0.0 0.0
0.5
1.0
BM HW y=0 HW y=0.5 HW y=1.6 BCC
1.5
2.0
Distance from initial crack tip [mm]
0.7
BM HW y=0 HW y=0.5 HW y=1.6 BCC
CT
0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.0
0.5
1.0
1.5
2.0
Distance from initial crack tip [mm]
Fig. 13. Distribution of stress triaxiality h (a) and equivalent plastic strain epl v (b), J = 273 N/mm.
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and weld metal. As expected the lowest triaxiality and the highest plastic strains are calculated in the BCC resulting from the crack size, shape and component dimension. The changes in the stress field resulting from the stress redistribution during the crack extension may cause a possible shift of failure mechanism from ductile to cleavage fracture, especially in components subjected to cyclic loading. This will be investigated in the future work. 9. Conclusions In this paper, the fracture behaviour of laser-hybrid weld is investigated by means of GTN damage model for steel S355. Except for element size ly, which is more than two times higher for BM compared to FZ, almost same parameters (e.g. initial and critical porosity) result both for BM and FZ. Although the higher element size ly should lead to the increase of fracture resistance, similar initiation level and R-curve for Da < 0.4 mm are obtained for CT specimens of homogenous BM and with HW. This is due to the lower yield strength and lower hardening level of BM. The initial porosity can be successfully derived from metallographical analyses of the non-metallic inclusions. Despite the microstructural differences between weld cap and weld root resulting from hybrid welding process the similar distribution and morphology of large inclusions are observed for both weld parts. However the smaller particles detected in the cap with dmid < 1 lm are not taken into account for modelling of ductile damage. It is assumed that the differences with respect to fracture toughness between acicular ferrite, which is mostly represented in the weld cap and Widmansta¨tten ferrite in the weld root, will play crucial role in the brittle and ductile-to-brittle transition temperature region. With critical porosity fc determined from unit cell calculations, a further damage parameter, viz. element size ly is calibrated and verified using test results from tests on the notched round bar and fracture mechanics specimens, respectively. Transferability of the identified parameters is demonstrated on the numerically obtained R-curves for SENB specimens. The same parameters are used for the local stress and strain analyses to explain the effect of experimentally observed crack path deviation. For this purpose, calculations are carried out for 2D and 3D models of CT specimen with initial crack located 0.5 mm away from the fusion line. In both models, the initiation of the second crack at the fusion line sets in before the first crack deviates from FZ into BM. With more progressed plastic zones development second crack appears at the lower load level in 3D than in 2D model. In contrast to mid-thickness plane of 3D model, no second crack initiation takes place at the specimen surface. This can be explained by the constraint loss when moving towards specimen surface. Similar local stress state exists at BM/FZ interface for 2D model with the initial crack located in the centre of the weld. Hence, high triaxiality level and certain amount of plastic strains are required for second crack initiation. Finally, CPD is a consequence of increasing plastic strains in the FZ between two cracks. Up to Da = 0.3 mm, higher level of fracture resistance is numerically obtained for BCC compared to the CT specimens with varying initial crack location. Crack resistance loss evident for Da > 0.3 mm is based on lower strains necessary to drive the crack because of the strain localisation in the rest ligament. References [1] Heyer J. Lokale Beanspruchung in angerissenen strahlgeschweißten Stahlbauteilen. RWTH Aachen PhD thesis, Verlag Shaker, Aachen; 2004. [2] Ne`gre P, Steglich D, Brocks W. Crack extension in aluminum welds: a numerical approach using the Gurson–Tvergaard–Neeldeman model. Engng Fract Mech 2004;71:2365–83. [3] Ne`gre P, Steglich D, Brocks W, Kocßak M. Numerical simulation of crack extension in aluminium welds. Comput Mater Sci 2003:723–31. [4] Tvergaard V, Needlemann A. Analysis of the cup-cone fracture in a round tensile bar. Acta Metall 1984;32(1):157–69. [5] Tvergaard V. On localisation in ductile materials containing spherical voids. Int J Fract 1982;18:237–52. [6] Gurson AL. Continuum theory of ductile rupture by void nucleation and growth: part I – yield criteria and flow rules for porous ductile media. J Engng Mater Tech 1977;99:2–15. [7] Siegmund T, Brocks W. Prediction of the work of separation and implications to modeling. Int J Fract 1999;99:97–116. [8] Steglich D, Brocks W. Micromechanical modelling of the behaviour of ductile materials including particles. Comput Mater Sci 1997;9:7–17. [9] Needleman A, Tvergaard V. An analysis of ductile rupture modes at a crack tip. J Mech Phys Solids 1987;35:151–83.
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[10] Tvergaard V, Needleman A. Effect of crack meandering on dynamic. J Mech Phys Solids 1992;40(2):447–71. [11] Needleman A, Rice JR. Limits to ductility set by plastic flow localisation. In: Koistinen DP, editor. Mechanics of sheet metal forming. New York: Plenum Press; 1978. p. 237–67. [12] Chu CC, Needleman A. Void nucleation effects in biaxially stretched sheets. J Engng Mater Tech 1980;102:249–55. [13] Tvergaard V. Ductile fracture by cavity nucleation between larger voids. J Mech Phys Solid 1982;30(4):265–86. [14] Hibbitt. Karlsson & Sorensen Inc. Abaqus Finite Element Program, Version 6.5, 2004. [15] Siegmund T, Brocks W. A user-material subroutine incorporating the Gurson–Tvergaard–Needleman model of porous metal plasticity into the ABAQUS finite element program, Technical report GKSS/WMG/97/2, Geesthacht: Institut fu¨r WerkstoffForschung, GKSS, 1997. [16] Philippa L Moore. Investigation into the microstructure and properties of laser and laser/arc hybrid welds in pipeline steels, University of Cambridge, PhD thesis; 2003. [17] Bernauer G, Brocks W. Micro-mechanical modelling of ductile damage and tearing-results of a European numerical round robin. Fatigue Fract Engng Mater Struct 2001;25:363–84. [18] Brocks W, Sun D, Ho¨nig A. Verification of micromechanical models for ductile fracture by cell model calculations. Comput Mater Sci 1996;7:235–41. [19] Brocks W, Sun D, Ho¨nig A. Verification of the transferability of micromechanical parameters by cell model calculations with viscoplastic materials. Int J Plast 1995;11(8):971–89. [20] Bajric A, Brocks W, Dahl W, Heyer J, Langenberg P. Investigations of failure behaviour of flawed steel specimens with electron beam welds (EBW). In: 11th International Congress on Fracture (ICF), Paper No. 4339, CD ROM, Turin (Italy); 2005.
Engineering Fracture Mechanics 75 (2008) 3251–3263 www.elsevier.com/locate/engfracmech
Numerical modelling of damage behaviour of laser-hybrid welds A. Nonn *, W. Dahl, W. Bleck Department of Ferrous Metallurgy, Aachen University, Intzestr. 1, 52072 Aachen, Germany Received 9 October 2006; received in revised form 1 October 2007; accepted 20 October 2007 Available online 4 November 2007
Abstract The effect of laser-hybrid welds on deformation and failure behaviour of fracture mechanics specimens is investigated in order to provide quantitative prediction of damage tolerance and residual strength. The simulation of crack initiation and crack extension in hybrid welds is performed by applying GTN damage model. The identification of damage parameters requires combined numerical and experimental analyses. The tendency to crack path deviation during crack growth depends strongly on the constraint development at the interface between base and weld metal. In order to quantify the influence of local stress state on the crack path deviation, the initial crack location is varied. Finally, the results from fracture mechanics tests are compared to real component, beam-column-connection, with respect to fracture resistance. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: GTN model; Laser-hybrid welds; Damage; Crack path deviation
1. Introduction Laser-hybrid welding process has been developed in 1970s, but the growing interest in different fields of manufacturing industry has been observed recently due to the widespread application of laser welding methods. The laser-hybrid welding benefits from advantages of both conventional arc beam and laser welding processes. Some of the advantages resulting from the laser welding are narrow weld seam and heat affected zone (HAZ), low thermal input and thus residual stresses, high welding speed, minimal work-piece distortion, etc. The advantages associated with conventional arc welding include high process stability, great tolerance to join gaps, deep welding penetration, ability to include filler wire, etc. Porosity, solidification cracks, gaps resulting from the lack of fusion and insufficient penetration are considered to be possible defects in the hybrid-laser welds. Porosity originates from dissolved gases in material, gas entrapment or gases from contaminated surfaces. On the other hand, the formation of solidification crack depends on the nature of precipitation as well as on weld geometry. With growing application of laser-hybrid welding, the demand for characterisation of mechanical and fracture behaviour under consideration of these defects increases constantly. The accurate fracture prediction is *
Corresponding author. Tel.: +49 241 8095841; fax: +49 241 8092253. E-mail address: [email protected] (A. Nonn).
0013-7944/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2007.10.015
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required for the utilisation of possible safety reserves thus providing economical and safe design of components. The objective of this paper is hence to investigate the mechanism responsible for crack initiation and crack propagation in order to predict crack resistance of hybrid welded components under quasi-static loading. For this purpose, the Gurson–Tvergaard–Needleman (GTN) damage model is applied in numerical analyses. This model, implemented in the finite element program ABAQUS, establishes the connection between stress state and the microstructural damage development and requires the identification of the adequate parameters for the analysed material. Since these parameters show no dependency on the specimen geometry and load type, they are applicable in the specimens and components with different geometrical configurations for the calculation of the ductile crack growth. However, combined experimental and numerical studies have to be performed for the identification of the damage parameter set. The results from the investigations on the laser weld with respect to fracture behaviour have been demonstrated in [1–3]. The quantification of the constraint effect before the crack initiation occurs is presented in [1]. Thus, the recommendations have been derived with respect to possible constraint corrections. The damage analyses performed in [2,3] on laser welded Al sheets by applying GTN model have shown the influence of the crack location on the stress state and fracture behaviour during stable crack growth. In the first part of the paper, the identification of damage parameters for a structural steel is presented. The fracture mechanics tests on the compact and bending specimens are used for the verification and calibration of the parameter set. The second part is concerned with the influence of the initial crack location on the crack growth. Depending on the crack configuration, the crack path deviation (CPD) is observed, which is characterised by the path change of the growing initial crack originally located in the fusion zone (FZ) towards base metal (BM). Regarding component safety, the CPD can be evaluated positively due to the higher component toughness resulting from higher energy consumption necessary for the crack propagation in the base metal. On the other hand, there is a risk of sudden load drop occurrence (pop-in’s) or even brittle fracture, since the crack could meet the brittle zones in the heterogeneous HAZ before it can reach the BM. This risk is strongly reduced once the crack reaches BM and continues to propagate there. The aim of numerical investigations is to quantitatively determine conditions under which the CPD occurs. At the end of the paper, the crack resistance of real components (beam-column-connection) is determined numerically and compared to the crack resistance from fracture mechanics testing. 2. The GTN damage model Based on the von Mises criterion extended with material damage parameters, the GTN damage model describes the formation, development and the coalescence of the voids during the stable crack growth by following equation [4]: r2eq 3q2 rm Uðrm ; req ; f Þ ¼ 2 þ 2q1 f cosh ð1Þ ð1 þ q3 f 2 Þ ¼ 0; r 2r where req is the von Mises equivalent stress, rm the hydrostatic stress and r the flow stress of the ‘‘voidless” matrix material, which is function of the accumulated plastic strain. The coefficients q1, q2, and q3 are phenomenologically based parameters, introduced by Tvergaard [5] to consider the interaction between adjacent voids. The original equation developed by Gurson [6] contains either coefficients q1, q2, and q3 nor the modification of void volume fraction. Tvergaard and Needleman [4] substituted the void volume fraction f in the original equation by modified void volume fraction f* in order to take into account the loss of the load bearing capacity due to the void coalescence. The function f*(f) (see Fig. 1) is given by 8 f 6 fc >
A. Nonn et al. / Engineering Fracture Mechanics 75 (2008) 3251–3263
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f* f u*
κ fc 1 fc
ff
f
Fig. 1. The modified void volume fraction f*(f).
The modified void volume fraction f* corresponds to the void volume fraction f, unless the critical porosity fc is exceeded. Subsequently, the accelerated void growth, quantified by the factor j, occurs until f and f* reach their final values ff and fu , respectively. For the unloaded case f* is equal to the initial void volume fraction f0, which can be derived from the microstructural analysis. The parameter at the onset of the void coalescence is the critical void volume fraction fc. One of the possibilities to determine the parameter fc is given by the unit cell model calculations, which can be performed using FE-analysis. The GTN model requires also the specification of the mesh size, viz. element height, which is normal to the direction of crack propagation. Once this parameter is chosen, it should be held constant regardless of the model dimensions. The element height corresponds to the dissipated deformation energy per crack extension and affects the level of the crack resistance curve. This energy depends on the metallographically determined ‘‘microstructural” length lc, defined as the average distance between void forming inclusions, and on the hardening behaviour of the material matrix. Therefore, the relation between the element height and lc is not directly proportional but related to dissipated energy [7]. The recommendation for the applied element height values results from the numerical simulations with GTN model for different materials. Accordingly, the element height should be estimated as a multiple of lc (factor 6–12) in order to obtain the correct Ji values [8]. Attention should be paid to the meshing of half of the specimen due to the symmetry condition. In this case, the actual damaged zone consists of two element rows and the element height is then 3– 6 times lc. The damage value fu is reached at the occurrence of the macroscopic fracture characterised by the loss of stress bearing capacity and it can be calculated as follows: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q1 þ q21 q3 fu ¼ : ð3Þ q3 With the chosen parameters q1 = 1.5, q2 = 1.0 and q3 ¼ q21 based on the recommendation by Tvergaard and Needleman [9,10], the corresponding damage value fu equals 1/q1. For the application of the GTN model, all parameters must be known, both for the BM and for FZ. The micromechanical processes in the microstructure can be described quantitatively by implementing the GTN model into the numerical analysis. The rate of the void volume fraction consists of terms for growth of existing voids and nucleation of new voids, proposed by Needleman et al. [11,12] f_ ¼ f_ growth þ f_ nucleation :
ð4Þ
The growth term is derived from the assumption of incompressible matrix material f_ growth ¼ ð1 f Þ e_ pl kk ;
with plastic volume dilatation rate e_ pl kk :
ð5Þ
The strain-dependent nucleation term, suggested by Tvergaard [13], describes the formation of secondary voids between primary voids during the damage development f_ nucleation ¼ Ae_ pl ;
ð6Þ
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where e_ pl is equivalent plastic strain rate of the matrix material and parameter A the intensity of the void nucleation: " 2 # fN 1 epl eN pffiffiffiffiffiffi exp A¼ : ð7Þ 2 SN S N 2p According to this equation, the new voids with volume fraction fN are nucleated at a mean equivalent plastic strain eN with the standard deviation SN following the Gaussian distribution. The numerical calculations are carried out using the finite element code ABAQUS standard [14] with GTN model as a user material subroutine (UMAT) [15] and large deformation theory. 3. Mechanical properties and microstructure A structural steel S355 with its widespread application in the field of building constructions is chosen for investigations. Hybrid welded butt joints are produced by using CO2 lasers for 12 mm (S355) thick steel plates. The first information about the weld heterogeneity along the cross section can be obtained from the hardness mapping, resulting from ultrasonic contact impedance (UCI) procedure. Fig. 2 shows macro views of a weld with three material zones for analysed steel with corresponding hardness measurements. These three zones are base metal (BM), HAZ and nail shaped fusion zone (FZ). The hardness values for BM range between 160 and 220HV. The fusion zone possesses hardness values between 200–260HV with increasing values towards ends of weld cap and weld root. The analysed weld is overmatched configurations with mismatch factor MF defined BM as a ratio RFZ el =Rel , which is equal to 1.68 for this steel. Tables 1 and 2 contain the results obtained from chemical analysis and mechanical properties of the analysed steel including yield strength Rel, mismatch factor MF, tensile strength Rm, uniform strain Ag, strain at rupture A. The true stress–strain curves, used as input for numerical calculations, are obtained from tensile tests on 8 40 and 3 15 round bar specimens for BM and FZ, respectively. The analysis of microstructure provides an indication of fracture toughness properties. Due to the low carbon content, the microstructure of BM is ferritic with a few intermediate pearlitic grains. The microstructure of weld metal consists of ferrite phases, such as Widmansta¨tten and acicular ferrite with good toughness properties. By using the MAG (Metal Active Gas) filler wire, which delivers inclusions for intragranular nucleation, the ratio of acicular ferrite increases at
Fig. 2. Macro view with hardness measurement for steel grades S355.
Table 1 Chemical composition of the investigated steels, mass contents in % Steel
C
Si
Mn
P
S
Cr
Mo
Ni
Al
Co
S355
0.054
0.018
1.085
0.011
<0.001
0.015
<0.005
0.019
0.021
<0.005
Table 2 Mechanical properties of the different zones of the welded specimen Steel
RBM (MPa) el
RFZ el (MPa)
MF
RBM m (MPa)
RFZ m (MPa)
ABM g
AFZ g
ABM
AFZ
S355
349
586
1.68
439
690
0.180
0.051
0.349
0.158
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the cap of the weld [16]. At the root, however, where weld looks more like laser weld, the ratio of acicular ferrite decreases and microstructure is characterised for the most part by Widmansta¨tten ferrite. 4. Parameter identification-metallographic analyses The identification of the damage parameters for the application of GTN model requires additional metallographic analyses on the polished specimens with analysed area of 100 mm2. The initial porosity f0 depends on the volume fraction of non-metallic inclusions, like sulphides and oxides, both for BM and FZ. In addition, shape, size and distribution of inclusions are quantified for BM, HAZ and FZ. The maximal, minimal and average distances (kmax, kmin and kmid) between neighbouring particles together with their size (maximal, minimal and average diameter, dmax, dmin and dmid), void aspect ratio (vmax and vmin), number of particles (Np) and volume fraction (Vp) are evaluated at five randomly selected independent locations at a magnification of 200. The metallographic data given in Table 3 are the average values resulting for these five locations. Void aspect ratio is defined as ratio between maximal and minimal diameter of inclusions, which returns a value of 1 for a perfectly spherical inclusions and values below 1 for varying degrees of circularity. Except for minimal neighbour distance kmin, the results for BM and FZ of steel S355 differ little from each other. The shape factor with maximal value of 1.0 indicates that most particles are oxides with almost spherical form. The initial porosity f0 for S355 is set equal to volume fraction of large particles both for BM and FZ. A further crucial parameter for damage calculation is ‘‘length-scale” parameter, which is considered by element size in FE modelling. This parameter is chosen on the basis of combined FE-analyses and microstructural measurements of the average distance between adjacent inclusions. 5. Parameter identification-tests on the notched round bar specimens Additional numerical and experimental analyses are required to determine further parameters, such as, critical porosity fcand element size. One way to determine the critical porosity fc is to perform different numerical calculations for the notched tensile specimens by varying parameter fc until the best agreement between numerical and experimental results is achieved with respect to the load drop occurrence [17]. However, this calibration of fc is only possible with already known element size. The second way involves the calculations of cylindrical unit cell with spherical cavity for different stress triaxialities [18]. The triaxiality is quantified by the parameter h defined as a ratio between hydrostatic and von Mises equivalent stresses. In these unit cell calculations, the parameter fc corresponds to the void volume fraction at which the cell collapse occurs, characterised by rapid stress reduction. For investigated steel S355, the critical porosity fc is obtained by using unit cell model calculations, which are performed in dependence of stress triaxiality. The resulting critical void volume fraction fc differs slightly, when evaluated for three h values (h = 1, h = 2 and h = 3). This result coincides with the conclusion obtained from cell model calculation in [19] that no strong dependence of the parameter fc is evident when the initial porosity is small enough and no secondary voids are taken into account. On the other hand, non-metallic inclusions of S355 possess high void aspect ratio, hence void shape effects on the critical porosity can be neglected. The fc values determined from these calculations are 0.027 and 0.030 for BM and FZ, respectively. With the fixed fc values, the element size can be estimated numerically by using the results from the tests on the notched round bar specimens. Fig. 3 shows the force vs. diameter reduction curves for S355, which are provided by the tests on the 8 40 notched round bar specimens of homogeneous BM and with laser-hybrid weld (HW).
Table 3 Results from metallographic investigations of the inclusion distributions at a magnification of 200 Steel
Zone
kmax (lm)
kmin (lm)
kmid (lm)
dmax (lm)
dmin (lm)
dmax (lm)
vmid
Np
Vp
S355-12
BM FZ
104 134
34 19
69 66
3.04 3.33
2.33 2.62
2.77 3.05
0.93 0.94
89 87
0.0006 0.0006
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14
Load F [kN]
12 10
8x40, DN=4.08 mm, ρ=1.0 mm
8 6 4
BM-Exp. BM-GTN HW-Exp. HW-GTN
2 0 0.0
0.2
0.4
0.6
0.8
1.0
1.2
Diameter reduction ΔD [mm] Fig. 3. Load vs. diameter reduction for S355.
The damage analyses are carried out on the 2D axisymmetric models of notched round bar specimens. The models are simplified by neglecting the effects of the weld geometry and weld material heterogeneity. The weld is presented as a rectangular strip of 3.2 mm width containing only FZ with homogeneous properties. The notch is located in the centre of the specimen as well as in the centre of the hybrid weld. Since the symmetry conditions to the ligament and length axis are considered, only fourth of model is meshed. The same notch geometry defined by notch radius q = 1.0 mm and net diameter DN = 4.0 mm is used for BM and HW. The first calculations with identified parameters f0 and fc (‘‘BM-GTN” and ‘‘HW-GTN” in Fig. 3) correspond well to the experimental results, if the size of the element in the centre of the symmetric specimen is set to 0.3 and 0.125 mm for BM and FZ respectively. It is important to point out, that the element size can only be estimated according to the results from the tensile tests. The final calibration of this parameter is achieved by fracture mechanics test results. Although the average distance between the particles is almost the same (see Table 3) the difference in the element size is based on the different hardening behaviour of BM and HW. The load drop observed in simulation corresponds to the loss of carrying capacity of the central element, when it reaches critical porosity. The relative diameter reduction DD/D0 in [%] at the fracture is with 13.4% for BM higher than for HW with 9.75%. The parameter sets resulting from the analyses on the notched round bar specimens are given in Table 4. The element size (lx ly lz) is expressed by the element length lx in ligament direction, height ly and thickness lz. 6. Fracture mechanics investigations Fracture mechanics tests are performed on 0.4CT specimens with an objective to verify determined parameter sets and, if required, to adjust element size. Furthermore, the influence of laser-hybrid welds on the local stress and strain fields during stable crack extension is quantified. The fracture surfaces with their characteristic honeycomb structure for ductile failure are presented in Fig. 4. Base metal of S355 (see Fig. 4a) shows large primary voids connected with distinctly smaller secondary dimples. It is assumed that large voids originate from the coarse particles, like oxides and sulphides. Compared to BM, the fracture surface of FZ (see Fig. 4b) contains higher fraction of fine secondary dimples especially at the cap. The fraction of large voids increases with decreasing width of FZ in direction of root.
Table 4 GTN parameter set used for the damage calculations Steel
Zone
f0
fn
en
sn
fc (h = 2)
j
q1
q2
lx ly lz (mm)
S355-12
BM FZ
0.0006 0.0006
0.000 0.000
0.0 0.0
0.0 0.0
0.027 0.030
6 6
1.5 1.5
1.0 1.0
0.30 0.60 0.60 0.25 0.25 0.60
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Fig. 4. Fracture surfaces for S355 BM (a) and FZ (b).
The results from the fracture mechanics tests with respect to load vs. displacement and fracture resistance curves are given in Fig. 5. Very good agreement is achieved between all experimental and numerical results. In contrast to specimens of homogenous BM, the uneven crack extension is evident across the thickness on the fracture surfaces of hybrid welded specimens with decreasing crack growth towards weld root. With numerous small inclusions, which act as nucleating sites for secondary microvoids, the crack extension at the FZ cap is distinctly faster compared to the root. Other asymmetric crack extension is observed, when the initial crack is located in the FZ near the fusion line. During crack growth, the stress triaxiality and plastic strains increase in the softer BM at the fusion line. Besides the increasing plastic strains, it is the high triaxiality level at the material interface, which triggers the second crack initiation in the BM. With increasing plastic zones between these two cracks, the first crack leaves the original path and starts to propagate in direction of BM. Due to the high mismatch and narrow weld seams, this phenomenon is specially pronounced for laser and electron beam welds [20]. The mechanisms behind crack path deviation (CPD) are examined in the following paragraph. The crack resistance curves for S355 are given in Fig. 5b both for BM and HW. Up to Da = 0.4 mm the level and the slope of the curves are almost identical for specimens with BM and HW. There is also no significant difference regarding level of crack initiation values with Ji = 202 N/mm for BM and Ji = 190.9 N/ mm for HW. However, BM shows better fracture behaviour with increasing crack growth compared to HW. In order to verify the transferability of GTN parameters and to investigate the geometry and the loading effect on the crack growth, the SENB specimens with the HW are also tested for S355, see Fig. 6. For FE-analysis the initial crack is located in the centre of the weld seam. The crack initiation and the level of crack resistance curve for BM and HW are very similar to those obtained for the CT samples. However, the crack initiation values are overestimated by numerical analysis due to the lower triaxiality compared to the CT
10
1400
9
1200
J-Integral [N/mm]
Load F [kN]
8 7 6 5 4 3
BM-Exp. BM-GTN HW-Exp. HW-GTN
2 1 2.0
4.0
1000 800 600 400 200
0 0.0
BM-Exp. BM-GTN HW-Exp. HW-GTN
6.0
Opening displacement VLL [mm]
0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
Crack growth Δa [mm]
Fig. 5. Load vs. opening displacement (a) and J-Integral vs. crack growth Da (b).
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1400 SENB 20x10
1200
4
J-Integral [N/mm]
Load F [kN]
5 SENB 10x10
3 2 1 0 0.0
BM-Exp. BM-GTN HW-Exp. HW-GTN
1000
BM-Exp. BM-GTN HW-Exp. HW-GTN
SENB 20x10
800 600 400 200
SENB 10x10
0 2.0
4.0
6.0
Deflection [mm]
8.0
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
Crack growth Δa [mm]
Fig. 6. Comparison between experiment and simulation with respect to load vs. deformation (a) and fracture resistance curves (b) for SENB specimens with hybrid weld and of homogenous base metal, S355.
specimens. In contrast, the experimentally determined initiation shows no strong dependency on the constraint effect.
7. Crack path deviation (CPD) The influence of the initial crack location at the crack growth is studied for the steel S355. Therefore, 3D models of 0.4CT specimens are generated and meshed with 8-noded brick elements (C3D8). The initial crack length is a0 = 10 mm, thus a0/W = 0.5. If the initial crack is located 0.5 mm away from the fusion line in the CT specimen, the crack path deviation (CPD) is observed. On the other hand, the crack growth is straight with initial crack in the centre of the weld. The minimal distance between initial crack and fusion line required for CPD occurrence is about 0.9 mm. Due to the different stress states, the crack extension varies along the specimen thickness, see Fig. 7. With the plane strain conditions in the mid-thickness plane, the high constraint at fusion line contributes to the initiation and propagation of second crack. The increasing plastic deformation between two cracks leads to the deviation of the first crack into the softer BM. At the surface no crack path deviation (CPD) occurs, since the triaxiality is low because of the plane stress conditions. The uneven fracture surface from numerical results coincides with the experimental findings. In order to understand the mechanisms, which control the crack growth and CPD, the 2D 0.4CT specimens are modelled with the initial crack located 1.6 and 0.5 mm away from fusion line. The local stress and strain state are quantified by the stress triaxiality h and equivalent plastic strain epl v for both configurations. The results are evaluated and compared for Da = 2 mm along the ligament in the FZ (path 1) and along the fusion line (path 2) in the BM (path 2-BM) and FZ (path 2-FZ). In the case of the straight crack growth for
Fig. 7. The generated 3D FE model of the 0.4CT specimen used for the numerical analysis (a) with crack propagation at the surface (b) and in the mid-thickness plane (c).
A. Nonn et al. / Engineering Fracture Mechanics 75 (2008) 3251–3263 3.5
3.5 y
Stress triaxiality h
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Initial crack
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Path 1-FZ Path 2-BM Path 2-FZ
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Fusion line y
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y=1.6 mm
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Path 2 Fusion line
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Initial crack
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FL-BM
Path 1-FZ Path 2-BM Path 2-FZ
2.0
0.5
0.0 0.0
3259
1.0
2.0
3.0
4.0
Distance from initial crack tip [mm]
Fig. 8. Distribution of stress triaxiality h for the cases when crack grows straight (a) and deviates (b) for Da = 2 mm.
0.80
0.80
y
0.70 0.60
y=0.5 mm
Fusion line pl
Path 2
y=1.6 mm
Equivalent plastic strain εv
Equivalent plastic strain εv
pl
0.90
Path 1 Initial crack
Path 1-FZ Path 2-BM Path 2-FZ
0.50 0.40
FZ
0.30 0.20 0.10
FL-BM FL-FZ
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1.0
2.0
3.0
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Distance from initial crack tip [mm]
Path 2
Fusion line
0.70
y FL-FZ
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Path 1
0.50 Path 1-FZ Path 2-BM Path 2-FZ
0.40 0.30
FL-BM
0.20
FZ
0.10 0.00 0.0
1.0
2.0
3.0
4.0
Distance from initial crack tip [mm]
Fig. 9. Distribution of equivalent plastic strain epl v for the cases when crack grows straight (a) and deviates (b), Da = 2 mm.
y = 1.6 mm (see Fig. 8a and 9a), the maximal h value in the BM is achieved 3 mm away from initial and 1 mm away from the current crack tip. It is almost 50% higher than the maximal h value in the FZ located directly behind the current crack tip. In contrast, the maximal plastic strain level achieved between initial and current crack tip is ca. 30% higher in the FZ than in the BM. On the other hand, the decrease of epl v values is stronger in FZ, thus leading to lower plastic strains ahead of the current tip. Neither h nor epl v values in the BM are sufficient to trigger the initiation of the second crack. The results for second configuration with y = 0.5 mm and CPD occurrence are depicted in Fig. 8b and 9b. Due to increasing stress triaxiality at the fusion line with increasing loading, the second crack initiates in BM after 1 mm extension of the first crack in the fusion zone. In 3D model, however, crack initiates already after 0.5 mm of first crack extension in mid-thickness plane. On the one hand, the stress triaxility at the interface between BM and FZ in 3D model is slightly reduced compared to the 2D model with maximal out-of-plane constraint. On the other hand, the plastic strain development is more pronounced in 3D than in the 2D model leading to the second crack appearance at lower global loading for 3D model. The local maxima of h and epl v values responsible for CPD are reached at the fusion line on the FZ side (path 2-FZ) ca. 1 mm away from the initial crack. These maximal values are almost three times higher than in the FZ (path 1). With epl v values of almost 0.7, it can be concluded that the CPD is controlled by the preceding high plastic strains at the fusion line. For the better understanding of mechanism behind the CPD, it is important to present the damage development which finally leads to CPD, besides the evaluation of the local stress and strains at the moment of
A. Nonn et al. / Engineering Fracture Mechanics 75 (2008) 3251–3263 0.030
0.030 0.025 EL3
0.020 EL4
0.015 0.010 0.005
EL3-FZ-1mm EL4-FZ-2mm
0.000 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 pl
Equivalent plastic strains εv
Modified void volume fraction f*
Modified void volume fraction f*
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EL1
0.025 EL 2
0.020 0.015 0.010 0.005
EL1-FL-BM
EL2-FL-FZ 0.000 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 pl
Equivalent plastic strains εv
Fig. 10. Damage development against equivalent plastic strain epl v for the cases when crack grows straight (a) and deviates (b).
CPD occurrence. Fig. 10a shows the damage development (modified void volume fraction against the equivalent plastic strain) for straight crack growth in the element containing current crack tip after 1mm (EL3-FZ1 mm) and 2 mm (EL4-FZ-2 mm) crack growth. It is obvious that the damage development is intensified with increasing crack growth. In case of CPD, the similar damage distribution in dependence of plastic strains can be found in the element (EL1-FL-BM) on the fusion line at the BM side, see Fig. 10b. Low triaxiality level and consequent strain localisation in the fusion zone element at the fusion line (EL2-FL-FZ) cause the shift of the damage development curve towards higher plastic strains. Thus, when the critical porosity is reached, the strains in this element (EL2-FL-BM) are 2.5 times higher than in the base metal element, see Fig. 10b. 8. Damage modelling of steel frame connections Besides ship building industry, a further application for the use of laser-hybrid welds is within the field of steel constructions. As regards beam-column-connections subjected to seismic loading, one advantage of the hybrid welding compared to conventional welding is the lower residual stresses and thus lower probability for cleavage fracture to occur. The 3D non-linear FE-analysis of the beam-column-connection (BCC) is conducted by using GTN damage model for S355, see Fig. 11. The element size given in Table 4 is used to mesh the weld and base metal area
Fig. 11. FE model of beam-column-connection (BCC) with semi-elliptical crack located in the centre of the flange.
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0.25
longitudinal strain load
0.20
buckling
800
1400
700
1200
500
crack initiation
0.15
gross section (beam) yielding
0.10
400 300
Load F [kN]
600
200 0.05
J-Integral [N/mm]
average longitudinal strain [-]
around semi-elliptical crack. As expected, the pre-calculation with elastic–plastic material properties shows the highest stress concentrations in area of geometric and material discontinuities, which can be found in the welded connection. Due to the possible metallographic defects during welding, e.g. solidification defects, a pre-existing crack is assumed and additionally included in the FE model as a surface semi-elliptical crack. The resulting curves showing load and strains vs. vertical deflection are presented in Fig. 12a. The gross section yielding in the beam is observed for small strains of ca. 2%. The crack initiation and thus failure occurs shortly before the load maximum is reached at the strain level of 10%. The results also show that the crack initiates before the buckling of the beam flange under pressure sets in. The crack initiation level with Ji = 273 N/mm is significantly higher compared to CT specimen with different crack configurations, see Fig. 12b. With Da > 0.3 mm the resistance level converges to R-curve for hybrid welded CT specimens. The crack initiation is observed under angle / = 60°, at the location of the highest stress triaxiality, see Fig. 12 Thereafter the crack extension continues in direction of the surface / = 0° and towards / = 90°. The quantified local stress and strain fields are given in Fig. 13 along the initial crack path for J = 273 N/ mm, which corresponds to J value at the crack initiation in BCC. The stress triaxiality and local plastic strains are evaluated at / = 60° for BCC. For the chosen crack tip loading, the crack extension in the FZ is about 0.2 mm and no second crack or CPD are evident in the CT specimen with y = 0.5 mm. Hence, the level and course of triaxiality and plastic strains for y = 0.5 and 1.6 mm are similar along the initial crack path. Stress triaxiality ahead of the crack for y = 0 mm is intensified by high mismatch at the interface between base
1000
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0.2
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800 60°
600 90°
400
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0 0
CT
200
100 0.00
BM HW y=0 HW y=0.5 HW y=1.6 BCC
0.6
Crack growth Δa [mm]
vertical deflection uv [m]
Fig. 12. Load and strains vs. vertical deflection curve (a), crack resistance curve (b) and crack propagation of semi-elliptical crack (c).
3.0
Equivalent plastic strain εv
pl
0.8
Stress triaxiality h
2.5 2.0 1.5 1.0 CT
0.5 0.0 0.0
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BM HW y=0 HW y=0.5 HW y=1.6 BCC
1.5
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Distance from initial crack tip [mm]
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BM HW y=0 HW y=0.5 HW y=1.6 BCC
CT
0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.0
0.5
1.0
1.5
2.0
Distance from initial crack tip [mm]
Fig. 13. Distribution of stress triaxiality h (a) and equivalent plastic strain epl v (b), J = 273 N/mm.
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and weld metal. As expected the lowest triaxiality and the highest plastic strains are calculated in the BCC resulting from the crack size, shape and component dimension. The changes in the stress field resulting from the stress redistribution during the crack extension may cause a possible shift of failure mechanism from ductile to cleavage fracture, especially in components subjected to cyclic loading. This will be investigated in the future work. 9. Conclusions In this paper, the fracture behaviour of laser-hybrid weld is investigated by means of GTN damage model for steel S355. Except for element size ly, which is more than two times higher for BM compared to FZ, almost same parameters (e.g. initial and critical porosity) result both for BM and FZ. Although the higher element size ly should lead to the increase of fracture resistance, similar initiation level and R-curve for Da < 0.4 mm are obtained for CT specimens of homogenous BM and with HW. This is due to the lower yield strength and lower hardening level of BM. The initial porosity can be successfully derived from metallographical analyses of the non-metallic inclusions. Despite the microstructural differences between weld cap and weld root resulting from hybrid welding process the similar distribution and morphology of large inclusions are observed for both weld parts. However the smaller particles detected in the cap with dmid < 1 lm are not taken into account for modelling of ductile damage. It is assumed that the differences with respect to fracture toughness between acicular ferrite, which is mostly represented in the weld cap and Widmansta¨tten ferrite in the weld root, will play crucial role in the brittle and ductile-to-brittle transition temperature region. With critical porosity fc determined from unit cell calculations, a further damage parameter, viz. element size ly is calibrated and verified using test results from tests on the notched round bar and fracture mechanics specimens, respectively. Transferability of the identified parameters is demonstrated on the numerically obtained R-curves for SENB specimens. The same parameters are used for the local stress and strain analyses to explain the effect of experimentally observed crack path deviation. For this purpose, calculations are carried out for 2D and 3D models of CT specimen with initial crack located 0.5 mm away from the fusion line. In both models, the initiation of the second crack at the fusion line sets in before the first crack deviates from FZ into BM. With more progressed plastic zones development second crack appears at the lower load level in 3D than in 2D model. In contrast to mid-thickness plane of 3D model, no second crack initiation takes place at the specimen surface. This can be explained by the constraint loss when moving towards specimen surface. Similar local stress state exists at BM/FZ interface for 2D model with the initial crack located in the centre of the weld. Hence, high triaxiality level and certain amount of plastic strains are required for second crack initiation. Finally, CPD is a consequence of increasing plastic strains in the FZ between two cracks. Up to Da = 0.3 mm, higher level of fracture resistance is numerically obtained for BCC compared to the CT specimens with varying initial crack location. Crack resistance loss evident for Da > 0.3 mm is based on lower strains necessary to drive the crack because of the strain localisation in the rest ligament. References [1] Heyer J. Lokale Beanspruchung in angerissenen strahlgeschweißten Stahlbauteilen. RWTH Aachen PhD thesis, Verlag Shaker, Aachen; 2004. [2] Ne`gre P, Steglich D, Brocks W. Crack extension in aluminum welds: a numerical approach using the Gurson–Tvergaard–Neeldeman model. Engng Fract Mech 2004;71:2365–83. [3] Ne`gre P, Steglich D, Brocks W, Kocßak M. Numerical simulation of crack extension in aluminium welds. Comput Mater Sci 2003:723–31. [4] Tvergaard V, Needlemann A. Analysis of the cup-cone fracture in a round tensile bar. Acta Metall 1984;32(1):157–69. [5] Tvergaard V. On localisation in ductile materials containing spherical voids. Int J Fract 1982;18:237–52. [6] Gurson AL. Continuum theory of ductile rupture by void nucleation and growth: part I – yield criteria and flow rules for porous ductile media. J Engng Mater Tech 1977;99:2–15. [7] Siegmund T, Brocks W. Prediction of the work of separation and implications to modeling. Int J Fract 1999;99:97–116. [8] Steglich D, Brocks W. Micromechanical modelling of the behaviour of ductile materials including particles. Comput Mater Sci 1997;9:7–17. [9] Needleman A, Tvergaard V. An analysis of ductile rupture modes at a crack tip. J Mech Phys Solids 1987;35:151–83.
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[10] Tvergaard V, Needleman A. Effect of crack meandering on dynamic. J Mech Phys Solids 1992;40(2):447–71. [11] Needleman A, Rice JR. Limits to ductility set by plastic flow localisation. In: Koistinen DP, editor. Mechanics of sheet metal forming. New York: Plenum Press; 1978. p. 237–67. [12] Chu CC, Needleman A. Void nucleation effects in biaxially stretched sheets. J Engng Mater Tech 1980;102:249–55. [13] Tvergaard V. Ductile fracture by cavity nucleation between larger voids. J Mech Phys Solid 1982;30(4):265–86. [14] Hibbitt. Karlsson & Sorensen Inc. Abaqus Finite Element Program, Version 6.5, 2004. [15] Siegmund T, Brocks W. A user-material subroutine incorporating the Gurson–Tvergaard–Needleman model of porous metal plasticity into the ABAQUS finite element program, Technical report GKSS/WMG/97/2, Geesthacht: Institut fu¨r WerkstoffForschung, GKSS, 1997. [16] Philippa L Moore. Investigation into the microstructure and properties of laser and laser/arc hybrid welds in pipeline steels, University of Cambridge, PhD thesis; 2003. [17] Bernauer G, Brocks W. Micro-mechanical modelling of ductile damage and tearing-results of a European numerical round robin. Fatigue Fract Engng Mater Struct 2001;25:363–84. [18] Brocks W, Sun D, Ho¨nig A. Verification of micromechanical models for ductile fracture by cell model calculations. Comput Mater Sci 1996;7:235–41. [19] Brocks W, Sun D, Ho¨nig A. Verification of the transferability of micromechanical parameters by cell model calculations with viscoplastic materials. Int J Plast 1995;11(8):971–89. [20] Bajric A, Brocks W, Dahl W, Heyer J, Langenberg P. Investigations of failure behaviour of flawed steel specimens with electron beam welds (EBW). In: 11th International Congress on Fracture (ICF), Paper No. 4339, CD ROM, Turin (Italy); 2005.