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Int. J. Materials and Product Technology, Vol. 48, Nos. 1/2/3/4, 2014
Fracture modelling of DP780 sheets using a hybrid experimental-numerical method and two-dimensional digital image correlation Keunhwan Pack, Kwanghyun Ahn, Hoon Huh* and Yanshan Lou School of Mechanical, Aerospace and Systems Engineering, KAIST, 291 Daehak-ro, Build. N7, Rm. 1203, Yuseong-gu, Daejeon 305-701, Korea E-mail:
[email protected] E-mail:
[email protected] E-mail:
[email protected] E-mail:
[email protected] *Corresponding author Abstract: This paper is concerned with the construction of fracture envelopes of DP780 sheets using two methods: a hybrid experimental-numerical method; two-dimensional digital image correlation (2D-DIC). For the hybrid method, four types of ductile fracture tests were carried out covering a wide range of stress states on specimens: with a central hole; two symmetric circular notches; flat grooved; and diagonally double-notched. Based on the fracture strain and loading paths identified with finite element simulation, a fracture envelope was obtained by employing the three-parameter modified Mohr-Coulomb fracture model. In addition, the fracture surface strain was directly measured using 2D-DIC. Loading histories of each test were extracted from a surface element of a three dimensional finite element model. The comparison of fracture envelopes constructed by the two methods reveals that there is little difference. Thus, it can be concluded that 2D-DIC is applicable to fracture modelling of DP780 sheets despite the assumption of the plane stress condition even after necking. Keywords: ductile fracture; hybrid experimental-numerical method; digital image correlation; DIC; fracture locus; fracture forming limit diagram. Reference to this paper should be made as follows: Pack, K., Ahn, K., Huh, H. and Lou, Y. (2014) ‘Fracture modelling of DP780 sheets using a hybrid experimental-numerical method and two-dimensional digital image correlation’, Int. J. Materials and Product Technology, Vol. 48, Nos. 1/2/3/4, pp.34–46. Biographical notes: Keunhwan Pack received his Bachelor’s and Master’s degrees in the Department of Mechanical Engineering at Korea Advanced Institute of Science and Technology (KAIST), South Korea, studying the effect of intermediate strain rates on the fracture loci of steel sheets. He and his team proposed a new ductile fracture criterion inspired by the microscopic mechanism to predict the fracture forming limit diagram. He is currently continuing his graduate studies at Massachusetts Institute of Technology for a PhD degree.
Copyright © 2014 Inderscience Enterprises Ltd.
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Kwanhyun Ahn received his Bachelor’s degree at Hanyang University, and Master’s degree at KAIST. He recently completed his PhD dissertation at KAIST on the strain rate-dependent hardening model for pure titanium with the consideration of the effect of deformation twinning. He has contributed to devising the buffer stopper for Korean railroad safety as an excellent experimentalist. He is now working at Samsung Heavy Industries. Hoon Huh is a Full Professor in the Department of Mechanical Engineering at KAIST and the Director of the Computational Solid Mechanics and Design Lab (CSMD) and the Center for High Speed Material Properties. He studied at Seoul National University, South Korea for his Bachelor’s and Master’s degrees and received his PhD degree from the University of Michigan. His major field is solid mechanics and numerical analysis with the emphasis on the strain rate effect for the enhancement of crashworthiness and auto-body design. He is the Editor-in-Chief of the International Journal of Automotive Technology and a Fellow of the American Society of Mechanical Engineers. Yanshan Lou graduated from Jiamusi University and Jilin University for his Bachelor’s and Master’s degrees, respectively. He joined CSMD in 2007 and completed his PhD dissertation in 2012 with the development of a new ductile fracture criterion and its application to various components made out of steel sheets. His interest was extended to general plasticity models. He is currently working at Swinburne University as a Postdoctoral Associate. This paper is a revised and expanded version of a paper entitled ‘Fracture modeling of DP780 sheets using a hybrid experimental-numerical method and two-dimensional digital image correlation’ presented at the 15th International Conference on Advanced in Materials & Processing Technologies (AMPT 2012), Wollongong, Australia, 23–26 September 2012.
1
Introduction
As the importance of an accurate prediction of ductile fracture increases in a variety of fields including the automobile industry, with numerical simulation widely used to save costs and time, many researchers spare no effort to develop an accurate fracture criterion which can be easily calibrated by simple tests and implemented to FE codes (Bai and Wierzbicki, 2010; Bao and Wierzbicki, 2004a, 2004b; Brozzo et al., 1972; Clift et al., 1990; Cockcroft and Latham, 1968; LeRoy et al., 1981; Lou et al., 2012; Oh et al., 1979; Wierzbicki et al., 2005). According to many previous studies, the ductility of metals is greatly influenced by two stress-related parameters: the stress triaxiality, defined as mean stress divided by von Mises equivalent stress; the Lode angle, expressed in terms of the third invariant of stress deviator and Mises stress (Bai and Wierzbicki, 2008; Bao and Wierzbicki, 2004b; Wierzbicki et al., 2005). Recently, the modified Mohr-Coulomb (MMC) fracture model has been suggested by Bai and Wierzbicki (2010) to capture these effects and verified by a series of experiments and corresponding finite element simulation (Beese et al., 2010; Dunand and Mohr, 2010, 2011; Li et al., 2010; Luo and Wierzibicki, 2009, 2010). This model was extended from the original Mohr-Coulomb model aimed at geo-materials since it can reflect the effects of both normal and shear stresses.
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Basically, four or five types of ductile fracture tests are performed covering a wide range of stress states, and the strain value at the onset of failure (fracture strain) should be defined as well as representative stress states in order to calibrate this model. There are largely two methods to determine fracture strain: 2D-DIC, where the fracture strain is calculated with measured in-plane strain components based on the definition of equivalent plastic strain in the plane stress condition; a hybrid experimental-numerical method, in which the fracture strain is defined as the maximum equivalent plastic strain predicted in finite element simulation when the same amount of displacement as in an actual test up until onset of failure is applied to the boundary of a finite element model for each type of specimen. In this study, the applicability of 2D-DIC to fracture modelling of DP780 sheets is investigated by comparing fracture envelopes constructed by both methods. For sheet metal application, fracture loci, which satisfy the plane stress condition, were taken out from each envelope, drawn in the space of the equivalent strain to fracture and the stress triaxiality, and compared with each other. Furthermore, the fracture loci were transformed into fracture forming limit diagrams (FFLD), which are a more extensively used description of ductility in sheet metal forming community.
2
Experiment
2.1 Material For the present study, DP780 steel sheets with 1.2 mm thickness and the minimum ultimate tensile strength of 780 MPa were provided by POSCO. A dual-phase steel in the class of advanced high strength steels has now come into wide use as an auto-body plate to enhance the crashworthiness and fuel efficiency. Its microstructure features scattered martensite islands in a soft ferrite matrix. The chemical composition of DP780 sheets analysed with the Electron Probe Micro Analyzer is summarised in Table 1. Table 1
Chemical composition of DP780 1.2 t in wt%
Element
Mn
Si
C
P
S
wt%
2.23
1.01
0.069
0.014
0.002
2.2 Plasticity model A series of uniaxial tensile tests were conducted using INSTRON 5583 with the intent of modelling the plastic behaviour of DP780 sheets (Huh et al., 2012). The geometry of a specimen is provided in Figure 1. To take into consideration the anisotropy of the sheets arising from the rolling process, the Hill 1948 quadratic yield function in the form of equation (1) was adopted and its coefficients were determined through uniaxial tensile tests of dog-bone (DB) specimens extracted from three different orientations (0°, 45°, and 90° to the rolling direction). F ( σ 22 − σ 33 ) + G ( σ 33 − σ11 ) + H ( σ11 − σ 22 ) 2
σ Hill =
+2 Lσ 232 + 2Mσ 312 + 2 Nσ12 2
2
2
(1)
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The Lankford ratios measured with DIC were r0 = 0.83, r45 = 0.99 and r90 = 0.92. Since out-of-plane shear tests are not feasible, L and M are assumed to be 1.5, which corresponds to isotropic condition. The hardening curve was also obtained in the rolling direction and fitted with Swift power law equation. Hill’s coefficients along with Swift parameters are given in Table 2. For a reliable numerical simulation even after necking, it should be an integral part to identify the stress-strain relation in the post necking region. To this end, a so-called inverse method was used, following the procedures by Dunand and Mohr (2010). Table 2
Hill’s parameters and Swift law parameters
Hill’s parameters F 0.49
Figure 1
2.3
Swift law parameters
G
H
L
M
N
A
ε0
n
0.55
0.45
1.50
1.50
1.55
1,299
0.0030
0.1742
Geometry of DB specimen for uniaxial tensile test
Ductile fracture test
In order to calibrate the MMC fracture model, four types of basic ductile fracture tests were performed on specimens; with a central hole (CH) of 8 mm in diameter; two symmetric circular notches (TSCN) of 10 mm radii; flat grooved (FG); diagonally double notched (DDN). These four specimens characterise the fracture property in uniaxial tension, condition between plane strain and biaxial tension, plane strain tension, and shear loading, respectively. Only rolling direction was considered, and each specimen was tested using INSTRON 5583, with load-displacement curves recorded in real time and deformation images captured all the way to fracture with a single high speed camera, Phantom v9.0. For the purpose of measuring fracture strain with 2D-DIC, CH specimens were replaced by DB specimens because DIC technique is not applicable to the point of fracture in CH specimens, which lie in the intersection of a hole and the transverse axis of symmetry. Figure 2 shows photographs of specimens before and after tests.
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Figure 2
Photograph of fractured specimen, (a) with a central hole (b) TSCN (c) FG (d) DDN (e) DB (see online version for colours)
(a)
(b)
(d)
3
(c)
(e)
Fracture strain and loading paths
3.1 Hybrid experimental-numerical method First, we employed a hybrid experimental-numerical technique, the way to determine the fracture strain at a material point inside of a specimen where the fracture is actually initiated with finite element simulation. Each specimen was modelled with eight-node three-dimensional brick elements with reduced integration, exploiting the geometrical symmetry to save computational time. Half thickness consists of eight layers of finite elements so that necking is verisimilarly described. To minimise spatial discretisation error (mesh size sensitivity), the side of elements around the point of onset of failure was set to be 0.1 mm, which turns out to calculate the converged strain value (Dunand and Mohr, 2010). As for FG specimens, the geometrical defect of 0.2% was introduced to surface nodes located on the transverse axis of symmetry to artificially trigger necking. None of other types of specimens need the artificial defect thanks to their geometric feature with no parallel gauge section. Abaqus v6.8 was used for simulation, and the
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obtained load-displacement curve was represented along with those from three experiments in Figure 3, which shows that the predicted load-displacement curve agrees well with experimental ones. The reliability of simulation was more strictly verified by comparing the evolution of the largest component of axial logarithmic strain on the surface of a specimen measured by DIC with the one obtained from simulation as demonstrated in Figure 4. When it comes to a DDN specimen, equivalent plastic strain was compared instead of axial logarithmic strain because a finite rotation of the gauge section in this type of specimen with deformation makes an axial component of strain meaningless, and the absence of recognisable necking in shear loading adds the dominance of planes stress condition. A blue curve without symbol in Figure 3 denotes the evolution of the equivalent plastic strain at the fracture initiation point. The change in the slope of the curve indicates the development of diffuse or localised necking, which accelerates deformation. Each specimen features a different necking phenomenon, and this is mainly due to the different stress states inside of it. The fracture strain was defined as the maximum equivalent plastic strain at the average of three displacements to fracture measured from experiments. In addition, to identify loading paths at the fracture initiation point, the histories of the stress triaxiality η and the normalised Lode angle θ defined as equation (2) were investigated. η= Figure 3
σm 6θ 2 ⎛ J ⎞ ,θ = 1− = 1 − cos −1 ⎜ 33 ⎟ σ π π ⎝ 2σ ⎠
(2)
Load-displacement curve from experiment and finite element simulation, and the evolution of equivalent plastic strain, (a) with a central hole (b) TSCN (c) FG (d) DDN (see online version for colours)
(a)
(b)
(c)
(d)
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Figure 4
Evolution of axial logarithmic strain and equivalent plastic train on the surface of each type specimen, (a) with a central hole (b) TSCN (c) FG (d) DDN (see online version for colours)
(a)
(b)
(c)
(d)
Even though all the specimens were designed such that stress states are kept as constant as possible, these two parameters vary with deformation, especially after necking. Thus, weighted averages during the entire process in the form of equation (3) were used as representative stress states for the calibration of the MMC model.
ηavg
∫ =
εf
0
η ( ε p ) dε p εf
, θavg
∫ =
εf
0
θ ( ε p ) dε p
(3)
εf
Calculated fracture strains and stress parameters are tabulated in Table 3. Table 3
Fracture strain and average stress states
Specimen type
Hybrid experimental-numerical method
Two-dimensional digital image correlation
CH
TSCN
FG
DDN
DB
TSCN
FG
DDN
Fracture strain, ε f
1.02
0.710
0.505
0.983
0.825
0.553
0.407
0.995
Stress triaxiality, η
0.320
0.624
0.628
0.0281
0.388
0.533
0.569
0.0287
Lode angle, θ
0.905
0.217
0.0696
0.0820
0.824
0.275
0.0787
0.101
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3.2 Two-dimensional digital image correlation Secondly, we used the optical method, 2D-DIC to directly measure in-plane strain components from speckle patterns coating the surface of a specimen. The commercial DIC tool, ARAMIS v6.3.0, was used to calculate the displacement field by assuming an affine transformation of the 25 × 25 pixels in the vicinity of points spaced five pixels apart. Since DIC is not available at the boundary of a specimen due to the absence of speckle patterns in the neighbourhood of the point of interest, CH specimens were substituted by DB specimens. Strictly speaking, a specimen undergoes tri-axial stress state after the localised necking develops, and thus three-dimensional digital image correlation (3D-DIC) requiring two high speed cameras should be applied for the better accuracy of measured strain. However, compared with conventional steels, DP780 in the class of advanced high strength steels shows a smaller amount of necking, and the assumption of the plane stress condition is quite suitable for sheet metals. Furthermore, the procedure of tests and analysis using 2D-DIC is much simpler than 3D-DIC. Figure 5 shows the distribution of strain on the surface of specimen analysed by ARAMIS. Deformation is concentrated in certain areas where necking prevails, and strain is on a steep gradient, which is a well-known phenomenon before fracture. Figure 5
Distribution of strain right before fracture, (a) DB (b) TSCN (c) FG (d) DDN (see online version for colours)
(a)
(b)
(c)
(d)
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The fracture surface strain was calculated by substituting measured in-plane strain components at the point of onset of failure until the first visible crack is detected into the work-conjugate equivalent plastic strain of the Hill 1948 yield function in the plane stress condition, which is derived as equation (4). dε p =
1 {( F + H )dε112 + ( H + G )dε222 + 2 Hdε11dε22 } + 2 dε122 FH + HG + GF N
(4)
To avoid the inaccuracy or uncertainty of measured values that DIC might intrinsically cause, only three significant figures were taken with round-off. Loading paths of the fracture initiation point were extracted from a surface element of finite element models used for the hybrid method because the strain was measured on the surface of a specimen. In the same way as the hybrid method, the average stress triaxiality and normalised Lode angle were calculated based on equation (3) and are written down in Table 3 along with the measured fracture surface strain. It is noted that fracture strains measured with 2D-DIC are smaller than the ones determined through the hybrid method except a diagonally double-notched specimen since they were measured on the surface of a specimen, not on the mid-plane where the actual fracture initiation point lies. However, they can still be a meaningful calibration points. Because their stress states satisfy the plane stress condition, which prevails in sheet metals, they can play an important role in predicting the onset of failure with shell finite elements. Figure 6 represents the tendency of increase in the equivalent plastic strain at the fracture initiation point for different types of specimens. For easier comparison, the displacement and the equivalent plastic strain were normalised by their maximum values, which are the displacement to fracture and the equivalent strain to fracture. For a TSCN specimen, the equivalent plastic strain evolves through two consecutive increases in the slope, which correspond to diffuse and localised necking, respectively. A FG specimen undergoes only one sudden rise in the strain rate caused by localised necking since adjacent materials prevent the centre point from being deformed in the transverse direction. A DDN specimen does not show any conspicuous change in the slope, and thus it can be confirmed that shear-dominated deformation is not accompanied by necking. These tendencies agree well with the result of finite element simulation exhibited in Figure 3. Figure 6
Evolution of the normalised equivalent plastic strain at the fracture initiation point for three kinds of specimen (see online version for colours)
Fracture modelling of DP780 sheets
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Fracture envelope
4.1 Fracture envelope In this study, the three-parameter MMC fracture model was chosen to quantitatively evaluate the ductility of DP780 sheets. The MMC model takes the form of equation (5) and describes the dependency of the onset of failure in ductile materials on stress states. ⎡ A ⎧⎪ ⎛ ⎛ θπ ⎞ ⎞ ⎫⎪ 3 ε f = ⎢ ⎨c3 + (1 − c3 ) ⎜ sec ⎜ ⎟ − 1⎟ ⎬ 2− 3 ⎝ ⎝ 6 ⎠ ⎠ ⎭⎪ ⎣⎢ c2 ⎩⎪
(5)
1
n ⎛ ⎪⎧ 1 + c12 ⎛ θπ ⎞ 1 ⎛ θπ ⎞ ⎞ ⎪⎫⎤ ⎨ cos ⎜ ⎟ + c1 ⎜ η + sin ⎜ ⎟ ⎟ ⎬⎥ 3 3 ⎝ 6 ⎠ ⎠ ⎪⎥ ⎝ 6 ⎠ ⎪⎩ ⎝ ⎭⎦
This model was extended from the original Mohr-Coulomb criterion, which was developed for geo-materials, due to its advantage of capturing the effects of both the stress triaxiality and the Lode angle on ductility. The equivalent plastic strain to fracture exponentially decays with the increasing stress triaxiality and shows asymmetric dependency on the normalised Lode angle. A and n are Swift law parameters, and c1, c2, and c3 should be determined based on the result of ductile fracture tests. A simple MATLAB code was developed such that the least square difference between equation (5) and test points in Table 3 is minimised. Figure 7 shows the finally constructed fracture envelopes in the space of the stress triaxiality, the normalised Lode angle, and the equivalent plastic strain to fracture. Each point on the fracture envelope indicates the maximum strain a material can resist in the proportional loading condition, where the stress parameters, η and θ , are kept constant. Two lines on each envelope correspond to the plane stress condition, for which the stress triaxiality has a unique relation with the normalised Lode angle written as equation (6). − Figure 7
27 ⎛ 2 1 ⎞ ⎛π ⎞ η ⎜ η − ⎟ = sin ⎜ θ ⎟ 2 ⎝ 3⎠ ⎝2 ⎠
(6)
Fracture envelopes constructed with the MMC fracture creation, (a) using the hybrid method (b) using 2D-DIC (see online version for colours)
(a)
(b)
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As opposed to Figure 7(a), all the calibration points in Figure 7(b) are placed on these lines, which substantiates that average loading paths of them satisfy the plane stress condition.
4.2 Fracture locus For direct comparison between two fracture envelopes, the two-dimensional fracture locus satisfying equation (6) was extracted from each envelope and represented in the space of the stress triaxiality and the equivalent plastic strain to fracture as illustrated in Figure 8(a). It is noteworthy that there is little difference between two fracture loci, which verifies the applicability of 2D-DIC to define the onset of failure for DP780. A fracture locus is not popular with the sheet metal forming community, so it was transformed into a FFLD, a better-known representation of strain limit in the space of major and minor strain. The term, ‘Fracture’ is added in front of the general ‘FLD’ because this limit of ductility was defined based on the onset of failure, not on necking. Fig. 8b shows transformed FFLDs. Five dotted lines in Fig 8a and Fig. 8b refer to representative loading paths: uniaxial compression, pure-shear, uniaxial tension, plane strain tension, equibiaxial tension. There are two additional branches from uniaxial compression to uniaxial tension, which does not appear in a conventional forming limit diagram. Great emphasis is put on these branches because some researchers report that fracture of advanced high strength steels sometimes occurs in these regions (Li et al., 2010). Again, two methods predict similar FFLDs with a trivial discrepancy in the state of plane strain tension, and thus it is concluded that 2D-DIC can be adopted for the purpose of modelling the fracture of DP780 sheets. Figure 8
(a) Fracture locus in the plane stress condition (b) Fracture forming limit diagram (see online version for colours)
(a)
5
(b)
Conclusions
In this study, four kinds of basic ductile fracture tests were carried out in the uniaxial loading frame to characterise the fracture behaviour of DP780 sheets. Two up-to-date
Fracture modelling of DP780 sheets
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techniques, the hybrid method and 2D-DIC, were used to define the fracture strain and identify loading histories all the way to fracture. Based on the established data, two fracture envelopes of DP780 were constructed with the three-parameter MMC model. For direct comparison, two-dimensional fracture loci were taken out from each envelope and transformed into FFLDs as an alternative representation, which show that they are almost identical. Thus, 2D-DIC without complicated procedures is shown to be applicable to fracture modelling of DP780 sheets. It is also expected that 2D-DIC can be used to investigate the dynamic effect on fracture thanks to its simplicity but reasonable accuracy.
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Luo, M. and Wierzbicki, T. (2010) ‘Numerical failure analysis of a stretch-bending test on dualphase steel sheets using a phenomenological fracture model’, International Journal of Solids and Structures, Vol. 47, Nos. 22–23, pp.3084–3102. Oh, S., Chen, C. and Kobayashi, S. (1979) ‘Ductile failure in axisymmetric extrusion and drawing, part 2, workability in extrusion and drawing’, Journal of Engineering for Industry, Vol. 101, No. 1, pp.36–44. Wierzbicki, T., Bao, Y., Lee, Y.W. and Bai, Y. (2005) ‘Calibration and evaluation of seven fracture models’, International Journal of Mechanical Sciences, Vol. 47, Nos. 4–5, pp.719–743.
