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Ductile fracture of AISI 304L stainless steel sheet in stretching
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Ductile fracture of AISI 304L stainless steel sheet in stretching Khadija. Ben Othmen , Nader. Haddar , Anthony Jegat , Pierre-Yves. Manach , Khaled. Elleuch PII: DOI: Reference:
S0020-7403(19)32869-3 https://doi.org/10.1016/j.ijmecsci.2019.105404 MS 105404
To appear in:
International Journal of Mechanical Sciences
Received date: Revised date: Accepted date:
4 August 2019 18 November 2019 20 December 2019
Please cite this article as: Khadija. Ben Othmen , Nader. Haddar , Anthony Jegat , Pierre-Yves. Manach , Khaled. Elleuch , Ductile fracture of AISI 304L stainless steel sheet in stretching, International Journal of Mechanical Sciences (2019), doi: https://doi.org/10.1016/j.ijmecsci.2019.105404
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Highlights
The prediction of the onset of rupture of austenitic stainless steel during a forming process has been carried out.
The fracture behavior of AISI 304L stainless steel, tensile and Erichsen tests were performed.
A 3D numerical simulation via Abaqus/Standard was used.
Four ductile fracture criteria are used to predict rupture during the Erichsen test.
1
Ductile fracture of AISI 304L stainless steel sheet in stretching Khadija. Ben Othmen1, Nader. Haddar1, Anthony. Jegat2, Pierre-Yves. Manach2, Khaled. Elleuch1 1) LGME, ENIS, Route Soukra Km 3.5 B.P. 1173 3038 Sfax, Université de Sfax, Tunisie,
2) Univ. Bretagne Sud, UMR CNRS 6027, IRDL, F-56100 Lorient, France [email protected], [email protected], [email protected], [email protected], [email protected]
Abstract The present study focuses upon the numerical investigation of fracture initiation of austenitic stainless steel, AISI 304L, during a forming process leading to an expansion strain path. Hence, a combined experimental and numerical approach was tested. Monotonic tensile tests up to fracture were carried out. Load-extensometer displacement curves and a major strain distribution were measured. Numerical simulations of the tensile test up to rupture were carried out to calibrate damage parameters of several macroscopic fracture criteria. Numerical simulations of the Erichsen test were also carried out to verify the validity of the determined parameters of the ductile fracture criteria. The numerical predictions of the strain field, the onset of fracture, and the fracture location were compared with the experimental results. The simulation results show that the Rice-Tracey or Brozzo fracture criteria are reliable predictors of the onset of fracture of AISI 304L in the Erichsen test. More precisely, the Rice-Tracey criterion is the most effective to predict the ductile fracture in this case.
Keywords: Onset of fracture; Austenitic stainless steel; Tensile test; Ductile fracture criteria; Erichsen test.
2
1. Introduction In recent years, formability has been viewed as one of the most important research topics, due to the development of sheet metal forming process modeling. Formability depends on several factors such as the material properties or the process parameters [1]. The manufacture of sheet metal into a required shape without defects, such as necking and fracture, is one of the main objectives for an accurate design and assessment. In other words, the occurrence of fracture or necking is generally regarded as one of the forming limitations of the sheet part. A ductile fracture is the most common failure problem encountered in sheets metal forming process ( [2], [3]) in particular, with austenitic stainless steel sheets. Austenitic stainless steel, particularly AISI 304L, is widely employed in several industrial applications such as the food industry, electronics, automobiles, and biomedicine, owing to its high strength, excellent corrosion resistance, and good formability ( [4], [5], [6]). In spite of their widespread use, few studies have focused on the prediction of the fracture of austenitic stainless steel sheets during forming processes. To optimize an efficient finite element analysis of manufacturing operations among those forming processes, reliable constitutive behavior and fracture properties of the sheet metal are required. The constitutive model of the AISI 304L sheet metal used in this study was identified in the previous work of Khadija et al. [7]. Hence, the accurate prediction of the fracture response is the main objective of the current work. In this sense, various ductile fracture criteria have been proposed in the literature and implemented in the finite element codes to predict the onset of fracture in sheet metal forming processes ( [8], [9], [10], [11]). These criteria may be classified into coupled and uncoupled categories [12]. The coupled category associates the fracture criterion into the constitutive equation of the material whereas for the uncoupled category, the damage evolution is independent of the constitutive law of the material. Despite their advantage to describe the physical mechanism of ductile fracture, coupled criteria are difficult to use to calibrate constitutive and damage models, which makes their industrial applications limited. Uncoupled criteria are preferred due to their simplicity and the few material constants to be experimentally calibrated ( [10], [13]). These uncoupled ductile fracture criteria have been widely employed to predict the fracture that occurred during forming processes ( [8], [9], [14], [15]). Many researchers have
3
suggested criteria that involve the history of stress and strain, reflecting the realistic conditions in sheet metal forming ( [16], [17], [18], [19] and [20]). Among these criteria, Freudenthal [21] argued that a critical fracture strain may be estimated by the total plastic deformation work. Cockcroft-Latham [16] developed a criterion that depends on the maximum principal stress (
. This criterion has been widely used, due to
its simplicity [22]. Oh et al. [17] suggested a normalized form of the Cockcroft-Latham criterion by the equivalent Mises stress ̅ to give an accurate prediction of the fracture strain. Brozzo et al. [18] empirically extended the Cockcroft-Latham criterion to introduce the effect of hydrostatic stress
for predicting the formability limit of sheet metal. This criterion was
successfully applied to predict the damage of bulk metals [23]. Later, Ayada et al. [24] proposed a criterion whereby the hydrostatic stress was normalized by the equivalent Mises stress. This ratio was called stress triaxiality µ
̅
. Rice and Tracey [25] adopted a single
spherical void growth criterion. They also introduced stress triaxiality through an exponential function. In this criterion, hydrostatic stress depends strongly on the equivalent plastic strain to fracture. Therefore, from an engineering and practical point-of-view, and in order to check a variety of stress tensor invariants, four of the most widely used ductile fracture criteria of Oh [17], Brozzo [18], Ayada [24], and Rice and Tracey [25] are selected. The main advantage of this choice is that only one material parameter is required to be identified. The previous criteria are calibrated with a hybrid experimental-numerical technique [8], using tensile test results. Subsequently, these criteria are used to predict fracture during the Erichsen test, which has been carefully instrumented in order to capture several strain effects up to fracture.
2. Ductile fracture criteria In uncoupled ductile fracture criteria, the fracture of the work piece occurs when the maximum damage value reaches a critical value
. These criteria with an integral form in
Eq. (1) express a function of the equivalent plastic strain ̅ . ∫ Where
̅
̅
(1)
is a function of the process parameter, ̅ is the equivalent plastic strain at fracture,
̅ is the equivalent plastic strain, and
is the damage value.
4
While there are numerous uncoupled criteria, only four criteria were chosen for this study. The selected criteria are shown as follows: Oh et al. [17] :
∫
̅
̅
̅
(2)
Brozzo et al. [18]:
∫
̅
̅
(3)
Ayada et al. [24]:
∫
̅
̅
̅
(4)
Rice-Tracey [25]: ∫
̅ ̅
̅
(5)
For all these criteria, the onset of fracture is attained once the damage value ( ) reaches the critical value
.
3. Material and experimental tests 3.1. Material test The material investigated is AISI 304L. It was received as cold-rolled sheets with a thickness of 1 mm. The chemical compositions (wt %) are as follows: 20.31% Cr, 7.87% Ni, 0.029% C, 0.384% Si, 1.85% Mn, 0.04% Nb, 0.196% Mo, and 68.5% Fe. The microstructure of the test samples, in the initial state, is shown in Fig. 1.
TD TD DT DL RD RD
10 µm
Fig. 1 Micrographic observations of AISI 304L at the initial state by optical microscopy. 5
3.2. Tensile tests Monotonic tensile tests up to fracture were conducted in the rolling direction using rectangular samples with a gauge area size of 150x20 mm2. Tests were carried out using an INSTRON 8803 machine with a maximum load capacity of 100 kN at room temperature with a strain rate ε˙= 1.2 10−3s-1. Both longitudinal
and transverse
logarithmic strains were
measured during the experiments with the Digital Image Correlation (DIC) system (Aramis) [26]. The displacement between two equidistant points, localized in the center of the sample and initially equal to 12.5mm, was measured. According to the recorded load-displacement curve up to rupture (Fig. 2 (a)), the necking point and the onset of fracture were used to extrapolate the hardening law in order to calibrate the fracture criteria. Using DIC, the image was captured every 1s. The longitudinal strain distribution ϵyy coincided with the moment just before the rupture, i.e. at tR-1, as illustrated in Fig. 2. (b). In addition, the average maximum local strain was calculated upon a square area of 3mm side length (Fig. 2. (b)). In this study, anisotropy of the material was not considered. However, the mechanical behavior was examined at 0°, 45°, and 90° to the rolling directions, as can be found in [7]. The tensile properties of the AISI 304L sheets are summarized in Table 1. The uniaxial tensile tests were carried out three times, and a representative curve was adopted.
14 12
Necking point Onset of fracture
Load (kN)
10 8 6 4 2 0 0
2
4
6
7.22
8
10
Extensometer displacement (mm)
(a)
6
28°
After rupture
Just before rupture
(b) Fig. 2 Experimental tensile test results measured with Aramis: (a) Load-extensometer displacement curve; (b) Longitudinal strain distribution ϵyy measured by DIC just before fracture and after fracture in tension (Maximum local strain calculated in the square area). Table 1 Tensile properties of the AISI 304L Angle with
Young's modulus
Tensile strength
Yield stress
Elongation
Anisotropic
RD (°)
E (GPa)
Rm (MPa)
Re (MPa)
A%
0
203.08
670
278
75
0.87
45
203
632
270
56
1.16
90
203.05
647
280
65
0.82
coefficient
3.3. Erichsen tests Erichsen tests were carried out on a Zwick-Roell BUP 200 sheet metal testing machine. The diameter of the blank was D1=120 mm. The diameter of the hemispherical punch was D2=60 mm. The die opening diameter was D3=65mm, and its radius was R=5 mm. The rolling direction (RD) and transverse direction (TD) were set along the x-axis and the y-axis, respectively. During tests, the blank was clamped between the blank holder and the die by applying a Blank Holder Force (BHF) of 158 KN.
7
These tests were carried out without lubrication and at a constant punch speed of 10 mm/min. The load vs. punch displacement curves were recorded during the test. As shown in Fig. 3 (a), results present good reproducibility. In order to follow the evolution of the strain field, local strain was measured by a non-contact ARAMIS DIC-3D system. Fig. 3(b) shows the major strain distribution for six different punch displacements, marked in Fig. 3(a). A localized band of major strain distribution, which coincides with the contact zone between the blank and the punch, occurred and then evolved radially. Similarly, the intensity of the strain field increased until the appearance of the fracture at the last point and the load reaches 88.8 KN for the punch displacement of 25 mm.
100
a 80
Punch force [KN]
6
Test 1 Test 2
5 4
60
3
40
20
2 1
0
0
5
10
15
20
25
30
Punch displacement [mm]
b
1
2 d=0 mm
3 d= 8 mm
d= 15 mm
8
4
5 d= 19 mm
6 d= 23 mm
d= 25 mm
Fig. 3 Experimental results of the Erichsen test: (a) Punch force-displacement curves, (b) Strain distribution for different tool displacements with DIC system measurements. 3.4. Fracture mode analysis The localized necking band of the tensile specimen is oriented at around 28° to the transverse axis (Fig. 2 (b)). Furthermore, the region near the fracture zone was analyzed by a scanning electron microscope (SEM). Regarding the tensile specimen, the fracture surface shows a high proportion of dimples, revealing a ductile fracture (Fig. 4). This failure mode is initiated by the formation of voids around non-metallic inclusions and/or second-phase particles in a metal matrix, being sustained to plastic deformation under an external load [27]. In a second step, growth and coalescence of the dimples lead to fracture of the specimen. Similar occurrences were observed on the fractured surface after the Erichsen tests (Fig. 5). Fig 5 (c) exhibits a fracture surface, with a large variation in size and shape of dimples, indicating clearly a ductile fracture facies. The fracture occurs after coalescence of voids leading to the appearance of macroscopic cracks which propagation conducts to the ligament separation.
(a)
(b)
Fig. 4 SEM observation of the fracture surface of the tensile test specimen: (a) General view; (b) Detail of a square area in (a) (
rounded inclusions) 9
a
b
c
Microvoids
Crack propagation
pprpppropagation
Fig. 5 (a) Cup drawn on the Erichsen test with fractured surface, SEM images of ductile fracture (b) fractured surface, (c) detail of the square area in (b). 4. Numerical calibration: Tensile test To calibrate constants of ductile fracture criteria, a tensile test up to rupture was simulated. Hence, the fracture calibration process is presented in this section. 4.1. Plastic hardening calibration To obtain an accurate FE simulation of the damaged material, a stress–strain relationship up to rupture was required. Then, this curve was divided into two portions at the necking point. For the first portion (before the necking) characterized by homogeneous deformations, the true stress-strain data were calculated from experimental loads and displacements (Fig. 6(a)). The stress-strain curve was expressed by the Voce hardening law [28], which can be defined through equation:
10
̅
)
(6)
is the yield stress at zero plastic strain, ̅ the equivalent plastic strain, while Q and
Where:
b represent the material parameters. In contrast, the second portion (beyond necking) was determined by numerous methods. Hence, in this study, a linear extrapolation of the hardening law was adopted, as shown in Fig. 6(b). The material parameters used as input data in the numerical simulation are summarized in Table 2. The necking point is determined at
=0.45 and Cauchy stress of 1054.3
MPa. 1200 1200
a 1000
Cauchy stress (MPa)
800
Stress (MPa)
b
1000
600
400
200
Nominal stress-strain True stress-strain
800
600
400
200
0
Extrapolated data Experiment data
0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Strain
Strain
Fig. 6 Tensile test: (a) Nominal and true stress-strain curves; (b) Experimental and extrapolated data.
Table 2 Material parameters of the Voce hardening law for the AISI 304L with the extrapolated portion after necking.
̅
̅
0.451
(MPa)
(MPa)
290
1170
b 2.3
̅ (MPa)
0.451
0.502
0.53
0.555
0.564
0.576
1054.3
1103.6
1129
1151
1158.8
1169
11
4.2. Finite element model The numerical simulation of the tensile test up to rupture was carried out using Abaqus/implicit FE code. Then, the mesh was structured and 3D 8-node linear brick elements with reduced integration (C3D8R) and with hourglass control were used. Fig. 7 illustrates the FE model with the boundary conditions. The specimens were rectangular with dimensions of 150 mm×20 mm and 1 mm thickness. Moreover, displacements of d and (-d) were applied at two extremities of the specimen along the y-direction. To promote the rupture at the center of the specimen, a slight imperfection was added to the edge of the central part. The local displacement was measured by a virtual extensometer between two points at the center of the specimen at a distance of 12.5 mm. The planar anisotropy is defined by
= 0.28 and the average normal
anisotropy ratio is given by ̅
. This value is close to 1 which
means that the normal strain anisotropy is weak. As the chosen validation test is close to biaxial expansion, in which normal anisotropy governs fracture, despite a slight planar anisotropy of the material, the anisotropy of the material was not considered in a first step. Hence, the von Mises yield criterion was used.
150 mm 12.5 mm d
20 mm
dx=0, dz=0
-d dx=0, dz=0
Fracture area
Fig. 7 Finite element model of tensile test
4.3. Mesh sensitivity To investigate the mesh sensitivity, four mesh sizes were examined (2, 1, 0.8, and 0.5 mm) for the central part and a size of 2 mm for the rest of the specimen. The comparison between experimental and numerical load-displacement curves shows that the mesh size of 0.8 mm is the most suitable, as illustrated in Fig. 8. Hence, a size of 0.8 mm of the central part was taken for the simulation of the tensile test up to rupture. Furthermore, the thickness of the specimen was divided into six elements. 12
14 12 10 8
12.8
Load (kN)
Load (kN)
13.0
6 4
12.6
12.4
Exp 2mm 1mm 0.8mm 0.5mm
12.2
12.0
2
8.0
8.5
9.0
9.5
10.0
Displacement (mm)
0 0
2
4
6
8
10
Displacement (mm)
Fig. 8 Experimental and numerical load-displacement curves using different mesh sizes (2, 1, 0.8, and 0.5 mm). 4.4. Calibration of ductile fracture criteria The results of the numerical simulation of the tensile test up to rupture were given in Fig. 9. As shown in Fig. 9(a), the numerical load-displacement curve presents a good correlation with the experimental one. It should also be noted that the onset of fracture was defined by a sudden drop in the load level reached for a critical extensometer displacement. Numerically, the same value has been obtained as in the experimental test (see the circle in Fig. 9(a)). The numerical strain distribution ϵyy in the tensile test just before fracture is presented in Fig. 9(b). Its maximum value is 0.78. On the other hand, the average of strain calculated in the critical zone (square area 3x3 mm2) is equal to 0.76, which is close to the experimental value (0.72) as represented in Fig. 9(c).
13
14 12
Fracture point
Load (kN)
10
a
8 6 4
Exp Num
2 0 0
2
4
6
8
10
Extensometer displacement (mm)
1.0
Experiment Simulation 0.8
0.6
yy
30°
0.4
0.2
0.0 0
2
4
6
8
10
12
Extensometer displacement (mm)
b
c
Fig. 9 Simulation results of tensile test up to rupture, for AISI 304L steel: (a) Comparison of load–extensometer displacement curves, (b) Numerical distribution of major tensile strain ϵyy. (c) Local longitudinal strain vs. extensometer displacement curves. The numerical longitudinal strain distribution just before fracture displays a cross shape whereas in the experiment, it shows a band shape (Fig. 2(b)). This result is similar to that published by Mishra [29] for the DP980 steel sheet. In addition, the numerical angle of the
14
localized band is at around 30° relative to the transverse axis (Fig .9(b)), which is very close to the experimental result. Hence, these results reveal that the finite element model provides sufficient accuracy for the simulation of the tensile test. Then, the validated numerical model was used to identify the critical value of the fracture criteria. The calibration of the fracture criteria was carried out by comparing the results of the numerical simulation with those measured experimentally. To evaluate the rupture, four macroscopic fracture criteria, which are presented in Eqs. (2 to 5), were implemented as a user subroutine USDFLD with the finite element code. As presented in Eq. (1), these criteria require only one parameter (
with i=1 to 4) to be calibrated, using
the numerical simulation of the tensile test up to rupture. Indeed, the damage parameter was calculated in each incremental time step of the numerical simulation until the fracture point (critical extensometer displacement) was reached. At this point, the damage parameter was defined as critical damage value, which corresponds with the fracture strain. At fracture point, the numerical damage distributions of the four ductile fracture criteria are shown in Fig. 10. The critical values of the damage parameter Oh,
for Brozzo,
for Ayada and
are equal to
for
for Rice and Tracey.
To verify the reliability of the calibrated fracture criteria, the Erichsen test was conducted; results are shown in the following section.
a
b b
15
c
d
b
b
Fig. 10 Numerical distribution of damage parameter Di in tensile tests: (a) Oh ( (b) Brozzo ( ); (c) Ayada ( ); (d) Rice and Tracey (
); ).
5. Numerical simulation of the Erichsen test The computational analysis of the Erichsen test was performed using the FE code Abaqus/Standard. For symmetry reasons, only one fourth of the blank was modeled as a deformable body, while the tools (punch, die, and blank holder) were considered as analytical rigid surfaces (Fig. 11(a)). The blank was meshed using the C3D8R element, which was the same as that selected for the numerical simulation of the tensile test. Fig. 11(b) shows the mesh size adopted for the FE simulation. In the center zone of the blank, fine meshe (0.8 mm) was considered, due to its contact with the top of the punch. Six layers of elements in the thickness are used.
a
16
b
Fig. 11 Erichsen test: (a) 3D finite element model; (b) Mesh size of the blank. The boundary conditions applied in numerical simulation were the symmetry along the x-axis and the z-axis. During simulation, the die was fixed, the punch was moved along the y-axis up to rupture of the sheet, and a blank holder force of 158 kN was applied. The friction coefficient, which characterizes the contact surfaces between the tools and the blank, was determined as 0.2 by fitting the experimental and numerical punch force-displacement curves. The same material parameters as for the simulation of the tensile test were considered. Fig. 12 shows a comparison between the load vs. punch displacement curves of the numerical simulation and experiments. The numerical results predict exactly the experimental one, except for the last portion when the displacement exceeds 20 mm. The gap is explained by the initiation and growth of the crack. Therefore, the good correlation of the simulation and experimental data proves the validation of the numerical model.
17
Exp Num R-Exp R-Oh R-Brozzo R-Ayada R-Rice-Tracey
120
100
Load (kN)
80
60
40
20
0 0
5
10
15
20
25
30
35
Punch displacement (mm)
Fig. 12 Comparison between experiments and numerical predictions of the Erichsen test (R: predicted onset of fracture)
6. Results and discussion 6.1. Application to the Erichsen test In order to verify the reliability of the damage parameters identified through the tensile test, the four fracture criteria were implemented via a user subroutine into the finite element code ABAQUS. The onset of fracture in the Erichsen test corresponds to the displacement when the critical damage value
, already calibrated from the tensile test, is reached (Fig. 12). The
value of the onset of the fracture is related to the selected ductile fracture criteria. The Oh fracture criterion reaches the highest punch displacement of 31.8 mm. By contrast, the Ayada fracture criterion reaches the lowest punch displacement of 23 mm. Similarly to the case of the onset of fracture, the Oh criterion reaches the maximum strain value at rupture of 0.45. However, the Ayada criterion reaches the minimum strain value at rupture of about 0.24. The numerical strain distribution of the Erichsen test at the onset of fracture is shown in Fig. 13(b), (c), (d) and (e). It is clear that for all the fracture criteria, the major strain is located along a circumference zone corresponding to the contact area with the punch radius, which is confirmed by the experimental distribution (Fig. 13 (a)). Fig 13 also indicates that the Oh fracture criterion reached the highest major strain value about 0.45 whereas Ayada criterion 18
reached lowest major strain value about 0.24. Table 3 illustrates the major strain values fracture in the Erichsen test matching the critical damage values
max
at
of the four fracture
criteria.
(a)
(b) Oh (LE max=0.45)
(c) Brozzo (LE max=0.28)
(d) Ayada (LE max=0.24)
(e) Rice-Tracey (LE max=0.3)
Fig. 13 (a) Experimental and (b), (c), (d) and (e) numerical strain distributions at rupture in the Erichsen test for different fracture criteria.
19
Table 3 Critical damage values test.
Fracture criterion
max
and major strain values
max
at fracture initiation during Erichsen
Oh
Brozzo
Ayada
Rice-Tracey
0.69
0.69
0.23
1.14
0.45
0.28
0.24
0.3
Fig. 14 displays the evolution of the damage parameters of the four fracture criteria vs. the punch displacement during the Erichsen test. These results show that the Oh criterion overestimates the critical displacement and strain at rupture in the Erichsen test by about 24% and 50%, respectively. The Ayada criterion underestimates the strain at rupture with an error equal to 20%. However, the Brozzo criterion slightly underestimates the strain at rupture by about 6%. A little overestimation of the critical displacement of about 4% is determined with the RiceTracey criterion. A comparison between each fracture criterion can also be achieved by plotting the absolute percentage error of the critical displacement and strain at rupture with reference to the experimental results (Fig. 15). Fig. 16 shows the numerical damage distributions at the rupture point. Results indicate that the maximum damage value of each criterion, except the Brozzo criterion, is situated along a circumference zone which is similar to the experimental result. However, for the Brozzo criterion, the maximum damage value is in the higher part of the dome. The little difference in maximum damage value for the Brozzo criterion is due to the underestimation of the strain at rupture.
20
1.2
Oh Brozzo Ayada Rice-Tracey
1.0
D
0.8
Experimental fracture displacement
0.6
0.4
0.2
0.0 0
5
10
15
20
25
30
Punch displacement (mm)
Fig. 14 Evolution of ductile damage parameters during the Erichsen test.
60
50
Absolute percentage (%)
50
Critical displacement Strain at rupture 40
30
24 20
20
10
6
6
1
0
0 1 Oh
2 Brozzo
4
3 Ayada
4 Rice-Tracey
Fig. 15 Comparison between the percentage errors of the different fracture criteria and the experimental results.
21
G
(a) Oh
(c) Ayada
(b) Brozzo
(d) Rice-Tracey
Fig. 16 Numerical damage distribution of the Erichsen test with the four ductile fracture criteria: (a) Oh criterion; (b) Brozzo criterion; (c) Ayada criterion; (d) Rice–Tracey criterion. 6.2. Stress triaxiality The variation in stress triaxiality in the tensile and Erichsen tests is presented in Fig. 17. It shows that stress triaxiality in the Erichsen test is higher than in the tension test and reaches a value of 0.66 (2/3), which indicates an equi-biaxial stress state, whereas in the tensile test, stress triaxiality is initially equal to 0.33 up to a plastic strain value of 0.57. Then, it increases and reaches a value of 0.45. Fig. 17 shows that the stress triaxiality values for the tension and Erichsen tests are almost equal at the fracture point [29], [30].
22
0.8
Stress Triaxiality
0.6
0.4
0.2
Tensile test Erichsen test 0.0 0.0
0.2
0.4
0.6
0.8
1.0
Plastic strain
Fig. 17 Stress triaxiality ratio in the tensile and Erichsen tests
7. Conclusion The present study aims to obtain a reliable prediction of the fracture strain of the AISI 304L steel sheets on the Erichsen test. Various fracture criteria (Oh, Brozzo, Ayada and RiceTracey), implemented in the finite element code Abaqus/Implicit via a user subroutine USDFLD, were investigated. To calibrate these criteria, an experimental-numerical approach was applied. In this respect, the numerical simulation of the tensile test up to rupture was modeled then compared to the experiment data, which were obtained by DIC system. Therefore, the numerical results gave a good description of the experimental ones and the critical parameters were determined. Then, the fracture criteria thus determined were introduced into the FE simulation of the Erichsen test. It is noted that a localized band of maximum damage distributions predicted by the finite element simulation of Erichsen specimen was observed along a circumferential area that appear in the similar zone in the experimental test. The onset of fracture and fracture location of the blank were well predicted during the Erichsen test.
23
Comparison between FE and experimental results indicate that both the Rice-Tracey criterion (errors lower than 4%) and the Brozzo criterion (errors lower than 6 %) can be used to predict fracture reasonably well in expansion stress states.
conflict of interest None author statements
We declare that this artcile is our research results
References
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Ductile fracture of AISI 304L stainless steel sheet in stretching Khadija. Ben Othmen , Nader. Haddar , Anthony Jegat , Pierre-Yves. Manach , Khaled. Elleuch PII: DOI: Reference:
S0020-7403(19)32869-3 https://doi.org/10.1016/j.ijmecsci.2019.105404 MS 105404
To appear in:
International Journal of Mechanical Sciences
Received date: Revised date: Accepted date:
4 August 2019 18 November 2019 20 December 2019
Please cite this article as: Khadija. Ben Othmen , Nader. Haddar , Anthony Jegat , Pierre-Yves. Manach , Khaled. Elleuch , Ductile fracture of AISI 304L stainless steel sheet in stretching, International Journal of Mechanical Sciences (2019), doi: https://doi.org/10.1016/j.ijmecsci.2019.105404
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Highlights
The prediction of the onset of rupture of austenitic stainless steel during a forming process has been carried out.
The fracture behavior of AISI 304L stainless steel, tensile and Erichsen tests were performed.
A 3D numerical simulation via Abaqus/Standard was used.
Four ductile fracture criteria are used to predict rupture during the Erichsen test.
1
Ductile fracture of AISI 304L stainless steel sheet in stretching Khadija. Ben Othmen1, Nader. Haddar1, Anthony. Jegat2, Pierre-Yves. Manach2, Khaled. Elleuch1 1) LGME, ENIS, Route Soukra Km 3.5 B.P. 1173 3038 Sfax, Université de Sfax, Tunisie,
2) Univ. Bretagne Sud, UMR CNRS 6027, IRDL, F-56100 Lorient, France [email protected], [email protected], [email protected], [email protected], [email protected]
Abstract The present study focuses upon the numerical investigation of fracture initiation of austenitic stainless steel, AISI 304L, during a forming process leading to an expansion strain path. Hence, a combined experimental and numerical approach was tested. Monotonic tensile tests up to fracture were carried out. Load-extensometer displacement curves and a major strain distribution were measured. Numerical simulations of the tensile test up to rupture were carried out to calibrate damage parameters of several macroscopic fracture criteria. Numerical simulations of the Erichsen test were also carried out to verify the validity of the determined parameters of the ductile fracture criteria. The numerical predictions of the strain field, the onset of fracture, and the fracture location were compared with the experimental results. The simulation results show that the Rice-Tracey or Brozzo fracture criteria are reliable predictors of the onset of fracture of AISI 304L in the Erichsen test. More precisely, the Rice-Tracey criterion is the most effective to predict the ductile fracture in this case.
Keywords: Onset of fracture; Austenitic stainless steel; Tensile test; Ductile fracture criteria; Erichsen test.
2
1. Introduction In recent years, formability has been viewed as one of the most important research topics, due to the development of sheet metal forming process modeling. Formability depends on several factors such as the material properties or the process parameters [1]. The manufacture of sheet metal into a required shape without defects, such as necking and fracture, is one of the main objectives for an accurate design and assessment. In other words, the occurrence of fracture or necking is generally regarded as one of the forming limitations of the sheet part. A ductile fracture is the most common failure problem encountered in sheets metal forming process ( [2], [3]) in particular, with austenitic stainless steel sheets. Austenitic stainless steel, particularly AISI 304L, is widely employed in several industrial applications such as the food industry, electronics, automobiles, and biomedicine, owing to its high strength, excellent corrosion resistance, and good formability ( [4], [5], [6]). In spite of their widespread use, few studies have focused on the prediction of the fracture of austenitic stainless steel sheets during forming processes. To optimize an efficient finite element analysis of manufacturing operations among those forming processes, reliable constitutive behavior and fracture properties of the sheet metal are required. The constitutive model of the AISI 304L sheet metal used in this study was identified in the previous work of Khadija et al. [7]. Hence, the accurate prediction of the fracture response is the main objective of the current work. In this sense, various ductile fracture criteria have been proposed in the literature and implemented in the finite element codes to predict the onset of fracture in sheet metal forming processes ( [8], [9], [10], [11]). These criteria may be classified into coupled and uncoupled categories [12]. The coupled category associates the fracture criterion into the constitutive equation of the material whereas for the uncoupled category, the damage evolution is independent of the constitutive law of the material. Despite their advantage to describe the physical mechanism of ductile fracture, coupled criteria are difficult to use to calibrate constitutive and damage models, which makes their industrial applications limited. Uncoupled criteria are preferred due to their simplicity and the few material constants to be experimentally calibrated ( [10], [13]). These uncoupled ductile fracture criteria have been widely employed to predict the fracture that occurred during forming processes ( [8], [9], [14], [15]). Many researchers have
3
suggested criteria that involve the history of stress and strain, reflecting the realistic conditions in sheet metal forming ( [16], [17], [18], [19] and [20]). Among these criteria, Freudenthal [21] argued that a critical fracture strain may be estimated by the total plastic deformation work. Cockcroft-Latham [16] developed a criterion that depends on the maximum principal stress (
. This criterion has been widely used, due to
its simplicity [22]. Oh et al. [17] suggested a normalized form of the Cockcroft-Latham criterion by the equivalent Mises stress ̅ to give an accurate prediction of the fracture strain. Brozzo et al. [18] empirically extended the Cockcroft-Latham criterion to introduce the effect of hydrostatic stress
for predicting the formability limit of sheet metal. This criterion was
successfully applied to predict the damage of bulk metals [23]. Later, Ayada et al. [24] proposed a criterion whereby the hydrostatic stress was normalized by the equivalent Mises stress. This ratio was called stress triaxiality µ
̅
. Rice and Tracey [25] adopted a single
spherical void growth criterion. They also introduced stress triaxiality through an exponential function. In this criterion, hydrostatic stress depends strongly on the equivalent plastic strain to fracture. Therefore, from an engineering and practical point-of-view, and in order to check a variety of stress tensor invariants, four of the most widely used ductile fracture criteria of Oh [17], Brozzo [18], Ayada [24], and Rice and Tracey [25] are selected. The main advantage of this choice is that only one material parameter is required to be identified. The previous criteria are calibrated with a hybrid experimental-numerical technique [8], using tensile test results. Subsequently, these criteria are used to predict fracture during the Erichsen test, which has been carefully instrumented in order to capture several strain effects up to fracture.
2. Ductile fracture criteria In uncoupled ductile fracture criteria, the fracture of the work piece occurs when the maximum damage value reaches a critical value
. These criteria with an integral form in
Eq. (1) express a function of the equivalent plastic strain ̅ . ∫ Where
̅
̅
(1)
is a function of the process parameter, ̅ is the equivalent plastic strain at fracture,
̅ is the equivalent plastic strain, and
is the damage value.
4
While there are numerous uncoupled criteria, only four criteria were chosen for this study. The selected criteria are shown as follows: Oh et al. [17] :
∫
̅
̅
̅
(2)
Brozzo et al. [18]:
∫
̅
̅
(3)
Ayada et al. [24]:
∫
̅
̅
̅
(4)
Rice-Tracey [25]: ∫
̅ ̅
̅
(5)
For all these criteria, the onset of fracture is attained once the damage value ( ) reaches the critical value
.
3. Material and experimental tests 3.1. Material test The material investigated is AISI 304L. It was received as cold-rolled sheets with a thickness of 1 mm. The chemical compositions (wt %) are as follows: 20.31% Cr, 7.87% Ni, 0.029% C, 0.384% Si, 1.85% Mn, 0.04% Nb, 0.196% Mo, and 68.5% Fe. The microstructure of the test samples, in the initial state, is shown in Fig. 1.
TD TD DT DL RD RD
10 µm
Fig. 1 Micrographic observations of AISI 304L at the initial state by optical microscopy. 5
3.2. Tensile tests Monotonic tensile tests up to fracture were conducted in the rolling direction using rectangular samples with a gauge area size of 150x20 mm2. Tests were carried out using an INSTRON 8803 machine with a maximum load capacity of 100 kN at room temperature with a strain rate ε˙= 1.2 10−3s-1. Both longitudinal
and transverse
logarithmic strains were
measured during the experiments with the Digital Image Correlation (DIC) system (Aramis) [26]. The displacement between two equidistant points, localized in the center of the sample and initially equal to 12.5mm, was measured. According to the recorded load-displacement curve up to rupture (Fig. 2 (a)), the necking point and the onset of fracture were used to extrapolate the hardening law in order to calibrate the fracture criteria. Using DIC, the image was captured every 1s. The longitudinal strain distribution ϵyy coincided with the moment just before the rupture, i.e. at tR-1, as illustrated in Fig. 2. (b). In addition, the average maximum local strain was calculated upon a square area of 3mm side length (Fig. 2. (b)). In this study, anisotropy of the material was not considered. However, the mechanical behavior was examined at 0°, 45°, and 90° to the rolling directions, as can be found in [7]. The tensile properties of the AISI 304L sheets are summarized in Table 1. The uniaxial tensile tests were carried out three times, and a representative curve was adopted.
14 12
Necking point Onset of fracture
Load (kN)
10 8 6 4 2 0 0
2
4
6
7.22
8
10
Extensometer displacement (mm)
(a)
6
28°
After rupture
Just before rupture
(b) Fig. 2 Experimental tensile test results measured with Aramis: (a) Load-extensometer displacement curve; (b) Longitudinal strain distribution ϵyy measured by DIC just before fracture and after fracture in tension (Maximum local strain calculated in the square area). Table 1 Tensile properties of the AISI 304L Angle with
Young's modulus
Tensile strength
Yield stress
Elongation
Anisotropic
RD (°)
E (GPa)
Rm (MPa)
Re (MPa)
A%
0
203.08
670
278
75
0.87
45
203
632
270
56
1.16
90
203.05
647
280
65
0.82
coefficient
3.3. Erichsen tests Erichsen tests were carried out on a Zwick-Roell BUP 200 sheet metal testing machine. The diameter of the blank was D1=120 mm. The diameter of the hemispherical punch was D2=60 mm. The die opening diameter was D3=65mm, and its radius was R=5 mm. The rolling direction (RD) and transverse direction (TD) were set along the x-axis and the y-axis, respectively. During tests, the blank was clamped between the blank holder and the die by applying a Blank Holder Force (BHF) of 158 KN.
7
These tests were carried out without lubrication and at a constant punch speed of 10 mm/min. The load vs. punch displacement curves were recorded during the test. As shown in Fig. 3 (a), results present good reproducibility. In order to follow the evolution of the strain field, local strain was measured by a non-contact ARAMIS DIC-3D system. Fig. 3(b) shows the major strain distribution for six different punch displacements, marked in Fig. 3(a). A localized band of major strain distribution, which coincides with the contact zone between the blank and the punch, occurred and then evolved radially. Similarly, the intensity of the strain field increased until the appearance of the fracture at the last point and the load reaches 88.8 KN for the punch displacement of 25 mm.
100
a 80
Punch force [KN]
6
Test 1 Test 2
5 4
60
3
40
20
2 1
0
0
5
10
15
20
25
30
Punch displacement [mm]
b
1
2 d=0 mm
3 d= 8 mm
d= 15 mm
8
4
5 d= 19 mm
6 d= 23 mm
d= 25 mm
Fig. 3 Experimental results of the Erichsen test: (a) Punch force-displacement curves, (b) Strain distribution for different tool displacements with DIC system measurements. 3.4. Fracture mode analysis The localized necking band of the tensile specimen is oriented at around 28° to the transverse axis (Fig. 2 (b)). Furthermore, the region near the fracture zone was analyzed by a scanning electron microscope (SEM). Regarding the tensile specimen, the fracture surface shows a high proportion of dimples, revealing a ductile fracture (Fig. 4). This failure mode is initiated by the formation of voids around non-metallic inclusions and/or second-phase particles in a metal matrix, being sustained to plastic deformation under an external load [27]. In a second step, growth and coalescence of the dimples lead to fracture of the specimen. Similar occurrences were observed on the fractured surface after the Erichsen tests (Fig. 5). Fig 5 (c) exhibits a fracture surface, with a large variation in size and shape of dimples, indicating clearly a ductile fracture facies. The fracture occurs after coalescence of voids leading to the appearance of macroscopic cracks which propagation conducts to the ligament separation.
(a)
(b)
Fig. 4 SEM observation of the fracture surface of the tensile test specimen: (a) General view; (b) Detail of a square area in (a) (
rounded inclusions) 9
a
b
c
Microvoids
Crack propagation
pprpppropagation
Fig. 5 (a) Cup drawn on the Erichsen test with fractured surface, SEM images of ductile fracture (b) fractured surface, (c) detail of the square area in (b). 4. Numerical calibration: Tensile test To calibrate constants of ductile fracture criteria, a tensile test up to rupture was simulated. Hence, the fracture calibration process is presented in this section. 4.1. Plastic hardening calibration To obtain an accurate FE simulation of the damaged material, a stress–strain relationship up to rupture was required. Then, this curve was divided into two portions at the necking point. For the first portion (before the necking) characterized by homogeneous deformations, the true stress-strain data were calculated from experimental loads and displacements (Fig. 6(a)). The stress-strain curve was expressed by the Voce hardening law [28], which can be defined through equation:
10
̅
)
(6)
is the yield stress at zero plastic strain, ̅ the equivalent plastic strain, while Q and
Where:
b represent the material parameters. In contrast, the second portion (beyond necking) was determined by numerous methods. Hence, in this study, a linear extrapolation of the hardening law was adopted, as shown in Fig. 6(b). The material parameters used as input data in the numerical simulation are summarized in Table 2. The necking point is determined at
=0.45 and Cauchy stress of 1054.3
MPa. 1200 1200
a 1000
Cauchy stress (MPa)
800
Stress (MPa)
b
1000
600
400
200
Nominal stress-strain True stress-strain
800
600
400
200
0
Extrapolated data Experiment data
0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Strain
Strain
Fig. 6 Tensile test: (a) Nominal and true stress-strain curves; (b) Experimental and extrapolated data.
Table 2 Material parameters of the Voce hardening law for the AISI 304L with the extrapolated portion after necking.
̅
̅
0.451
(MPa)
(MPa)
290
1170
b 2.3
̅ (MPa)
0.451
0.502
0.53
0.555
0.564
0.576
1054.3
1103.6
1129
1151
1158.8
1169
11
4.2. Finite element model The numerical simulation of the tensile test up to rupture was carried out using Abaqus/implicit FE code. Then, the mesh was structured and 3D 8-node linear brick elements with reduced integration (C3D8R) and with hourglass control were used. Fig. 7 illustrates the FE model with the boundary conditions. The specimens were rectangular with dimensions of 150 mm×20 mm and 1 mm thickness. Moreover, displacements of d and (-d) were applied at two extremities of the specimen along the y-direction. To promote the rupture at the center of the specimen, a slight imperfection was added to the edge of the central part. The local displacement was measured by a virtual extensometer between two points at the center of the specimen at a distance of 12.5 mm. The planar anisotropy is defined by
= 0.28 and the average normal
anisotropy ratio is given by ̅
. This value is close to 1 which
means that the normal strain anisotropy is weak. As the chosen validation test is close to biaxial expansion, in which normal anisotropy governs fracture, despite a slight planar anisotropy of the material, the anisotropy of the material was not considered in a first step. Hence, the von Mises yield criterion was used.
150 mm 12.5 mm d
20 mm
dx=0, dz=0
-d dx=0, dz=0
Fracture area
Fig. 7 Finite element model of tensile test
4.3. Mesh sensitivity To investigate the mesh sensitivity, four mesh sizes were examined (2, 1, 0.8, and 0.5 mm) for the central part and a size of 2 mm for the rest of the specimen. The comparison between experimental and numerical load-displacement curves shows that the mesh size of 0.8 mm is the most suitable, as illustrated in Fig. 8. Hence, a size of 0.8 mm of the central part was taken for the simulation of the tensile test up to rupture. Furthermore, the thickness of the specimen was divided into six elements. 12
14 12 10 8
12.8
Load (kN)
Load (kN)
13.0
6 4
12.6
12.4
Exp 2mm 1mm 0.8mm 0.5mm
12.2
12.0
2
8.0
8.5
9.0
9.5
10.0
Displacement (mm)
0 0
2
4
6
8
10
Displacement (mm)
Fig. 8 Experimental and numerical load-displacement curves using different mesh sizes (2, 1, 0.8, and 0.5 mm). 4.4. Calibration of ductile fracture criteria The results of the numerical simulation of the tensile test up to rupture were given in Fig. 9. As shown in Fig. 9(a), the numerical load-displacement curve presents a good correlation with the experimental one. It should also be noted that the onset of fracture was defined by a sudden drop in the load level reached for a critical extensometer displacement. Numerically, the same value has been obtained as in the experimental test (see the circle in Fig. 9(a)). The numerical strain distribution ϵyy in the tensile test just before fracture is presented in Fig. 9(b). Its maximum value is 0.78. On the other hand, the average of strain calculated in the critical zone (square area 3x3 mm2) is equal to 0.76, which is close to the experimental value (0.72) as represented in Fig. 9(c).
13
14 12
Fracture point
Load (kN)
10
a
8 6 4
Exp Num
2 0 0
2
4
6
8
10
Extensometer displacement (mm)
1.0
Experiment Simulation 0.8
0.6
yy
30°
0.4
0.2
0.0 0
2
4
6
8
10
12
Extensometer displacement (mm)
b
c
Fig. 9 Simulation results of tensile test up to rupture, for AISI 304L steel: (a) Comparison of load–extensometer displacement curves, (b) Numerical distribution of major tensile strain ϵyy. (c) Local longitudinal strain vs. extensometer displacement curves. The numerical longitudinal strain distribution just before fracture displays a cross shape whereas in the experiment, it shows a band shape (Fig. 2(b)). This result is similar to that published by Mishra [29] for the DP980 steel sheet. In addition, the numerical angle of the
14
localized band is at around 30° relative to the transverse axis (Fig .9(b)), which is very close to the experimental result. Hence, these results reveal that the finite element model provides sufficient accuracy for the simulation of the tensile test. Then, the validated numerical model was used to identify the critical value of the fracture criteria. The calibration of the fracture criteria was carried out by comparing the results of the numerical simulation with those measured experimentally. To evaluate the rupture, four macroscopic fracture criteria, which are presented in Eqs. (2 to 5), were implemented as a user subroutine USDFLD with the finite element code. As presented in Eq. (1), these criteria require only one parameter (
with i=1 to 4) to be calibrated, using
the numerical simulation of the tensile test up to rupture. Indeed, the damage parameter was calculated in each incremental time step of the numerical simulation until the fracture point (critical extensometer displacement) was reached. At this point, the damage parameter was defined as critical damage value, which corresponds with the fracture strain. At fracture point, the numerical damage distributions of the four ductile fracture criteria are shown in Fig. 10. The critical values of the damage parameter Oh,
for Brozzo,
for Ayada and
are equal to
for
for Rice and Tracey.
To verify the reliability of the calibrated fracture criteria, the Erichsen test was conducted; results are shown in the following section.
a
b b
15
c
d
b
b
Fig. 10 Numerical distribution of damage parameter Di in tensile tests: (a) Oh ( (b) Brozzo ( ); (c) Ayada ( ); (d) Rice and Tracey (
); ).
5. Numerical simulation of the Erichsen test The computational analysis of the Erichsen test was performed using the FE code Abaqus/Standard. For symmetry reasons, only one fourth of the blank was modeled as a deformable body, while the tools (punch, die, and blank holder) were considered as analytical rigid surfaces (Fig. 11(a)). The blank was meshed using the C3D8R element, which was the same as that selected for the numerical simulation of the tensile test. Fig. 11(b) shows the mesh size adopted for the FE simulation. In the center zone of the blank, fine meshe (0.8 mm) was considered, due to its contact with the top of the punch. Six layers of elements in the thickness are used.
a
16
b
Fig. 11 Erichsen test: (a) 3D finite element model; (b) Mesh size of the blank. The boundary conditions applied in numerical simulation were the symmetry along the x-axis and the z-axis. During simulation, the die was fixed, the punch was moved along the y-axis up to rupture of the sheet, and a blank holder force of 158 kN was applied. The friction coefficient, which characterizes the contact surfaces between the tools and the blank, was determined as 0.2 by fitting the experimental and numerical punch force-displacement curves. The same material parameters as for the simulation of the tensile test were considered. Fig. 12 shows a comparison between the load vs. punch displacement curves of the numerical simulation and experiments. The numerical results predict exactly the experimental one, except for the last portion when the displacement exceeds 20 mm. The gap is explained by the initiation and growth of the crack. Therefore, the good correlation of the simulation and experimental data proves the validation of the numerical model.
17
Exp Num R-Exp R-Oh R-Brozzo R-Ayada R-Rice-Tracey
120
100
Load (kN)
80
60
40
20
0 0
5
10
15
20
25
30
35
Punch displacement (mm)
Fig. 12 Comparison between experiments and numerical predictions of the Erichsen test (R: predicted onset of fracture)
6. Results and discussion 6.1. Application to the Erichsen test In order to verify the reliability of the damage parameters identified through the tensile test, the four fracture criteria were implemented via a user subroutine into the finite element code ABAQUS. The onset of fracture in the Erichsen test corresponds to the displacement when the critical damage value
, already calibrated from the tensile test, is reached (Fig. 12). The
value of the onset of the fracture is related to the selected ductile fracture criteria. The Oh fracture criterion reaches the highest punch displacement of 31.8 mm. By contrast, the Ayada fracture criterion reaches the lowest punch displacement of 23 mm. Similarly to the case of the onset of fracture, the Oh criterion reaches the maximum strain value at rupture of 0.45. However, the Ayada criterion reaches the minimum strain value at rupture of about 0.24. The numerical strain distribution of the Erichsen test at the onset of fracture is shown in Fig. 13(b), (c), (d) and (e). It is clear that for all the fracture criteria, the major strain is located along a circumference zone corresponding to the contact area with the punch radius, which is confirmed by the experimental distribution (Fig. 13 (a)). Fig 13 also indicates that the Oh fracture criterion reached the highest major strain value about 0.45 whereas Ayada criterion 18
reached lowest major strain value about 0.24. Table 3 illustrates the major strain values fracture in the Erichsen test matching the critical damage values
max
at
of the four fracture
criteria.
(a)
(b) Oh (LE max=0.45)
(c) Brozzo (LE max=0.28)
(d) Ayada (LE max=0.24)
(e) Rice-Tracey (LE max=0.3)
Fig. 13 (a) Experimental and (b), (c), (d) and (e) numerical strain distributions at rupture in the Erichsen test for different fracture criteria.
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Table 3 Critical damage values test.
Fracture criterion
max
and major strain values
max
at fracture initiation during Erichsen
Oh
Brozzo
Ayada
Rice-Tracey
0.69
0.69
0.23
1.14
0.45
0.28
0.24
0.3
Fig. 14 displays the evolution of the damage parameters of the four fracture criteria vs. the punch displacement during the Erichsen test. These results show that the Oh criterion overestimates the critical displacement and strain at rupture in the Erichsen test by about 24% and 50%, respectively. The Ayada criterion underestimates the strain at rupture with an error equal to 20%. However, the Brozzo criterion slightly underestimates the strain at rupture by about 6%. A little overestimation of the critical displacement of about 4% is determined with the RiceTracey criterion. A comparison between each fracture criterion can also be achieved by plotting the absolute percentage error of the critical displacement and strain at rupture with reference to the experimental results (Fig. 15). Fig. 16 shows the numerical damage distributions at the rupture point. Results indicate that the maximum damage value of each criterion, except the Brozzo criterion, is situated along a circumference zone which is similar to the experimental result. However, for the Brozzo criterion, the maximum damage value is in the higher part of the dome. The little difference in maximum damage value for the Brozzo criterion is due to the underestimation of the strain at rupture.
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1.2
Oh Brozzo Ayada Rice-Tracey
1.0
D
0.8
Experimental fracture displacement
0.6
0.4
0.2
0.0 0
5
10
15
20
25
30
Punch displacement (mm)
Fig. 14 Evolution of ductile damage parameters during the Erichsen test.
60
50
Absolute percentage (%)
50
Critical displacement Strain at rupture 40
30
24 20
20
10
6
6
1
0
0 1 Oh
2 Brozzo
4
3 Ayada
4 Rice-Tracey
Fig. 15 Comparison between the percentage errors of the different fracture criteria and the experimental results.
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G
(a) Oh
(c) Ayada
(b) Brozzo
(d) Rice-Tracey
Fig. 16 Numerical damage distribution of the Erichsen test with the four ductile fracture criteria: (a) Oh criterion; (b) Brozzo criterion; (c) Ayada criterion; (d) Rice–Tracey criterion. 6.2. Stress triaxiality The variation in stress triaxiality in the tensile and Erichsen tests is presented in Fig. 17. It shows that stress triaxiality in the Erichsen test is higher than in the tension test and reaches a value of 0.66 (2/3), which indicates an equi-biaxial stress state, whereas in the tensile test, stress triaxiality is initially equal to 0.33 up to a plastic strain value of 0.57. Then, it increases and reaches a value of 0.45. Fig. 17 shows that the stress triaxiality values for the tension and Erichsen tests are almost equal at the fracture point [29], [30].
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0.8
Stress Triaxiality
0.6
0.4
0.2
Tensile test Erichsen test 0.0 0.0
0.2
0.4
0.6
0.8
1.0
Plastic strain
Fig. 17 Stress triaxiality ratio in the tensile and Erichsen tests
7. Conclusion The present study aims to obtain a reliable prediction of the fracture strain of the AISI 304L steel sheets on the Erichsen test. Various fracture criteria (Oh, Brozzo, Ayada and RiceTracey), implemented in the finite element code Abaqus/Implicit via a user subroutine USDFLD, were investigated. To calibrate these criteria, an experimental-numerical approach was applied. In this respect, the numerical simulation of the tensile test up to rupture was modeled then compared to the experiment data, which were obtained by DIC system. Therefore, the numerical results gave a good description of the experimental ones and the critical parameters were determined. Then, the fracture criteria thus determined were introduced into the FE simulation of the Erichsen test. It is noted that a localized band of maximum damage distributions predicted by the finite element simulation of Erichsen specimen was observed along a circumferential area that appear in the similar zone in the experimental test. The onset of fracture and fracture location of the blank were well predicted during the Erichsen test.
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Comparison between FE and experimental results indicate that both the Rice-Tracey criterion (errors lower than 4%) and the Brozzo criterion (errors lower than 6 %) can be used to predict fracture reasonably well in expansion stress states.
conflict of interest None author statements
We declare that this artcile is our research results
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