1)†
Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology Science town, Daejeon 305-701, Korea
2)POSCO Technical Research Laboratories, 699, Cumho-dong, Gwangyang-si, Jeonnam, 545-090, Korea
Abstract
A hole expansion process is an important process in producing a hub-hole in a wheel disc of a vehicle. In this process, the main parameter is the formability of a material, which is expressed as the hole expansion ratio. As the hole is expanded during the process, a crack tends to occur in the upper edge of a hole. Since prediction of the forming limit by hole expansion experiments needs tremendous time and effort, an appropriate fracture criterion has to be developed for finite element analysis to define the forming limit of materials.
This paper newly proposes a modified ductile fracture criterion to consider the deformation characteristics of a material accurately in a hole expansion process. In order to verify the validity of the proposed criterion, the hole expansion process of a hub-hole is studied by finite element analysis with ABAQUS/standard considering several ductile fracture criteria. The fracture mode and hole expanding ratio are compared with respect to the various facture criteria. While existing criteria do not predict the fracture mode or hole expansion ratio adequately and show deviation from experimental results of hole expansion, the newly proposed fracture criterion predicts the fracture mode and the hole expanding ratio accurately.
Keywords: Finite element analysis, Ductile fracture criterion, Hole expanding ratio, Hub-hole expanding process
1. Introduction
A hub-hole in a wheel disc, which is manufactured by hole expanding process, affects the fatigue ability and vehicle safety critically. The hub-hole is produced by a punching process of a plate and an expanding process of the hole. Defects of the hole during these processes decrease the capacity and life time of the wheel abruptly. Therefore, the determination of the forming limit of a hole and the prediction of the material fracture in the hole expanding process are important to improve the capacity of the wheel and enable the safety design of the vehicle. Because the prediction of the forming limit of the material in the hole expanding experiment is varied by various forming condition of the hole and material property, the experiment approach can be hardly achieved from a view-point of time and cost. The fracture prediction using finite element method is an easy and efficient way to apply various ductile fracture criteria and determine the forming condition as well as the material.
General methods to predict the material fracture in the process of forming analysis make use of the fracture strain of the specimen obtained by the simple tension test, the limit strain based on plastic instability theory [1], and a ductile fracture criterion [2]. The existing ductile fracture criteria show higher accuracy to predict the material
fracture in general forming process[3] than other methods. But the hub-hole expanding process is different from general forming process because the side face of the hole has no constraints and the fracture propagates through the side face of a thick plate during the hole expanding process. Therefore, the existing ductile fracture criteria of the hub-hole expanding process need to be carefully examined and a more accurate ductile fracture criterion should be suggested through the comparison with experimental results.
2. Hub-hole expanding experiment and finite element analysis
2.1 Hub-hole expanding experiment
The die and specimens for hub-hole expanding experiments were prepared as shown in fig. 1. The material used in the experiment is SAPH440 which is hot-rolled high strength steel. Eight specimens were chosen from each of three coils. Hole expanding experiments were carried out using the specimens. The hole expanding ratio(HER) was then calculated by equation (1) which represents the ratio of the initial hole diameter to variation of the hole diameter. Average HER of 6 specimens is demonstrated in Tab. 1 eliminating two specimens with the maximum and minimum HER.
= − ×100 (%)
initial initial final
D D
HER D (1)
Tab. 1Results of HER test
Fig. 1 Specimen and tool of hole expanding experiment Fig. 2Finite element model of a specimen 2.2 Analysis condition for finite element method
Finite element model of the specimen for finite element analysis is shown in fig. 2. The boundary of the blank is fixed to prevent the inflow of a material from the blank holder in the hole expanding analysis and one eighth of the domain is modeled for the sake of symmetry in order to simulate it efficiently. The mesh system consists of three regions of rough meshes, moderate meshes and fine meshes around the hole in order to accurately evaluate the stress state around the hole. Cracks actually occur in the side face of the hub-hole due to the punching process to make the
Coil 1 137.0 %
Coil 2 204.7 %
Coil 3 162.1 %
Average 167.9 %
hole in the hub-hole manufacturing process. It is, however, assumed that a specimen does not have the initial cracks since a specimen is made by a machining process. ABAQUS/Standard is utilized to simulate the hole expanding process with the flow stress equation of the material as shown in equation (2).
σ =832.85(0.0078+εp)0.182 MPa (2).
2.3 Application of ductile fracture criteria in finite element analysis
Ductile fracture criteria suggested by Oyane[4], Brozzo[5], Cockcroft[6] and Rice[7] are applied to ABAQUS/Standard(UMAT) to predict the material fracture in the hole expanding process and the analysis process.
The fracture criteria are applied to calculate in each incremental time step. These criteria are expressed in equation (3)~(6).
∫
+= f mean
Oyane C d
I C
ε ε
σ σ
0 1
2
)
1 ( (3) =
∫
f −mean
Brozzo d
I C
ε ε
σ σ
σ
0 max
max 3
) 3(
2
1 (4)
ICockcroft=C
∫
0εfσσ dε max 41 (5) IRice=C
∫
0εfeCσmeanσ dε 5 61 (6)
In these equation, σ , σm, σmax denote the effective stress, the mean stress and the maximum principal stress respectively and dε , εf represent the effective strain increment and the fracture strain. Material constants, C1~C6
are determined by the result based on the simple tension test and the plane strain test. Tab. 2 shows the result of the simple tension test and the plane strain test. From this result, the material constants are calculated as shown in Tab. 3.
The material fracture is estimated when these criteria meet the fracture condition in which the value of I calculated by each ductile fracture criterion within an element reaches to the unit value.
Tab. 2Fracture strain in the tensile direction of the material Tab. 3 Coefficients of ductile fracture criteria
εf(uniaxial) 0.7156
εf (plane strain) 0.5686
Tab. 4 HER results for ductile fracture criteria
Experiment Oyane Brozzo Cockcroft Rice
167.9% 106.9% 135.6% 120.9% 106.9%
(a) (b) (c) (d) (f)
Fig. 3 Distribution of the field variable: (a) effective strain; (b) Oyane criterion; (c) Brozzo criterion; (d) Cockcroft criterion; (f) Rice criterion
3. Analysis result applying ductile fracture criteria
3.1 Result of finite element analysis
C1 C2 C3 C4 C5 C6
1.9438 2.3831 0.7955 0.7369 0.9418 0.9794
Cracks occur in the upper part of the side face of the hole during the hub-hole manufacturing process. Fig 3(a) shows the distribution of the effective strain in the deformed shape. The effective strain distribution represents the maximum value in the lower part of the side face of the hole due to concentration of deformation by the compressive force at the contact surface. For that reason, a fracture criterion by only the effective strain can not accurately predict the fracture mode which appears in the hub-hole expanding process. Fig. 3(b)~(f) show the distributions of the ductile fracture criteria calculated by finite element analysis. The distributions of criteria suggested by Oyane and Rice et al. show a difference with real fracture mode because the distribution represents the maximum value at the lower part of the side face of the hole. Brozzo and Cockcroft’s criteria predict the fracture mode correctly as they show the maximum value at the upper part of the side face of the hole. In this paper, the final fracture of the specimen is decided at the onset when the fracture condition is satisfied in all elements along the thickness direction in the side face of the hole. Tab. 4 shows HER results calculated at the final fracture. The comparison demonstrates that a new ductile fracture criterion is necessary in order to achieve a correct prediction since HER result predicted by each ductile fracture criterion has large deviation from the experimental result of 167.9%.
4. Modification of ductile fracture criterion
4.1 Consideration of ductility curve
The ductility curve represents the variation of the fracture strain. A general ductility curve is shown in fig. 4 as a function of triaxiality ratio(TR) defined by the ratio of the mean stress to the effective stress as shown in equation (7). If a positive mean stress is imposed to a ductile material, the fracture strain is diminished due to expansion of voids in the material and a negative one has the fracture strain to increase by shrinking voids in the material. Hence the variation of the fracture strain of the material by the effect of TR is a factor to be considered necessarily in a ductile fracture criterion.
TR = σ σmean
(7)
-1.0 -0.5 0.0 0.5 1.0
Tension commpression
Effective plastic strain at fracture
Triaxiality ratio
Ductility Curve
Fig. 4 Ductility curve of common ductile material Fig. 5 Strain path around the hole edge 4.2 Consideration of deformation characteristics in hole expanding process
Fig. 5 shows the strain path obtained from the side face of the hole in finite element analysis. The major strain is strain along the circumferential direction and the minor strain is one along the radial direction. As shown in the figure, the deformation state around the hole can be regarded as the simple tension state because the strain path follows an asymptote of the path of the simple tension state. This result concludes that the stress in the circumferential direction, that is, the maximum principal stress is an important factor in the hole expanding process.
4.3 Suggestion of a new ductile fracture criterion
A ductile fracture criterion is newly suggested as equation (8) in order to consider the fracture characteristics of a ductile material by the TR effect and the deformation characteristics at the side face of the hole together in the hole expanding process. This equation consists of f(σ) term to consider the deformation characteristics and
) / (σmean σ
w as a weighting function to consider the effect of TR. The former considers the damage by the maximum principal stress referring to a ductile fracture criterion suggested by Cockcroft as shown in equation (9). The latter considers the weighting function of the ductile fracture criteria suggested by Oyane, Rice and Brokken[8]. Fig. 6 shows each weight function applying the material constant. In particular Brokken’s weight function prevents the accumulated damage from diminishing when the weighting value is negative. In this paper, the weighting function suggested by Brokken is applied in a new ductile fracture criterion as shown in equation (10). In this equation, Cnew represents the material constant and has 1.7514 for SAPH440. The analysis result with the new ductile fracture criterion is shown in fig. 7 and HER result calculated by each ductile fracture criterion at the onset of the final fracture is shown in fig. 8. The new ductile fracture criterion not only describes the fracture mode correctly but also represents high accuracy in HER result compared to the measured one by the hole expanding experiment.
∫
= f mean
New f w d
I D
ε ε
σ σ σ
2 0
) ( )
1 ( (8)
σ σ) σmax
( =
f (9)
∫
+= f mean
New
New d
I C
ε ε
σ σ σ σ
0
max 1 3
1 where, <x>=0 for x≤0 (10) 0
for x x
x>= >
<
Fig. 6 Weight functions for ductile fracture criterion Fig. 7 Distribution of the new ductile fracture criterion
Fig. 8 HER results for ductile fracture criteria Fig. 9 Flow stress curves for CT440 and FB590 4.4 Application of new ductile fracture criterion for other materials
The new ductile fracture criterion needs to be applied to various materials in order to verify the accuracy and reliability of the criterion. Other two materials which is CT440 and FB590 are considered in finite element analysis with the new criterion. The flow stress curves of these materials are shown in fig. 9. The results of the simple
tension test for the materials are demonstrated in Tab. 5 and the material constants calculated from the experiment are shown in Tab. 6. Fig. 10 shows the analysis result of CT440 and FB590 for the hole expanding process respectively. The results demonstrate that the new ductile fracture criterion predicts the correct fracture mode of the specimen. HER results obtained by the experiment are compared with ones calculated by finite element analysis with the new ductile fracture criterion in fig 11. It is noted from these results that the new criterion proposed shows high accuracy for CT440 and FB590 as well as SAPH440.
Tab. 5Fracture strain in the tensile direction of the materials Tab. 6 coefficients of ductile fracture criterion
0 40 80 120 160 200 240
HER(%)
SAPH440 CT440 FB590 Experimental Result Analysis Result
Fig. 10 Distributions of new ductile fracture criterion Fig. 11 HER results for ductile fracture criterion
5. Conclusion
In this paper, the material fracture in the hole expanding process is predicted by finite element analysis with a newly proposed ductile fracture criterion and the result is compared with the experimental result. In the case of a hub-hole expanding process, the existing ductile fracture criteria show the large deviation from the experimental result. As a remedy, a new ductile fracture criterion is proposed in order to overcome this problem considering the fracture characteristics of the ductile material and the deformation characteristics of the hole expanding process.
Three materials of high strength steel SAPH440, CT440 and FB590 are used in the analysis to obtain the HER for the verification of the criterion. The proposed criterion demonstrates the sharper accuracy than those of the existing ductile fracture criteria compared to the experimental result.
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εf (uniaxial) 0.911 0.814
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