MATERIALS TRANSACTIONS
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Numerical Simulation of Cut Surface Shape and Residual Stress Distribution in Shearing Process
Masaru FukumuraYoshiaki ZaizenTakeshi OmuraKunihiro SendaYoshihiko Oda
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2019 Volume 60 Issue 9 Pages 1996-2002

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Abstract

Many analyses relating to shearing of sheet products have already been reported, and an increasing number of elasto-plastic analyses evaluating the residual stress inside sheets have also appeared in recent years. However, because only a few papers have presented measured residual stress distributions, very few in-depth comparisons between calculated stress levels and experimental results are available. In this study, two simulations of the shearing process were carried out, considering the ductile fracture conditions of thin steel sheets. The simulation results of the cut surface shape and residual stress distribution in the vicinity of the cut surface were in good agreement with the experimental results. As another example, after punching a round hole to about half the sheet thickness, i.e., extruding a cylindrical protrusion, a round interlock was formed by subsequently stacking another sheet on the previous sheets. When compared with the X-ray diffraction data, it was found that a simple four-layer interlock simulation model could predict the residual stress distributions with accuracy close to the measured results. The abovementioned simulations are expected to be promising tools benefitting the performance of steel products by reducing residual stress levels.

 

This Paper was Originally Published in Japanese in J. JSTP 59(688) (2018) 65–70.

1. Introduction

Because shearing of thin steel sheets is a widely used process and is carried out on a routine basis, many simulations of the shearing process have been reported.1,2) However, almost all of those simulations only considered the cut surface shape, such as the ratio of the sheared surface to the fractured one, and most were rigid-plastic analyses.3) Recently, elasto-plastic analyses have been performed using general-purpose analysis codes, and, for example, comparisons of several different ductile fracture criteria have been carried out.4)

Regarding the accuracy of these simulations, a large number of comparisons of the cut surface shape have been carried out,5) as the above mentioned, and evaluation of the plastic strain distribution near the cut surface6) has also been compared with the measured hardness.7) However, in residual stress predictions, first, a rigid-plastic analysis can not evaluate residual stress, and most previous elasto-plastic analyses only presented simulation results.5,79) Thus, only a few reports have compared calculated stress results with actual measured data.1013)

Accurate residual stress estimation is very important, as it is well known that residual stress after shearing affects fatigue strength10,14) and also influences the deterioration of electromagnetic characteristics.69,11,15) In previous reports, for example, Matsuno et al.12,13) compared the simulation and experimental results for the cross-sectional shape of a pierced hole and the residual stress around the hole. In those reports, the authors fitted ductile fracture parameters according to the occurrence of cracks from the punch and the die corner, but the burr shape on the blank bottom surface was different from the actual one. It also seems questionable that the residual stress level in that region, where the shape is quite different, would generally agree with the results of X-ray measurement. Moreover, the equivalent plastic strain in that region reaches as much as 3.12) Since there is a longstanding argument concerning X-ray measurement data for regions where plastic deformation occurs,16) certain considerations are required, such as the dependence on dislocation, the diffraction surface and so on.

In this report, as a start toward prediction of the fatigue or electromagnetic characteristics of steel sheets in the near future, first, verifications of shearing simulations were carried out in order to accurately estimate the sheared surface shape,7) plastic strain distribution and residual stress distribution around cut surfaces. For example, a piercing simulation was performed using a ductile facture criterion which could be easily identified based on simple and more accessible tensile test results of the material.

If the void volume ratio3) is used as a ductile fracture criterion, it is necessary to investigate void nucleation, growth and development in detail beforehand. Furthermore, as ductile fracture criteria usually depend on stress triaxiality,17) considerable time is required merely to obtain the ductile fracture parameters. On the other hand, mechanical properties inevitably fluctuate, even in stably-produced mass production materials. Moreover, in view of the fact that some material standards do not specify yield strength, etc., and a great amount of labor is required in order to identify the ductile fracture criteria accurately, it is considered difficult to accumulate and utilize this information as a general purpose material database. Therefore, one purpose of the present study is to use a simple ductile fracture criterion, which can be identified even when only tensile test results for each material are available, in simulations of the practical shearing processes, and to investigate the valid range for that criterion.

First, a simulation of the piercing process for a hot-rolled steel sheet having a thickness of 4 mm was performed, and the cross-sectional shape of the cut surface was compared, as this is a very typical problem in actual manufacturing processes. Next, considering electromagnetic deterioration in motor cores, shearing of cold-rolled steel sheets with thicknesses from 0.3 mm to 0.35 mm was analyzed, and the residual stress distribution after shearing was compared. Finally, a simulation of the formation of an interlock was performed. When assembling laminated cores consisting of multiple electrical steel sheets, interlocks are applied to the sheets by forming a protrusion of a certain height without fracture, and after arranging the positions of the protrusions, laminating additional sheared sheets one by one. In this study, the residual stress distribution after lamination was estimated using a simple four-layer model and compared with the measured results, and the practical availability of the analysis method was demonstrated.

2. Criterion of Ductile Fracture and Simulation of Tensile Test

In this report, ductile fracture simulations were carried out using eq. (1), which was proposed by Cockcroft.18) In eq. (1), when tensile tests results such as the stress-strain curve and total elongation are known, it is only necessary to identify the ductile fracture criterion parameter C. Because uniaxial tensile stress is prevalent during a tensile test, eq. (1) using the maximum principal stress and plastic strain limit until fracture occurrence is easily understood.   

\begin{equation} I = \int^{\varepsilon_{f}}\sigma_{1}d\bar{\varepsilon}/C\geq 1 \end{equation} (1)
Here, I: ductile fracture index, σ1: maximum principal stress, $\bar{\varepsilon }$: equivalent plastic strain, εf: critical $\bar{\varepsilon }$ at fracture limit and C: ductile fracture parameter. When I ≧ 1, it is assumed that fracture occurs. However, individual verification of applicability is required for cases in which the direction of maximum principal stress varies during the forming process, such that fracture occurs after multi-processes.

During identification of the ductile fracture limit parameter C, numerical simulations of the tensile test, as shown in Fig. 1, are performed repeatedly. Figure 1 shows the case of the JIS No. 5 test specimen. In a numerical simulation of a tensile test, after the index I in eq. (1) in an element reaches or exceeds 1, the relevant element is removed and the stress at that element is relieved, and this process is repeated until finally the specimen separates into two parts.

Fig. 1

Result of simulation of tensile test (JIS No. 5 test specimen).

In this report, an isotropic elasto-plastic material model was used, and a conservative estimation with smaller elongation was employed for an anisotropic material. It should be noted that accuracy can presumably be improved by considering the stress triaxiality dependency of the ductile fracture criterion.

In the simulation of the tensile test shown in Fig. 1, a piecewise linear approximation is applied to the definition of the stress-strain curve in order to evaluate the tensile load history, especially the highest peak value and the drop after the peak. In the higher strain range, where a tensile test can not predict the corresponding true stress due to localized deformation, an extrapolation from the data for the final two points is employed. During identification of the local elongation after the maximum load points, an iterative trial-and-error process is usually necessary, and in the present case, the work hardening behavior was defined in a range that would avoid a negative increment in the true stress-strain curve. After identifying the stress–strain curve, the value of the ductile fracture limit parameter C was finally calculated from the breaking point of the specimen in the tensile test.

Because iterative 3D analyses are required in the abovementioned identification, in this report, finite element meshes which could emulate local section area reduction were employed in the tensile test simulations from the practical viewpoint. For example, in case of a 4 mm thick blank sheet, hexahedral elements having a representative size of about 0.5 mm were used. Now, if the ductile fracture criterion depends on the strain level, it is well known that C displays gauge length dependency.19) It is desirable that meshes of the same size are applied to both the tensile test and shearing simulations. However, for example, in the case of the 0.3 mm thick steel sheet mentioned later, meshes with a minimum element length of 1.25 µm were employed in the shearing simulation, and it would not be practical to perform the tensile testing simulations with meshes of the same size. Even though the identification of ductile fracture criterion does not always use the same size meshes as the following shearing simulation, those are basically based on only the tensile test results. However, in cases where actual shearing results are available, it is practical to apply the measured information, such as the forming load or cut surface shape, to simulations.

3. Simulation of Piercing and Shearing

3.1 Piercing simulation of hot-rolled steel sheet

Round hole piercing of a hot-rolled steel sheet was simulated. Table 1 shows the mechanical properties of the 4 mm thick hot-rolled steel sheet investigated here. This material is S35C for machine structural use, where S35C means approximately 0.35 mass% carbon steel. Tensile test simulations (Fig. 2), as described in the previous section, were performed using this material. Because the displacement gauge was removed before completion of the tensile test to prevent damage of the gauge, the final displacement of the initial 50 mm gauge length is unknown. Therefore, in Fig. 2, the horizontal axis shows the displacement of the chuck of the JIS 5 test specimen, while the vertical axis indicates nominal stress. The ductile fracture criteria for S35C material were identified using the same indexes, and the true stress-strain curve (Fig. 2) and parameter C = 587 MPa were obtained. These material parameters were used in the shearing simulations.

Table 1 Typical mechanical properties of S35C hot-rolled steel sheet.
Fig. 2

Stress and chuck device displacement curve and stress-strain curve of 4 mm thick steel sheet in Table 1.

Next, a round hole having a diameter of 10 mm was pierced. In the experiment, the clearance condition was 20% of the blank thickness of 4 mm, and the punch diameter was 10 mm, so the die cavity diameter was set to 11.6 mm, and a round hole was punched under these conditions.

This experimental round hole piercing was simulated. The general-purpose finite element program LS-DYNA was used in this report, and the dynamic explicit method was applied to the simulation. Utilizing the symmetry of the process, a two-dimensional axisymmetric analysis was performed. The tools were assumed to be rigid, and both the punch and the die had right-angle corners. The location of the blank holder was fixed and the distance of 4 mm from the die was maintained, corresponding to the initial blank thickness. The Coulomb friction coefficient μ was calculated as μ = 0.1 based on an actual friction test of the material. During piercing, finite elements in which the ductile fracture index reached or exceeded the critical value were removed, and the forming process was continued. The remeshing technique was not used in this report. In meshing, in order to describe the sheared cut surface, the initial size of the mesh was set about 30 µm for the radial direction and about 60 µm for the thickness direction.

Figures 3 and 4 show the deformed blank and tools halfway through the piercing process, as well as the maximum principal stress distribution. Figure 3 shows the state just before blank separation; more than 600 MPa tensile stress exists in the band region from the punch corner to the die corner. In Fig. 4, the tensile stress level decreased immediately after separation of the blank, and it appears that the punch bottom surface is not in full contact with the blank sheet. In Fig. 3, a region where the positive maximum principal stress exists under the punch bottom surface up to about 2.4 mm in the radial direction, except near the axisymmetric axis. Although not clear in Fig. 3, it is thought that there might be a narrow gap between the punch and the blank. In such a case, the contact regions and the respective contact lengths would vary during the piercing process; this means that a static analysis is not appropriate for solving the transient load balance, and convergent solutions may not be obtained in some cases. It is noted that the dynamic explicit method is suitable for simulations of the abovementioned contact conditions. However, since stress fluctuation is unavoidable in a dynamic explicit analysis, in this report, the unloading procedure and removal of the tools were carried out before evaluating residual stress.

Fig. 3

Result of simulation of shearing process (σ1, maximum principal stress distribution).

Fig. 4

Result of simulation of cut surface immediately after separation (σ1, maximum principal stress distribution).

Figure 5(a) is a photo of the cut section of the blank without a round disc. Both the sheared surface and the fractured surface can be seen clearly in the cut section shape. In this case, the visible outline at the zooming rate in Fig. 5(a) seems to consist of two roughly straight lines. Figure 5(b) shows the simulated result superimposed on the photo of the experimental result, and compares the shapes of the two cut surfaces. Although there is a very slight difference, particularly in the fractured surface shape, the ratio of the shear surface length to the fracture surface length is in good agreement. Thus, it can be said that the results of the simulation using the ductile fracture criterion based on tensile test data are sufficiently accurate.

Fig. 5

(a) Photo of section shape of cut surface and (b) superposition of experimental and simulated cut surface shapes (t = 4 mm, ϕ = 10 mm, 20% clearance).

As a detailed comparison, the slope of the fractured surface region is slightly steeper than the experimental result, especially around the bottom corner of the cut surface shape. In Fig. 3, the compressive maximum principal stress seems to exist around the die corner, suggesting that stress triaxiality may affect the abovementioned difference in the cut surface shape.

3.2 Shearing simulation of cold-rolled steel sheets

The previous section examined the accuracy of the cut surface shape. The next issue is estimation of residual stress after shearing, which is especially important for motor core characteristics. Therefore, a simulation of thin sheet shearing was attempted. Cold-rolled steel sheets with a thickness of 0.3 mm and the material properties shown in Table 2 were used. The tensile test results and true stress-strain relationship are shown in Fig. 6. The friction coefficient μ was set to 0.3 based on the result of a sliding test.

Table 2 Mechanical properties of cold-rolled sheet.
Fig. 6

Stress-strain curve of 0.3 mm thick steel sheet in Table 2.

Next, according to Refs. 7) and 11), a steel sheet sample with a length of 180 mm and width of 30 mm was prepared and then cut into 6 pieces with the same width of 5 mm each (Fig. 7).7,11) The analysis simulated the first 5 mm width shearing process among these 5 cutting operations. The clearance was 15 µm, which was 5% of the blank thickness, and the holder force was 98 N, in accordance with the experimental condition. The other conditions were the same as in the previous section.

Fig. 7

Schematic views of shearing: (a) initial 30 mm wide steel sheet sample, (b) six 5 mm wide specimens cut from (a).

As a result, during shearing, the blank holder rose by a maximum of about 0.1 mm in the thickness direction. After the shearing analysis, all tools were removed, and a springback analysis was carried out to eliminate the stress fluctuation by a dynamic analysis.

Figure 8 shows the stress distribution obtained by the simulation and the outline of the blank cross-sectional shape in an experiment. The direction of the stress represented in the contour in Fig. 8 is perpendicular to this paper, which is equivalent to a magnetic flux direction in the motor core tooth region, so it is important for the magnetic characteristics of a motor. In Fig. 8, the slope that can be seen at the cut surface is inclined in the direction opposite the clearance of 15 µm.7) This is due to the rise of the holder, by about 0.1 mm, during the shearing process. In Fig. 8, there is a difference of about 18 µm at maximum between the experimental shape and the simulated result in the region from the thickness center to the burr. However, because the shear droop shape and the uneven blank bottom surface shape are in very good agreement, the difference in the cut surface shape is considered to be due to the blank rotation effect induced by the rise of the holder during shearing.

Fig. 8

Residual stress distribution after shearing.

Next, the stress perpendicular to the paper plane shown in Fig. 8 was measured at the synchrotron radiation facility at SPring-8,11) which has a high permeation capability. Synchrotron radiation permeated through the blank sheet, and the diffraction at thickness center was detected for evaluation of lattice strain while gradually changing the measured position along the arrow in Fig. 8. Figure 9 shows a comparison of the simulation results and the residual stress distribution converted from the measured strain assuming an elastic deformation field. In Fig. 9, the horizontal axis shows the distance from the cut surface. Regardless of the beam diameter (slit width: 30 × 110 µm) used in the measurement, the simulated results were plotted on the thickness center line without averaging. In the results converted from the measured strain, a white square mark, which means the dislocation effect is included, is plotted in case plastic deformation exists, referring to the simulated equivalent plastic strain at that position. As a result, the residual stress level and the extent of the affected region where residual stress exists could be roughly simulated, despite the difference in the plastic deformation region and the difference of 18 µm in the cut surface shape in Fig. 8. It was found that residual stress remains up to about 0.6 mm from the cut surface, which is double the blank thickness of 0.3 mm.

Fig. 9

Residual stress distribution.

4. Residual Stress Simulation around Interlock

4.1 Round interlock

In this section, the round interlocks shown in Fig. 10 were investigated. Figure 11 shows a typical example of the cross-sectional shape of an interlock. This kind of interlock is mainly applied to motor cores and is used to fix multiple layers of cut electrical steel sheets.20) Because the stress distribution that exists around interlocks, and particularly compressive stress, has a negative effect on the electromagnetic characteristics of a motor core,20) it is important to reduce the residual stress level and the extent where compressive stress exists. Figure 10 shows a schematic view of a ring sample for an interlock experiment. In this experiment, the number of layers is 20, and there are six interlocks in a ring sample.

Fig. 10

Schematic view of experimental specimen with round interlocks.

Fig. 11

Cross-sectional shape of round interlock (three dark lines indicate the surface of each sheet.).

Again, Fig. 11 is a photo of the cross-sectional shape of a round interlock and illustrates the layered system of a motor core around one of the interlocks shown in Fig. 10. In Fig. 11, dark lines have been drawn to show the boundary lines between some sheets more clearly. In the forming processes, after a flat ring sample is sheared, six cylindrical protrusions are formed by punching to a height of about half the sheet thickness. Similarly, another ring sample is formed, the positions of the six protrusions of the first two samples, or the additional ring and the already-layered ones, are aligned, and the samples are interlocked by pressing in the sheet thickness direction. In the experiment, protrusion forming, stacking and interlocking were carried out one sheet at a time until a total of 20 ring samples were laminated. Here, a cold-rolled steel sheet made on an experimental basis was employed to enable X-ray measurement of residual stress by intensifying the diffraction peak. The mechanical properties of this material are shown in Table 3 and Fig. 12. Since this material was used in the example in Fig. 11, it should be noted that this is an experimental result using a material that was trial-produced in the laboratory.

Table 3 Mechanical properties of sample cold-rolled sheet prepared in laboratory.
Fig. 12

Stress-strain curve of 0.35 mm thick steel sheet in Table 3.

A round interlock simulation was performed through the above mentioned forming processes. A simplified 2-dimensional axisymmetric simulation was adopted, focusing on one of the six interlocks in Fig. 10. The modeled region was narrowed to around the interlock shown in Fig. 11, and the symmetric axis location is set to the center of the round interlock.

First, using a 3 mm diameter punch, protrusion forming of a flat blank sheet was simulated. The clearance was set to 1%, and the forming stroke was 75% of the blank thickness. Multiple copies of the achieved protrusion result were prepared, and a simulation of the interlocking process was carried out by arranging the sheets at the prescribed distance and stacking the sheets one by one (Fig. 13). Figure 13 is an illustration of the lamination process. It should be noted that the die and holder are only shown in the final stage.

Fig. 13

Axisymmetric analysis of laminating process for a round interlock and radial stress distribution.

In this report, the interlocking process was only modeled up to lamination of four sheets in order to save simulation time. Figure 13 shows the bottom dead center condition, when the tools are in contact with the top and the bottom surfaces of the layered part from both sides. Figure 14 shows the condition after unloading, that is, after the upper tool was released and separated from the part. The contour map in Fig. 14 shows the radial stress distribution from the interlock center axis, which is the same direction as that of the X-ray stress measurement.

Fig. 14

Radial stress distribution around an interlock after release of upper tool.

As a representative stress distribution in this simplified model, focusing on the second layer below the top layer, the residual stress level was evaluated along the arrow line at the thickness center of the blank. The horizontal axis of Fig. 15 starts from the wall of a protrusion and indicates the distance from that point along the arrows in Figs. 13 and 14. In the X-ray evaluation, after embedding an interlocked sample in resin, the sample was cut at the center of a round interlock as shown in Fig. 11 and chemically polished, and then the radial residual stress along the half thickness line from the wall of a protrusion was measured. Figure 15 shows the experimental and simulation results. Partly because the analysis model has only 4 layers, the stress state just after bottom dead center, when constraint by the tools is strong, is much closer to the X-ray measured data than that after the upper tool was released. In the near future, it is expected to be possible to estimate the stress distribution with accuracy by increasing the number of layers in the analysis model and thereby imposing stronger constraint by the interlocks. However, even with the relatively simple analysis model in this section, it was possible to make a rough estimation of the residual stress around an interlock which was close to the measured results.

Fig. 15

Comparison of residual stress distribution.

5. Conclusion

A piercing simulation of a hot-rolled steel sheet, a shearing simulation and interlocking simulation of cold-rolled sheets were carried out, and the results were compared with the experimental results. Even though this was a simple elasto-plastic simulation based on the results of a uniaxial tensile test of the material and identification of the ductile fracture criterion proposed by Cockcroft, it was found that the cut surface shape and residual stress distribution were in good agreement with the experimental results in all cases.

Based on a more extensive and detailed investigation of the effects of forming conditions and material properties, this analysis method is expected to become a powerful tool for reducing the plastic strain level and residual stresses in the near future.

Finally, there are also remaining tasks for improving the accuracy of the above-mentioned simulations; these include consideration of stress triaxiality and an increase in the number of interlocked layers. Considering the cost of manual data preparation and computational time, improvement of this analysis method and fine tuning of the analysis conditions should be carried out in accordance with the goals of actual simulations.

REFERENCES
 
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