Experimental and Numerical Investigation on the Influence of Process

A. M. Goijaerts L. E. Govaert e-mail: [email protected]

F. P. T. Baaijens Materials Technology, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands

1

Experimental and Numerical Investigation on the Influence of Process Speed on the Blanking Process In a previous work a numerical tool was presented which accurately predicted both process force and fracture initiation for blanking of a ferritic stainless steel in various blanking geometries. This approach was based on the finite element method, employing a rate-independent elasto-plastic constitutive model combined with a fracture criterion which accounts for the complete loading history. In the present investigation this work is extended with respect to rate-dependence by employing an elasto-viscoplastic constitutive model in combination with the previously postulated fracture criterion for ferritic stainless steel. Numerical predictions are compared to experimental data over a large range of process speeds. The rate-dependence of the process force is significant and accurately captured by the numerical simulations at speeds ranging from 0.001 to 10 mm/s. Both experiments and numerical simulations show no influence of punch velocity on fracture initiation. 关DOI: 10.1115/1.1445152兴

Introduction

In an earlier publication 关1兴 an elasto-plastic constitutive model using Von Mises plasticity had been successfully used for a stainless steel 共X30Cr13, DIN 17006兲 for the prediction of ductile fracture initiation in the blanking process. The use of an elastoplastic constitutive model, however, implies that strain rate effects are neglected. Hence, the possible influence of blanking speed was not modelled. Tilsley and Howard 关2兴 found empirically that with a higher blanking speed the deformation becomes more concentrated, which was claimed to result in a better quality of the blanked product. Johnson and Slater 关3兴 stated that high blanking velocities are not desired because of the larger process force, resulting in an increase of consumed energy. They also cited some contradicting publications concerning the change of product quality due to varying process speeds. Therefore, further research is needed on this issue of product shape as a function of blanking velocity. In this paper, the effect of strain rate on the blanking process is investigated numerically using an elasto-viscoplastic constitutive model with a Bodner-Partom viscosity and compared to experimental data. Effects on both process force and shape of the blanked edge will be considered. Punch speeds ranging from 1 ␮m/s up to 1 m/s will be considered. To the authors’ knowledge, the maximum blanking speeds employed in industrial applications are of the order of 50 mm/s 共corresponding to a particular blanking application of approximately 600 parts per minute兲. The methods required for this study are presented in section 2. In section 3 the effect of strain rate on the blanking process is investigated both experimentally and numerically, considering both punch force and ductile fracture initiation. The results will be discussed in section 4.

2

Methods

2.1 Constitutive Modeling. To incorporate viscous effects into the constitutive description the enhanced compressible LeContributed by the Manufacturing Engineering Division for publication in the JOURNAL OF MANUFACTURING SCIENCE AND ENGINEERING. Manuscript received March 2000; Revised May 2001. Associate Editor: K. Stelson.

416 Õ Vol. 124, MAY 2002

onov model is used 关4,5,6兴. The chosen viscosity function is an adaptation of the definition in the original Bodner-Partom model 关7兴, suggested by Rubin 关8兴:

␩ 共 ¯␴ ,¯⑀ 兲 ⫽

¯␴

冑12⌫ 0

exp

冉冋 册冊 1 Z 共 ¯␧ p 兲 2 ¯␴

2n

(1)

In this formulation ¯␴ is the equivalent stress, according to the Von Mises definition in Eq. 共2兲:

冑

¯␴ ⫽

3 d d ␴ :␴ 2

(2)

with ␴d the deviatoric part of the Cauchy stress tensor and ¯␧ p is the equivalent strain, defined as: ␧ថ p ⫽

冑

2 D :D 3 p p

(3)

with Dp the plastic deformation rate tensor: Dp ⫽

␴d 2␩

(4)

The material parameters ⌫ 0 and n are related to the strain rate sensitivity. The function Z(␧ ¯ p ) is the resistance to plastic flow and influences the yield stress. In section 2.2 the function Z(␧ ¯ p ) is elaborated. The enhanced compressible Leonov model, including the Bodner-Partom viscosity definition, was implemented in the updated Lagrange environment of the commercial finite element package MARC 关9兴 by Van der Aa et al. 关10兴. This implementation is based on an implicit time integration procedure proposed by Rubin 关11兴. 2.2 Material Characterization. A ferritic stainless steel 共X30Cr13, DIN 17006兲 is used in this study. The constitutive model of section 2.1 is able to incorporate strain rate effects. Therefore, to quantify the material characteristics for X30Cr13 using this constitutive model, tensile tests are performed at three different strain rates. Young’s modulus E and ␯ are taken from an earlier publication 关12兴. As it is common for metals, ⌫ 0 is assumed to be 108 关 s⫺2 兴

Copyright © 2002 by ASME

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Table 1 Material parameters of X30Cr13 for the elasto-viscoplastic model with Z „ ␧¯ p … according to Eq. „7…

关7,13,14兴. What is left to be determined is parameter n and function Z(␧ ¯ p ) in the viscosity formulation 关Eq. 共1兲兴. To arrive at an equation that is able to fit the remaining parameters, relation 共1兲 is substituted into Eq. 共4兲 and the result is elaborated for a bar in tension. Neglecting elastic effects and knowing that in a tensile test the axial component of Dp equals ␧ថ p and the axial component of ␴d equals 2/3¯␴ , the following relation between n, ¯␴ , ¯␧ p and ␧ថ p can be derived:

冉 冉 冑 冊冊

2n ln共 ¯␴ 兲 ⫺2n ln共 Z 共 ¯␧ p 兲兲 ⫹ln ⫺2 ln

1 2

3 ថ␧ ⌫0 p

⫽0

(5)

Having the stress versus strain relations for three tensile tests at different strain rates 共assuming the strain rates are constant in each single tensile test兲, it is possible, using Eq. 共5兲, to fit all parameters in a least square estimation procedure, adopting a particular expression for Z(␧ ¯ p ). In earlier work 关12,15兴 the yield stress was determined as a function of equivalent plastic strain using tensile tests on prerolled specimens to acquire this relation for large strains. For an elasto-plastic constitutive model this relation was determined up to strains of 3 for a strain rate of approximately 0.002关s⫺1兴: ¯␴ ⫽420⫹133共 1⫺exp共 ⫺␧ ¯ p /0.0567兲兲 ⫹406冑¯␧ p ⫹70.7␧ ¯ p (6) ¯ p ) is posed, inspired by the above formuNow, a function for Z(␧ lation: Z⫽Z 1 ⫹ 共 Z 0 ⫺Z 1 兲 exp共 ⫺Z 2¯␧ p 兲 ⫹Z 3 冑¯␧ p ⫹Z 4¯␧ p

(7)

A least square estimator is used in combination with Eq. 共5兲 to fit all parameters involved and the results are presented in Table 1 and Fig. 1. It can be observed that the model describes the mechanical behavior of X30Cr13 very well for different strain rates. 共The bold solid line in Fig. 1 represents the stress-strain relation of Eq. 共6兲.兲 It is noted that the behavior up to strains of 3 corresponds very well with the stress-strain behavior of Eq. 共6兲 关15兴.

Fig. 1 Performance of constitutive model with the BodnerPartom viscosity function using Z „ ␧¯ p … according to Eq. „7…. The markers represent the experimentally found stress-strain relations out of the tensile tests for different strain rates. The bold solid line represents the stress-strain relation of Eq. „6….

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2.3 Experimental Setup. An axisymmetric blanking setup with a die-hole diameter of 10.00 mm, including a blankholder with constant pressure, was built with a punch of 9.80 mm in diameter resulting in a clearance of 10 percent of the sheet thickness of 1 mm. To avoid exorbitant simulation times, the cutting radii of the punch and die are enlarged and measured to be 0.098 mm and 0.136 mm, respectively. The cutting radius of the punch is somewhat smaller than that of the die, to assure fracture will initiate at the punch and grow to the die. For a more elaborate description one is referred to 关15兴. 2.4 Numerical Model. We simulated the blanking process using a two-dimensional, axisymmetric finite element model, described by Brokken et al. 关16兴 and Brokken 关17兴. The material behavior is described by the isotropic elasto-viscoplastic constitutive model of section 2.1 with isotropic hardening. Linear quadrilateral elements are used, which become smaller as they approach either the die radius or the punch radius. Near those radii the element proportions need to be in the range of 10 ␮m, resulting in up to 1400 elements in the entire mesh. This element size is not necessary to predict the process force correctly, but it will be vital to accurately describe the field variables, needed to precisely predict ductile fracture initiation. The punch moves down and penetrates the specimen, resulting in constantly changing boundary conditions. To deal with these difficult boundary conditions and the localized large deformations, the finite element application that we used, combines three numerical procedures: the commercial implicit finite element package MARC 关9兴 共using an updated Lagrange formulation兲, an Arbitrary Lagrange-Euler approach 关18,19兴 and an automatic adaptive remeshing algorithm 关17,20兴. A typical mesh used is shown in Fig. 2. All dimensions are in agreement with the experimental setup described in section 2.3. This model was experimentally validated up to fracture on rollover zones, deformation fields—using Digital Image Correlation—and process forces, using a planar blanking setup

Fig. 2 A typical mesh produced by the adaptive automatic remeshing algorithm

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Fig. 4 Comparison for experimental punch displacements at fracture and numerical predictions using the criteria of Table 2 as a function of blanking velocities. The error bars represent twice the standard-deviations for the experiments „95 percent interval…. Fig. 3 Comparison of experimental maximum blanking forces with numerical predictions using the compressible Leonov model with Bodner-Partom viscosity over a large range of punching velocities.

关12,15,21兴. Therefore, the deformation history in the blanking process can be calculated adequately, which is a prerequisite for the local modeling of ductile fracture.

3

Results

In section 3.1 results are presented considering process forces. In section 3.2 the issue of fracture initiation in the blanking process is investigated. 3.1 Effect of Punch Speed on Process Force. For the experimental determination of the maximum blanking force the axisymmetric setup of section 2.3 is used. Two tensile testing machines are utilized with the experimental setup to obtain information over a large range of punch velocities; with a universal tensile testing machine, slow punch speeds are accomplished up to 10 mm/s; punch speeds up to 1 m/s are realized using a hydraulic tensile testing machine. The results for the maximum punch forces averaged over five experiments for every punch velocity, are presented in Fig. 3. The error bars represent twice the standard-deviation 共95 percent interval兲. Numerical predictions of the maximum blanking forces, using the aforementioned numerical model, are also presented in Fig. 3. This figure shows that for blanking speeds up to and including 1 mm/s the numerical predictions agree well with the experiments. However, for the larger blanking velocities the numerical predictions start to deviate from the experiments. This may be caused by the material parameters that were quantified by performing different tensile tests with strain rates varying from 0.032 关s⫺1兴 to 13 关s⫺1兴. These values compare well with the maximum strain rates observed in the numerical simulations with punch velocities from 0.001 mm/s to 1 mm/s, respectively. In order to achieve better numerical results for larger blanking speeds, the material parameters need quantification at the proper strain rates. Another explanation for the observed deviation is thermal softening. For metals in general, an increase in temperature will cause a decrease of the actual yield stress. During deformation, plastic work will generate heat. At low deformation velocities, the local increase of temperature will not be excessive due to thermal conduction, resulting in a nonsignificant local temperature change. However, when large speeds are applied, the temperature rise is not negligible anymore and a local drop in the yield stress will result in a smaller blanking force. 418 Õ Vol. 124, MAY 2002

3.2 Effect of Punch Speed on Ductile Fracture Initiation In our blanking setup, six experiments were performed for every blanking speed. The shear zone or burnish 共b in Fig. 4兲 is measured afterwards at eight positions over the circumference of the blanked products, and averaged to justify for the misalignment of the punch. The values averaged over six experiments with twice the standard-deviations 共95 percent interval兲 are presented in Fig. 4 up to and including blanking speeds of 10 mm/s. It is shown that strain rate has no influence on ductile fracture initiation for these blanking velocities. To evaluate the predictive abilities of the numerical model on ductile fracture initiation for different strain rates, three different ductile fracture initiation criteria 共criteria of Table 2, valid for the quasi-static blanking process 关1兴兲 are applied for different punch velocities. Ductile fracture is predicted to initiate as soon as the integral formulation of such a criterion reaches the critical material parameter C during the plastic deformation in the blanking process. All three criteria show a large influence of the triaxiality 共hydrostatic stress divided by equivalent stress: ␴ h / ¯␴ ) on ductile fracture initiation. The critical values of the material parameters C RT , C O and C G were adopted from previous work 关1兴. C RT and C O were determined in a blanking experiment to be 2.76 and 2.38 respectively. C G was determined in a tensile test to be 3.53. Ductile fracture initiation determines the numerical predictions for the punch displacement at fracture initiation. Results are presented in Fig. 4 along with the experimentally determined punch displacements at ductile fracture initiation (a⫹b⫹c, which can all be determined afterwards兲. The error bars correspond with twice the standard-deviation 共95 percent interval兲. All criteria for the prediction of ductile fracture initiation perform within the experimental error.

Table 2 Three ductile fracture initiation criteria, valid for the quasi-static blanking process †1‡

Data from Refs. †1‡, †22‡, and †23‡ included in table.

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4

Conclusion

Up to blanking velocities of 10 mm/s 共which is within the range of nearly every industrial application兲 the blanking speed has no influence on the punch displacement at ductile fracture initiation for X30Cr13 共Fig. 4兲. This is an important conclusion, because it shows that blanking processes with punch velocities smaller than 10 mm/s can be described by the elasto-plastic constitutive model using Von Mises plasticity 关1兴, when ductile fracture initiation is considered. When process forces are considered, the effect of the strain rate is small but noticeable. Therefore, if the correct prediction of the blanking force as a function of the strain rate is not an issue, the constitutive model using Von Mises plasticity will suffice.

Acknowledgments This research was funded by the Dutch Innovative Research Program 共IOP-C.94.702.TU.WB兲.

References 关1兴 Goijaerts, A. M., Govaert, L. E., and Baaijens, F. P. T., 2000, ‘‘Prediction of Ductile Fracture in Metal Blanking,’’ ASME J. Manuf. Sci. Eng., 122, pp. 476 – 483. 关2兴 Tilsley, R., and Howard, F., 1958, ‘‘Recent Investigations into the Blanking and Piercing of Sheet Materials,’’ Machinery, 93, pp. 151–158. 关3兴 Johnson, W., and Slater, R. A. C., 1967, ‘‘A Survey of the Slow and Fast Blanking of Metals at Ambient and High Temperatures,’’ Proceedings of the International Conference of Manufacturing Technology pp. 773– 851, Michigan. 关4兴 Leonov, A. I., 1976, ‘‘Non-equilibrium Thermodynamics and Rheology of Viscoelastic Polymer Media,’’ Rheol. Acta, 15, pp. 85–98. 关5兴 Baaijens, F. P. T., 1991, ‘‘Calculation of Residual Stresses in Injection Molded Products,’’ Rheol. Acta, 30, pp. 284 –299. 关6兴 Tervoort, T. A., Smit, R. J. M., Brekelmans, W. A. M., and Govaert, L. E., 1998, ‘‘A Constitutive Equation for the Elasto-Viscoplastic Deformation of Glassy Polymers,’’ Mech. Time-Depend. Mater., 1, pp. 269–291. 关7兴 Bodner, S. R., and Partom, Y., 1975, ‘‘Constitutive Equations for ElastoViscoplastic Strain-Hardening Materials,’’ ASME J. Appl. Mech., 42, pp. 385–389.

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关8兴 Rubin, M. B., 1987, ‘‘An Elasto-Viscoplastic Model Exhibiting Continuity of Solid and Fluid States,’’ Int. J. Eng. Sci., 25, pp. 1175–1191. 关9兴 MARC Manuals: Volume A to F, 1997, Version K7, Palo Alto, California, U.S.A. 关10兴 Van der Aa, M. A. H., Schreurs, P. J. G., and Baaijens, F. P. T., 2001, ‘‘Modelling of the Wall Ironing Process of Polymer Coated Sheet Metal,’’ Mech. Mater., 33, pp. 555–572. 关11兴 Rubin, M. B., 1989, ‘‘A Time Integration Procedure for Plastic Deformation in Elasto-Viscoplastic Metals,’’ Journal of Applied Mathematics and Physics, 40, pp. 846 – 871. 关12兴 Stegeman, Y. W., Goijaerts, A. M., Brokken, D., Brekelmans, W. A. M., Govaert, L. E., and Baaijens, F. P. T., 1999, ‘‘An Experimental and Numerical Study of a Planar Blanking Process,’’ J. Mater. Process. Technol., 87, pp. 266 –276. 关13兴 Kolkailah, F. A., and McPhate, A. J., 1990, ‘‘Bodner-Partom Constitutive Model and Non-Linear Finite Element Analysis,’’ J. Eng. Mater. Technol., 112, pp. 287–291. 关14兴 van der Aa, M. A. H., 1999, ‘‘Wall Ironing of Polymer Coated Sheet Metal,’’ Ph.D. thesis, Eindhoven University of Technology, The Netherlands. 关15兴 Goijaerts, A. M., 1999, ‘‘Prediction of Ductile Fracture in Metal Blanking,’’ Ph.D. thesis, Eindhoven University of Technology, The Netherlands. http:// wfwweb.wfw.wtb.tue.nl/mate/pdfs/54.pdf 关16兴 Brokken, D., Brekelmans, W. A. M., and Baaijens, F. P. T., 1998, ‘‘Numerical Modelling of the Metal Blanking Process,’’ J. Mater. Process. Technol. 83, pp. 192–199. 关17兴 Brokken, D., 1999, ‘‘Numerical Modelling of Ductile Fracture in Blanking,’’ Ph.D. thesis, Eindhoven University of Technology, The Netherlands. 关18兴 Schreurs, P. J. G., Veldpaus, F. E., and Brekelmans, W. A. M., 1986, ‘‘Simulation of Forming Processes Using the Arbitrary Eulerian-Lagrangian Formulation,’’ Comput. Methods Appl. Mech. Eng., 58, pp. 19–36. 关19兴 Baaijens, F. P. T., 1993, ‘‘An U-ALE Formulation of 3-D Unsteady Viscoelastic Flow,’’ Int. J. Numer. Methods Eng., 36, pp. 1115–1143. 关20兴 Brokken, D., Goijaerts, A. M., Brekelmans, W. A. M., Oomens, C. W. J., and Baaijens, F. P. T., 1997, ‘‘Modelling of the Blanking Process,’’ D. R. J. Owen, ˜ ate, and E. Hinton, Computational Plasticity, Fundamentals and AppliE. On cations, Vol. 2, pp. 1417–1424. CIMNE, Barcelona. 关21兴 Goijaerts, A. M., Stegeman, Y. W., Govaert, L. E., Brokken, D., Brekelmans, W. A. M., and Baaijens, F. P. T., 2000, ‘‘Can a New and Experimental and Numerical Study Improve Metal Blanking?’’ J. Mater. Process. Technol., 103, pp. 44 –50. 关22兴 Rice, J. R., and Tracey, D. M., 1969, ‘‘On the Ductile Enlargement of Voids in Triaxial Stress Fields,’’ J. Mech. Phys. Solids, 17, pp. 201–217. 关23兴 Oyane, M., Sato, T., Okimoto, K., and Shima, S., 1980, ‘‘Criteria for Ductile Fracture and Their Applications,’’ J. Mech. Work. Technol. 4, pp. 65– 81.

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