- Email: [email protected]
Prediction of plastic flow localization with shell element in thick AHSS sheets
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Procedia Manufacturing 00 (2018) 000–000 Available online atatwww.sciencedirect.com Available www.sciencedirect.com Procediaonline Manufacturing 00 (2018) 000–000
ScienceDirect ScienceDirect
www.elsevier.com/locate/procedia www.elsevier.com/locate/procedia
Procedia Manufacturing 15 (2018) 861–868 Procedia Manufacturing 00 (2017) 000–000 www.elsevier.com/locate/procedia
17th International Conference on Metal Forming, Metal Forming 2018, 16-19 September 2018, 17th International Conference on MetalToyohashi, Forming, Metal Japan Forming 2018, 16-19 September 2018, Toyohashi, Japan
Prediction of plastic flow localization with shell element in thick
Prediction Engineering of plasticSociety flow International localization with shell element thick Manufacturing Conference 2017, MESIC 2017,in 28-30 June AHSS sheets 2017, Vigo (Pontevedra), Spain AHSS sheets Minsu Wi, Jae Hyun Choi, Fredéric Barlat* Costing models for capacity optimization in Industry 4.0: Trade-off Minsu Wi, Jae Hyun Choi, Fredéric Barlat* Graduate Institute of Ferrous Technology, Pohang University of Science and Technology, 77 Cheongam-Ro, Nam-Gu, Pohang 37673, between usedPohang capacity operational efficiency Graduate Institute of Ferrous Technology, University ofand Science and Technology, 77 Cheongam-Ro, Nam-Gu, Pohang 37673, Gyeongbuk, Korea Gyeongbuk, Korea
Abstract Abstract
A. Santana , P. Afonsoa,*, A. Zaninb, R. Wernkeb a
a
University of Minho, 4800-058 Guimarães, Portugal
b The aim of this study was to investigate the differences between shellChapecó, and solid Unochapecó, 89809-000 SC,elements Brazil in the numerical simulation of a 2.85 thick The aim of this study was to investigate the differences shell and solid in the from numerical simulationafter of a maximum 2.85 thick DP780 sheet. It was observed the stress-strain responsebetween with shell elements waselements very different the experiment DP780 sheet. It was observed the stress-strain response with shell elements was very different from the experiment after maximum load while, solid element showed a much better agreement. For shell elements, stress triaxiality had the tendency to abruptly load while, solid element showed a much better agreement. For shell elements, stress triaxiality the tendency abruptly converge at certain value while, with solid elements, the triaxiality was continuously increasing. To had compensate for thetolimitation converge at certain with curve solid elements, the triaxiality was continuously increasing. To compensate for the limitation Abstract of shell element, thevalue strainwhile, hardening was modified after maximum load. The method was validated with plane strain tension of shell element, the strain hardening curve was modified after maximum load. The method was validated with plane strain tension simulation. simulation. Under the concept of "Industry 4.0", production processes will be pushed to be increasingly interconnected, information based on a real time basis and, necessarily, much more efficient. In this context, capacity optimization © 2018 The © 2018 2018 The The Authors. Authors. Published Published by by Elsevier Elsevier B.V. B.V. © Authors. Published by Elsevier B.V. Peer-review under responsibility of the scientific committee of the the 17thInternational International Conference onMetal Metal Forming. and value. goes beyond the traditional aim of capacity maximization, contributing also for organization’s profitability Peer-review under responsibility of the scientific committee of 17th Conference on Forming. Peer-review under responsibilityand of the scientific committee of the 17th International Conference on Metal Forming. instead of Indeed, lean management continuous improvement approaches suggest capacity optimization
Keywords: Hor-rolled element; Plastic flow localization; Stress triaxility; distortionresearch topic that deserves maximization. TheAHSS; studyShell of finite capacity optimization and costing models is Element an important Keywords: Hor-rolled AHSS; Shell finite element; Plastic flow localization; Stress triaxility; Element distortion contributions from both the practical and theoretical perspectives. This paper presents and discusses a mathematical model for capacity management based on different costing models (ABC and TDABC). A generic model has been 1. Introduction developed and it was used to analyze idle capacity and to design strategies towards the maximization of organization’s 1. Introduction value. The trade-off capacity maximization vs operational efficiency is highlighted and it is shown that capacity Advanced high-strength steel (AHSS) sheets have been increasingly used in the automotive industry but their high optimization might hide operational inefficiency. Advanced high-strength steel (AHSS) sheets have increasingly in the automotive industry theirthese high strength results in many forming problems as been springback, plasticused localization and fracture. To but predict © 2017 The Authors. Published by Elsevier B.V. such strength results in many forming problems such as springback, plastic localization and fracture. To predict these problems efficiently, numerical approaches shell of elements are widelyEngineering used to save computational Because Peer-review under responsibility of the scientific with committee the Manufacturing Society Internationalcost. Conference problems efficiently, numerical shell elements widely used savesheets computational shell elements are based on theapproaches plane stresswith assumption, this isare very useful fortothin like cold cost. rolledBecause sheets. 2017. shell elements arerolled basedsheets, on the about plane 3stress assumption, very usefulleads for thin sheets like cold rolled sheets. However, for hot mm thick or over,this thisisassumption to an inaccurate prediction of the However,Cost forModels; hot rolled 3 Management; mm thick or thisOperational assumption leads to an inaccurate prediction of the Keywords: ABC; sheets, TDABC;about Capacity Idleover, Capacity; Efficiency
1. Introduction
* Corresponding author. Tel.: +82-54-279-9022; fax: +82-54-279-9299. * E-mail Corresponding Tel.: +82-54-279-9022; fax: +82-54-279-9299. address:author. [email protected] The cost of idle capacity is a fundamental information for companies and their management of extreme importance E-mail address: [email protected]
in modern©production systems. In general, it isB.V. defined as unused capacity or production potential and can be measured 2351-9789 2018 The Authors. Published by Elsevier 2351-9789 2018 Authors. Published Elsevier B.V.hours of the Peer-review underThe responsibility of theby scientific committee 17th International on Metal Forming. in several©ways: tons of production, available manufacturing, etc.Conference The management of the idle capacity Peer-review under responsibility thefax: scientific committee * Paulo Afonso. Tel.: +351 253 510of 761; +351 253 604 741 of the 17th International Conference on Metal Forming. E-mail address: [email protected]
2351-9789 © 2017 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the scientific committee of the Manufacturing Engineering Society International Conference 2017. 2351-9789 © 2018 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the scientific committee of the 17th International Conference on Metal Forming. 10.1016/j.promfg.2018.07.182
862 2
Minsu Wi et al. / Procedia Manufacturing 15 (2018) 861–868 Author name / Procedia Manufacturing 00 (2018) 000–000
tensile engineering stress-strain curve after deformations beyond the maximum force [1]. This is because stress triaxiality cannot be well predicted in the necking zone with shell elements. In contrast, the stress-strain curve computed using solid elements is in better agreement with the experiment because the triaxiality is fully considered with a 3-dimensional stress state. In this study, a method proposed by Luo and Wierzbicki [2] to compensate for stress triaxiality effects in shell elements is investigated for uniaxial and plane strain tension simulations. Nomenclature 𝜎𝜎𝜎𝜎𝑚𝑚𝑚𝑚 𝜎𝜎𝜎𝜎� 𝑇𝑇𝑇𝑇 𝐾𝐾𝐾𝐾, 𝜀𝜀𝜀𝜀0 , 𝑛𝑛𝑛𝑛 𝐾𝐾𝐾𝐾 ′ , 𝜀𝜀𝜀𝜀0 ′ , 𝑛𝑛𝑛𝑛′ 𝜀𝜀𝜀𝜀𝑚𝑚𝑚𝑚
Mean stress Equivalent stress Stress triaxility (𝜎𝜎𝜎𝜎𝑚𝑚𝑚𝑚 ⁄𝜎𝜎𝜎𝜎�) Coefficients for swift hardening law Coefficients for another swift hardening law Equivalent plastic strain at maximum load point
1.1. Triaxial stress state in necking During post-uniform deformation in uniaxial tension, a hydrostatic stress field builds up inside a material, such as explained by Bridgeman [3] for round specimens, which helps load transmission from on specimen end to the other and delays plastic flow localization. As a result, the stress triaxility parameter 𝑇𝑇𝑇𝑇 = 𝜎𝜎𝜎𝜎𝑚𝑚𝑚𝑚 ⁄𝜎𝜎𝜎𝜎� increases during necking. The stress triaxiality evolution is fully considered with solid elements but not with shell elements due to the plane stress assumption. This leads to a significant deviation in the stress-strain curves predicted with shell and solid elements. Fig. 1 shows the triaxiality evolution for uniaxial and plane strain tension tests for a 2.85 mm thick DP780 specimen. It was found that the stress triaxiality computed with the shell elements had the tendency to abruptly converge to a value between 0.55 and 0.58 during plastic flow localization. Stress triaxiality of 0.58 value corresponds to plane strain tension for an isotropic material. In addition, when the triaxiality of the shell element saturates, the deformation occurs only in one layer of elements, which indicates that the simulation is no longer valid. Compared to shell elements, solid elements provide a more reasonable prediction of the stress triaxiality, which increases continuously as a function of the strain.
Fig. 1. Stress triaxiality evolution for DP780 AHSS steel; a) as function of gauge length in uniaxial tension and; (b) as function of true strain at center of notched specimen for plane strain tension.
Minsu Wi et al. / Procedia Manufacturing 15 (2018) 861–868 Author name / Procedia Manufacturing 00 (2018) 000–000
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1.2. Corrected hardening law In order to compensate for the stress triaxiality evolution in shell elements, there are several possibilities such as non-local models, strain rate correction method and hardening correction method. In this study, the hardening correction method, which was developed in [2] was adapted for this work. Hardening laws are usually fitted from unixaxial tension data with strain ranging from zero of uniform elongation. After the uniform elongation, the hardening laws are extrapolated based on the hardening law selected; however, this method is not highly reliable. Thus, in order to extrapolate, one idea is to develop a correction of the hardening after maximum that such that the predicted engineering stress-strain curve is in good agreement with the experimental curve. This method was used with solid element and led to a flow curve in good agreement with the experimental data [2]. By adapting this method for shell element, the hardening law can be corrected in order to reproduce either the predicted curve from solid elements or the experimental curve. For validation, this correction is then applied to the case of plane strain tension. Of course, this is not a rigorous approach but neither is plane stress itself for a real problem and this correction is simpler to implement. 2. Experiments 2.1. Material A hot-rolled dual-phase (DP) steel is considered, namely, a 2.85 mm thick DP780 steel sheet from POSCO (hot rolled advanced high strength steel, AHSS). 2.2. Uniaxial tension test The uniaxial tension properties were measured with a 500 kN MTS tensile testing machine. The specimens were prepared according to the ASTM E8 standards along the rolling direction. Two mechanical extensometers were used to measure the changes of gauge length and width. The force was calculated from the load cell data during the test. The tests were conducted at a constant strain rate of 0.002/s and at room temperature. The load – displacement curve is plotted in Fig. 2 (a) and the true stress – true strain curve was approximated with the Swift hardening law as shown in Fig. 2 (b). The variables for the Swift hardening law are listed in Table 1.
Fig. 2. (a) Load-displacement curves for DP780 AHSS steel; (b) true stress-true plastic strain curve with Swift hardening law approximation.
Minsu Wi et al. / Procedia Manufacturing 15 (2018) 861–868 Author name / Procedia Manufacturing 00 (2018) 000–000
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Table 1. Swift hardening variables for DP780. Swift hardening law 𝜎𝜎𝜎𝜎� = 𝐾𝐾𝐾𝐾 ∙ (𝜀𝜀𝜀𝜀̅𝑝𝑝𝑝𝑝 + 𝜀𝜀𝜀𝜀0 )𝑛𝑛𝑛𝑛
𝐾𝐾𝐾𝐾
1282 MPa
𝑛𝑛𝑛𝑛
0.126
𝜀𝜀𝜀𝜀0
0.001
2.3. Plane strain tension test The tension test of a wide notched specimen was conducted on the MTS machine to simulate plane strain tension. The digital image correlation technique was adopted to measure the strain on the surface of the specimen. The strain rate was 0.002/s. A picture of the specimen and its specific dimensions are shown in Fig. 3 [4].
Fig. 3. Plane strain specimen with speckle patterns and its dimensions.
Fig.4 represents the strain distribution during the test. The upper figure corresponds to the minor strain distribution and the lower figure the major strain distribution for DP780. Since the strain distribution varies from the center to the edge of the specimen, an area of interest of 10 mm x 5 mm was selected for the digital image correlation images.
Fig. 4. Minor and major strain maps in area of interest from digital image correlation images during notched tension test of DP780 steel.
Fig. 5 displays the load as a function of the major strain (5a) and strain history (minor strain to major strain ratio) of the central point during the test before maximum load (5b). The major strains were extracted from area of interest
Minsu Wi et al. / Procedia Manufacturing 15 (2018) 861–868 Author name / Procedia Manufacturing 00 (2018) 000–000
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as defined in Fig. 4. The strain ratio starts with a positive slope in the elastic region and changes to a negative slope in the plastic range. This can be explained with the yield surface. In the elastic region, the strain is roughly proportional to the stress. In tension of a wide notched specimen, the direction of strain ratio is therefore positive due to Hooke’s law in elasticity. However, in the plastic deformation region, the strain increment is defined with the associated flow rule, that is, normal to the yield surface, which leads to a negative strain ratio for this test. Nevertheless, the amount of minor strain is very small compared to the major strain, which indicate that the present test simulates the plane strain behavior well.
Fig.5. (a) Load-major strain curve for DP780 AHSS steel; (b) major-minor strain ratio during plane strain tension before maximum load.
3. Simulation results 3.1. Hardening law correction The Swift hardening law was employed with the shell elements using the original coefficients before maximum load 𝜎𝜎𝜎𝜎� = 𝐾𝐾𝐾𝐾 ∙ (𝜀𝜀𝜀𝜀̅𝑝𝑝𝑝𝑝 + 𝜀𝜀𝜀𝜀0 )𝑛𝑛𝑛𝑛 .
(1)
After maximum load, the hardening was corrected using another set of coefficients in the Swift law ′
𝜎𝜎𝜎𝜎� = 𝐾𝐾𝐾𝐾 ′ ∙ (𝜀𝜀𝜀𝜀̅𝑝𝑝𝑝𝑝 + 𝜀𝜀𝜀𝜀0 ′ )𝑛𝑛𝑛𝑛 .
(2)
Since the hardening curve should be continuous and differentiable at all plastic strain, two conditions were applied at maximum load where the coefficients are changing, namely, ′
𝐾𝐾𝐾𝐾 ∙ (𝜀𝜀𝜀𝜀𝑚𝑚𝑚𝑚 + 𝜀𝜀𝜀𝜀0 )𝑛𝑛𝑛𝑛 = 𝐾𝐾𝐾𝐾 ′ ∙ (𝜀𝜀𝜀𝜀𝑚𝑚𝑚𝑚 + 𝜀𝜀𝜀𝜀0 ′ )𝑛𝑛𝑛𝑛 ,
(3) ′
𝑛𝑛𝑛𝑛 ∙ 𝐾𝐾𝐾𝐾 ∙ (𝜀𝜀𝜀𝜀𝑚𝑚𝑚𝑚 + 𝜀𝜀𝜀𝜀0 )𝑛𝑛𝑛𝑛−1 = 𝑛𝑛𝑛𝑛′ ∙ 𝐾𝐾𝐾𝐾 ′ ∙ (𝜀𝜀𝜀𝜀𝑚𝑚𝑚𝑚 + 𝜀𝜀𝜀𝜀0 ′ )𝑛𝑛𝑛𝑛 −1 .
(4)
From Eqs. (3) and (4), the following relationship is obtained,
𝜀𝜀𝜀𝜀𝑚𝑚𝑚𝑚 + 𝜀𝜀𝜀𝜀0 𝑛𝑛𝑛𝑛
=
𝜀𝜀𝜀𝜀𝑚𝑚𝑚𝑚 + 𝜀𝜀𝜀𝜀0 ′ 𝑛𝑛𝑛𝑛′
.
(5)
In Eq. (5), 𝜀𝜀𝜀𝜀𝑚𝑚𝑚𝑚 , 𝑛𝑛𝑛𝑛 and 𝜀𝜀𝜀𝜀0 were already determined from the original Swift hardening law and 𝐾𝐾𝐾𝐾 ′ , 𝜀𝜀𝜀𝜀0 ′ and 𝑛𝑛𝑛𝑛′ were optimized based on the engineering uniaxial tension curve. Several value of 𝑛𝑛𝑛𝑛′ were selected and 𝐾𝐾𝐾𝐾 ′ and 𝜀𝜀𝜀𝜀0 ′ were
Minsu Wi et al. / Procedia Manufacturing 15 (2018) 861–868 Author name / Procedia Manufacturing 00 (2018) 000–000
866 6
calculated from Eqs. (3) and (5), respectively. To consider plastic anisotropy, Yld2000-2d and Yld2004-18p were adopted as the yield functions for shell and solid elements, respectively. Fig. 6(a) shows the corrected hardening curves and Fig. 6(b) the corresponding uniaxial load-displacement curves for each 𝑛𝑛𝑛𝑛′ values. The shell elements and hardening law with a 𝑛𝑛𝑛𝑛′ value of 0.25 led to an engineering flow curve in good agreement with that predicted with solid elements and constant hardening coefficients. 𝑛𝑛𝑛𝑛′ values of 0.35 and 0.40 led to a good approximation of the experimental curve. The coefficients of the corrected hardening curves are listed in Table 2.
Fig. 6. (a) Corrected hardening curves for each 𝑛𝑛𝑛𝑛′ value for DP780 AHSS steel; (b) uniaxial tension load-displacement curves for each 𝑛𝑛𝑛𝑛′ value. Table 2. Corrected swift hardening variables. Swift hardening law ′ 𝜎𝜎𝜎𝜎� = 𝐾𝐾𝐾𝐾 ′ ∙ (𝜀𝜀𝜀𝜀̅𝑝𝑝𝑝𝑝 + 𝜀𝜀𝜀𝜀0 ′ )𝑛𝑛𝑛𝑛
𝐾𝐾𝐾𝐾 ′
𝑛𝑛𝑛𝑛′
𝜀𝜀𝜀𝜀0′
n′ = 0.35
1437 MPa
0.35
0.218
1438 MPa
0.40
0.266
n′ = 0.25 ′
n = 0.40
1402 MPa
0.25
0.121
3.2. Plane strain tension Tension of the wide notched specimen was simulated with shell and solid elements using the anisotropic Yld20002d and Yld2004-18p yield functions for shell and solid elements, respectively. The experimental and predicted strains were extracted in a consistent manner with the AOI in Fig. 5. Fig. 7 represents the load as a function of the major strain for the shell and solid elements. The black solid line corresponds to the prediction with solid elements, the red dash line with shell and original Swift hardening and the others to corrected hardening cases with shell elements. All shell and solid elements provide similar load predictions until maximum load. However, beyond this point, a sudden load drop associated with plastic low localization through thickness occurs with the shell elements, which makes predictions very far from the real behavior. However, this sudden load drop is significantly attenuated with an increase of the n^' value. Among all the corrected hardening cases, a n^' value of 0.4 led to an excellent agreement with the experiments.
Minsu Wi / et al. / Procedia Manufacturing 15 (2018) 861–868 Author name Procedia Manufacturing 00 (2018) 000–000
8677
Fig. 7. Plane strain load-major strain curves for experiment, each 𝑛𝑛𝑛𝑛′ value of shell elements and solid element for DP780 AHSS steel.
4. Discussion With shell elements, compared to the original Swift hardening law, the corrected hardening cases lead to much better prediction of the load-strain curves after maximum load. The corrected hardening law postponed the plastic flow localization and this made shell behaves closer to the solid elements in terms of load prediction. However, although the shell elements with corrected hardening led to a load prediction in good agreement with that of the solid elements in uniaxial tension, this was not as satisfactory in plane strain tension. This means that other approaches like strain rate dependency and strain gradient plasticity should be investigated to find out whether shell elements can behave more like solids in multiple deformation modes. Acknowledgments The authors are very grateful to POSCO for generous support and for providing material. Appendix A. Yld2000 parameters for DP780An example appendix α1
1.064
α2
α3
0.840
α4
1.072
α5
1.053
α6
1.038
α7
1.122
1.029
α8
0.958
Appendix B. Yld2004 parameters for DP780 α1
1.000
α2
1.000
α3
1.096
α4
0.999
α5
0.728
α6
0.263
α7
0.684
α8
0.857
1.024
α10
α11
α12
α13
α14
α15
α16
α17
α18
0.846
0.762
0.775
0.599
1.281
1.338
1.239
1.076
α9
0.998
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Minsu Wi et al. / Procedia Manufacturing 15 (2018) 861–868 Author name / Procedia Manufacturing 00 (2018) 000–000
References [1] K.H. Pack, D. Mohr, Combined necking & fracture model to predict ductile failure with shell finite element, Engineering Fracture Mechanics, 182 (2017) 325–51. [2] M. Luo, T. Wierzbicki, Numerical failure analysis of a stretch-bending test on dual-phase steel sheets using a phenomenological fracture model, International Journal of Solids and Structures, 47 (2010) 3084–3102. [3] P.W. Bridgman, Studies in large plastic flow and fracture with special emphasis on the effects of hydrostatic pressure, McGraw-Hill, (1952). [4] M.D. Xavier, R.L. Plaut, C.G. Schön, Uniaxial near plena strain tensile tests applied to the determination of the FLC0 formability parameter, Materials Research, 17 (2014).
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Procedia Manufacturing 00 (2018) 000–000 Available online atatwww.sciencedirect.com Available www.sciencedirect.com Procediaonline Manufacturing 00 (2018) 000–000
ScienceDirect ScienceDirect
www.elsevier.com/locate/procedia www.elsevier.com/locate/procedia
Procedia Manufacturing 15 (2018) 861–868 Procedia Manufacturing 00 (2017) 000–000 www.elsevier.com/locate/procedia
17th International Conference on Metal Forming, Metal Forming 2018, 16-19 September 2018, 17th International Conference on MetalToyohashi, Forming, Metal Japan Forming 2018, 16-19 September 2018, Toyohashi, Japan
Prediction of plastic flow localization with shell element in thick
Prediction Engineering of plasticSociety flow International localization with shell element thick Manufacturing Conference 2017, MESIC 2017,in 28-30 June AHSS sheets 2017, Vigo (Pontevedra), Spain AHSS sheets Minsu Wi, Jae Hyun Choi, Fredéric Barlat* Costing models for capacity optimization in Industry 4.0: Trade-off Minsu Wi, Jae Hyun Choi, Fredéric Barlat* Graduate Institute of Ferrous Technology, Pohang University of Science and Technology, 77 Cheongam-Ro, Nam-Gu, Pohang 37673, between usedPohang capacity operational efficiency Graduate Institute of Ferrous Technology, University ofand Science and Technology, 77 Cheongam-Ro, Nam-Gu, Pohang 37673, Gyeongbuk, Korea Gyeongbuk, Korea
Abstract Abstract
A. Santana , P. Afonsoa,*, A. Zaninb, R. Wernkeb a
a
University of Minho, 4800-058 Guimarães, Portugal
b The aim of this study was to investigate the differences between shellChapecó, and solid Unochapecó, 89809-000 SC,elements Brazil in the numerical simulation of a 2.85 thick The aim of this study was to investigate the differences shell and solid in the from numerical simulationafter of a maximum 2.85 thick DP780 sheet. It was observed the stress-strain responsebetween with shell elements waselements very different the experiment DP780 sheet. It was observed the stress-strain response with shell elements was very different from the experiment after maximum load while, solid element showed a much better agreement. For shell elements, stress triaxiality had the tendency to abruptly load while, solid element showed a much better agreement. For shell elements, stress triaxiality the tendency abruptly converge at certain value while, with solid elements, the triaxiality was continuously increasing. To had compensate for thetolimitation converge at certain with curve solid elements, the triaxiality was continuously increasing. To compensate for the limitation Abstract of shell element, thevalue strainwhile, hardening was modified after maximum load. The method was validated with plane strain tension of shell element, the strain hardening curve was modified after maximum load. The method was validated with plane strain tension simulation. simulation. Under the concept of "Industry 4.0", production processes will be pushed to be increasingly interconnected, information based on a real time basis and, necessarily, much more efficient. In this context, capacity optimization © 2018 The © 2018 2018 The The Authors. Authors. Published Published by by Elsevier Elsevier B.V. B.V. © Authors. Published by Elsevier B.V. Peer-review under responsibility of the scientific committee of the the 17thInternational International Conference onMetal Metal Forming. and value. goes beyond the traditional aim of capacity maximization, contributing also for organization’s profitability Peer-review under responsibility of the scientific committee of 17th Conference on Forming. Peer-review under responsibilityand of the scientific committee of the 17th International Conference on Metal Forming. instead of Indeed, lean management continuous improvement approaches suggest capacity optimization
Keywords: Hor-rolled element; Plastic flow localization; Stress triaxility; distortionresearch topic that deserves maximization. TheAHSS; studyShell of finite capacity optimization and costing models is Element an important Keywords: Hor-rolled AHSS; Shell finite element; Plastic flow localization; Stress triaxility; Element distortion contributions from both the practical and theoretical perspectives. This paper presents and discusses a mathematical model for capacity management based on different costing models (ABC and TDABC). A generic model has been 1. Introduction developed and it was used to analyze idle capacity and to design strategies towards the maximization of organization’s 1. Introduction value. The trade-off capacity maximization vs operational efficiency is highlighted and it is shown that capacity Advanced high-strength steel (AHSS) sheets have been increasingly used in the automotive industry but their high optimization might hide operational inefficiency. Advanced high-strength steel (AHSS) sheets have increasingly in the automotive industry theirthese high strength results in many forming problems as been springback, plasticused localization and fracture. To but predict © 2017 The Authors. Published by Elsevier B.V. such strength results in many forming problems such as springback, plastic localization and fracture. To predict these problems efficiently, numerical approaches shell of elements are widelyEngineering used to save computational Because Peer-review under responsibility of the scientific with committee the Manufacturing Society Internationalcost. Conference problems efficiently, numerical shell elements widely used savesheets computational shell elements are based on theapproaches plane stresswith assumption, this isare very useful fortothin like cold cost. rolledBecause sheets. 2017. shell elements arerolled basedsheets, on the about plane 3stress assumption, very usefulleads for thin sheets like cold rolled sheets. However, for hot mm thick or over,this thisisassumption to an inaccurate prediction of the However,Cost forModels; hot rolled 3 Management; mm thick or thisOperational assumption leads to an inaccurate prediction of the Keywords: ABC; sheets, TDABC;about Capacity Idleover, Capacity; Efficiency
1. Introduction
* Corresponding author. Tel.: +82-54-279-9022; fax: +82-54-279-9299. * E-mail Corresponding Tel.: +82-54-279-9022; fax: +82-54-279-9299. address:author. [email protected] The cost of idle capacity is a fundamental information for companies and their management of extreme importance E-mail address: [email protected]
in modern©production systems. In general, it isB.V. defined as unused capacity or production potential and can be measured 2351-9789 2018 The Authors. Published by Elsevier 2351-9789 2018 Authors. Published Elsevier B.V.hours of the Peer-review underThe responsibility of theby scientific committee 17th International on Metal Forming. in several©ways: tons of production, available manufacturing, etc.Conference The management of the idle capacity Peer-review under responsibility thefax: scientific committee * Paulo Afonso. Tel.: +351 253 510of 761; +351 253 604 741 of the 17th International Conference on Metal Forming. E-mail address: [email protected]
2351-9789 © 2017 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the scientific committee of the Manufacturing Engineering Society International Conference 2017. 2351-9789 © 2018 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the scientific committee of the 17th International Conference on Metal Forming. 10.1016/j.promfg.2018.07.182
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Minsu Wi et al. / Procedia Manufacturing 15 (2018) 861–868 Author name / Procedia Manufacturing 00 (2018) 000–000
tensile engineering stress-strain curve after deformations beyond the maximum force [1]. This is because stress triaxiality cannot be well predicted in the necking zone with shell elements. In contrast, the stress-strain curve computed using solid elements is in better agreement with the experiment because the triaxiality is fully considered with a 3-dimensional stress state. In this study, a method proposed by Luo and Wierzbicki [2] to compensate for stress triaxiality effects in shell elements is investigated for uniaxial and plane strain tension simulations. Nomenclature 𝜎𝜎𝜎𝜎𝑚𝑚𝑚𝑚 𝜎𝜎𝜎𝜎� 𝑇𝑇𝑇𝑇 𝐾𝐾𝐾𝐾, 𝜀𝜀𝜀𝜀0 , 𝑛𝑛𝑛𝑛 𝐾𝐾𝐾𝐾 ′ , 𝜀𝜀𝜀𝜀0 ′ , 𝑛𝑛𝑛𝑛′ 𝜀𝜀𝜀𝜀𝑚𝑚𝑚𝑚
Mean stress Equivalent stress Stress triaxility (𝜎𝜎𝜎𝜎𝑚𝑚𝑚𝑚 ⁄𝜎𝜎𝜎𝜎�) Coefficients for swift hardening law Coefficients for another swift hardening law Equivalent plastic strain at maximum load point
1.1. Triaxial stress state in necking During post-uniform deformation in uniaxial tension, a hydrostatic stress field builds up inside a material, such as explained by Bridgeman [3] for round specimens, which helps load transmission from on specimen end to the other and delays plastic flow localization. As a result, the stress triaxility parameter 𝑇𝑇𝑇𝑇 = 𝜎𝜎𝜎𝜎𝑚𝑚𝑚𝑚 ⁄𝜎𝜎𝜎𝜎� increases during necking. The stress triaxiality evolution is fully considered with solid elements but not with shell elements due to the plane stress assumption. This leads to a significant deviation in the stress-strain curves predicted with shell and solid elements. Fig. 1 shows the triaxiality evolution for uniaxial and plane strain tension tests for a 2.85 mm thick DP780 specimen. It was found that the stress triaxiality computed with the shell elements had the tendency to abruptly converge to a value between 0.55 and 0.58 during plastic flow localization. Stress triaxiality of 0.58 value corresponds to plane strain tension for an isotropic material. In addition, when the triaxiality of the shell element saturates, the deformation occurs only in one layer of elements, which indicates that the simulation is no longer valid. Compared to shell elements, solid elements provide a more reasonable prediction of the stress triaxiality, which increases continuously as a function of the strain.
Fig. 1. Stress triaxiality evolution for DP780 AHSS steel; a) as function of gauge length in uniaxial tension and; (b) as function of true strain at center of notched specimen for plane strain tension.
Minsu Wi et al. / Procedia Manufacturing 15 (2018) 861–868 Author name / Procedia Manufacturing 00 (2018) 000–000
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1.2. Corrected hardening law In order to compensate for the stress triaxiality evolution in shell elements, there are several possibilities such as non-local models, strain rate correction method and hardening correction method. In this study, the hardening correction method, which was developed in [2] was adapted for this work. Hardening laws are usually fitted from unixaxial tension data with strain ranging from zero of uniform elongation. After the uniform elongation, the hardening laws are extrapolated based on the hardening law selected; however, this method is not highly reliable. Thus, in order to extrapolate, one idea is to develop a correction of the hardening after maximum that such that the predicted engineering stress-strain curve is in good agreement with the experimental curve. This method was used with solid element and led to a flow curve in good agreement with the experimental data [2]. By adapting this method for shell element, the hardening law can be corrected in order to reproduce either the predicted curve from solid elements or the experimental curve. For validation, this correction is then applied to the case of plane strain tension. Of course, this is not a rigorous approach but neither is plane stress itself for a real problem and this correction is simpler to implement. 2. Experiments 2.1. Material A hot-rolled dual-phase (DP) steel is considered, namely, a 2.85 mm thick DP780 steel sheet from POSCO (hot rolled advanced high strength steel, AHSS). 2.2. Uniaxial tension test The uniaxial tension properties were measured with a 500 kN MTS tensile testing machine. The specimens were prepared according to the ASTM E8 standards along the rolling direction. Two mechanical extensometers were used to measure the changes of gauge length and width. The force was calculated from the load cell data during the test. The tests were conducted at a constant strain rate of 0.002/s and at room temperature. The load – displacement curve is plotted in Fig. 2 (a) and the true stress – true strain curve was approximated with the Swift hardening law as shown in Fig. 2 (b). The variables for the Swift hardening law are listed in Table 1.
Fig. 2. (a) Load-displacement curves for DP780 AHSS steel; (b) true stress-true plastic strain curve with Swift hardening law approximation.
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Table 1. Swift hardening variables for DP780. Swift hardening law 𝜎𝜎𝜎𝜎� = 𝐾𝐾𝐾𝐾 ∙ (𝜀𝜀𝜀𝜀̅𝑝𝑝𝑝𝑝 + 𝜀𝜀𝜀𝜀0 )𝑛𝑛𝑛𝑛
𝐾𝐾𝐾𝐾
1282 MPa
𝑛𝑛𝑛𝑛
0.126
𝜀𝜀𝜀𝜀0
0.001
2.3. Plane strain tension test The tension test of a wide notched specimen was conducted on the MTS machine to simulate plane strain tension. The digital image correlation technique was adopted to measure the strain on the surface of the specimen. The strain rate was 0.002/s. A picture of the specimen and its specific dimensions are shown in Fig. 3 [4].
Fig. 3. Plane strain specimen with speckle patterns and its dimensions.
Fig.4 represents the strain distribution during the test. The upper figure corresponds to the minor strain distribution and the lower figure the major strain distribution for DP780. Since the strain distribution varies from the center to the edge of the specimen, an area of interest of 10 mm x 5 mm was selected for the digital image correlation images.
Fig. 4. Minor and major strain maps in area of interest from digital image correlation images during notched tension test of DP780 steel.
Fig. 5 displays the load as a function of the major strain (5a) and strain history (minor strain to major strain ratio) of the central point during the test before maximum load (5b). The major strains were extracted from area of interest
Minsu Wi et al. / Procedia Manufacturing 15 (2018) 861–868 Author name / Procedia Manufacturing 00 (2018) 000–000
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as defined in Fig. 4. The strain ratio starts with a positive slope in the elastic region and changes to a negative slope in the plastic range. This can be explained with the yield surface. In the elastic region, the strain is roughly proportional to the stress. In tension of a wide notched specimen, the direction of strain ratio is therefore positive due to Hooke’s law in elasticity. However, in the plastic deformation region, the strain increment is defined with the associated flow rule, that is, normal to the yield surface, which leads to a negative strain ratio for this test. Nevertheless, the amount of minor strain is very small compared to the major strain, which indicate that the present test simulates the plane strain behavior well.
Fig.5. (a) Load-major strain curve for DP780 AHSS steel; (b) major-minor strain ratio during plane strain tension before maximum load.
3. Simulation results 3.1. Hardening law correction The Swift hardening law was employed with the shell elements using the original coefficients before maximum load 𝜎𝜎𝜎𝜎� = 𝐾𝐾𝐾𝐾 ∙ (𝜀𝜀𝜀𝜀̅𝑝𝑝𝑝𝑝 + 𝜀𝜀𝜀𝜀0 )𝑛𝑛𝑛𝑛 .
(1)
After maximum load, the hardening was corrected using another set of coefficients in the Swift law ′
𝜎𝜎𝜎𝜎� = 𝐾𝐾𝐾𝐾 ′ ∙ (𝜀𝜀𝜀𝜀̅𝑝𝑝𝑝𝑝 + 𝜀𝜀𝜀𝜀0 ′ )𝑛𝑛𝑛𝑛 .
(2)
Since the hardening curve should be continuous and differentiable at all plastic strain, two conditions were applied at maximum load where the coefficients are changing, namely, ′
𝐾𝐾𝐾𝐾 ∙ (𝜀𝜀𝜀𝜀𝑚𝑚𝑚𝑚 + 𝜀𝜀𝜀𝜀0 )𝑛𝑛𝑛𝑛 = 𝐾𝐾𝐾𝐾 ′ ∙ (𝜀𝜀𝜀𝜀𝑚𝑚𝑚𝑚 + 𝜀𝜀𝜀𝜀0 ′ )𝑛𝑛𝑛𝑛 ,
(3) ′
𝑛𝑛𝑛𝑛 ∙ 𝐾𝐾𝐾𝐾 ∙ (𝜀𝜀𝜀𝜀𝑚𝑚𝑚𝑚 + 𝜀𝜀𝜀𝜀0 )𝑛𝑛𝑛𝑛−1 = 𝑛𝑛𝑛𝑛′ ∙ 𝐾𝐾𝐾𝐾 ′ ∙ (𝜀𝜀𝜀𝜀𝑚𝑚𝑚𝑚 + 𝜀𝜀𝜀𝜀0 ′ )𝑛𝑛𝑛𝑛 −1 .
(4)
From Eqs. (3) and (4), the following relationship is obtained,
𝜀𝜀𝜀𝜀𝑚𝑚𝑚𝑚 + 𝜀𝜀𝜀𝜀0 𝑛𝑛𝑛𝑛
=
𝜀𝜀𝜀𝜀𝑚𝑚𝑚𝑚 + 𝜀𝜀𝜀𝜀0 ′ 𝑛𝑛𝑛𝑛′
.
(5)
In Eq. (5), 𝜀𝜀𝜀𝜀𝑚𝑚𝑚𝑚 , 𝑛𝑛𝑛𝑛 and 𝜀𝜀𝜀𝜀0 were already determined from the original Swift hardening law and 𝐾𝐾𝐾𝐾 ′ , 𝜀𝜀𝜀𝜀0 ′ and 𝑛𝑛𝑛𝑛′ were optimized based on the engineering uniaxial tension curve. Several value of 𝑛𝑛𝑛𝑛′ were selected and 𝐾𝐾𝐾𝐾 ′ and 𝜀𝜀𝜀𝜀0 ′ were
Minsu Wi et al. / Procedia Manufacturing 15 (2018) 861–868 Author name / Procedia Manufacturing 00 (2018) 000–000
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calculated from Eqs. (3) and (5), respectively. To consider plastic anisotropy, Yld2000-2d and Yld2004-18p were adopted as the yield functions for shell and solid elements, respectively. Fig. 6(a) shows the corrected hardening curves and Fig. 6(b) the corresponding uniaxial load-displacement curves for each 𝑛𝑛𝑛𝑛′ values. The shell elements and hardening law with a 𝑛𝑛𝑛𝑛′ value of 0.25 led to an engineering flow curve in good agreement with that predicted with solid elements and constant hardening coefficients. 𝑛𝑛𝑛𝑛′ values of 0.35 and 0.40 led to a good approximation of the experimental curve. The coefficients of the corrected hardening curves are listed in Table 2.
Fig. 6. (a) Corrected hardening curves for each 𝑛𝑛𝑛𝑛′ value for DP780 AHSS steel; (b) uniaxial tension load-displacement curves for each 𝑛𝑛𝑛𝑛′ value. Table 2. Corrected swift hardening variables. Swift hardening law ′ 𝜎𝜎𝜎𝜎� = 𝐾𝐾𝐾𝐾 ′ ∙ (𝜀𝜀𝜀𝜀̅𝑝𝑝𝑝𝑝 + 𝜀𝜀𝜀𝜀0 ′ )𝑛𝑛𝑛𝑛
𝐾𝐾𝐾𝐾 ′
𝑛𝑛𝑛𝑛′
𝜀𝜀𝜀𝜀0′
n′ = 0.35
1437 MPa
0.35
0.218
1438 MPa
0.40
0.266
n′ = 0.25 ′
n = 0.40
1402 MPa
0.25
0.121
3.2. Plane strain tension Tension of the wide notched specimen was simulated with shell and solid elements using the anisotropic Yld20002d and Yld2004-18p yield functions for shell and solid elements, respectively. The experimental and predicted strains were extracted in a consistent manner with the AOI in Fig. 5. Fig. 7 represents the load as a function of the major strain for the shell and solid elements. The black solid line corresponds to the prediction with solid elements, the red dash line with shell and original Swift hardening and the others to corrected hardening cases with shell elements. All shell and solid elements provide similar load predictions until maximum load. However, beyond this point, a sudden load drop associated with plastic low localization through thickness occurs with the shell elements, which makes predictions very far from the real behavior. However, this sudden load drop is significantly attenuated with an increase of the n^' value. Among all the corrected hardening cases, a n^' value of 0.4 led to an excellent agreement with the experiments.
Minsu Wi / et al. / Procedia Manufacturing 15 (2018) 861–868 Author name Procedia Manufacturing 00 (2018) 000–000
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Fig. 7. Plane strain load-major strain curves for experiment, each 𝑛𝑛𝑛𝑛′ value of shell elements and solid element for DP780 AHSS steel.
4. Discussion With shell elements, compared to the original Swift hardening law, the corrected hardening cases lead to much better prediction of the load-strain curves after maximum load. The corrected hardening law postponed the plastic flow localization and this made shell behaves closer to the solid elements in terms of load prediction. However, although the shell elements with corrected hardening led to a load prediction in good agreement with that of the solid elements in uniaxial tension, this was not as satisfactory in plane strain tension. This means that other approaches like strain rate dependency and strain gradient plasticity should be investigated to find out whether shell elements can behave more like solids in multiple deformation modes. Acknowledgments The authors are very grateful to POSCO for generous support and for providing material. Appendix A. Yld2000 parameters for DP780An example appendix α1
1.064
α2
α3
0.840
α4
1.072
α5
1.053
α6
1.038
α7
1.122
1.029
α8
0.958
Appendix B. Yld2004 parameters for DP780 α1
1.000
α2
1.000
α3
1.096
α4
0.999
α5
0.728
α6
0.263
α7
0.684
α8
0.857
1.024
α10
α11
α12
α13
α14
α15
α16
α17
α18
0.846
0.762
0.775
0.599
1.281
1.338
1.239
1.076
α9
0.998
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References [1] K.H. Pack, D. Mohr, Combined necking & fracture model to predict ductile failure with shell finite element, Engineering Fracture Mechanics, 182 (2017) 325–51. [2] M. Luo, T. Wierzbicki, Numerical failure analysis of a stretch-bending test on dual-phase steel sheets using a phenomenological fracture model, International Journal of Solids and Structures, 47 (2010) 3084–3102. [3] P.W. Bridgman, Studies in large plastic flow and fracture with special emphasis on the effects of hydrostatic pressure, McGraw-Hill, (1952). [4] M.D. Xavier, R.L. Plaut, C.G. Schön, Uniaxial near plena strain tensile tests applied to the determination of the FLC0 formability parameter, Materials Research, 17 (2014).