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Constitutive Equations Based on Non-associated Flow Rule for the Analysis of Forming of Anisotropic Sheet Metals

  • Boxun WuEmail author
  • Koichi Ito
  • Naomichi Mori
  • Tetsuo Oya
  • Tom Taylor
  • Jun Yanagimoto
Regular Paper
  • 4 Downloads

Abstract

In this study, an anisotropic constitutive model based on the non-associated flow rule was developed for anisotropic sheet metals. This model was defined in the quadratic form of the Hill’s anisotropic function under a general three-dimensional stress condition. The anisotropic parameters for the yield function were identified using the directional planar yield stresses, bulge yield stress and shear yield stress, while those for the plastic potential function were identified using the directional r-values. A full expression related to the non-associated flow rule was applied and the model was implemented into the finite element code ABAQUS. A static-implicit analysis and the solid element were applied. Capability of the developed model for predicting the anisotropic behavior of sheet metal was investigated by considering two different sheet metal forming processes: cylindrical cup drawing of AA2090-T3, A6061P-T6 and SPCE; and hole expansion forming test of A6016-O. Cup heights and through-thickness strain distributions obtained from the simulations were compared with the experimental data. Results demonstrate that the developed material model considering 3D condition can improve accuracy of predicting the anisotropic behaviors. Furthermore, the simple formulations are efficient and user-friendly for computational analyses and solving the common industrial sheet metal forming problems.

Keywords

Sheet metal forming Anisotropic material Constitutive behavior Finite element 

List of Symbols

\(\varvec{d\sigma }\)

Stress tensor increment

\(\varvec{d\varepsilon }\)

Strain tensor increment

\(\varvec{d\varepsilon }^{{e}}\)

Elastic strain increment

\(\varvec{d\varepsilon }^{{p}}\)

Plastic strain increment

\(d\overline{\varepsilon }^{p}\)

Equivalent plastic strain increment

\(d\lambda\)

Plastic multiplier increment

\(dW\)

Plastic work increment

\(\varvec{D}^{e}\)

Elastic stiffness tensor

\(\varvec{D}^{ep}\)

Elastic–plastic tangent modulus

\(\phi\)

Yield criterion

\(\bar{\varepsilon }^{p}\)

Equivalent plastic strain

\(f\)

Yield stress function

\(g\)

Plastic potential function

\(h\)

Isotropic hardening function

\(F,G,H,L,M,N\)

Parameters of the yield function

\(F^{*} ,G^{*} ,H^{*} ,L^{*} ,M^{*} ,N^{*}\)

Parameters of the potential function

\(\varvec{m}\)

First order gradient of the yield function

\(\varvec{n}\)

First order gradient of the potential function

\(P\)

Ratio of the potential function to the yield function

\(\bar{\sigma }_{f}\)

Equivalent stress of the yield function

\(\bar{\sigma }_{g}\)

Equivalent stress of the potential function

\(\sigma_{Y}\)

Equivalent stress at uniaxial tension

\(\varvec{\sigma}\)

Cauchy stress tensor

\(\varvec{\sigma}^{{{tri}}}\)

Trial stress tensor

\(\varvec{\sigma}^{{c}}\)

Stress tensor after plastic correction

\(\varvec{X }:\varvec{Y}\)

Double contraction of tensors X and Y

\(\varvec{X} \otimes \varvec{Y}\)

Tensor product of tensors X and Y

Notes

Acknowledgements

This investigation was conducted as part of a corporate social program (Based Technologies for Future Robots) supported by NIDEC Corporation.

Compliance with Ethical Standards

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

References

  1. 1.
    Park, H. S., Nguyen, T. T., & Dang, X. P. (2016). Energy-efficient optimization of forging process considering the manufacturing history. International Journal of Precision Engineering and Manufacturing Green Technology, 3, 147–154.CrossRefGoogle Scholar
  2. 2.
    Jang, D. Y., Jung, J., & Seok, J. (2016). Modeling and parameter optimization for cutting energy reduction in MQL milling process. International Journal of Precision Engineering and Manufacturing Green Technology, 3, 5–12.CrossRefGoogle Scholar
  3. 3.
    Hill, R. (1948). A theory of the yielding and plastic flow of anisotropic metals. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 193, 281–297.MathSciNetzbMATHGoogle Scholar
  4. 4.
    Pearce, R. (1968). Some aspects of anisotropic plasticity in sheet metals. International Journal of Mechanical Sciences, 10, 995–1004.CrossRefGoogle Scholar
  5. 5.
    Hill, R. (1979). Theoretical plasticity of textured aggregates. Mathematical Proceedings of the Cambridge Philosophical Society, 85, 179–191.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Hill, R. (1990). Constitutive modelling of orthotropic plasticity in sheet metals. Journal of the Mechanics and Physics of Solids, 38, 405–417.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Banabic, D., Kuwabara, T., Balan, T., Comsa, D. S., & Julean, D. (2003). Non-quadratic yield criterion for orthotropic sheet metals under plane-stress conditions. International Journal of Mechanical Sciences, 45, 797–811.CrossRefzbMATHGoogle Scholar
  8. 8.
    Banabic, D., Aretz, H., Comsa, D. S., & Paraianu, L. (2005). An improved analytical description of orthotropy in metallic sheets. International Journal of Plasticity, 21, 493–512.CrossRefzbMATHGoogle Scholar
  9. 9.
    Barlat, F., Lege, D. J., & Brem, J. C. (1991). A six-component yield function for anisotropic materials. International Journal of Plasticity, 7, 693–712.CrossRefGoogle Scholar
  10. 10.
    Barlat, F., et al. (1997). Yielding description for solution strengthened aluminum alloys. International Journal of Plasticity, 13, 385–401.CrossRefGoogle Scholar
  11. 11.
    Barlat, F., et al. (2003). Plane stress yield function for aluminum alloy sheets—part 1: Theory. International Journal of Plasticity, 19, 1297–1319.CrossRefzbMATHGoogle Scholar
  12. 12.
    Barlat, F., et al. (2005). Linear transfomation-based anisotropic yield functions. International Journal of Plasticity, 21, 1009–1039.CrossRefzbMATHGoogle Scholar
  13. 13.
    Barlat, F., Yoon, J. W., & Cazacu, O. (2007). On linear transformations of stress tensors for the description of plastic anisotropy. International Journal of Plasticity, 23, 876–896.CrossRefzbMATHGoogle Scholar
  14. 14.
    Hu, W. (2005). An orthotropic yield criterion in a 3-D general stress state. International Journal of Plasticity, 21, 1771–1796.CrossRefzbMATHGoogle Scholar
  15. 15.
    Hu, W. (2007). Constitutive modeling of orthotropic sheet metals by presenting hardening-induced anisotropy. International Journal of Plasticity, 23, 620–639.CrossRefzbMATHGoogle Scholar
  16. 16.
    Spitzig, W. A., & Richmond, O. (1984). The effect of pressure on the flow stress of metals. Acta Metallurgica, 32, 457–463.CrossRefGoogle Scholar
  17. 17.
    Stoughton, T. B. (2002). A non-associated flow rule for sheet metal forming. International Journal of Plasticity, 18, 687–714.CrossRefzbMATHGoogle Scholar
  18. 18.
    Cvitanić, V., Vlak, F., & Lozina, Ž. (2008). A finite element formulation based on non-associated plasticity for sheet metal forming. International Journal of Plasticity, 24, 646–687.CrossRefzbMATHGoogle Scholar
  19. 19.
    Stoughton, T. B., & Yoon, J. W. (2004). A pressure-sensitive yield criterion under a non-associated flow rule for sheet metal forming. International Journal of Plasticity, 20, 705–731.CrossRefzbMATHGoogle Scholar
  20. 20.
    Stoughton, T. B., & Yoon, J. W. (2006). Review of Drucker’s postulate and the issue of plastic stability in metal forming. International Journal of Plasticity, 22, 391–433.CrossRefzbMATHGoogle Scholar
  21. 21.
    Stoughton, T. B., & Yoon, J. W. (2008). On the existence of indeterminate solutions to the equations of motion under non-associated flow. International Journal of Plasticity, 24, 583–613.CrossRefzbMATHGoogle Scholar
  22. 22.
    Taherizadeh, A., Green, D. E., Ghaei, A., & Yoon, J.-W. (2010). A non-associated constitutive model with mixed iso-kinematic hardening for finite element simulation of sheet metal forming. International Journal of Plasticity, 26, 288–309.CrossRefzbMATHGoogle Scholar
  23. 23.
    Taherizadeh, A., Green, D. E., & Yoon, J. W. (2011). Evaluation of advanced anisotropic models with mixed hardening for general associated and non-associated flow metal plasticity. International Journal of Plasticity, 27, 1781–1802.CrossRefzbMATHGoogle Scholar
  24. 24.
    Taherizadeh, A., Green, D. E., & Yoon, J. W. (2015). A non-associated plasticity model with anisotropic and nonlinear kinematic hardening for simulation of sheet metal forming. International Journal of Solids and Structures, 69–70, 370–382.CrossRefGoogle Scholar
  25. 25.
    Yoshida, F., Hamasaki, H., & Uemori, T. (2013). A user-friendly 3D yield function to describe anisotropy of steel sheets. International Journal of Plasticity, 45, 119–139.CrossRefGoogle Scholar
  26. 26.
    Tang, S. C., Pan, J. (2007). Mechanics modeling of sheet metal forming. Warrendale: SAE International.  https://doi.org/10.4271/r-321. ISBN 978-0-7680-0896-8.
  27. 27.
    Safaei, M., Yoon, J. W., & De Waele, W. (2014). Study on the definition of equivalent plastic strain under non-associated flow rule for finite element formulation. International Journal of Plasticity, 58, 219–238.CrossRefGoogle Scholar
  28. 28.
    Stoughton, T. B., & Yoon, J. W. (2009). Anisotropic hardening and non-associated flow in proportional loading of sheet metals. International Journal of Plasticity, 25, 1777–1817.CrossRefzbMATHGoogle Scholar
  29. 29.
    Safaei, M., Zang, S. L., Lee, M. G., & De Waele, W. (2013). Evaluation of anisotropic constitutive models: Mixed anisotropic hardening and non-associated flow rule approach. International Journal of Mechanical Sciences, 73, 53–68.CrossRefGoogle Scholar
  30. 30.
    Cardoso, R. P. R., & Yoon, J. W. (2009). Stress integration method for a nonlinear kinematic/isotropic hardening model and its characterization based on polycrystal plasticity. International Journal of Plasticity, 25, 1684–1710.CrossRefzbMATHGoogle Scholar
  31. 31.
    Ito, K., Mori, N., Uemura, G., Oya, T., & Yanagimoto, J. (2013). Developement of the stress rate dependence constitutive model to plastic anisotropy. Proceedings of IDDRG, 2013, 107–112.Google Scholar
  32. 32.
    Oya, T., Yanagimoto, J., Ito, K., Uemura, G., & Mori, N. (2014). Material model based on non-associated flow rule with higher order yield function for anisotropic metals. ICTP 2014 Procedia Engineering, 81, 1210–1215.Google Scholar
  33. 33.
    Park, T., & Chung, K. (2012). Non-associated flow rule with symmetric stiffness modulus for isotropic-kinematic hardening and its application for earing in circular cup drawing. International Journal of Solids and Structures, 49, 3582–3593.CrossRefGoogle Scholar
  34. 34.
    Simo, J. C., & Hughes, T. J. R. (1998). Computational inelasticity, interdisciplinary applied mathematics. New York: Springer. ISBN 0387975209.Google Scholar
  35. 35.
    Safaei, M., Lee, M. G., & De Waele, W. (2015). Evaluation of stress integration algorithms for elastic–plastic constitutive models based on associated and non-associated flow rules. Computer Methods in Applied Mechanics and Engineering, 295, 414–445.MathSciNetCrossRefGoogle Scholar
  36. 36.
    Yoon, J. W., Yang, D. Y., & Chung, K. (1999). Elasto-plastic finite element method based on incremental deformation theory and continuum based shell elements for planar anisotropic sheet materials. Computer Methods in Applied Mechanics and Engineering, 174, 23–56.CrossRefzbMATHGoogle Scholar
  37. 37.
    Yoon, J. W., Barlat, F., Dick, R. E., Chung, K., & Kang, T. J. (2004). Plane stress yield function for aluminum alloy sheets—Part II: FE formulation and its implementation. International Journal of Plasticity, 20, 495–522.CrossRefzbMATHGoogle Scholar
  38. 38.
    ISO 6892-1:2016 Metallic materials—tensile testing—Part 1: Method of test at room temperature.Google Scholar
  39. 39.
    Merklein, M., & Biasutti, M. (2011). Forward and reverse simple shear test experiments for material modeling in forming simulations. In Special edition: 10th international conference on technology of plasticity, ICTP 2011 (pp. 702–707).Google Scholar
  40. 40.
    Yoon, J. W., & Barlat, F. (2006). Modeling and simulation of the forming of aluminum sheet alloys. In ASM Handbook, Metalworking: Sheet Forming, Vol. 14B (pp. 792–826). Materials Park: ASM International.Google Scholar
  41. 41.
    Yoon, J. W., Barlat, F., Dick, R. E., & Karabin, M. E. (2006). Prediction of six or eight ears in a drawn cup based on a new anisotropic yield function. International Journal of Plasticity, 22, 174–193.CrossRefzbMATHGoogle Scholar
  42. 42.
    Yoon, J. W., Barlat, F., Chung, K., Pourboghrat, F., & Yang, D. Y. (2000). Earing predictions based on asymmetric nonquadratic yield function. International Journal of Plasticity, 16, 1075–1104.CrossRefzbMATHGoogle Scholar
  43. 43.
    Kuwabara, T., Mori, T., Asano, M., Hakoyama, T., & Barlat, F. (2017). Material modeling of 6016-O and 6016-T4 aluminum alloy sheets and application to hole expansion forming simulation. International Journal of Plasticity, 93, 164–186.CrossRefGoogle Scholar

Copyright information

© Korean Society for Precision Engineering 2019

Authors and Affiliations

  1. 1.Graduate School of EngineeringThe University of TokyoTokyoJapan
  2. 2.M&M Research Inc. (Professor Emeritus, Tohoku University)TokyoJapan
  3. 3.M&M Research Inc.TokyoJapan
  4. 4.Department of System Design EngineeringKeio UniversityTokyoJapan
  5. 5.Institute of Industrial ScienceThe University of TokyoTokyoJapan

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