Skip to main content
SpringerLink
Log in

A method of analysis for planar ideal plastic flows of anisotropic materials

  • Original Paper
  • Published:
Acta Mechanica Aims and scope Submit manuscript

Abstract

The objective of the present paper is to provide an efficient method for finding steady planar ideal plastic flows of anisotropic materials. The method consists of determining two mappings between coordinate systems. One of these mappings is between principal lines-based and characteristics-based coordinate systems, and the other is between Cartesian- and characteristics-based coordinate systems. Thus, the mapping between the Cartesian- and principal lines-based coordinate systems is given in parametric form. It is shown that the boundary value problem of finding the mapping between the principal lines-based and characteristics-based coordinate systems can be reduced to the solution of a telegraph equation where two families of characteristics are curved and to the evaluation of ordinary integrals where one family of characteristics is straight. In either case, after solving this problem the problem of finding the mapping between the Cartesian- and characteristics- based coordinate systems can be reduced to the evaluation of ordinary integrals.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Richmond, O., Alexandrov, S.: The theory of general and ideal plastic deformations of Tresca solids. Acta Mech. 158, 33–42 (2002)

    Article  MATH  Google Scholar 

  2. Hill, R.: Ideal forming operations for perfectly plastic solids. J. Mech. Phys. Solids 15, 223–227 (1967)

    Article  Google Scholar 

  3. Richmond, O.: Theory of streamlined dies for drawing and extrusion. In: Rimrott, F.P.J., Schwaighofer, J. (eds.) Mechanics of the Solid State, pp. 154–167. University of Toronto Press, Toronto (1968)

    Google Scholar 

  4. Richmond, O., Devenpeck, M.L.: A die profile for maximum efficiency in strip drawing. In: Rosenberg, R.M. (ed.) Proceedings of 4th U.S. National Congress on Applied Mechanics, vol. 2, pp. 1053–1057. ASME, New York (1962)

  5. Richmond, O., Morrison, H.L.: Streamlined wire drawing dies of minimum length. J. Mech. Phys. Solids 15, 195–203 (1967)

    Article  Google Scholar 

  6. Chung, K., Alexandrov, S.: Ideal flow in plasticity. Appl. Mech. Rev. 60, 316–335 (2007)

    Article  Google Scholar 

  7. Barlat, F., Chung, K., Richmond, O.: Anisotropic potentials for polycrystals and application to the design of optimum blank shapes in sheet forming. Metall. Mater. Trans. A 25, 1209–1216 (1994)

    Article  Google Scholar 

  8. Chung, K., Barlat, F., Brem, J.C., Lege, D.J., Richmond, O.: Blank shape design for a planar anisotropic sheet based on ideal sheet forming design theory and FEM analysis. Int. J. Mech. Sci. 39, 105–120 (1997)

    Article  MATH  Google Scholar 

  9. Park, S.H., Yoon, J.W., Yang, D.Y., Kim, Y.H.: Optimum blank design in sheet metal forming by the deformation path iteration method. Int. J. Mech. Sci. 41, 1217–1232 (1999)

    Article  MATH  Google Scholar 

  10. Alexandrov, S., Mustafa, Y., Lyamina, E.: Steady planar ideal flow of anisotropic materials. Meccanica 51, 2235–2241 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  11. Collins, I.F., Meguid, S.A.: On the influence of hardening and anisotropy on the plane-strain compression of thin metal strip. ASME J. Appl. Mech. 44, 271–278 (1977)

    Article  MATH  Google Scholar 

  12. Rice, J.R.: Plane strain slip line theory for anisotropic rigid/plastic materials. J. Mech. Phys. Solids. 21, 63–74 (1973)

    Article  MATH  Google Scholar 

  13. Sadowsky, M.A.: Equiareal pattern of stress trajectories in plane plastic strain. ASME J. Appl. Mech. 63, A74–A76 (1941)

    MATH  MathSciNet  Google Scholar 

  14. Alexandrov, S., Harris, D.: Geometry of principal stress trajectories for a Mohr--Coulomb material under plane strain. ZAMM (2017). doi:10.1002/zamm.201500284

    Google Scholar 

  15. Hill, R.: The Mathematical Theory of Plasticity. Clarendon Press, Oxford (1950)

    MATH  Google Scholar 

  16. Chakrabarty, J.: Theory of Plasticity. McGraw-Hill, Singapore (1987)

    Google Scholar 

  17. Druyanov, B.A., Nepershin, R.I.: Problems of Technological Plasticity. Elsevier, Amsterdam (1994)

    Google Scholar 

Download references

Acknowledgements

This research was made possible by the Grant RSCF-16-49-02026.

Author information

Authors and Affiliations

  1. Institute for Problems in Mechanics, Russian Academy of Sciences, 101-1 Prospect Vernadskogo, Moscow, Russia, 119526

    Sergei Alexandrov

  2. Department of Materials Science and Engineering, Seoul National University, 1 Gwanak-ro, Gwanak-gu, Seoul, 08826, Republic of Korea

    Woncheol Jeong

Authors
  1. Sergei Alexandrov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Woncheol Jeong
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Woncheol Jeong.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Alexandrov, S., Jeong, W. A method of analysis for planar ideal plastic flows of anisotropic materials. Acta Mech 228, 3839–3846 (2017). https://doi.org/10.1007/s00707-017-1915-3

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-017-1915-3

Search

Navigation