Skip to main content
SpringerLink
Log in

Generalization of the Prandtl solution to the case of axisymmetric deformation of materials obeying the double shear model

  • Published:
Mechanics of Solids Aims and scope Submit manuscript

Abstract

A semianalytic solution of the problem on the compression of an annular layer of a plastic material obeying the double shear model on a cylindrical mandrel is obtained. The approximate statement of boundary conditions, which cannot be satisfied exactly in the framework of the constructed solution, is based on the same assumptions as the statement of the classical plasticity problem of compression of a material layer between rough plates (Prandtl’s problem). It is assumed that the maximum friction law is satisfied on the inner surface of the layer. The solution is singular near this surface. The strain rate intensity factor is calculated, and its dependence on the process and material parameters is shown.

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. R. Hill, TheMathematical Theory of Plasticity (Oxford Univ. Press, Oxford, 1950; Gostekhizdat,Moscow, 1956).

    Google Scholar 

  2. L.M. Kachanov, Theory of Creep The Theory of Creep (Fizmatgiz, Moscow, 1960; Nat. Lending Lib., 1967).

    Google Scholar 

  3. A. I. Kuznetsov, “The Problem on the Inhomogeneous Layer,” Arch.Mech. Stos. 12(2), 163–172 (1960).

    MathSciNet  MATH  Google Scholar 

  4. V. N. Dudukalenko and D. D. Ivlev, “Compression of Bands of Plastic Material which can be Strengthened with Rigid Rough Plates,” Dokl. Akad. Nauk SSSR 153(5), 1024–1026 (1963) [Sov. Phys. Dokl. (Engl. Transl.) 8, 1252–1254 (1964)].

    Google Scholar 

  5. J. Najar, “Inertia Effects in the Problem of Compression of a Perfectly Plastic Layer between Two Rigid Plates,” Arch.Mech. Stos. 19(1), 129–149 (1967).

    Google Scholar 

  6. E. A. Marshall, “The Compression of a Slab of Ideal Soil between Rough Plates,” Acta Mech. 3, 82–92 (1967).

    Article  Google Scholar 

  7. I. F. Collins and S. A. Mequid, “On the Influence of Hardening and Anisotropy on the Plane-Strain Compression of Thin Metal Strip,” Trans.ASME. J. Appl. Mech. 44(2), 271–278 (1977).

    Article  ADS  MATH  Google Scholar 

  8. M. J. Adams, B. J. Briscoe, G. M. Corfield, et al., “An Analysis of the Plane-Strain Compression of Viscoplastic Materials,” Trans. ASME. J. Appl. Mech. 64(2), 420–424 (1997).

    Article  ADS  MATH  Google Scholar 

  9. R. I. Nepershin, “Influence of Heat Transfer on the Nonisothermic Plane Flow in Compression of a Thin Blank between Plane Punches,” Probl. Mashinostr. Nadezh. Mashin, No. 1, 96–103 (1997).

  10. S. E. Aleksandrov, I. D. Baranova, and G. Mishuris, “Compression of a Viscoelastic Layer between Rough Parallel Plates,” Izv. Akad. Nauk. Mekh. Tverd. Tela, No. 6, 33–39 (2008) [Mech. Solids (Engl. Transl.) 43 (6), 863–869 (2008)].

  11. A. J.M. Spencer, “A Theory of the Kinematics of Ideal Soils under Plane Strain Conditions,” J.Mech. Phys. Solids 12, 337–351 (1964).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  12. A. J.M. Spencer, “Deformation of Ideal Granular Materials,” in Mechanics of Solids, The Rodney Hill 60th Anniversary Volume, Ed. by H. G. Hopkins and M. J. Sewell (Pergamon Press, Oxford, 1982), pp. 607–652.

    Google Scholar 

  13. W. A. Spitzig, R. J. Sober, and O. Richmond, “The Effect of Hydrostatic Pressure on the Deformation Behavior of Maraging and HY-80 Steels and Its Implications for Plasticity Theory,” Metallurg. Trans. 7A(11), 1703–1710 (1976).

    Article  ADS  Google Scholar 

  14. A. J. M. Spencer, “A Theory of the Failure of Ductile Materials Reinforced by Elastic Fibres,” Int. J. Mech. Sci. 7, 197–209 (1965).

    Article  Google Scholar 

  15. S. E. Aleksandrov and Y.-R. Jeng, “A Generalization of Prandtl’s and Spencer’s Solutions on Axisymmetric Viscous Flow,” Arch. Appl. Mech. 81(4), 437–449 (2011).

    Article  MathSciNet  ADS  Google Scholar 

  16. S. E. Aleksandrov and E. A. Lyamina, “Singular Solutions for Plane Plastic Flow of Pressure-Dependent Materials,” Dokl. Ross. Akad. Nauk 383(4), 492–495 (2002) [Dokl. Phys. (Engl. Transl.) 47 (4), 308–311 (2002)].

    MathSciNet  Google Scholar 

  17. S. E. Aleksandrov, “Singular Solutions in an Axisymmetric Flow of a Medium Obeying the Double Shear Model,” Zh. Prikl.Mekh. Tekhn. Fiz. 46(5), 180–186 (2005) [J. Appl.Mech. Tech. Phys. (Engl. Transl.) 46 (5), 766–771 (2005)].

    ADS  MATH  Google Scholar 

  18. T. A. Trunina and E. A. Kokovkin, “Formation of Highly Dispersed Structure in Surface Layers of Steel under Combined Treatment with Hydraulic Forging,” Probl. Mashinostr. Nadezh. Mashin, No. 2, 71–74 (2008).

  19. S. E. Aleksandrov, D. Z. Grabko, and O. A. Shikimaka, “To the Determination of Intensive Strain Layer Thickness near the Friction Surface in Metal Forming Processes,” Probl. Mashinostr. Nadezh. Mashin, No. 3, 72–78 (2009).

  20. S. E. Aleksandrov and E. A. Lyamina, “Prediction of Fracture in the Vicinity of Friction Surfaces in Metal Forming Processes,” Zh. Prikl.Mekh. Tekhn. Fiz. 47(5), 169–174 (2006) [J. Appl.Mech. Tech. Phys. (Engl. Transl.) 47 (5), 757–761 (2006)].

    ADS  Google Scholar 

  21. E. A. Lyamina and S. E. Aleksandrov, “Application of the Strain Rate Intensity Factor to Modeling Material Behavior in the Vicinity of Frictional Interfaces,” in Lecture Notes in Applied and Computational Mechanics, Vol. 58: Trends in Computational Contact Mechanics, Ed. by Giorgio Zavarise and Peter Wriggers (Springer, 2011), pp. 291–320.

  22. S. E. Aleksandrov and E. A. Lyamina, “On Constructing the Theory of Ductile Fracture near Friction Surfaces,” Zh. Prikl.Mekh. Tekhn. Fiz. 52(4), 183–190 (2011) [J. Appl. Mech. Tech. Phys. (Engl. Transl.) 52 (4), 657–663 (2011)].

    Google Scholar 

  23. S. E. Aleksandrov, “The Strain Rate Intensity Factor and Its Applications: A Review,” Mater. Sci. Forum 623, 1–20 (2009).

    Article  Google Scholar 

  24. S. E. Aleksandrov and E. A. Lyamina, Strain Rate Intensity Factor in Double Shear Model (LAP Lambert Academic Publishing GmbH&Co, Saarbrücken, 2011).

    Google Scholar 

  25. S. E. Aleksandrov and O. Richmond, “Singular Plastic Flow Fields near Surfaces of Maximum Friction Stress,” Int. J. Nonlin. Mech. 36(1), 1–11 (2001).

    Article  MathSciNet  Google Scholar 

  26. R. Z. Valiev, R. K. Islamgaliev, and I. V. Aleksandrov, “Bulk Nanostructured Materials from Severe Plastic Deformation,” Prog.Mater. Sci. 45, 103–189 (2000).

    Article  Google Scholar 

  27. S. E. Aleksandrov and R. V. Goldstein, “Kinetic Equation for the Grain Size in Processes of Intense Plastic Deformation,” Dokl. Ross. Akad. Nauk 429(6), 754–757 (2009) [Dokl. Phys. (Engl. Transl.) 54 (12), 553–556 (2009)].

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences, pr-t Vernadskogo 101, str. 1, Moscow, 119526, Russia

    S. E. Aleksandrov & R. V. Goldstein

Authors
  1. S. E. Aleksandrov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. R. V. Goldstein
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. E. Aleksandrov.

Additional information

Original Russian Text © S.E. Aleksandrov, R.V. Goldstein, 2012, published in Izvestiya Akademii Nauk. Mekhanika Tverdogo Tela, 2012, No. 6, pp. 67–79.

About this article

Cite this article

Aleksandrov, S.E., Goldstein, R.V. Generalization of the Prandtl solution to the case of axisymmetric deformation of materials obeying the double shear model. Mech. Solids 47, 654–664 (2012). https://doi.org/10.3103/S0025654412060076

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S0025654412060076

Keywords

Search

Navigation