On the Main Hypotheses of the General Mathematical Theory of Plasticity and the Limits of Their Applicability

Abstract—

The article discusses and analyzes the issues of applicability and the limits of applicability of some of the main hypotheses of the general mathematical theory of plasticity. In the theory of elastoplastic deformation processes, this is the postulate of the isotropy of initially isotropic bodies, in which the invariance of orthogonal transformations of the process images is established when a relationship between stresses and deformations is established. In the theory of flow, this is a hypothesis about the decomposition of total deformations into elastic and plastic deformations and the influence on its relationship between stresses and deformations under complex loading.

This is a preview of subscription content, access via your institution.

REFERENCES

  1. 1

    A. A. Il’yushin, Selected Works, Vol. 2: Plasticity (1946–1966) (Fizmatlit, Moscow, 2004) [in Russian].

  2. 2

    A. A. Il’yushin, Plasticity. Foundations of the General Mathematical Theory (Izd-vo AN SSSR, Moscow, 1963) [in Russian].

    Google Scholar 

  3. 3

    A. A. Il’yushin, Continuum Mechanics (MGU, Moscow, 1990) [in Russian].

    Google Scholar 

  4. 4

    A. A. Il’yushin, “Once again on the isotropy postulate,” Izv. Akad. Nauk SSSR, Mekh. Mashinostr. No. 1, 201–204 (1962).

    Google Scholar 

  5. 5

    A. Yu. Ishlinskii and D. D. Ivlev, Mathematical Theory of Plasticity (Fizmatlit, Moscow, 2001)[in Russian].

    Google Scholar 

  6. 6

    D. D. Ivlev, “On the postulate of isotropy in the theory of plasticity,” Izv. Akad. Nauk SSSR, Mekh. Mashiriostr. No. 2, 125–127 (1960).

    Google Scholar 

  7. 7

    Theory of Plasticity, Eb. by Yu. N. Rabotnov, (Izdat. Inostr. Lit., Moscow, 1948) [in Russian].

  8. 8

    A. Nadai, Theory of Flow and Fracture of Solids (McGraw-Hill, New York, 1950).

    Google Scholar 

  9. 9

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

    Google Scholar 

  10. 10

    I. S. Sokol’nikov, Tensor Analysis (Nauka, Moscow, 1971) [in Russian].

    Google Scholar 

  11. 11

    V. G. Zubchaninov, Mechanics of Processes of Plastic Media (Fizmatlit, Moscow, 2010) [in Russian].

    Google Scholar 

  12. 12

    V.G. Zubchaninov, “The general mathematical theory of plasticity and the Il’yushin postulates of macroscopic definability and isotropy,” Moscow Univ. Mech. Bull. 73, 101–116 (2018).

    Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to V. G. Zubchaninov.

Additional information

Translated by I. K. Katuev

About this article

Verify currency and authenticity via CrossMark

Cite this article

Zubchaninov, V.G. On the Main Hypotheses of the General Mathematical Theory of Plasticity and the Limits of Their Applicability. Mech. Solids 55, 820–826 (2020). https://doi.org/10.3103/S0025654420060163

Download citation

Keywords:

  • elasticity
  • plasticity
  • stress and strain tensors
  • invariants
  • complex loading
  • stress and strain vectors
  • isotropy postulate
  • deformation decomposition hypothesis