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The construction of constitutive relations for isotropic strain-hardening elastoplastic materials of the differential type of complexity n. Part 1. Finite strains

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Using approaches in rational continuum mechanics, we have developed a mathematical theory of strict construction and specialization of constitutive equations for simple (in Noll’s sense) isotropic strain-hardening elastoplastic materials of the differential type of complexity n (analogs of the Rivlin–Ericksen solids of complexity n) as the most important representatives of the materials with infinitesimal fading path shape memory (a fading path shape memory on an arbitrarily small interval of the “past”). The strains are assumed to be arbitrary. The hierarchy of the constitutive relations is constructed according to the level of complexity of the material response to deformation.

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References

  1. 1.

    P. P. Lepikhin, “Simulation of the fading memory of form of the trajectory in the theory of simple materials with elastoplastic behavior. Part 1. Finite strains,” Strength Mater., 36, No. 5, 494–503 (2004).

  2. 2.

    P. P. Lepikhin, “Simulation of fading path shape memory in the theory of simple elastoplastically deforming materials,” in: Abstracts of the Euromech (European Mechanics Society) Colloquium 458 “Advanced Methods in Validation and Identification of Nonlinear Constitutive Equations in Solid Mechanics,” Russian Foundation for Basic Research, Lomonosov Moscow State University, Institute of Mechanics, Moscow University Press, Moscow (2004).

  3. 3.

    P. P. Lepikhin, “Modeling of fading path shape memory in the theory of Noll-simple materials with elastoplastic behavior,” in: Advanced Equipment and Processes in Mechanical Engineering, Instrument Engineering, and Welding Fabrication, Proc. Int. Conf. Dedicated to the 100th Anniversary of the Mechanical Engineering and 50th Anniversary of the Welding Engineering (May 25–28, 1998) [in Russian], National Technical University of Ukraine “KPI,” Kiev (1998), Vol. 3, pp. 105–109.

  4. 4.

    P. P. Lepikhin, “Simulation of the fading memory of the form of trajectory in the theory of simple materials with elastoplastic behavior. Part 2. Infinitely small strains,” Strength Mater., 36, No. 6, 612–620 (2004).

  5. 5.

    P. P. Lepikhin, “Simulation of fading path shape memory in the theory of simple materials with elastoplastic behavior and initial loading surface,” Strength Mater., 39, No. 4, 339–348 (2007).

  6. 6.

    V. T. Troshchenko, A. Ya. Krasovskii, V. V. Pokrovskii, et al., Resistance of Materials to Deformation and Fracture. Handbook [in Russian], Vol. 1, Naukova Dumka, Kiev (1993).

  7. 7.

    V. T. Troshchenko, A. Ya. Krasovskii, V. V. Pokrovskii, et al., Resistance of Materials to Deformation and Fracture. Handbook [in Russian], Vol. 2, Naukova Dumka, Kiev (1993).

  8. 8.

    C. Truesdell, A First Course in Rational Continuum Mechanics, The Johns Hopkins University, Baltimore (1972).

  9. 9.

    C. Truesdell and W. Noll, The Non-Linear Field Theories of Mechanics, Springer, New York (1992).

  10. 10.

    P. P. Lepikhin, Theoretical Construction of Constitutive Relations for Simple Initially Isotropic Inelastic Solid Materials. Finite Strains [in Russian], Academy of Sciences of Ukraine, Institute of Problems of Strength, Preprint, Kiev (1993).

  11. 11.

    P. P. Lepikhin, Structure of Constitutive Relations for Viscoelastic-Viscoplastic State of Materials [in Russian], Author’s Abstract of the Doctor Degree Thesis (Phys&Math. Sci.), Kiev (1997).

  12. 12.

    P. P. Lepikhin, “Simulation of the proportional deformation of elastoplastic continua simple in Noll’s sense. Part 1. Defining relations,” Strength Mater., 30, No. 5, 497–506 (1998).

  13. 13.

    A. A. Pozdeev, P. V. Trusov, and Yu. I. Nyashin, Large Elastoplastic Strains: Theory, Algorithms, and Applications [in Russian], Nauka, Moscow (1986).

  14. 14.

    E. Spencer, Theory of Invariants [Russian translation], Mir, Moscow (1974).

  15. 15.

    R. S. Rivlin and J. L. Ericksen, “Stress-deformation relations for isotropic materials,” J. Rat. Mech. Analysis, 4, No. 2, 323–425 (1955).

  16. 16.

    B. E. Pobedrya, Lectures on the Tensor Analysis. Textbook [in Russian], 3rd edition, Moscow University Publ., Moscow (1986).

  17. 17.

    R. S. Rivlin, “Further remarks of the stress-deformation relations for isotropic materials,” J. Rat. Mech. Analysis, 4, No. 5, 681–702 (1955).

  18. 18.

    V. V. Novozhilov, “On the forms of relationship between stresses and strains in initially isotropic inelastic bodies (geometric aspects),” Prikl. Matem. Mekh., 27, Issue 5, 794–812 (1963).

  19. 19.

    A. Freidental and H. Geiringer, A Mathematical Theory of Inelastic Continuous Medium [Russian translation], GIFML, Moscow (1962).

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Correspondence to P. P. Lepikhin.

Additional information

Translated from Problemy Prochnosti, No. 2, pp. 27–42, March–April, 2009.

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Lepikhin, P.P. The construction of constitutive relations for isotropic strain-hardening elastoplastic materials of the differential type of complexity n. Part 1. Finite strains. Strength Mater 41, 135–146 (2009). https://doi.org/10.1007/s11223-009-9119-2

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Keywords

  • mathematical theory
  • constitutive relation
  • simple (in Noll’s sense) strain-hardening elastoplastic material of the differential type of complexity n
  • finite strains
  • isotropy