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Correlation between Geometrically Nonlinear Elastoviscoplastic Constitutive Relations Formulated in Terms of the Actual and Unloaded Configurations for Crystallites

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Abstract

A correct description of severe plastic deformation requires the use of nonlinear kinematic and constitutive relations. The relations should meet certain criteria, such as the frame-independence, closed stress cycles and the absence of energy dissipation under elastic cyclic loading. A most complicated issue in formulating these relations is the decomposition of motion into quasi-rigid and strain-induced motions, which is an extremely difficult problem with no unambiguous solution. For the majority of structural metals and alloys, this decomposition can be done in a physically justified way on the level of crystallites. Earlier, using a multilevel approach we introduced a corotational coordinate system for crystallites responsible for quasi-rigid motion. This allowed us to formulate frame-independent mesoscopic constitutive relations in the actual configuration and to perform test calculations to show that the above criteria are satisfied with high accuracy. A new way of motion decomposition was proposed which is a multiplicative representation of the deformation gradient with an explicit extraction of the corotational frame motion. Elastoviscoplastic constitutive relations satisfying the above criteria were formulated in terms of an unloaded "lattice" configuration. However, it is more preferable to formulate nonlinear boundary value problems in terms of the actual configuration in rates. This paper is aimed to study correlation between the earlier derived constitutive relations formulated in terms of the actual configuration and in terms of the unloaded "lattice" configuration. Example problems are solved to demonstrate the closeness of results obtained with these constitutive relations.

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References

  1. Segal, V.M., Reznikov, V.I., Drobyshevskii, A.E., and Kopylov, V.I., Plastic Working of Metals by Simple Shear, Russian Metallurgy (Metally), 1981, no. 1, pp. 99–105.

    Google Scholar 

  2. Valiev, R.Z. and Alexandrov, I.V., Nanostructured Materials Produced by Severe Plastic Deformation, Moscow: Logos, 2000.

    Google Scholar 

  3. Truesdell, C.A., A First Course in Rational Continuum Mechanics, Boston: Academic Press, 1991.

    MATH  Google Scholar 

  4. Lurie, A.I., Nonlinear Theory of Elasticity, Amsterdam, N.Y.: North-Holland, 1990.

    MATH  Google Scholar 

  5. Korobeinikov, S.N., Nonlinear Deformation of Solids, Novosibirsk: Izd-vo SORAN, 2000.

    MATH  Google Scholar 

  6. Levitas, V.I., Large Deformation of Materials with Complex Rheological Properties at Normal and High Pressure, Commack, N.Y.: Nova Publishers, 1996.

    Google Scholar 

  7. Pozdeev, A.A., Trusov, P.V., and Nyashin, Yu.I., Large Elastic-Plastic Deformations: Theory, Algorithms, Applications, Moscow: Nauka, 1986.

    MATH  Google Scholar 

  8. Makarov, P.V., Smolin, I.Yu., Stefanov, Yu.P., Kuznetsov, P.V., Trubitsyn, A.A., Trubitsyna, N.V., Voroshilov, S.P., and Voroshilov, Ya.S., Nonlinear Mechanics of Geomaterials and Geomedia, Novosibirsk: GEO, 2007.

    Google Scholar 

  9. Trusov, P.V., Shveikin, A.I., and Yanz, A.Yu., Motion Decomposition, Frame-Indifferent Derivatives, and Constitutive Relations at Large Displacement Gradients from the Viewpoint of Multilevel Modeling, Phys. Mesomech., 2017, vol. 20, no. 4, pp. 357–376.

    Article  Google Scholar 

  10. Trusov, P.V. and Shveikin, A.I., On Motion Decomposition and Constitutive Relations in Geometrically Nonlinear Elastoviscoplasticity of Crystallites, Phys. Mesomech., 2017, vol. 20, no. 4, pp. 377–391.

    Article  Google Scholar 

  11. Kolarov, D., Baltov, A., and Bonceva, N., Mechanika Plasticeskich Sred, Moscow: Mir, 1979.

    Google Scholar 

  12. Kondaurov, V.I. and Nikitin, L.V., Theoretical Foundations of Rheology of Geomaterials, Moscow: Nauka, 1990.

    Google Scholar 

  13. Zienkiewicz, O.C., The Finite Element Method in Engineering Science, London: McGraw-Hill, 1971.

    MATH  Google Scholar 

  14. Zienkiewicz, O.C. and Morgan, K., Finite Elements and Approximation, Mineola, N.Y.: Dover Publications, 2006.

    MATH  Google Scholar 

  15. Oden, J.T., Finite Elements of Nonlinear Continua, Mineola, N.Y.: Dover Publications, 2006.

    MATH  Google Scholar 

  16. Zaremba, S., Sur une Forme Perfectionnee de la Theorie de la Relaxation, Bull. Int. Acad. Sci. Cracovie, 1903, pp. 595–614.

    Google Scholar 

  17. Jaumann, G., Geschlossenes System physikalischer und chemischer Differential-gesetze, Sitzber. Akad. Wiss. Wien. Abt. IIa, 1911, vol. 120, pp. 385–530.

    MATH  Google Scholar 

  18. Green, A.E. and Naghdi, P.M., A General Theory of an Elasto-Plastic Continuum, Arch. Rat. Mech. Anal., 1965, vol. 18, pp. 251–281.

    Article  MATH  Google Scholar 

  19. Oldroyd, J.G., On the Formulation of Rheological Equations of State, Proc. R. Soc. Lond. A, 1950, vol. 200, pp. 523–541.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. Cotter, B.A. and Rivlin, R.S., Tensors Associated with Time-Dependent Stress, Q. Appl. Math., 1955, vol. 13, pp. 177–182.

    Article  MathSciNet  MATH  Google Scholar 

  21. Makarov, P.V., Microdynamic Theory of Plasticity and Failure of Structurally Inhomogeneous Media, Russ. Phys. J., 1992, vol. 35, no. 4, pp. 334–346.

    Article  Google Scholar 

  22. Makarov, P.V., Simulation of Mesoscale Elastoplastic Deformation and Fracture of Heterogeneous Media, Phys. Mesomech., 2003, vol. 6, no. 4, pp. 99–112.

    Google Scholar 

  23. Trusov, P.V., Ashikhmin, V.N., Volegov, P.S., and Shveykin, A.I., Mathematical Modeling of the Evolution of Polycrystalline Materials Structure under Elastoplastic Deformation, Uch. Zap. Kazan. Univ. Ser. Fiz.-Mat. Nauki, 2010, vol. 152, no. 4, pp. 225–237.

    MathSciNet  Google Scholar 

  24. Xiao, H., Bruhns, O.T., and Meyers, A., Logarithmic Strain, Logarithmic Spin and Logarithmic Rate, Acta Mech., 1997, vol. 124, pp. 89–105.

    Article  MathSciNet  MATH  Google Scholar 

  25. Bruhns, O.T., Xiao, H., and Meyers, A., New Results for the Spin of the Eulerian Triad and the Logarithmic Spin and Rate, Acta Mech., 2002, vol. 155, pp. 95–109.

    Article  MATH  Google Scholar 

  26. Meyers, A., Xiao, H., and Bruhns, O., Elastic Stress Ratcheting and Corotational Stress Rates, Tech. Mech., 2003, vol. 23, no. 2–4, pp. 92–102.

    Google Scholar 

  27. Xiao, H., Bruhns, O.T., and Meyers, A., The Choice of Objective Rates in Finite Elastoplasticity: General Results on the Uniqueness of the Logarithmic Rate, Proc. R. Soc. Lond. A, 2000, vol. 456, pp. 1865–1882.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  28. Trusov, P.V., Kondratev, N.S., and Shveykin, A.I., About Geometrically Nonlinear Constitutive Relations for Elastic Material, PNRPU Mech. Bull., 2015, no. 3, pp. 182–200.

    Google Scholar 

  29. Xiao, H., Bruhns, O.T., and Meyers, A., A Natural Generalization of Hypoelasticity and Eulerian Rate Type Formulation of Hyperelasticity, J. Elasticity, 1999, vol. 56, pp. 59–93.

    Article  MathSciNet  MATH  Google Scholar 

  30. Xiao, H., Bruhns, O.T., and Meyers, A., A Consistent Finite Elastoplasticity Theory Combining Additive and Multiplicative Decomposition of the Stretching and the Deformation Gradient, Int. J. Plasticity, 2000, vol. 16, pp. 143–177.

    Article  MATH  Google Scholar 

  31. Rybin, V.V., High Plastic Strains and Fracture of Metals, Moscow: Metallurgiya, 1986.

    Google Scholar 

  32. Honeycombe, R.W.K., The Plastic Deformation of Metals, London: E. Arnold, 1984.

    Google Scholar 

  33. Mandel, J., Equations Constitutives et Directeurs dans les Milieux Plastiques et Viscoplastiques, Int. J. Solids Struct., 1973, vol. 9, pp. 725–740.

    Article  MATH  Google Scholar 

  34. Rubin, M.B., On the Treatment of Elastic Deformation in Finite Elastic-Viscoplastic Theory, Int. J. Plasticity, 1996, vol. 12, no. 7, pp. 951–965.

    Article  MATH  Google Scholar 

  35. Rubin, M.B., Physical Reasons for Abandoning Plastic Deformation Measures in Plasticity and Viscoplasticity Theory, Arch. Mech., 2001, vol. 53, no. 4–5, pp. 519–539.

    MATH  Google Scholar 

  36. Rubin, M.B., Plasticity Theory Formulated in Terms of Physically Based Microstructural Variables. Part I. Theory, Int. J. Solids Struct., 1994, vol. 31, no. 19, pp. 2615–2634.

    Article  MATH  Google Scholar 

  37. Rubin, M.B., Plasticity Theory Formulated in Terms of Physically Based Microstructural Variables. Part II.Examples, Int. J. Solids Struct., 1994, vol. 31, no. 19, pp. 2635–2652.

    Article  MATH  Google Scholar 

  38. Trusov, P.V., Shveykin, A.I., Nechaeva, E.S., and Volegov, P.S., Multilevel Models of Inelastic Deformation of Materials and Their Application for Description of Internal Structure Evolution, Phys. Mesomech., 2012, vol. 15, no. 3–4, pp. 155–175.

    Article  Google Scholar 

  39. Trusov, P.V. and Shveykin, A.I., Multilevel Crystal Plasticity Models of Single- and Polycrystals. Statistical Models, Phys. Mesomech., 2013, vol. 16, no. 1, pp. 23–33.

    Article  Google Scholar 

  40. Trusov, P.V. and Shveykin, A.I., Multilevel Crystal Plasticity Models of Single- and Polycrystals. Direct Models, Phys. Mesomech., 2013, vol. 16, no. 2, pp. 99–124.

    Article  Google Scholar 

  41. Shermergor, T.D., Theory of Elasticity of Microheterogeneous Media, Moscow: Nauka, 1977.

    Google Scholar 

  42. Kondratev, N.S. and Trusov, P.V., A Mathematical Model for Deformation of BCC Single Crystals Taking into Consideration the Twinning Mechanism, Comp. Cont. Mech., 2011, vol. 4, no. 4, pp. 20–33.

    Article  Google Scholar 

  43. Shveykin, A.I. and Sharifullina, E.R., Analysis of Constitutive Relations for Intragranular Dislocation Sliding Description within Two-Level Elasto-Viscoplastic Model of FCC-Polycrystals, Tambov Univ. Reports, 2013, vol. 18, no. 4, pp. 1665–1666.

    Google Scholar 

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  1. Perm National Research Polytechnic University, Perm, 614990, Russia

    A. I. Shveikin & P. V. Trusov

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  1. A. I. Shveikin
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Original Russian Text © A.I. Shveikin, P.V. Trusov, 2016, published in Fizicheskaya Mezomekhanika, 2016, Vol. 19, No. 5, pp. 48–57.

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Shveikin, A.I., Trusov, P.V. Correlation between Geometrically Nonlinear Elastoviscoplastic Constitutive Relations Formulated in Terms of the Actual and Unloaded Configurations for Crystallites. Phys Mesomech 21, 193–202 (2018). https://doi.org/10.1134/S1029959918030025

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  • DOI: https://doi.org/10.1134/S1029959918030025

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