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Modeling of Localized Inelastic Deformation at the Mesoscale with Account for the Local Lattice Curvature in the Framework of the Asymmetric Cosserat Theory

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Abstract

In the paper, inelastic strain localization in homogeneous specimens and mesovolumes of a polycrystalline material is modeled based on the asymmetric theory of an elastoplastic Cosserat continuum in a two-dimensional formulation for plane strain. It is assumed that rotational deformation in loaded materials occurs due to the development of localized plastic deformation as well as bending and torsion of the material lattice at the micro- and nanoscale levels. For this reason, the parameters of the micropolar model are considered as functions of inelastic strain for each local mesovolume of the continuum. It is shown that the observed parabolic hardening can be attributed to a large extent to the development of rotational deformation modes, bending and torsion, and appearance of couple stresses in the loaded material. The modeling results indicate that if rotational deformation is stopped in the loaded material, its accommodation capacity decreases, the local and macroscopic inelastic strains sharply increase, leading to a much more rapid formation of fracture structures. Conversely, the formation of meso- and nanoscale substructures with high lattice curvature in materials promotes the activation of rotational deformation modes, reduction of localized strains, and relaxation of stress concentrators.

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References

  1. Panin, V.E., Surikova, N.S., Smirnova, A.S., and Pochivalov, Yu.I., Mesoscopic Structural States in Plastically Deformed Nanostructured Metal Materials, Phys. Mesomech., 2018, vol. 21, no. 5, pp. 396–400. doi https://doi.org/10.1134/S102995991805003X

    Article  Google Scholar 

  2. Panin, V.E., Derevyagina, L.S., Lebedev, M.P., Syromyatnikova, A.S., Surikova, N.S., Pochivalov, Yu.I., and Ovechkin, B.B., Scientific Basis for Cold Brittleness of Structural BCC Steels and Their Structural Degradation at Below Zero Temperatures, Phys. Mesomech., 2017, vol. 20, no. 2, pp. 125–133. doi https://doi.org/10.1134/S1029959917020023

    Article  Google Scholar 

  3. Surikova, N.S., Panin, V.E., Narkevich, N.A., Mishin, I.P., and Gordienko, A.I., Formation of a Multilevel Hierarchical Mesosubstructure by Cross Rolling and Its Effect on the Mechanical Behavior of Austenitic Steel, Phys. Mesomech, 2018, vol. 21, no. 5, pp. 430–440. doi https://doi.org/10.1134/S1029959918050077

    Article  Google Scholar 

  4. Tyumentsev A.N., Ditenberg I.A., Korotaev A.D. and Denisov K.I., Lattice Curvature Evolution in Metal Materials on Meso- and Nanostructural Scales of Plastic Deformation, Phys. Mesomech., 2013, vol. 16, no. 4, pp. 319–334. doi https://doi.org/10.1134/S1029959913040061

    Article  Google Scholar 

  5. Vince, S.A., Ditenberg, I.A., Tyumentsev, A.N., and Korznikov, A.V., Evolution of the Microstructure and Mechanical Properties of Mo-47% Re Alloy Depending on the Strain Degree under High-Pressure Torsion, Izv. Vuzov. Fiz., 2009, vol. 52, no. 12–2, pp. 31–36.

    Google Scholar 

  6. Tyumentsev, A.N., Ditenberg, I.A., Grinyaev, K.V., Chernov, V.M., and Potapenko, M.M., Multi-Directional Forge Molding as a Promising Method of Enhancement of Mechanical Properties of V-4Ti-4Cr Alloys, J. Nucl. Mater., 2011, vol. 413, no. 2, pp. 103–106.

    Article  ADS  Google Scholar 

  7. Sadovskii, V.M., Guzev, M.A., Sadovskaya, O.V., and Qi, Ch., Modeling of Plastic Deformation Based on the Theory of an Orthotropic Cosserat Continuum, Fiz. Mezomekh., 2019, vol. 22, no. 2, pp. 59–66. doi https://doi.org/10.24411/1683-805X-2019-12005

    Google Scholar 

  8. Sadovskii, V.M. and Sadovskaya, O.V., Modeling of Elastic Waves in a Blocky Medium Based on Equations of the Cosserat Continuum, Wave Motion, 2015, vol. 52, pp. 138–150. doi https://doi.org/10.1016/j.wavemoti.2014.09.008

    Article  MathSciNet  Google Scholar 

  9. Makarov, P.V., Evolutionary Nature of Structure Formation in Lithospheric Material: Universal Principle for Fractality of Solids, Russ. Geol. Geophys., 2007, vol. 48, pp. 558–574.

    Article  ADS  Google Scholar 

  10. Makarov, P.V., Resonance Structure and Inelastic Strain and Defect Localization in Loaded Media, Phys. Mesomech., 2011, vol. 14, no. 5–6, pp. 297–307.

    Article  Google Scholar 

  11. Cosserat, E. and Cosserat, F., Théorie des Corps Déformables. Chwolson’s Traité Physique, Paris: Librairie Scientifique A. Hermann et Fils, 1909, pp. 953–1173.

    MATH  Google Scholar 

  12. Nowacki, W., Theory of Micropolar Elasticity, Vienna: Springer-Verlag, 1970.

    Book  Google Scholar 

  13. Aero, E.L. and Kuvshinsii, E.V., Fundamental Equations of the Theory of Elastic Media with Rotationally Interacting Particles, Sov. Phys. Solid State, 1960, no. 2, pp. 1272–1281.

  14. Kuvshinskii, E.V. and Aero, E.L., Continuum Theory of Asymmetric Elasticity—the Problem of Internal Rotation, Sov. Phys. Solid State, 1963, vol. 5, no. 5, pp. 1892–1897.

    MathSciNet  Google Scholar 

  15. Palmov, V.A., The Plane Problem of the Theory of Asymmetric Elasticity, Prikl. Mat. Mekh., 1964, vol. 2, no. 6, pp. 1117–1120.

    Google Scholar 

  16. Palmov, V.A., Basic Equations of the Theory of Asymmetric Elasticity, Prikl. Mat. Mekh., 1964, vol. 28, no. 3, pp. 401–408.

    MathSciNet  Google Scholar 

  17. Günther, W., Zur Statik und Kinematik des Cosseratchen Kontinuum, Abh. Braunschweigischen Wissenschaftlichen Gesellschaft, 1958, vol. 10, pp. 195–213.

    MATH  Google Scholar 

  18. Toupin, R.A., Elastic Materials with Couple Stresses, Arch. Rat. Mech. Anal., 1962, no. 11, pp. 385–414.

  19. Grekova, E., Kulesh, M., Herman, G., and Shardakov, I., Modeling of the Propagation of Seismic Waves in Non-Classical Media: Reduced Cosserat Continuum, American Geophysical Union Fall Meeting 2006 Abstracts, EOS Trans. AGU, 2006, p. B151.

  20. Green, A.E. and Rivlin, R.S., Multipolar Continuum Mechanics, Arch. Ration. Mech. Anal., 1964, vol. 17, pp. 113–147.

    Article  MathSciNet  Google Scholar 

  21. Lakes, R., Experimental Methods for Study of Cosserat Elastic Solids and Other Generalized Elastic Continua, Continuum Models for Materials with Microstructure. Ch. 1, Mühlhaus, H., Ed., New York: J. Wiley, 1995, pp. 1–22.

    MATH  Google Scholar 

  22. Gauthier, R.D. and Jahsman, W.E., A Quest for Micropolar Elastic Constants. Part 1, Trans. ASME. J. Appl. Mech., 1975, vol. 97, no. 2, pp. 369–374.

    Article  Google Scholar 

  23. Gauthier, R.D. and Jahsman, W.E., A Quest for Micropolar Elastic Constants. Part 2, Arch. Mech., 1981, vol. 33, no. 5, pp. 717–737.

    MATH  Google Scholar 

  24. De Borst, R., Simulation of Strain Localization: a Reappraisal of the Cosserat Continuum, Eng. Comput., 1991, vol. 8, no. 4, pp. 317–332.

    Article  Google Scholar 

  25. Bay, B., Hansen, N., Hughes, D., and Kuhlman-Wilsdorf, D., Evolution of FCC Deformation Structures in Polyslip, Acta Met. Mater., 1992, vol. 40, no. 2, pp. 205–219.

    Article  Google Scholar 

  26. Pavlov, I.S., Granular Medium with Particle Rotation. Two-Dimensional Model, Probl. Prochn. Plast., 2003, no. 65, pp. 53–64.

  27. Forest, S., Barbe, F., and Cailletaud G., Cosserat Modelling of Size Effects in the Mechanical Behaviour of Polycrystals and Multi-Phase Materials, Int. J. Solids Struct., 2000, vol. 37, pp. 7105–7126.

    Article  Google Scholar 

  28. Makarov, P.V., Microdynamic Theory of Plasticity and Fracture of Structurally Heterogeneous Materials, Russ. Phys. J., 1992, vol. 35, pp. 334–346.

    Article  Google Scholar 

  29. Makarov, P.V., Mathematical Multilevel Model of Elastic-Plastic Deformation of Structurally Inhomogeneous Media, Doctoral (Phys.-Math.) Dissertation, Tomsk, 1995.

  30. Makarov, P.V., Modeling of Deformation and Fracture at the Mesoscale, Izv. RAN. MTT, 1999, no. 5, pp. 109–131.

  31. Bakeev, R.A., Modeling of Deformation of Solids at the Mesolevel with Consideration for Independent Rotations, Cand. Sci. (Phys.-Math.) Dissertation, Tomsk, 2010.

  32. Ostoja-Starzewski, M. and Jasiuk, I., Stress Invariance in Planar Cosserat Elasticity, Proc. Roy. Soc. Lond. A, 1995, vol. 451, no. 1942, pp. 453–470.

    Article  ADS  MathSciNet  Google Scholar 

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Funding

This work was supported by the Russian Science Foundation (project No. 19-17-00122).

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Authors and Affiliations

  1. National Research Tomsk State University, Tomsk, 634050, Russia

    P. V. Makarov, R. A. Bakeev & I. Yu. Smolin

  2. Institute of Strength Physics and Materials Science, Siberian Branch, Russian Academy of Sciences, Tomsk, 634055, Russia

    P. V. Makarov, R. A. Bakeev & I. Yu. Smolin

Authors
  1. P. V. Makarov
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  2. R. A. Bakeev
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  3. I. Yu. Smolin
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Correspondence to P. V. Makarov.

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Russian Text © The Author(s), 2019, published in Fizicheskaya Mezomekhanika, 2019, Vol. 22, No. 4, pp. 29–38.

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Makarov, P.V., Bakeev, R.A. & Smolin, I.Y. Modeling of Localized Inelastic Deformation at the Mesoscale with Account for the Local Lattice Curvature in the Framework of the Asymmetric Cosserat Theory. Phys Mesomech 22, 392–401 (2019). https://doi.org/10.1134/S1029959919050060

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