K. P. Frolova, E. N. Vilchevskaya

COMPARISON OF THE YIELD FUNCTIONS FOR MICROPOLAR MEDIA USING A CYLINDRICAL SAMPLE MODEL

The article discusses different yield functions extended to the case of micropolar media in order to investigate the effect of local plasticity in metals. The paper is concerned with the inhomogeneous stress-strain state obtained within the frame of the micropolar continuum approach for a cylindrical specimen under distributed couple stress on its lateral surface. The case corresponding to specimen tension is considered. Plastic strains prove to arise near the border of the specimen, while the material continues to behave elastically. According to the results, it is necessary to take into account couple stresses to clarify what happens in the area near the border when the size of this area is comparable to the size of the structure inhomogeneity.

References:

1. Fizicheskaya mezomekhanika i kompyuternoe konstruirovanie [Panin V.E., ed., Physical Mesomechanics and the Computer Design of Materials]. Novosibirsk, Nauka Publ., 1995. (In Russian).

2. Panin V.E., Grinyaev Yu.V., Psakhye S.G. Two decades of developments in physical mesomechanics: achievements, problems and prospects. Fizicheskaya Mezomekhanika, 2004, vol. 7, no. 4, pp. 93–113. (In Russian).

3. Tyumentsev A.N., Korotayev A.D., Pinzhin Yu.P. Highly Defective Structural States, Fields of Local Internal Stresses and Cooperative Mesoscopic Mechanisms of Crystal Deformation and Reorientation in Nanostructured Metal Materials. Phys. Mesomech., 2004, vol. 7, nos. 3–4, pp. 31–46.

4. Panin V.E., Panin A.V. Effect of the Surface Layer in a Solid under. Fizicheskaya Mezomekhanika, 2005, vol. 8, nos. 5–6, pp. 7–14. (In Russian).

5. Kuznetsov P.V. and Panin V.E., Direct Observation of Flows of Defects and of nm-Range Localization of Deformation on Duralumin Surface with the Aid of Scanning Tunnel and Atom Force Microscopes. Phys. Mesomech., 2000, vol. 3, no. 2, pp. 85–91.

6. Panin V.E., Yelsukova T.F., Popkova Yu.F., Pochivalov Yu.I. Role of curvature of the surface layer of titanium polycrystals in generation and development of fatigue fracture. Problems of Atomic Science and Technology, 2015, vol. 96, no. 2, pp. 148–153. (In Russian).

7. Panin V.E., Egorushkin V.E., Panin A.V., Moiseenko D.D. On the nature of plastic strain localization in solids. Technical Physics, 2017, vol. 52, iss. 8, pp 1024–1030. DOI: 10.1134/S1063784207080105.

8. Eremeyev V.A., Lebedev L.P., Altenbach H. Foundations of Micropolar Mechanics, Springer Science & Business Media, 2012.

9. Eringen A.C. Theory of micropolar elasticity. In: Microcontinuum Field Theories, Springer, New York, NY, 1999, pp. 101–248. DOI: 10.1007/978-1-4612-0555-5_5.

10. Mindlin R.D. Micro-structure in linear elasticity. Archive for Rational Mechanics and Analysis, 1964, vol. 16, no. 1, pp. 51–78. DOI: 10.1007/BF00248490.

11. Mindlin R.D., Tiersten H.F. Effects of couple-stresses in linear elasticity. Archive for Rational Mechanics and analysis, 1962, vol. 11, no. 1, pp. 415–448.

12. Kozlov E.V., Koneva N.A., and Zhdanov A.N. Structure and Strain Resistance of FCC Ultrafine-Grained Metals and Alloys. Fizicheskaya Mezomekhanika, 2004, vol. 7, no. 4, pp. 93–113. (In Russian).

13. Forest S., Sievert R. Elastoviscoplastic constitutive frameworks for generalized continua. Acta Mechanica, 2003, vol. 160, nos. 1–2, pp. 71–111. DOI: 10.1007/s00707-002-0975-0.

14. Smolin I.Yu. The use of the micropolar models for the description of the plastic deformations at the mesoscale level. Vestnik Permskogo natsyonalnogo issledovatelskogo politekhnicheskogo universiteta: Mekhanika, 2006, no. 14, pp. 189–205. (In Russian).

15. Lippmann H. Eine Cosserat-Theorie des plastischen Fließens. Acta Mechanica, 1969, vol. 8, nos. 3–4, pp. 255–284. DOI: 10.1007/BF01182264.

16. Sawczuk A. On the yielding of Cosserat continua. Archives of Mechanics, 1967, vol. 19, no. 3, pp. 471–480.

17. Besdo D. Ein beitrag zur nichtlinearen theorie des Cosserat-kontinuums. Acta Mechanica, 1974, vol. 20, nos. 1–2, pp. 105–131. DOI: 10.1007/BF01374965.

18. Mühlhaus H.B., Vardoulakis I. The thickness of shear bands in granular materials. Geotechnique, 1987, vol. 37, no. 3, pp. 271–283. DOI: 10.1680/geot.1987.37.3.271.

19. De Borst R. Simulation of strain localization: a reappraisal of the Cosserat continuum. Engineering Computations, 1991, vol. 8, no. 4, pp. 317–332. DOI: 10.1108/eb023842.

20. De Borst R. A generalisation of J2-flow theory for polar continua. Computer Methods in Applied Mechanics and Engineering, 1993, vol. 103, no. 3, pp. 347–362. DOI: 10.1016/0045-7825(93)90127-J.

21. Lippmann H. Cosserat Plasticity and Plastic Spin. Appl. Mech. Rev., 1995, vol. 48, no. 11. DOI: 10.1115/1.3005091.

22. Altenbach H., Eremeyev V.A. Strain rate tensors and constitutive equations of inelastic micropolar materials. International Journal of Plasticity, 2014, vol. 63, pp. 3–17. DOI: 10.1016/j.ijplas.2014.05.009.

23. Dietsche A., Steinmann P., Willam K. Micropolar elastoplasticity and its role in localization. International Journal of Plasticity, 1993, vol. 9, no. 7, pp. 813–831. DOI: 10.1016/0749-6419(93)90053-S.

24. Steinmann P. A micropolar theory of finite deformation and finite rotation multiplicative elastoplasticity. International Journal of Solids and Structures, 1994, vol. 31, no. 8, pp. 1063–1084. DOI: 10.1016/0020-7683(94)90164-3.

25. Kachanov L.M. Osnovy teorii plastichnosti [Fundamentals of Plasticity Theory]. Moscow, Nauka Publ., 1969, 420 p. (In Russian).

26. Mase G.E. Theory and Problems of Continuum, McGraw-Hill Book Company, New York, 1970.

27. Vardoulakis I. Shear-banding and liquefaction in granular materials on the basis of a Cosserat continuum theory. Ingenieur-Archiv., 1989, vol. 59, no. 2, pp. 106–113. DOI: 10.1007/BF00538364.

28. Salehi S.H., Salehi M. Numerical investigation of nanoindentation size effect using micropolar theory. Acta Mechanica, 2014, vol. 225, no. 12, pp. 3365–3376. DOI: 10.1007/s00707-014-1116-2.

29. Neuber H. Über Probleme der Spannungskonzentration im Cosserat-Körper. Acta Mechanica, 1966, vol. 2, no. 1, pp. 48–69. DOI: 10.1007/BF01176729.

30. Groen A.E., Schellekens J.C.J., De Borst R. Three-dimensional finite element studies of failure in soil bodies using a Cosserat continuum. In: Computer Methods and Advances in Geomechanics: Proceedings of the 8th International Conference on Computer Methods and Advances in Geomechanics, Morgantown, West Virginia, USA, 22–28 May 1994, Balkema, 1994, pp. 581–586.

31. Frolova K., Vilchevskaya E., Polyanskiy V., Alekseeva E. Modelling of a Hydrogen Saturated Layer Within the Micropolar Approach. In: New Achievements in Continuum Mechanics and Thermodynamics, Springer, Cham, 2019, pp. 117–128. DOI: 10.1007/978-3-030-13307-8_9.

32. Adomeit G. Determination of elastic constants of a structured material. In: Mechanics of Generalized Continua, Springer, Berlin, Heidelberg, 1968, pp. 80–82. DOI: 10.1007/978-3-662-30257-6_8.

33. Lakes R.S. Size effects and micromechanics of a porous solid. Journal of Materials Science, 1983, vol. 18, no. 9, pp. 2572–2580. DOI: 10.1007/BF00547573.

34. Yang J.F.C., Lakes R.S. Transient study of couple stress effects in compact bone: torsion. Journal of Biomechanical Engineering, 1981, vol. 103, no. 4, pp. 275–279.

35. Alekseeva E., Belyaev A., Zegzhda A., Polyanskiy A., Polyanskiy V., Frolova K., Yakovlev Y. Boundary layer influence on the distribution of hydrogen concentrations during hydrogen-induced cracking test of steels. Diagnostics, Resource and Mechanics of materials and structures, 2018, iss. 3, pp. 43–57. DOI: 10.17804/2410-9908.2018.3.043-057.

36. Belyaev A.K., Polyanskiy V.A., Yakovlev Y.A., Mansyrev D.E., Polyanskiy A.M. Surface effect of the waves of plastic deformation and hydrogen distribution in metals. In: Days on Diffraction: IEEE, 2017, pp. 45–50. DOI: 10.1109/DD.2017.8167993.

37. Polyanskiy V.A., Belyaev A.K. Alekseeva E.L., Polyanskiy A.M., Tretyakov D.A., Yakovlev Yu.A. Phenomenon of skin effect in metals due to hydrogen absorption. Continuum Mech. Thermodyn., 2019, vol. 31, iss. 6, pp. 1–15. DOI: 10.1007/s00161-019-00839-2.

38. Martinsson A., Sandstrom R. Hydrogen depth profile in phosphorus-doped, oxygen-free copper after cathodic charging. Journal of Materials Science, 2012, vol. 47, no. 19, pp. 6768–6776. DOI: 10.1007/s10853-012-6592-y.

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