Relaxation Element Method in Mechanics of Deformable Solid

  • Ye. Ye. Deryugin
  • G. V. Lasko
  • S. Schmauder
Living reference work entry


In this chapter a new method – the relaxation element method is justified. The definition of the changing of stress fields in solids under loading as a result of the change of elastic energy in a local volume, undergoing plastic deformation, is laid down at the basis of the method.

The Relaxation Element Method (REM) solves effectively two problems of a deforming solid (DS):
  1. 1.

    The construction of the different distributions of plastic deformation in local regions of various geometrical shape.

  2. 2.

    Modelling of the consequent involvement of separate structural elements into plastic deformation, operating on the principle of an inverse task of mechanics of deforming solids.


With this method a stress-strain state of the elastic plane with the sites of plastic deformation in the form of a circle, rectangle, and a localized shear band is analytically described. Examples of the construction of the sites of plastic deformation with gradients are given. The stress-strain state of a plane with a round inclusion is considered.

Examples of the simulations by the REM of the effects of Lüders band formation and interrupted flow in polycrystals are given. The analysis of the influence of rigidity of a testing device on qualitative and quantitative characteristics of the loading diagram is presented.

The effect of the gradients of plastic deformation on the stress of Lüders band initiation is analyzed. It is shown that the dependence of the stress of Lüders band initiation on grain sizes is the consequence of the independency of the gradient of plastic deformation under the changing of grain sizes.

A modified model of Griffith crack surrounded by a layer of plastically deformed material is proposed. Plastic deformation is shown to eliminate the singularity at the crack tip. The maximum stresses are observed at the boundary of the plastic zone in an elastically deformed matrix. The stress concentration increases as the thickness of the plastic layer decreases.

The obtained results testify to high predictable possibilities of the developed method. They are in a good agreement with known experimental data.


  1. 1.
    Sutton AP, Balluffi RW. Interfaces in crystalline materials. Oxford: Clarendon Press; 1995.Google Scholar
  2. 2.
    Panin VE. Modern problems of physical mesomechanics. In: Proceedings of the International Conference of MESO-MECHANICS 2000: Role of Mechanics for Development of Science and Technology, June 13–16, 2000, vol. 1. Beijing: Tsinghua University Press; 2000. p. 127–142.Google Scholar
  3. 3.
    Panin VE, Grinyaev YV. Physical mesomechanics – new paradigm at the junction of physics and mechanics. Phys Mesomech. 2003;6(4):9–36.Google Scholar
  4. 4.
    Panin VE, Egorushkin ВЕ, Panin AV. Physical mechanics of deformed solid as multi-level system. Phys Mesomech. 2006;9(3):9–22.Google Scholar
  5. 5.
    Estrin Y, Kubin LP. Spatial coupling and propagative plastic instabilities, Chapter 2. In: Mühlhaus H-B, editor. Continuum models of materials with microstructure. Chichester: John Wiley & Sons Ltd; 1995. p. 395–450.Google Scholar
  6. 6.
    Klose FB, Ziegenbein A, Weidenmüller J, Neuhäuser H, Hähner P. Portevin-Le Chatelier effect in strain and stress controlled tensile tests. Comput Mater Sci. 2003;26:80–6.CrossRefGoogle Scholar
  7. 7.
    Zhang Q, Jiang Z, Jiang H, Chen Z, Wu X. On the propagation and pulsation of Portevin-Le Chatelier deformation bands: an experimental study with digital specle pattern metrology. Int J Plast. 2005;21:2150–73.CrossRefGoogle Scholar
  8. 8.
    Krishtal ММ. Instability and mesoscopic inhomogeneity of plastic deformation. Phys Mesomech. 2004;7(5):5–45.Google Scholar
  9. 9.
    Deryugin YY, Panin VE, Schmauder S, Storozhenko IV. Effects of strain localization in the composites on the basis of Al with Al2O3. Phys Mesomech. 2001;4(3):35–47.Google Scholar
  10. 10.
    Deryugin YY. Relaxation Element method. Novosibirsk: Nauka, Siberian publishing House RAN; 1998.Google Scholar
  11. 11.
    Deryugin YY, Lasko GV. Relaxation element method in the problems of mesomechanics and calculations of band structures. In: Panin VE, editor. Physical Mesomechanics and computer-aided design of materials, vol. 1. Novosibirsk: Nauka; 1995. p. 130–61.Google Scholar
  12. 12.
    Deryugin YY, Lasko GV, Schmauder S. Relaxation element method. Comput Mater Sci. 1998;11(3):189–203.CrossRefzbMATHGoogle Scholar
  13. 13.
    Deryugin YY. Relaxation element method in calculations of stress state of elastic plane with the plastic deformation band. Comput Mater Sci. 1999;19:53–68.CrossRefGoogle Scholar
  14. 14.
    Deryugin YY, Lasko G, Schmauder S. Relaxation Element Method In Mechanics of Deformed Solid. In: Oster WU, editor. Computational Materials. Hauppauge: Nova Science Publishers, Inc; 2009. p. 479–545.Google Scholar
  15. 15.
    Eshelby JD. Continuum theory of defects. In: Seitz F, Turnbull D, editors. Solid State Physics, vol. 3. New York: Academic Press; 1956. p. 79.Google Scholar
  16. 16.
    de Wit R. (1970) Linear theory of static disclinations. In: Fundamental aspects of dislocation. Ed. by J.A. Simons, R. de Wit, R. Bullough, Nat. Bur. Stand. (US) Spec. Publ. 317, vol. I 651-673.Google Scholar
  17. 17.
    Likhachev VА, Volkov АЕ, Shudegov VЕ. Continuum theory of defects. Leningrad: Leningrad University Publishing House; 1986.Google Scholar
  18. 18.
    Rybin VV. Large plastic deformations and damage of metalls. Moscow: Mettallurgia; 1986. 224 p.Google Scholar
  19. 19.
    McHugh PE, Asaro RJ, Sich CF. Cristal plasticity models. In: Suresh S, editor. Fundamental of metal matrix composites. Boston: Butherworth-Heinman; 1993.Google Scholar
  20. 20.
    Harder J. Simulation lokaler Fliessvorgänge in Polykristallen. Braunschweiger Schriften zur Mechanik. B. 28. Braunschweig; 1997.Google Scholar
  21. 21.
    Balokhonov RR, Romanova VА. Numerical simulation of thermo-mechanical behaviour of steels with accounting of Luders band propagation. Appl Mech Tech Phys. 2007;48(5):146–55.Google Scholar
  22. 22.
    Dudarev EF. Microplastic deformation and yield stress of polycrystals. Tomsk: Publishing House of Tomsk University; 1988. 256p.Google Scholar
  23. 23.
    Dudarev EF, Deryugin YeYe. Microplastic deformation and yield stress of polycrystals. Izv Vyzov Fizika. 1982;(6):43–56.Google Scholar
  24. 24.
    Iricibar R, Mazza I, Cabo A. The microscopic strain profile of a propagating. Lüders band front in mild steel. Scr Metall. 1975;9(10):1051–8.CrossRefGoogle Scholar
  25. 25.
    Andronov VM, Gvozdikov AM. Stress state at the front of Chernov-Lüders band and unstable plastic flow of crystals. Physica Metals Metallovedenie. 1987;63(6):1212–9.Google Scholar
  26. 26.
    Danilov VI, Zuev LB, Letahova EV, Orlova DV, Оhrimenko IА. Types of localization of plastic deformation and the stages of loading diagrams of metallic materials with different crystalline structure. Appl Mech Tech Phys. 2006;47(2):176–84.CrossRefGoogle Scholar
  27. 27.
    Barannikova SА, Danilov VI, Zuev LB. Localization of plastic deformation in mono- and polycrystals of Fe−3%Si –alloy under tension. J Tech Phys. 2004;74(10):52–6.Google Scholar
  28. 28.
    Iricibar R, Mazza I, Cabo A. The microscopic strain profile of a propagating Lüders band front in mild steel. Scr Metall. 1975;9(10):1051–8.CrossRefGoogle Scholar
  29. 29.
    Sakui S, Sakai Т. The effect of strain rate, temperature and grain size on the lower yield stress and flow stress of pure iron. Proc Int Conf Sci Technol Iron Steel, Tokyo Part 2 Tokyo (1971). 1970:989–91.Google Scholar
  30. 30.
    Bell JF. In: Filin AP, editor. Experimental basis of mechanics of deformed solids. М.: Nauka; 1984. 600 p.Google Scholar
  31. 31.
    Wirtman J, Wirtman JР. Mechanical properties, insignificantly depending on the temperature. In: Han RU, Haasen P-T, editors. Phizicheskoe metallovedenie. Мoscow: Metallurgiya; 1987. 663 p.Google Scholar
  32. 32.
    Stress relaxation testing. In: Fox A, editor. Symposium on Mechanical Testing, American Society for Testing and Materials, Kansas City, Mo, 24, 25 May 1978. Philadelphia: American Society for Testing and Materials; 1979. p. 216S.Google Scholar
  33. 33.
    Oding IA, editor. Creep and stress relaxation in metals. Academy of Sciences of the U.S.S.R. All-Union Institute of Scientific and Technical Information. Edinburgh: Oliver & Boyd, X; 1965. p. 377S.Google Scholar
  34. 34.
    Kirsch G. Die Theorie der Elastizitat und die Bedurfnisse der Festigkeitslehre. Zantralblatt: Berlin Deutscher Ingenieure. 1898;42:797–807.Google Scholar
  35. 35.
    Timoshenko SP, Goodier JN. Theory of elasticity. 3rd ed. New York: McGraw Hill; 1970.Google Scholar
  36. 36.
    Ziegenbein А, Plessing, Näuhauser X. Investigation of the mesolevel of deformation at the formation of Luder’s band in monocrystals of concentrated alloys on the basis of Iron. Phys Mesomech. 1998;1(2):5–20.Google Scholar
  37. 37.
    Malin A, Hubert J, Hatherley M. The microstructure of rolled copper single crystals. Zs Metallk. 1981;72(5):310–7.Google Scholar
  38. 38.
    Nakayama Y, Morii K. Microstructure and shear band formation in cold single crystals of Al-Mg alloy. Acta Metall. 1987;35(7 (2)):1747–56.CrossRefGoogle Scholar
  39. 39.
    Zasimchuk EЭ, Markashova LI. 1998. Microbands in nickel monocrystals, deformed by rolling. Kiev: АNUkr SSR, Institute of Metallophysics, Preparing No. 23. 36p.Google Scholar
  40. 40.
    Gould H, Tobochnik Y. An introduction to computer simulation methods. Pt. 2: Application to physical systems. Moscow: Mir; 1990.Google Scholar
  41. 41.
    Smolin АY. Development of the method of movable cellular automata for the simulaton of the deformation and fracture of the media with accounting to their structure. Habilitation Thesis, Томск, 2009, 285 p.Google Scholar
  42. 42.
    Crouch SL, Starfield AM. Boundary element methods in solid mechanics. London: George Allen & Unwin.Google Scholar
  43. 43.
    Panin VЕ. Physical mesomechanics of the surface layers in solids. 1999;2(6): 5–23.Google Scholar
  44. 44.
    Physical values: handbook. Moscow: Energoatomizdat; 1991.Google Scholar
  45. 45.
    Koneva NА, Kozlov EV. Physical nature of the stages of plastic deformation. Izv Vyzov, Filika. 1990;33(2):89–106.Google Scholar
  46. 46.
    Kelly А. High-strength materials. Мoscow: Mir; 1976. 264p.Google Scholar
  47. 47.
    Peterson IV. Stress intensity factors. Мoscow: Mir. 1977. 304p.Google Scholar
  48. 48.
    Thompson A. Substructure strengthening mechanisms. Met Trans. 1977;6:833–42.CrossRefGoogle Scholar
  49. 49.
    Muskhelishvili NI. Some basic problems of the mathematical theory of elasticity. Moscow: Nauka. 708 p.Google Scholar
  50. 50.
    Utyashev FZ. Compatibility condition of plastic deformation and the ultimate refinement of the grains in metals. Fizika i technika vysokih davlenii. 2010;20(4):10–20.Google Scholar
  51. 51.
    Rybin VV. Regularities of Mesostructures Development in Metals in the Course of Plastic Deformation. Prob Mater Sci. 2003;33(1):9–28.Google Scholar
  52. 52.
    Hertzberg RW. Deformation and fracture mechanics of engineering materials. New York: Wiley; 1976.Google Scholar
  53. 53.
    Broek D. Elementary engineering fracture mechanics. Leiden: Noordhoff; 1974.zbMATHGoogle Scholar
  54. 54.
    Stepanova LV. Mathematical methods of fracture mechanics. Moscow: FIZMATLIT; 2009. 336 p.Google Scholar
  55. 55.
    Dugdale DS. Yielding of steel sheets containing slits. J Mech Phys Solids. 1960;8(2):100–8.CrossRefGoogle Scholar
  56. 56.
    Slepyan LI. Mechanics of cracks. Leningrad: Sudostroenie; 1990. 296 p.Google Scholar

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© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Institute of Strength Physics and Materials ScienceSB RASTomskRussia
  2. 2.Institut für Materialprüfung, Werskstoffkunde und FestigkeitslehreUniversität StuttgartStuttgartGermany

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