Skip to main content
SpringerLink
Log in

On a Model of Heterogenous Deformation of Elastic Bodies by the Mechanism of Multiple Appearance of New Phase Layers

  • Published:
Meccanica Aims and scope Submit manuscript

Abstract

A model describing a possible scenario of martensite type phase transformations is examined. A new phase is supposed to nucleate in the form of plane parallel layers. As the boundary condition, average strains are imposed. Then, the governing parameters of the two-phase structure are the concentration of new phase layers, their orientation and also the orientation of anisotropy axes. The parameters depend on the average strains and are determined by the requirement to minimize the average Helmholtz free-energy function. Once a general procedure has been discussed, average strain–stress diagrams are constructed for two cases. In the first case, for the simplicity sake, both phases are assumed to be isotropic. In the second case anisotropy is produced by a non-spherical phase transformation strain tensor. For both cases phase transition zones (PTZs) are constructed. The PTZ is formed in the space of strains by those which can exist on equilibrium interfaces. Loading and unloading paths, corresponding to uniaxial stretching and plane stretching/compression, are examined and related with the PTZ. Effects of internal stresses induced by the nucleation of new phase areas and the anisotropy of new phase are discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. R. Abeyaratne (1983) ArticleTitle‘An admissibility condition for equilibrium shocks in finite elasticity’ J. Elasticity 13 175–184 Occurrence Handle0545.73032 Occurrence Handle711132

    MATH  MathSciNet  Google Scholar 

  2. J.M. Ball R.D. James (1992) ArticleTitle‘Proposed experimental tests of a theory of fine microstructure, and the two-well problem’ Phil. Trans. R Soc. Lond. A. 338 389–450 Occurrence Handle0758.73009 Occurrence Handle1992RSPTA.338..389B

    MATH  ADS  Google Scholar 

  3. K. Bhattacharya (1999) ArticleTitle‘Phase boundary propagation in a heterogeneous body’ Proc. R. Soc. Lond. A. 455 757–766 Occurrence Handle0990.74047 Occurrence Handle1999RSPSA.455..757B Occurrence Handle10.1098/rspa.1999.0333

    Article  MATH  ADS  Google Scholar 

  4. K. Bhattacharya B. Li M. Luskin (1999) ArticleTitle‘The simply laminated microstructure in martensitic crystals that undergo a cubic to orthorhombic phase transformation’ Arch. Rat. Mech. Anal. 149 123–154 Occurrence Handle0942.74056 Occurrence Handle1719149 Occurrence Handle10.1007/s002050050170

    Article  MATH  MathSciNet  Google Scholar 

  5. Bhattacharya, K. and Schlomerkemper, A., ‘Transformation Yield Surface of Shape-Memory Alloys’, J. de Phys. IV (Proc.). 115 , (June 2004), 155–163.

  6. Eremeev, V. A., Freidin, A. B. and Sharipova, L. L., ‘Nonuniqueness and Stability in Problems of Equilibrium of Elastic Two-Phase Bodies’, Dokl. Phys. 48 (7) (2003) 359–363. Translated from Dokl Akad. Nauk. 391 (2) (2003) 189–193.

  7. J.L. Ericksen (1975) ArticleTitle‘Equilibrium of bars’ J. Elasticity 5 191–201 Occurrence Handle0324.73067 Occurrence Handle471528 Occurrence Handle10.1007/BF00126984

    Article  MATH  MathSciNet  Google Scholar 

  8. J.D. Eshelby (1975) ArticleTitle‘The elastic energy-momentum tensor’ J. Elasticity 5 321–335 Occurrence Handle0323.73011 Occurrence Handle489190 Occurrence Handle10.1007/BF00126994

    Article  MATH  MathSciNet  Google Scholar 

  9. F.D. Fischer M. Berveiller K. Tanaka E.R. Oberaigner (1994) ArticleTitle‘Continuum mechanical aspects of phase transformations in solids’ Arc. Appl. Mech. 64 54–85 Occurrence Handle0790.73010

    MATH  Google Scholar 

  10. F.D. Fischer E.R. Oberaigner K. Tanaka F. Nishimura (1998) ArticleTitle‘Transformation induced plasticity Revised an updated formulation’. Int. J. Solids Structures 35 IssueID18 2209–2227 Occurrence Handle0949.74506 Occurrence Handle10.1016/S0020-7683(97)00134-0

    Article  MATH  Google Scholar 

  11. R.L. Fosdick B. Hertog (1989) ArticleTitle‘The Maxwell relation and Eshelby’s conservation law for minimizers in elasticity theory’ J. Elasticity 22 193–200 Occurrence Handle0698.73008 Occurrence Handle1040759 Occurrence Handle10.1007/BF00041111

    Article  MATH  MathSciNet  Google Scholar 

  12. Freidin, A.B., ‘Crazes and shear bands in glassy polymer as layers of a new phase’, Mech. Comp. Mater. 1 (1989) 1–7. Translated from Mekhanika Kompozitnih Materialov 1 (1989) 3–10.

  13. Freidin, A.B. and Chiskis, A.M., ‘Regions of phase transitions in nonlinear-elastic isotropic materials. Part 1: Basic relations.’ Mech. Solid, 29 (4) (1994), 91–109. Translated from Izvestia RAN, Mekhanika Tverdogo Tela 4 (1994) 91–109.

  14. Freidin, A.B. and Chiskis, A.M., ‘Regions of phase transitions in nonlinear-elastic isotropic materials. Part 2: Incompressible materials with a potential depending on one of strain invariants’, Mech Solids, 29 (4) (1994) 46–58. Translated from Izvestia RAN, Mekhanika Tverdogo Tela 5 (1994) 49–61.

  15. Freidin, A.B., ‘Heterogeneous deformation of elastic solids due to the multiple appearance of new phase layers’, in: Proceedings of the 1st International Seminar, “Actual Problems of Strength”, 15–18 October, 1997. Novgorod, Russia Vol. 1. pp. 236–240 (in Russian).

  16. Freidin, A.B., ‘Small strains approach to the theory on solid-solid phase transformations under the process of deformation’, Studies on Elasticity and Plasticity (St. Petersburg State University) 18 (1999) 266–290 (in Russian).

  17. Freidin, A.B., Eremeyev, V.A. and Sharipova, L.L., ‘On nonuniqueness and stability of centrally symmetric two phase deformations’, in: The proceedings of the XXIX Summer School “Advanced Problems in Mechanics”. - SPb.: IPME RAS 2002, pp. 73–83.

  18. A.B. Freidin E.N. Vilchevskay L.L. Sharipova (2002) ArticleTitle‘Two-phase deformations within the framework of phase transition zones’ Theor. Apll. Mech. 28-29 149–172

    Google Scholar 

  19. Freidin, A.B. and Vilchevskaya, E.N., ‘Phase transition zones for one class of nonlinear elastic materials’, in:Proceedings of the XXIX Summer School “Actual Problems in Mechanics” APM 2002, St.Petersburg, IPME RAS, pp. 43–44.

  20. Freidin, A.B. and Sharipova, L.L. ‘Equilibrium two-phase deformations and phase transitions zones within framework of small strains’, Nonlinear problems of continuum mech. Izvestia Vuzov. North-Caucasus region. Special issue (2003) 291–299 (in Russian).

  21. Freidin, A.B., Sharipova, L.L. and Vilchevskaya, E.N., ‘Phase transition zones in relation with constitutive equations of elastic solids’, in: Proceedings of the XXXII Summer School “Actual Problems in Mechanics” (APM-2004), St.Petersburg, IPME RAS, pp. 140–150.

  22. G.H. Goldsztein (2001) ArticleTitle‘The effective energy and laminated microstructures in martensitic phase transformations’ J. Mech. Phys. Solids. 49 899–925 Occurrence Handle1017.74050 Occurrence Handle10.1016/S0022-5096(00)00057-0 Occurrence Handle2001JMPSo..49..899G

    Article  MATH  ADS  Google Scholar 

  23. M.A. Grinfeld (1980) ArticleTitle‘On conditions of thermodynamic equilibrium of the phases of a nonlinear elastic material’ Dokl. Acad. Nauk SSSR 251 824–827

    Google Scholar 

  24. M.E. Gurtin (1983) ArticleTitle‘Two–phase deformations of elastic solids’ Arch. Rat. Mech. Anal. 84 1–29 Occurrence Handle0525.73054 Occurrence Handle713116 Occurrence Handle10.1007/BF00251547

    Article  MATH  MathSciNet  Google Scholar 

  25. V. Idesman A. I. Levitas V. L. Preston D. J.-Y. Cho (2004) ArticleTitle‘Finite element simulations of martensitic phase transitions and microstructure based on strain softening model’ J. Mech. Phys. Solids. 53 495–523 Occurrence Handle10.1016/j.jmps.2004.10.001 Occurrence Handle2005JMPSo..53..495I

    Article  ADS  Google Scholar 

  26. R.D. James (1981) ArticleTitle‘Finite deformation by mechanical twinning’ Arch. Rat. Mech. Anal. 77 143–177 Occurrence Handle0537.73031 Occurrence Handle10.1007/BF00250621

    Article  MATH  Google Scholar 

  27. J.K. Knowles E. Sternberg (1978) ArticleTitle‘On the failure of ellipticity and the emergence of discontinuous deformation gradients in plane finite elastostatics’ J. Elasticity 8 329–379 Occurrence Handle0422.73038 Occurrence Handle516599 Occurrence Handle10.1007/BF00049187

    Article  MATH  MathSciNet  Google Scholar 

  28. J.K. Knowles (1979) ArticleTitle‘On the dissipation associated with equilibrium shocks in finite elasticity’ J. Elasticity. 9 131–158 Occurrence Handle0407.73037 Occurrence Handle535274 Occurrence Handle10.1007/BF00041322

    Article  MATH  MathSciNet  Google Scholar 

  29. R.V. Kohn (1991) ArticleTitle‘The relaxation of a double-well energy’.Continuum Mech Thermodyn Historical Arch. 3 IssueID3 193–236 Occurrence Handle0825.73029 Occurrence Handle1122017

    MATH  MathSciNet  Google Scholar 

  30. V.I. Kondaurov L.V. Nikitin (1982) ArticleTitle‘Phase transformations of the first kind in nonlinear elastic media’ Dokl. Acad. Nauk SSSR. 262 IssueID6 1348–1351 Occurrence Handle1982DoSSR.262.1348K

    ADS  Google Scholar 

  31. L.B. Kublanov A.B. Freidin (1988) ArticleTitle‘Solid phase seeds in a deformable material’ J. Appl. Math. Mech.PMM USSR 52 382–389 Occurrence Handle0731.73006 Occurrence Handle981315 Occurrence Handle10.1016/0021-8928(88)90094-9

    Article  MATH  MathSciNet  Google Scholar 

  32. Kunin, I.A., Elastic Media with Microstructure II. Springer-Verlag, Berlin, New York, etc., 1983.

  33. D.C. Lagoudas Z. Bo (1999) ArticleTitle‘Thermomechanical modelling of polycrystalline SMAs under cyclic loading’ IJES 37 1089–1249 Occurrence Handle1695186 Occurrence Handle10.1016/S0020-7225(98)00113-X

    Article  MathSciNet  Google Scholar 

  34. I. Levitas V. V. Idesman A. D. Preston (2004) ArticleTitle‘Microscale simulation of evolution of martensitic microstructure’ Phys. Rev. Lett. 93 105701 Occurrence Handle10.1103/PhysRevLett.93.105701 Occurrence Handle2004PhRvL..93j5701L

    Article  ADS  Google Scholar 

  35. I.R. MorozovN.F. Nazyrov A.B. Freidin (1996) ArticleTitle‘One-dimensional problem of phase transformation of an elastic ball’ Dokl. Akad. Nauk (Proc. Russ. Acad. Sci). 346 IssueID2 188–191

    Google Scholar 

  36. N.F. Morozov A.B. Freidin (1998) ArticleTitle‘Phase transition zones and phase transformations of elastic solids under different stress states’ Proc. Steklov Mathe. Inst 223 220–232 Occurrence Handle01523080 Occurrence Handle1727291

    MATH  MathSciNet  Google Scholar 

  37. Nazyrov, I.R. and Freidin, A.B., ‘Phase transformation of deformable solids in a model problem on an elastic sphere’, Izvestia RAN, Mekhanika Tverdogo Tela 33(5) (1998) 52–71. Translated into English in Mechanics of Solids.

  38. Osmolovsky, V.G., Variational problem on phase transitions in mechanics of solids. St. Petersburg State University, St. Petersburg 2000 (in Russian).

  39. K. Tanaka R. Iwasaki (1985) ArticleTitle‘A phenomenological theory of transformation superplasticity’ Eng. Frac. Mec. 21 709–720 Occurrence Handle10.1016/0013-7944(85)90080-3

    Article  Google Scholar 

  40. L. Truskinovsky (1982) ArticleTitle‘Equilibrium interphase boundaries’ Dokl. Acad. Nauk SSSR 265 306–310

    Google Scholar 

  41. A.J. Zak M.P. Cartmell Ostachowicz M. Wiercigroch (2003) ArticleTitleOne-dimensional shape memory alloy models for use with reinforced composite structures’ Smart Materials and Structures 12 338–346 Occurrence Handle10.1088/0964-1726/12/3/304

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mechanical Engineering Problems, Russian Academy of Sciences, Bolshoy Prospect, 61, V.O.St. Petersburg, 199178, Russia

    Alexander B. Freidin & Leah L. Sharipova

Authors
  1. Alexander B. Freidin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Leah L. Sharipova
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexander B. Freidin.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Freidin, A.B., Sharipova, L.L. On a Model of Heterogenous Deformation of Elastic Bodies by the Mechanism of Multiple Appearance of New Phase Layers. Meccanica 41, 321–339 (2006). https://doi.org/10.1007/s11012-005-5901-9

Download citation

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11012-005-5901-9

Keywords

Search

Navigation