Archive for Rational Mechanics and Analysis

, Volume 196, Issue 3, pp 715–751 | Cite as

Stress-Induced Phase Transformations in Shape-Memory Polycrystals

  • Kaushik BhattacharyaEmail author
  • Anja Schlömerkemper
Open Access


Shape-memory alloys undergo a solid-to-solid phase transformation involving a change of crystal structure. We examine model problems in the scalar setting motivated by the situation when this transformation is induced by the application of stress in a polycrystalline material made of numerous grains of the same crystalline solid with varying orientations. We show that the onset of transformation in a granular polycrystal with homogeneous elasticity is in fact predicted accurately by the so-called Sachs bound based on the ansatz of uniform stress. We also present a simple example where the onset of phase transformation is given by the Sachs bound, and the extent of phase transformation is given by the constant strain Taylor bound. Finally we discuss the stress–strain relations of the general problem using Milton–Serkov bounds.


Austenite Martensite Critical Stress Transformation Strain Recoverable Strain 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



We are delighted to acknowledge various enlightening and enjoyable discussions with Pierre Suquet and Christian Lexcellent during the course of this work. This work started when Anja Schlömerkemper held a postdoctoral position at the University of Oxford and was continued while Anja Schlömerkemper was affiliated to the Institute for Analysis, Dynamics and Modeling at the University of Stuttgart. The work was completed during Kaushik Bhattacharya’s sabbatical leave of absence at the Cavendish Laboratory and Clare Hall of Cambridge. We gratefully acknowledge the financial support of the EU TMR Network (FMRX-CT 98-0229, AS), the US Department of Energy (PSAAP, KB), UK EPSRC ( KB).

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.


  1. 1.
    Ball J.M., James R.D.: Proposed experimental tests of a theory of fine microstructure, and the two-well problem. Philos. Trans. R. Soc. Lond. A 338, 389–450 (1992)zbMATHCrossRefADSGoogle Scholar
  2. 2.
    Bhattacharya K.: Microstructure of martensite. Oxford Series of Materials Modelling. Oxford University Press, New York (2003)Google Scholar
  3. 3.
    Bhattacharya K., Kohn R.V.: Symmetry, texture and the recoverable strain of shape-memory polycrystals. Acta Mater. 44, 529–542 (1996)CrossRefGoogle Scholar
  4. 4.
    Bhattacharya K., Kohn R.V.: Elastic energy minimization and the recoverable strains of polycrystalline shape-memory materials. Arch. Rational Mech. Anal. 139, 99–180 (1997)zbMATHCrossRefMathSciNetADSGoogle Scholar
  5. 5.
    Bhattacharya K., Schlömerkemper A.: Transformation yield surface of shape-memory alloys. J. Phys. IV France 115, 155–162 (2004)CrossRefGoogle Scholar
  6. 6.
    Bhattacharya K., Schlömerkemper A.: On the Sachs bound in stress-induced phase transformations in polycrystalline scalar shape-memory alloys. PAMM Proc. Appl. Math. Mech. 8, 10569–10570 (2009)Google Scholar
  7. 7.
    Bhattacharya K., Suquet P.: A model problem concerning recoverable strains of shape-memory polycrystals. Proc. R. Soc. Lond. A 461, 2797–2816 (2005)zbMATHCrossRefMathSciNetADSGoogle Scholar
  8. 8.
    Brenner R., Lebensohn R.L., Castelnau O.: Elastic anisotropy and yield surface estimates of polycrystals. Int. J. Solids Struct. 46(16), 3018–3026 (2009)zbMATHCrossRefGoogle Scholar
  9. 9.
    Brinson L.C., Schmidt I., Lammering R.: Stress-induced transformation behavior of a polycrystalline NiTi shape memory alloy: micro and macromechanical investigations via in situ optical microscopy. J. Mech. Phys. Solids 52, 1549–1571 (2004)zbMATHCrossRefADSGoogle Scholar
  10. 10.
    Daly S., Ravichandran G., Bhattacharya K.: Stress-induced martensitic phase transformation in thin sheets of Nitinol. Acta Mater. 55, 3593–3600 (2007)CrossRefGoogle Scholar
  11. 11.
    Dacorogna B.: Direct Methods in the Calculus of Variations. Springer, New York (1989)zbMATHGoogle Scholar
  12. 12.
    DeSimone A., James R.D.: A constrained theory of magnetoelasticity. J. Mech. Phys. Solids 50, 283–320 (2002)zbMATHCrossRefMathSciNetADSGoogle Scholar
  13. 13.
    Goldsztein G.H.: Rigid perfectly plastic two-dimensional polycrystals. Proc. R. Soc. Lond. A 457, 2789–2798 (2001)zbMATHCrossRefADSGoogle Scholar
  14. 14.
    Hutchinson J.W.: Elastic-plastic behavior of polycrystalline metals and composites. Proc. R. Soc. Lond. A 319, 247–272 (1970)CrossRefADSGoogle Scholar
  15. 15.
    James R.D.: Displacive phase transformations in solids. J. Mech. Phys. Solids 34, 359–394 (1986)zbMATHCrossRefADSGoogle Scholar
  16. 16.
    Kohn R.V., Little T.D.: Some model problems of polycrystal plasticity with deficient basic crystals. . SIAM J. Appl. Math. 59, 172–197 (1998)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Kohn R.V., Niethammer B.: Geometrically nonlinear shape-memory polycrystals made from a two-variant material. ESIAM: Math. Model. Num. Anal. 2, 377–398 (2000)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Lexcellent C., Vivet A., Bouvet C., Calloch S., Blanc P.: Experimental and numerical determinations of the initial surface of phase transformation under biaxial loading in some polycrystalline shape-memory alloys. J. Mech. Phys. Solids 50, 2717–2735 (2002)zbMATHCrossRefADSGoogle Scholar
  19. 19.
    Lexcellent C., Schlömerkemper A.: Comparison of several models of the determination of the phase transformation yield surface in shape-memory alloys with experimental data. Acta Mater. 55, 2995–3006 (2007)CrossRefGoogle Scholar
  20. 20.
    Milton G.W., Serkov S.K.: Bounding the current in nonlinear conduction composites. J. Mech. Phys. Solids 48, 1295–1324 (2000)zbMATHCrossRefMathSciNetADSGoogle Scholar
  21. 21.
    Schlömerkemper A.: On a Sachs bound for stress-induced phase transformations in polycrystalline shape memory alloys. PAMM Proc. Appl. Math. Mech. 6, 507–508 (2006)CrossRefGoogle Scholar
  22. 22.
    Shield T.W.: Orientation dependence of the pseudoelastic behavior of single-crystals of Cu-Al-Ni in tension. J. Mech. Phys. Solids 43, 869–895 (1995)CrossRefADSGoogle Scholar
  23. 23.
    Shu Y.C., Bhattacharya K.: The influence of texture on the shape-memory effect in polycrystals. Acta Mater. 46, 5457–5473 (1998)CrossRefGoogle Scholar
  24. 24.
    Šittner P., Novák V.: Anisotropy of martensitic transformations in modeling of shape memory alloy polycrystals. Int. J. Plast. 16, 1243–1268 (2000)zbMATHCrossRefGoogle Scholar
  25. 25.
    Zhang Z., James R.D., Müller S.: Energy barriers and hysteresis in martensitic phase transformations. Acta Materialia 57(15), 4332–4352 (2009)CrossRefGoogle Scholar

Copyright information

© The Author(s) 2009

Authors and Affiliations

  1. 1.Division of Engineering and Applied ScienceCalifornia Institute of TechnologyPasadenaUSA
  2. 2.Max Planck Institute for Mathematics in the SciencesLeipzigGermany

Personalised recommendations