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Continuum Mechanics and Thermodynamics

, Volume 5, Issue 3, pp 205–242 | Cite as

Comparison of the geometrically nonlinear and linear theories of martensitic transformation

  • K. Bhattacharya
Original

Abstract

Over the last few years, a continuum model based on finite or nonlinear thermoelasticity has been developed and successfully used to study crystalline solids that undergo a martensitic phase transformation. A geometrically linear version of this model was developed independently and has been widely used in the materials science literature. This paper presents the two theories and evaluates them by comparing and contrasting the results in various problems. It is established that in analyzing particular microstructures, the linear theory does not offer significant simplifications and misses important details. However, in more general situations where the particular microstructure is unknown and may involve stress, the linear theory can address certain problems which are currently beyond the capabilities of the nonlinear theory. Such analysis can yield valuable qualitative information. Finally, an example where the two theories differ dramatically is presented.

Keywords

Microstructure Phase Transformation Science Literature Material Science Martensitic Transformation 
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.

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References

  1. 1.
    Ball, J. M.; James, R. D.: Fine phase mixtures as minimizers of energy, Arch. Rat. Mech. Anal. 100 (1987) 13–52Google Scholar
  2. 2.
    Ball, J. M.; James, R. D.: Proposed experimental tests of a theory of fine microstructure and the two well problem, Phil. Trans. Royal Soc. London A 338 (1992) 389–450Google Scholar
  3. 3.
    Bhattacharya, K.: Wedge-like microstructure in martensites, Acta Metall. 39 (1991) 2431–2444Google Scholar
  4. 4.
    Bhattacharya, K.: Self-accommodation in martensite, Arch. Rat. Mech. Anal. (1992)Google Scholar
  5. 5.
    Chipot, M.; Kinderlehrer, D.: Equilibrium configurations of crystals, Arch. Rat. Mech. Anal. 103 (1988) 237–277Google Scholar
  6. 6.
    Chu, C.; James, R. D.: Detwinning thermoelastic martensites, in preparationGoogle Scholar
  7. 7.
    Ericksen, J. L.: Nonlinear elasticity of diatomic crystals, Int. J. Solids Struc. 6 (1970) 951–957Google Scholar
  8. 8.
    Ericksen, J. L.: Special topics in elastostatics, Adv. Appl. Mech. 7, Academic Press (1977) 189–243Google Scholar
  9. 9.
    Ericksen, J.: On the symmetry and stability of thermoelastic solids, J. Appl. Mech. 45 (1978) 740–744Google Scholar
  10. 10.
    Ericksen, J.: On the symmetry of deformable crystals, Arch. Rat. Mech. Anal. 72 (1979) 1–13Google Scholar
  11. 11.
    Ericksen, J. L.: Some phase transitions in crystals, Arch. Rat. Mech. Anal. 73 (1980) 99–124Google Scholar
  12. 12.
    Ericksen, J. L.: The Cauchy and Born hypotheses for crystals, Phase transformations and material instabilities in solids (ed. Gurtin, M.), Academic Press (1984) 61–78Google Scholar
  13. 13.
    Ericksen, J. L.: Constitutive theory for some constrained elastic crystals, Int. J. Solids Struc. 22 (1986) 951–964Google Scholar
  14. 14.
    Ericksen, J. L.: Weak martensitic transformation in Bravais lattices, Arch. Rat. Mech. Anal. 107 (1988) 23–36Google Scholar
  15. 15.
    Ericksen, J. L.: Local bifurcation theory for thermoelastic Bravais lattices, Preprint (1991)Google Scholar
  16. 16.
    James, R. D.: Displacive phase transformations in solids, J. Mech. Phys. Solids 34 (1986) 359–394Google Scholar
  17. 17.
    James, R. D.: The stability and metastability of quartz, Metastability and incompletely posed problems (ed. Antman, S., Ericksen, J. L., Kinderlehrer, D. and Muller, I.), IMA Vol 3 Springer-Verlag (1987) 147–176Google Scholar
  18. 18.
    James, R. D.; Kinderlehrer, D.: Theory of diffusionless phase transitions, PDEs and Continuum Models of Phase Trans. (ed. Rascle, M., Serre, D. and Slemrod, M.), Lecture Notes in Physics 344, Springer-Verlag (1989) 51–84Google Scholar
  19. 19.
    Pitteri, M.: Reconciliation of local and global symmetries of crystals, J. Elasticity 14 (1984) 175–190Google Scholar
  20. 20.
    Pitteri, M.: On ν+1-lattices, J. Elasticity 15 (1985) 3–25Google Scholar
  21. 21.
    Chen, L. Q.; Wang, W.; Khachaturyan, A. G.: Transformation-induced elastic strain effect on the precipitation kinetics of ordered intermetallics, Phil. Mag. Letters 64 (1991) 241–251Google Scholar
  22. 22.
    Kosenko, N. S.; Roytburd, A. L.; Khandros, L. G.: Thermodynamics and morphology of martensitic transformations under external stresses, Phys Met. Metall. 5 (1977) 48–55Google Scholar
  23. 23.
    Kaganova, I. M.; Roytburd, A. L.: Equilibrium between elastically-interacting phases, Sov. Phys. JETP 67 (1988) 1173–1183Google Scholar
  24. 24.
    Khachaturyan, A. G.: Some questions concerning the theory of phase transformations in solids, Sov. Phys-Solid State 8 (1967) 2163–2168Google Scholar
  25. 25.
    Khachaturyan, A. G.: Theory of structural transformations in solids, John Wiley and Sons (1983)Google Scholar
  26. 26.
    Khachaturyan, A. G.; Shatalov, G. A.: Theory of macroscopic periodicity for a phase transition in the solid state, Sov. Phys. JETP 29 (1969) 557–561Google Scholar
  27. 27.
    Kostlan, E.; Morris, J. W.: The preferred habit of a coherent thin-plate inclusion in an anisotropic elastic solid. Acta Metall. 35 (1987) 2167–2175Google Scholar
  28. 28.
    Libman, M. A.; Roytburd, A. L.: Influence of stresses on the equilibrium domain structure and phase transition temperature in ordering alloys and ferroelastics, Sov. Phys.-Cryst. 32 (1987) 5–10Google Scholar
  29. 29.
    Roytburd, A. L.: Domain structure caused by internal stresses in heterophase solids, Phys. Stat. Sol. (A) 16 (1973) 329–339Google Scholar
  30. 30.
    Roytburd, A. L.: Martensitic transformation as a typical phase transformation in solids, Solid State Physics 33 (1978) 317–390Google Scholar
  31. 31.
    Roytburd, A. L.: Equilibrium and phase diagrams of coherent phases in solids, Sov. Phys. — Solid State 26 (1984) 1229–1233Google Scholar
  32. 32.
    Roytburd, A. L.; Pankova, M. N.: Effect of external stresses on habitus orientation and substructure of stress-induced martensite plates in ferrous alloys, Phys. Met. Metall. 59 (1985) 131–140Google Scholar
  33. 33.
    Roytburd, A. L.; Kosenko, N. S.: Orientational dependence of the elastic energy of a plane interlayer in a system of coherent phases. Phys. Stat. Sol. (A) 35 (1976) 735–746Google Scholar
  34. 34.
    Semenovskaya, S.; Khachaturyan, A. G.: Kinetics of strain-related morphology transformation in YBa2Cu3O7−x, Phys. Rev. Letters 67 (1991) 2223–2226Google Scholar
  35. 35.
    Wen, S. H.; Khachaturyan, A. G.; Morris, J. W.: Computer simulation of a tweed-transformation in an idealized elastic crystal, Metall. Trans. A 12A (1981) 581–587Google Scholar
  36. 36.
    Wen, S. H.; Kostlan, E.; Hong, M.; Khachaturyan, A. G.; Morris, J. W.: The preferred habit plane of a tetragonal inclusion in a cubic matrix, Acta Metall. 29 (1981) 1247–1254Google Scholar
  37. 37.
    Wert, J. A.: The strain energy of a disc-shaped GP zone, Acta Metall. 24 (1976) 65–71Google Scholar
  38. 38.
    Eshelby, J. D.: The determination of the elastic field of an ellipsoidal inclusion and related problems, Proc. Royal Soc. London A 241 (1957) 376–396Google Scholar
  39. 39.
    Kohn, R. V.: The relaxation of a double-well energy, Cont. Mech. Thermodynamics 3 (1991) 193–236Google Scholar
  40. 40.
    Dacarogna, B.: Direct methods in the calculus of variations, Springer-Verlag (1989)Google Scholar
  41. 41.
    Evans, L. C.: Weak Convergence Methods for Nonlinear Partial Differential Equations, CBMS 74, Am. Math. Soc. (1990)Google Scholar
  42. 42.
    Bowles, J. S.; MacKenzie, J. K.: The crystallography of martensitic transformations I and II, Acta Metall. 2 (1954), 129–137 and 138–147Google Scholar
  43. 43.
    Wechsler, M. S.; Lieberman, D. S.; Read, T. A.: On the theory of the formation of martensite, J. Metals. Trans. AIME 197 (1953) 1503–1515Google Scholar
  44. 44.
    Bhattacharya, K.: Korn's inequality for sequences. Proc. Royal Soc. London A 434 (1991) 479–484Google Scholar
  45. 45.
    Kondratiev, V. A.; Oleinik, O. A.: Boundary — value problems for the system of elasticity theory in unbounded domains. Korn's inequalities, Russ. Math. Surveys 43:5 (1988) 65–119Google Scholar
  46. 46.
    Payne, L. E.; Weinberger, H. F.: On Korn's inequality, Arch. Rat. Mech. Anal. 8 (1961) 89–98Google Scholar
  47. 47.
    Van Tendeloo, G.; Amelinckx, S.: Group theoretical considerations concerning domain formation in ordered alloys, Acta Cryst. A30 (1974) 431–440Google Scholar
  48. 48.
    Chakravorty, S.; Wayman, C. M.: The thermoelastic martensitic transformation in β′ Ni−Al alloys: I. Crystallography and morphology and II. Electron microscopy. Metall. Trans. A 7A (1976), 555–568, 569–582Google Scholar
  49. 49.
    Pipkin, A. C.: Elastic materials with two preferred states, Quart. J. Mech. Appl. Math. 44 (1991) 1–15Google Scholar
  50. 50.
    Ball, J. M.: A version of the fundamental theorem for Young measures, PDEs and Cont. Models of Phase Trans. (ed. Rascle, M., Serre, D. and Slemrod, M.), Lecture Notes in Physics 344, Springer-Verlag (1989) 207–215Google Scholar
  51. 51.
    Kinderlehrer, D.; Pedregal, P.: Characterizations of Young measures, Arch. Rat. Mech. Anal. 115 (1991) 329–365Google Scholar
  52. 52.
    Tartar, L.: Compensated compactness and applications to partial differential equations, Nonlinear Analysis and Mechanics (ed. Knops, R. J.), Res. Notes 39 (1978), Pittman, 136–212Google Scholar
  53. 53.
    Young, L. C.: Lectures on the calculus of variations and optimal control theory, W. B. Saunders Co. (1969)Google Scholar
  54. 54.
    Billington, E. W.; Tate, A.: The Physics of Deformation and Flow, Mc-Graw Hill (1960)Google Scholar
  55. 55.
    Guttman, L.: Crystal structures and transformations in Indium-Thallium solid solutions, J. Metals, Trans. AIME 188 (1950) 1472–1477Google Scholar
  56. 56.
    Okamoto, K.; Ichinose, S.; Morri, I.; Otsuka, K.; Shimizu, K.: Crystallography of β1−γ1 stress induced martensitic transformation in a Cu−Al−Ni alloy, Acta Metall. 34 (1986) 2065–2073Google Scholar
  57. 57.
    Otsuka, K.; Shimizu, K.: Morphology and crystallography of thermoelastic Cu−Al−Ni martensite analyzed by the phenomenological theory, Trans. Jap. Inst. Metals 15 (1974) 103–108Google Scholar
  58. 58.
    Knowles, K. M.; Smith, D. A.: The crystallography of the martensitic transformation in equiatomic Nickel-Titanium. Acta Metall. 29 (1981) 101–110Google Scholar
  59. 59.
    Knowles, J. K.; Sternberg, E.: Discontinuous deformation gradients near the tip of a crack in finite anti-plane shear: an example, J. Elasticity 10 (1980) 81–110Google Scholar
  60. 60.
    Knowles, J. K.; Sternberg, E.: Anti-plane shear fields with discontinuous gradients near the tip of a crack in finite elastostatics, J. Elasticity 11 (1981) 129–164Google Scholar
  61. 61.
    Abeyaratne, R.: Discontinuous gradients away from the tip of a crack in anti-plane shear, J. Elasticity 11 (1981) 373–393Google Scholar
  62. 62.
    Silling, S. A.: Consequences of the Maxwell relation for anti-plane shear deformations of an elastic solid, J. Elasticity 19 (1988) 241–284Google Scholar
  63. 63.
    Rosakis, P.: Compact zones of shear transformation in an anisotropic solid. J. Mech. Phys. Solids 40 (1992) 1163–1195Google Scholar
  64. 64.
    Fosdick, R.; Zhang, Y.: The torsion problem for a convex stored energy function, IMA Preprint #941 (1992)Google Scholar
  65. 65.
    Liu, X.; James, R. D.: Stability of fiber networks under biaxial stretching, in preparationGoogle Scholar
  66. 66.
    Collins, C.; Luskin, M.; The computation of the austenitic-martensitic phase transition, PDEs and Cont. Models of Phase Trans. (ed. Rascle, M., Serre, D. and Slemrod, M.), Lecture Notes in Physics 344, Springer-Verlag (1989) 34–50Google Scholar
  67. 67.
    Nicolaides, R. A.; Walkington, N. J.: Numerical minimization of free energy functionals for diffusionless phase transitions, Recent Advances in Adaptive and Sensory Materials and their Applications (ed. Rogers, C. A. and Rogers, R. C.), Technomic Publishing Co. (1992) 131–141Google Scholar
  68. 68.
    Landau, L. D.; Lifshitz, E. M.: Statistical Physics, Pergamon Press (1958)Google Scholar

Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • K. Bhattacharya
    • 1
  1. 1.Cóurant Institute of Mathematical SciencesNew YorkUSA

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