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Analysis of a Moving Mask Hypothesis for Martensitic Transformations

  • Francesco Della PortaEmail author
Article
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Abstract

In this work we introduce a moving mask hypothesis to describe the dynamics of austenite-to-martensite phase transitions at a continuum level. In this framework, we prove a new type of Hadamard jump condition, from which we deduce that the deformation gradient \(\nabla \mathbf {y}\) must satisfy the differential constraint \({{\,\mathrm{\mathsf {cof}}\,}}(\nabla \mathbf {y} -\mathsf {1}) = \mathsf {0}\) a.e. in the martensite phase. This provides a selection mechanism for physically relevant energy-minimizing microstructures and is useful to better understand the complex microstructures and the formation of curved interfaces between phases in new ultra-low hysteresis alloys such as Zn45Au30Cu25. In particular, we use the new type of Hadamard jump condition to deduce a rigidity theorem for the two-well problem. The latter provides more insight on the cofactor conditions, particular conditions of supercompatibility between phases believed to influence reversibility of martensitic transformations.

Keywords

Martensitic phase transitions Selection mechanism Microstructures Moving mask Cofactor conditions 

Mathematics Subject Classification

74A50 74N05 74N10 74N15 74N20 

Notes

Acknowledgements

This work was supported by the Engineering and Physical Sciences Research Council [EP/L015811/1]. The author would like to thank John Ball for his helpful suggestions and feedback which greatly improved this work, as well as Richard James, Giacomo Canevari and Xian Chen for the useful discussions. The author would like to acknowledge the two anonymous referees for carefully reading this paper and improving it with their comments.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

References

  1. Alberti, G., Bianchini, S., Crippa, G.: Structure of level sets and Sard-type properties of Lipschitz maps. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 12(4), 863–902 (2013)MathSciNetzbMATHGoogle Scholar
  2. Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York (2000)zbMATHGoogle Scholar
  3. Ball, J.M., Carstensen, C.: Nonclassical austenite-martensite interfaces. Le J. Phys. IV 7(C5), C5–35 (1997)Google Scholar
  4. Ball, J.M., Carstensen, C.: Compatibility conditions for microstructures and the austenite–martensite transition. Mater. Sci. Eng. A 273, 231–236 (1999)CrossRefGoogle Scholar
  5. Ball, J.M., Carstensen, C.: Hadamard’s compatibility condition for microstructures (personal communication)Google Scholar
  6. Ball, J.M., James, R.D.: Fine phase mixtures as minimizers of energy. Arch. Ration. Mech. Anal. 100(1), 13–52 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  7. Ball, J.M., James, R.D.: A characterization of plane strain. Proc. R. Soc. Lond. Ser. A 432(1884), 93–99 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 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(1650), 389–450 (1992)CrossRefzbMATHGoogle Scholar
  9. Ball, J.M., Koumatos, K.: An investigation of non-planar austenite–martensite interfaces. Math. Models Methods Appl. Sci. 24(10), 1937–1956 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. Bhattacharya, K.: Self-accommodation in martensite. Arch. Ration. Mech. Anal. 120(3), 201–244 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  11. Bhattacharya, K.: Microstructure of Martensite: Why It Forms and How it Gives Rise to the Shape-Memory Effect, vol. 2. Oxford University Press, Oxford (2003)zbMATHGoogle Scholar
  12. Chen, X., Srivastava, V., Dabade, V., James, R.D.: Study of the cofactor conditions: conditions of supercompatibility between phases. J. Mech. Phys. Solids 61(12), 2566–2587 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  13. Chen, X., Tamura, N., MacDowell, A., James, R.D.: In-situ characterization of highly reversible phase transformation by synchrotron X-ray laue microdiffraction. Appl. Phys. Lett. 108(21), 211902 (2016)CrossRefGoogle Scholar
  14. Chluba, C., Ge, W., de Miranda, R.L., Strobel, J., Kienle, L., Quandt, E., Wuttig, M.: Ultralow-fatigue shape memory alloy films. Science 348(6238), 1004–1007 (2015)CrossRefGoogle Scholar
  15. Della Porta, F.: A model for the evolution of highly reversible martensitic transformations Math. Model. Method. Appl. Sci. 29(03), 493–530 (2019a).  https://doi.org/10.1142/S0218202519500143 MathSciNetCrossRefGoogle Scholar
  16. Della Porta, F.: On the cofactor conditions and further conditions of supercompatibility between phases. J. Mech. Phys. Solids 122, 27–53 (2019b)MathSciNetCrossRefGoogle Scholar
  17. Dolzmann, G.: Variational Methods for Crystalline Microstructure—Analysis and Computation. Volume 1803 of Lecture Notes in Mathematics. Springer, Berlin (2003)Google Scholar
  18. Dolzmann, G., Müller, S.: Microstructures with finite surface energy: the two-well problem. Arch. Ration. Mech. Anal. 132(2), 101–141 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  19. Evans, L.C.: Partial Differential Equations, Volume 19 of Graduate Studies in Mathematics, 2nd edn. American Mathematical Society, Providence (2010)Google Scholar
  20. Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics. CRC Press, Boca Raton (1992)zbMATHGoogle Scholar
  21. Federer, H.: Geometric Measure Theory. Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer, New York (1969)Google Scholar
  22. Flanders, H.: Differentiation under the integral sign. Am. Math. Mon. 80, 615–627 (1973); correction, ibid. 81 145 (1974)Google Scholar
  23. Girault, V., Raviart, P.A.: Finite Element Approximation of the Navier–Stokes Equations. Lecture Notes in Mathematics, vol. 749. Springer, Berlin (1979)CrossRefzbMATHGoogle Scholar
  24. Halkin, H.: Implicit functions and optimization problems without continuous differentiability of the data. SIAM J. Control 12, 229–236 (1974). Collection of articles dedicated to the memory of Lucien W. NeustadtGoogle Scholar
  25. Iwaniec, T., Verchota, G.C., Vogel, A.L.: The failure of rank-one connections. Arch. Ration. Mech. Anal. 163(2), 125–169 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  26. James, R.D.: Taming the temperamental metal transformation. Science 348(6238), 968–969 (2015)CrossRefGoogle Scholar
  27. James, R.D.: Materials from mathematics. http://www.ams.org/CEB-2018-Master.pdf (2018)
  28. Morgan, F.: Geometric Measure Theory. A Beginner’s Guide, 2nd edn. Academic Press, San Diego (1995)zbMATHGoogle Scholar
  29. Müller, S.: Variational models for microstructure and phase transitions. In: Hildebrandt, S., Struwe, M. (eds.) Calculus of Variations and Geometric Evolution Problems (Cetraro, 1996). Lecture Notes in Mathematics, vol. 1713, pp. 85–210. Springer, Berlin (1999)CrossRefGoogle Scholar
  30. Pedregal, P.: Parametrized Measures and Variational Principles. Progress in Nonlinear Differential Equations and their Applications, vol. 30. Birkhäuser Verlag, Basel (1997)zbMATHGoogle Scholar
  31. Pitteri, M., Zanzotto, G.: Generic and non-generic cubic-to-monoclinic transitions and their twins. Acta Mater. 46(1), 225–237 (1998)CrossRefGoogle Scholar
  32. Song, Y., Chen, X., Dabade, V., Shield, T.W., James, R.D.: Enhanced reversibility and unusual microstructure of a phase-transforming material. Nature 502(7469), 85 (2013)CrossRefGoogle Scholar
  33. Wechsler, M.S., Lieberman, D.S., Read, T.A.: On the theory of the formation of martensite. Trans. AIME 197, 1503–1515 (1953)Google Scholar
  34. Zhang, Z., James, R.D., Müller, S.: Energy barriers and hysteresis in martensitic phase transformations. Acta Mater. 57(15), 4332–4352 (2009)CrossRefGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Mathematical InstituteUniversity of OxfordOxfordUK
  2. 2.Max-Planck Institute for Mathematics in the SciencesLeipzigGermany

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