Skip to main content
SpringerLink
Log in

Extension of the classical theory for types I and II twinning

  • Article
  • Multiscale Materials Modeling of Interface-mediated Thermomechanical Behavior
  • Published:
Journal of Materials Research Aims and scope Submit manuscript

Abstract

A plane-displacement diagram showing the four twinning elements, planes and directions, is fundamental to the classical theory of twinning. One aspect of the classical theory of type I and II twinning is shown to be inapplicable when the twin rotation is large. We employ the topological model with certain nonlinear characteristics to deduce a modified set of twinning elements. For twinning associated with a small rotation, both the classical theory and the topological model for type I and II twinning are shown, which give the same set of twinning elements. However, only the topological model is applicable for the large rotation case. As for the classical model, the twin plane in the type II twinning case is irrational unless it, and the type I twin is compound. Often, this irrational plane is close to a low-index orientation for a given orientation relationship. Then it can be favorable for the interface to break up into low-index, rational facets, separated by disconnections. This occurs without changing the orientation relationship. We apply the topological model to describe both the irrational type II twins and faceting in NiTi. The results agree with TEM observations.

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

Figure 1
Figure 2
Figure 3
Figure 4
Figure 5
Figure 6
Figure 7
Figure 8

Similar content being viewed by others

References

  1. O.B.M. Hardouin, Duparc: A review of some elements for the history of mechanical twinning centred on its German origins until Otto Mügge’s K1 and K2 invariant plane notation. J. Mater. Sci. 52, 4182 (2016)

    Article  Google Scholar 

  2. O. Mügge, Uber homogene deformationen (einfache schiebungen) an den triklinen doppelsalzen BaCdCl4.4aq. Neues Jahrbuch für Mineral. Geol. Palaeontol. Beilage. 6, 274 (1889)

    Google Scholar 

  3. R.W. Cahn, Plastic deformation of alpha-uranium; twinning and slip. Acta Metall. 1, 49 (1953)

    Article  CAS  Google Scholar 

  4. J.P. Hirth, R.C. Pond, R.G. Hoagland, X.Y. Liu, J. Wang, Interface defects, reference spaces and the Frank-Bilby equation. Prog. Mater. Sci. 58, 749 (2013)

    Article  Google Scholar 

  5. J.P. Hirth, J. Wang, C.N. Tomé, Disconnections and other defects associated with twin interfaces. Prog. Mater. Sci. 83, 417 (2016)

    Article  Google Scholar 

  6. A. Ostapovets, A. Serra, Review of non-classical features of deformation twinning in hcp metals and their description by disconnection mechanisms. Crystals (Metals) 10, 1134 (2020)

    Article  CAS  Google Scholar 

  7. J.W. Christian, S. Mahajan, Deformation twinning. Prog. Mater. Sci. 39, 1 (1995)

    Article  Google Scholar 

  8. B.A. Bilby, A.G. Crocker, The theory of the crystallography of deformation twinning. Proc. Roy. Soc. A. 288(1413), 240 (1965)

    CAS  Google Scholar 

  9. M. Bevis, A.G. Crocker, Twinning shears in lattices. Proc. Roy. Soc. A 304, 123 (1968)

    Google Scholar 

  10. F.C. Frank, Crystal dislocations-elementary concepts and definitions. Phil. Mag. 42, 809 (1951)

    Article  CAS  Google Scholar 

  11. J.P. Hirth, R.C. Pond, Steps, dislocations and disconnections as interface defects relating to structure and phase transformations. Acta Mater 44, 4749 (1996)

    Article  CAS  Google Scholar 

  12. R.C. Pond, J.P. Hirth, Topological model of type II deformation twinning. Acta Mater 151, 229 (2018)

    Article  CAS  Google Scholar 

  13. R.C. Pond, J.P. Hirth, K.M. Knowles, Topological model of type II deformation twinning in NiTi martensite. Phil. Mag. 99, 1619 (2019)

    Article  CAS  Google Scholar 

  14. D.Y. Xie, G. Hirth, J.P. Hirth, J. Wang, Defects in deformation twins in plagioclase minerals. Phys. Chem. Minerals 46, 959 (2019)

    Article  CAS  Google Scholar 

  15. R. Bullough, The dislocation content of a large angle tilt boundary. Phil. Mag. 5, 1139 (1965)

    Article  Google Scholar 

  16. A. Serra, D.J. Bacon, A new model for 1012 twin growth in hcp metals. Phil. Mag. A 73, 333 (1996)

    Article  CAS  Google Scholar 

  17. J.P. Hirth, J. Wang, G. Hirth, A topological model for defects and interfaces in complex crystal structures. Am. Mineral. 104, 966 (2019)

    Article  Google Scholar 

  18. J.P. Hirth, R.C. Pond, J. Lothe, Spacing defects and disconnections in grain boundaries. Acta Mater. 55, 5428 (2007)

    Article  CAS  Google Scholar 

  19. A.S.K. Mohammed, H. Sehitoglu, Modeling the interface structure of type II twin boundary in B19’ NiTi from an atomistic and topological standpoint. Acta Mater. 183, 93 (2020)

    Article  CAS  Google Scholar 

  20. K.M. Knowles, D.A. Smith, The crystallography of the martensitic transformation in equiatomic nickel-titanium. Acta Metall. 29, 101 (1981)

    Article  CAS  Google Scholar 

  21. M.Y. Gong, J.P. Hirth, Y. Liu, Y. Shen, J. Wang, Interface structures and twinning mechanisms of twins in HCP metals. Mater. Res. Lett. 5, 449 (2017)

    Article  CAS  Google Scholar 

  22. J.W. Christian, Chapter 20—Deformation twinning, in The Theory of Transformations in Metals and Alloys, ed. by J.W. Christian (Pergamon, Oxford, 2002)

    Google Scholar 

  23. K.M. Knowles, A high-resolution electron microscope study of nickel-titanium martensite. Phil. Mag. 45A, 357 (1982)

    Article  Google Scholar 

  24. Z.L. Xie, Y. Liu, HRTEM study of <011> type II twins in NiTi shape memory alloy. Phil. Mag. 84, 3497 (2004)

    Article  CAS  Google Scholar 

  25. P.M. Anderson, J.P. Hirth, J. Lothe, Theory of Dislocations (Cambridge University Press, Cambridge, 2017)

    Google Scholar 

  26. J.P. Hirth, Stabilization of strained multilayers by thin films. J. Mater. Res. 8, 1572 (1993)

    Article  Google Scholar 

  27. A. Serra, R.C. Pond, D.J. Bacon, Computer Simulation of the structure and mobility of twinning dislocations in hcp metals. Acta Metall. Mater. 39, 1469 (1991)

    Article  CAS  Google Scholar 

  28. P. Müllner, Twinning stress of type I and type II deformation twins. Acta Mater. 176, 211 (2019)

    Article  Google Scholar 

Download references

Acknowledgments

This work was financially supported by the U.S. National Science Foundation (NSF) (CMMI-1661686). We thank R.C. Pond for helpful comments.

Author information

Authors and Affiliations

  1. Green Valley, USA

    J. P. Hirth

  2. Mechanical and Materials Engineering Department, University of Nebraska-Lincoln, Lincoln, NE, 68588, USA

    J. Wang

Authors
  1. J. P. Hirth
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. Wang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hirth, J.P., Wang, J. Extension of the classical theory for types I and II twinning. Journal of Materials Research 36, 2615–2622 (2021). https://doi.org/10.1557/s43578-020-00003-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1557/s43578-020-00003-6

Keywords

Search

Navigation