Skip to main content

Thank you for visiting You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

History-independent cyclic response of nanotwinned metals


Nearly 90 per cent of service failures of metallic components and structures are caused by fatigue at cyclic stress amplitudes much lower than the tensile strength of the materials involved1. Metals typically suffer from large amounts of cumulative, irreversible damage to microstructure during cyclic deformation, leading to cyclic responses that are unstable (hardening or softening)2,3,4 and history-dependent5,6,7,8. Existing rules for fatigue life prediction, such as the linear cumulative damage rule1,9, cannot account for the effect of loading history, and engineering components are often loaded by complex cyclic stresses with variable amplitudes, mean values and frequencies10,11, such as aircraft wings in turbulent air. It is therefore usually extremely challenging to predict cyclic behaviour and fatigue life under a realistic load spectrum1,11. Here, through both atomistic simulations and variable-strain-amplitude cyclic loading experiments at stress amplitudes lower than the tensile strength of the metal, we report a history-independent and stable cyclic response in bulk copper samples that contain highly oriented nanoscale twins. We demonstrate that this unusual cyclic behaviour is governed by a type of correlated ‘necklace’ dislocation consisting of multiple short component dislocations in adjacent twins, connected like the links of a necklace. Such dislocations are formed in the highly oriented nanotwinned structure under cyclic loading and help to maintain the stability of twin boundaries and the reversible damage, provided that the nanotwins are tilted within about 15 degrees of the loading axis. This cyclic deformation mechanism is distinct from the conventional strain localizing mechanisms associated with irreversible microstructural damage in single-crystal12,13, coarse-grained1,14, ultrafine-grained and nanograined metals4,15,16.

This is a preview of subscription content, access via your institution

Relevant articles

Open Access articles citing this article.

Access options

Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.

Figure 1: Microstructure of NT-Cu before and after cyclic deformation.
Figure 2: History-independent cyclic deformation behaviour of NT-Cu.
Figure 3: Cyclic deformation features of NT-Cu.
Figure 4: CND morphology in cyclically deformed NT-Cu.


  1. Suresh, S. Fatigue of Materials 2nd edn (Cambridge Univ. Press, 1998)

  2. Smith, R. W ., Hirschberg, M. H. & Manson, S. S. Fatigue Behavior of Materials under Strain Cycling in Low and Intermediate Life Range (NASA, 1963)

  3. Polakowski, N. H. Softening of cold-worked metals under cyclic strains. Nature 171, 173 (1953)

    ADS  Google Scholar 

  4. Mughrabi, H. & Höppel, H. W. Cyclic deformation and fatigue properties of very fine-grained metals and alloys. Int. J. Fatigue 32, 1413–1427 (2010)

    CAS  Google Scholar 

  5. Feltner, C. E. & Laird, C. Cyclic stress–strain response of FCC metals and alloys 1. Phenomenological experiments. Acta Metall. 15, 1621–1632 (1967)

    CAS  Google Scholar 

  6. Laird, C., Finney, J. M., Schwartzman, A. & Veaux, R. D. L. History dependence in cyclic stress-strain response of wavy slip materials. J. Test. Eval. 3, 435–441 (1975)

    CAS  Google Scholar 

  7. Mecke, K. TEM investigations of cyclic stress strain behavior and formation of persistent slip bands in fatigued single-crystals of nickel using changing amplitude tests. Phys. Status Solidi A 25, K93–K96 (1974)

    ADS  CAS  Google Scholar 

  8. Lukáš, P. & Klesnil, M. Cyclic stress–strain response and fatigued life of metals in low amplitude region. Mater. Sci. Eng. 11, 345–356 (1973)

    Google Scholar 

  9. May, A. N. Fatigue under random loads. Nature 192, 158–159 (1961)

    ADS  Google Scholar 

  10. Head, A. K. & Hooke, F. H. Fatigue of metals under random loads. Nature 177, 1176–1177 (1956)

    ADS  Google Scholar 

  11. Schijve, J. Fatigue of Structures and Materials 2nd edn (Springer, 2009)

  12. Mughrabi, H. Cyclic hardening and saturation behavior of copper single-crystals. Mater. Sci. Eng. 33, 207–223 (1978)

    CAS  Google Scholar 

  13. Li, P., Li, S. X., Wang, Z. G. & Zhang, Z. F. Fundamental factors on formation mechanism of dislocation arrangements in cyclically deformed fcc single crystals. Prog. Mater. Sci. 56, 328–377 (2011)

    CAS  Google Scholar 

  14. Peralta, P. & Laird, C. in Physical Metallurgy 5th edn (eds Laughlin, D. E. & Hono, K. ) 1765–1880 (Elsevier, 2014)

  15. Hanlon, T., Kwon, Y. N. & Suresh, S. Grain size effects on the fatigue response of nanocrystalline metals. Scr. Mater. 49, 675–680 (2003)

    CAS  Google Scholar 

  16. Pineau, A., Amine Benzerga, A. & Pardoen, T. Failure of metals III. Fracture and fatigue of nanostructured metallic materials. Acta Mater. 107, 508–544 (2016)

    CAS  Google Scholar 

  17. You, Z. S., Lu, L. & Lu, K. Tensile behavior of columnar grained Cu with preferentially oriented nanoscale twins. Acta Mater. 59, 6927–6937 (2011)

    CAS  Google Scholar 

  18. Pan, Q. S. & Lu, L. Strain-controlled cyclic stability and properties of Cu with highly oriented nanoscale twins. Acta Mater. 81, 248–257 (2014)

    ADS  CAS  Google Scholar 

  19. Lu, K., Lu, L. & Suresh, S. Strengthening materials by engineering coherent internal boundaries at the nanoscale. Science 324, 349–352 (2009)

    ADS  CAS  PubMed  Google Scholar 

  20. Wang, J. et al. Detwinning mechanisms for growth twins in face-centered cubic metals. Acta Mater. 58, 2262–2270 (2010)

    CAS  Google Scholar 

  21. Shute, C. J. et al. Detwinning, damage and crack initiation during cyclic loading of Cu samples containing aligned nanotwins. Acta Mater. 59, 4569–4577 (2011)

    ADS  CAS  Google Scholar 

  22. Yoo, B.-G. et al. Quantitative damage and detwinning analysis of nanotwinned copper foil under cyclic loading. Acta Mater. 81, 184–193 (2014)

    CAS  Google Scholar 

  23. Wang, Y. M. et al. Defective twin boundaries in nanotwinned metals. Nat. Mater. 12, 697–702 (2013)

    ADS  CAS  PubMed  Google Scholar 

  24. Jang, D. C., Li, X. Y., Gao, H. J. & Greer, J. R. Deformation mechanisms in nanotwinned metal nanopillars. Nat. Nanotech. 7, 594–601 (2012)

    ADS  CAS  Google Scholar 

  25. Zhu, T. & Gao, H. J. Plastic deformation mechanism in nanotwinned metals: an insight from molecular dynamics and mechanistic modeling. Scr. Mater. 66, 843–848 (2012)

    CAS  Google Scholar 

  26. Zhou, H. F., Li, X. Y., Qu, S. X., Yang, W. & Gao, H. J. A jogged dislocation governed strengthening mechanism in nanotwinned metals. Nano Lett. 14, 5075–5080 (2014)

    ADS  CAS  PubMed  Google Scholar 

  27. Williams, D. B . & Carter, C. B. Transmission Electron Microscopy: A Textbook for Materials Science 2nd edn (Springer Science/Business Media, 2009)

  28. Lu, Q. H., You, Z. S., Huang, X., Hansen, N. & Lu, L. Dependence of dislocation structure on orientation and slip systems in highly oriented nanotwinned Cu. Acta Mater. 127, 85–97 (2017)

    CAS  Google Scholar 

  29. Bufford, D. C., Wang, Y. M., Liu, Y. & Lu, L. Synthesis and microstructure of electrodeposited and sputtered nanotwinned face-centered-cubic metals. MRS Bull. 41, 286–291 (2016)

    CAS  Google Scholar 

  30. Hodge, A. M., Wang, Y. M. & Barbee, T. W. Mechanical deformation of high-purity sputter-deposited nano-twinned copper. Scr. Mater. 59, 163–166 (2008)

    CAS  Google Scholar 

  31. Zhang, X. et al. High-strength sputter-deposited Cu foils with preferred orientation of nanoscale growth twins. Appl. Phys. Lett. 88, 173116 (2006)

    ADS  Google Scholar 

  32. Li, Y. S., Tao, N. R. & Lu, K. Microstructural evolution and nanostructure formation in copper during dynamic plastic deformation at cryogenic temperatures. Acta Mater. 56, 230–241 (2008)

    CAS  Google Scholar 

  33. Agnew, S. R., Vinogradov, A. Y., Hashimoto, S. & Weertman, J. R. Overview of fatigue performance of Cu processed by severe plastic deformation. J. Electron. Mater. 28, 1038–1044 (1999)

    ADS  CAS  Google Scholar 

  34. Höppel, H. W ., Brunnbauer, M . & Mughrabi, H. in Proceedings of Materials Week 2000 Ch. 25, 1–8 (Werkstoffwoche-Partnerschaft, 2000)

  35. Abel, A. Fatigue of copper single crystals at low constant plastic strain amplitudes. Mater. Sci. Eng. 36, 117–124 (1978)

    CAS  Google Scholar 

  36. Polák, J., Obrtlik, K. & Helesic, J. Cyclic strain localization in polycrystalline copper at room-temperature and low-temperatures. Mater. Sci. Eng. A 132, 67–76 (1991)

    Google Scholar 

  37. Wang, R. & Mughrabi, H. Secondary cyclic hardening in fatigued copper monocrystals and polycrystals. Mater. Sci. Eng. 63, 147–163 (1984)

    CAS  Google Scholar 

  38. Li, X. W., Wang, Z. G., Li, G. Y., Wu, S. D. & Li, S. X. Cyclic stress-strain response and surface deformation features of [011] multiple-slip-oriented copper single crystals. Acta Mater. 46, 4497–4505 (1998)

    CAS  Google Scholar 

  39. Lepistö, T. K. & Kettunen, P. O. Comparison of the cyclic stress–strain behavior of single-and <111> multiple-slip-oriented copper single-crystals. Mater. Sci. Eng. 83, 1–15 (1986)

    Google Scholar 

  40. Christ, H. J. & Mughrabi, H. Cyclic stress-strain response and microstructure under variable amplitude loading. Fatigue Fract. Eng. Mater. Struct. 19, 335–348 (1996)

    CAS  Google Scholar 

  41. Morrow, J. D. in Internal Friction, Damping, and Cyclic Plasticity (ed. Lazan, B. J. ) 45–87 (Special Technical Publication 378, American Society for Testing of Materials, 1965)

  42. Nosé, S. A unified formation of the constant temperature molecular-dynamic methods. J. Chem. Phys. 81, 511–519 (1984)

    ADS  Google Scholar 

  43. Hoover, W. G. Constant-pressure equations of motion. Phys. Rev. A 34, 2499–2500 (1986)

    ADS  CAS  Google Scholar 

  44. Mishin, Y., Mehl, M. J., Papaconstantopoulos, D. A., Voter, A. F. & Kress, J. D. Structural stability and lattice defects in copper: ab initio, tight-binding, and embedded-atom calculations. Phys. Rev. B 63, 224106 (2001)

    ADS  Google Scholar 

  45. Li, X. Y., Wei, Y. J., Lu, L., Lu, K. & Gao, H. J. Dislocation nucleation governed softening and maximum strength in nano-twinned metals. Nature 464, 877–880 (2010)

    ADS  CAS  PubMed  Google Scholar 

  46. Faken, D. & Jonsson, H. Systematic analysis of local atomic structure combined with 3D computer graphics. Comput. Mater. Sci. 2, 279–286 (1994)

    CAS  Google Scholar 

  47. Zhou, H. F. & Gao, H. J. A plastic deformation mechanism by necklace dislocations near crack-like defects in nanotwinned metals. J. Appl. Mech. 82, 071015 (2015)

    ADS  Google Scholar 

  48. Zhu, Y. X., Li, Z. H., Huang, M. S. & Liu, Y. Strengthening mechanisms of the nanolayered polycrystalline metallic multilayers assisted by twins. Int. J. Plast. 72, 168–184 (2015)

    CAS  Google Scholar 

  49. Chowdhury, P. B., Sehitoglu, H. & Rateick, R. G. Predicting fatigue resistance of nano-twinned materials: Part I. Role of cyclic slip irreversibility and Peierls stress. Int. J. Fatigue 68, 277–291 (2014)

    CAS  Google Scholar 

  50. Lu, K. Stabilizing nanostructures in metals using grain and twin boundary architectures. Nat. Rev. Mater. 1, 16019 (2016)

    ADS  CAS  Google Scholar 

Download references


L.L. acknowledges support by the National Natural Science Foundation of China (NSFC) (grant numbers 51371171, 51471172 and U1608257) and the Key Research Program of Frontier Science, Chinese Academy of Sciences. H.G. acknowledges support by the US National Science Foundation through grant DMR-1709318. L.L. and H.G. acknowledge support by an international collaboration grant from NSFC (51420105001). The simulations reported were performed on resources provided by the Extreme Science and Engineering Discovery Environment (XSEDE) through grant MS090046. We thank H. Mughrabi for discussions and S. Jin for sample preparation.

Author information

Authors and Affiliations



L.L. and H.G. designed and supervised the project; Q.P., Q.L. and L.L. performed the experiments; H.Z. performed the atomistic simulations; Q.P., H.Z., H.G. and L.L. analysed data, developed models, discussed the results and wrote the paper.

Corresponding authors

Correspondence to Huajian Gao or Lei Lu.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

Additional information

Reviewer Information Nature thanks J. Wang and the other anonymous reviewer(s) for their contribution to the peer review of this work.

Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Extended data figures and tables

Extended Data Figure 1 Engineering stress–strain curves of NT-Cu, UFG-Cu and CG-Cu samples in uniaxial tensile experiments.

At least three repeated tensile tests were performed on each sample.

Extended Data Figure 2 Long-cycle symmetric tension–compression loading procedure imposed on NT-Cu samples.

Under strain control, a triangular wave loading profile was used. The applied plastic strain amplitude Δεpl/2 increases stepwise from 0.02% to 0.04%, 0.06% and 0.09%, and then decreases back to 0.02%. N0 = 1,500 cycles, N1 = 500 cycles and Nf is the number of cycles up to final failure.

Extended Data Figure 3 History-independent cyclic response of NT-Cu.

Cyclic stress response as a function of the cumulative plastic strain for NT-Cu-A (a, b) and NT-Cu-B (c, d) cyclically deformed by increasing Δεpl/2 step-by-step from 0.02% to 0.09% and decreasing back to 0.02%. Cyclic stress response of NT-Cu-A (e, f), UFG-Cu (h, i) and CG-Cu (k, l) cyclically deformed by increasing Δεpl/2 step-by-step from 0.05% to 0.25% and decreasing back to 0.05%, with cyclic numbers of 70 at each Δεpl/2. Their corresponding hysteresis loops (at the 50th cycle of each Δεpl/2) are shown in g, j and m, respectively.

Extended Data Figure 4 Atomistic simulation setup.

a, Perspective view of the initial configuration of NT-Cu. b, Cross sectional view of a typical NT grain G4. Each grain in a has the same TB distribution. c, View of the NT-Cu sample in x–y projection. The misorientation angle θi (i = 1, 2, 3 and 4) of the ith grain represents the angle between its [110] direction and the y axis. Colours are assigned to atoms according to their local crystal structure. d, Simulated tensile loading and unloading stress–strain curves of NT-Cu.

Extended Data Figure 5 History-independent cyclic response of NT-Cu demonstrated by MD simulations.

a, Three-step cyclic loading scheme. b, c, Simulated Δσ/2 versus N relation and hysteresis loops. df, Snapshots of NT-Cu captured at three sequential cycles (that is, N = 1, 10 and 20) at Δεt/2 = 1% (corresponding to Δεpl/2 = 0.25%). Colours are assigned to atoms based on their spatial coordinates. g, CNDs formed by tail-linkage or TB transmission (indicated by black arrows) during cyclic loading. See Extended Data Fig. 7 for details on the formation mechanism of CNDs. Colours are assigned to atoms based on their local crystal structure. h, The dislocation density reaches a plateau after ten cycles at Δεt/2 = 1%.

Extended Data Figure 6 Stability of NT structure under cyclic deformation.

The grain size (a, c, e, g) and twin thickness (b, d, f, h) distributions of both NT-Cu-A and NT-Cu–B samples before (ad) and after being cyclically deformed to failure (eh), showing no detectable changes in the mean grain size and twin thickness between the as-deposited and cyclically deformed states. In each panel, the value given for d or λ is the mean of the values plotted.

Extended Data Figure 7 Formation mechanism of CNDs in NT-Cu under cyclic loading.

a, Double Thompson tetrahedron representation of the slip systems in matrix (upper) and twin (lower). b, c, Atomic configurations of a typical CND projected along the and crystallographic directions. Atoms in fcc structure are made transparent for clarity. d, Perspective view of the same CND with hcp atoms hidden to emphasize the necklace-like feature of the dislocation. The inset shows the Burgers vectors of an extended stair-rod component of a CND which bends from the matrix to the twin plane. eh, Schematics showing the formation of CNDs through the linking of threading dislocation tails in adjacent twin layers under cyclic loading. ik, Sometimes threading tails can also slip across TBs.

Extended Data Figure 8 Deformation patterns in grains labelled as G1–G4 in Extended Data Fig. 4 after 20 tension–compression cycles with Δεt2 = 1%.

a–d, A single slip system is activated in G1 (a), whereas double slip systems are activated in G2–G4 (b–d) during cyclic deformation. e, Parallel slip traces observed in G1. f, Numerous dislocation locks are observed in G4. g–i, Snapshots showing a typical CND (indicated by black arrows) emerging from a grain boundary as a single unit in cyclically deformed NT-Cu sample (see details in Methods). Colours are assigned to atoms based on their spatial coordinates.

Extended Data Figure 9 Molecular dynamics simulations showing history-independent cyclic response of NT-Cu containing tilted TBs with respect to the loading axis.

a, NT-Cu containing tilted TBs with respect to the loading axis. bd, CNDs in NT-Cu samples with tilt angles of 5° (b), 10° (c) and 15° (d) under cyclic loading, respectively. e, Typical CND structure observed in tilted TBs. CND tails connecting threading segments in neighbouring twin layers are indicated by black arrows. f, Relationship Δσ/2 and N for tilt angle of 15°; g, the corresponding hysteresis loops. h, Cyclic deformation of NT-Cu with TBs tilted 20° with respect to the loading axis is governed by the three types of dislocation mechanism shown. i, Simulated NT-Cu sample containing 27 grains with randomly oriented TBs. j, Relationship between Δσ/2 and N demonstrating history-dependent cyclic response of randomly oriented NT sample. k, The corresponding hysteresis loops for different Δεt/2.

Extended Data Table 1 Activated slip systems in NT-Cu during the cyclic experiments illustrated with double Thompson tetrahedron

Supplementary information

Experimental details

This file includes three major parts: 1) Preparation of NT-Cu samples for cyclic loading tests, 2) Fatigue explanatory and fatigue test recording, 3) Microstructure characterization of NT-Cu before, during and after cyclic deformation.

A perspective view of correlated necklace dislocations (CNDs) in grain G1 (Extended Data Fig. 4).

CNDs are gliding parallel to the TBs of the simulated NT-Cu sample under cyclic deformation at a total strain amplitude of 1%.

A lateral view of correlated necklace dislocations (CNDs) in grain G1 (Extended Data Fig. 4).

CNDs are gliding parallel to the TBs of the simulated NT-Cu sample under cyclic deformation at a total strain amplitude of 1%.

A lateral view of the uncorrelated dislocation activities in grain G4 (Extended Data Fig. 4).

The simulated NT-Cu sample is under cyclic deformation at a total strain amplitude of 1%.

A perspective view of correlated necklace dislocations (CNDs) in tilted TBs with a tilt angle of 5° relative to the loading axis.

CNDs are gliding parallel to the tilted TBs under cyclic deformation at an applied strain amplitude Δεt/ of 1.3%

A perspective view of correlated necklace dislocations (CNDs) in tilted TBs with a tilt angle of 10° relative to the loading axis.

CNDs are gliding parallel to the tilted TBs under cyclic deformation at an applied strain amplitude Δεt/ of 1.3%

A perspective view of correlated necklace dislocations (CNDs) in tilted TBs with a tilt angle of 15° relative to the loading axis.

CNDs are gliding parallel to the TBs under cyclic deformation at an applied strain amplitude Δεt/ of 1.3%

A perspective view of the cyclic deformation of a simulated NT-Cu sample containing tilted TBs with a tilt angle of 20° relative to the loading axis.

The applied strain amplitude Δεt/2 = 1.3%.

PowerPoint slides

Source data

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Pan, Q., Zhou, H., Lu, Q. et al. History-independent cyclic response of nanotwinned metals. Nature 551, 214–217 (2017).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:

This article is cited by


By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.


Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing