Research Papers

A Simple Physically Based Phenomenological Model for the Strengthening/Softening Behavior of Nanotwinned Copper

[+] Author and Article Information
X. Zhang

School of Mechanics and Engineering,
Southwest Jiaotong University,
Chengdu 610031, China
e-mail: xuzhang85@126.com

A. E. Romanov

International Laboratory of Modern
Functional Materials,
ITMO University,
St. Petersburg 199034, Russia
e-mail: alexey.romanov@niuitmo.ru

E. C. Aifantis

School of Mechanics and Engineering,
Southwest Jiaotong University,
Chengdu 610031, China;
International Laboratory of Modern
Functional Materials,
ITMO University,
St. Petersburg 199034, Russia;
Michigan Technological University,
Houghton, MI 49931;
Laboratory of Mechanics and Materials,
Aristotle University,
Thessaloniki 54124, Greece
e-mail: mom@mom.gen.auth.gr

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received July 21, 2015; final manuscript received August 11, 2015; published online September 10, 2015. Editor: Yonggang Huang.

J. Appl. Mech 82(12), 121005 (Sep 10, 2015) (8 pages) Paper No: JAM-15-1377; doi: 10.1115/1.4031291 History: Received July 21, 2015; Revised August 11, 2015

A robust phenomenological model based on a modified size-dependent Voce-type constitutive equation is proposed to describe the dependence of strength on twin thickness, for nanotwinned copper (nt-Cu) polycrystals, in agreement with experiments and related atomistic simulations. A gradient plasticity argument is employed to determine the critical nanotwin thickness where the transition from Hall–Petch (HP) hardening to inverse Hall–Petch (IHP) softening occurs. Strain rate and temperature effects are also discussed. The proposed constitutive equation may be used for engineering design purposes by controlling the interplay between grain size and twin thickness.

Copyright © 2015 by ASME
Your Session has timed out. Please sign back in to continue.


Lu, L. , Chen, X. , Huang, X. , and Lu, K. , 2009, “ Revealing the Maximum Strength in Nanotwinned Copper,” Science, 323(5914), pp. 607–610. [CrossRef] [PubMed]
Wei, Y. , Li, Y. , Zhu, L. , Liu, Y. , Lei, X. , Wang, G. , Wu, Y. , Mi, Z. , Liu, J. , Wang, H. , and Gao, H. , 2014, “ Evading the Strength–Ductility Trade-Off Dilemma in Steel Through Gradient Hierarchical Nanotwins,” Nat. Commun., 5, pp. 1–8.
Li, X. , Wei, Y. , Lu, L. , Lu, K. , and Gao, H. , 2010, “ Dislocation Nucleation Governed Softening and Maximum Strength in Nano-Twinned Metals,” Nature, 464(7290), pp. 877–880. [CrossRef] [PubMed]
Lu, L. , Dao, M. , Zhu, T. , and Li, J. , 2009, “ Size Dependence of Rate-Controlling Deformation Mechanisms in Nanotwinned Copper,” Scr. Mater., 60(12), pp. 1062–1066. [CrossRef]
Lu, L. , Schwaiger, R. , Shan, Z. W. , Dao, M. , Lu, K. , and Suresh, S. , 2005, “ Nano-Sized Twins Induce High Rate Sensitivity of Flow Stress in Pure Copper,” Acta Mater., 53(7), pp. 2169–2179. [CrossRef]
Yamakov, V. , Wolf, D. , Phillpot, S. R. , and Gleiter, H. , 2003, “ Dislocation–Dislocation and Dislocation–Twin Reactions in Nanocrystalline Al by Molecular Dynamics Simulation,” Acta Mater., 51(14), pp. 4135–4147. [CrossRef]
Jin, Z. H. , Gumbsch, P. , Albe, K. , Ma, E. , Lu, K. , Gleiter, H. , and Hahn, H. , 2008, “ Interactions Between Non-Screw Lattice Dislocations and Coherent Twin Boundaries in Face-Centered Cubic Metals,” Acta Mater., 56(5), pp. 1126–1135. [CrossRef]
Jin, Z. H. , Gumbsch, P. , Ma, E. , Albe, K. , Lu, K. , Hahn, H. , and Gleiter, H. , 2006, “ The Interaction Mechanism of Screw Dislocations With Coherent Twin Boundaries in Different Face-Centred Cubic Metals,” Scr. Mater., 54(6), pp. 1163–1168. [CrossRef]
Zhu, Y. T. , Wu, X. L. , Liao, X. Z. , Narayan, J. , Kecskés, L. J. , and Mathaudhu, S. N. , 2011, “ Dislocation–Twin Interactions in Nanocrystalline FCC Metals,” Acta Mater., 59(2), pp. 812–821. [CrossRef]
Li, N. , Wang, J. , Huang, J. Y. , Misra, A. , and Zhang, X. , 2011, “ Influence of Slip Transmission on the Migration of Incoherent Twin Boundaries in Epitaxial Nanotwinned Cu,” Scr. Mater., 64(2), pp. 149–152. [CrossRef]
Stukowski, A. , Albe, K. , and Farkas, D. , 2010, “ Nanotwinned FCC Metals: Strengthening Versus Softening Mechanisms,” Phys. Rev. B, 82(22), p. 224103.
Wang, Y. B. , Wu, B. , and Sui, M. L. , 2008, “ Dynamical Dislocation Emission Processes From Twin Boundaries,” Appl. Phys. Lett., 93(4), p. 041906. [CrossRef]
Frøseth, A. G. , Derlet, P. M. , and Van Swygenhoven, H. , 2006, “ Vicinal Twin Boundaries Providing Dislocation Sources in Nanocrystalline Al,” Scr. Mater., 54(3), pp. 477–481.
Zhou, H. , Qu, S. , and Yang, W. , 2010, “ Toughening by Nano-Scaled Twin Boundaries in Nanocrystals,” Modell. Simul. Mater. Sci. Eng., 18(6), p. 065002. [CrossRef]
Zhou, H. F. , Li, X. Y. , Qu, S. X. , Yang, W. , and Gao, H. J. , 2014, “ A Jogged Dislocation Governed Strengthening Mechanism in Nanotwinned Metals,” Nano Lett., 14(9), pp. 5075–5080. [CrossRef] [PubMed]
Zhu, L. , Ruan, H. , Li, X. , Dao, M. , Gao, H. , and Lu, J. , 2011, “ Modeling Grain Size Dependent Optimal Twin Spacing for Achieving Ultimate High Strength and Related High Ductility in Nanotwinned Metals,” Acta Mater., 59(14), pp. 5544–5557. [CrossRef]
Zhang, X. , Romanov, A. E. , and Aifantis, E. C. , 2011, “ On Gradient Nanomechanics: Plastic Flow in Nanopolycrystals,” 5th International Conference on Nanomaterials by Severe Plastic Deformation, NanoSPD5, pp. 991–996.
Zhang, X. , and Aifantis, K. E. , 2015, “ Examining the Evolution of the Internal Length as a Function of Plastic Strain,” Mater. Sci. Eng. A, 631, pp. 27–32. [CrossRef]
Zhang, X. , Aifantis, K. E. , Senger, J. , Weygand, D. , and Zaiser, M. , 2014, “ Internal Length Scale and Grain Boundary Yield Strength in Gradient Models of Polycrystal Plasticity: How Do They Relate to the Dislocation Microstructure?” J. Mater. Res., 29(18), pp. 2116–2128. [CrossRef]
Voyiadjis, G. Z. , and Abu Al-Rub, R. , 2004, “ Determination of the Material Intrinsic Length Scale of Gradient Plasticity Theory,” IUTAM Symposium on Multiscale Modeling and Characterization of Elastic–Inelastic Behavior of Engineering Materials, Marrakech, Morocco, Oct. 20–25, Springer, Dordrecht, Vol. 114, pp. 167–174.
Nix, W. D. , and Gao, H. , 1998, “ Indentation Size Effects in Crystalline Materials: A Law for Strain Gradient Plasticity,” J. Mech. Phys. Solids, 46(3), pp. 411–425. [CrossRef]
Zhao, J. , Zhang, X. , Konstantinidis, A. A. , and Kang, G. , 2015, “ Correlating the Internal Length in Strain Gradient Plasticity Theory With the Microstructure of Material,” Philos. Mag. Lett., 95(6), pp. 340–349. [CrossRef]
Zhang, X. , and Aifantis, K. , 2015, “ Interpreting the Internal Length Scale in Strain Gradient Plasticity,” Rev. Adv. Mater. Sci., 41(1), pp. 72–83.
Aifantis, K. E. , and Konstantinidis, A. A. , 2009, “ Yielding and Tensile Behavior of Nanocrystalline Copper,” Mater. Sci. Eng. A, 503(1–2), pp. 198–201. [CrossRef]
Aifantis, K. E. , and Willis, J. R. , 2005, “ The Role of Interfaces in Enhancing the Yield Strength of Composites and Polycrystals,” J. Mech. Phys. Solids, 53(5), pp. 1047–1070. [CrossRef]
Aifantis, K. E. , Soer, W. A. , De Hosson, J. T. M. , and Willis, J. R. , 2006, “ Interfaces Within Strain Gradient Plasticity: Theory and Experiments,” Acta Mater., 54(19), pp. 5077–5085. [CrossRef]
Estrin, Y. , and Mecking, H. , 1984, “ A Unified Phenomenological Description of Work Hardening and Creep Based on One-Parameter Models,” Acta Metall., 32(1), pp. 57–70. [CrossRef]
Hirth, J. P. , and Lothe, J. , 1982, Theory of Dislocations, Wiley, New York.
Meyers, M. A. , Mishra, A. , and Benson, D. J. , 2006, “ Mechanical Properties of Nanocrystalline Materials,” Prog. Mater Sci., 51(4), pp. 427–556. [CrossRef]
Evers, L. P. , Parks, D. M. , Brekelmans, W. A. M. , and Geers, M. G. D. , 2002, “ Crystal Plasticity Model With Enhanced Hardening by Geometrically Necessary Dislocation Accumulation,” J. Mech. Phys. Solids, 50(11), pp. 2403–2424. [CrossRef]
Armstrong, R. , Codd, I. , Douthwaite, R. M. , and Petch, N. J. , 1962, “ The Plastic Deformation of Polycrystalline Aggregates,” Philos. Mag., 7(73), pp. 45–58. [CrossRef]
Zhang, X. , and Aifantis, K. E. , 2011, “ Interpreting the Softening of Nanomaterials Through Gradient Plasticity,” J. Mater. Res., 26(11), pp. 1399–1405. [CrossRef]
Kocks, U. F. , and Mecking, H. , 2003, “ Physics and Phenomenology of Strain Hardening: The FCC Case,” Prog. Mater Sci., 48(3), pp. 171–273. [CrossRef]
Lu, L. , You, Z. S. , and Lu, K. , 2012, “ Work Hardening of Polycrystalline Cu With Nanoscale Twins,” Scr. Mater., 66(11), pp. 837–842. [CrossRef]
Ma, A. , and Roters, F. , 2004, “ A Constitutive Model for FCC Single Crystals Based on Dislocation Densities and Its Application to Uniaxial Compression of Aluminium Single Crystals,” Acta Mater., 52(12), pp. 3603–3612. [CrossRef]
Konstantinidis, A. A. , Aifantis, K. E. , and De Hosson, J. T. M. , 2014, “ Capturing the Stochastic Mechanical Behavior of Micro and Nanopillars,” Mater. Sci. Eng. A, 597, pp. 89–94. [CrossRef]
Zhang, X. , Aifantis, K. E. , and Zaiser, M. , 2013, “ Material vs. Discretization Length Scales in Plasticity Simulations of Solid Foams,” Rev. Adv. Mater. Sci., 35(1), pp. 39–47.
Aifantis, K. E. , Konstantinidis, A. , and Forest, S. , 2010, “ Modeling Strain Localization Bands in Metal Foams,” J. Comput. Theor. Nanosci., 7(2), pp. 360–366. [CrossRef]
Mayeur, J. R. , Beyerlein, I . J. , Bronkhorst, C. A. , and Mourad, H. M. , 2015, “ Incorporating Interface Affected Zones Into Crystal Plasticity,” Int. J. Plast., 65, pp. 206–225. [CrossRef]
Zhu, L. , Qu, S. , Guo, X. , and Lu, J. , 2015, “ Analysis of the Twin Spacing and Grain Size Effects on Mechanical Properties in Hierarchically Nanotwinned Face-Centered Cubic Metals Based on a Mechanism-Based Plasticity Model,” J. Mech. Phys. Solids, 76, pp. 162–179. [CrossRef]
Dao, M. , Lu, L. , Shen, Y. F. , and Suresh, S. , 2006, “ Strength, Strain-Rate Sensitivity and Ductility of Copper With Nanoscale Twins,” Acta Mater., 54(20), pp. 5421–5432. [CrossRef]
Purohit, Y. , Sun, L. , Irving, D. L. , Scattergood, R. O. , and Brenner, D. W. , 2010, “ Computational Study of the Impurity Induced Reduction of Grain Boundary Energies in Nano- and Bi-Crystalline Al–Pb Alloys,” Mater. Sci. Eng. A, 527(7–8), pp. 1769–1775. [CrossRef]
Murdoch, H. A. , and Schuh, C. A. , 2013, “ Stability of Binary Nanocrystalline Alloys Against Grain Growth and Phase Separation,” Acta Mater., 61(6), pp. 2121–2132. [CrossRef]
Mishin, Y. , Mehl, M. , Papaconstantopoulos, D. , Voter, A. , and Kress, J. , 2001, “ Structural Stability and Lattice Defects in Copper: Ab Initio, Tight-Binding, and Embedded-Atom Calculations,” Phys. Rev. B, 63(22), p. 224106. [CrossRef]
You, Z. , Li, X. , Gui, L. , Lu, Q. , Zhu, T. , Gao, H. , and Lu, L. , 2013, “ Plastic Anisotropy and Associated Deformation Mechanisms in Nanotwinned Metals,” Acta Mater., 61(1), pp. 217–227. [CrossRef]
Wei, Y. , 2011, “ Scaling of Maximum Strength With Grain Size in Nanotwinned FCC Metals,” Phys. Rev. B, 83(13), p. 132104.
Wei, Y. , 2011, “ The Kinetics and Energetics of Dislocation Mediated De-Twinning in Nano-Twinned Face-Centered Cubic Metals,” Mater. Sci. Eng. A, 528(3), pp. 1558–1566. [CrossRef]
Lu, K. , Lu, L. , and Suresh, S. , 2009, “ Strengthening Materials by Engineering Coherent Internal Boundaries at the Nanoscale,” Science, 324(5925), pp. 349–352. [CrossRef] [PubMed]
Asaro, R. J. , and Suresh, S. , 2005, “ Mechanistic Models for the Activation Volume and Rate Sensitivity in Metals With Nanocrystalline Grains and Nano-Scale Twins,” Acta Mater., 53(12), pp. 3369–3382. [CrossRef]
Carsley, J. E. , Ning, J. , Milligan, W. W. , Hackney, S. A. , and Aifantis, E. C. , 1995, “ A Simple, Mixtures-Based Model for the Grain Size Dependence of Strength in Nanophase Metals,” Nanostruct. Mater., 5(4), pp. 441–448. [CrossRef]
Konstantinidis, D. A. , and Aifantis, E. C. , 1998, “ On the ‘Anomalous' Hardness of Nanocrystalline Materials,” Nanostruct. Mater., 10(7), pp. 1111–1118. [CrossRef]
Kim, H. S. , and Estrin, Y. , 2005, “ Phase Mixture Modeling of the Strain Rate Dependent Mechanical Behavior of Nanostructured Materials,” Acta Mater., 53(3), pp. 765–772. [CrossRef]
Aifantis, E. C. , 2011, “ Gradient Nanomechanics: Applications to Deformation, Fracture, and Diffusion in Nanopolycrystals,” Metall. Mater. Trans. A, 42(10), pp. 2985–2998. [CrossRef]
Ovid'ko, I. , and Aifantis, E. C. , 2013, “ Nanocrystals & Nanomechanics: Mechanisms & Models. A Selective Review,” Rev. Adv. Mater. Sci., 35(1–2), pp. 1–24.


Grahic Jump Location
Fig. 1

The fit of the various size-dependent true stress–strain curves for nt-Cu specimens with varying twin thicknesses. The modified size-dependent Voce model employed was based on Eqs. (7) and (8) with the unique set of the parameter values used for all curves given in Table 1: (a) twin thickness is the same as that reported in Ref. [1] and (b) slightly modified twin thickness.

Grahic Jump Location
Fig. 2

Schematic illustration of the different deformed zones of a representative twin lamella comprised of a TBAZ, a PTL, and an EC

Grahic Jump Location
Fig. 3

Comparison between the theoretical predictions and experimental measurements for the size-dependent initial yield stress of nt-Cu specimens with varying twin thicknesses

Grahic Jump Location
Fig. 4

Comparison between the theoretical predictions and results obtained from the experiments (d = 500 nm) and molecular dynamic simulations (d = 10 nm, 20 nm, and 70 nm) for the twin thickness-dependent yield strength of nt-Cu specimens



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In