Skip to main content
SpringerLink
Log in
Book cover

Encyclopedia of Nanotechnology pp 2147–2154Cite as

Plasticity Theory at Small Scales

  • Reference work entry

Synonyms

Higher-order plasticity theory; Strain gradient plasticity theory

Definition

Plasticity theory is the mathematical formalism that describes the constitutive model of a material undergoing permanent deformation upon loading. For polycrystalline metals at low temperature and strain rate, the J 2 theory is the simplest adequate model. Classic plasticity theory does not include any explicit length scale, and as a result, the constitutive behavior is independent of the sample dimensions. As the characteristic length of a sample is reduced to the micro (and nano) scale, careful experimental observations clearly reveal the presence of a size effect that is not accounted for by the classical theory. Strain gradient plasticity is a formalism devised to extend plasticity theory to these smaller scales. For most metals, strain gradient plasticity is intended to apply to objects in the range from roughly 100 nm to 100 μm. Above 100 μm, the theory converges with the classical theory and...

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   1,299.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. Evans, A.G., Hutchinson, J.: A critical assessment of theories of strain gradient plasticity. Acta Mater. 57, 1675–1688 (2009)

    Article  CAS  Google Scholar 

  2. Nix, W.D., Gao, H.: Indentation size effects in crystalline materials: a law for strain gradient plasticity. J. Mech. Phys. Solids. 46(3), 411–425 (1998)

    Article  CAS  Google Scholar 

  3. Stolken, J., Evans, A.: A microbend test method for measuring the plasticity length scale. Acta. Mater. 46(14), 5109–5115 (1998)

    Article  CAS  Google Scholar 

  4. Fleck, N., et al.: Strain gradient plasticity – Theory and experiments. Acta. Metall. Mater. 42(2), 475–487 (1994)

    Article  CAS  Google Scholar 

  5. Lubliner, J.: Plasticity theory. Dover Publications, New York (2008)

    Google Scholar 

  6. Hill, R.: The mathematical theory of plasticity. Oxford University Press, Oxford (1998)

    Google Scholar 

  7. Mises, R.V.: Mechanik der festen Körper im plastisch deformablen Zustand. Göttin. Nachr. Math. Phys. 1, 582–592 (1913)

    Google Scholar 

  8. Toupin, R.A.: Elastic materials with couple stresses. Arch. Ration. Mech. An. 11(5), 385–414 (1963)

    Google Scholar 

  9. Mindlin, R.D.: Micro-structure in linear elasticity. Arch. Ration. Mech. An. 16(1), 51–78 (1964)

    Article  Google Scholar 

  10. Fleck, N., Hutchinson, J.: Strain gradient plasticity. Adv. appl. Mech. 33, 295–361 (1997)

    Article  Google Scholar 

  11. Niordson, C.F., Hutchinson, J.W.: Basic strain gradient plasticity theories with application to constrained film deformation. J. Mech. Mater. Struct. 6(1–4), 395–416 (2011)

    Google Scholar 

  12. Zibb, H., Aifantis, E.: On the gradient-dependent theory of plasticity and shear banding. Acta. Mech. 92(1–4), 209–225 (1992)

    Article  Google Scholar 

  13. Fleck, N.A., Willis, J.R.: A mathematical basis for strain-gradient plasticity theory. Part II: Tensorial plastic multiplier. J. Mech. Phys. Solids. 57(7), 1045–1057 (2009)

    Article  CAS  Google Scholar 

  14. Fleck, N.A., Willis, J.R.: A mathematical basis for strain-gradient plasticity theory-Part I: Scalar plastic multiplier. J. Mech. Phys. Solids. 57(1), 161–177 (2009)

    Article  Google Scholar 

  15. Gudmundson, P.: A unified treatment of strain gradient plasticity. J. Mech. Phys. Solids. 52(6), 1379–1406 (2004)

    Article  Google Scholar 

  16. Gurtin, M., Anand, L.: A theory of strain-gradient plasticity for isotropic, plastically irrotational materials. Part I: small deformations. J. Mech. Phys. Solids. 53(7), 1624–1649 (2005)

    Article  Google Scholar 

  17. Ashby, M.F.: Deformation of plastically non-homogeneous materials. Phil. Mag. 21(170), 399 (1970)

    Article  CAS  Google Scholar 

  18. Deshpande, V.S., Needleman, A., Van der Giessen, E.: Plasticity size effects in tension and compression of single crystals. J. Mech. Phys. Solids. 53(12), 2661–2691 (2005)

    Article  CAS  Google Scholar 

  19. Tang, H., Schwarz, K.W., Espinosa, H.D.: Dislocation-source shutdown and the plastic behavior of single-crystal micropillars. Phys. Rev. Lett. 100(18), 185503 (2008)

    Article  CAS  Google Scholar 

  20. Van der Giessen, E., Needleman, A.: Discrete dislocation plasticity – A simple planar model. Model. Simul. Mater. Sci. Eng. 3(5), 689–735 (1995)

    Article  Google Scholar 

  21. Uchic, M.D., Shade, P.A., Dimiduk, D.M.: Plasticity of micrometer-scale single crystals in compression. Annu. Rev Mater. Res. 39(1), 1–23 (2009)

    Article  CAS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical and Aerospace Engineering, University of California, 4227 Engineering Gateway, Irvine, CA, 92697-3975, USA

    Prof. Lorenzo Valdevit

  2. School of Engineering and Applied Sciences, Harvard University, 29 Oxford Street, Cambridge, MA, 02138, USA

    Prof. John W. Hutchinson

Authors
  1. Prof. Lorenzo Valdevit
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Prof. John W. Hutchinson
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lorenzo Valdevit .

Editor information

Editors and Affiliations

  1. Ohio Eminent Scholar and The Howard D. Winbigler Professor, Director, Nanoprobe Laboratory for Bio- & Nanotechnology and Biomimetics (NLB2), Ohio State University, 201 W. 19th Avenue, Columbus, Ohio, 43210-1142, USA

    Bharat Bhushan

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer Science+Business Media B.V.

About this entry

Cite this entry

Valdevit, L., Hutchinson, J.W. (2012). Plasticity Theory at Small Scales. In: Bhushan, B. (eds) Encyclopedia of Nanotechnology. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-9751-4_272

Download citation

Publish with us

Policies and ethics

Search

Navigation