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...
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Evans, A.G., Hutchinson, J.: A critical assessment of theories of strain gradient plasticity. Acta Mater. 57, 1675–1688 (2009)
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)
Stolken, J., Evans, A.: A microbend test method for measuring the plasticity length scale. Acta. Mater. 46(14), 5109–5115 (1998)
Fleck, N., et al.: Strain gradient plasticity – Theory and experiments. Acta. Metall. Mater. 42(2), 475–487 (1994)
Lubliner, J.: Plasticity theory. Dover Publications, New York (2008)
Hill, R.: The mathematical theory of plasticity. Oxford University Press, Oxford (1998)
Mises, R.V.: Mechanik der festen Körper im plastisch deformablen Zustand. Göttin. Nachr. Math. Phys. 1, 582–592 (1913)
Toupin, R.A.: Elastic materials with couple stresses. Arch. Ration. Mech. An. 11(5), 385–414 (1963)
Mindlin, R.D.: Micro-structure in linear elasticity. Arch. Ration. Mech. An. 16(1), 51–78 (1964)
Fleck, N., Hutchinson, J.: Strain gradient plasticity. Adv. appl. Mech. 33, 295–361 (1997)
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)
Zibb, H., Aifantis, E.: On the gradient-dependent theory of plasticity and shear banding. Acta. Mech. 92(1–4), 209–225 (1992)
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)
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)
Gudmundson, P.: A unified treatment of strain gradient plasticity. J. Mech. Phys. Solids. 52(6), 1379–1406 (2004)
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)
Ashby, M.F.: Deformation of plastically non-homogeneous materials. Phil. Mag. 21(170), 399 (1970)
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)
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)
Van der Giessen, E., Needleman, A.: Discrete dislocation plasticity – A simple planar model. Model. Simul. Mater. Sci. Eng. 3(5), 689–735 (1995)
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)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights 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
DOI: https://doi.org/10.1007/978-90-481-9751-4_272
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-9750-7
Online ISBN: 978-90-481-9751-4
eBook Packages: Chemistry and Materials ScienceReference Module Physical and Materials ScienceReference Module Chemistry, Materials and Physics