Encyclopedia of Continuum Mechanics

Living Edition
| Editors: Holm Altenbach, Andreas Öchsner

Strain Gradient Plasticity

  • Lorenzo Bardella
Living reference work entry
DOI: https://doi.org/10.1007/978-3-662-53605-6_110-1

Synonyms

Definitions

Strain gradient plasticity (SGP) is a theory of continuum solid mechanics which aims at modeling the irreversible mechanical behavior of materials, with specific focus on metals and on their response at appropriately small size, typically on the order of micrometers or less. For such small metallic components, a variation in size leads to a peculiar effect, denoted as “smaller being stronger.”

Background

The term plasticity refers to the irreversible mechanical behavior of materials, with particular reference to metals. This behavior occurs when the stress state is large enough for the material to yield, thus leading to a permanent deformation, denoted as plastic deformation. Such deformation can be observed, and the inherent plastic strain measured, after removing a suitable, monotonically applied load which enables yielding. In simple tests, such as uniaxial tension, the yield stressis...

This is a preview of subscription content, log in to check access.

References

  1. Aifantis EC (1984) On the microstructural origin of certain inelastic models. J Eng Mater Tech-T ASME 106:326–330CrossRefGoogle Scholar
  2. Arsenlis A, Parks DM (1999) Crystallographic aspects of geometrically-necessary and statistically-stored dislocation density. Acta Mater 47:1597–1611CrossRefGoogle Scholar
  3. Ashby MF (1970) The deformation of plastically non-homogeneous materials. Philos Mag 21:399–424CrossRefGoogle Scholar
  4. Bardella L (2009) A comparison between crystal and isotropic strain gradient plasticity theories with accent on the role of the plastic spin. Eur J Mech A-Solid 28:638–646CrossRefzbMATHGoogle Scholar
  5. Bardella L (2010) Size effects in phenomenological strain gradient plasticity constitutively involving the plastic spin. Int J Eng Sci 48:550–568MathSciNetCrossRefzbMATHGoogle Scholar
  6. Bardella L, Panteghini A (2015) Modelling the torsion of thin metal wires by distortion gradient plasticity. J Mech Phys Solids 78:467–492MathSciNetCrossRefzbMATHGoogle Scholar
  7. Burgers JM (1939) Some considerations of the field of stress connected with dislocations in a regular crystal lattice. K Ned Akad Van Wet 42:293–325 (Part 1), 378–399 (Part 2)Google Scholar
  8. Carstensen C, Ebobisse F, McBride AT, Reddy BD, Steinmann P (2017) Some properties of the dissipative model of strain-gradient plasticity. Philos Mag 97: 693–717CrossRefGoogle Scholar
  9. Chiricotto M, Giacometti L, Tomassetti G (2016) Dissipative scale effects in strain-gradient plasticity: the case of simple shear. SIAM J Appl Math 76:688–704MathSciNetCrossRefzbMATHGoogle Scholar
  10. Del Piero G (2009) On the method of virtual power in continuum mechanics. J Mech Mater Struct 4:281–292CrossRefGoogle Scholar
  11. Dillon OW J, Kratochvíl J (1970) A strain gradient theory of plasticity. Int J Solids Struct 6:1513–1533CrossRefzbMATHGoogle Scholar
  12. Fleck NA, Hutchinson JW (1997) Strain gradient plasticity. Adv Appl Mech 33:295–361CrossRefzbMATHGoogle Scholar
  13. Fleck NA, Hutchinson JW (2001) A reformulation of strain gradient plasticity. J Mech Phys Solids 49:2245–2271CrossRefzbMATHGoogle Scholar
  14. Fleck NA, Willis JR (2009) A mathematical basis for strain-gradient plasticity theory. Part II: tensorial plastic multiplier. J Mech Phys Solids 57:1045–1057zbMATHGoogle Scholar
  15. Fleck NA, Willis JR (2015) Strain gradient plasticity: energetic or dissipative? Acta Mech Sinica 31:465–472MathSciNetCrossRefzbMATHGoogle Scholar
  16. Fleck NA, Muller GM, Ashby MF, Hutchinson JW (1994) Strain gradient plasticity: theory and experiments. Acta Metall Mater 42:475–487CrossRefGoogle Scholar
  17. Fleck NA, Hutchinson JW, Willis JR (2014) Strain gradient plasticity under non-proportional loading. Proc R Soc Lond A 470:20140267CrossRefGoogle Scholar
  18. Fleck NA, Hutchinson JW, Willis JR (2015) Guidelines for constructing strain gradient plasticity theories. J Appl Mech-T ASME 82:1–10CrossRefGoogle Scholar
  19. Forest S, Guéninchault N (2013) Inspection of free energy functions in gradient crystal plasticity. Acta Mech Sinica 29:763–772MathSciNetCrossRefzbMATHGoogle Scholar
  20. Forest S, Sievert R (2003) Elastoviscoplastic constitutive frameworks for generalized continua. Acta Mech 160:71–111CrossRefzbMATHGoogle Scholar
  21. Groma I, Györgyi G, Kocsis B (2007) Dynamics of coarse grained dislocation densities from an effective free energy. Philos Mag 87:1185–1199CrossRefGoogle Scholar
  22. Gudmundson P (2004) A unified treatment of strain gradient plasticity. J Mech Phys Solids 52:1379–1406MathSciNetCrossRefzbMATHGoogle Scholar
  23. Gurtin ME (2004) A gradient theory of small-deformation isotropic plasticity that accounts for the Burgers vector and for dissipation due to plastic spin. J Mech Phys Solids 52:2545–2568MathSciNetCrossRefzbMATHGoogle Scholar
  24. Gurtin ME, Anand L (2009) Thermodynamics applied to gradient theories involving the accumulated plastic strain: the theories of Aifantis and Fleck & Hutchinson and their generalization. J Mech Phys Solids 57: 405–421MathSciNetCrossRefzbMATHGoogle Scholar
  25. Gurtin ME, Needleman A (2005) Boundary conditions in small-deformation, single-crystal plasticity that account for the Burgers vector. J Mech Phys Solids 53: 1–31MathSciNetCrossRefzbMATHGoogle Scholar
  26. Gurtin ME, Fried E, Anand L (2010) The mechanics and thermodynamics of continua. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  27. Hall EO (1951) The deformation and ageing of mild steel: III discussion of results. Proc Phys Soc B 64:747–753CrossRefGoogle Scholar
  28. Hayden W, Moffatt WG, Wulff J (1965) The structure and properties of materials: vol III, mechanical behavior. Wiley, New YorkGoogle Scholar
  29. Huang Y, Gao H, Nix WD, Hutchinson JW (2000) Mechanism-based strain gradient plasticity – II. Analysis. J Mech Phys Solids 48:99–128MathSciNetCrossRefzbMATHGoogle Scholar
  30. Hull D, Bacon DJ (2001) Introduction to dislocations, 4th edn. Butterworth-Heinemann, OxfordGoogle Scholar
  31. Kröner E (1962) Dislocations and continuum mechanics. Appl Mech Rev 15:599–606Google Scholar
  32. Ma Q, Clarke DR (1995) Size dependent hardness in silver single crystals. J Mater Res 10:853–863CrossRefGoogle Scholar
  33. Martínez-Pañeda E, Niordson CF, Bardella L (2016) A finite element framework for distortion gradient plasticity with applications to bending of thin foils. Int J Solids Struct 96:288–299CrossRefGoogle Scholar
  34. Nielsen KL, Niordson CF (2014) A numerical basis for strain-gradient plasticity theory: rate-independent and rate-dependent formulations. J Mech Phys Solids 63:113–127MathSciNetCrossRefzbMATHGoogle Scholar
  35. Nye JF (1953) Some geometrical relations in dislocated crystals. Acta Metall 1:153–162CrossRefGoogle Scholar
  36. Panteghini A, Bardella L (2016) On the finite element implementation of higher-order gradient plasticity, with focus on theories based on plastic distortion incompatibility. Comput Method Appl M 310:840–865MathSciNetCrossRefGoogle Scholar
  37. Petch NJ (1953) The cleavage strength of polycrystals. J Iron Steel Inst 174:25–28Google Scholar
  38. Poh LH, Peerlings RHJ (2016) The plastic rotation effect in an isotropic gradient plasticity model for applications at the meso scale. Int J Solids Struct 78–79:57–69CrossRefGoogle Scholar
  39. Polizzotto C (2009) A link between the residual-based gradient plasticity theory and the analogous theories based on the virtual work principle. Int J Plasticity 25:2169–2180CrossRefGoogle Scholar
  40. Stölken JS, Evans AG (1998) A microbend test method for measuring the plasticity length scale. Acta Mater 46:5109–5115CrossRefGoogle Scholar
  41. Svendsen B, Bargmann S (2010) On the continuum thermodynamic rate variational formulation of models for extended crystal plasticity at large deformation. J Mech Phys Solids 58:1253–1271MathSciNetCrossRefzbMATHGoogle Scholar
  42. Valdevit L, Hutchinson JW (2012) Plasticity theory at small scales. In: Bhushan B (ed) Encyclopedia of nanotechnology. Springer, Dordrecht, pp 3319–3327Google Scholar
  43. Wulfinghoff S, Forest S, Böhlke T (2015) Strain gradient plasticity modelling of the cyclic behaviour of laminate microstructures. J Mech Phys Solids 79:1–20MathSciNetCrossRefzbMATHGoogle Scholar
  44. Zbib HM, Aifantis EC (1992) On the gradient-dependent theory of plasticity and shear banding. Acta Mech 92:209–225CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2018

Authors and Affiliations

  1. 1.Department of Civil, Environmental, Architectural Engineering and MathematicsUniversity of BresciaBresciaItaly

Section editors and affiliations

  • Samuel Forest
    • 1
  1. 1.Centre des Matériaux UMR 7633Mines ParisTech CNRSEvry CedexFrance