Shakedown Analysis Within the Framework of Strain Gradient Plasticity

  • Castrenze PolizzottoEmail author


A class of rate-independent material models is addressed within the framework of isotropic strain gradient plasticity. These models exhibit a size dependence through the strengthening effects (Hall–Petch effects), whereby the yield stress is related to the effective plastic strain by a suitable second-order partial differential equation with related boundary conditions. For a perfectly plastic material with strengthening effects, the classical concepts of plastic and shakedown limit analysis do hold, which lead to size dependent plastic and shakedown limit loads according to the dictum: smaller is stronger. In the perspective of a development of direct methods for applications to small-scale structures within micro/nano technologies, a shakedown theory for perfectly plastic materials with strengthening effects, previously elaborated by the present author [51], is presented and discussed. Apart from the inevitable mathematical complications carried in by the more complex constitutive behavior of the material herein considered, the overall conceptual architecture of the shakedown theorems remain within the classical Melan and Koiter theoretical framework. Further research efforts are needed to develop specific numerical procedures for the computation of the plastic and shakedown limit loads and the concomitant collapse mechanisms.


  1. 1.
    Abu Al-Rub RK (2008) Interfacial gradient plasticity governs scale-dependent yield strength and strain hardening rates in micro/nano structural metals. Int J Plast 24:1277–1306CrossRefzbMATHGoogle Scholar
  2. 2.
    Abu Al-Rub RK, Voyiadjis GZ, Aifantis EC (2009) On the thermodynamics of higher-order gradient plasticity for size effects at the micron and submicron length scale. Int J Mater Prod Technol 34:172–187CrossRefGoogle Scholar
  3. 3.
    Aifantis EC (2003) Update on a class of gradient theories. Mech Mater 35:259–280CrossRefGoogle Scholar
  4. 4.
    Aifantis KE, Soer WA, De Hosson JThM, Willis JR (2006) Interfaces within strain gradient plasticity: theory and experiments. Acta Materialia 54:5077–5085CrossRefGoogle Scholar
  5. 5.
    Anand L, Gurtin ME, Lele SP, Gething C (2005) A one-dimensional theory of strain-gradient plasticity: formulation, analysis, numerical results. J Mech Phys Solids 53:1789–1826CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Borg U (2007) A strain gradient crystal plasticity analysis of strain gradient effects in polycrystals. Eur J Mech A/Solids 26:313–324CrossRefzbMATHGoogle Scholar
  7. 7.
    Borino G, Polizzotto C (2007) A thermodynamically consistent gradient plasticity theory and comparison with other formulations. Model Mater Sci Eng 15:23–35CrossRefGoogle Scholar
  8. 8.
    Ceradini G (1980) Dynamic shakedown in elastic-plastic bodies. J Eng Mech Div, ASCE 106:481–499Google Scholar
  9. 9.
    Débordes O, Nayroles B (1976) Sur la théorie et le calcul à l’adaptation des structures élastoplastiques. J Mécanique 15:1–53zbMATHGoogle Scholar
  10. 10.
    Espinosa HD, Prorok BC, Peng B (2004) Plasticity size effects in free-standing submicron polycrystalline FCC films subjected to pure tension. J Mech Phys Solids 52:667–689CrossRefGoogle Scholar
  11. 11.
    Feng X-Q, Sun Q (2007) Shakedown analysis of shape memory structures. Int J Plast 23:183–206CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Fleck NA, Hutchinson JW (1997) Strain gradient plasticity. Adv Appl Mech 33:295–361CrossRefGoogle Scholar
  13. 13.
    Fleck NA, Muller GM, Ashby MF, Hutchinson JW (1994) Strain gradient plasticity: theory and experiments. Acta Metall Mater 42:475–487CrossRefGoogle Scholar
  14. 14.
    Fleck NA, Willis JR (2008) A mathematical basis for strain-gradient plasticity theory—Part I: scalar plastic multiplier. J Mech Phys Solids 57:161–177CrossRefMathSciNetGoogle Scholar
  15. 15.
    Fredriksson P, Gudmundson P (2005) Size-dependent yield strength of thin films. Int J Plast 21:1834–1854CrossRefzbMATHGoogle Scholar
  16. 16.
    Fuschi P, Polizzotto C (1995) The shakedown load boundary of an elastic-perfectly plastic structure. Meccanica 30:155–174CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Gokhfeld DA, Cherniavsky DF (1980) Limit analysis of structures at thermal cycling. Sijthoff and Noordhoff, Alphen aan der RijnGoogle Scholar
  18. 18.
    Gudmundson P (2004) A unified treatment of strain gradient plasticity. J Mech Phys Solids 52:1377–1406CrossRefMathSciNetGoogle Scholar
  19. 19.
    Gurtin ME, Anand L (2005) A theory of strain-gradient plasticity for isotropic, plastically irrotational materials. Part I: small deformation. J Mech Phys Solids 53:1624–1649CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Gurtin ME, Anand L, Lele SP (2007) Gradient single-crystal plasticity with free energy dependent on dislocation density. J Mech Phys Solids 55:1853–1878CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Gurtin ME, Fried E, Anand L (2010) The mechanics and thermodynamics of continua. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  22. 22.
    Halphen B (1979) Adaptation of elasto-visco-plastic structures. In: Matériax et Structures Sous Chargement Cyclique Association Amicale des Ingénieurs Ancien El\(\grave{e}\)ves. Ponts et Chaussés, Paris, pp 203–229Google Scholar
  23. 23.
    Halphen B, Nguyen QS (1979) Sur les matériaux standards dénéralisés. J Mécanique 14:39–63Google Scholar
  24. 24.
    Hansen N (2004) Hall-Petch relation and boundary strengthening. Scripta Mater 51:801–806CrossRefGoogle Scholar
  25. 25.
    Haque MA, Saif MTA (2003) Strain gradient effects in nanoscale thin films. Acta Mater 51:3053–3061CrossRefGoogle Scholar
  26. 26.
    Huang H, Spaepan F (2000) Tensile testing of free-standing Cu, Ag and Al thin films and Ag/Cu multilayers. Acta Mater 48:3261–3269CrossRefGoogle Scholar
  27. 27.
    Hutchinson JW (2000) Plasticity at micron scale. Int J Solids Struct 37:225–238CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Kamenjarzh J, Merzljakov A (1994) On kinematic method in shakedown theory: I. Duality of extremum problems; II. Modified kinematic method. Int J Plast 10:363–392CrossRefzbMATHGoogle Scholar
  29. 29.
    Koiter WT (1960) General theorems for elastic-plastic solids. In: Hill R, Sneddon I (eds) Progress in solid mechanics, I. North-Holland, The Netherlands, pp 167–221Google Scholar
  30. 30.
    König JA (1987) Shakedown of elastic plastic structures. Elsevier, AmsterdamGoogle Scholar
  31. 31.
    König JA, Maier G (1981) Shakedown analysis of elastoplastic structures: a review of recent developments. Nucl Eng Des 66:81–95CrossRefGoogle Scholar
  32. 32.
    Lele SP, Anand L (2008) A small-deformation strain-gradient theory for isotropic viscoplastic materials. Phil Mag 88:3655–3689CrossRefGoogle Scholar
  33. 33.
    Lemaitre J, Chaboche J-L (1990) Mechanics of solid materials. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  34. 34.
    Maier G (2001) On some issues of shakedown analysis. J Appl Mech, ASME 68:799–808CrossRefzbMATHGoogle Scholar
  35. 35.
    Maier G, Carvelli V, Cocchetti G (2000) On direct methods for shakedown and limit analysis. Eur J Mech A/Solids 19:79–100Google Scholar
  36. 36.
    Martin JB (1975) Plasticity: fundamentals and general results. MTI Press, CambridgeGoogle Scholar
  37. 37.
    Mróz Z, Weichert D, Dorosz S (eds) (1995) Inelastic behaviour of structures under repeated loads. Kluwer Academic Publishers, DordrechtGoogle Scholar
  38. 38.
    Pham DC (1996) Dynamic shakedown and a reduced kinematic theorem. Int J Plast 12:1055–1068CrossRefzbMATHGoogle Scholar
  39. 39.
    Pham DC (2003) Shakedown theory for elastic-perfectly plastic bodies revisited. Int J Mech Sci 45:1011–1027CrossRefzbMATHGoogle Scholar
  40. 40.
    Pham DC (2005) Shakedown static and kinematic theorems for elastic-plastic limited linear kinematic hardening solids. Eur J Mech A/Solids 24:35–45CrossRefzbMATHMathSciNetGoogle Scholar
  41. 41.
    Polizzotto C (1982) A unified treatment of shakedown theory and related bounding techniques. Solid Mech Arch 7:19–75zbMATHGoogle Scholar
  42. 42.
    Polizzotto C (1984) On shakedown of structures under dynamic agencies. In: Sawczuk A, Polizzotto C (eds) Inelastic analysis under variable repeated loads. Cogras, PalermoGoogle Scholar
  43. 43.
    Polizzotto C (1993) On the conditions to prevent plastic shakedown: Part I-Theory; Part II-The plastic shakedown limit load. J Appl Mech, ASME 60:15–25CrossRefzbMATHMathSciNetGoogle Scholar
  44. 44.
    Polizzotto C (2003) Unified thermodynamic framework for nonlocal/gradient continuum theories. Eur J Mech A/Solids 22:651–668CrossRefzbMATHMathSciNetGoogle Scholar
  45. 45.
    Polizzotto C (2007) Strain-gradient elastic-plastic material models and assessment of the higher order boundary conditions. Eur J Mech A/Solids 26:189–211CrossRefzbMATHMathSciNetGoogle Scholar
  46. 46.
    Polizzotto C (2008) Thermodynamics-based gradient plasticity theories with an application to interface models. Int J Solids Struct 45:4820–4834Google Scholar
  47. 47.
    Polizzotto C (2008) Shakedown theorems for elastic-plastic solids in the framework of gradient plasticity. Int J Plast 24:218–241Google Scholar
  48. 48.
    Polizzotto C (2009) Interfacial energy effects in the framework of strain gradient plasticity. Int J Solids Struct 46:1685–1694Google Scholar
  49. 49.
    Polizzotto C (2009) A link between the residual-based gradient plasticity theory and the analogous theories based on the virtual work principle. Int J Plast 25:2169–2180Google Scholar
  50. 50.
    Polizzotto C (2010) Strain gradient plasticity, strengthening effects and plastic limit analysis. Int J Solids Struct 47:100–112Google Scholar
  51. 51.
    Polizzotto C (2010) Shakedown analysis for a class of strengthening materials within the framework of gradient plasticity. Int J Plast 26:1050–1069Google Scholar
  52. 52.
    Polizzotto C (2011) A unified residual-based thermodynamic framework for strain gradient theories of plasticity. Int J Plast 27:388–413CrossRefzbMATHGoogle Scholar
  53. 53.
    Polizzotto C, Borino G (1998) A thermodynamics-based formulation of gradient dependent plasticity. Eur J Mech A/Solids 17:741–761CrossRefzbMATHMathSciNetGoogle Scholar
  54. 54.
    Polizzotto C, Borino G (2014) Shakedown under thermo-mechanical loads. In: Hetnarski RB (ed) Encyclopedia of thermal stresses, vol 8. Springer, Dordrecht, pp 4317–4333CrossRefGoogle Scholar
  55. 55.
    Polizzotto C, Borino G, Fuschi P (2000) Shakedown of cracked bodies with nonlocal elasticity. In: CD Proceedings of ECCOMAS, Barcelona, September 2000Google Scholar
  56. 56.
    Ponter ARS (1983) Shakedown and ratchetting below the creep range commission of the European communities, Report EUR 8702 EN, Brussels, BelgiumGoogle Scholar
  57. 57.
    Ponter ARS (2002) A linear matching method for shakedown analysis. In: Weichert D, Maier G (eds) Inelastic behaviour of structures under variable repeated loads, Springer, WienGoogle Scholar
  58. 58.
    Ponter ARS, Karadenitz S (1985) An extended shakedown theory for structures that suffer cyclic thermal loadings. Part I-Theory. J Appl Mech 52:877–882CrossRefGoogle Scholar
  59. 59.
    Stein E, Zhang G, König JA (1992) Shakedown with nonlinear strain hardening including structural computation using finite element method. Int J Plast 8:1–31CrossRefzbMATHGoogle Scholar
  60. 60.
    Stölken JS, Evans AG (1998) A microbend test method for measuring the plasticity length-scale. Acta Mater 46:5109–5115CrossRefGoogle Scholar
  61. 61.
    Weichert D, Maier G (eds) (2000) Inelastic analysis of structures under variable repeated loads, theory and engineering applications. Kluwer, DordrechtGoogle Scholar
  62. 62.
    Weichert D, Maier G (eds) (2002) Inelastic behaviour of structures under variable repeated loads. Springer, WienGoogle Scholar
  63. 63.
    Zarka J, Casier J (1979) Cyclic loading on elastic-plastic structures. In: Nemat-Nasser S (ed) Mechanics to-day. Pergamon Press, Oxford, pp 93–198Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Dipartimento di Ingegneria Civile Ambientale Aerospaziale e Dei Materiali, Viale Delle ScienzeUniversità di PalermoPalermoItaly

Personalised recommendations