Advertisement

Strain Gradient Plasticity: Theory and Implementation

  • Lorenzo BardellaEmail author
  • Christian F. Niordson
Chapter
  • 14 Downloads
Part of the CISM International Centre for Mechanical Sciences book series (CISM, volume 600)

Abstract

This chapter focuses on the foundation and development of various higher-order strain gradient plasticity theories, and it also provides the basic elements for their finite element implementation. To this aim, we first refer to experiments exhibiting size-effects in metals and explain them by resorting to the concept of geometrically necessary dislocations. We then bring this concept to the continuum level by introducing Nye’s dislocation density tensor and by postulating the existence of higher-order stresses associated with dislocation densities. This provides the motivation for the development of higher-order strain gradient plasticity theories. For this purpose, we adopt the generalized principle of virtual work, initially illustrated for conventional crystal plasticity and subsequently extended to both crystal and phenomenological strain gradient plasticity theories.

Keywords

Micron-scale metal plasticity Geometrically necessary dislocations Dislocation density tensor Strain gradient crystal plasticity Strain gradient plasticity 

References

  1. Acharya, A., & Bassani, J. L. (2000). Lattice incompatibility and a gradient theory of crystal plasticity. Journal of the Mechanics and Physics of Solids, 48, 1565–1595.MathSciNetzbMATHGoogle Scholar
  2. Anand, L., Gurtin, M. E., Lele, S. P., & Gething, C. (2005). A one-dimensional theory of strain-gradient plasticity: Formulation, analysis, numerical results. Journal of the Mechanics and Physics of Solids, 53, 1789–1826.MathSciNetzbMATHGoogle Scholar
  3. Arsenlis, A., & Parks, D. M. (1999). Crystallographic aspects of geometrically-necessary and statistically-stored dislocation density. Acta Materialia, 47(5), 1597–1611.Google Scholar
  4. Asaro, R. J. (1975). Elastic-plastic memory and kinematic-type hardening. Acta Metallurgica, 23, 1255–1265.Google Scholar
  5. Ashby, M. F. (1970). The deformation of plastically non-homogeneous materials. Philosophical Magazine, 21, 399–424.Google Scholar
  6. Bardella, L. (2006). A deformation theory of strain gradient crystal plasticity that accounts for geometrically necessary dislocations. Journal of the Mechanics and Physics of Solids, 54, 128–160.MathSciNetzbMATHGoogle Scholar
  7. Bardella, L. (2007). Some remarks on the strain gradient crystal plasticity modelling, with particular reference to the material length scales involved. International Journal of Plasticity, 23, 296–322.zbMATHGoogle Scholar
  8. Bardella L. (2020). Strain Gradient Plasticity. In H. Altenbach & A. Öchsner (Eds), Encyclopedia of Continuum Mechanics. Springer-Verlag, Berlin Heidelberg, pp. 2330–2341.  https://doi.org/10.1007/978-3-662-55771-6_110
  9. Bardella, L., & Giacomini, A. (2008). Influence of material parameters and crystallography on the size effects describable by means of strain gradient plasticity. Journal of the Mechanics and Physics of Solids, 56(9), 2906–2934.MathSciNetzbMATHGoogle Scholar
  10. Bardella, L., & Panteghini, A. (2015). Modelling the torsion of thin metal wires by distortion gradient plasticity. Journal of the Mechanics and Physics of Solids, 78, 467–492.MathSciNetzbMATHGoogle Scholar
  11. Bardella, L., Segurado, J., Panteghini, A., & Llorca, J. (2013). Latent hardening size effect in small-scale plasticity. Modelling and Simulation in Materials Science and Engineering, 21(5), 055009.Google Scholar
  12. Bishop, J. F. W., & Hill, R. (1951). A theory of plastic distortion of a polycrystalline aggregate under combined stress. Philosophical Magazine, 42, 414–427.MathSciNetzbMATHGoogle Scholar
  13. Bittencourt, E., Needleman, A., Gurtin, M. E., & van der Giessen, E. (2003). A comparison of nonlocal continuum and discrete dislocation plasticity predictions. Journal of the Mechanics and Physics of Solids, 51, 281–310.MathSciNetzbMATHGoogle Scholar
  14. Borg, U., & Fleck, N. A. (2007). Strain gradient effects in surface roughening. Modelling and Simulation in Materials Science and Engineering, 15, 1–12.Google Scholar
  15. Burgers, J. M. (1939). Some considerations of the field of stress connected with dislocations in a regular crystal lattice. Koninklijke Nederlandse Akademie Van Wetenschappen, 42, 293–325 (Part 1), 378–399 (Part 2).Google Scholar
  16. Chiricotto, M., Giacometti, L., & Tomassetti, G. (2012). Torsion in strain-gradient plasticity: Energetic scale effects. SIAM Journal on Applied Mathematics, 72(4), 1169–1191.MathSciNetzbMATHGoogle Scholar
  17. Chiricotto, M., Giacometti, L., & Tomassetti, G. (2016). Dissipative scale effects in strain-gradient plasticity: The case of simple shear. SIAM Journal on Applied Mathematics, 76(2), 688–704.MathSciNetzbMATHGoogle Scholar
  18. Conti, S., & Ortiz, M. (2005). Dislocation microstructures and the effective behavior of single crystals. Archive for Rational Mechanics and Analysis, 176, 103–147.MathSciNetzbMATHGoogle Scholar
  19. Danas, K., Deshpande, V. S., & Fleck, N. A. (2012). Size effects in the conical indentation of an elasto-plastic solid. Journal of the Mechanics and Physics of Solids, 60(9), 1605–1625.Google Scholar
  20. Del Piero, G. (2009). On the method of virtual power in continuum mechanics. Journal of Mechanics of Materials and Structures, 4(2), 281–292.Google Scholar
  21. El-Naaman, S. A., Nielsen, K. L., & Niordson, C. F. (2019). An investigation of back stress formulations under cyclic loading. Mechanics of Materials, 130, 76–87.Google Scholar
  22. Ertürk, I., van Dommelen, J. A. W., & Geers, M. G. D. (2009). Energetic dislocation interactions and thermodynamical aspects of strain gradient crystal plasticity theories. Journal of the Mechanics and Physics of Solids, 57(11), 1801–1814.MathSciNetzbMATHGoogle Scholar
  23. Evans, A. G., & Hutchinson, J. W. (2009). A critical assessment of theories of strain gradient plasticity. Acta Materialia, 57(5), 1675–1688.Google Scholar
  24. Evers, L. P., Brekelmans, W. A. M., & Geers, M. G. D. (2004). Non-local crystal plasticity model with intrinsic SSD and GND effects. Journal of the Mechanics and Physics of Solids, 52(10), 2379–2401.zbMATHGoogle Scholar
  25. Fleck, N. A., & Hutchinson, J. W. (1997). Strain gradient plasticity. Advances in Applied Mechanics, 33, 295–361.zbMATHGoogle Scholar
  26. Fleck, N. A., & Hutchinson, J. W. (2001). A reformulation of strain gradient plasticity. Journal of the Mechanics and Physics of Solids, 49(10), 2245–2271.zbMATHGoogle Scholar
  27. Fleck, N. A., & Willis, J. R. (2009a). A mathematical basis for strain-gradient plasticity theory. Part I: Scalar plastic multiplier. Journal of the Mechanics and Physics of Solids, 57, 161–177.Google Scholar
  28. Fleck, N. A., & Willis, J. R. (2009b). A mathematical basis for strain-gradient plasticity theory. Part II: Tensorial plastic multiplier. Journal of the Mechanics and Physics of Solids, 57, 1045–1057.Google Scholar
  29. Fleck, N. A., Muller, G. M., Ashby, M. F., & Hutchinson, J. W. (1994). Strain gradient plasticity: Theory and experiments. Acta Metallurgica et Materialia, 42, 475–487.Google Scholar
  30. Fleck, N. A., Hutchinson, J. W., & Willis, J. R. (2014). Strain gradient plasticity under non-proportional loading. Proceedings of the Royal Society of London A, 470, 20140267.MathSciNetGoogle Scholar
  31. Fleck, N. A., Hutchinson, J. W., & Willis, J. R. (2015). Guidelines for constructing strain gradient plasticity theories. Journal of Applied Mechanics, Transactions ASME, 82, 071002, 1–10.Google Scholar
  32. Forest, S., & Guéninchault, N. (2013). Inspection of free energy functions in gradient crystal plasticity. Acta Mechanica Sinica, 29(6), 763–772.MathSciNetzbMATHGoogle Scholar
  33. Forest, S., & Sievert, R. (2003). Elastoviscoplastic constitutive frameworks for generalized continua. Acta Mechanica, 160, 71–111.zbMATHGoogle Scholar
  34. Fredriksson, P., & Gudmundson, P. (2005). Size-dependent yield strength of thin films. International Journal of Plasticity, 21, 1834–1854.zbMATHGoogle Scholar
  35. Gambin, W. (1992). Refined analysis of elastic-plastic crystals. International Journal of Solids and Structures, 29, 2013–2021.zbMATHGoogle Scholar
  36. Gao, H., Huang, Y., Nix, W. D., & Hutchinson, J. W. (1999). Mechanism-based strain gradient plasticity — I. Theory. Journal of the Mechanics and Physics of Solids, 47, 1239–1263.Google Scholar
  37. Garroni, A., Leoni, G., & Ponsiglione, M. (2010). Gradient theory for plasticity via homogenization of discrete dislocations. Journal of the European Mathematical Society, 12(5), 1231–1266.MathSciNetzbMATHGoogle Scholar
  38. Groma, I., Csikor, F. F., & Zaiser, M. (2003). Spatial correlations and higher-order gradient terms in a continuum description of dislocation dynamics. Acta Materialia, 51, 1271–1281.Google Scholar
  39. Groma, I., Györgyi, G., & Kocsis, B. (2007). Dynamics of coarse grained dislocation densities from an effective free energy. Philosophical Magazine, 87(8–9), 1185–1199.Google Scholar
  40. Gudmundson, P. (2004). A unified treatment of strain gradient plasticity. Journal of the Mechanics and Physics of Solids, 52, 1379–1406.MathSciNetzbMATHGoogle Scholar
  41. Gurtin, M. E. (2002). A gradient theory of single-crystal viscoplasticity that accounts for geometrically necessary dislocations. Journal of the Mechanics and Physics of Solids, 50, 5–32.MathSciNetzbMATHGoogle Scholar
  42. Gurtin, M. E. (2004). A gradient theory of small-deformation isotropic plasticity that accounts for the Burgers vector and for dissipation due to plastic spin. Journal of the Mechanics and Physics of Solids, 52, 2545–2568.MathSciNetzbMATHGoogle Scholar
  43. Gurtin, M. E., & Anand, L. (2005). A gradient theory of strain-gradient plasticity for isotropic, plastically irrotational materials. Part I: Small deformations. Journal of the Mechanics and Physics of Solids, 53, 1624–1649.Google Scholar
  44. Gurtin, M. E., & Anand, L. (2007). A gradient theory for single-crystal plasticity. Modelling and Simulation in Materials Science and Engineering, 15, 263–270.Google Scholar
  45. Gurtin, M. E., & Anand, L. (2009). Thermodynamics applied to gradient theories involving the accumulated plastic strain: The theories of Aifantis and Fleck & Hutchinson and their generalization. Journal of the Mechanics and Physics of Solids, 57, 405–421.MathSciNetzbMATHGoogle Scholar
  46. Gurtin, M. E., & Needleman, A. (2005). Boundary conditions in small-deformation, single-crystal plasticity that account for the Burgers vector. Journal of the Mechanics and Physics of Solids, 53, 1–31.MathSciNetzbMATHGoogle Scholar
  47. Gurtin, M. E., & Ohno, N. (2011). A gradient theory of small-deformation, single crystal plasticity that accounts for GND-induced interactions between slip systems. Journal of the Mechanics and Physics of Solids, 59, 320–343.MathSciNetzbMATHGoogle Scholar
  48. Gurtin, M. E., & Reddy, B. D. (2014). Gradient single-crystal plasticity within a Mises-Hill framework based on a new formulation of self- and latent-hardening. Journal of the Mechanics and Physics of Solids, 68, 134–160.MathSciNetzbMATHGoogle Scholar
  49. Gurtin, M. E., Anand, L., & Lele, S. P. (2007). Gradient single-crystal plasticity with free energy dependent on dislocation densities. Journal of the Mechanics and Physics of Solids, 55, 1853–1878.MathSciNetzbMATHGoogle Scholar
  50. Hall, E. O. (1951). The deformation and ageing of mild steel: III discussion of results. Proceedings of the Physical Society B, 64, 747–753.Google Scholar
  51. Hull, D., & Bacon, D. J. (2001). Introduction to dislocations (4th ed.). Oxford: Butterworth-Heinemann.Google Scholar
  52. Hutchinson, J. W. (2000). Plasticity at the micron scale. International Journal of Solids and Structures, 37, 225–238.MathSciNetzbMATHGoogle Scholar
  53. Kraft, O., Hommel, M., & Arzt, E.: X-ray diffraction as a tool to study the mechanical behaviour of thin films. Materials Science and Engineering: A, 288, 209–216.Google Scholar
  54. Kröner, E. (1962). Dislocations and continuum mechanics. Applied Mechanics Reviews, 15, 599–606.Google Scholar
  55. Kuroda, M., & Tvergaard, V. (2008). On the formulations of higher-order strain gradient crystal plasticity models. Journal of the Mechanics and Physics of Solids, 56, 1591–1608.MathSciNetzbMATHGoogle Scholar
  56. Lanczos, C. (1970). The variational principles of mechanics (4th ed.). Toronto: University of Toronto Press.zbMATHGoogle Scholar
  57. Liu, D., He, Y., Dunstan, D. J., Zhang, B., Gan, Z., Hu, P., et al. (2013). Anomalous plasticity in cyclic torsion of micron scale metallic wires. Physical Review Letters, 110, 244301.Google Scholar
  58. Ma, Q., & Clarke, D. R. (1995). Size dependent hardness in silver single crystals. Journal of Materials Research, 10, 853–863.Google Scholar
  59. Malvern, L. E. (1969). Introduction to the mechanics of a continuous medium. New Jersey: Prentice-Hall.Google Scholar
  60. Martínez-Pañeda, E., Niordson, C. F., & Bardella, L. (2016). A finite element framework for distortion gradient plasticity with applications to bending of thin foils. International Journal of Solids and Structures, 96, 288–299.Google Scholar
  61. Martínez-Pañeda, E., Deshpande, V. S., Niordson, C. F., & Fleck, N. A. (2019). The role of plastic strain gradients in the crack growth resistance of metals. Journal of the Mechanics and Physics of Solids, 126, 136–150.MathSciNetGoogle Scholar
  62. McMeeking, R. M., & Rice, J. R. (1975). Finite-element formulations for problems of large elastic-plastic deformation. International Journal of Solids and Structures, 11, 601–616.zbMATHGoogle Scholar
  63. Moreau, P., Raulic, M., P’ng, M. Y., Gannaway, G., Anderson, P., Gillin, W. P., et al. (2005). Measurement of the size effect in the yield strength of nickel foils. Philosophical Magazine Letters, 85(7), 339–343.Google Scholar
  64. Needleman, A. (1988). Material rate dependence and mesh sensitivity in localization problems. Computer Methods in Applied Mechanics and Engineering, 67, 69–85.zbMATHGoogle Scholar
  65. Nicola, L., van der Giessen, E., & Gurtin, M. E. (2005). Effect of defect energy on strain-gradient predictions of confined single-crystal plasticity. Journal of the Mechanics and Physics of Solids, 53, 1280–1294.MathSciNetzbMATHGoogle Scholar
  66. Nielsen, K. L., & Niordson, C. F. (2014). A numerical basis for strain-gradient plasticity theory: Rate-independent and rate-dependent formulations. Journal of the Mechanics and Physics of Solids, 63, 113–127.MathSciNetzbMATHGoogle Scholar
  67. Nielsen, K. L., & Niordson, C. F. (2019). Print share a finite strain FE-implementation of the Fleck-Willis gradient theory: Rate-independent versus visco-plastic formulation. European Journal of Mechanics/A Solids, 75, 389–398.MathSciNetzbMATHGoogle Scholar
  68. Niordson, C. F., & Hutchinson, J. W. (2011). Basic gradient plasticity theories with application to constrained film deformation. Journal of Mechanics of Materials and Structures, 6(1–4), 395–416.Google Scholar
  69. Niordson, C. F., & Legarth, B. N. (2010). Strain gradient effects on cyclic plasticity. Journal of the Mechanics and Physics of Solids, 58, 542–557.MathSciNetzbMATHGoogle Scholar
  70. Niordson, C. F., & Tvergaard, V. (2018). A homogenized model for size-effects in porous metals. Journal of the Mechanics and Physics of Solids, 123, 222–233.MathSciNetGoogle Scholar
  71. Niordson, C. F., & Tvergaard, V. (2019). A homogenized model for size-effects in porous metals. Journal of the Mechanics and Physics of Solids, 123, 222–233.MathSciNetGoogle Scholar
  72. Nye, J. F. (1953). Some geometrical relations in dislocated crystals. Acta Metallurgica, 1, 153–162.Google Scholar
  73. Ohno, N., & Okumura, D. (2007). Higher-order stress and grain size effects due to self-energy of geometrically necessary dislocations. Journal of the Mechanics and Physics of Solids, 55, 1879–1898.MathSciNetzbMATHGoogle Scholar
  74. Ortiz, M., & Popov, E. P. (1985). Accuracy and stability of integration algorithms for elastoplastic constitutive relations. International Journal for Numerical Methods in Engineering, 21(9), 1561–1576.MathSciNetzbMATHGoogle Scholar
  75. Panteghini, A., & Bardella, L. (2016). On the finite element implementation of higher-order gradient plasticity, with focus on theories based on plastic distortion incompatibility. Computer Methods in Applied Mechanics and Engineering, 310, 840–865.MathSciNetzbMATHGoogle Scholar
  76. Panteghini, A., & Bardella, L. (2018). On the role of higher-order conditions in distortion gradient plasticity. Journal of the Mechanics and Physics of Solids, 118, 293–321. ISSN 0022-5096.Google Scholar
  77. Panteghini, A., Bardella, L., & Niordson, C. F. (2019). A potential for higher-order phenomenological strain gradient plasticity to predict reliable response under non-proportional loading. Proceedings of the Royal Society of London A, 475(2229), 20190258.MathSciNetGoogle Scholar
  78. Peirce, D., Asaro, R. J., & Needleman, A. (1983). Material rate dependence and localized deformation in crystalline solids. Acta Metallurgica, 31(12), 1951–1976.Google Scholar
  79. Petch, N. J. (1953). The cleavage strength of polycrystals. Journal of the Iron and Steel Institute, 174, 25–28.Google Scholar
  80. Poh, L. H., & Peerlings, R. H. J. (2016). The plastic rotation effect in an isotropic gradient plasticity model for applications at the meso scale. International Journal of Solids and Structures, 78–79, 57–69.Google Scholar
  81. Polizzotto, C. (2009). A link between the residual-based gradient plasticity theory and the analogous theories based on the virtual work principle. International Journal of Plasticity, 25, 2169–2180.Google Scholar
  82. Polizzotto, C., Borino, G., & Fuschi, P. (1998). A thermodynamically consistent formulation of nonlocal and gradient plasticity. Mechanics Research Communications, 25(1), 75–82.MathSciNetzbMATHGoogle Scholar
  83. Stelmashenko, N. A., Walls, M. G., Brown, L. M., & Milman, Y. V. (1993). Microindentations on W and Mo oriented single crystals: An STM study. Acta Metallurgica et Materialia, 41, 2855–2865.Google Scholar
  84. Stölken, J. S., & Evans, A. G. (1998). A microbend test method for measuring the plasticity length scale. Acta Materialia, 46, 5109–5115.Google Scholar
  85. Svendsen, B., & Bargmann, S. (2010). On the continuum thermodynamic rate variational formulation of models for extended crystal plasticity at large deformation. Journal of the Mechanics and Physics of Solids, 58, 1253–1271.MathSciNetzbMATHGoogle Scholar
  86. Willis, J. R. (2019). Some forms and properties of models of strain-gradient plasticity. Journal of the Mechanics and Physics of Solids, 123, 348–356.MathSciNetGoogle Scholar
  87. Wulfinghoff, S., Forest, S., & Böhlke, T. (2015). Strain gradient plasticity modelling of the cyclic behaviour of laminate microstructures. Journal of the Mechanics and Physics of Solids, 79, 1–20.MathSciNetzbMATHGoogle Scholar
  88. Xiang, Y., & Vlassak, J. J. (2006). Bauschinger and size effects in thin-film plasticity. Acta Materialia, 54, 5449–5460.Google Scholar

Copyright information

© CISM International Centre for Mechanical Sciences, Udine 2020

Authors and Affiliations

  1. 1.Department of Civil, Environmental, Architectural Engineering and MathematicsUniversity of BresciaBresciaItaly
  2. 2.Department of Mechanical EngineeringTechnical University of DenmarkLyngbyDenmark

Personalised recommendations