Advertisement

Computational Mechanics

, Volume 55, Issue 4, pp 755–769 | Cite as

A computational investigation of a model of single-crystal gradient thermoplasticity that accounts for the stored energy of cold work and thermal annealing

  • A. McBride
  • S. Bargmann
  • B. D. Reddy
Original Paper

Abstract

A theory of single-crystal gradient thermoplasticity that accounts for the stored energy of cold work and thermal annealing has recently been proposed by Anand et al. (Int J Plasticity 64:1–25, 2015). Aspects of the numerical implementation of the aforementioned theory using the finite element method are detailed in this presentation. To facilitate the implementation, a viscoplastic regularization of the plastic evolution equations is performed. The weak form of the governing equations and their time-discrete counterparts are derived. The theory is then elucidated via a series of three-dimensional numerical examples where particular emphasis is placed on the role of the defect-flow relations. These relations govern the evolution of a measure of the glide and geometrically necessary dislocation densities which is associated with the stored energy of cold work.

Keywords

Gradient single crystal plasticity Thermoplasticity  Finite element method Cold work Annealing 

Notes

Acknowledgments

A.M. and B.D.R. acknowledge the support provided by the National Research Foundation through the South African Research Chair in Computational Mechanics. A part of this work was undertaken while A.M. was visiting the Hamburg University of Technology. A.M. also acknowledges the support provided by the University Research Committee of the University of Cape Town.

References

  1. 1.
    Anand L, Gurtin ME, Reddy BD (2015) The stored energy of cold work, thermal annealing, and other thermodynamic issues in single crystal plasticity at small length scales. Int J Plasticity 64:1–25CrossRefGoogle Scholar
  2. 2.
    Armero F, Simo JC (1992) A new unconditionally stable fractional step method for non-linear coupled thermomechanical problems. Int J Numer Methods Eng 35(4):737–766. ISSN 0029-5981Google Scholar
  3. 3.
    Arsenlis AP, Parks DM (1999) Crystallographic aspects of geometrically-necessary and statistically-stored dislocation density. Acta Mater 47:1597–1611CrossRefGoogle Scholar
  4. 4.
    Bargmann S, Ekh M (2013) Microscopic temperature field prediction during adiabatic loading using gradient extended crystal plasticity. Int J Solids Struct 50(6):899–906CrossRefGoogle Scholar
  5. 5.
    Bargmann S, Reddy BD (2011) Modeling of polycrystals using a gradient crystal plasticity theory that includes dissipative micro-stresses. Eur J Mech A 30(5):719–730CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Bargmann S, Reddy BD, Klusemann B (2014) A computational study of a model of single-crystal strain-gradient viscoplasticity with an interactive hardening relation. Int J Solids Struct 51(15–16):2754–2764CrossRefGoogle Scholar
  7. 7.
    Bayley CJ, Brekelmans WAM, Geers MGD (2006) A comparison of dislocation induced back stress formulations in strain gradient crystal plasticity. Int J Solids Struct 43(24):7268–7286CrossRefzbMATHGoogle Scholar
  8. 8.
    Benzerga AA, Bréchet Y, Needleman A, van der Giessen E (2005) The stored energy of cold work: predictions from discrete dislocation plasticity. Acta Mater 53(18):4765–4779CrossRefGoogle Scholar
  9. 9.
    Bever MB, Holt DL, Titchener AL (1973) The stored energy of cold work. Prog Mater Sci 17:833–849CrossRefGoogle Scholar
  10. 10.
    Bittencourt E, Needleman A, Gurtin ME, van der Giessen E (2003) A comparison of nonlocal continuum and discrete dislocation plasticity predictions. J Mech Phys Solids 51:281–310CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Cleveringa HHM, van Der Giessen E, Needleman A (1997) Comparison of discrete dislocation and continuum plasticity predictions for a composite material. Acta Mater 45(8):3163–3179CrossRefGoogle Scholar
  12. 12.
    Coleman BD, Noll W (1963) The thermodynamics of elastic materials with heat conduction and viscosity. Arch Ration Mech Anal 13(1):167–178CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Dederichs PH, Leibfri G (1969) Elastic Green’s function for anisotropic cubic crystals. Phys Rev 188(3):1175–1183CrossRefMathSciNetGoogle Scholar
  14. 14.
    Ebobisse F, Reddy BD (2004) Some mathematical problems in perfect plasticity. Comput Methods Appl Mech Eng 193(48–51):5071–5094CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Ekh M, Grymer M, Runesson K, Svedberg T (2007) Gradient crystal plasticity as part of the computational modelling of polycrystals. Int J Numer Methods Eng 72(2):197–220CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Ekh M, Bargmann S, Grymer M (2011) Influence of grain boundary conditions on modeling of size-dependence in polycrystals. Acta Mech 218(1–2):103–113 ISSN 0001–5970CrossRefzbMATHGoogle Scholar
  17. 17.
    Ertürk İ, van Dommelen JAW, Geers MGD (2009) Energetic dislocation interactions and thermodynamical aspects of strain gradient crystal plasticity theories. J Mech Phys Solids 57(11):1801–1814CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Evers LP, Parks DM, Brekelmans WAM, Geers MGD (2002) Crystal plasticity model with enhanced hardening by geometrically necessary dislocation accumulation. J Mech Phys Solids 50(11):2403–2424CrossRefzbMATHGoogle Scholar
  19. 19.
    Evers LP, Brekelmans WAM, Geers MGD (2004a) Scale dependent crystal plasticity framework with dislocation density and grain boundary effects. Int J Solids Struct 41:5209–5230CrossRefzbMATHGoogle Scholar
  20. 20.
    Evers LP, Brekelmans WAM, Geers MGD (2004b) Non-local crystal plasticity model with intrinsic SSD and GND effects. J Mech Phys Solids 52(10):2379–2401CrossRefzbMATHGoogle Scholar
  21. 21.
    Gurtin ME (2000) On the plasticity of single crystals: free energy, microforces, plastic-strain gradients. J Mech Phys Solids 48(5):989–1036CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Gurtin ME (2002) A gradient theory of single-crystal plasticity that accounts for geometrically necessary dislocations. J Mech Phys Solids 50(1):5–32CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Gurtin ME (2006) The Burgers vector and the flow of screw and edge dislocations in finite-deformation single-crystal plasticity. J Mech Phys Solids 54(9):1882–1898CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Gurtin ME (2008) A finite-deformation, gradient theory of single-crystal plasticity with free energy dependent on densities of geometrically necessary dislocations. Int J Plasticity 24(4):702–725CrossRefzbMATHGoogle Scholar
  25. 25.
    Gurtin ME (2010) A finite-deformation, gradient theory of single-crystal plasticity with free energy dependent on the accumulation of geometrically necessary dislocations. Int J Plasticity 26(8):1073–1096CrossRefzbMATHGoogle Scholar
  26. 26.
    Gurtin ME, Anand L (2005) A theory of strain-gradient plasticity for isotropic, plastically irrotational materials, Part I: small deformations. J Mech Phys Solids 53(7):1624–1649Google Scholar
  27. 27.
    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):1–31CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
  29. 29.
    Intel. Math Kernel Library (2014) http://developer.intel.com/software/products/mkl/
  30. 30.
    Korelc J (2002) Multi-language and multi-environment generation of nonlinear finite element codes. Eng Comput 18(4):312–327CrossRefGoogle Scholar
  31. 31.
    Lele SP, Anand L (2008) A small-deformation strain-gradient theory for isotropic viscoplastic materials. Philos Mag 88(30–32):3655–3689CrossRefGoogle Scholar
  32. 32.
    Miehe C, Mauthe S, Hildebrand FE (2014) Variational gradient plasticity at finite strains, Part III: local-global updates and regularization techniques in multiplicative plasticity for single crystals. Comput Methods Appl Mech Eng 268:735–762CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    Niordson CF, Kysar JW (2014) Computational strain gradient crystal plasticity. J Mech Phys Solids 62:31–47CrossRefMathSciNetGoogle Scholar
  34. 34.
    Reddy BD (2011a) The role of dissipation and defect energy in variational formulations of problems in strain-gradient plasticity, Part 2: single-crystal plasticity. Contin Mech Thermodyn 23(6):551–572CrossRefzbMATHMathSciNetGoogle Scholar
  35. 35.
    Reddy BD (2011b) The role of dissipation and defect energy in variational formulations of problems in strain-gradient plasticity, Part 1: polycrystalline plasticity. Contin Mech Thermodyn 23(6):527–549CrossRefzbMATHMathSciNetGoogle Scholar
  36. 36.
    Reddy BD, Wieners C, Wohlmuth B (2012) Finite element analysis and algorithms for single-crystal strain-gradient plasticity. Int J Numer Methods Eng 90(6):784–804CrossRefzbMATHMathSciNetGoogle Scholar
  37. 37.
    Rosakis P, Rosakis AJ, Ravichandran G, Hodowany J (2000) A thermodynamic internal variable model for the partition of plastic work into heat and stored energy in metals. J Mech Phys Solids 48(3):581–607CrossRefzbMATHMathSciNetGoogle Scholar
  38. 38.
    Rudraraju S, van der Ven A, Garikipati K (2014) Three-dimensional isogeometric solutions to general boundary value problems of Toupin’s gradient elasticity theory at finite strains. Comput Methods Appl Mech Eng 278:705–728 ISSN 0045–7825CrossRefGoogle Scholar
  39. 39.
    Schmidt-Baldassari M (2003) Numerical concepts for rate-independent single crystal plasticity. Comput Methods Appl Mech Eng 192:1261–1280CrossRefzbMATHGoogle Scholar
  40. 40.
    Schröder J, Miehe C (1997) Aspects of computational rate-independent crystal plasticity. Comput Mater Sci 9:168–176CrossRefGoogle Scholar
  41. 41.
    Taylor GI, Quinney H (1934) The latent energy remaining in a metal after cold working. Proc R Soc Lond A 143:307–326CrossRefGoogle Scholar
  42. 42.
    Taylor GI, Quinney H (1937) The latent energy remaining in a metal after cold working. Proc R Soc Lond A 163:157–181CrossRefGoogle Scholar
  43. 43.
    Wriggers P (2008) Nonlinear finite element methods. Springer, BerlinzbMATHGoogle Scholar
  44. 44.
    Wulfinghoff S, Böhlke T (2012) Equivalent plastic strain gradient enhancement of single crystal plasticity: theory and numerics. Proc R Soc Lond A 468(2145):2682–2703CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Centre for Research in Computational and Applied MechanicsUniversity of Cape TownRondeboschSouth Africa
  2. 2.Institute of Continuum Mechanics and Materials Mechanics, Helmholtz-Zentrum GeesthachtHamburg University of Technology & Institute of Materials ResearchGeesthachtGermany

Personalised recommendations