A framework for geometrically non-linear gradient extended crystal plasticity coupled to heat conduction and damage

  • Ekh, Magnus (Division of Material and Computational Mechanics, Department of Applied Mechanics, Chalmers University of Technology) ;
  • Bargmann, Swantje (Institute of Continuum Mechanics and Material Mechanics, Hamburg University of Technology)
  • Received : 2015.09.01
  • Accepted : 2015.12.15
  • Published : 2016.04.25


Gradient enhanced theories of crystal plasticity enjoy great research interest. The focus of this work is on thermodynamically consistent modeling of grain size dependent hardening effects. In this contribution, we develop a model framework for damage coupled to gradient enhanced crystal thermoplasticity. The damage initiation is directly linked to the accumulated plastic slip. The theoretical setting is that of finite strains. Numerical results on single-crystalline metal showing the development of damage conclude the paper.


  1. Acharya, A., Bassani, J.L. and Beaudoin, A. (2003), "Geometrically necessary dislocations, hardening, and a simple gradient theory of crystal plasticity", Scripta Materialia, 48(2), 167-172.
  2. Aslan, O. and Forest, S. (2011), "The micromorphic versus phase field approach to gradient plasticity and damage with application to cracking in metal single crystals", Multiscale Methods in Computational Mechanics, Springer, Netherlands.
  3. Bargmann, S., Ekh, M., Runesson, K. and Svendsen, B. (2010), "Modeling of polycrystals with gradient crystal plasticity: A comparison of strategies", Philosoph. Magaz., 90(10), 1263-1288.
  4. Bargmann, S., Svendsen, B. and Ekh, M. (2011), "An extended crystal plasticity model for latent hardening in polycrystals", Comput. Mech., 48(6), 631-645.
  5. Bargmann, S. and Ekh, M. (2013), "Microscopic temperature field prediction during adiabatic loading in a gradient extended crystal plasticity theory", Int. J. Solid. Struct., 50(6), 899-906.
  6. Bayley, C., Brekelmans, W. and Geers, M. (2006), "A comparison of dislocation induced back stress formulations in strain gradient crystal plasticity", Int. J. Solid. Struct., 43(24), 7268-7286.
  7. Bazant, Z. and Lin, F.-B. (1988), "Non-local yield limit degradation", Int. J. Numer. Meth. Eng., 26(8), 1805-1823.
  8. Bodelot, L., Charkaluk, E., Sabatier, L. and Dufrenoy, P. (2011), "Experimental study of heterogeneities in strain and temperature fields at the microstructural level of polycrystalline metals through fully-coupled full-field measurements by digital image correlation and infrared thermography", Mech. Mater., 43(11), 654-670.
  9. Borg, U. (2007), "A strain gradient crystal plasticity analysis of grain size effects in polycrystals", Eur. J. Mech. Solid., 26(2), 313-324.
  10. Cermelli, P. and Gurtin, M.E. (2001), "On the characterization of the geometrically necessary dislocations in finite plasticity", J. Mech. Phys. Solid., 49(7), 1539-1568.
  11. Clayton, J. and McDowell, D. (2004), "Homogenized finite elastoplasticity and damage: theory and computations", Mech. Mater., 36(9), 799-824.
  12. Dimitrijevic, B.J. and Hackl, K. (2011), "A regularization framework for damage-plasticity models via gradient enhancement of the free energy", Int. J. Numer. Meter. Biol. Eng., 27(8), 1199-1210.
  13. Dunne, F.P.E., Wilkinson, A.J. and Allen, R. (2007), "Experimental and computational studies of low cycle fatigue crack nucleation in a polycrystal", Int. J. Plast., 23(2), 273-295.
  14. Ekh, M., Lillbacka, R. and Runesson, K. (2004), "A model framework for anisotropic damage coupled to crystal (visco)plasticity", Int. J. Plast., 20(12), 2143-2159.
  15. Ekh, M., Grymer, M., Runesson, K. and Svedberg, T. (2007), "Gradient crystal plasticity as part of the computational modeling of polycrystals", Int. J. Numer. Meter. Eng., 72(2), 197-220.
  16. Ekh, M., Bargmann, S. and Grymer, M. (2011), "Influence of grain boundary conditions on modeling of size-dependence in polycrystals", Acta Mechanica, 218(1-2), 103-113.
  17. Evers, L.P., Brekelmanns, W.A.M. and Geers, M.G.D. (2004), "Non-local crystal plasticity model with intrinsic ssd and gnd effects", J. Mech. Phys. Solid., 52(10), 2379-2401.
  18. Evers, L.P., Brekelmanns, W.A.M. and Geers, M.G.D. (2004a), "Scale dependent crystal plasticity framework with dislocation density and grain boundary effects", Int. J. Solid. Struct., 41(18), 5209-5230.
  19. Fleck, N.A., Muller, G.M., Ashby, M.F. and Hutchinson, J.W. (1994), "Strain gradient plasticity: theory and experiment", Acta Metallurgica et Materialia, 42(2), 475-487.
  20. Fleck, N.A. and Hutchinson, J.W. (1997), "Strain gradient plasticity", Adv. Appl. Mech., 33, 295-361.
  21. 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", J. Mech. Phys. Solid., 52(11), 2545-2568.
  22. Hakansson, P., Wallin, M. and Ristinmaa, M. (2008), "Prediction of stored energy in polycrystalline materials during cyclic loading", Int. J. Solid. Struct., 45(6), 1570-1586.
  23. Heino, S. and Karlsson, B. (2001), "Cyclic deformation and fatigue behavior of 7Mo-0.5N superaustenitic stainless steel characteristics and development of the dislocation structures", Acta Materialia, 49(2), 353-363.
  24. Horstemeyer, M., Ramaswamy, S. and Negrete, M. (2003), "Using a micromechanical finite element parametric study to motivate a phenomenological macroscale model for void/crack nucleation in aluminum with a hard second phase", Mech. Mater., 35(7), 675-687.
  25. Hou, N., Wen, Z. and Yue, Z. (2009), "Creep behavior of single crystal superalloy specimen under temperature gradient condition", Mater. Sci. Eng., A510, 42-45.
  26. Husser, E., Lilleodden, E. and Bargmann, S. (2014), "Computational modeling of intrinsically induced strain gradients during compression of c-axis oriented magnesium single crystal", Acta Materialia, 71, 206-219.
  27. Kroner, E. (1960), "Allgemeine kontinuumstheorie der versetzungen und eigenspannungen", Archiv. Ration. Mech. Anal., 4(1), 273-334.
  28. Kuroda, M. and Tvergaard, V. (2006), "Studies of scale dependent crystal viscoplasticity models", J. Mech. Phys. Solid., 54(9), 1789-1810.
  29. Kuroda, M. and Tvergaard, V. (2008), "On the formulations of higher-order strain gradient crystal plasticity models", J. Mech. Phys. Solid., 56(4), 1591-1608.
  30. Kuroda, M. and Tvergaard, V. (2008a), "A finite deformation theory of higher-order gradient crystal plasticity", J. Mech. Phys. Solid., 56(8), 2573-2584.
  31. Lammer, H. and Tsakmakis, C. (2000), "Discussion of coupled elastoplasticity and damage constitutive equations for small and finite deformations", Int. J. Plast., 16(5), 495-523.
  32. Lemaȋtre, J. (1992), A Course on Damage Mechanics.
  33. Levkovitch, V. and Svendsen, B. (2006), "On the large-deformation-and continuum-based formulation of models for extended crystal plasticity", Int. J. Solid. Struct., 43(24), 7246-7267.
  34. McBride, A., Bargmann, S. and Reddy, D. (2015), "A computational investigation of a model of singlecrystal gradient thermoplasticity that accounts for the stored energy of cold work and thermal annealing", Compos. Mech., 55(4), 755-769.
  35. Ohno, N. and Okumura, D. (2007), "Higher-order stress and grain size effects due to self-energy of geometrically necessary dislocations", J. Mech. Phys. Solid., 55(9), 1879-1898.
  36. Parisot, R., Forest, S., Pineau, A., Grillon, F., Demonet, X. and Mataigne, J.-M. (2004), "Deformation and damage mechanisms of zinc coatings on hot-dip galvanized steel sheets: Part II. Damage modes", Metal. Mater. Trans. A, 35(3), 813-823.
  37. Peerlings, R., Poh, L. and Geers, M. (2012), "An implicit gradient plasticity-damage theory for predicting size effects in hardening and softening", Eng. Fract. Mech., 95, 2-12.
  38. Rice, J. (1971), "Inelastic constitutive relations for solids: an internal-variable theory and its application to metal plasticity", J. Mech. Phys. Solid., 19(6), 433-455.
  39. Roters, F., Eisenlohr, P., Hantcherli, L., Tjahjanto, D., Bieler, T. and Raabe, D. (2010), "Overview of constitutive laws, kinematics, homogenization and multiscale methods in crystal plasticity finite-element modeling: Theory, experiments, applications", Acta Materialia, 58(4), 1152-1211.
  40. Vrech, S.M. and Etse, G. (2007), "FE approach for thermodynamically consistent gradient-dependent plasticity", Latt. Am. Appl. Res., 37(2), 127-132.
  41. Welschinger, F. (2011), "A variational framework for gradient-extended dissipative continua. Application to damage mechanics, fracture, and plasticity", Ph.D. thesis, University of Stuttgart, Germany.
  42. Yefimov, S., Groma, I. and Giessen, E. van der (2004), "A comparison of a statistical-mechanics based plasticity model with discrete dislocation plasticity calculations", J. Mech. Phys. Solid., 52(2), 279-300.
  43. Yefimov, S. and Giessen, E. van der (2005), "Multiple slip in a strain-gradient plasticity model motivated by a statistical-mechanics description of dislocations", Int. J. Solid. Struct., 42(11), 3375-3394.