Continuum Mechanics and Thermodynamics

, Volume 29, Issue 6, pp 1207–1217 | Cite as

Micromorphic approach for gradient-extended thermo-elastic–plastic solids in the logarithmic strain space

Original Article


The coupled thermo-mechanical strain gradient plasticity theory that accounts for microstructure-based size effects is outlined within this work. It extends the recent work of Miehe et al. (Comput Methods Appl Mech Eng 268:704–734, 2014) to account for thermal effects at finite strains. From the computational viewpoint, the finite element design of the coupled problem is not straightforward and requires additional strategies due to the difficulties near the elastic–plastic boundaries. To simplify the finite element formulation, we extend it toward the micromorphic approach to gradient thermo-plasticity model in the logarithmic strain space. The key point is the introduction of dual local–global field variables via a penalty method, where only the global fields are restricted by boundary conditions. Hence, the problem of restricting the gradient variable to the plastic domain is relaxed, which makes the formulation very attractive for finite element implementation as discussed in Forest (J Eng Mech 135:117–131, 2009) and Miehe et al. (Philos Trans R Soc A Math Phys Eng Sci 374:20150170, 2016).


Size effects Finite gradient plasticity Micromorphic regularization Thermo-mechanical processes 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Miehe, C., Welschinger, F., Aldakheel, F.: Variational gradient plasticity at finite strains. Part II: local–global updates and mixed finite elements for additive plasticity in the logarithmic strain space. Comput. Methods Appl. Mech. Eng. 268, 704–734 (2014)ADSMathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Forest, S.: Micromorphic approach for gradient elasticity, viscoplasticity, and damage. J. Eng. Mech. 135, 117–131 (2009)CrossRefGoogle Scholar
  3. 3.
    Miehe, C., Teichtmeister, S., Aldakheel, F.: Phase-field modeling of ductile fracture: a variational gradient-extended plasticity-damage theory and its micromorphic regularization. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 374, 20150170 (2016). doi: 10.1098/rsta.2015.0170
  4. 4.
    Fleck, N.A., Muller, G.M., Ashby, M.F., Hutchinson, J.W.: Strain gradient plasticity: theory and experiment. Acta Mater. 42, 475–487 (1994)CrossRefGoogle Scholar
  5. 5.
    de Borst, R., Mühlhaus, H.B.: Gradient-dependent plasticity: formulation and algorithmic aspects. Int. J. Numer. Methods Eng. 35, 521–539 (1992)CrossRefMATHGoogle Scholar
  6. 6.
    Liebe, T., Steinmann, P.: Theory and numerics of a thermodynamically consistent framework for geometrically linear gradient plasticity. Int. J. Numer. Methods Eng. 51, 1437–1467 (2001)CrossRefMATHGoogle Scholar
  7. 7.
    Engelen, R.A.B., Geers, M.G.D., Baaijens, F.P.T.: Nonlocal implicit gradient-enhanced elasto-plasticity for the modelling of softening behavior. Int. J. Plast. 19, 403–433 (2003)CrossRefMATHGoogle Scholar
  8. 8.
    Gurtin, E.: A finite-deformation, gradient theory of single-crystal plasticity with free energy dependent on densities of geometrically necessary dislocations. Int. J. Plast. 24, 702–725 (2008)CrossRefMATHGoogle Scholar
  9. 9.
    Svendsen, B., Bargmann, S.: On the continuum thermodynamic rate variational formulation of models for extended crystal plasticity at large deformation. J. Mech. Phys. Solids 58, 1253–1271 (2010)ADSMathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Wulfinghoff, S., Böhlke, T.: Equivalent plastic strain gradient enhancement of single crystal plasticity: theory and numerics. In: Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences (2012)Google Scholar
  11. 11.
    Klusemann, B., Yalcinkaya, T.: Plastic deformation induced microstructure evolution through gradient enhanced crystal plasticity based on a non-convex helmholtz energy. Int. J. Plast. 48, 168–188 (2013)CrossRefGoogle Scholar
  12. 12.
    Miehe, C., Mauthe, S., Hildebrand, F.E.: 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, 735–762 (2014)ADSMathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Forest, S., Sievert, R.: Elastoviscoplastic constitutive frameworks for generalized continua. Acta Mech. 160, 71–111 (2003)CrossRefMATHGoogle Scholar
  14. 14.
    Gudmundson, P.: A unified treatment of strain gradient plasticity. J. Mech. Phys. Solids 52, 1379–1406 (2004)ADSMathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Anand, L., Aslan, O., Chester, S.A.: A large-deformation gradient theory for elastic–plastic materials: strain softening and regularization of shear bands. Int. J. Plast. 30–31, 116–143 (2012)CrossRefGoogle Scholar
  16. 16.
    Reddy, B., Ebobisse, F., McBride, A.: Well-posedness of a model of strain gradient plasticity for plastically irrotational materials. Int. J. Plast. 24, 55–73 (2008)CrossRefMATHGoogle Scholar
  17. 17.
    Fleck, N.A., Willis, J.R.: A mathematical basis for strain-gradient plasticity theory. Part I: scalar plastic multiplier. J. Mech. Phys. Solids 57, 161–177 (2009a)ADSMathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Fleck, N.A., Willis, J.R.: A mathematical basis for strain-gradient plasticity theory. Part II: tensorial plastic multiplier. J. Mech. Phys. Solids 57, 1045–1057 (2009b)ADSMathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Polizzotto, C.: A nonlocal strain gradient plasticity theory for finite deformations. Int. J. Plast. 25, 1280–1300 (2009)CrossRefMATHGoogle Scholar
  20. 20.
    Forest, S.: Nonlinear regularization operators as derived from the micromorphic approach to gradient elasticity, viscoplasticity and damage. In: Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 472 (2016)Google Scholar
  21. 21.
    Voyiadjis, G.Z., Pekmezi, G., Deliktas, B.: Nonlocal gradient-dependent modeling of plasticity with anisotropic hardening. Int. J. Plast. 26, 1335–1356 (2010)CrossRefMATHGoogle Scholar
  22. 22.
    Kuroda, M., Tvergaard, V.: An alternative treatment of phenomenological higher-order strain-gradient plasticity theory. Int. J. Plast. 26, 507–515 (2010)CrossRefMATHGoogle Scholar
  23. 23.
    Miehe, C., Aldakheel, F., Mauthe, S.: Mixed variational principles and robust finite element implementations of gradient plasticity at small strains. Int. J. Numer. Methods Eng. 94, 1037–1074 (2013)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Wriggers, P., Miehe, C., Kleiber, M., Simo, J.: On the coupled thermomechanical treatment of necking problems via finite element methods. Int. J. Numer. Methods Eng. 33, 869–883 (1992)CrossRefMATHGoogle Scholar
  25. 25.
    Anand, L., Ames, N.M., Srivastava, V., Chester, S.A.: A thermo-mechanically coupled theory for large deformations of amorphous polymers. Part I: formulation. Int. J. Plast. 25, 1474–1494 (2009)CrossRefMATHGoogle Scholar
  26. 26.
    Canadija, M., Mosler, J.: On the thermomechanical coupling in finite strain plasticity theory with non-linear kinematic hardening by means of incremental energy minimization. Int. J. Solids Struct. 48, 1120–1129 (2011)CrossRefMATHGoogle Scholar
  27. 27.
    Yang, Q., Stainier, L., Ortiz, M.: A variational formulation of the coupled thermo-mechanical boundary-value problem for general dissipative solids. J. Mech. Phys. Solids 54, 401–424 (2006)ADSMathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Stainier, L., Ortiz, M.: Study and validation of thermomechanical coupling in finite strain visco-plasticity. Int. J. Solids Struct. 47, 704–715 (2010)CrossRefMATHGoogle Scholar
  29. 29.
    Voyiadjis, Z., Faghihi, D.: Thermo-mechanical strain gradient plasticity with energetic and dissipative length scales. Int. J. Plast. 30–31, 218–247 (2012)CrossRefGoogle Scholar
  30. 30.
    Faghihi, D., Voyiadjis, Z., Park, T.: Coupled thermomechanical modeling of small volume fcc metals. J. Eng. Mater. Technol. 135, 1–17 (2013)CrossRefGoogle Scholar
  31. 31.
    Forest, S., Aifantis, E.: Some links between recent gradient thermo-elasto-plasticity theories and the thermomechanics of generalized continua. Int. J. Solids Struct. 47, 3367–3376 (2010)CrossRefMATHGoogle Scholar
  32. 32.
    Bertram, A., Forest, S.: The thermodynamics of gradient elastoplasticity. Contin. Mech. Thermodyn. 26, 269–286 (2014)ADSMathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Wcislo, B., Pamin, J.: Local and non-local thermomechanical modeling of elastic–plastic materials undergoing large strains. Int. J. Numer. Methods Eng. 109, 102–124 (2016)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Miehe, C., Aldakheel, F., Teichtmeister, S.: Phase-field modeling of ductile fracture at finite strains. A robust variational-based numerical implementation of a gradient-extended theory by micromorphic regularization. Int. J. Numer. Methods Eng. (2016). doi: 10.1002/nme.5484 MATHGoogle Scholar
  35. 35.
    Aldakheel, F.: Mechanics of nonlocal dissipative solids: gradient plasticity and phase field modeling of ductile fracture. Ph.D. thesis, Institute of Applied Mechanics (CE), Chair I, University of Stuttgart (2016). doi: 10.18419/opus-8803
  36. 36.
    Aldakheel, F., Miehe, C.: Coupled thermomechanical response of gradient plasticity. Int. J. Plast. 91, 1–24 (2017)Google Scholar
  37. 37.
    Miehe, C., Apel, N., Lambrecht, M.: Anisotropic additive plasticity in the logarithmic strain space. Modularkinematic formulation and implementation based on incremental minimization principles for standard materials. Comput. Methods Appl. Mech. Eng. 191, 5383–5425 (2002)ADSCrossRefMATHGoogle Scholar
  38. 38.
    Geers, M.G.D., Peerlings, R.H.J., Brekelmans, W.A.M., de Borst, R.: Phenomenological nonlocal approaches based on implicit gradient-enhanced damage. Acta Mech. 144, 1–15 (2000)CrossRefMATHGoogle Scholar
  39. 39.
    Peerlings, R.H.J., Geers, M.G.D., de Borst, R., Brekelmans, W.A.M.: A critical comparison of nonlocal and gradient-enhanced softening continua. Int. J. Solids Struct. 38, 7723–7746 (2001)CrossRefMATHGoogle Scholar
  40. 40.
    Peerlings, R.H.J., Massart, T.J., Geers, M.G.D.: A thermodynamically motivated implicit gradient damage framework and its application to brick masonry cracking. Comput. Methods Appl. Mech. Eng. 193, 3403–3417 (2004)ADSCrossRefMATHGoogle Scholar
  41. 41.
    Simó, J., Miehe, C.: Associative coupled thermoplasticity at finite strains: formulation, numerical analysis and implementation. Comput. Methods Appl. Mech. Eng. 98, 41–104 (1992)ADSCrossRefMATHGoogle Scholar
  42. 42.
    Boyce, M.C., Montagut, E.L., Argon, A.S.: The effects of thermomechanical coupling on the cold drawing process of glassy polymers. Polym. Eng. Sci. 32, 1073–1085 (1992)CrossRefGoogle Scholar
  43. 43.
    Miehe, C., Lambrecht, M.: Algorithms for computation of stresses and elasticity moduli in terms of Seth–Hill’s family of generalized strain tensors. Commun. Numer. Methods Eng. 17, 337–353 (2001)CrossRefMATHGoogle Scholar
  44. 44.
    Hallquist, J.O.: Nike 2D: An implicit, finite deformation, finite element code for analyzing the static and dynamic response of two-dimensional solids. Rept. UCRL-52678, Lawrence Livermore National Laboratory, University of California, Livermore, CA (1984)Google Scholar
  45. 45.
    Simó, J.C.: A framework for finite strain elastoplasticity based on maximum plastic dissipation and the multiplicative decomposition. Part II: computational aspects. Comput. Methods Appl. Mech. Eng. 68, 1–31 (1988)ADSCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Institute of Applied Mechanics (Civil Engineering), Chair IUniversity of StuttgartStuttgartGermany

Personalised recommendations