Micromorphic approach to crystal plasticity and phase transformation

  • Samuel Forest
  • Kais Ammar
  • Benoît Appolaire
  • Nicolas Cordero
  • Anaïs Gaubert
Chapter
Part of the CISM International Centre for Mechanical Sciences book series (CISM, volume 550)

Abstract

Continuum crystal plasticity models are extended to incorporate the effect of the dislocation density tensor on material hardening. The approach is based on generalized continuum mechanics including strain gradient plasticity, Cosserat and micromorphic media. The applications deal with the effect of precipitate size in two–phase single crystals and to the Hall-Petch grain size effect in polycrystals. Some links between the micromorphic approach and phase field models are established. A coupling between phase field approach and elastoviscoplasticity constitutive equations is then presented and applied to the prediction of the influence of viscoplasticity on the kinetics of diffusive precipitate growth and morphology changes.

Keywords

Slip System Crystal Plasticity Kinematic Hardening Strain Gradient Plasticity Virtual Power 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. Guillaume Abrivard. A coupled crystal plasticity–phase field formulation to describe microstructural evolution in polycrystalline aggregates. PhD, Mines ParisTech, 2009.Google Scholar
  2. E.C. Aifantis. On the microstructural origin of certain inelastic models. Journal of Engineering Materials and Technology, 106:326–330, 1984.CrossRefGoogle Scholar
  3. K. Ammar, B. Appolaire, G. Cailletaud, F. Feyel, and F. Forest. Finite element formulation of a phase field model based on the concept of generalized stresses. Computational Materials Science, 45:800–805, 2009a.CrossRefGoogle Scholar
  4. K. Ammar, B. Appolaire, G. Cailletaud, and S. Forest. Combining phase field approach and homogenization methods for modelling phase transformation in elastoplastic media. European Journal of Computational Mechanics, 18:485–523, 2009b.Google Scholar
  5. K. Ammar, B. Appolaire, G. Cailletaud, and S. Forest. Phase field modeling of elasto-plastic deformation induced by diffusion controlled growth of a misfitting spherical precipitate. Philosophical Magazine Letters, 91:164–172, 2011.CrossRefGoogle Scholar
  6. B. Appolaire, E. Aeby-Gautier, J. D. Teixeira, M. Dehmas, and S. Denis. Non-coherent interfaces in diffuse interface models. Philosophical Magazine, 90:461–483, 2010.CrossRefGoogle Scholar
  7. R.J. Asaro. Crystal plasticity. J. Appl. Mech., 50:921–934, 1983.CrossRefMATHGoogle Scholar
  8. O. Aslan and S. Forest. Crack growth modelling in single crystals based on higher order continua. Computational Materials Science, 45:756–761, 2009.CrossRefGoogle Scholar
  9. O. Aslan and S. Forest. The micromorphic versus phase field approach to gradient plasticity and damage with application to cracking in metal single crystals. In R. de Borst and E. Ramm, editors, Multiscale Methods in Computational Mechanics, pages 135–154. Lecture Notes in Applied and Computational Mechanics 55, Springer, 2011.CrossRefGoogle Scholar
  10. O. Aslan, N. M. Cordero, A. Gaubert, and S. Forest. Micromorphic approach to single crystal plasticity and damage. International Journal of Engineering Science, 49:1311–1325, 2011.MathSciNetCrossRefGoogle Scholar
  11. V.P. Bennett and D.L. McDowell. Crack tip displacements of microstructurally small surface cracks in single phase ductile polycrystals. Engineering Fracture Mechanics, 70(2):185–207, 2003.CrossRefGoogle Scholar
  12. J. Besson, G. Cailletaud, J.-L. Chaboche, S. Forest, and M. Blétry. Non–Linear Mechanics of Materials. Series: Solid Mechanics and Its Applications, Vol. 167, Springer, ISBN: 978-90-481-3355-0, 433 p., 2009.Google Scholar
  13. B.A. Bilby, R. Bullough, L.R.T. Gardner, and E. Smith. Continuous distributions of dislocations iv: Single glide and plane strain. Proc. Roy. Soc. London, A236:538–557, 1957.Google Scholar
  14. P. Cermelli and M.E. Gurtin. On the characterization of geometrically necessary dislocations in finite plasticity. Journal of the Mechanics and Physics of Solids, 49:1539–1568, 2001.CrossRefMATHGoogle Scholar
  15. Y. Chen and J.D. Lee. Connecting molecular dynamics to micromorphic theory. (I) instantaneous and averaged mechanical variables. Physica A, 322:359–376, 2003a.CrossRefMATHGoogle Scholar
  16. Y. Chen and J.D. Lee. Connecting molecular dynamics to micromorphic theory. (II) balance laws. Physica A, 322:377–392, 2003b.CrossRefMATHGoogle Scholar
  17. Y. Chen and J.D. Lee. Determining material constants in micromorphic theory through phonon dispersion relations. International Journal of Engineering Science, 41:871–886, 2003c.CrossRefGoogle Scholar
  18. W.D. Claus and A.C. Eringen. Three dislocation concepts and micromorphic mechanics. In Developments in Mechanics, Vol. 6. Proceedings of the 12th Midwestern Mechanics Conference, pages 349–358, 1969.Google Scholar
  19. W.D. Claus and A.C. Eringen. Dislocation dispersion of elastic waves. International Journal of Engineering Science, 9:605–610, 1971.CrossRefMATHGoogle Scholar
  20. J.D. Clayton, D.J. Bamman, and D.L. McDowell. A geometric framework for the kinematics of crystals with defects. Philosophical Magazine, 85: 3983–4010, 2005.CrossRefGoogle Scholar
  21. N.M. Cordero, A. Gaubert, S. Forest, E. Busso, F. Gallerneau, and S. Kruch. Size effects in generalised continuum crystal plasticity for two–phase laminates. Journal of the Mechanics and Physics of Solids, 58:1963–1994, 2010.MathSciNetCrossRefMATHGoogle Scholar
  22. W. Ehlers and W. Volk. On theoretical and numerical methods in the theory of porous media based on polar and non–polar elasto–plastic solid materials. Int. J. Solids Structures, 35:4597–4617, 1998.CrossRefMATHGoogle Scholar
  23. R.A.B. Engelen, M.G.D. Geers, and F.P.T. Baaijens. Nonlocal implicit gradient-enhanced elasto-plasticity for the modelling of softening behaviour. International Journal of Plasticity, 19:403–433, 2003.CrossRefMATHGoogle Scholar
  24. A.C. Eringen and W.D. Claus. A micromorphic approach to dislocation theory and its relation to several existing theories. In J.A. Simmons, R. de Wit, and R. Bullough, editors, Fundamental Aspects of Dislocation Theory, pages 1023–1062. Nat. Bur. Stand. (US) Spec. Publ. 317, II, 1970.Google Scholar
  25. A.C. Eringen and E.S. Suhubi. Nonlinear theory of simple microelastic solids. Int. J. Engng Sci., 2:189–203, 389–404, 1964.MathSciNetCrossRefMATHGoogle Scholar
  26. Y. Estrin. Dislocation density related constitutive modelling. In Unified Constitutive Laws of Plastic Deformation, pages 69–106. Academic Press, 1996.CrossRefGoogle Scholar
  27. A. Finel, Y. Le Bouar, A. Gaubert, and U. Salman. Phase field methods: Microstructures, mechanical properties and complexity. Comptes Rendus Physique, 11:245–256, 2010.CrossRefGoogle Scholar
  28. M. Fivel and S. Forest. Plasticité cristalline et transition d’échelle : cas du monocristal. Techniques de l’Ingénieur, M4016, 23 pages, 2004a.Google Scholar
  29. M. Fivel and S. Forest. Plasticité cristalline et transition d’échelle : cas du polycristal. Techniques de l’Ingénieur, M4017, 11 pages, 2004b.Google Scholar
  30. S. Forest. The micromorphic approach for gradient elasticity, viscoplasticity and damage. ASCE Journal of Engineering Mechanics, 135:117–131, 2009.CrossRefGoogle Scholar
  31. S. Forest. Some links between cosserat, strain gradient crystal plasticity and the statistical theory of dislocations. Philosophical Magazine, 88: 3549–3563, 2008.CrossRefGoogle Scholar
  32. S. Forest and E. C. Aifantis. Some links between recent gradient thermoelasto-plasticity theories and the thermomechanics of generalized continua. International Journal of Solids and Structures, 47:3367–3376, 2010.CrossRefMATHGoogle Scholar
  33. S. Forest and R. Sedláček. Plastic slip distribution in two–phase laminate microstructures: Dislocation–based vs. generalized–continuum approaches. Philosophical Magazine A, 83:245–276, 2003.CrossRefGoogle Scholar
  34. S. Forest and R. Sievert. Elastoviscoplastic constitutive frameworks for generalized continua. Acta Mechanica, 160:71–111, 2003.CrossRefMATHGoogle Scholar
  35. S. Forest and R. Sievert. Nonlinear microstrain theories. International Journal of Solids and Structures, 43:7224–7245, 2006.MathSciNetCrossRefMATHGoogle Scholar
  36. S. Forest, F. Barbe, and G. Cailletaud. Cosserat modelling of size effects in the mechanical behaviour of polycrystals and multiphase materials. International Journal of Solids and Structures, 37:7105–7126, 2000.MathSciNetCrossRefMATHGoogle Scholar
  37. S. Forest, F. Pradel, and K. Sab. Asymptotic analysis of heterogeneous Cosserat media. International Journal of Solids and Structures, 38:4585–4608, 2001.MathSciNetCrossRefMATHGoogle Scholar
  38. M. Frémond and B. Nedjar. Damage, gradient of damage and principle of virtual power. Int. J. Solids Structures, 33:1083–1103, 1996.CrossRefMATHGoogle Scholar
  39. E. Fried and M.E. Gurtin. Continuum theory of thermally induced phase transitions based on an order parameter. Physica D, 68:326–343, 1993.MathSciNetCrossRefMATHGoogle Scholar
  40. A. Gaubert, A. Finel, Y. Le Bouar, and G. Boussinot. Viscoplastic phase field modellling of rafting in ni base superalloys. In Continuum Models and Discrete Systems CMDS11, pages 161–166. Mines Paris Les Presses, 2008.Google Scholar
  41. A. Gaubert, Y. Le Bouar, and A. Finel. Coupling phase field and viscoplasticity to study rafting in ni-based superalloys. Philosophical Magazine, 90:375–404, 2010.CrossRefGoogle Scholar
  42. P. Germain. La méthode des puissances virtuelles en mécanique des milieux continus, première partie : théorie du second gradient. J. de Mécanique, 12:235–274, 1973a.MathSciNetMATHGoogle Scholar
  43. P. Germain. The method of virtual power in continuum mechanics. part 2 : Microstructure. SIAM J. Appl. Math., 25:556–575, 1973b.CrossRefMATHGoogle Scholar
  44. P. Grammenoudis and Ch. Tsakmakis. Micromorphic continuum Part I: Strain and stress tensors and their associated rates. International Journal of Non–Linear Mechanics, 44:943–956, 2009.CrossRefGoogle Scholar
  45. P. Grammenoudis, Ch. Tsakmakis, and D. Hofer. Micromorphic continuum Part II: Finite deformation plasticity coupled with damage. International Journal of Non–Linear Mechanics, 44:957–974, 2009.CrossRefGoogle Scholar
  46. W. Günther. Zur Statik und Kinematik des Cosseratschen Kontinuums. Abhandlungen der Braunschweig. Wiss. Ges., 10:195–213, 1958.MATHGoogle Scholar
  47. M.E. Gurtin. A gradient theory of single–crystal viscoplasticity that accounts for geometrically necessary dislocations. Journal of the Mechanics and Physics of Solids, 50:5–32, 2002.MathSciNetCrossRefMATHGoogle Scholar
  48. M.E. Gurtin. Generalized Ginzburg–Landau and Cahn–Hilliard equations based on a microforce balance. Physica D, 92:178–192, 1996.MathSciNetCrossRefMATHGoogle Scholar
  49. M.E. Gurtin and L. Anand. Nanocrystalline grain boundaries that slip and separate: A gradient theory that accounts for grain-boundary stress and conditions at a triple-junction. Journal of the Mechanics and Physics of Solids, 56:184–199, 2008.MathSciNetCrossRefMATHGoogle Scholar
  50. M.E. Gurtin and L. Anand. 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, 2009.MathSciNetCrossRefMATHGoogle Scholar
  51. C.B. Hirschberger and P. Steinmann. Classification of concepts in thermodynamically consistent generalized plasticity. ASCE Journal of Engineering Mechanics, 135:156–170, 2009.CrossRefGoogle Scholar
  52. William C. Johnson and J. Iwan D. Alexander. Interfacial conditions for thermomechanical equilibrium in two-phase crystals. Journal of Applied Physics, 9:2735–2746, 1986.CrossRefGoogle Scholar
  53. A.G. Khachaturyan. Theory of structural transformations in solids. John Wiley & Sons, 1983.Google Scholar
  54. S.G. Kim, W.T. Kim, and T Suzuki. Interfacial compositions of solid and liquid in a phase–field model with finite interface thickness for isothermal solidification in binary alloys. Physical Review E, 58(3):3316–3323, 1998.CrossRefGoogle Scholar
  55. S.G. Kim, W.T. Kim, and T Suzuki. Phase–field model for binary alloys. Physical Review E, 60(6):7186–7197, 1999.CrossRefGoogle Scholar
  56. E. Kröner. On the physical reality of torque stresses in continuum mechanics. Int. J. Engng. Sci., 1:261–278, 1963.CrossRefGoogle Scholar
  57. E. Kröner. Initial studies of a plasticity theory based upon statistical mechanics. In M.F. Kanninen, W.F. Adler, A.R. Rosenfield, and R.I. Jaffee, editors, Inelastic Behaviour of Solids, pages 137–147. McGraw-Hill, 1969.Google Scholar
  58. E. Kröner and C. Teodosiu. Lattice defect approach to plasticity and viscoplasticity. In A. Sawczuk, editor, Problems of Plasticity, International Symposium on Foundations of Plasticity, Warsaw. Noordhoff International Publishing Leyden, 1972.Google Scholar
  59. M. Lazar and G.A. Maugin. Dislocations in gradient micropolar–I: screw dislocation. Journal of the Mechanics and Physics of Solids, 52:2263–2284, 2004.MathSciNetCrossRefMATHGoogle Scholar
  60. M. Lazar and G.A. Maugin. On microcontinuum field theories: the eshelby stress tensor and incompatibility conditions. Philosophical Magazine, 87: 3853–3870, 2007.CrossRefGoogle Scholar
  61. M. Lazar, G.A. Maugin, and E.C. Aifantis. Dislocations in second strain gradient elasticity. International Journal of Solids and Structures, 43: 1787–1817, 2006.MathSciNetCrossRefMATHGoogle Scholar
  62. J.D. Lee and Y. Chen. Constitutive relations of micromorphic thermoplasticity. International Journal of Engineering Science, 41:387–399, 2003.MathSciNetCrossRefMATHGoogle Scholar
  63. J. Mandel. Une généralisation de la théorie de la plasticité de W.T. Koiter. International Journal of Solids and Structures, 1:273–295, 1965.CrossRefGoogle Scholar
  64. J. Mandel. Plasticité classique et viscoplasticité. CISM Courses and Lectures No. 97, Udine, Springer Verlag, Berlin, 1971.Google Scholar
  65. J. Mandel. Equations constitutives et directeurs dans les milieux plastiques et viscoplastiques. Int. J. Solids Structures, 9:725–740, 1973.CrossRefMATHGoogle Scholar
  66. G.A. Maugin. The method of virtual power in continuum mechanics: Application to coupled fields. Acta Mechanica, 35:1–70, 1980.MathSciNetCrossRefMATHGoogle Scholar
  67. J.R. Mayeur, D.L. McDowell, and D.J. Bammann. Dislocation–based micropolar single crystal plasticity: Comparison if multi– and single cristerion theories. Journal of the Mechanics and Physics of Solids, 59:398–422, 2011.MathSciNetCrossRefMATHGoogle Scholar
  68. M. Mazière, J. Besson, S. Forest, B. Tanguy, H. Chalons, and F. Vogel. Numerical aspects in the finite element simulation of the portevin-le chatelier effect. Computer Methods in Applied Mechanics and Engineering, 199:734–754, 2010.MathSciNetCrossRefMATHGoogle Scholar
  69. L. Méric, P. Poubanne, and G. Cailletaud. Single crystal modeling for structural calculations. Part 1: Model presentation. J. Engng. Mat. Technol., 113:162–170, 1991.CrossRefGoogle Scholar
  70. C. Miehe. A multifield incremental variational framework for gradienttype standard dissipative solids, in press. Journal of the Mechanics and Physics of Solids, 2011.Google Scholar
  71. C. Miehe, F. Welchinger, and M. Hofacker. Thermodynamically–consistent phase field models of fracture: Variational principles and multifield FE implementations. International Journal for Numerical Methods in Engineering, 83:1273–1311, 2010a.CrossRefMATHGoogle Scholar
  72. C. Miehe, F. Welchinger, and M. Hofacker. A phase field model of electromechanical fracture. Journal of the Mechanics and Physics of Solids, 58:1716–1740, 2010b.CrossRefMATHGoogle Scholar
  73. R.D. Mindlin. Micro–structure in linear elasticity. Arch. Rat. Mech. Anal., 16:51–78, 1964.MathSciNetCrossRefMATHGoogle Scholar
  74. A. I. Murdoch. A thermodynamical theory of elastic material interfaces. Q. J. Mech. appl. Math., 29:245–275, 1978.MathSciNetCrossRefGoogle Scholar
  75. J.F. Nye. Some geometrical relations in dislocated crystals. Acta Metall., 1:153–162, 1953.CrossRefGoogle Scholar
  76. R.H.J. Peerlings, M.G.D. Geers, R. de Borst, and W.A.M. Brekelmans. A critical comparison of nonlocal and gradient–enhanced softening continua. Int. J. Solids Structures, 38:7723–7746, 2001.CrossRefMATHGoogle Scholar
  77. A. Rajagopal, P. Fischer, E. Kuhl, and P. Steinmann. Natural element analysis of the cahn-hilliard phase-field model. Computational Mechanics, 46: 471–493, 2010.MathSciNetCrossRefMATHGoogle Scholar
  78. R.A. Regueiro. On finite strain micromorphic elastoplasticity. International Journal of Solids and Structures, 47:786–800, 2010.CrossRefMATHGoogle Scholar
  79. C. Sansour, S. Skatulla, and H. Zbib. A formulation for the micromorphic continuum at finite inelastic strains. Int. J. Solids Structures, 47:1546–1554, 2010.CrossRefMATHGoogle Scholar
  80. Schäfer,H. Eine Feldtheorie der Versetzungen im Cosserat-Kontinuum. ZAMP, 20:891–899, 1969.CrossRefMATHGoogle Scholar
  81. I. Steinbach and M. Apel. Multi phase field model for solid state transformation with elastic strain. Physica D, 217:153–160, 2006.CrossRefMATHGoogle Scholar
  82. P. Steinmann. Views on multiplicative elastoplasticity and the continuum theory of dislocations. International Journal of Engineering Science, 34: 1717–1735, 1996.CrossRefMATHGoogle Scholar
  83. B. Svendsen. Continuum thermodynamic models for crystal plasticity including the effects of geometrically-necessary dislocations. J. Mech. Phys. Solids, 50:1297–1329, 2002.MathSciNetCrossRefMATHGoogle Scholar
  84. C. Teodosiu. A dynamic theory of dislocations and its applications to the theory of the elastic-plastic continuum. In J.A. Simmons, R. de Wit, and R. Bullough, editors, Fundamental Aspects of Dislocation Theory, pages 837–876. Nat. Bur. Stand. (US) Spec. Publ. 317, II, 1970.Google Scholar
  85. C. Teodosiu and F. Sidoroff. A theory of finite elastoviscoplasticity of single crystals. Int. J. of Engng Science, 14:165–176, 1976.CrossRefMATHGoogle Scholar
  86. R.L.J.M. Ubachs, P.J.G. Schreurs, and M.G.D. Geers. A nonlocal diffuse interface model for microstructure evolution of tin–lead solder. Journal of the Mechanics and Physics of Solids, 52:1763–1792, 2004.CrossRefMATHGoogle Scholar
  87. R.L.J.M. Ubachs, P.J.G. Schreurs, and M.G.D. Geers. Elasto-viscoplastic nonlocal damage modelling of thermal fatigue in anisotropic lead-free solder. Mechanics of Materials, 39:685–701, 2007.CrossRefGoogle Scholar
  88. Y. Wang, L.-Q. Chen, and A.G. Khachaturyan. Kinetics of strain-induced morphological transformation in cubic alloys with a miscibility gap. Acta Metallurgica et Materialia, 41:279–296, 1993.CrossRefGoogle Scholar
  89. A. Zeghadi, S. Forest, A.-F. Gourgues, and O. Bouaziz. Ensemble averaging stress–strain fields in polycrystalline aggregates with a constrained surface microstructure–Part 2: Crystal plasticity. Philosophical Magazine, 87:1425–1446, 2007.CrossRefGoogle Scholar

Copyright information

© CISM, Udine 2014

Authors and Affiliations

  • Samuel Forest
    • 1
  • Kais Ammar
    • 1
  • Benoît Appolaire
    • 2
  • Nicolas Cordero
    • 1
  • Anaïs Gaubert
    • 3
  1. 1.Centre des matériaux, CNRS UMR 7633MINES ParisTechEvryFrance
  2. 2.Laboratoire d'étude des microstructuresCNRS/ONERAChâtillonFrance
  3. 3.Dept. of Metallic Materials and StructuresONERAChâtillonFrance

Personalised recommendations