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.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
Keywords
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.
Bibliography
Guillaume Abrivard. A coupled crystal plasticity–phase field formulation to describe microstructural evolution in polycrystalline aggregates. PhD, Mines ParisTech, 2009.
E.C. Aifantis. On the microstructural origin of certain inelastic models. Journal of Engineering Materials and Technology, 106:326–330, 1984.
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.
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.
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.
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.
R.J. Asaro. Crystal plasticity. J. Appl. Mech., 50:921–934, 1983.
O. Aslan and S. Forest. Crack growth modelling in single crystals based on higher order continua. Computational Materials Science, 45:756–761, 2009.
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.
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.
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.
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.
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.
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.
Y. Chen and J.D. Lee. Connecting molecular dynamics to micromorphic theory. (I) instantaneous and averaged mechanical variables. Physica A, 322:359–376, 2003a.
Y. Chen and J.D. Lee. Connecting molecular dynamics to micromorphic theory. (II) balance laws. Physica A, 322:377–392, 2003b.
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.
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.
W.D. Claus and A.C. Eringen. Dislocation dispersion of elastic waves. International Journal of Engineering Science, 9:605–610, 1971.
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.
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.
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.
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.
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.
A.C. Eringen and E.S. Suhubi. Nonlinear theory of simple microelastic solids. Int. J. Engng Sci., 2:189–203, 389–404, 1964.
Y. Estrin. Dislocation density related constitutive modelling. In Unified Constitutive Laws of Plastic Deformation, pages 69–106. Academic Press, 1996.
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.
M. Fivel and S. Forest. Plasticité cristalline et transition d’échelle : cas du monocristal. Techniques de l’Ingénieur, M4016, 23 pages, 2004a.
M. Fivel and S. Forest. Plasticité cristalline et transition d’échelle : cas du polycristal. Techniques de l’Ingénieur, M4017, 11 pages, 2004b.
S. Forest. The micromorphic approach for gradient elasticity, viscoplasticity and damage. ASCE Journal of Engineering Mechanics, 135:117–131, 2009.
S. Forest. Some links between cosserat, strain gradient crystal plasticity and the statistical theory of dislocations. Philosophical Magazine, 88: 3549–3563, 2008.
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.
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.
S. Forest and R. Sievert. Elastoviscoplastic constitutive frameworks for generalized continua. Acta Mechanica, 160:71–111, 2003.
S. Forest and R. Sievert. Nonlinear microstrain theories. International Journal of Solids and Structures, 43:7224–7245, 2006.
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.
S. Forest, F. Pradel, and K. Sab. Asymptotic analysis of heterogeneous Cosserat media. International Journal of Solids and Structures, 38:4585–4608, 2001.
M. Frémond and B. Nedjar. Damage, gradient of damage and principle of virtual power. Int. J. Solids Structures, 33:1083–1103, 1996.
E. Fried and M.E. Gurtin. Continuum theory of thermally induced phase transitions based on an order parameter. Physica D, 68:326–343, 1993.
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.
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.
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.
P. Germain. The method of virtual power in continuum mechanics. part 2 : Microstructure. SIAM J. Appl. Math., 25:556–575, 1973b.
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.
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.
W. Günther. Zur Statik und Kinematik des Cosseratschen Kontinuums. Abhandlungen der Braunschweig. Wiss. Ges., 10:195–213, 1958.
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.
M.E. Gurtin. Generalized Ginzburg–Landau and Cahn–Hilliard equations based on a microforce balance. Physica D, 92:178–192, 1996.
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.
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.
C.B. Hirschberger and P. Steinmann. Classification of concepts in thermodynamically consistent generalized plasticity. ASCE Journal of Engineering Mechanics, 135:156–170, 2009.
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.
A.G. Khachaturyan. Theory of structural transformations in solids. John Wiley & Sons, 1983.
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.
S.G. Kim, W.T. Kim, and T Suzuki. Phase–field model for binary alloys. Physical Review E, 60(6):7186–7197, 1999.
E. Kröner. On the physical reality of torque stresses in continuum mechanics. Int. J. Engng. Sci., 1:261–278, 1963.
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.
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.
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.
M. Lazar and G.A. Maugin. On microcontinuum field theories: the eshelby stress tensor and incompatibility conditions. Philosophical Magazine, 87: 3853–3870, 2007.
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.
J.D. Lee and Y. Chen. Constitutive relations of micromorphic thermoplasticity. International Journal of Engineering Science, 41:387–399, 2003.
J. Mandel. Une généralisation de la théorie de la plasticité de W.T. Koiter. International Journal of Solids and Structures, 1:273–295, 1965.
J. Mandel. Plasticité classique et viscoplasticité. CISM Courses and Lectures No. 97, Udine, Springer Verlag, Berlin, 1971.
J. Mandel. Equations constitutives et directeurs dans les milieux plastiques et viscoplastiques. Int. J. Solids Structures, 9:725–740, 1973.
G.A. Maugin. The method of virtual power in continuum mechanics: Application to coupled fields. Acta Mechanica, 35:1–70, 1980.
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.
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.
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.
C. Miehe. A multifield incremental variational framework for gradienttype standard dissipative solids, in press. Journal of the Mechanics and Physics of Solids, 2011.
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.
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.
R.D. Mindlin. Micro–structure in linear elasticity. Arch. Rat. Mech. Anal., 16:51–78, 1964.
A. I. Murdoch. A thermodynamical theory of elastic material interfaces. Q. J. Mech. appl. Math., 29:245–275, 1978.
J.F. Nye. Some geometrical relations in dislocated crystals. Acta Metall., 1:153–162, 1953.
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.
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.
R.A. Regueiro. On finite strain micromorphic elastoplasticity. International Journal of Solids and Structures, 47:786–800, 2010.
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.
Schäfer,H. Eine Feldtheorie der Versetzungen im Cosserat-Kontinuum. ZAMP, 20:891–899, 1969.
I. Steinbach and M. Apel. Multi phase field model for solid state transformation with elastic strain. Physica D, 217:153–160, 2006.
P. Steinmann. Views on multiplicative elastoplasticity and the continuum theory of dislocations. International Journal of Engineering Science, 34: 1717–1735, 1996.
B. Svendsen. Continuum thermodynamic models for crystal plasticity including the effects of geometrically-necessary dislocations. J. Mech. Phys. Solids, 50:1297–1329, 2002.
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.
C. Teodosiu and F. Sidoroff. A theory of finite elastoviscoplasticity of single crystals. Int. J. of Engng Science, 14:165–176, 1976.
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.
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.
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.
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.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 CISM, Udine
About this chapter
Cite this chapter
Forest, S., Ammar, K., Appolaire, B., Cordero, N., Gaubert, A. (2014). Micromorphic approach to crystal plasticity and phase transformation. In: Schröder, J., Hackl, K. (eds) Plasticity and Beyond. CISM International Centre for Mechanical Sciences, vol 550. Springer, Vienna. https://doi.org/10.1007/978-3-7091-1625-8_3
Download citation
DOI: https://doi.org/10.1007/978-3-7091-1625-8_3
Publisher Name: Springer, Vienna
Print ISBN: 978-3-7091-1624-1
Online ISBN: 978-3-7091-1625-8
eBook Packages: EngineeringEngineering (R0)