Hostname: page-component-745bb68f8f-b95js Total loading time: 0 Render date: 2025-01-12T09:21:24.278Z Has data issue: false hasContentIssue false
  • English
  • Français

Continuum modeling of dislocation plasticity: Theory, numerical implementation, and validation by discrete dislocation simulations

Published online by Cambridge University Press:  18 March 2011

Stefan Sandfeld ,
Thomas Hochrainer ,
Michael Zaiser  and
Peter Gumbsch
Stefan Sandfeld*
Affiliation:
Karlsruher Institut für Technologie, IZBS—Institut für Zuverlässigkeit von Bauteilen und Systemen, 76131 Karlsruhe, Germany
Thomas Hochrainer
Affiliation:
Department of Scientific Computing, Florida State University, Tallahassee, Florida 32310
Michael Zaiser
Affiliation:
The University of Edinburgh, Center for Materials Science and Engineering, Edinburgh EH93JL, United Kingdom
Peter Gumbsch
Affiliation:
Karlsruher Institut für Technologie, IZBS—Institut für Zuverlässigkeit von Bauteilen und Systemen, 76131 Karlsruhe, Germany; and Fraunhofer IWM, 79108 Freiburg, Germany
*
a)Address all correspondence to this author. e-mail: Stefan.sandfeld@kit.edu
Get access

Abstract

Miniaturization of components and devices calls for an increased effort on physically motivated continuum theories, which can predict size-dependent plasticity by accounting for length scales associated with the dislocation microstructure. An important recent development has been the formulation of a Continuum Dislocation Dynamics theory (CDD) that provides a kinematically consistent continuum description of the dynamics of curved dislocation systems [T. Hochrainer, et al., Philos. Mag.87, 1261 (2007)]. In this work, we present a brief overview of dislocation-based continuum plasticity models. We illustrate the implementation of CDD by a numerical example, bending of a thin film, and compare with results obtained by three-dimensional discrete dislocation dynamics (DDD) simulation.

Keywords

CrystalDislocationsDefects
Type
Invited Feature Paper
Information
Journal of Materials Research , Volume 26 , Issue 5 , 14 March 2011 , pp. 623 - 632
Copyright
Copyright © Materials Research Society 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

1.Acharya, A.: A model of crystal plasticity based on the theory of continuously distributed dislocation. J. Mech. Phys. Solids 49(4), 761 (2001).CrossRefGoogle Scholar
2.Acharya, A.: Driving forces and boundary conditions in continuum dislocation mechanics. Proc. R. Soc. London, Ser. A 459(2034), 1343 (2003).CrossRefGoogle Scholar
3.Aifantis, E.C.: The physics of plastic deformation. Int. J. Plast. 3, 211 (1987).CrossRefGoogle Scholar
4.Aifantis, K.E., Senger, J., Weygand, D., and Zaiser, M.: Discrete dislocation dynamics simulation and continuum modeling of plastic boundary layers in tricrystal micropillars. IOP Conf. Ser.: Mat. Sci. Eng. 3, 012025 (2009).CrossRefGoogle Scholar
5.Arsenlis, A., Cai, W., Tang, M., Rhee, M., Oppelstrup, T., Hommes, G., Pierce, T.G., and Bulatov, V.V.: Enabling strain hardening simulations with dislocation dynamics. Modell. Simul. Mater. Sci. Eng. 15, 553 (2007).CrossRefGoogle Scholar
6.Arsenslis, A., Parks, D., Becker, R., and Bulatov, V.: On the evolution of crystallographic dislocation density in non-homogeneously deforming crystals. J. Mech. Phys. Solids 52, 1213 (2004).CrossRefGoogle Scholar
7.Arzt, E., Dehm, G., Gumbsch, P., Kraft, O., and Weiss, D.: Interface controlled plasticity in metals: Dispersion hardening and thin film deformation. Prog. Mater. Sci. 46(3–4), 283 (2001).CrossRefGoogle Scholar
8.Bilby, B.A., Bullough, R., and Smith, E.: Continuous distributions of dislocations: A new application of the methods of non-Riemannian geometry. Proc. R. Soc. London, Ser. A 231, 263 (1955).Google Scholar
9.Bittencourt, E., Needleman, A., Gurtin, M.E., and van der Giessen, E.: A comparison of nonlocal continuum and discrete dislocation plasticity predictions. J. Mech. Phys. Solids 51, 281 (2003).CrossRefGoogle Scholar
10.Bulatov, V.V. and Cai, W.: Nodal effects in dislocation mobility. Phys. Rev. Lett. 89(11), 115501 (2002).CrossRefGoogle ScholarPubMed
11.Cosserat, E. and Cosserat, F.: Theorie des corps deformables (Libraire scientifique, A. Hermann & Fils, Paris, France, 1909).Google Scholar
12.Devincre, B., Kubin, L., Lemarchand, C., and Madec, R.: Mesoscopic simulations of plastic deformation. Mater. Sci. Eng., A 309–310, 211 (2001).CrossRefGoogle Scholar
13.El-Azab, A.: Statistical mechanics treatment of the evolution of dislocation distributions in single crystals. Phys. Rev. B 61(18), 11,956 (2000).CrossRefGoogle Scholar
14.Evans, A.G. and Hutchinson, J.W.: A critical assessment of theories of strain gradient plasticity. Acta Mater. 57(5), 1675 (2009).CrossRefGoogle Scholar
15.Fivel, M., Verdier, M., and Ganova, G.: 3D simulation of a nanoindentation test at a mesoscopic scale. Mater. Sci. Eng., A 234–236, 923 (1997).CrossRefGoogle Scholar
16.Fleck, N.A. and Hutchinson, J.W.: A phenomenological theory for strain gradient effects in plasticity. J. Mech. Phys. Solids 41(12), 1825 (1993).CrossRefGoogle Scholar
17.Forest, S.: Generalized continuum modeling of single and polycrystal plasticity, Continuum Scale Simulation of Engineering Materials, edited by Raabe, D., Roters, F., Barlat, F., and Chen, L. (Wiley-VCH Verlag GmbH & Co., KGaA, Weinhem, Germany, 2005).Google Scholar
18.Ghoniem, N.M. and Sun, L.Z.: Fast-sum method for the elastic field of three-dimensional dislocation ensembles. Phys. Rev. B 60(1), 128 (1999).CrossRefGoogle Scholar
19.Groma, I.: Link between the microscopic and mesoscopic length-scale description of the collective behavior of dislocations. Phys. Rev. B 56, 5807 (1997).CrossRefGoogle Scholar
20.Groma, I., Csikor, F.F., and Zaiser, M.: Spatial correlations and higher-order gradient terms in a continuum description of dislocation dynamics. Acta Mater. 51, 1271 (2003).CrossRefGoogle Scholar
21.Gurtin, M.E.: A gradient theory of single-crystal viscoplasticity that accounts for geometrically necessary dislocations. J. Mech. Phys. Solids 50(1), 5 (2002).CrossRefGoogle Scholar
22.Gurtin, M.E.: A theory of grain boundaries that accounts automatically for grain misorientation and grain-boundary orientation. J. Mech. Phys. Solids 56(2), 640 (2008).CrossRefGoogle Scholar
23.Hochrainer, T.: Evolving systems of curved dislocations: Mathematical foundations of a statistical theory. Ph.D. Thesis, Universität Karlsruhe (TH), Shaker Verlag, Aachen, Germany, 2006.Google Scholar
24.Hochrainer, T., Gumbsch, P., and Zaiser, M.: A non-linear multiple slip theory in continuum dislocation dynamics, in Proc. of the 4th Int. Conf. on Multiscale Materials Modeling, pp. 115–118 (2008).Google Scholar
25.Hochrainer, T., Zaiser, M., and Gumbsch, P.: A three-dimensional continuum theory of dislocations: Kinematics and mean field formulation. Philos. Mag. 87, 1261 (2007).CrossRefGoogle Scholar
26.Hochrainer, T., Zaiser, M., and Gumbsch, P.: Dislocation transport and line length increase in averaged descriptions of dislocations. AIP Conf. Proc. 1168(1), 1133 (2009).CrossRefGoogle Scholar
27.Kondo, K.: On the geometrical and physical foundations of the theory of yielding, in Proc. 2nd Japan Nat. Congress of Appl. Mech., pp. 41–47 (1952).Google Scholar
28.Kosevich, A.: Crystal dislocations and the theory of elasticity, Dislocations in Solids, Vol. 1: The elastic theory, edited by Nabarro, F.R.N., (North-Holland, Amsterdam, 1979).Google Scholar
29.Kratochvíl, J. and Sedláček, R.: Statistical foundation of continuum dislocation plasticity. Phys. Rev., B 77, 134102/1 (2008).CrossRefGoogle Scholar
30.Kröner, E.: Kontinuumstheorie der Versetzungen und Eigenspannungen (Springer-Verlag, Berlin, 1958).CrossRefGoogle Scholar
31.Kröner, E.: Benefits and shortcomings of the continuous theory of dislocations. Int. J. Solids Struct. 38(6–7), 1115 (2001).CrossRefGoogle Scholar
32.Kubin, L. and Canova, G.: The modeling of dislocation patterns. Scr. Metall. Mater. 27(8), 957 (1992).CrossRefGoogle Scholar
33.Mecking, H. and Kocks, U.: Kinetics of flow and strain-hardening. Acta Metall. 29(11), 1865 (1981).CrossRefGoogle Scholar
34.Motz, C., Weygand, D., Senger, J., and Gumbsch, P.: Micro-bending tests: A comparison between three-dimensional discrete dislocation dynamics simulations and experiments. Acta Mater. 56, 1942 (2008).CrossRefGoogle Scholar
35.Motz, C., Weygand, D., Senger, J., and Gumbsch, P.: Initial dislocation structures in 3-D discrete dislocation dynamics and their influence on microscale plasticity. Acta Mater. 57(6), 1744 (2009).CrossRefGoogle Scholar
36.Nix, W.D. and Gao, H.: Indentation size effects in crystalline materials: A law for strain gradient plasticity. J. Mech. Phys. Solids 46(3), 411 (1998).CrossRefGoogle Scholar
37.Nye, J.F.: Some geometrical relations in dislocated crystals. Acta Metall. 1, 153 (1953).CrossRefGoogle Scholar
38.Mindlin, R.D.: Microstructure in linear elasticity. Arch. Ration. Mech. Anal. 16, 51 (1964).CrossRefGoogle Scholar
39.Reese, S. and Svendsen, B.: Continuum thermodynamic modeling and simulation of additional hardening due to deformation incompatibility, Kluwer Series on Solid Mechanics and Its Application, Vol. 108, edited by Miehe, C., (Dordrecht, The Netherlands, 2003), pp. 141–150.Google Scholar
40.Roters, F. and Raabe, D.: A dislocation density based constitutive model for crystal plasticity FEM including geometrically necessary dislocations. Acta Mater. 54, 2169 (2006).Google Scholar
41.Sandfeld, S.: The evolution of dislocation density in a higher-order continuum theory of dislocation plasticity. Ph.D. Thesis, Shaker Verlag, Aachen, Germany, 2010.Google Scholar
42.Sandfeld, S., Hochrainer, T., Zaiser, M., and Gumbsch, P.: Numerical implementation of a 3D continuum theory of dislocation dynamics and application to microbending. Philos. Mag. 90(27–28), 3697–3728 (2010).CrossRefGoogle Scholar
43.Sandfeld, S., Hochrainer, T., and Zaiser, M.: Application of a 3D-continuum theory of dislocations to problems of constrained plastic flow: Microbending of a thin film, in Mechanical Behavior at Small Scales—Experiments and Modeling, edited by Lou, J., Lilleodden, E., Boyce, B.L., Lu, L., Derlet, P.M., Weygand, D., Li, J., Uchic, M., and Le Bourhis, E., (Mater. Res. Soc. Symp. Proc., 1224, Warrendale, PA, 2010), p. 143.Google Scholar
44.Sandfeld, S., Zaiser, M., and Hochrainer, T.: Expansion of quasi-discrete dislocation loops in the context of a 3D continuum theory of curved dislocations. AIP Conf. Proc. 1168(1), 1148 (2009).CrossRefGoogle Scholar
45.Sedláček, R., Kratochvíl, J., and Werner, E.: The importance of being curved: Bowing dislocations in a continuum description. Philos. Mag. 83(31–34), 3735 (2003).CrossRefGoogle Scholar
46.Senger, J., Weygand, D., Gumbsch, P., and Kraft, O.: Discrete dislocation simulations of the plasticity of micro-pillars under uniaxial loading. Scr. Mater. 58(7), 587 (2008).CrossRefGoogle Scholar
47.Weygand, D., Friedman, L.H., van der Giessen, E., and Needleman, A.: Aspects of boundary-value problem solutions with three-dimensional dislocation dynamics. Modell. Simul. Mater. Sci. Eng. 10, 437 (2002).CrossRefGoogle Scholar
48.Weygand, D. and Gumbsch, P.: Study of dislocation reactions and rearrangements under different loading conditions. Mater. Sci. Eng., A 400–401, 158 (2005).CrossRefGoogle Scholar
49.Weygand, D., Senger, J., Motz, C., Augustin, W., Heuveline, V., and Gumbsch, P.: High performance computing and discrete dislocation dynamics: Plasticity of micrometer sized specimens, High Performance Computing in Science and Engineering ’08, edited by Nagel, W.E. (Springer, Berlin, Germany 2009), pp. 507–523.CrossRefGoogle Scholar
50.Yefimov, S. and van der Giessen, E.: Size effects in single crystal thin films: Nonlocal crystal plasticity simulations. Eur. J. Mech. A Solids 24(2), 183 (2005).CrossRefGoogle Scholar
51.Zaiser, M., Nikitas, N., Hochrainer, T., and Aifantis, E.: Modeling size effects using 3D density- based dislocation dynamics. Philos. Mag. 87, 1283 (2007).CrossRefGoogle Scholar