Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-27T11:05:23.325Z Has data issue: false hasContentIssue false
  • English
  • Français

Upscaling of dislocation walls in finite domains

Published online by Cambridge University Press:  28 August 2014

P. VAN MEURS ,
A. MUNTEAN  and
M. A. PELETIER
P. VAN MEURS
Affiliation:
Centre for Analysis, Scientific computing and Applications (CASA), Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands email: p.j.p.v.meurs@tue.nl
A. MUNTEAN
Affiliation:
Centre for Analysis, Scientific computing and Applications (CASA), Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands email: p.j.p.v.meurs@tue.nl Institute for Complex Molecular Systems (ICMS), Eindhoven University of Technology, The Netherlands
M. A. PELETIER
Affiliation:
Centre for Analysis, Scientific computing and Applications (CASA), Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands email: p.j.p.v.meurs@tue.nl Institute for Complex Molecular Systems (ICMS), Eindhoven University of Technology, The Netherlands
Get access Rights & Permissions [Opens in a new window]

Abstract

We wish to understand the macroscopic plastic behaviour of metals by upscaling the micro-mechanics of dislocations. We consider a highly simplified dislocation network, which allows our discrete model to be a one dimensional particle system, in which the interactions between the particles (dislocation walls) are singular and non-local. As a first step towards treating realistic geometries, we focus on finite-size effects rather than considering an infinite domain as typically discussed in the literature. We derive effective equations for the dislocation density by means of Γ-convergence on the space of probability measures. Our analysis yields a classification of macroscopic models, in which the size of the domain plays a key role.

Keywords

PlasticityMultiscaleStraight edge-dislocationsDiscrete-to-continuum limitΓ-convergence74Q05, 74C05, 82B21, 49J45, 82D35
Type
Papers
Information
European Journal of Applied Mathematics , Volume 25 , Issue 6 , December 2014 , pp. 749 - 781
Copyright
Copyright © Cambridge University Press 2014 

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

[1]Alicandro, R., De Luca, L., Garroni, A., & Ponsiglione, M. (2013) Metastability and dynamics of discrete topological singularities in two dimensions: a γ-convergence approach. In: Archive for Rational Mechanics and Analysis, pp. 1–62.Google Scholar
[2]Cacace, S. & Garroni, A. (2009) A multi-phase transition model for dislocations with interfacial microstructure. Interfaces Free Bound 11, 291–316.CrossRefGoogle Scholar
[3]Cai, W., Arsenlis, A., Weinberger, C. R. & Bulatov, V. V. (2006) A non-singular continuum theory of dislocations. J. Mech. Phys. Solids 54 (3), 561–587.CrossRefGoogle Scholar
[4]Callister, W. D. (2007) Materials Science and Engineering, An Introduction, New York, USA: John Wiley & Sons.Google Scholar
[5]Cermelli, P. & Leoni, G. (2006) Renormalized energy and forces on dislocations. SIAM J. Math. Anal. 37 (4), 1131–1160.Google Scholar
[6]Deng, J. & El-Azab, A. (2007) Dislocation pair correlations from dislocation dynamics simulations. J. Comput.-Aided Mater. Des. 14 (1), 295–307.Google Scholar
[7]Deng, J. & El-Azab, A. (2009) Mathematical and computational modelling of correlations in dislocation dynamics. Modelling Simul. Mater. Sci. Eng. 17, 075010.CrossRefGoogle Scholar
[8]Dogge, M. (To appear) Mechanics of Phase Boundaries, PhD thesis, Eindhoven University of Technology.Google Scholar
[9]Duong, M. H., Laschos, V. and Renger, M. (2013) Wasserstein gradient flows from large deviations of many-particle limits. ESAIM: Control, Optimisation Calculus Variations 19 (4), 1166–1188.Google Scholar
[10]Evers, L. P., Brekelmans, W. A. M. & Geers, M. G. D. (2004) Scale dependent crystal plasticity framework with dislocation density and grain boundary effects. Int. J. Solids Structures 41 (18–19), 5209–5230.CrossRefGoogle Scholar
[11]Focardi, M. & Garroni, A. (2007) A 1D macroscopic phase field model for dislocations and a second order Γ-limit. Multiscale Model. Simul. 6 (4), 1098–1124.Google Scholar
[12]Forcadel, N., Imbert, C. & Monneau, R. (2008) On the Notions of Solutions to Nonlinear Elliptic Problems: Results and Developments, chapter Viscosity solutions for particle systems and homogenization of dislocation dynamics. Department of Mathematics of the Seconda Universita di Napoli.Google Scholar
[13]Forcadel, N., Imbert, C. & Monneau, R. (2009) Homogenization of the dislocation dynamics and of some particle systems with two-body interactions. Discrete Continuous Dyn. Syst. A 23 (3), 785–826.Google Scholar
[14]Garroni, A., Leoni, G. & Ponsiglione, M. (2010) Gradient theory for plasticity via homogenization of discrete dislocations. J. Eur. Math. Soc. 12 (5), 1231–1266.Google Scholar
[15]Garroni, A. & Müller, S. (2005) Γ-limit of a phase-field model of dislocations. SIAM J. Math. Anal. 36 (6), 1943–1964.Google Scholar
[16]Geers, M. G. D., Peerlings, R. H. J., Peletier, M. A., & Scardia, L. (2013) Asymptotic behaviour of a pile-up of infinite walls of edge dislocations. Arch. Ration. Mech. Anal. 209, 495–539.Google Scholar
[17]Groma, I. (1997) Link between the microscopic and mesoscopic length-scale description of the collective behavior of dislocations. Phys. Rev. B 56 (10), 5807–5813.Google Scholar
[18]Groma, I., Csikor, F. F. & Zaiser, M. (2003) Spatial correlations and higher-order gradient terms in a continuum description of dislocation dynamics. Acta Mater. 51 (5), 1271–1281.CrossRefGoogle Scholar
[19]El Hajj, A., Ibrahim, H. & Monneau, R. (2009) Homogenization of dislocation dynamics. In IOP Conferences Series: Materials Science and Engineering.Google Scholar
[20]Hall, C. L. (2010) Asymptotic expressions for the nearest and furthest dislocations in a pile-up against a grain boundary. Phil. Mag. 90 (29), 3879–3890.Google Scholar
[21]Hall, C. L. (2011) Asymptotic analysis of a pile-up of regular edge dislocation walls. Mater. Sci. Eng.: A 530, 144–148.Google Scholar
[22]Hall, C. L., Chapman, S. J. & Ockendon, J. R. (2010) Asymptotic analysis of a system of algebraic equations arising in dislocation theory. SIAM J. Appl. Math. 70 (7), 2729–2749.Google Scholar
[23]Hirth, J. P. & Lothe, J. (1982) Theory of Dislocations, John Wiley & Sons.Google Scholar
[24]Hudson, T. & Ortner, C. (2014) Existence and stability of screw dislocation configurations with arbitrary net burgers vector. arXiv: 1403.0518.Google Scholar
[25]Hull, D. & Bacon, D. J. (2001) Introduction to Dislocations, Butterworth Heinemann, Oxford.Google Scholar
[26]Koslowski, M. & Ortiz, M. (2004) A multi-phase field model of planar dislocation networks. Modelling Simul. Mater. Sci. Eng. 12 (6), 1087.CrossRefGoogle Scholar
[27]Limkumnerd, S. & Van der Giessen, E. (2008) Statistical approach to dislocation dynamics: From dislocation correlations to a multiple-slip continuum theory of plasticity. Phys. Rev. B 77 (18), 184111.Google Scholar
[28]Dal Maso, G. (1993) An Introduction to Γ-Convergence, Boston, USA: Birkhäuser Boston.CrossRefGoogle Scholar
[29]Ponsiglione, M. (2007) Elastic energy stored in a crystal induced by screw dislocations: From discrete to continuous. SIAM J. Math. Anal. 39 (2), 449–469.Google Scholar
[30]Portegies, J. (2013) Non-equidistant Dislocation Walls, Technical report, Eindhoven University of Technology. To appear.Google Scholar
[31]Roy, A., Peerlings, R. H. J., Geers, M. G. D. & Kasyanyuk, Y. (2008) Continuum modeling of dislocation interactions: Why discreteness matters? Mater. Sci. Eng.: A, 486, 653–661.CrossRefGoogle Scholar
[32]Scardia, L., Peerlings, R. H. J., Peletier, M. A. & Geers, M. G. D. (2014) Mechanics of dislocation pile-ups: a unification of scaling regimes. J. Mech. Phys. Solids 70, 42–61.Google Scholar
[33]Serfaty, S. (2011) Gamma-convergence of gradient flows on hilbert and metric spaces and applications. Discrete Continuous Dyn. Syst. A 31, 1427–1451.Google Scholar
[34]Voskoboinikov, R. E., Chapman, S. J., McLeod, J. B. & Ockendon, J. R. (2009) Asymptotics of edge dislocation pile-up against a bimetallic interface. Math. Mech. Solids 14, 284–295.CrossRefGoogle Scholar
[35]Yefimov, S., Groma, I. & Van der Giessen, E. (2004) A comparison of a statistical-mechanics based plasticity model with discrete dislocation plasticity calculations. J. Mech. Phys. Solids 52 (2), 279–300.Google Scholar
[36]Zaiser, M. & Groma, I. (2011) Some limitations of dislocation walls as models for plastic boundary layers. arXiv: 1109.2216.Google Scholar
[37]Zaiser, M., Miguel, M. C. & Groma, I. (2001) Statistical dynamics of dislocation systems: The influence of dislocation-dislocation correlations. Phys. Rev. B 64 (22), 224102.Google Scholar