# Modelling and Numerical Studies of Discrete Dislocation Dynamics

## Main Article Content

## Abstract

We investigate a possible inaccuracy in discrete dislocation dynamics (DDD) simulations. As a model problem we consider two distinct dislocations of the opposite signs, gliding and bowing out in parallel slip planes in a channel of persistent slip band (PSB). Dislocations are pushed by the applied stress and when overlap, they either pass or form a dipole. The objective of our study is to determine the lower and upper estimate of the passing stress needed to escape each other. In our simulations, we consider two loading regimes - the stress controlled and the total strain controlled regime. The motion law is described by the mean curvature flow of planar curves and treated by the parametric method. Results of our numerical experiments indicate that the upper and lower estimate of the passing stress differ less than 10%.

## Article Details

How to Cite

KOLÁŘ, Miroslav; BENEŠ, Michal; KRATOCHVÍL, Jan.
Modelling and Numerical Studies of Discrete Dislocation Dynamics.

**Proceedings of the Conference Algoritmy**, [S.l.], p. 302-311, feb. 2016. Available at: <http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/algoritmy/article/view/419>. Date accessed: 22 sep. 2017.
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## References

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[8] J. Křišťan, st J. Kratochvíl, V. Minárik, M. Beneš, Numerical simulation of interacting dislocations glide in a channel of a persistent slip band, Modelling and Simulation in Materials Science and Engineering, 17 045009 (2009).

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[11] M. Beneš, J. Kratochv ́ıl, J. Křišťan, st V. Minárik and P. Pauš, A parametric simulation method for discrete dislocation dynamics, European Physical Journal ST, 177 (2009), 177– 192.

[12] P. Pauš and M. Beneš, Direct Approach to Mean-Curvature Flow with Topological Changes, Kybernetika, 45 (2009), pp. 591–604.

[13] P. Pauš, J. Kratochvíl and M. Beneš, A dislocation dynamics analysis of the critical crossslip annihilation distance and the cyclic saturation stress in fcc single crystals at different temperatures, Acta Materialia, Vol. 61 (2013), Issue 20, pp. 7917-7923.

[14] M. Kolář, M. Beneš, D. Ševčovič and J. Kratochvíl, Mathematical Model and Computational Studies of Discrete Dislocation Dynamics, IAENG International Journal of Applied Mathematics, 45 (2015), no. 3, pp. 198–207.

[15] J. Kratochvíl and R. Sedláček, Statistical foundation of continuum dislocation plasticity, Physical Review B, 77 (2008), p. 134102.

[16] K. Deckelnick, Parametric mean curvature evolution with a dirichlet boundary condition, Journal f ̈ ur die reine und angewandte Mathematik, 459 (1995), 37–60.

[17] G. Dziuk, K. Deckelnick and C. M. Elliott, Computation of geometric partial differential equations and mean curvature flow, Acta Numerica, 14 (2005), 139–232.

[18] D. ševčovič and S. Yazaki, Evolution of plane curves with a curvature adjusted tangential velocity, Japan Journal of Industrial and Applied Mathematics, 28 (2011), 413–442.

[19] B. Devincre, Three dimensional stress field expression for straight dislocation segment, JSolid State Communications, 93 (1995), p. 875

[20] M. Peach and J. S. Koehler, The forces exerted on dislocations and the stress fields produced by them, Physical Review, (1950).

[21] M. Kolář, M. Beneš, J. Kratochvíl and P. Pauš, Numerical Simulations of Glide Dislocations in Persistent Slip Band, Acta Physica Polonica A, 128 (2015), no. 3, 506–509.

[22] V. Minárik, M. Beneš and J. Kratochvíl, Simulation of dynamical interaction between dislocations and dipolar loops, Journal of Applied Physics, 77 (2008), 177–192.

[23] K. Mikula and D. Ševčovič, Computational and qualitative aspects of evolution of curves driven by curvature and external force, Computing and Visualization in Science, 6 (2004), 211–225.

[24] H. Mughrabi and F. Pschenitzka, Constrained glide and interaction of bowed-out screw dislocations in confined channels, Philosophical Magazine 85 (2005), 3029.

[2] T. Mura, Micromechanics of Defects in Solids, Kluwer Academic Publishers Group, Netherlands, 1987.

[3] L. P. Kubin, Dislocations, Mesoscale Simulations and Plastic Flow, Oxford University Press, 2013.

[4] V. V. Bulatov and w. Cai, Computer Simulations of Dislocations, Oxford University Press, 2006.

[5] L. P. Kubin, The modelling of dislocation patterns, Scripta Metallurgica et Materialia, 27 (1992), pp. 957–962.

[6] B. Devincre and L. P. Kubin, Mesoscopic simulations of dislocations and plasticicity, Materials Science and Engineering, A234-236:8 (1997), pp. 8–14.

[7] A. Vattré́, B. Devincre, F. Feyel, R. Gatti, S. Groh, O. Jamond and A. Roos, Modelling crystal plasticity by 3D dislocation dynamics and the finite element method: The discretecontinuous model revisited, Journal of the Mechanics and Physics of Solids, 63 (2014), pp. 491–505.

[8] J. Křišťan, st J. Kratochvíl, V. Minárik, M. Beneš, Numerical simulation of interacting dislocations glide in a channel of a persistent slip band, Modelling and Simulation in Materials Science and Engineering, 17 045009 (2009).

[9] N. M. Ghoniem and L. Z. Sun, Fast sum method for the elastic field of 3-D dislocation ensembles, Physical Review B, 60:1 (1999).

[10] J. Huang, N. M. Ghoniem and J. Kratochvíl, On the sweeping mechanism of dipolar dislocation loops under fatigue conditions, Modelling and Simulation in Materials Science and Engineering, 12 (2004), pp. 917–928.

[11] M. Beneš, J. Kratochv ́ıl, J. Křišťan, st V. Minárik and P. Pauš, A parametric simulation method for discrete dislocation dynamics, European Physical Journal ST, 177 (2009), 177– 192.

[12] P. Pauš and M. Beneš, Direct Approach to Mean-Curvature Flow with Topological Changes, Kybernetika, 45 (2009), pp. 591–604.

[13] P. Pauš, J. Kratochvíl and M. Beneš, A dislocation dynamics analysis of the critical crossslip annihilation distance and the cyclic saturation stress in fcc single crystals at different temperatures, Acta Materialia, Vol. 61 (2013), Issue 20, pp. 7917-7923.

[14] M. Kolář, M. Beneš, D. Ševčovič and J. Kratochvíl, Mathematical Model and Computational Studies of Discrete Dislocation Dynamics, IAENG International Journal of Applied Mathematics, 45 (2015), no. 3, pp. 198–207.

[15] J. Kratochvíl and R. Sedláček, Statistical foundation of continuum dislocation plasticity, Physical Review B, 77 (2008), p. 134102.

[16] K. Deckelnick, Parametric mean curvature evolution with a dirichlet boundary condition, Journal f ̈ ur die reine und angewandte Mathematik, 459 (1995), 37–60.

[17] G. Dziuk, K. Deckelnick and C. M. Elliott, Computation of geometric partial differential equations and mean curvature flow, Acta Numerica, 14 (2005), 139–232.

[18] D. ševčovič and S. Yazaki, Evolution of plane curves with a curvature adjusted tangential velocity, Japan Journal of Industrial and Applied Mathematics, 28 (2011), 413–442.

[19] B. Devincre, Three dimensional stress field expression for straight dislocation segment, JSolid State Communications, 93 (1995), p. 875

[20] M. Peach and J. S. Koehler, The forces exerted on dislocations and the stress fields produced by them, Physical Review, (1950).

[21] M. Kolář, M. Beneš, J. Kratochvíl and P. Pauš, Numerical Simulations of Glide Dislocations in Persistent Slip Band, Acta Physica Polonica A, 128 (2015), no. 3, 506–509.

[22] V. Minárik, M. Beneš and J. Kratochvíl, Simulation of dynamical interaction between dislocations and dipolar loops, Journal of Applied Physics, 77 (2008), 177–192.

[23] K. Mikula and D. Ševčovič, Computational and qualitative aspects of evolution of curves driven by curvature and external force, Computing and Visualization in Science, 6 (2004), 211–225.

[24] H. Mughrabi and F. Pschenitzka, Constrained glide and interaction of bowed-out screw dislocations in confined channels, Philosophical Magazine 85 (2005), 3029.