Abstract
The need for predicting the behavior of crystalline materials on small-scales has led to the development of physically based descriptions of the motion of dislocations. Several dislocation-based continuum theories have been introduced, but only recently rigorous techniques have been developed for performing meaningful averages over systems of moving, curved dislocations, yielding evolution equations based on a dislocation density tensor. Those evolution equations provide a physically based framework for describing the motion of curved dislocations in three-dimensional systems. However, a meaningful description of internal interfaces and the complex mechanistic interaction of dislocations and interfaces in a dislocation based continuum model is still an open task. In this paper, we address the conflict between the need for a mechanistic modeling of the involved physical mechanisms and a reasonable reduction of model complexity and numerical effort. We apply a dislocation density based continuum formulation to systems which are strongly affected by internal interfaces, i.e. grain boundaries and interfaces in composite materials, and focus on the physical and numerical realization. Particularly, the interplay between physical and numerical accuracy is pointed out and discussed.
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References
H. Cleveringa, E. Van der Giessen, A. Needleman, A discrete dislocation analysis of bending. Int. J. Plast. 15, 837–868 (1999)
H.H.M. Cleveringa, E. VanderGiessen, A. Needleman, Comparison of discrete dislocation and continuum plasticity predictions for a composite material. Acta Mater. 45(8), 3163–3179 (1997)
L. Friedman, D. Chrzan, Continuum analysis of dislocation pile-ups: influence of sources. Phil. Mag. A 77(5), 1185–1204 (1998)
S. Groh, B. Devincre, L. Kubin, A. Roos, F. Feyel, J.L. Chaboche, Size effects in metal matrix composites. Mater. Sci. Eng. A 400, 279–282 (2005)
I. Groma, F. Csikor, M. Zaiser, Spatial correlations and higher-order gradient terms in a continuum description of dislocation dynamics. Acta Mater. 51, 1271–1281 (2003)
C. Hirschberger, R. Peerlings, W. Brekelmans, M. Geers, On the role of dislocation conservation in single-slip crystal plasticity. Model. Simul. Mater. Sci. Eng. 19(085002) (2011)
J. Hirth, J. Lothe, Theory of Dislocations (Wiley, New York, 1982)
T. Hochrainer, Thermodynamically consistent continuum dislocation dynamics. J. Mech. Phys. Solids 88, 12–22 (2016)
T. Hochrainer, S. Sandfeld, M. Zaiser, P. Gumbsch, Continuum dislocation dynamics: towards a physical theory of crystal plasticity. J. Mech. Phys. Solids 63, 167–178 (2014)
E. Kröner, Kontinuumstheorie der Versetzungen und Eigenspannungen (Springer, 1958)
E. Kröner, Benefits and shortcomings of the continuous theory of dislocations. Int. J. Solids Struct. 38, 1115–1134 (2001). https://doi.org/10.1016/S0020-7683(00)00077-9
D. Liu, Y. He, B. Zhang, Towards a further understanding of dislocation pileups in the presence of stress gradients. Phil. Mag. 1–23 (2013). https://doi.org/10.1080/14786435.2013.774096
J. Nye, Some geometrical relations in dislocated crystals. Acta Metall. 1, 153–162 (1953)
T. Richeton, G. Wang, C. Fressengeas, Continuity constraints at interfaces and their consequences on the work hardening of metal-matrix composites. J. Mech. Phys. Solids 59(10), 2023–2043 (2011)
Y. Saad, M.H. Schultz, Gmres: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7(3), 856–869 (1986)
S. Sandfeld, E. Thawinan, C. Wieners, A link between microstructure evolution and macroscopic response in elasto-plasticity: formulation and numerical approximation of the higher-dimensional continuum dislocation dynamics theory. Int. J. Plast. 72, 1–20 (2015)
S. Schmitt, P. Gumbsch, K. Schulz, Internal stresses in a homogenized representation of dislocation microstructures. J. Mech. Phys. Solids 84, 528–544 (2015)
K. Schulz, D. Dickel, S. Schmitt, S. Sandfeld, D. Weygand, P. Gumbsch, Analysis of dislocation pile-ups using a dislocation-based continuum theory. Model. Simul. Mater. Sci. Eng. 22(2), 025,008 (2014)
K. Schulz, S. Schmitt, Discrete-continuum transition: a discussion of the continuum limit. Tech. Mech. 38(1), 126–134 (2018)
K. Schulz, M. Sudmanns, P. Gumbsch, Dislocation-density based description of the deformation of a composite material. Model. Simul. Mater. Sci. Eng. 25(6), 064,003 (2017)
K. Schulz, L. Wagner, C. Wieners, A mesoscale approach for dislocation density motion using a runge-kutta discontinuous Galerkin method. PAMM 16(1), 403–404 (2016)
C. Schwarz, R. Sedláček, E. Werner, Plastic deformation of a composite and the source-shortening effect simulated by the continuum dislocation-based model. Model. Simul. Mater. Sci. Eng. 15, S37–S49 (2007)
M. Stricker, J. Gagel, S. Schmitt, K. Schulz, D. Weygand, P. Gumbsch, On slip transmission and grain boundary yielding. Meccanica 51(2), 271–278 (2016)
G.I. Taylor, The mechanism of plastic deformation of crystals. Part I. Theoretical. Proc. R. Soc. Lond. Ser. A, Containing Papers of a Mathematical and Physical Character 145(855), 362–387 (1934)
E. Van der Giessen, A. Needleman, Discrete dislocation plasticity: a simple planar model. Model. Simul. Mater. Sci. Eng. 3, 689–735 (1995). https://doi.org/10.1088/0965-0393/3/5/008
C. Wieners, A geometric data structure for parallel finite elements and the application to multigrid methods with block smoothing. Comput. Vis. Sci. 13(4), 161–175
C. Wieners, Distributed point objects. a new concept for parallel finite elements, in Domain Decomposition Methods in Science and Engineering (Springer, 2005), pp. 175–182
S. Yefimov, I. Groma, E. van der Giessen, A comparison of a statistical-mechanics based plasticity model with discrete dislocation plasticity calculations. J. Mech. and Phys. Solids 52(2), 279–300 (2004). https://doi.org/10.1016/S0022-5096(03)00094-2, http://www.sciencedirect.com/science/article/B6TXB-49JPKTK-1/2/5df57c08baa877d5ebfd601f33e50933
Acknowledgements
The Financial support for the research group FOR1650 Dislocation based Plasticity funded by the German Research Foundation (DFG) under the contract number GU367/36-2 as well as the support by the European Social Fund and the state of Baden-Württemberg is gratefully acknowledged. This work was performed on the computational resource ForHLR II funded by the Ministry of Science, Research and the Arts Baden-Württemberg and DFG (“Deutsche Forschungsgemeinschaft”).
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Schulz, K., Sudmanns, M. (2019). Mesoscale Simulation of Dislocation Microstructures at Internal Interfaces. In: Nagel, W., Kröner, D., Resch, M. (eds) High Performance Computing in Science and Engineering ' 18. Springer, Cham. https://doi.org/10.1007/978-3-030-13325-2_7
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DOI: https://doi.org/10.1007/978-3-030-13325-2_7
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