On a continuum theory of dislocation equilibrium. (English) Zbl 1423.74147
Summary: A continuum theory of dislocations is suggested which is capable of predicting the equilibrium distributions of a large number of screw dislocations in anisotropic beams with arbitrary cross-section. The theory leads to a boundary value problem with unknown boundary. The problem is solved for isotropic beams with circular cross-sections and thin rectangular cross-sections. The solution for circular cross-sections is compared with the results of numerical simulations by C. R. Weinberger [Int. J. Plast. 27, No. 9, 1391–1408 (2011; Zbl 1452.74006)]. An extension of the theory to equilibrium of edge dislocations is discussed.
MSC:
| 74C05 | Small-strain, rate-independent theories of plasticity (including rigid-plastic and elasto-plastic materials) |
| 74K10 | Rods (beams, columns, shafts, arches, rings, etc.) |
| 74A60 | Micromechanical theories |
| 82B21 | Continuum models (systems of particles, etc.) arising in equilibrium statistical mechanics |
Keywords:
continuum theory of dislocations; dislocation equilibrium; plastic deformation of twisted beamsCitations:
Zbl 1452.74006References:
| [1] | Bardella, L.; Panteghini, A., Modelling the torsion of thin metal wires by distortion gradient plasticity, Journal of the Mechanics and Physics of Solids, 78, 467-492, (2015) · Zbl 1349.74072 |
| [2] | Berdichevsky, V. L., On the energy of an elastic beam, Journal of Applied Mathematics and Mechanics (PMM), 45, 4, 518-529, (1981) · Zbl 0519.73043 |
| [3] | Berdichevsky, V. L., Thermodynamics of Chaos and Order, (1997), Addison Wesley Harlow: Longman · Zbl 0892.70001 |
| [4] | Berdichevsky, V. L., Homogenization in micro-plasticity, Journal of the Mechanics and Physics of Solids, 53, 2459-2469, (2005) · Zbl 1176.74146 |
| [5] | Berdichevsky, V. L., Continuum theory of dislocation revisited, Continuum Mechanics and Thermodynamics, 18, 195-222, (2006) · Zbl 1160.74309 |
| [6] | Berdichevsky, V. L., On thermodynamics of crystal plasticity, Scripta Materialia, 54, 711-716, (2006) |
| [7] | Berdichevsky, V. L., Variational Principles of Continuum Mechanics, (2009), Springer-Verlag · Zbl 1183.49002 |
| [8] | Berdichevsky, V. L., Energy of dislocation networks, International Journal of Engineering Science, 103,, 35-44, (2016) · Zbl 1423.74146 |
| [9] | El-Azab, A., Statistical mechanics treatment of the evolution of dislocation distributions in single crystals, Physical Review B, 61, 11956, (2000) |
| [10] | Eshelby, J. D., Screw dislocations in thin rods, Journal of Applied Physics, 24, 176-179, (1953) |
| [11] | Gao, H.; Huang, Y.; Hutchinson, J. W., Mechanism-based strain gradient plasticity - 1, Journal of the Mechanics and Physics of Solids, 47, 1239-1263, (1999) · Zbl 0982.74013 |
| [12] | Geers, M. G.D.; Peerlings, R. H.J.; Peletier, M. A.; Scardia, L., Asymptotic behavior of a pile-up of infinite walls of edge dislocations, Archive for Rational Mechanics and Analysis, 495-539, (2013) · Zbl 1282.74067 |
| [13] | Ghoniem, N. M.; Tong, S. H.; Sun, L. Z., Parametric dislocation dynamics: A thermodynamics-based approach to investigations of mesoscopic plastic deformation, Physics Review B, 61, 2, 913-927, (2000) |
| [14] | Groma, I.; Csikor, F. F.; Zaiser, M., Spatial correlations and higher-order gradient terms in a continuum description of dislocation dynamics, Acta Materialia, 51, 1271-1281, (2003) |
| [15] | Hiang, Y.; Gao, H.; Nix, W. D.; Hutchinson, J. W., Mechanism-based strain gradient plasticity - 2, Analysis, Journal of the Mechanics and Physics of Solids, 48, 99-128, (2000) · Zbl 0990.74016 |
| [16] | Hochrainer, T., Multipole expansion of continuum dislocations dynamics in terms of aligment tensor, Philosophical Magazine, 95, 12, 1321-1367, (2015) |
| [17] | Hochrainer, T., Thermodynamically consistent continuum dislocation dynamics, Journal of the Mechanics and Physics of Solids, 88,, 12-22, (2016) |
| [18] | Hochrainer, T.; Sandfeld, S.; Zaiser, M.; Gumbsch, P., Continuum dislocation dynamics: toward a physical theory of crystal plasticity, Journal of the Mechanics and Physics of Solids, 167-178, (2014) |
| [19] | Kaluza, M.; Le, K. C., On torsion of a single crystal rod, International Journal of Plasticity, 27, 460-469, (2011) · Zbl 1426.74184 |
| [20] | Le, K. C., Self-energy of dislocations and dislocation pileups, International Journal of Engineering Science, 100, 1-7, (2016) · Zbl 1423.82007 |
| [21] | Limkumnerd, S.; Giessen, V.d., Statistical approach to dislocation dynamics: from dislocation correlations to multiple-slip continuum theory of plasticity, Physics Review B, 77, 184111, (2008) |
| [22] | Mesarovic, S. D.; Baskaran, R.; Panchenko, A., Thermodynamic coarsening of dislocation mechanics and the size-dependent continuum crystal plasticity, Journal of the Mechanics and Physics of Solids, 311-329, (2010) · Zbl 1193.74016 |
| [23] | Mohamed, M. S.; Larson, B. C.; Tischler, J. Z.; El-Azab, A., A statistical analysis of the elastic distortion and dislocation density fields in deformed crystals, Journal of Mechanics of Solids, 82, 32-47, (2015) |
| [24] | O’Neil, K. A.; Redner, R. A., On the limiting distribution of pair-summable potential functions in many particle systems, Journal of Statistical Physics, 62, 399-410, (1991) |
| [25] | Park, H.; Fellinger, M. R.; Lenovsky, T. J.; Tipton, W. W.; Trincle, D. R.; Rudin, S. P., Ab inito based empirical potential used to study the mechanical properties of molybdenum, Physics Review B, 85, 214121, (2012) |
| [26] | Poh, L. H.; Peerlings, R. H.J.; Geers, M. G.D.; Swaddiwudhipong, S., Homogenization towards a grain-size dependent plasticity theory for single slip, Journal of the Mechanics and Physics of Solids, 61, 913-927, (2013) |
| [27] | Poh, L. H.; Peerlings, R. H.J.; Geers, M. G.D.; Swaddiwudhipong, S., Towards a homogenized plasticity theory which predict structural and microstructural size effects, Journal of the Mechanics and Physics of Solids, 61, 2240-2259, (2013) |
| [28] | Weinberger, C. R., The structure and energetics of, and the plasticity caused by, eshelby dislocations, International Journal of Plasticity, 27, 1391-1408, (2011) · Zbl 1452.74006 |
| [29] | Zaiser, M., Local density distribution for the energy functional of a three-dimensional dislocation system, Physics Review B, 92, 17, 174120, (2015) |
| [30] | Zaiser, M.; Hochrainer, T., Some steps toward a continuum representation of 3d dislocation systems, Scripta Materialia, 54, 717-721, (2006) |
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