Abstract
This article is concerned with the development of a discrete theory of crystal elasticity and dislocations in crystals. The theory is founded upon suitable adaptations to crystal lattices of elements of algebraic topology and differential calculus such as chain complexes and homology groups, differential forms and operators, and a theory of integration of forms. In particular, we define the lattice complex of a number of commonly encountered lattices, including body-centered cubic and face-centered cubic lattices. We show that material frame indifference naturally leads to discrete notions of stress and strain in lattices. Lattice defects such as dislocations are introduced by means of locally lattice-invariant (but globally incompatible) eigendeformations. The geometrical framework affords discrete analogs of fundamental objects and relations of the theory of linear elastic dislocations, such as the dislocation density tensor, the equation of conservation of Burgers vector, Kröner's relation and Mura's formula for the stored energy. We additionally supply conditions for the existence of equilibrium displacement fields; we show that linear elasticity is recovered as the Γ-limit of harmonic lattice statics as the lattice parameter becomes vanishingly small; we compute the Γ-limit of dilute dislocation distributions of dislocations; and we show that the theory of continuously distributed linear elastic dislocations is recovered as the Γ-limit of the stored energy as the lattice parameter and Burgers vectors become vanishingly small.
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Abraham, F.F., Schneider, D., Land, B., Lifka D., Skovira, J., Gerner, J., Rosenkrantz, M.: Instability dynamics in the 3-dimensional fracture - an atomistic simulation. Journal of the Mechanics and Physics of Solids 45, 1461–1471 (1997)
Abraham, R., Marsden, J.E., Ratiu, T.: Manifolds, Tensor Analysis and Applications. Addison-Wesley, London, 1983
Bacon, D.J., Barnett, D.M., Scattergood, R.O.: Anisotropic Continuum Theory of Lattice Defects. Progress in Material Sciences 23, 51–262 (1979)
Born, M., Huang, K.: Dynamical theory of crystal lattices. Oxford University Press, London, 1954
Bott, R., Tu, L.W.: Differential Forms in Algebraic Topology. Springer-Verlag, 1982
Bradley, C.J., Cracknell, A.P.: The Mathematical Theory of Symmetry in Solids. Clarendon Press, Oxford, 1972
Braides, A., Gelli, M.S.: The passage from discrete to continuous variational problems: a nonlinear homogenization process. In: P.Ponte Castaneda, editor, Nonlinear Homogenization and its Applications to Composites, Polycrystals and Smart Materials. Kluwer, 2004
Cioranescu, D., Donato, P.: An Introduction to Homogeneization. Oxford University Press, 1999
Cuitiño, A.M., Ortiz, M.: Computational modeling of single-crystals. Modelling and Simulation in Materials Science and Engineering 1, 225–263 (1993)
Dal Maso, G.: An Introduction to Γ-Convergence. Birkhauser, Boston, 1993
Daw, M.S.: The embedded atom method: A review. In Many-Atom Interactions in Solids, of Springer Proceedings in Physics, Springer-Verlag, Berlin, 48, pp. 49–63 1990
Ericksen, J.L.: On the symmetry of deformable crystrals. Archive for Rational Mechanics and Analysis 72, 1–13 (1979)
Finnis, M.W., Sinclair, J.E.: A simple empirical n-body potential for transition- metals. Philosophical Magazine A-Physics of Condensed Matter Structure Defects and Mechanical Properties 50, 45–55 (1984)
Garroni, A., Müller, S.: Γ-limit of a phase-field model of dislocations. Preprint 92, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, Germany, 2003
Garroni, A., Müller, S.: A variational model for dislocations in the line tension limit. Preprint 76, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, Germany, 2004
Hamermesh, M.: Group Theory and its Applications to Physical Problems. Dover Publications, New York, 1962
Hansen, N., Kuhlmann-Wilsdorff, D.: Low Energy Dislocation Structures due to Unidirectional Deformation at Low Temperatures. Materials Science and Engineering 81, 141–161 (1986)
Hirani, A.: Discrete Exterior Calculus. PhD thesis, California Institute of Technology, 2003
Hirth, J.P., Lothe, J.: Theory of Dislocations. McGraw-Hill, New York, 1968
Holz, A.: Topological properties of linked disclinations in anisotropic liquids. Journal of Physics A 24, L1259–L1267 (1991)
Holz, A.: Topological properties of linked disclinations and dislocations in solid continua. Journal of Physics A 25, L1–L10 1992
Holz, A.: Topological properties of static and dynamic defect configurations in ordered liquids. Physica A 182, 240–278 (1992)
Kleman, M., Michel, L., Toulouse, G.: Classification of topologically stable defects in ordered media. Journal de Physique 38, L195–L197 (1977)
Koslowski, M., Cuiti no, A.M., Ortiz, M.: A phase-field theory of dislocation dynamics, strain hardening and hysteresis in ductile single crystals. Journal of the Mechanics and Physics of Solids 50, 2597–2635 (2002)
Koslowski, M., Ortiz, M.: A multi-phase field model of planar dislocation networks. Modeling and Simulation in Materials Science and Engineering 12, 1087–1097 (2004)
Kröner, E.: Berechnung der elastischen Konstanten des Vielkristalls aus den Konstanten des Einkristalls. Zeitung der Physik 151, 504–518 (1958)
Kuhlmann-Wilsdorf, D.: Theory of plastic deformation: properties of low energy dislocation structures. Materials Science and Engineering A113, 1 (1989)
Leok, M.: Foundations of Computational Geometric Mechanics. PhD thesis, California Institute of Technology, 2004
Lubarda, V.A., Blume, J.A., Needleman, A.: An Analysis of Equilibrium Dislocation Distributions. Acta Metallurgica et Materialia 41, 625–642 (1993)
Mermin, N.D.: The topological theory of defects in ordered media. Reviews of Modern Physics 51, 591–648 (1979)
Morgan, F.: Geometric Measure Theory. Academic Press, London, 2000
Moriarty, J.A.: Angular forces and melting in bcc transition-metals – a case-study of molybdenum. Physical Review B 49, 12431–12445 (1994)
Mughrabi, H.: Description of the Dislocation Structure after Unidirectional Deformation at Low Temperatures. In A.S. Argon, editor, Constitutive Equations in Plasticity, Cambridge, Mass, 1975. MIT Press pp. 199–250
Munkres, J.R.: Elements of Algebraic Topology. Perseus Publishing, 1984
Mura, T.: Continuous distribution of moving dislocations. Philosophical Magazine 8, 843 (1963)
Mura, T.: Micromechanics of defects in solids. Kluwer Academic Publishers, Boston, 1987
Neumann, P.: Low Energy Dislocation Configurations: A Possible Key to the Understanding of Fatigue. Materials Science and Engineering 81, 465–475 (1986)
Nye, J.F.: Some geometrical relations in dislocated crystals. Acta Metallurgica 1, 153–162 (1953)
Ortiz, M., Phillips, R.: Nanomechanics of defects in solids. Advances in Applied Mechanics 36, 1–79 (1999)
Peierls, R.E.: The Size of a Dislocation. Proceedings of the Royal Society of London A52, 34 (1940)
Pettifor, D.G., Oleinik, I.I., Nguyen-Manh, D., Vitek, V.: Bond-order potentials: bridging the electronic to atomistic modelling hierarchies. Computational Materials Science 23, 33–37 (2002)
Rudin, W.: Functional Analysis. McGraw-Hill, 1991
Sarkar, S.K., Sengupta, S.: On born-huang invariance conditions. Phys. Status Solidi (b) 83, 263–271 (1977)
Schwarzenberger, R.L.E.: Classification of crystal lattices. Proceedings of the Cambridge Philosophical Society 72, 325–349 (1972)
Sengupta, S.: Lattice Theory of Elastic Constants. Trans Tech Publications, Aedermannsdorf, Switzerland, 1988
Stillinger, F.H., Weber, T.A.: Computer simulation of local order in condensed phases of silicon. Phys. Rev. B 31, 5262–5271 (1985)
Toulouse, G., Kleman, M.: Principles of a classification of defects in ordered media. Journal de Physique 37, L149–L151 (1976)
Trebin, H.R.: The topology of non-uniform media in condensed matter physics. Advances in Physics 31, 195–254 (1982)
Wang, C.C.: On representations for isotropic functions .i. isotropic functions of symmetric tensors and vectors. Archive for Rational Mechanics and Analysis 33, 249 (1969)
Yuan, X.Y., Takahashi, K., Ouyang, Y.F., Onzawa, A.: Development of a modified embedded atom method for bcc transition metals. Journal of Physics-Condensed Matter 15, 8917–8926 (2003)
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Ariza, M., Ortiz, M. Discrete Crystal Elasticity and Discrete Dislocations in Crystals. Arch. Rational Mech. Anal. 178, 149–226 (2005). https://doi.org/10.1007/s00205-005-0391-4
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DOI: https://doi.org/10.1007/s00205-005-0391-4