Abstract
A nonlinear continuum theory of material bodies with continuously distributed dislocations is presented, based on a gauge theoretical approach. Firstly, we derive the canonical conservation laws that correspond to the group of translations and rotations in the material space using Noether’s theorem. These equations give us the canonical Eshelby stress tensor as well as the total canonical angular momentum tensor. The canonical Eshelby stress tensor is neither symmetric nor gauge-invariant. Based on the Belinfante-Rosenfeld procedure, we obtain the gauge-invariant Eshelby stress tensor which can be symmetric relative to the reference configuration only for isotropic materials. The gauge-invariant angular momentum tensor is obtained as well. The decomposition of the gauge-invariant Eshelby stress tensor in an elastic and in a dislocation part gives rise to the derivation of the famous Peach-Koehler force.
Article PDF
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.Avoid common mistakes on your manuscript.
References
Agiasofitou, E.K., Lazar, M.: Conservation and balance laws in linear elasticity of grade three. J. Elast. 94, 69–85 (2009)
Bluman, G.W., Kumei, S.: Symmetries and Differential Equations. Springer, Berlin (1989)
Edelen, D.G.B.: Applied Exterior Calculus. Wiley, New York (1985)
Edelen, D.G.B., Lagoudas, D.C.: Gauge theory and defects in solids. In: Sih, G.C. (ed.) Mechanics and Physics of Discrete System, vol. 1. North-Holland, Amsterdam (1988)
Eshelby, J.D.: The elastic energy-momentum tensor. J. Elast. 5, 321–335 (1975)
Epstein, M., Elżanowski, M.: Material Inhomogeneities and Their Evolution. Springer, Berlin (2007)
Epstein, M., Maugin, G.A.: The energy-momentum tensor and material uniformity in finite elasticity. Acta Mech. 83, 127–133 (1990)
Hehl, F.W., McCrea, J.D., Mielke, E.W., Ne’eman, Y.: Metric-affine gauge theory of gravity: field equations, Noether identities, world spinors, and breaking of dilation invariance. Phys. Rep. 258, 1–171 (1995)
Kadić, A., Edelen, D.G.B.: A Yang-Mills type minimal coupling theory for materials with dislocations and disclinations. Int. J. Eng. Sci. 20, 433–438 (1982)
Kröner, E.: Dislocations in crystals and in continua: a confrontation. Int. J. Eng. Sci. 33, 2127–2135 (1995)
Lazar, M.: Dislocation theory as a 3-dimensional translation gauge theory. Ann. Phys. (Leipz.) 9, 461–473 (2000)
Lazar, M., Anastassiadis, C.: The gauge theory of dislocations: conservation and balance laws. Philos. Mag. 88, 1673–1699 (2008)
Lazar, M., Anastassiadis, C.: The gauge theory of dislocations: static solutions of screw and edge dislocations. Philos. Mag. 89, 199–231 (2009)
Le, K.C., Stumpf, H.: Nonlinear continuum theory of dislocations. Int. J. Eng. Sci. 34, 339–358 (1996)
Le, K.C., Stumpf, H.: A model of elastoplastic bodies with continuously distributed dislocations. Int. J. Plast. 12, 611–627 (1996)
Lovelock, D., Rund, H.: Tensors, Differential Forms and Variational Principles. Wiley, New York (1975)
Maugin, G.A.: Material Inhomogeneities in Elasticity. Chapman & Hall, London (1993)
Noether, E.: Invariante variationspropleme. Goettinger Nachr. Math.-Phys. Kl. 2, 235–256 (1918); [English translation: Invariant variation problems. Transp. Theory Stat. Phys. 1, 186–207 (1971)]
Olver, P.J.: Applications of Lie Groups to Differential Equations. Springer, New York (1986)
Peach, M.O., Koehler, J.S.: Forces extended on dislocations and the stress fields produced by them. Phys. Rev. 80, 436–439 (1950)
Rogula, D.: Forces in material space. Arch. Mech. 29, 705–713 (1977)
Sfyris, D., Charalambakis, N., Kalpakides, V.K.: Variational arguments and Noether’s theorem on the nonlinear continuum theory of dislocations. Int. J. Eng. Sci. 44, 501–512 (2006)
Soper, D.E.: Classical Field Theory. Wiley, New York (1976)
Steinmann, P.: On spatial and material settings of hyperelastostatic crystal defects. J. Mech. Phys. Solids 50, 1743–1766 (2002)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Agiasofitou, E., Lazar, M. On the Nonlinear Continuum Theory of Dislocations: A Gauge Field Theoretical Approach. J Elast 99, 163–178 (2010). https://doi.org/10.1007/s10659-009-9238-9
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10659-009-9238-9