September  2010, 2(3): 303-320. doi: 10.3934/jgm.2010.2.303

The non-Riemannian dislocated crystal: A tribute to Ekkehart Kröner (1919-2000)

1. 

Universidade de Lisboa, Faculdade de Ciências, Departamento de Matemática, Centro de Matemática e Aplicações Fundamentais, Av. Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal

Received  April 2010 Revised  October 2010 Published  November 2010

This expository paper is a tribute to Ekkehart Kröner's results on the intrinsic non-Riemannian geometrical nature of a single crystal filled with point and/or line defects. A new perspective on this old theory is proposed, intended to contribute to the debate around the still open Kröner's question: "what are the dynamical variables of our theory?"
Citation: Nicolas Van Goethem. The non-Riemannian dislocated crystal: A tribute to Ekkehart Kröner (1919-2000). Journal of Geometric Mechanics, 2010, 2 (3) : 303-320. doi: 10.3934/jgm.2010.2.303
References:
[1]

L. Ambrosio, N. Fusco and D. Palara, "Functions of Bounded Variation and Free Discontinuity Problems," Oxford Mathematical Monographs, Oxford, 2000.

[2]

K. H. Anthony, Die Reduktion von nichteuklidischen geometrischen Objekten in eine euklidische Form und physikalische Deutung der Reduktion durch Eigenspannungszustände in Kristallen, Arch. Rational Mech. Anal., 37 (1970), 43-88. doi: 10.1007/BF00281418.

[3]

K. H. Anthony, Die theorie der Disklinationen, Arch. Rational Mech. Anal., 39 (1970), 161-180. doi: 10.1007/BF00281475.

[4]

S. Ben-Abraham, Generalized stress and non-Riemannian geometry, in "Fundam. Aspects of Dislocation Theory" (Nat. Bur. Stand. (U.S.)), Spec. Publ 317, II, (1970), 943-962.

[5]

V. L. Berdichevsky, Continuum theory of dislocations revisited, Continuum Mech. Thermodyn., 18 (2006), 195-222. doi: 10.1007/s00161-006-0024-7.

[6]

B. A. Bilby, R. Bullough and E. Smith, Continuous distribution of dislocations: A new application of the methods of non-Riemannian geometry, Proc. Roy. Soc. London A, 231 (1955), 263-273. doi: 10.1098/rspa.1955.0171.

[7]

J.-P. Bourguignon, Transport parallèle et connexions en Géométrie et en Physique. (French) [Parallel transport and connections in geometry and physics], in "1830-1930: A Century of Geometry" (Paris, 1989), Lecture Notes in Phys. 402, Springer, Berlin, (1992), 150-164.

[8]

I. Bucataru and M. Epstein, Geometrical theory of dislocations in bodies with microstructure, Journal of Geometry and Physics, 52 (2004), 57-73. doi: 10.1016/j.geomphys.2004.01.006.

[9]

E. Cartan, Sur une generalisation de la notion de courbure de Riemann et les espaces a torsion, C. R. Acad. Sci. Paris, 174 (1922), 593-597.

[10]

M. de León and M. Epstein, The geometry of uniformity in second-grade elasticity, Acta Mechanica, 114 (1996), 217-224. doi: 10.1007/BF01170405.

[11]

M. de León and M. Epstein, Geometrical theory of uniform Cosserat media, Journal of Geometry and Physics, 26 (1998), 127-170. doi: 10.1016/S0393-0440(97)00042-9.

[12]

B. A. Dubrovin, A. T. Fomenko and S. P. Novikov, "Modern Geometry - Methods and Applications," 2nd edn., Springer-Verlag, New York, 1992.

[13]

M. Epstein, "The Geometrical Language of Continuum Mechanics," Cambridge University Press, Cambridge, 2010.

[14]

M. Epstein and G. A. Maugin, The energy-momentum tensor and material uniformity in finite elasticity, Acta Mech., 83 (1990), 127-133. doi: 10.1007/BF01172974.

[15]

H. Kleinert, "Gauge Fields in Condensed Matter, Vol. 1," World Scientific Publishing, Singapore, 1989.

[16]

H. Kleinert, "Multivalued Fields. In Condensed Matter, Electromagnetism, and Gravitation," World Scientific Publishing, Singapore, 2008.

[17]

K. Kondo, On the geometrical and physical foundations of the theory of yielding, in "Proc. 2nd Japan Nat. Congr. Applied Mechanics," Tokyo, (1952), 41-47.

[18]

K. Kondo, Non-Riemannian geometry of the imperfect crystal from a macroscopic viewpoint, in "RAAG Memoirs of the Unifying Study of Basic Problems in Engineering Sciences by Means of Geometry," Vol. 1, Division D, Gakuyusty Bunken Fukin-Day, Tokyo, (1955), 458-469.

[19]

E. Kröner, Die spannungsfunktionen der dreidimensionalen anisotropen elastizitätstheorie, Z. Physik, 140 (1955), 386-398.

[20]

E. Kröner, Allgemeine kontinuumstheorie der versetzungen und eigenspannungen, Arch. Rat. Mech. Anal., 4 (1960), 273-334. doi: 10.1007/BF00281393.

[21]

E. Kröner, Continuum theory of defects, in "Physiques des Défauts" (ed. R. Balian), Les Houches session XXXV, Course 3, (1980), 219-315.

[22]

E. Kröner, The differential geometry of elementary point and line defects in Bravais crystals, Int. J. Theor. Phys., 29 (1990), 1219-1237. doi: 10.1007/BF00672933.

[23]

E. Kröner, The internal mechanical state of solids with defects, Int. J. Solids and Structures, 29 (1992), 1849-1257. doi: 10.1016/0020-7683(92)90176-T.

[24]

E. Kröner, Dislocations in crystals and in continua: A confrontation, Int. J. Engng Sci., 33 (1995), 2127-2135. doi: 10.1016/0020-7225(95)00061-2.

[25]

E. Kröner, Dislocation theory as a physical field theory, Meccanica, 31 (1996), 577-587. doi: 10.1007/BF00420827.

[26]

E. Kröner, Benefits and shortcomings of the continuous theory of dislocations, Int. J. Solids Struc., 38 (2001), 1115-1134. doi: 10.1016/S0020-7683(00)00077-9.

[27]

M. Lazar and G. Maugin, Dislocations in gradient elasticity revisited, Proc. R. Soc. A, 462 (2006), 3465-3480. doi: 10.1098/rspa.2006.1699.

[28]

G. Maugin, Geometry and thermomechanics of structural rearrangements: Ekkehart Kröner's legacy, ZAMM, 83 (2003), 75-84. doi: 10.1002/zamm.200310007.

[29]

W. Noll, Materially uniform bodies with inhomogeneities, Arch. Rational Mech. Anal., 27 (1967), 1-32. doi: 10.1007/BF00276433.

[30]

J. F. Nye, Some geometrical relations in dislocated crystals, Acta Metall, 1 (1953), 153-162. doi: 10.1016/0001-6160(53)90054-6.

[31]

M. Palombaro and S. Müller, Existence of minimizers for a polyconvex energy in a crystal with dislocations, Calc. Var., 31 (2008), 473-482. doi: 10.1007/s00526-007-0120-y.

[32]

J. Philibert, "Atom Movements, Diffusion and Mass Transport in Solids," Monographies de physique., Les éditions de physique, les Ulis, France, 1988.

[33]

J. A. Schouten, "Ricci-Calculus," 2nd edn., Springer-Verlag, Berlin, 1954.

[34]

N. Van Goethem, A. de Potter, N. Van den Bogaert and F. Dupret, Dynamic prediction of point defects in Czochralski silicon growth. An attempt to reconcile experimental defect diffusion coefficients with the $V/G$ criterion, J. Phys. Chem. Solids, 69 (2008), 320-324. doi: 10.1016/j.jpcs.2007.07.129.

[35]

N. Van Goethem and F. Dupret, A distributional approach to the geometry of $2D$ dislocations at the meso-scale.Parts A and Part B, preprints (2009), arXiv:1003.6021.

[36]

N. Van Goethem, "Strain Incompatibility in Single Crystals: Kröner's Formula Revisited," J. Elast., Springer Netherlands, 2010. doi: 10.1007/s10659-010-9275-4.

[37]

N. Van Goethem, A multiscale model for dislocations: From mesoscopic elasticity to macroscopic plasticity, (in preparation).

[38]

N. Van Goethem, Incompatibility of dislocation clusters, (in preparation).

[39]

C. C. Wang, On the geometric structure of simple bodies, a mathematical foundation for the theory of continuous distributions of dislocations, Arch. Rational Mech. Anal., 27 (1967), 33-94. doi: 10.1007/BF00276434.

show all references

References:
[1]

L. Ambrosio, N. Fusco and D. Palara, "Functions of Bounded Variation and Free Discontinuity Problems," Oxford Mathematical Monographs, Oxford, 2000.

[2]

K. H. Anthony, Die Reduktion von nichteuklidischen geometrischen Objekten in eine euklidische Form und physikalische Deutung der Reduktion durch Eigenspannungszustände in Kristallen, Arch. Rational Mech. Anal., 37 (1970), 43-88. doi: 10.1007/BF00281418.

[3]

K. H. Anthony, Die theorie der Disklinationen, Arch. Rational Mech. Anal., 39 (1970), 161-180. doi: 10.1007/BF00281475.

[4]

S. Ben-Abraham, Generalized stress and non-Riemannian geometry, in "Fundam. Aspects of Dislocation Theory" (Nat. Bur. Stand. (U.S.)), Spec. Publ 317, II, (1970), 943-962.

[5]

V. L. Berdichevsky, Continuum theory of dislocations revisited, Continuum Mech. Thermodyn., 18 (2006), 195-222. doi: 10.1007/s00161-006-0024-7.

[6]

B. A. Bilby, R. Bullough and E. Smith, Continuous distribution of dislocations: A new application of the methods of non-Riemannian geometry, Proc. Roy. Soc. London A, 231 (1955), 263-273. doi: 10.1098/rspa.1955.0171.

[7]

J.-P. Bourguignon, Transport parallèle et connexions en Géométrie et en Physique. (French) [Parallel transport and connections in geometry and physics], in "1830-1930: A Century of Geometry" (Paris, 1989), Lecture Notes in Phys. 402, Springer, Berlin, (1992), 150-164.

[8]

I. Bucataru and M. Epstein, Geometrical theory of dislocations in bodies with microstructure, Journal of Geometry and Physics, 52 (2004), 57-73. doi: 10.1016/j.geomphys.2004.01.006.

[9]

E. Cartan, Sur une generalisation de la notion de courbure de Riemann et les espaces a torsion, C. R. Acad. Sci. Paris, 174 (1922), 593-597.

[10]

M. de León and M. Epstein, The geometry of uniformity in second-grade elasticity, Acta Mechanica, 114 (1996), 217-224. doi: 10.1007/BF01170405.

[11]

M. de León and M. Epstein, Geometrical theory of uniform Cosserat media, Journal of Geometry and Physics, 26 (1998), 127-170. doi: 10.1016/S0393-0440(97)00042-9.

[12]

B. A. Dubrovin, A. T. Fomenko and S. P. Novikov, "Modern Geometry - Methods and Applications," 2nd edn., Springer-Verlag, New York, 1992.

[13]

M. Epstein, "The Geometrical Language of Continuum Mechanics," Cambridge University Press, Cambridge, 2010.

[14]

M. Epstein and G. A. Maugin, The energy-momentum tensor and material uniformity in finite elasticity, Acta Mech., 83 (1990), 127-133. doi: 10.1007/BF01172974.

[15]

H. Kleinert, "Gauge Fields in Condensed Matter, Vol. 1," World Scientific Publishing, Singapore, 1989.

[16]

H. Kleinert, "Multivalued Fields. In Condensed Matter, Electromagnetism, and Gravitation," World Scientific Publishing, Singapore, 2008.

[17]

K. Kondo, On the geometrical and physical foundations of the theory of yielding, in "Proc. 2nd Japan Nat. Congr. Applied Mechanics," Tokyo, (1952), 41-47.

[18]

K. Kondo, Non-Riemannian geometry of the imperfect crystal from a macroscopic viewpoint, in "RAAG Memoirs of the Unifying Study of Basic Problems in Engineering Sciences by Means of Geometry," Vol. 1, Division D, Gakuyusty Bunken Fukin-Day, Tokyo, (1955), 458-469.

[19]

E. Kröner, Die spannungsfunktionen der dreidimensionalen anisotropen elastizitätstheorie, Z. Physik, 140 (1955), 386-398.

[20]

E. Kröner, Allgemeine kontinuumstheorie der versetzungen und eigenspannungen, Arch. Rat. Mech. Anal., 4 (1960), 273-334. doi: 10.1007/BF00281393.

[21]

E. Kröner, Continuum theory of defects, in "Physiques des Défauts" (ed. R. Balian), Les Houches session XXXV, Course 3, (1980), 219-315.

[22]

E. Kröner, The differential geometry of elementary point and line defects in Bravais crystals, Int. J. Theor. Phys., 29 (1990), 1219-1237. doi: 10.1007/BF00672933.

[23]

E. Kröner, The internal mechanical state of solids with defects, Int. J. Solids and Structures, 29 (1992), 1849-1257. doi: 10.1016/0020-7683(92)90176-T.

[24]

E. Kröner, Dislocations in crystals and in continua: A confrontation, Int. J. Engng Sci., 33 (1995), 2127-2135. doi: 10.1016/0020-7225(95)00061-2.

[25]

E. Kröner, Dislocation theory as a physical field theory, Meccanica, 31 (1996), 577-587. doi: 10.1007/BF00420827.

[26]

E. Kröner, Benefits and shortcomings of the continuous theory of dislocations, Int. J. Solids Struc., 38 (2001), 1115-1134. doi: 10.1016/S0020-7683(00)00077-9.

[27]

M. Lazar and G. Maugin, Dislocations in gradient elasticity revisited, Proc. R. Soc. A, 462 (2006), 3465-3480. doi: 10.1098/rspa.2006.1699.

[28]

G. Maugin, Geometry and thermomechanics of structural rearrangements: Ekkehart Kröner's legacy, ZAMM, 83 (2003), 75-84. doi: 10.1002/zamm.200310007.

[29]

W. Noll, Materially uniform bodies with inhomogeneities, Arch. Rational Mech. Anal., 27 (1967), 1-32. doi: 10.1007/BF00276433.

[30]

J. F. Nye, Some geometrical relations in dislocated crystals, Acta Metall, 1 (1953), 153-162. doi: 10.1016/0001-6160(53)90054-6.

[31]

M. Palombaro and S. Müller, Existence of minimizers for a polyconvex energy in a crystal with dislocations, Calc. Var., 31 (2008), 473-482. doi: 10.1007/s00526-007-0120-y.

[32]

J. Philibert, "Atom Movements, Diffusion and Mass Transport in Solids," Monographies de physique., Les éditions de physique, les Ulis, France, 1988.

[33]

J. A. Schouten, "Ricci-Calculus," 2nd edn., Springer-Verlag, Berlin, 1954.

[34]

N. Van Goethem, A. de Potter, N. Van den Bogaert and F. Dupret, Dynamic prediction of point defects in Czochralski silicon growth. An attempt to reconcile experimental defect diffusion coefficients with the $V/G$ criterion, J. Phys. Chem. Solids, 69 (2008), 320-324. doi: 10.1016/j.jpcs.2007.07.129.

[35]

N. Van Goethem and F. Dupret, A distributional approach to the geometry of $2D$ dislocations at the meso-scale.Parts A and Part B, preprints (2009), arXiv:1003.6021.

[36]

N. Van Goethem, "Strain Incompatibility in Single Crystals: Kröner's Formula Revisited," J. Elast., Springer Netherlands, 2010. doi: 10.1007/s10659-010-9275-4.

[37]

N. Van Goethem, A multiscale model for dislocations: From mesoscopic elasticity to macroscopic plasticity, (in preparation).

[38]

N. Van Goethem, Incompatibility of dislocation clusters, (in preparation).

[39]

C. C. Wang, On the geometric structure of simple bodies, a mathematical foundation for the theory of continuous distributions of dislocations, Arch. Rational Mech. Anal., 27 (1967), 33-94. doi: 10.1007/BF00276434.

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