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, (2000).   Google Scholar

[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.  doi: 10.1007/BF00281418.  Google Scholar

[3]

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

[4]

S. Ben-Abraham, Generalized stress and non-Riemannian geometry,, in, 317 (1970), 943.   Google Scholar

[5]

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

[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.  doi: 10.1098/rspa.1955.0171.  Google Scholar

[7]

J.-P. Bourguignon, Transport parallèle et connexions en Géométrie et en Physique. (French) [Parallel transport and connections in geometry and physics],, in, 402 (1992), 1830.   Google Scholar

[8]

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

[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.   Google Scholar

[10]

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

[11]

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

[12]

B. A. Dubrovin, A. T. Fomenko and S. P. Novikov, "Modern Geometry - Methods and Applications,", 2nd edn., (1992).   Google Scholar

[13]

M. Epstein, "The Geometrical Language of Continuum Mechanics,", Cambridge University Press, (2010).   Google Scholar

[14]

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

[15]

H. Kleinert, "Gauge Fields in Condensed Matter, Vol. 1,", World Scientific Publishing, (1989).   Google Scholar

[16]

H. Kleinert, "Multivalued Fields. In Condensed Matter, Electromagnetism, and Gravitation,", World Scientific Publishing, (2008).   Google Scholar

[17]

K. Kondo, On the geometrical and physical foundations of the theory of yielding,, in, (1952), 41.   Google Scholar

[18]

K. Kondo, Non-Riemannian geometry of the imperfect crystal from a macroscopic viewpoint,, in, 1 (1955), 458.   Google Scholar

[19]

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

[20]

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

[21]

E. Kröner, Continuum theory of defects,, in, (1980), 219.   Google Scholar

[22]

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

[23]

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

[24]

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

[25]

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

[26]

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

[27]

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

[28]

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

[29]

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

[30]

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

[31]

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

[32]

J. Philibert, "Atom Movements, Diffusion and Mass Transport in Solids,", Monographies de physique., (1988).   Google Scholar

[33]

J. A. Schouten, "Ricci-Calculus,", 2nd edn., (1954).   Google Scholar

[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.  doi: 10.1016/j.jpcs.2007.07.129.  Google Scholar

[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), (2009).   Google Scholar

[36]

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

[37]

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

[38]

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

[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.  doi: 10.1007/BF00276434.  Google Scholar

show all references

References:
[1]

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

[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.  doi: 10.1007/BF00281418.  Google Scholar

[3]

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

[4]

S. Ben-Abraham, Generalized stress and non-Riemannian geometry,, in, 317 (1970), 943.   Google Scholar

[5]

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

[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.  doi: 10.1098/rspa.1955.0171.  Google Scholar

[7]

J.-P. Bourguignon, Transport parallèle et connexions en Géométrie et en Physique. (French) [Parallel transport and connections in geometry and physics],, in, 402 (1992), 1830.   Google Scholar

[8]

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

[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.   Google Scholar

[10]

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

[11]

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

[12]

B. A. Dubrovin, A. T. Fomenko and S. P. Novikov, "Modern Geometry - Methods and Applications,", 2nd edn., (1992).   Google Scholar

[13]

M. Epstein, "The Geometrical Language of Continuum Mechanics,", Cambridge University Press, (2010).   Google Scholar

[14]

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

[15]

H. Kleinert, "Gauge Fields in Condensed Matter, Vol. 1,", World Scientific Publishing, (1989).   Google Scholar

[16]

H. Kleinert, "Multivalued Fields. In Condensed Matter, Electromagnetism, and Gravitation,", World Scientific Publishing, (2008).   Google Scholar

[17]

K. Kondo, On the geometrical and physical foundations of the theory of yielding,, in, (1952), 41.   Google Scholar

[18]

K. Kondo, Non-Riemannian geometry of the imperfect crystal from a macroscopic viewpoint,, in, 1 (1955), 458.   Google Scholar

[19]

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

[20]

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

[21]

E. Kröner, Continuum theory of defects,, in, (1980), 219.   Google Scholar

[22]

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

[23]

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

[24]

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

[25]

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

[26]

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

[27]

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

[28]

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

[29]

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

[30]

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

[31]

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

[32]

J. Philibert, "Atom Movements, Diffusion and Mass Transport in Solids,", Monographies de physique., (1988).   Google Scholar

[33]

J. A. Schouten, "Ricci-Calculus,", 2nd edn., (1954).   Google Scholar

[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.  doi: 10.1016/j.jpcs.2007.07.129.  Google Scholar

[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), (2009).   Google Scholar

[36]

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

[37]

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

[38]

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

[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.  doi: 10.1007/BF00276434.  Google Scholar

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