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Interlude: Smooth Distributions of Defects

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Foundations of Geometric Continuum Mechanics

Part of the book series: Advances in Mechanics and Mathematics ((ACM,volume 49))

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Abstract

This short interlude presents an example of an application of smooth analysis on manifolds to the geometric description of smooth distributions of dislocations in a solid body.

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References

  1. R. Abraham, J.E. Marsden, and T. Ratiu. Manifolds, Tensor Analysis. and Applications. Springer, 1988.

    Google Scholar 

  2. P. Cermelli. Material symmetry and singularities in solids. Proceedings of the Royal Society of London, A 455:299–322, 1999.

    Google Scholar 

  3. S. Chandrasekhar. Liquid Crystals. Cambridge University Press, 1977.

    Google Scholar 

  4. C. Davini and G. Parry. A complete list of invariants for defective crystals. Proceedings of the Royal Society of London, 432:341–365, 1991.

    MathSciNet  MATH  Google Scholar 

  5. A.C. Eringen and W.D. Claus. A micromorphic approach to dislocation theory and its relation to several existing theories. In J.A. Simmons, R. de Wit, and R. Bullough, editors, Fundamental Aspects of Dislocation Theory, pages 1023–1040. U.S. National Bureau of Standards, 1970.

    Google Scholar 

  6. M. Elzanowski and M. Epstein. Material Inhomogeneities and their Evolution. Springer, 2007.

    MATH  Google Scholar 

  7. M. Epstein and R. Segev. Geometric aspects of singular dislocations. Mathematics and Mechanics of Solids, 19:335–347, 2014. https://doi.org/10.1177/1081286512465222.

    Article  MathSciNet  MATH  Google Scholar 

  8. M. Epstein and R. Segev. Differential geometry and continuum mechanics. volume 137 of Springer Proceedings in Mathematics and Statistics, chapter 7: On the geometry and kinematics of smoothly distributed and singular defects, pages 203–234. Springer, 2015.

    Google Scholar 

  9. M. Epstein and R. Segev. Regular and Singular Dislocations. In R. Segev and M. Epstein, editors, Geometric Continuum Mechanics, Advances in Mechanics and Mathematics, pages 223–265. Springer, 2020.

    Google Scholar 

  10. F.C. Frank. 1. Liquid crystals. On the theory of liquid crystals. Discussions of the Faraday Society, 25:19–28, 1958.

    Google Scholar 

  11. E. Kroner and K.H. Anthony. Dislocations and disclinations in material structures: The basic topological concepts. Annual Review of Material Science, 5:43–72, 1975.

    Article  Google Scholar 

  12. K. Kondo. Geometry of Elastic Deformation and incompatibility. Tokyo Gakujutsu Benken Fukyu-Kai, IC, 1955.

    Google Scholar 

  13. S.A. Lurie and A.L. Kalamkarov. General theory of defects in continous media. International Journal of Solids and Structures, 43:91–111, 2006.

    Article  MathSciNet  MATH  Google Scholar 

  14. W. Noll. Materially uniform bodies with inhomogeneities. Archive for Rational mechanics and Analysis, 27:1–32, 1967.

    Article  MathSciNet  MATH  Google Scholar 

  15. D. Sahoo. Elastic continuum theories of lattice defects: a review. Bulletin of Materials Science, 6:775–798, 1984.

    Article  Google Scholar 

  16. S. Sternberg. Lectures on Differential Geometry. American Mathematical Society, 1983.

    MATH  Google Scholar 

  17. R.A. Toupin. Dislocated and oriented media. In Continuum Theory of Inhomogeneities in Simple Bodies, pages 9–24. Springer, 1968.

    Google Scholar 

  18. C.-C. Wang. On the geometric structure of simple bodies, a mathematical foundation for the theory of continuous distributions of dislocations. Archive for Rational Mechanics and Analysis, 27:33–94, 1967.

    Article  MathSciNet  MATH  Google Scholar 

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Segev, R. (2023). Interlude: Smooth Distributions of Defects. In: Foundations of Geometric Continuum Mechanics. Advances in Mechanics and Mathematics(), vol 49. Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-35655-1_7

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