Abstract
A continuum mechanical framework for the description of the geometry and kinematics of defects in material structure is proposed. The setting applies to a body manifold of any dimension which is devoid of a Riemannian or a parallelism structure. In addition, both continuous distributions of defects as well as singular distributions are encompassed by the theory. In the general case, the material structure is specified by a de Rham current \(T\) and the associated defects are given by its boundary \(\partial T\). For a motion of defects associated with a family of diffeomorphisms of a material body, it is shown that the rate of change of the distribution of defects is given by the dual of the Lie derivative operator.
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References
Abraham R, Marsden J, Ratiu T (1988) Manifolds, tensor analysis and applications. Springer, New York
Cermelli P (1999) Material symmetry and singularities in solids. Proc R Soc Lond A 455:299–322
Chandrasekhar S (1977) Liquid crystals. Cambridge University Press, Cambridge
Davini C, Parry G (1991) A complete list of invariants for defective crystals. Proc R Soc Lond A 432:341–365
de Rham G (1984) Differentiable manifolds. Springer, New York
Elzanowski M, Epstein M (2007) Material inhomogeneities and their evolution. Springer, Berlin
Epstein M, Segev R (2012) Geometric aspects of singular dislocations. Mathematics and Mechanics of Solids. doi:10.1177/1081286512465222
Eringen A, Claus W (1970) A micromorphic approach to dislocation theory and its relation to several existing theories. In: Simmons J, de Wit R, Bullough R (eds) Fundamental aspects of dislocation theory. U.S. National Bureau of Standards, pp 1023–1040
Falach L, Segev R (2014) Reynolds transport theorem for smooth deformations of currents on manifolds. Mathematics and Mechanics of Solids, 2014. Volume for the occasion of R. Ogden’s 70th anniversary, doi:10.1177/1081286514551503
Federer H (1969) Geometric measure theory. Springer, Berlin
Frank F (1958) 1. Liquid crystals. On the theory of liquid crystals. Discuss Faraday Soc 25:19–28
Giaquinta M, Modica G, Soucek J (1998) Cartesian currents in the calculus of variation I. Springer, Berlin
Kondo K (1955) Geometry of elastic deformation and incompatibility. In: Kondo K (ed) Memoirs of the unifying study of the basic problems in engineering science by means of geometry, vol 1. Division C-1. Gakujutsu Bunken, Fukyo-kai, 5–17 (=361–373), Tokyo
Kröner E, Anthony K (1975) Dislocations and disclinations in material structures: the basic topological concepts. Ann Rev Mater Sci 5:43–72
Lurie S, Kalamkarov A (2006) General theory of defects in continous media. Int J Solids Struct 43:91–111
Michor P (2008) Topics in differential geometry. American Mathematical Society, Providence
Noll W (1967) Materially uniform bodies with inhomogeneities. Arch Ration Mech Anal 27:1–32
Sahoo D (1984) Elastic continuum theories of lattice defects: a review. Bull Mater Sci 6:775–798
Segev R, Rodnay G (2003) Worldlines and body points associated with an extensive property. Int J Non-Linear Mech 38:1–9
Sternberg S (1983) Lectures on differential geometry. AMS Chelsea, Providence
Toupin R (1968) Dislocated and oriented media. Continuum theory of inhomogeneities in simple bodies. Springer, New York, pp 9–24. doi:10.1007/978-3-642-85992-2
Wang CC (1967) On the geometric structure of simple bodies, a mathematical foundation for the theory of continuous distributions of dislocations. Arch Ration Mech Anal 27:33–94
Whitney H (1957) Geometric integration theory. Princeton University Press, Princeton
Acknowledgments
This work was partially supported by the Perlstone Center for Aeronautical Engineering Studies and the H. Greenhill Chair for Theoretical and Applied Mechanics at Ben-Gurion University and by the Natural Sciences and Engineering Research Council of Canada.
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Epstein, M., Segev, R. (2015). On the Geometry and Kinematics of Smoothly Distributed and Singular Defects. In: Chen, GQ., Grinfeld, M., Knops, R. (eds) Differential Geometry and Continuum Mechanics. Springer Proceedings in Mathematics & Statistics, vol 137. Springer, Cham. https://doi.org/10.1007/978-3-319-18573-6_7
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