×

A kinematic model for continuous distributions of dislocations. (English) Zbl 0977.74003

Summary: In continuum theory of defects the notion of a flat connection is employed. This paper gives a characterization of these connections via injective \(\mathbb{R}^3\)-valued differential forms. For material structures with continuous distributions of dislocations, a configuration space in the sense of global analysis is introduced and analysed. A kinematics for these dislocations is formulated which generalizes from elasticity.

MSC:

74A60 Micromechanical theories
58A14 Hodge theory in global analysis
74M25 Micromechanics of solids
53B50 Applications of local differential geometry to the sciences
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Abraham, R.; Marsden, J. E.; Ratiu, T., Manifolds, Tensor Analysis, and Applications, (Applied Mathematical Sciences, Vol. 75 (1988), Springer: Springer New York) · Zbl 0875.58002
[2] Binz, E.; Śniatycki, J.; Fischer, H., Geometry of Classical Fields, (Mathematics Studies, Vol. 154 (1988), North-Holland: North-Holland Amsterdam) · Zbl 0675.53065
[3] Binz, E.; Fischer, H. R., One-forms on spaces of embeddings: A frame work for constitutive laws in elasticity, Note Mat., XI, 21-48 (1991) · Zbl 0802.58011
[4] Binz, E.; Schwarz, G., The principle of work and a symplectic reduction of non-local continuum mechanics, Rep. Math. Phys., 32, 49-69 (1994) · Zbl 0797.58022
[5] Bilby, B. A.; Bullough, R.; Smith, E., Continuous distributions of dislocations: A new application of the method of non-riemannian geometry, (Proc. Roy. Soc. London Ser. A, 231 (1955)), 263-273
[6] Davini, C., A complete list of invariants for defective crystals, (Proc. Roy. Soc. London Ser. A, 432 (1991)), 341-365 · Zbl 0726.73032
[7] Elżanowski, M.; Epstein, M.; Śniatycki, J., G-Structures and material homogeneity, J. Elasticity, 13, 167-180 (1990) · Zbl 0709.73002
[8] Greub, W.; Halperin, S.; Vanstone, R., (Connections, Curvature and Cohomology, Vol. II (1973), Academic Press: Academic Press New York)
[9] Kondo, K., On the geometrical and physical foundation of the theory of yielding, (Proc. 2nd Japan Nat. Congr. Appl. Mech.. Proc. 2nd Japan Nat. Congr. Appl. Mech., Tokyo (1952))
[10] Kröner, E., Continuum theory of defects, (Balian, R.; etal., Les Houches, Session XXXV, 1980 — Physics of Defects (1981), North-Holland: North-Holland Amsterdam) · Zbl 0802.73065
[11] Marsden, J. E.; Hughes, J. R., Mathematical Foundations of Elasticity (1983), Prentice Hall: Prentice Hall Eaglewood Cliffs, NJ · Zbl 0545.73031
[12] Noll, W., Materially uniform simple bodies with inhomogeneities, Arch. Rat. Mech. Anal., 27, 1-32 (1967) · Zbl 0168.45701
[13] Nye, J. F., Some geometrical relations in dislocated crystals, Acta Met., 1, 153-162 (1953)
[14] Schwarz, G., Hodge Decomposition — A Method for Solving Boundary Value Problems (1995), Springer: Springer Heidelberg · Zbl 0828.58002
[15] Spivak, M., (Differential Geometry, Vols. I-V (1979), Publish or Perish: Publish or Perish Berkeley)
[16] Taylor, G. I., The mechanism of plastic deformation of crystals, parts I and II, (Proc. Roy. Soc. London Ser. A, 145 (1934)), 388-404 · JFM 60.0713.01
[17] Wang, C.-C., On the geometric structures of simple bodies, a mathematical foundation for the theory of continuous distributions of dislocations, Arch. Rat. Mech. Anal., 27, 33-94 (1967) · Zbl 0187.48802
[18] Wenzelburger, J., Die Hodge-Zerlegung in der Kontinuumstheorie von Defekten, (Dissertation (1994), Universität Mannheim, Verlag Shaker: Universität Mannheim, Verlag Shaker Aachen)
[19] Wenzelburger, J., On the smooth deformation of Hilbert space decompositions, Appendix in G. Schwarz. Hodge Decomposition — A Method for Solving Boundary Value Problems, (Lecture Notes in Mathematics, Vol. 1607 (1995), Springer: Springer Heidelberg), 107-112
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.