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DIRECT EXPRESSION OF INCOMPATIBILITY IN CURVILINEAR SYSTEMS

Part of: Classical differential geometry Elastic materials Generalities, axiomatics, foundations of continuum mechanics of solids Elliptic equations and systems

Published online by Cambridge University Press:  08 July 2016

NICOLAS VAN GOETHEM [Opens in a new window]
NICOLAS VAN GOETHEM
Affiliation:
Universidade de Lisboa, Faculdade de Ciências, Departamento de Matemática, CMAF+CIO, Alameda da Universidade, C6, 1749-016 Lisboa, Portugal email vangoeth@fc.ul.pt
Corresponding
E-mail address:
vangoeth@fc.ul.pt
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Abstract

We would like to present a method to compute the incompatibility operator in any system of curvilinear coordinates (components). The procedure is independent of the metric in the sense that the expression can be obtained by means of the basis vectors only, which are first defined as normal or tangential to the domain boundary, and then extended to the whole domain. It is an intrinsic method, to some extent, since the chosen curvilinear system depends solely on the geometry of the domain boundary. As an application, the in-extenso expression of incompatibility in a spherical system is given.

Keywords

incompatibility dislocations elasticity differential geometry curvilinear basis spherical system

MSC classification

Primary: 35J48: Higher-order elliptic systems
Secondary: 35J58: Boundary value problems for higher-order elliptic systems 53A05: Surfaces in Euclidean space 74A45: Theories of fracture and damage 74B99: None of the above, but in this section
Type
Research Article
Information
The ANZIAM Journal , Volume 58 , Issue 1 , July 2016 , pp. 33 - 50
Copyright
© 2016 Australian Mathematical Society 

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References

Amstutz, S. and Van Goethem, N., “Analysis of the incompatibility operator and application in intrinsic elasticity with dislocations”, SIAM J. Math. Anal. 48 (2016) 320–348 doi:10.1137/15M1020113.CrossRefGoogle Scholar
Brézis, H., “Functional analysis. Theory and applications. (Analyse fonctionnelle. Théorie et applications)”, in: Collection Mathématiques Appliquées pour la Maîtrise (Masson, Paris, 1994).Google Scholar
Ciarlet, P. G., “An introduction to differential geometry with applications to elasticity”, J. Elasticity 78–79 (2005) 3–201; doi:10.1007/s10659-005-4738-8.Google Scholar
Darboux, G., Leçons sur la théorie générale des surfaces et les applications géométriques du calcul infinitésimal. I. Généralités. Coordonnées curvilignes. Surfaces minima (Gauthier-Villars, Paris, 1941).Google Scholar
Delfour, M. C. and Zolésio, J.-P., Shapes and geometries, Volume 4 of Advances in Design and Control (Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2001).Google Scholar
Dubrovin, B. A., Fomenko, A. T. and Novikov, S. P., Modern geometry — methods and applications, Part 1, Volume 93 of Graduate Texts in Mathematics, 2nd edn (Springer-Verlag, New York, 1992).CrossRefGoogle Scholar
Guggenheimer, H. W., Differential geometry, McGraw-Hill Series in Higher Mathematics, 1st edn (McGraw-Hill, New York, 1963).Google Scholar
Hackl, K. and Fischer, F. D., “On the relation between the principle of maximum dissipation and inelastic evolution given by dissipation potentials”, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 464 (2008) 117–132; doi:10.1098/rspa.2007.0086.CrossRefGoogle Scholar
Hoger, A. and Johnson, B. E., “Linear elasticity for constrained materials: Incompressibility”, J. Elasticity 38 (1995) 69–93; doi:10.1007/BF00121464.CrossRefGoogle Scholar
Kröner, E., Continuum theory of defects, Physiques des défauts, Les Houches session XXXV (Course 3) (ed. Balian, R.), (North-Holland, Amsterdam, 1980).Google Scholar
Lelong-Ferrand, J., Elements de géomtrie différentielle (Cours de Sorbonne, Centre de Documentation Universitaire, Paris, 1959).Google Scholar
Maggiani, G., Scala, R. and Van Goethem, N., “A compatible-incompatible decomposition of symmetric tensors in $L^{p}$ with application to elasticity”, Math. Methods Appl. Sci. 38 (2015) 5217–5230; doi:10.1002/mma.3450.CrossRefGoogle Scholar
Malvern, L. E., Introduction to the mechanics of a continuous medium, Prentice-Hall Series in Engineering of the Physical Sciences (Prentice-Hall, Upper Saddle River, NJ, 1969).Google Scholar
Scala, R. and Van Goethem, N., “Constraint reaction and the Peach–Koehler force for dislocation networks”, Preprint, 2016; doi:10.1177/1081286516642817.Google Scholar
Van Goethem, N., “Fields of bounded deformation for mesoscopic dislocations”, Math. Mech. Solids 19 (2014) 579–600; doi:10.1177/1081286513479196.CrossRefGoogle Scholar
Van Goethem, N., “Incompatibility-governed singularities in linear elasticity with dislocations”, Math. Mech. Solids (2016); doi:10.1177/1081286516642817.Google Scholar
Van Goethem, N., de Potter, A., Van den Bogaert, N. and Dupret, F., “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.CrossRefGoogle Scholar
Van Goethem, N. and Dupret, F., “A distributional approach to $2D$ Volterra dislocations at the continuum scale”, European J. Appl. Math. 23 (2012) 417–439 doi:10.1017/S0956792512000010.CrossRefGoogle Scholar

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