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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.



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[1] Amstutz, S. and Van Goethem, N., “Analysis of the incompatibility operator and application in intrinsic elasticity with dislocations”, SIAM J. Math. Anal. 48 (2016) 320348 doi:10.1137/15M1020113.
[2] Brézis, H., “Functional analysis. Theory and applications. (Analyse fonctionnelle. Théorie et applications)”, in: Collection Mathématiques Appliquées pour la Maîtrise (Masson, Paris, 1994).
[3] Ciarlet, P. G., “An introduction to differential geometry with applications to elasticity”, J. Elasticity 78–79 (2005) 3201; doi:10.1007/s10659-005-4738-8.
[4] 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).
[5] 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).
[6] 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).
[7] Guggenheimer, H. W., Differential geometry, McGraw-Hill Series in Higher Mathematics, 1st edn (McGraw-Hill, New York, 1963).
[8] 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) 117132; doi:10.1098/rspa.2007.0086.
[9] Hoger, A. and Johnson, B. E., “Linear elasticity for constrained materials: Incompressibility”, J. Elasticity 38 (1995) 6993; doi:10.1007/BF00121464.
[10] Kröner, E., Continuum theory of defects, Physiques des défauts, Les Houches session XXXV (Course 3) (ed. Balian, R.), (North-Holland, Amsterdam, 1980).
[11] Lelong-Ferrand, J., Elements de géomtrie différentielle (Cours de Sorbonne, Centre de Documentation Universitaire, Paris, 1959).
[12] 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) 52175230; doi:10.1002/mma.3450.
[13] 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).
[14] Scala, R. and Van Goethem, N., “Constraint reaction and the Peach–Koehler force for dislocation networks”, Preprint, 2016; doi:10.1177/1081286516642817.
[15] Van Goethem, N., “Fields of bounded deformation for mesoscopic dislocations”, Math. Mech. Solids 19 (2014) 579600; doi:10.1177/1081286513479196.
[16] Van Goethem, N., “Incompatibility-governed singularities in linear elasticity with dislocations”, Math. Mech. Solids (2016); doi:10.1177/1081286516642817.
[17] 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) 320324 doi:10.1016/j.jpcs.2007.07.129.
[18] Van Goethem, N. and Dupret, F., “A distributional approach to $2D$ Volterra dislocations at the continuum scale”, European J. Appl. Math. 23 (2012) 417439 doi:10.1017/S0956792512000010.
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