# American Institute of Mathematical Sciences

December  2016, 8(4): 391-411. doi: 10.3934/jgm.2016013

## The Frank tensor as a boundary condition in intrinsic linearized elasticity

 1 Universidade de Lisboa, Faculdade de Ciências, Departamento de Matemática, CMAF+CIO, Alameda da Universidade, C6, 1749-016 Lisboa, Portugal

Received  December 2015 Revised  September 2016 Published  November 2016

The Frank tensor plays a crucial role in linear elasticity, and in particular in the presence of dislocation lines, since its curl is exactly the elastic strain incompatibility. Furthermore, the Frank tensor also appears in Cesaro decomposition, and in Volterra theory of dislocations and disclinations, since its jump is the Frank vector around the defect line. The purpose of this paper is to show to which functional space the compatible strain $e$ belongs in order to imply a homogeneous boundary conditions for the induced displacement field on a portion $\Gamma_0$ of the boundary. This will allow one to define the homogeneous, or even the mixed problem of linearized elasticity in a variational setting involving the strain $e$ in place of displacement $u$. With other purposes, this problem was originaly treated by Ph. Ciarlet and C. Mardare, and termed the intrinsic formulation. In this paper we propose alternative conditions on $e$ expressed in terms of $e$ and the Frank tensor Curl$^t$ $e$ only, yielding a clear physical understanding and showing as equivalent to Ciarlet-Mardare boundary condition.
Citation: Nicolas Van Goethem. The Frank tensor as a boundary condition in intrinsic linearized elasticity. Journal of Geometric Mechanics, 2016, 8 (4) : 391-411. doi: 10.3934/jgm.2016013
##### References:
 [1] S. Amstutz and N. Van Goethem, Analysis of the incompatibility operator and application in intrinsic elasticity with dislocations, SIAM J. Math. Anal., 48 (2016), 320-348. doi: 10.1137/15M1020113. [2] R. Carroll, G. Duff, J. Friberg, J. Gobert, P. Grisvard, J. Nečas and R. Seeley, Équations Aux Dérivées Partielles, Séminaire de Mathématiques Supérieures. 19. Montréal: Les Presses de l'Université de Montréal, 1966. [3] P. G. Ciarlet, An introduction to differential geometry with applications to elasticity, J. Elasticity, 78/79 (2005), iv+215 pp. doi: 10.1007/s10659-005-4738-8. [4] P. G. Ciarlet, Three-Dimensional Elasticity, Vol.1, North-Holland, 1994. [5] P. G. Ciarlet and C. Mardare, Intrinsic formulation of the displacement-traction problem in linearized elasticity, Math. Models Methods Appl. Sci., 24 (2014), 1197-1216. doi: 10.1142/S0218202513500814. [6] G. Dal Maso, An Introduction to G-Convergence, Progress in Nonlinear Differential Equations and Their Applications. Birkhäuser Boston, 1993. doi: 10.1007/978-1-4612-0327-8. [7] M. C. Delfour and J.-P. Zolésio, Shapes and Geometries, volume 4 of Advances in Design and Control, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2001. [8] B. A. Dubrovin, A. T. Fomenko, and S. P. Novikov, Modern Geometry - Methods and Applications, Part 1 (2nd edn), Cambridge studies in advanced mathematics. Springer-Verlag, New-York, 1992. doi: 10.1007/978-1-4612-4398-4. [9] M. Epstein, The Geometrical Language of Continuum Mechanics, Cambridge University Press, 2010. doi: 10.1017/CBO9780511762673. [10] M. Epstein and M. Elzanowski, Material Inhomogeneities and their Evolution: A Geometric Approach, Interaction of Mechanics and Mathematics. Springer Berlin Heidelberg, 2007. [11] H. Kleinert, Gauge Fields in Condensed Matter, Vol.1, World Scientific Publishing, Singapore, 1989. [12] E. Kröner, Continuum theory of defects, In R. Balian, editor, Physiques des défauts, Les Houches session XXXV (Course 3), North-Holland, Amsterdam, 1980. [13] A. E. H. Love, A Treatise on the Mathematical Theory of Elasticity, Number vol. 1 in A Treatise on the Mathematical Theory of Elasticity. Cambridge University Press, 2013. [14] G. Maggiani, R. Scala and N. Van Goethem, A compatible-incompatible decomposition of symmetric tensors in $L^p$ with application to elasticity, Math. Meth. Appl. Sci, 38 (2015), 5217-5230. doi: 10.1002/mma.3450. [15] R. Scala and N. Van Goethem, Analytic and geometric properties of dislocation singularities, https://hal.archives-ouvertes.fr/hal-01297917, 2016. [16] R. Scala and N. Van Goethem, Currents and dislocations at the continuum scale, Methods Appl. Anal., 23 (2016), 1-34. doi: 10.4310/MAA.2016.v23.n1.a1. [17] J. A. Schouten, Ricci-Calculus (2nd edn), Springer Verlag, New York, 1978. [18] N. Van Goethem, The non-Riemannian dislocated crystal: A tribute to Ekkehart Kröner's (1919-2000), J. Geom. Mech., 2 (2010), 303-320. doi: 10.3934/jgm.2010.2.303. [19] N. Van Goethem, Direct expression of incompatibility in curvilinear systems, The ANZIAM J., 58 (2016), 33-50. doi: 10.1017/S1446181116000158. [20] N. Van Goethem, Incompatibility-governed singularities in linear elasticity with dislocations, Math. Mech. Solids, 2017. doi: 10.1177/1081286516642817.

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##### References:
 [1] S. Amstutz and N. Van Goethem, Analysis of the incompatibility operator and application in intrinsic elasticity with dislocations, SIAM J. Math. Anal., 48 (2016), 320-348. doi: 10.1137/15M1020113. [2] R. Carroll, G. Duff, J. Friberg, J. Gobert, P. Grisvard, J. Nečas and R. Seeley, Équations Aux Dérivées Partielles, Séminaire de Mathématiques Supérieures. 19. Montréal: Les Presses de l'Université de Montréal, 1966. [3] P. G. Ciarlet, An introduction to differential geometry with applications to elasticity, J. Elasticity, 78/79 (2005), iv+215 pp. doi: 10.1007/s10659-005-4738-8. [4] P. G. Ciarlet, Three-Dimensional Elasticity, Vol.1, North-Holland, 1994. [5] P. G. Ciarlet and C. Mardare, Intrinsic formulation of the displacement-traction problem in linearized elasticity, Math. Models Methods Appl. Sci., 24 (2014), 1197-1216. doi: 10.1142/S0218202513500814. [6] G. Dal Maso, An Introduction to G-Convergence, Progress in Nonlinear Differential Equations and Their Applications. Birkhäuser Boston, 1993. doi: 10.1007/978-1-4612-0327-8. [7] M. C. Delfour and J.-P. Zolésio, Shapes and Geometries, volume 4 of Advances in Design and Control, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2001. [8] B. A. Dubrovin, A. T. Fomenko, and S. P. Novikov, Modern Geometry - Methods and Applications, Part 1 (2nd edn), Cambridge studies in advanced mathematics. Springer-Verlag, New-York, 1992. doi: 10.1007/978-1-4612-4398-4. [9] M. Epstein, The Geometrical Language of Continuum Mechanics, Cambridge University Press, 2010. doi: 10.1017/CBO9780511762673. [10] M. Epstein and M. Elzanowski, Material Inhomogeneities and their Evolution: A Geometric Approach, Interaction of Mechanics and Mathematics. Springer Berlin Heidelberg, 2007. [11] H. Kleinert, Gauge Fields in Condensed Matter, Vol.1, World Scientific Publishing, Singapore, 1989. [12] E. Kröner, Continuum theory of defects, In R. Balian, editor, Physiques des défauts, Les Houches session XXXV (Course 3), North-Holland, Amsterdam, 1980. [13] A. E. H. Love, A Treatise on the Mathematical Theory of Elasticity, Number vol. 1 in A Treatise on the Mathematical Theory of Elasticity. Cambridge University Press, 2013. [14] G. Maggiani, R. Scala and N. Van Goethem, A compatible-incompatible decomposition of symmetric tensors in $L^p$ with application to elasticity, Math. Meth. Appl. Sci, 38 (2015), 5217-5230. doi: 10.1002/mma.3450. [15] R. Scala and N. Van Goethem, Analytic and geometric properties of dislocation singularities, https://hal.archives-ouvertes.fr/hal-01297917, 2016. [16] R. Scala and N. Van Goethem, Currents and dislocations at the continuum scale, Methods Appl. Anal., 23 (2016), 1-34. doi: 10.4310/MAA.2016.v23.n1.a1. [17] J. A. Schouten, Ricci-Calculus (2nd edn), Springer Verlag, New York, 1978. [18] N. Van Goethem, The non-Riemannian dislocated crystal: A tribute to Ekkehart Kröner's (1919-2000), J. Geom. Mech., 2 (2010), 303-320. doi: 10.3934/jgm.2010.2.303. [19] N. Van Goethem, Direct expression of incompatibility in curvilinear systems, The ANZIAM J., 58 (2016), 33-50. doi: 10.1017/S1446181116000158. [20] N. Van Goethem, Incompatibility-governed singularities in linear elasticity with dislocations, Math. Mech. Solids, 2017. doi: 10.1177/1081286516642817.
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