# 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.  doi: 10.1137/15M1020113.  Google Scholar [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).   Google Scholar [3] P. G. Ciarlet, An introduction to differential geometry with applications to elasticity,, J. Elasticity, 78/79 (2005).  doi: 10.1007/s10659-005-4738-8.  Google Scholar [4] P. G. Ciarlet, Three-Dimensional Elasticity, Vol.1,, North-Holland, (1994).   Google Scholar [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.  doi: 10.1142/S0218202513500814.  Google Scholar [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.  Google Scholar [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), (2001).   Google Scholar [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, (1992).  doi: 10.1007/978-1-4612-4398-4.  Google Scholar [9] M. Epstein, The Geometrical Language of Continuum Mechanics,, Cambridge University Press, (2010).  doi: 10.1017/CBO9780511762673.  Google Scholar [10] M. Epstein and M. Elzanowski, Material Inhomogeneities and their Evolution: A Geometric Approach,, Interaction of Mechanics and Mathematics. Springer Berlin Heidelberg, (2007).   Google Scholar [11] H. Kleinert, Gauge Fields in Condensed Matter, Vol.1,, World Scientific Publishing, (1989).   Google Scholar [12] E. Kröner, Continuum theory of defects,, In R. Balian, (1980).   Google Scholar [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).   Google Scholar [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.  doi: 10.1002/mma.3450.  Google Scholar [15] R. Scala and N. Van Goethem, Analytic and geometric properties of dislocation singularities,, https://hal.archives-ouvertes.fr/hal-01297917, (2016).   Google Scholar [16] R. Scala and N. Van Goethem, Currents and dislocations at the continuum scale,, Methods Appl. Anal., 23 (2016), 1.  doi: 10.4310/MAA.2016.v23.n1.a1.  Google Scholar [17] J. A. Schouten, Ricci-Calculus (2nd edn),, Springer Verlag, (1978).   Google Scholar [18] N. Van Goethem, The non-Riemannian dislocated crystal: A tribute to Ekkehart Kröner's (1919-2000),, J. Geom. Mech., 2 (2010), 303.  doi: 10.3934/jgm.2010.2.303.  Google Scholar [19] N. Van Goethem, Direct expression of incompatibility in curvilinear systems,, The ANZIAM J., 58 (2016), 33.  doi: 10.1017/S1446181116000158.  Google Scholar [20] N. Van Goethem, Incompatibility-governed singularities in linear elasticity with dislocations,, Math. Mech. Solids, (2017).  doi: 10.1177/1081286516642817.  Google Scholar

show all references

##### 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.  doi: 10.1137/15M1020113.  Google Scholar [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).   Google Scholar [3] P. G. Ciarlet, An introduction to differential geometry with applications to elasticity,, J. Elasticity, 78/79 (2005).  doi: 10.1007/s10659-005-4738-8.  Google Scholar [4] P. G. Ciarlet, Three-Dimensional Elasticity, Vol.1,, North-Holland, (1994).   Google Scholar [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.  doi: 10.1142/S0218202513500814.  Google Scholar [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.  Google Scholar [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), (2001).   Google Scholar [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, (1992).  doi: 10.1007/978-1-4612-4398-4.  Google Scholar [9] M. Epstein, The Geometrical Language of Continuum Mechanics,, Cambridge University Press, (2010).  doi: 10.1017/CBO9780511762673.  Google Scholar [10] M. Epstein and M. Elzanowski, Material Inhomogeneities and their Evolution: A Geometric Approach,, Interaction of Mechanics and Mathematics. Springer Berlin Heidelberg, (2007).   Google Scholar [11] H. Kleinert, Gauge Fields in Condensed Matter, Vol.1,, World Scientific Publishing, (1989).   Google Scholar [12] E. Kröner, Continuum theory of defects,, In R. Balian, (1980).   Google Scholar [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).   Google Scholar [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.  doi: 10.1002/mma.3450.  Google Scholar [15] R. Scala and N. Van Goethem, Analytic and geometric properties of dislocation singularities,, https://hal.archives-ouvertes.fr/hal-01297917, (2016).   Google Scholar [16] R. Scala and N. Van Goethem, Currents and dislocations at the continuum scale,, Methods Appl. Anal., 23 (2016), 1.  doi: 10.4310/MAA.2016.v23.n1.a1.  Google Scholar [17] J. A. Schouten, Ricci-Calculus (2nd edn),, Springer Verlag, (1978).   Google Scholar [18] N. Van Goethem, The non-Riemannian dislocated crystal: A tribute to Ekkehart Kröner's (1919-2000),, J. Geom. Mech., 2 (2010), 303.  doi: 10.3934/jgm.2010.2.303.  Google Scholar [19] N. Van Goethem, Direct expression of incompatibility in curvilinear systems,, The ANZIAM J., 58 (2016), 33.  doi: 10.1017/S1446181116000158.  Google Scholar [20] N. Van Goethem, Incompatibility-governed singularities in linear elasticity with dislocations,, Math. Mech. Solids, (2017).  doi: 10.1177/1081286516642817.  Google Scholar
 [1] Helmut Abels, Andreas Marquardt. On a linearized Mullins-Sekerka/Stokes system for two-phase flows. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020467 [2] Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248 [3] Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020340 [4] Lingfeng Li, Shousheng Luo, Xue-Cheng Tai, Jiang Yang. A new variational approach based on level-set function for convex hull problem with outliers. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020070 [5] Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264 [6] Fabio Camilli, Giulia Cavagnari, Raul De Maio, Benedetto Piccoli. Superposition principle and schemes for measure differential equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020050 [7] Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020317 [8] Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020047 [9] Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $q$-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440 [10] Fathalla A. Rihan, Hebatallah J. Alsakaji. Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020468

2019 Impact Factor: 0.649