# 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] Manh Hong Duong, Hoang Minh Tran. On the fundamental solution and a variational formulation for a degenerate diffusion of Kolmogorov type. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3407-3438. doi: 10.3934/dcds.2018146 [2] Xiaojun Chen, Guihua Lin. CVaR-based formulation and approximation method for stochastic variational inequalities. Numerical Algebra, Control & Optimization, 2011, 1 (1) : 35-48. doi: 10.3934/naco.2011.1.35 [3] Chjan C. Lim, Junping Shi. The role of higher vorticity moments in a variational formulation of Barotropic flows on a rotating sphere. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 717-740. doi: 10.3934/dcdsb.2009.11.717 [4] Philippe G. Ciarlet, Liliana Gratie, Cristinel Mardare. Intrinsic methods in elasticity: a mathematical survey. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 133-164. doi: 10.3934/dcds.2009.23.133 [5] Yuri B. Suris. Variational formulation of commuting Hamiltonian flows: Multi-time Lagrangian 1-forms. Journal of Geometric Mechanics, 2013, 5 (3) : 365-379. doi: 10.3934/jgm.2013.5.365 [6] George Avalos, Thomas J. Clark. A mixed variational formulation for the wellposedness and numerical approximation of a PDE model arising in a 3-D fluid-structure interaction. Evolution Equations & Control Theory, 2014, 3 (4) : 557-578. doi: 10.3934/eect.2014.3.557 [7] François Gay-Balmaz, Tudor S. Ratiu. Clebsch optimal control formulation in mechanics. Journal of Geometric Mechanics, 2011, 3 (1) : 41-79. doi: 10.3934/jgm.2011.3.41 [8] Matthew M. Dunlop, Andrew M. Stuart. The Bayesian formulation of EIT: Analysis and algorithms. Inverse Problems & Imaging, 2016, 10 (4) : 1007-1036. doi: 10.3934/ipi.2016030 [9] Haiyang Wang, Jianfeng Zhang. Forward backward SDEs in weak formulation. Mathematical Control & Related Fields, 2018, 8 (3&4) : 1021-1049. doi: 10.3934/mcrf.2018044 [10] Mehdi Badsi, Martin Campos Pinto, Bruno Després. A minimization formulation of a bi-kinetic sheath. Kinetic & Related Models, 2016, 9 (4) : 621-656. doi: 10.3934/krm.2016010 [11] Azmy S. Ackleh, Ben G. Fitzpatrick, Horst R. Thieme. Rate distributions and survival of the fittest: a formulation on the space of measures. Discrete & Continuous Dynamical Systems - B, 2005, 5 (4) : 917-928. doi: 10.3934/dcdsb.2005.5.917 [12] André Nachbin, Roberto Ribeiro-Junior. A boundary integral formulation for particle trajectories in Stokes waves. Discrete & Continuous Dynamical Systems - A, 2014, 34 (8) : 3135-3153. doi: 10.3934/dcds.2014.34.3135 [13] Lorenzo Brasco, Filippo Santambrogio. An equivalent path functional formulation of branched transportation problems. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 845-871. doi: 10.3934/dcds.2011.29.845 [14] Xiaoying Han, Jinglai Li, Dongbin Xiu. Error analysis for numerical formulation of particle filter. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1337-1354. doi: 10.3934/dcdsb.2015.20.1337 [15] Andaluzia Matei, Mircea Sofonea. Dual formulation of a viscoplastic contact problem with unilateral constraint. Discrete & Continuous Dynamical Systems - S, 2013, 6 (6) : 1587-1598. doi: 10.3934/dcdss.2013.6.1587 [16] Francesco Demontis, Cornelis Van der Mee. Novel formulation of inverse scattering and characterization of scattering data. Conference Publications, 2011, 2011 (Special) : 343-350. doi: 10.3934/proc.2011.2011.343 [17] Qiang Du, Manlin Li. On the stochastic immersed boundary method with an implicit interface formulation. Discrete & Continuous Dynamical Systems - B, 2011, 15 (2) : 373-389. doi: 10.3934/dcdsb.2011.15.373 [18] Wenjun Xia, Jinzhi Lei. Formulation of the protein synthesis rate with sequence information. Mathematical Biosciences & Engineering, 2018, 15 (2) : 507-522. doi: 10.3934/mbe.2018023 [19] Lorena Bociu, Steven Derochers, Daniel Toundykov. Linearized hydro-elasticity: A numerical study. Evolution Equations & Control Theory, 2016, 5 (4) : 533-559. doi: 10.3934/eect.2016018 [20] Paolo Baroni, Agnese Di Castro, Giampiero Palatucci. Intrinsic geometry and De Giorgi classes for certain anisotropic problems. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 647-659. doi: 10.3934/dcdss.2017032

2018 Impact Factor: 0.525