The Frank tensor as a boundary condition in intrinsic linearized elasticity

Previous Article
The Tulczyjew triple in mechanics on a Lie group
JGM Home
This Issue
Next Article
No monodromy in the champagne bottle, or singularities of a superintegrable system

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

The Frank tensor as a boundary condition in intrinsic linearized elasticity

Nicolas Van Goethem ^1,

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

Abstract
Related Papers

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.

Keywords: linearized elasticity, Intrinsic formulation, Frank tensor, incompatibility, variational formulation, boudary condition, differential geometry..

Mathematics Subject Classification: Primary: 35J48, 35J58; Secondary: 53A05, 74A45, 74B9.

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.

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), 320-348. 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.

[1]	Manh Hong Duong, Hoang Minh Tran. On the fundamental solution and a variational formulation for a degenerate diffusion of Kolmogorov type. Discrete and Continuous Dynamical Systems, 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 and Optimization, 2011, 1 (1) : 35-48. doi: 10.3934/naco.2011.1.35
[3]	Jianxun Liu, Shengjie Li, Yingrang Xu. Quantitative stability of the ERM formulation for a class of stochastic linear variational inequalities. Journal of Industrial and Management Optimization, 2021 doi: 10.3934/jimo.2021083
[4]	Chjan C. Lim, Junping Shi. The role of higher vorticity moments in a variational formulation of Barotropic flows on a rotating sphere. Discrete and Continuous Dynamical Systems - B, 2009, 11 (3) : 717-740. doi: 10.3934/dcdsb.2009.11.717
[5]	Philippe G. Ciarlet, Liliana Gratie, Cristinel Mardare. Intrinsic methods in elasticity: a mathematical survey. Discrete and Continuous Dynamical Systems, 2009, 23 (1&2) : 133-164. doi: 10.3934/dcds.2009.23.133
[6]	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
[7]	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 and Control Theory, 2014, 3 (4) : 557-578. doi: 10.3934/eect.2014.3.557
[8]	Evelyn Herberg, Michael Hinze. Variational discretization of one-dimensional elliptic optimal control problems with BV functions based on the mixed formulation. Mathematical Control and Related Fields, 2022 doi: 10.3934/mcrf.2022013
[9]	Odo Diekmann, Francesca Scarabel, Rossana Vermiglio. Pseudospectral discretization of delay differential equations in sun-star formulation: Results and conjectures. Discrete and Continuous Dynamical Systems - S, 2020, 13 (9) : 2575-2602. doi: 10.3934/dcdss.2020196
[10]	Haiyang Wang, Jianfeng Zhang. Forward backward SDEs in weak formulation. Mathematical Control and Related Fields, 2018, 8 (3&4) : 1021-1049. doi: 10.3934/mcrf.2018044
[11]	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
[12]	Matthew M. Dunlop, Andrew M. Stuart. The Bayesian formulation of EIT: Analysis and algorithms. Inverse Problems and Imaging, 2016, 10 (4) : 1007-1036. doi: 10.3934/ipi.2016030
[13]	Lorena Bociu, Steven Derochers, Daniel Toundykov. Linearized hydro-elasticity: A numerical study. Evolution Equations and Control Theory, 2016, 5 (4) : 533-559. doi: 10.3934/eect.2016018
[14]	Mehdi Badsi, Martin Campos Pinto, Bruno Després. A minimization formulation of a bi-kinetic sheath. Kinetic and Related Models, 2016, 9 (4) : 621-656. doi: 10.3934/krm.2016010
[15]	Azmy S. Ackleh, Ben G. Fitzpatrick, Horst R. Thieme. Rate distributions and survival of the fittest: a formulation on the space of measures. Discrete and Continuous Dynamical Systems - B, 2005, 5 (4) : 917-928. doi: 10.3934/dcdsb.2005.5.917
[16]	André Nachbin, Roberto Ribeiro-Junior. A boundary integral formulation for particle trajectories in Stokes waves. Discrete and Continuous Dynamical Systems, 2014, 34 (8) : 3135-3153. doi: 10.3934/dcds.2014.34.3135
[17]	Xiaoying Han, Jinglai Li, Dongbin Xiu. Error analysis for numerical formulation of particle filter. Discrete and Continuous Dynamical Systems - B, 2015, 20 (5) : 1337-1354. doi: 10.3934/dcdsb.2015.20.1337
[18]	Lorenzo Brasco, Filippo Santambrogio. An equivalent path functional formulation of branched transportation problems. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 845-871. doi: 10.3934/dcds.2011.29.845
[19]	Andaluzia Matei, Mircea Sofonea. Dual formulation of a viscoplastic contact problem with unilateral constraint. Discrete and Continuous Dynamical Systems - S, 2013, 6 (6) : 1587-1598. doi: 10.3934/dcdss.2013.6.1587
[20]	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

2020 Impact Factor: 0.857

Tools

Metrics

Other articles
by authors

on AIMS
Nicolas Van Goethem
on Google Scholar
Nicolas Van Goethem

[Back to Top]