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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 |
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. |
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