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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.
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).
doi: 10.1007/s10659-005-4738-8. |
[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. |
[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), (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, (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, (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. |
[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. |
[17] |
J. A. Schouten, Ricci-Calculus (2nd edn),, Springer Verlag, (1978).
|
[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. |
[19] |
N. Van Goethem, Direct expression of incompatibility in curvilinear systems,, The ANZIAM J., 58 (2016), 33.
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.
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).
doi: 10.1007/s10659-005-4738-8. |
[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. |
[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), (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, (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, (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. |
[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. |
[17] |
J. A. Schouten, Ricci-Calculus (2nd edn),, Springer Verlag, (1978).
|
[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. |
[19] |
N. Van Goethem, Direct expression of incompatibility in curvilinear systems,, The ANZIAM J., 58 (2016), 33.
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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