 | | Mechanics of Solids
A Journal of Russian Academy of Sciences | | Founded
in January 1966
Issued 6 times a year
Print ISSN 0025-6544
Online ISSN 1934-7936 |
Archive of Issues
| Total articles in the database: | | 12804 |
| In Russian (Èçâ. ÐÀÍ. ÌÒÒ): | | 8044
|
| In English (Mech. Solids): | | 4760 |
|
| << Previous article | Volume 41, Issue 5 / 2006 | Next article >> |
| M. C. Delfour, "Intrinsic differential geometric methods in the asymptotic analysis of linear thin shells," Mech. Solids. 41 (5), 66-129 (2006) |
| Year |
2006 |
Volume |
41 |
Number |
5 |
Pages |
66-129 |
| Title |
Intrinsic differential geometric methods in the asymptotic analysis of linear thin shells |
| Author(s) |
M. C. Delfour (Montréal, Canada) |
| Abstract |
In earlier papers a completely intrinsic differential calculus on C1,1 submanifolds of
codimension one in ℝN has been developed by a marriage of tangential derivatives and the
oriented distance function. Its potential has been illustrated by investigating some linear
models of thin shells based on truncated series expansions with respect to the variable
normal to the midsurface. In this paper we analyze the asymptotic behavior of three models
for an arbitrary constitutive law. Given a midsurface with Lipschitzian boundary in a
C1,1 submanifold of ℝN, we show that solutions of the intrinsic linear P(1,1),
P(2n,1) and P(2,1) models of thin shells converge to solutions of asymptotic shell models which consist of a coupled system of two variational equations. The first is
the asymptotic P(1,0) model which yields the generally accepted classical membrane shell
equation and the Love-Kirchhoff terms. The second is a generalized bending equation. In the
bending dominated case and for the special constitutive law specified by two Lamé
constants, the quadratic term of the second equation of the asymptotic P(2n,1) and
P(2,1) models is the classical bending dominated equation. Also Naghdi's model is an
approximation of the reduced P(2,1) model and Koiter's is a projection of Naghdi's. A
detailed analysis of the three asymptotic models is given: existence and spaces of
solutions, decomposition of the equations, and construction of the associated effective
constitutive laws. Strong/weak convergence is established in the natural spaces and norms
under an assumption on the asymptotic behavior of the constant of continuity of the
right-hand side for shells without boundary or shells with homogeneous Neumann boundary
conditions (quotient space) or homogeneous Dirichlet boundary conditions on a part of the
boundary. |
| References |
| 1. | E. Acerbi, G. Buttazzo, and D. Percivale, "A variational definition for the strain energy
of an elastic string," J. Elasticity, Vol. 25, pp. 137-148, 1991. |
| 2. | S. M. Alessandrini,
Some Two-Dimensional Plate
Models: Derivation, Asymptotic Properties, and Numerical Approximation. Ph.D. Thesis,
Rutgers University, New Brunswick, New Jersey, 1991. |
| 3. | S. M. Alessandrini, D. N. Arnold, R. S. Falk, and A. L. Madureira,
"Derivation and justification of
plate models by variational methods," in M. Fortin (Editor),
"Plates and Shells", CRM Proc. Lect. Notes ser., Vol. 21, pp. 1-20, AMS
Publications, Providence, R.I., 1999. |
| 4. | M. Bernadou, Méthodes d'éléments finis pour les problèmes de coques minces, Masson,
Paris, Milan, Barcelone, 1994. |
| 5. | M. Bernadou, Ph. G. Ciarlet, and B. Miara, "Existence theorems for two-dimensional linear
shell theories," J. Elasticity, Vol. 34, pp. 111-138, 1994. |
| 6. | A. Blouza, "Existence et unicité pour le modèle de Naghdi pour une coque peu
régulière," C. R. Acad. Sci. Paris Sér. I Math., Vol. 324, pp. 839-844, 1997. |
| 7. | A. Blouza and H. Le Dret, "Existence and uniqueness for the linear Koiter model for shells
with little regularity," Quart. Appl. Math., Vol. 57, No. 2, pp. 317-337, 1999. |
| 8. | F. Bourquin, Ph. G. Ciarlet, G. Geymonat, and A. Raoult, "Γ-convergence et analyse
asymptotique des plaques minces," C. R. Acad. Sci. Paris Sér. I Math., Vol. 315,
pp. 1017-1024, 1992. |
| 9. | F. Brezzi and M. Fortin,
Mixed and Hybrid Finite
Element Methods, Springer-Verlag, New York, Berlin, Heidelberg, 1991. |
| 10. | D. Chapelle and K.-J. Bathe,
"Fundamental considerations for the
finite element analysis of shell structures," Computers and structures, 1997 (submitted). |
| 11. | C. Chen, Asymptotic Convergence Rates for the Kirchhoff Plate Model. Ph.D. Thesis,
University Park, Pennsylvania, Penn. State, 1995. |
| 12. | D. Chenais and J.-C. Paumier,
"On the locking phenomenon for a class of elliptic
problems," Numer. Math., Vol. 67, pp. 427-440, 1994. |
| 13. | Ph.G. Ciarlet,
Plates and Junctions in Elastic
Multi-Structures: an Asymptotic Analysis, Springer-Verlag, Berlin, New York, 1990. |
| 14. | Ph. G. Ciarlet and V. Lods, "Ellipticité des équations membranaires d'une coque
uniformément elliptique," C. R. Acad. Sci. Paris Sér. I Math., Vol. 318, pp. 195-200,
1994. |
| 15. | Ph. G. Ciarlet and V. Lods, "Analyse asymptotique des coques linéairement élastique.
I. Coques membranaires," C. R. Acad. Sci. Paris Sér. I Math., Vol. 318,
pp. 863-868, 1994. |
| 16. | Ph. G. Ciarlet and V. Lods, "Analyse asymptotique des coques linéairement élastique.
III. Une justification du modèle de Koiter," C. R. Acad. Sci. Paris Sér. I
Math., Vol. 319, pp. 299-304, 1994. |
| 17. | Ph. G. Ciarlet and V. Lods, "On the ellipticity of linear membrane shell equations," J.
Math. Pures Appl., Vol. 75, pp. 107-124, 1996. |
| 18. | Ph. G. Ciarlet and V. Lods, "Asymptotic analysis of linearly elastic shells. III.
Justification of Koiter's shell equations," Arch. Rational Mech. Anal., Vol. 136,
pp. 191-200, 1996. |
| 19. | Ph. G. Ciarlet and V. Lods, "Asymptotic analysis of linearly elastic shells. Generalized
membrane shells," J. Elasticity, Vol. 43, pp. 147-188, 1996. |
| 20. | Ph. G. Ciarlet and V. Lods, "Asymptotic analysis of linearly elastic shells. I.
Justification of membrane shell equations," Arch. Rational Mech. Anal., Vol. 136,
pp. 116-161, 1996. |
| 21. | Ph. G. Ciarlet, V. Lods, and B. Miara, "Analyse asymptotique des coques linéairement
élastique. II. Coques en flexion," C. R. Acad. Sci. Paris Sér. I Math.,
Vol. 319, pp. 95-100, 1994. |
| 22. | Ph. G. Ciarlet, V. Lods, and B. Miara, "Asymptotic analysis of linearly elastic shells.
II. Justification of flexural shell equations," Arch. Rational Mech. Anal.,
Vol. 136, pp. 163-190, 1996. |
| 23. | Ph. G. Ciarlet and É. Sanchez-Palencia, "Un théorème d'existence et d'unicité pour
les équations des coques membranaires," C. R. Acad. Sci. Paris Sér. I Math., Vol. 317,
pp. 801-805, 1993. |
| 24. | Ph. G. Ciarlet and É. Sanchez-Palencia, "An existence and uniqueness theorem for the
two-dimensional linear membrane shell equations," J. Math. Pures Appl., Vol. 75,
pp. 51-67, 1996. |
| 25. | M. Dauge, Complete Asymptotics in Thin Elastic Plates and Optimal Estimates for
Kirchhoff-Love Model. Prépublication 95-06, Institut de Recherche Mathématique de
Rennes, Université de Rennes, février 1995. |
| 26. | M. Dauge and I. Gruais, "Développement asymptotique d'ordre arbitraire pour une plaque
élastique mince encastrée," C. R. Acad. Sc. Paris Sér. I Math., Vol. 321,
pp. 375-380, 1995. |
| 27. | M. C. Delfour,
"Intrinsic P(2,1) thin shell model and Naghdi's models without a
priori assumption on the stress tensor,"
in K. H. Hoffmann, G. Leugering, and F. Tröltzsch (Editors),
Proc International Conference on Optimal Control of Partial Differential Equations.
Int. Ser. Of Numerical Mathematics, Vol. 133, pp. 99-113, Birkhäuser Verlag, Basel, 1999. |
| 28. | M. C. Delfour,
"Membrane shell equation: characterization of the space of
solutions," in Shuping Chen, Xunjing Li, Jiongmin Yong, and Xun Yu Zhou (Editors),
Control of Distributed Parameter and Stochastic Systems, pp. 21-29, Chapman and Hall,
New York, 1999. |
| 29. | M. C. Delfour,
"Characterization of the space of the membrane shell equation
for arbitrary C1,1 midsurfaces," Control and Cybernetics, Vol. 28, No. 3,
pp. 481-501, 1999. |
| 30. | M. C. Delfour,
"Tangential differential calculus and functional analysis
on a C1,1 submanifold,"
in R. Gulliver, W. Littman and R. Triggiani (Editors), Differential-Geometric Methods in the
Control of Partial Differential Equations. Contemporary Mathematics, AMS Publications, in
press. |
| 31. | M. C. Delfour and J.-P. Zolésio, "Shape analysis via distance functions," J. Funct.
Anal., Vol. 123, pp. 129-201, 1994. |
| 32. | M. C. Delfour and J.-P. Zolésio, "On a variational equation for thin shells," Control
and Optimal Design of Distributed Parameter Systems (J. Lagnese, D. L. Russell, and
L. White, eds.), Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, pp. 25-37, 1994. |
| 33. | M. C. Delfour and J.-P. Zolésio, "A boundary differential equation for thin shells," J.
Differential Equations, Vol. 119, pp. 426-449, 1995. |
| 34. | M. C. Delfour and J.-P. Zolésio, "Tangential differential equations for dynamical
thin/shallow shells," J. Differential Equations, Vol. 128, pp. 125-167, 1996. |
| 35. | M. C. Delfour and J.-P. Zolésio, "Differential equations for linear shells:
comparison between intrinsic and classical models," in Luc Vinet (Editor), Advances in
Mathematical Sciences-CRM's 25 years. CRM Proc. Lecture Notes, pp. 42-124, AMS
Publications, Providence, RI, 1997. |
| 36. | M. C. Delfour and J.-P. Zolésio, "Shape analysis via distance functions: local
theory," in M. Delfour (Editor), Boundaries, interfaces and transitions. CRM Proc. Lect.
Notes Ser., pp. 91-123, AMS Publications, Providence, R.I., 1998. |
| 37. | M. C. Delfour and J.-P. Zolésio, "On the design and control of systems governed by
differential equations on submanifolds," Control Cybernet., Vol. 25, pp. 497-514, 1996. |
| 38. | M. C. Delfour and J.-P. Zolésio, "Hidden boundary smoothness for some classes of
differential equations on submanifolds," in S. Cox and I. Lasiecka (Editors), Optimization
Methods in Partial Differential Equations. Contemp. Math., Vol. 209, pp. 59-73, AMS
Publications, Providence, R.I., 1997. |
| 39. | M. C. Delfour and J.-P. Zolésio, Intrinsic Differential Geometry and Theory of Thin
Shells. Lecture Notes. Version 1.0, Scuola Normale Superiore, Pisa (Italy), August 1996. |
| 40. | M. C. Delfour and J.-P. Zolésio, "Convergence to the asymptotic model for linear thin
shells," in S. Cox and I. Lasiecka (Editors), Optimization Methods in Partial Differential
Equations Contemp. Math., Vol. 209, AMS Publications, Providence, R.I., 1977. |
| 41. | M. C. Delfour and J.-P. Zolésio, "Convergence of the linear P(1,1) and P(2,1) thin
shells to asymptotic shells," in M. Fortin (Editor), Plates and Shells. CRM Proc. Lect.
Notes Ser., Vol. 21, pp. 125-158, AMS Publications, Providence, R.I., 1999. |
| 42. | Ph. Destuynder, Sur la Justification des Modèles de Plaques et de Coques par les
Méthodes Asymptotiques. Doctoral dissertation, Université Pierre et Marie Curie, 1980. |
| 43. | Ph. Destuynder, Modélisation des Coques Minces Élastiques, Masson, Paris, Milan,
Barcelone, 1990. |
| 44. | Ph. Destuynder, Une Théorie Asymptotique des Plaques Minces en Elasticité Linéaire,
Masson, Paris, Milan, Barcelone, 1986. |
| 45. | H. Federer, "Curvature measures," Trans. Amer. Math. Soc., Vol. 93, pp. 418-419, 1959. |
| 46. | D. D. Fox, A. Raoult, and
J. C. Simo, "A justification of nonlinear properly invariant plate theories," Arch.
Rational Mech. Anal., Vol. 124, pp. 157-199, 1993. |
| 47. | D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,
Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1983. |
| 48. | H. Le Dret and A. Raoult, "The nonlinear membrane model as variational limit of nonlinear
three-dimensional elasticity," J. Math. Pures Appl., Vol. 74, pp. 549-578, 1995. |
| 49. | H. Le Dret and A. Raoult, "The membrane shell model in nonlinear elasticity: a
variational asymptotic derivation," J. Nonlinear Sci., Vol. 6, pp. 59-84, 1996. |
| 50. | K. H. Lo, R. M. Christensen, and E. M. Wu, "A high-order theory of plate deformations," J.
Appl. Mech., Vol. 46, pp. 663-676, 1977. |
| 51. | C. Mardare, "Modèles bi-dimensionnels de coques linéairement élastiques:
estimations de l'écart entre leurs solutions," C. R. Acad. Sc. Paris Sér. I Math.,
Vol. 322, pp. 793-796, 1996. |
| 52. | C. Mardare, "Estimation d'erreur dans l'analyse asymptotique des coques linéairement
élastiques," C. R. Acad. Sc. Paris Sér. I Math., Vol. 322, pp. 895-898, 1996. |
| 53. | D. Morgenstern, "Herleitung der Plattentheorie aus der dreidimensionalen
Elastizitätstheorie," Arch. Rational Mech. Anal., Vol. 4, pp. 145-152, 1959. |
| 54. | P. M. Naghdi, "Foundations of elastic theory," Progress in Solid Mechanics, Amsterdam,
North-Holland, Vol. 4, pp. 1-90, 1963. |
| 55. | P. M. Naghdi, "The theory of shells and plates," Handbuch des Physik, Springer-Verlag,
Berlin, Vol. VI a-2, pp. 425-640, 1972. |
| 56. | J.-C. Paumier and A. Raoult, "Asymptotic consistency of the polynomial approximation in the
linearized plate theory. Application to the Reissner-Mindlin model," in Élasticité,
viscoélasticité et contrôle optimal (Lyon, 1995), ESAIM Proc., 2, Soc. Math. Appl.
Indust., Paris, pp. 203-213 (electronic), 1997. |
| 57. | J. Piila, "Characterization of the membrane theory of a clamped shell. The hyperbolic
case," Math. Models and Methods Appl. Sci., Vol. 6, pp. 169-194, 1996. |
| 58. | J. Pitkaranta, "The problem of membrane locking in finite element analysis of cylindrical
shells," Numer. Math., Vol. 61, pp. 523-542, 1992. |
| 59. | É. Sanchez-Palencia,
"Statique et dynamique des coques
minces. I. Cas de flexion pure non inhibée," C. R. Acad. Sc. Paris Sér. I
Math., Vol. 309, pp. 411-417, 1989. |
| 60. | É. Sanchez-Palencia, "Statique et dynamique des coques minces. II. Cas de
flexion pure inhibée - Approximation membranaire," C. R. Acad. Sc. Paris Sér. I Math.,
Vol. 309, pp. 531-537, 1989. |
| 61. | É. Sanchez-Palencia, "Surfaces et coques élastiques minces: problèmes et
défis," La Vie des Sciences, Vol. 12, No. 3, pp. 239-258, 1995. |
| 62. | J. Sanchez-Hubert and É. Sanchez-Palencia, Coques Élastiques Minces, propriétés
asymptotiques, Masson, Paris, 1997. |
| 63. | R. Valid, The Nonlinear Theory of Shells Through Variational Principles, Wiley, Chichester,
1995. |
|
| Received |
15 January 2005 |
| << Previous article | Volume 41, Issue 5 / 2006 | Next article >> |
|
 If you find a misprint on a webpage, please help us correct it promptly - just highlight and press Ctrl+Enter
|
|
Russian 
English