A General Linear Theory of Elastic Plates and its Variational Validation

Danilo Percivale; Paolo Podio-Guidugli

Bollettino dell'Unione Matematica Italiana (2009)

  • Volume: 2, Issue: 2, page 321-341
  • ISSN: 0392-4041

Abstract

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We provide a variational justification for shearable-plate models that generalize the classic Reissner-Mindlin model. Firstly, we give an argument leading to choose a fairly general linearly elastic monoclinic material response. Secondly, we prove that, for materials in such constitutive class, the variational limit of certain suitably scaled 3D energies is a functional whose minimum over a maximal subspace of admissible functions coincides with the minimum of the generalized Reissner-Mindlin functional.

How to cite

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Percivale, Danilo, and Podio-Guidugli, Paolo. "A General Linear Theory of Elastic Plates and its Variational Validation." Bollettino dell'Unione Matematica Italiana 2.2 (2009): 321-341. <http://eudml.org/doc/290561>.

@article{Percivale2009,
abstract = {We provide a variational justification for shearable-plate models that generalize the classic Reissner-Mindlin model. Firstly, we give an argument leading to choose a fairly general linearly elastic monoclinic material response. Secondly, we prove that, for materials in such constitutive class, the variational limit of certain suitably scaled 3D energies is a functional whose minimum over a maximal subspace of admissible functions coincides with the minimum of the generalized Reissner-Mindlin functional.},
author = {Percivale, Danilo, Podio-Guidugli, Paolo},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {6},
number = {2},
pages = {321-341},
publisher = {Unione Matematica Italiana},
title = {A General Linear Theory of Elastic Plates and its Variational Validation},
url = {http://eudml.org/doc/290561},
volume = {2},
year = {2009},
}

TY - JOUR
AU - Percivale, Danilo
AU - Podio-Guidugli, Paolo
TI - A General Linear Theory of Elastic Plates and its Variational Validation
JO - Bollettino dell'Unione Matematica Italiana
DA - 2009/6//
PB - Unione Matematica Italiana
VL - 2
IS - 2
SP - 321
EP - 341
AB - We provide a variational justification for shearable-plate models that generalize the classic Reissner-Mindlin model. Firstly, we give an argument leading to choose a fairly general linearly elastic monoclinic material response. Secondly, we prove that, for materials in such constitutive class, the variational limit of certain suitably scaled 3D energies is a functional whose minimum over a maximal subspace of admissible functions coincides with the minimum of the generalized Reissner-Mindlin functional.
LA - eng
UR - http://eudml.org/doc/290561
ER -

References

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  1. BRAIDES, A., Γ -convergence for beginners. Oxford University Press, 2002. MR1968440DOI10.1093/acprof:oso/9780198507840.001.0001
  2. DAL MASO, G., An Introduction to Γ -convergence. Birkhäuser, 1993. MR1201152DOI10.1007/978-1-4612-0327-8
  3. GURTIN, M. E., The Linear Theory of Elasticity. In Handbuch der PhysikVIa/2, C. Truesdell Ed., Springer1972. 
  4. LANCIONI, G. - PODIO-GUIDUGLI, P., Dynamics of incoherent elastic slabs. Fortcoming (2008). 
  5. LEMBO, M. - PODIO-GUIDUGLI, P., How to use reactive stresses to improve plate-theory approximations of the stress field in a linearly elastic plate-like body. Int. J. Sol. & Struct.44 (5) (2007), 1337-1369. Zbl1126.74027MR1916966DOI10.1023/A:1016109822140
  6. MIARA, B. - PODIO-GUIDUGLI, P., Une approche formelle unifiée des théories de plaques et poutres linéairement élastiques. C.R. Acad. Sci. Paris, Ser. I, 343, (2006), 675-678. Zbl1108.74036MR2271746DOI10.1016/j.crma.2006.09.035
  7. MIARA, B. - PODIO-GUIDUGLI, P., Deduction by scaling: a unified approach to classic plate and rod theories. Asympt. Anal., 51 (2) (2007), 113-131. Zbl1121.35036MR2311156
  8. PARONI, R. - PODIO-GUIDUGLI, P. - TOMASSETTI, G., The Reissner-Mindlin plate theory via Γ -convergence. C.R. Acad. Sci. Paris, Ser. I, 343 (2006), 437-440. Zbl1102.74031MR2259887DOI10.1016/j.crma.2006.08.006
  9. PARONI, R. - PODIO-GUIDUGLI, P. - TOMASSETTI, G., A justification of the Reissner-Mindlin plate theory through variational convergence. Anal.5 (2) (2007), 165-182. Zbl1109.74031MR2311707DOI10.1142/S0219530507000936
  10. PODIO-GUIDUGLI, P., An exact derivation of the thin plate equation. J. Elasticity, 22 (1989), 121-133. Zbl0692.73049MR1040755DOI10.1007/BF00041107
  11. PODIO-GUIDUGLI, P., Some recent results on Saint-Venant Problem. Pp. 35-45 of Atti dei Convegni Lincei N. 140, Giornata Lincea su "Il Problema di de Saint-Venant: Aspetti Teorici e Applicativi" (Roma, 6 March 1997). MR1783020

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