For a plate subject to stress boundary condition, the deformation determined by the Reissner-Mindlin plate bending model could be bending dominated, transverse shear dominated, or neither (intermediate), depending on the load. We show that the Reissner-Mindlin model has a wider range of applicability than the Kirchhoff-Love model, but it does not always converge to the elasticity theory. In the case of bending domination, both the two models are accurate. In the case of transverse shear domination, the Reissner-Mindlin model is accurate but the Kirchhoff-Love model totally fails. In the intermediate case, while the Kirchhoff-Love model fails, the Reissner-Mindlin solution also has a relative error comparing to the elasticity solution, which does not decrease when the plate thickness tends to zero. We also show that under the conventional definition of the resultant loading functional, the well known shear correction factor in the Reissner-Mindlin model should be replaced by . Otherwise, the range of applicability of the Reissner-Mindlin model is not wider than that of Kirchhoff-Love’s.
Mots clés : Reissner-Mindlin plate, shear correction factor, stress boundary condition
@article{M2AN_2006__40_2_269_0,
author = {Zhang, Sheng},
title = {On the accuracy of {Reissner-Mindlin} plate model for stress boundary conditions},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {269--294},
publisher = {EDP-Sciences},
volume = {40},
number = {2},
year = {2006},
doi = {10.1051/m2an:2006014},
zbl = {1137.74397},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an:2006014/}
}
TY - JOUR AU - Zhang, Sheng TI - On the accuracy of Reissner-Mindlin plate model for stress boundary conditions JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2006 SP - 269 EP - 294 VL - 40 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an:2006014/ DO - 10.1051/m2an:2006014 LA - en ID - M2AN_2006__40_2_269_0 ER -
%0 Journal Article %A Zhang, Sheng %T On the accuracy of Reissner-Mindlin plate model for stress boundary conditions %J ESAIM: Modélisation mathématique et analyse numérique %D 2006 %P 269-294 %V 40 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an:2006014/ %R 10.1051/m2an:2006014 %G en %F M2AN_2006__40_2_269_0
Zhang, Sheng. On the accuracy of Reissner-Mindlin plate model for stress boundary conditions. ESAIM: Modélisation mathématique et analyse numérique, Tome 40 (2006) no. 2, pp. 269-294. doi : 10.1051/m2an:2006014. http://www.numdam.org/articles/10.1051/m2an:2006014/
[1] ,, and, Derivation and justification of plate models by variational methods, Centre de Recherches Math., CRM Proc. Lecture Notes (1999). | MR | Zbl
[2] and, Asymptotic analysis of the boundary layer for the Reissner-Mindlin plate model. SIAM J. Math. Anal. 27 (1996) 486-514. | Zbl
[3] and, Asymptotic estimates of hierarchical modeling. Math. Mod. Meth. Appl. S. 13 (2003). | MR | Zbl
[4] , and, On the range of applicability of the Reissner-Mindlin and Kirchhoff-Love plate bending models, J. Elasticity 67 (2002) 171-185. | Zbl
[5] and, Interpolation space: An introduction, Springer-Verlag (1976). | MR | Zbl
[6] , Asymptotic convergence rates for the Kirchhoff plate model, Ph.D. Thesis, Pennsylvania State University (1995).
[7] , Mathematical elasticity, Vol II: Theory of plates. North-Holland (1997). | MR | Zbl
[8] , and, Full Asymptotic expansions for thin elastic free plates, C.R. Acad. Sci. Paris Sér. I. 326 (1998) 1243-1248. | Zbl
[9] , and, The influence of lateral boundary conditions on the asymptotics in thin elastic plates. SIAM J. Math. Anal. 31 (1999) 305-345. | Zbl
[10] , and, Plates and shells: Asymptotic expansions and hierarchical models, in Encyclopedia of computational mechanics, E. Stein, R. de Borst, T.J.R. Hughes Eds., John Wiley & Sons, Ltd. (2004).
[11] and, A boundary-layer theory for elastic plates. Comm. Pure Appl. Math. XIV (1961) 1-33. | Zbl
[12] , The finite element method: Linear static and dynamic finite element analysis. Prentice-Hall, Englewood Cliffs (1987). | MR | Zbl
[13] , On the foundations of linear theory of thin elastic shells. Proc. Kon. Ned. Akad. Wetensch. B73 (1970) 169-195. | Zbl
[14] , A treatise on the mathematical theory of elasticity. Cambridge University Press (1934). | JFM
[15] , Herleitung der Plattentheorie aus der dreidimensionalen Elastizitatstheorie. Arch. Rational Mech. Anal. 4 (1959) 145-152. | Zbl
[16] , The theory of shells and plates, in Handbuch der Physik, Springer-Verlag, Berlin, VIa/2 (1972) 425-640.
[17] and, Approximations in elasticity based on the concept of function space. Q. J. Math. 5 (1947) 241-269. | Zbl
[18] , Reflections on the theory of elastic plates. Appl. Mech. Rev. 38 (1985) 1453-1464.
[19] ,, and, On the mathematical foundation of the -plate model. Int. J. Solids Structures 36 (1999) 2143-2168. | Zbl
[20] and, Coques élastiques minces: Propriétés asymptotiques, Recherches en mathématiques appliquées, Masson, Paris (1997). | Zbl
[21] ,, Finite Element analysis. Wiley, New York (1991). | MR | Zbl
[22] , Stress boundary conditions for plate bending. Int. J. Solids Structures 40 (2003) 4107-4123. | Zbl
[23] , Equivalence estimates for a class of singular perturbation problems. C.R. Acad. Sci. Paris, Ser. I 342 (2006) 285-288. | Zbl
Cité par Sources :
