On the accuracy of Reissner-Mindlin plate model for stress boundary conditions
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 40 (2006) no. 2, p. 269-294

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 $5/6$ in the Reissner-Mindlin model should be replaced by $1$. Otherwise, the range of applicability of the Reissner-Mindlin model is not wider than that of Kirchhoff-Love’s.

DOI : https://doi.org/10.1051/m2an:2006014
Classification:  73C02,  73K10
Keywords: 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: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {40},
number = {2},
year = {2006},
pages = {269-294},
doi = {10.1051/m2an:2006014},
zbl = {1137.74397},
language = {en},
url = {http://www.numdam.org/item/M2AN_2006__40_2_269_0}
}

Zhang, Sheng. On the accuracy of Reissner-Mindlin plate model for stress boundary conditions. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 40 (2006) no. 2, pp. 269-294. doi : 10.1051/m2an:2006014. http://www.numdam.org/item/M2AN_2006__40_2_269_0/

[1] S.M. Alessandrini, D.N. Arnold, R.S. Falk and A.L. Madureira, Derivation and justification of plate models by variational methods, Centre de Recherches Math., CRM Proc. Lecture Notes (1999). | MR 1696513 | Zbl 0958.74033

[2] D.N. Arnold and R.S. Falk, Asymptotic analysis of the boundary layer for the Reissner-Mindlin plate model. SIAM J. Math. Anal. 27 (1996) 486-514. | Zbl 0846.73027

[3] D.N. Arnold and A. Mardureira, Asymptotic estimates of hierarchical modeling. Math. Mod. Meth. Appl. S. 13 (2003). | MR 2005646 | Zbl 1046.35024

[4] D.N. Arnold, A. Mardureira and S. Zhang, On the range of applicability of the Reissner-Mindlin and Kirchhoff-Love plate bending models, J. Elasticity 67 (2002) 171-185. | Zbl 1089.74595

[5] J. Bergh and J. Lofstrom, Interpolation space: An introduction, Springer-Verlag (1976). | MR 482275 | Zbl 0344.46071

[6] C. Chen, Asymptotic convergence rates for the Kirchhoff plate model, Ph.D. Thesis, Pennsylvania State University (1995).

[7] P.G. Ciarlet, Mathematical elasticity, Vol II: Theory of plates. North-Holland (1997). | MR 1477663 | Zbl 0953.74004

[8] M. Dauge, I. Djurdjevic and A. Rössle, Full Asymptotic expansions for thin elastic free plates, C.R. Acad. Sci. Paris Sér. I. 326 (1998) 1243-1248. | Zbl 0916.73021

[9] M. Dauge, I. Gruais and A. Rössle, The influence of lateral boundary conditions on the asymptotics in thin elastic plates. SIAM J. Math. Anal. 31 (1999) 305-345. | Zbl 0958.74034

[10] M. Dauge, E. Faou and Z. Yosibash, 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] K.O. Friedrichs and R.F. Dressler, A boundary-layer theory for elastic plates. Comm. Pure Appl. Math. XIV (1961) 1-33. | Zbl 0096.40001

[12] T.J.R. Hughes, The finite element method: Linear static and dynamic finite element analysis. Prentice-Hall, Englewood Cliffs (1987). | MR 1008473 | Zbl 0634.73056

[13] W.T. Koiter, On the foundations of linear theory of thin elastic shells. Proc. Kon. Ned. Akad. Wetensch. B73 (1970) 169-195. | Zbl 0213.27002

[14] A.E.H. Love, A treatise on the mathematical theory of elasticity. Cambridge University Press (1934). | JFM 53.0752.01

[15] D. Morgenstern, Herleitung der Plattentheorie aus der dreidimensionalen Elastizitatstheorie. Arch. Rational Mech. Anal. 4 (1959) 145-152. | Zbl 0126.20605

[16] P.M. Naghdi, The theory of shells and plates, in Handbuch der Physik, Springer-Verlag, Berlin, VIa/2 (1972) 425-640.

[17] W. Prager and J.L. Synge, Approximations in elasticity based on the concept of function space. Q. J. Math. 5 (1947) 241-269. | Zbl 0029.23505

[18] E. Reissner, Reflections on the theory of elastic plates. Appl. Mech. Rev. 38 (1985) 1453-1464.

[19] A. Rössle, M. Bischoff, W. Wendland and E. Ramm, On the mathematical foundation of the $\left(1,1,2\right)$-plate model. Int. J. Solids Structures 36 (1999) 2143-2168. | Zbl 0962.74036

[20] J. Sanchez-Hubert and E. Sanchez-Palencia, Coques élastiques minces: Propriétés asymptotiques, Recherches en mathématiques appliquées, Masson, Paris (1997). | Zbl 0881.73001

[21] B. Szabó, I. Babuska, Finite Element analysis. Wiley, New York (1991). | MR 1164869 | Zbl 0792.73003

[22] F.Y.M. Wan, Stress boundary conditions for plate bending. Int. J. Solids Structures 40 (2003) 4107-4123. | Zbl 1039.74031

[23] S. Zhang, Equivalence estimates for a class of singular perturbation problems. C.R. Acad. Sci. Paris, Ser. I 342 (2006) 285-288. | Zbl 1083.74034