# On the accuracy of Reissner–Mindlin plate model for stress boundary conditions

ESAIM: Mathematical Modelling and Numerical Analysis (2006)

- Volume: 40, Issue: 2, page 269-294
- ISSN: 0764-583X

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topZhang, Sheng. "On the accuracy of Reissner–Mindlin plate model for stress boundary conditions." ESAIM: Mathematical Modelling and Numerical Analysis 40.2 (2006): 269-294. <http://eudml.org/doc/249689>.

@article{Zhang2006,

abstract = {
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.
},

author = {Zhang, Sheng},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Reissner–Mindlin plate; shear correction factor; stress boundary
condition.; Reissner-Mindlin plate; stress boundary condition},

language = {eng},

month = {6},

number = {2},

pages = {269-294},

publisher = {EDP Sciences},

title = {On the accuracy of Reissner–Mindlin plate model for stress boundary conditions},

url = {http://eudml.org/doc/249689},

volume = {40},

year = {2006},

}

TY - JOUR

AU - Zhang, Sheng

TI - On the accuracy of Reissner–Mindlin plate model for stress boundary conditions

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2006/6//

PB - EDP Sciences

VL - 40

IS - 2

SP - 269

EP - 294

AB -
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.

LA - eng

KW - Reissner–Mindlin plate; shear correction factor; stress boundary
condition.; Reissner-Mindlin plate; stress boundary condition

UR - http://eudml.org/doc/249689

ER -

## References

top- 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). Zbl0958.74033
- 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. Zbl0846.73027
- D.N. Arnold and A. Mardureira, Asymptotic estimates of hierarchical modeling. Math. Mod. Meth. Appl. S.13 (2003).
- D.N. Arnold, A. Mardureira and S. Zhang, On the range of applicability of the Reissner–Mindlin and Kirchhoff–Love plate bending models, J. Elasticity67 (2002) 171–185. Zbl1089.74595
- J. Bergh and J. Lofstrom, Interpolation space: An introduction, Springer-Verlag (1976). Zbl0344.46071
- C. Chen, Asymptotic convergence rates for the Kirchhoff plate model, Ph.D. Thesis, Pennsylvania State University (1995).
- P.G. Ciarlet, Mathematical elasticity, Vol II: Theory of plates. North-Holland (1997). Zbl0888.73001
- 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. Zbl0916.73021
- 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. Zbl0958.74034
- 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).
- K.O. Friedrichs and R.F. Dressler, A boundary-layer theory for elastic plates. Comm. Pure Appl. Math.XIV (1961) 1–33. Zbl0096.40001
- T.J.R. Hughes, The finite element method: Linear static and dynamic finite element analysis. Prentice-Hall, Englewood Cliffs (1987). Zbl0634.73056
- W.T. Koiter, On the foundations of linear theory of thin elastic shells. Proc. Kon. Ned. Akad. Wetensch.B73 (1970) 169–195. Zbl0213.27002
- A.E.H. Love, A treatise on the mathematical theory of elasticity. Cambridge University Press (1934).
- D. Morgenstern, Herleitung der Plattentheorie aus der dreidimensionalen Elastizitatstheorie. Arch. Rational Mech. Anal.4 (1959) 145–152. Zbl0126.20605
- P.M. Naghdi, The theory of shells and plates, in Handbuch der Physik, Springer-Verlag, Berlin, VIa/2 (1972) 425–640.
- W. Prager and J.L. Synge, Approximations in elasticity based on the concept of function space. Q. J. Math.5 (1947) 241–269. Zbl0029.23505
- E. Reissner, Reflections on the theory of elastic plates. Appl. Mech. Rev.38 (1985) 1453–1464.
- A. Rössle, M. Bischoff, W. Wendland and E. Ramm, On the mathematical foundation of the (1,1,2)-plate model. Int. J. Solids Structures36 (1999) 2143–2168. Zbl0962.74036
- J. Sanchez-Hubert and E. Sanchez-Palencia, Coques élastiques minces: Propriétés asymptotiques, Recherches en mathématiques appliquées, Masson, Paris (1997).
- B. Szabó, I. Babuska, Finite Element analysis. Wiley, New York (1991). Zbl0792.73003
- F.Y.M. Wan, Stress boundary conditions for plate bending. Int. J. Solids Structures40 (2003) 4107–4123. Zbl1039.74031
- S. Zhang, Equivalence estimates for a class of singular perturbation problems. C.R. Acad. Sci. Paris, Ser. I342 (2006) 285–288. Zbl1083.74034

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