On the solutions of clamped Reissner-Mindlin plates under transverse loads
Authors:
Thomas C. Assiff and David H. Y. Yen
Journal:
Quart. Appl. Math. 45 (1987), 679-690
MSC:
Primary 73K10
DOI:
https://doi.org/10.1090/qam/917017
MathSciNet review:
917017
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Abstract: The governing equations in the Reissner—Mindlin theory may be written in a form such that a small parameter $\epsilon$ is involved. This parameter $\epsilon$ depends on a combination of the shear modulus and the plate thickness. The governing equations are singularly perturbed with respect to $\epsilon$. However, as $\epsilon \to 0$ one does recover the biharmonic equation of the classical plate theory. In a previous work of ours [1] the behavior of solutions for clamped Reissner—Mindlin plates as $\epsilon \to 0$ was studied and it was shown there that these solutions tend continuously, in various functional norms, to their corresponding solutions in the classical plate theory. This paper deals with two specific questions concerning the detailed dependence of these solutions on $\epsilon$ as $\epsilon \to 0$. We shall show the nonexistence of regular asymptotic expansions of the solutions in integral powers of $\epsilon$ for general clamped Reissner—Mindlin plates. We shall also construct an exact solution for a circular plate which exhibits dependence on fractional powers of $\epsilon$. This latter solution shows a boundary layer phenomenon, decaying away from the boundary, often encountered in singular perturbation problems.
- Thomas C. Assiff and David H. Y. Yen, On a penalty-perturbation theory for plate problems, IMA J. Appl. Math. 34 (1985), no. 2, 121–136. MR 795507, DOI https://doi.org/10.1093/imamat/34.2.121
- Eric Reissner, The effect of transverse shear deformation on the bending of elastic plates, J. Appl. Mech. 12 (1945), A-69–A-77. MR 0012579
R. D. Mindlin, Influence of ratatory inertia and shear on flexural motions of isotropic elastic plates, J. Appl. Mech. 18, 31–38 (1951)
M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover Inc., New York (1965)
- B. M. Fraeijs de Veubeke, A course in elasticity, Applied Mathematical Sciences, vol. 29, Springer-Verlag, New York-Berlin, 1979. Translated from the French by F. A. Ficken. MR 533738
D. R. Westbrook, A variational principle with applications in finite elements, J. Inst. Math. Applics. 14, 79–82 (1974)
T. C. Assiff and D. H. Y. Yen, On a penalty-perturbation finite element method for boundary value problems for elastic plates, Proceedings of the International Conference on Accuracy Estimates and Adaptive Refinements in Finite Element Computations, Lisbon, Portugal 1, 189–198 (1984)
- R. D. Gregory and F. Y. M. Wan, On plate theories and Saint-Venant’s principle, Internat. J. Solids Structures 21 (1985), no. 10, 1005–1024. MR 807846, DOI https://doi.org/10.1016/0020-7683%2885%2990052-6
T. C. Assiff and D. H. Y. Yen, On a penalty-perturbation theory for plate problems, IMA J. Appl. Math. 34, 121–136 (1985)
E. Reissner, The effect of transverse shear deformation on the bending of elastic plates, J. Appl. Mech. 12, Trans. ASME 67, A-69—A-77, 7–42 (1945)
R. D. Mindlin, Influence of ratatory inertia and shear on flexural motions of isotropic elastic plates, J. Appl. Mech. 18, 31–38 (1951)
M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover Inc., New York (1965)
B. Fraeijs de Veubeke, A Course in Elasticity, Springer-Verlag, New York (1979)
D. R. Westbrook, A variational principle with applications in finite elements, J. Inst. Math. Applics. 14, 79–82 (1974)
T. C. Assiff and D. H. Y. Yen, On a penalty-perturbation finite element method for boundary value problems for elastic plates, Proceedings of the International Conference on Accuracy Estimates and Adaptive Refinements in Finite Element Computations, Lisbon, Portugal 1, 189–198 (1984)
R. D. Gregory and F. Y. M. Wan, On plate theories and Saint-Venant’s principle, Internat. J. Solids Structures 21, 1005–1024 (1985)
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© Copyright 1987
American Mathematical Society