Abstract
In this paper we derive and mathematically justify models of micropolar rods and plates from the equations of linearized micropolar elasticity. Derivation is based on the asymptotic techniques with respect to the small parameter being the thickness of the elastic body we consider. Justification of the models is obtained through the convergence result for the displacement and microrotation fields when the thickness tends to zero. The limiting microrotation is then related to the macrorotation of the cross–section (transversal segment) and the model is rewritten in terms of macroscopic unknowns. The obtained models are recognized as being either the Reissner–Mindlin plate or the Timoshenko beam type.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aganović, I., Tambača, J., Tutek, Z.: On te asymptotic analysis of elastic rods, submitted to Mathematics and Mechanics of Solids
Aganović, I., Tambača, J., Tutek, Z.: On the asymptotic analysis of micropolar elastic rods, submitted for publication
I. Aganović and Z. Tutek, A justification of the one-dimensional linear model of elastic beam. Math. Methods Appl. Sci. 8, 1–14 (1986)
Bermudez, A., Viaño, J.M.: A justification of thermoelastic equations for variable-section beams by asymptotic methods. RAIRO Anal. Numér. 18, 347–376 (1984)
Ciarlet, P.G.: Mathematical Elasticity. Vol. II. Theory of Plates. North-Holland, Amsterdam (1997)
Ciarlet, P.G., Destuynder, P.: A justification of the two dimensional linear plate model. J. Méc. 18, 315–344 (1979)
Erbay, H.A.: An asymptotic theory of thin micropolar plates. Int. J. Eng. Sci. 38, 1497–1516 (2000)
Eringen, A.C.: Microcontinuum Field Theories, I. Foundations and Solids. Springer, New York (1999)
Goldenveizer, A.L.: Derivation of an approximate theory of bending a plate by a method of asymptotic integration of the equations in the theory of elasticity. J. Appl. Math. 26, 1000–1025 (1963)
Green, A.E., Naghdi, P.M.: Micropolar and director theories of plates. Q. J. Mech. Appl. Math. 20, 183–199 (1967)
Miara, B.: Justification of the asymptotic analysis of elastic plates. I. The linear case. Asymptot. Anal. 9, 47–60 (1994)
Naghdi, P.M.: The Theory of Shells and Plates. In: Handbuch der Physik Vol. VIa/2. Springer, Berlin Heidelberg New York (1972)
Trabucho, L., Viańo, J.M.: Mathematical Modelling of Rods. In: Handbook of Numerical Analysis, Vol. IV, P.G. Ciarlet & J.L. Lions, Eds., North-Holland (1996)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Aganović, I., Tambača, J. & Tutek, Z. Derivation and Justification of the Models of Rods and Plates From Linearized Three-Dimensional Micropolar Elasticity. J Elasticity 84, 131–152 (2006). https://doi.org/10.1007/s10659-006-9060-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10659-006-9060-6