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Derivation and justification of the models of rods and plates from linearized three-dimensional micropolar elasticity. (English) Zbl 1158.74401
Summary: 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.

74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74K20 Plates
74A35 Polar materials
Full Text: DOI
[1] Aganović, I., Tambača, J., Tutek, Z.: On te asymptotic analysis of elastic rods, submitted to Mathematics and Mechanics of Solids · Zbl 0603.73056
[2] Aganović, I., Tambača, J., Tutek, Z.: On the asymptotic analysis of micropolar elastic rods, submitted for publication
[3] I. Aganović and Z. Tutek, A justification of the one-dimensional linear model of elastic beam. Math. Methods Appl. Sci. 8, 1–14 (1986) · Zbl 0617.35136 · doi:10.1002/mma.1670080102
[4] Bermudez, A., Viaño, J.M.: A justification of thermoelastic equations for variable-section beams by asymptotic methods. RAIRO Anal. Numér. 18, 347–376 (1984) · Zbl 0572.73053
[5] Ciarlet, P.G.: Mathematical Elasticity. Vol. II. Theory of Plates. North-Holland, Amsterdam (1997)
[6] Ciarlet, P.G., Destuynder, P.: A justification of the two dimensional linear plate model. J. Méc. 18, 315–344 (1979) · Zbl 0415.73072
[7] Erbay, H.A.: An asymptotic theory of thin micropolar plates. Int. J. Eng. Sci. 38, 1497–1516 (2000) · Zbl 1210.74109 · doi:10.1016/S0020-7225(99)00118-4
[8] Eringen, A.C.: Microcontinuum Field Theories, I. Foundations and Solids. Springer, New York (1999) · Zbl 0953.74002
[9] 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)
[10] Green, A.E., Naghdi, P.M.: Micropolar and director theories of plates. Q. J. Mech. Appl. Math. 20, 183–199 (1967) · Zbl 0148.19302 · doi:10.1093/qjmam/20.2.183
[11] Miara, B.: Justification of the asymptotic analysis of elastic plates. I. The linear case. Asymptot. Anal. 9, 47–60 (1994) · Zbl 0806.73029
[12] Naghdi, P.M.: The Theory of Shells and Plates. In: Handbuch der Physik Vol. VIa/2. Springer, Berlin Heidelberg New York (1972) · Zbl 0283.73034
[13] 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) · Zbl 0873.73041
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