Closed-form Saint-Venant solutions in the Koiter theory of shells. (English) Zbl 1440.74203
Summary: In this paper we investigate the deformation of cylindrical linearly elastic shells using the Koiter model. We formulate and solve the relaxed Saint-Venant’s problem for thin cylindrical tubes made of isotropic and homogeneous elastic materials. To this aim, we adapt a method established previously in the three-dimensional theory of elasticity. We present a general solution procedure to determine closed-form solutions for the extension, bending, torsion and flexure problems. We remark the analogy and formal resemblance of these solutions to the classical Saint-Venant’s solutions for solid cylinders. The special case of circular cylindrical shells is also discussed.
MSC:
| 74K25 | Shells |
| 74G05 | Explicit solutions of equilibrium problems in solid mechanics |
| 74B05 | Classical linear elasticity |
Keywords:
Koiter shells; Saint-Venant’s problem; extension; bending; torsion; flexure; closed-form solution; cylindrical elastic shellsReferences:
| [1] | Altenbach, H.; Zhilin, P., A general theory of elastic simple shells, Usp. Mekh., 11, 107-148 (1988) |
| [2] | Altenbach, H.; Zhilin, P.; Kienzler, R.; Altenbach, H.; Ott, I., The theory of simple elastic shells, Theories of Plates and Shells. Critical Review and New Applications, Euromech Colloquium, 1-12 (2004), Heidelberg: Springer, Heidelberg · Zbl 1182.74142 |
| [3] | Berdichevsky, V.; Armanios, E.; Badir, A., Theory of anisotropic thin-walled closed-cross-section beams, Comput. Eng., 2, 411-432 (1992) |
| [4] | Berdichevsky, V.; Misyura, V., Effect of accuracy loss in classical shell theory, J. Appl. Mech., 59, S217-S223 (1992) · Zbl 0770.73047 · doi:10.1115/1.2899492 |
| [5] | Bîrsan, M., The solution of Saint-Venant’s problem in the theory of Cosserat shells, J. Elast., 74, 185-214 (2004) · Zbl 1058.74057 · doi:10.1023/B:ELAS.0000039621.94195.f0 |
| [6] | Bîrsan, M., On Saint-Venant’s problem for anisotropic, inhomogeneous, cylindrical Cosserat elastic shells, Int. J. Eng. Sci., 47, 21-38 (2009) · Zbl 1213.74142 · doi:10.1016/j.ijengsci.2008.06.015 |
| [7] | Bîrsan, M., Thermal stresses in cylindrical Cosserat elastic shells, Eur. J. Mech. A, Solids, 28, 94-101 (2009) · Zbl 1155.74377 · doi:10.1016/j.euromechsol.2008.03.001 |
| [8] | Bîrsan, M.; Altenbach, H.; Altenbach, H.; Eremeyev, V., Analysis of the deformation of multi-layered orthotropic cylindrical elastic shells using the direct approach, Shell-Like Structures: Non-classical Theories and Applications, 29-52 (2011), Berlin: Springer, Berlin |
| [9] | Bîrsan, M.; Sadowski, T.; Pietras, D., Thermoelastic deformations of cylindrical multi-layered shells using a direct approach, J. Therm. Stresses, 36, 749-789 (2013) · doi:10.1080/01495739.2013.764802 |
| [10] | Ciarlet, P., Mathematical Elasticity, Vol. III: Theory of Shells (2000), Amsterdam: North-Holland, Amsterdam · Zbl 0953.74004 |
| [11] | Ciarlet, P., An Introduction to Differential Geometry with Applications to Elasticity (2005), Dordrecht: Springer, Dordrecht · Zbl 1100.53004 |
| [12] | Goldenveizer, A., Theory of Elastic Thin Shells (1961), New York: Pergamon Press, New York |
| [13] | Ieşan, D., On Saint-Venant’s problem, Arch. Ration. Mech. Anal., 91, 363-373 (1986) · Zbl 0602.73005 · doi:10.1007/BF00282340 |
| [14] | Ieşan, D., Saint-Venant’s Problem (1987), New York: Springer, New York · Zbl 0625.73030 |
| [15] | John, F., Estimates for the derivatives of the stresses in a thin shell and interior shell equations, Commun. Pure Appl. Math., 18, 235-267 (1965) · doi:10.1002/cpa.3160180120 |
| [16] | Koiter, W.; Koiter, W., A consistent first approximation in the general theory of thin elastic shells, The Theory of Thin Elastic Shells, IUTAM Symposium Delft 1960, 12-33 (1960), Amsterdam: North-Holland, Amsterdam · Zbl 0109.43002 |
| [17] | Koiter, W., On the foundations of the linear theory of thin elastic shells, Proc. K. Ned. Akad. Wet., Ser. B, 73, 169-195 (1970) · Zbl 0213.27002 |
| [18] | Koiter, W., On the mathematical foundation of shell theory, Proc. Int. Congr. Math. Nice, 123-130 (1970), Paris: Gauthier-Villars, Paris · Zbl 0232.73094 |
| [19] | Koiter, W.; Simmonds, J.; Becker, E., Foundations of shell theory, Theoretical and Applied Mechanics, 150-176 (1973), Heidelberg: Springer, Heidelberg · Zbl 0286.73068 |
| [20] | Ladevèze, P., Justification de la théorie linéaire des coques élastiques, J. Méc., 15, 813-856 (1976) · Zbl 0356.73067 |
| [21] | Ladevèze, P.; Sanchez, P.; Simmonds, J., Beamlike (Saint-Venant) solutions for fully anisotropic elastic tubes of arbitrary cross section, Int. J. Solids Struct., 41, 1925-1944 (2004) · Zbl 1140.74485 · doi:10.1016/j.ijsolstr.2003.11.006 |
| [22] | Love, A., A Treatise on the Mathematical Theory of Elasticity (1944), New York: Dover, New York · Zbl 0063.03651 |
| [23] | Lurie, A., Statics of Thin-Walled Elastic Shells (1947), Moscow - Leningrad: Gostekhizdat, Moscow - Leningrad |
| [24] | Morgenstern, D., Mathematische Begründung der Scheibentheorie (zweidimensionale Elastizitätstheorie), Arch. Ration. Mech. Anal., 3, 91-96 (1959) · Zbl 0084.40502 · doi:10.1007/BF00284167 |
| [25] | Novozhilov, V., The Theory of Thin Shells (1959), Groningen: Noordhoff, Groningen · Zbl 0085.18803 |
| [26] | Reissner, E., On torsion of thin cylindrical shells, J. Mech. Phys. Solids, 7, 157-162 (1959) · Zbl 0212.57503 · doi:10.1016/0022-5096(59)90002-X |
| [27] | Reissner, E.; Tsai, W., Pure bending, stretching, and twisting of anisotropic cylindrical shells, J. Appl. Mech., 39, 148-154 (1972) · Zbl 0229.73083 · doi:10.1115/1.3422604 |
| [28] | de Saint-Venant, B., Mémoire sur le calcul des pièces solides à simple ou à double courbure, et prenant simultanément en considération les divers efforts auxquelles elles peuvent être soumises dans tous les sens, C. R. Acad. Sci. Paris, 17, 942-954 (1843) |
| [29] | Sokolnikoff, I., Mathematical Theory of Elasticity (1956), New York: McGraw-Hill, New York · Zbl 0070.41104 |
| [30] | Steigmann, D., Koiter’s shell theory from the perspective of three-dimensional nonlinear elasticity, J. Elast., 111, 91-107 (2013) · Zbl 1315.74016 · doi:10.1007/s10659-012-9393-2 |
| [31] | Timoshenko, S.; Goodier, J., Theory of Elasticity (1951), New York: McGraw-Hill, New York · Zbl 0045.26402 |
| [32] | Ventsel, E.; Krauthammer, T., Thin Plates and Shells: Theory, Analysis, and Applications (2001), New York: Taylor & Francis, New York |
| [33] | Vlasov, V., General Theory of Shells and Its Application in Engineering (1949), Moscow: Gostekhizdat, Moscow |
| [34] | Zhilin, P., Mechanics of deformable directed surfaces, Int. J. Solids Struct., 12, 635-648 (1976) · doi:10.1016/0020-7683(76)90010-X |
| [35] | Zhilin, P., Applied Mechanics - Foundations of Shell Theory (2006), Sankt Petersburg: State Polytechnical University Publisher, Sankt Petersburg |
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