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Justification of the asymptotic expansion method for homogeneous isotropic beams by comparison with de Saint-Venant’s solutions. (English) Zbl 1355.74041

Summary: The formal asymptotic expansion method is an attractive mean to derive simplified models for problems exhibiting a small parameter, such as the elastic analysis of beam-like structures. Usually this method is rigorously justified using convergence theorems [W. Yu and D. H. Hodges, J. Appl. Mech. 71, No. 1, 15–23 (2004; Zbl 1111.74734)]. In this paper it is illustrated how the Saint-Venant’s solution naturally arises from the lowest order terms of an asymptotic expansion of the elastic state for the case of homogeneous isotropic beams. It is also highlighted that the Saint-Venant solutions corresponding to pure traction, bending and torsion involve the solution of the first-order microscopic problems, while for the simple bending problem, the solution of the second-order microscopic problems is needed. The second-order problems provide therefore a way to characterize the transverse shear behavior and the cross-sectional warping of the beam.

MSC:

74G05 Explicit solutions of equilibrium problems in solid mechanics
74G10 Analytic approximation of solutions (perturbation methods, asymptotic methods, series, etc.) of equilibrium problems in solid mechanics
74K10 Rods (beams, columns, shafts, arches, rings, etc.)

Citations:

Zbl 1111.74734
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References:

[1] Bamberger, Y.: Mécanique de l’ingénieur III-solides déformables. Hermann étude des science et des arts, Paris (1997) · Zbl 1111.74734
[2] Berdichevsky, V.L.: Variational Principles of Continuum Mechanics. II. Applications. Springer, Berlin (2009) · Zbl 1189.49002
[3] Boutin, C.: Microstructural effects in elastic composites. Int. J. Solids Struct. 33(7), 1023-1051 (1996) · Zbl 0920.73282 · doi:10.1016/0020-7683(95)00089-5
[4] Buannic, N.: Analyse Asymptotique de Poutres Elastique Hétérogènes. Ph.D. thesis, Ecole Centrale de Nantes (2000)
[5] Buannic, N.; Cartraud, P., Higher-order asymptotic model for a heterogeneous beam, including corrections due to end effects (2000), Atlanta
[6] Buannic, N., Cartraud, P.: Higher-order effective modeling of periodic heterogeneous beams. I. Asymptotic expansion method. Int. J. Solids Struct. 38(40-41), 7139-7161 (2001) · Zbl 0998.74042 · doi:10.1016/S0020-7683(00)00422-4
[7] Buannic, N., Cartraud, P.: Higher-order effective modeling of periodic heterogeneous beams. II. Derivation of the proper boundary conditions for the interior asymptotic solution. Int. J. Solids Struct. 38, 7163-7180 (2001) · Zbl 0998.74042 · doi:10.1016/S0020-7683(00)00423-6
[8] Carrera, E., Petrolo, M.: On the effectiveness of higher-order terms in refined beam theories. J. Appl. Mech. 78(2), 021013 (2011) · Zbl 1448.74064 · doi:10.1115/1.4002207
[9] Cimetière, A., Geymonat, G., Le Dret, H., Raoult, A., Tutek, Z.: Asymptotic theory and analysis for displacements and stress distribution in nonlinear elastic straight slender rods. J. Elast. 19(2), 111-161 (1988). doi:10.1007/BF00040890 · Zbl 0653.73010 · doi:10.1007/BF00040890
[10] Dauge, M., Yosibash, Z.: Boundary layer realization in thin elastic three-dimensional domains and two-dimensional hierarchic plate models. Int. J. Solids Struct. 37(17), 2443-2471 (2000) · Zbl 0973.74043 · doi:10.1016/S0020-7683(99)00004-9
[11] Hodges, D.H., Atilgan, A.R., Cesnik, C.E., Fulton, M.V.: On a simplified strain energy function for geometrically nonlinear behaviour of anisotropic beams. Compos. Eng. 2(5), 513-526 (1992) · doi:10.1016/0961-9526(92)90040-D
[12] Horgan, C.O.: Recent developments concerning saint-Venant’s principle: a second update. Appl. Mech. Rev. 42(11), 295-303 (1989) · doi:10.1115/1.3152414
[13] Karama, M., Afaq, K.S., Mistou, S.: Mechanical behaviour of laminated composite beam by the new multi-layered laminated composite structures model with transverse shear stress continuity. Int. J. Solids Struct. 40(6), 1525-1546 (2003) · Zbl 1087.74579 · doi:10.1016/S0020-7683(02)00647-9
[14] Kim, J.S., Wang, K.W.: On the asymptotic boundary conditions of an anisotropic beam via virtual work principle. Int. J. Solids Struct. 48(16-17), 2422-2431 (2011). doi:10.1016/j.ijsolstr.2011.04.016 · doi:10.1016/j.ijsolstr.2011.04.016
[15] Kolpakov, A.G.: Calculation of the characteristics of thin elastic rods with a periodic structure. J. Appl. Math. Mech. 55(3), 358-365 (1991) · Zbl 0787.73038 · doi:10.1016/0021-8928(91)90039-W
[16] Kolpakov, A.G.: Stressed Composite Structures: Homogenized Models for Thin-Walled Nonhomogeneous Structures with Initial Stresses. Springer, Berlin (2004) · Zbl 1428.74002 · doi:10.1007/978-3-540-45211-9
[17] Marigo, J.J., Ghidouche, H., Sedkaoui, Z.: Des poutres flexibles aux fils extensibles : une hiérarchie de modéles asymptotiques. C. R. Acad. Sci. 326(2), 79-84 (1998) · Zbl 0924.73109
[18] Marur, S.R., Kant, T.: Free vibration analysis of fiber reinforced composite beams using higher order theories and finite element modelling. J. Sound Vib. 194(3), 337-351 (1996) · doi:10.1006/jsvi.1996.0362
[19] Miara, B.: Justification des mises à l’échelle et des hypothèses sur les données dans l’analyse asymptotique des modèles bidimensionnels de plaques minces élastiques. I : Le cas non linéaire. C. R. Acad. Sci. 314(1), 687-690 (1992) · Zbl 0754.73056
[20] Sanchez-Hubert, J., Sanchez-Palencia, E.: Introduction aux méthodes asymptotiques et à l’homogénéisation. Masson, Paris (1992)
[21] Trabucho, L.; Viaño, J. M.; Ciarlet, P. G. (ed.); Lions, J. L. (ed.), Mathematical modelling of rods, No. IV, 487-974 (1996), Amsterdam · Zbl 0873.73041
[22] Yu, W., Hodges, D.H.: Elasticity solutions versus asymptotic sectional analysis of homogeneous, isotropic, prismatic beams. J. Appl. Mech. 71(1), 15 (2004). doi:10.1115/1.1640367 · Zbl 1111.74734 · doi:10.1115/1.1640367
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