# American Institute of Mathematical Sciences

October  2019, 12(6): 1547-1588. doi: 10.3934/dcdss.2019107

## Formal asymptotic analysis of elastic beams and thin-walled beams: A derivation of the Vlassov equations and their generalization to the anisotropic heterogeneous case

 1 Sorbonne Université, Centre National de la Recherche Scientifique, UMR 7190, Institut Jean le Rond ∂'Alembert, F-75005 Paris, France 2 Université Paris-Est, Noisy-le-Grand Cedex, France

* Corresponding author: Patrick Ballard

Received  January 2018 Revised  June 2018 Published  November 2018

The modelling of ordinary beams and thin-walled beams is rigorously obtained from a formal asymptotic analysis of three-dimensional linear elasticity. In the case of isotropic homogeneous elasticity, ordinary beams yield the Navier-Bernoulli beam model, thin-walled beams with open profile yield the Vlassov beam model and thin-walled beams with closed profile the Navier-Bernoulli beam model. The formal asymptotic analysis is also extensively performed in the case of the most general anisotropic transversely heterogeneous material (meaning the heterogeneity is the same in every cross-section), delivering the same qualitative results. We prove, in particular, the non-intuitive fact that the warping function appearing in the Vlassov model for general anisotropic transversely heterogeneous material, is the same as the one appearing in the isotropic homogeneous case. In the general case of anisotropic transversely heterogeneous material, the analysis provides a rigorous and systematic constructive procedure for calculating the reduced elastic moduli, both in Navier-Bernoulli and Vlassov theories.

Citation: Patrick Ballard, Bernadette Miara. Formal asymptotic analysis of elastic beams and thin-walled beams: A derivation of the Vlassov equations and their generalization to the anisotropic heterogeneous case. Discrete and Continuous Dynamical Systems - S, 2019, 12 (6) : 1547-1588. doi: 10.3934/dcdss.2019107
##### References:
 [1] C. Davini, L. Freddi and R. Paroni, Linear Models for Composite Thin-Walled Beams by $Γ$-Convergence. Part Ⅰ: Open Cross Sections, SIAM Journal on Mathematical Analysis, 46 (2014), 3296-3331.  doi: 10.1137/140951473. [2] C. Davini, L. Freddi and R. Paroni, Linear Models for Composite Thin-Walled Beams by $Γ$-Convergence. Part Ⅱ: Closed Cross Sections, SIAM Journal on Mathematical Analysis, 46 (2014), p. 3332-3360.  doi: 10.1137/140964321. [3] L. Freddi, A. Morassi and R. Paroni, Thin-Walled Beams: The Case of the Rectangular Cross-Section, Journal of Elasticity, 76 (2004), 45-66.  doi: 10.1007/s10659-004-7193-z. [4] L. Freddi, A. Morassi and R. Paroni, Thin-walled beams: A derivation of Vlassov theory via $Γ$-convergence, Journal of Elasticity, 86 (2007), 263-296.  doi: 10.1007/s10659-006-9094-9. [5] L. Freddi, F. Murat and R. Paroni, Anisotropic Inhomogeneous Rectangular Thin-walled Beams, SIAM Journal on Mathematical Analysis, 40 (2009), 1923-1951.  doi: 10.1137/080720279. [6] L. Grillet, A. Hamdouni and C. Allery, Modèle asymptotique linéaire de poutres voiles fortement courbés Comptes Rendus de l'Académie des Sciences, Paris, Série IIb, 328 (2000), 587-592. [7] A. Hamdouni and O. Millet, An asymptotic linear thin-walled rod model coupling twist and bending, International Applied Mechanics, 46 (2011), 1072-1092.  doi: 10.1007/s10778-011-0400-2. [8] V. A. Kondrat'ev and O. A. Oleinik, On the dependence of the constant in Korn's inequality on parameters characterizing the geometry of the region, Russian Mathematical Surveys, 44 (1989), 187-195. [9] H. Le Dret, Convergence of displacements and stresses in linearly elastic slender rods as the thickness goes to zero, Asymptotic Analysis, 10 (1995), 367-402. [10] J. L. Lions, Perturbations Singulières dans les Problèmes aux Limites et en Contrȏle Optimal, Lecture Notes in Mathematics, Vol. 323, Springer-Verlag, Berlin, 1973. [11] B. Miara and E. Sanchez-Palencia, Asymptotic analysis of linearly elastic shells, Asymptotic Analysis, 12 (1996), 41-54. [12] D. Percivale, Thin elastic beams: The variational approach to St. Venant's problem, Asymptotic Analysis, 20 (1999), 39-59. [13] J. Sanchez-Hubert and E. Sanchez-Palencia, Introduction aux Méthodes Asymptotiques et à L'homogéneisation, Masson, Paris, 1992. [14] I. S. Sokolnikoff, Mathematical Theory of Elasticity, McGraw-Hill, 1956. [15] S. P. Timoshenko, De la stabilité à la flexion plane d ne poutre en double té, Nouvelles de l'Institut Polytechnique de Saint-Pétesbourg, T. Ⅳ-Ⅴ (1905-1906). [16] L. Trabucho and J. M. Viaño, Mathematical modelling of rods, In Handbook of Numerical Analysis, Volume IV, Elsevier, Amsterdam, 1996, 487-974. [17] B. Z. Vlassov, Pièces Longues en Voiles Minces, Éditions Eyrolles, Paris, 1962. French translation of the Russian original book.

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##### References:
 [1] C. Davini, L. Freddi and R. Paroni, Linear Models for Composite Thin-Walled Beams by $Γ$-Convergence. Part Ⅰ: Open Cross Sections, SIAM Journal on Mathematical Analysis, 46 (2014), 3296-3331.  doi: 10.1137/140951473. [2] C. Davini, L. Freddi and R. Paroni, Linear Models for Composite Thin-Walled Beams by $Γ$-Convergence. Part Ⅱ: Closed Cross Sections, SIAM Journal on Mathematical Analysis, 46 (2014), p. 3332-3360.  doi: 10.1137/140964321. [3] L. Freddi, A. Morassi and R. Paroni, Thin-Walled Beams: The Case of the Rectangular Cross-Section, Journal of Elasticity, 76 (2004), 45-66.  doi: 10.1007/s10659-004-7193-z. [4] L. Freddi, A. Morassi and R. Paroni, Thin-walled beams: A derivation of Vlassov theory via $Γ$-convergence, Journal of Elasticity, 86 (2007), 263-296.  doi: 10.1007/s10659-006-9094-9. [5] L. Freddi, F. Murat and R. Paroni, Anisotropic Inhomogeneous Rectangular Thin-walled Beams, SIAM Journal on Mathematical Analysis, 40 (2009), 1923-1951.  doi: 10.1137/080720279. [6] L. Grillet, A. Hamdouni and C. Allery, Modèle asymptotique linéaire de poutres voiles fortement courbés Comptes Rendus de l'Académie des Sciences, Paris, Série IIb, 328 (2000), 587-592. [7] A. Hamdouni and O. Millet, An asymptotic linear thin-walled rod model coupling twist and bending, International Applied Mechanics, 46 (2011), 1072-1092.  doi: 10.1007/s10778-011-0400-2. [8] V. A. Kondrat'ev and O. A. Oleinik, On the dependence of the constant in Korn's inequality on parameters characterizing the geometry of the region, Russian Mathematical Surveys, 44 (1989), 187-195. [9] H. Le Dret, Convergence of displacements and stresses in linearly elastic slender rods as the thickness goes to zero, Asymptotic Analysis, 10 (1995), 367-402. [10] J. L. Lions, Perturbations Singulières dans les Problèmes aux Limites et en Contrȏle Optimal, Lecture Notes in Mathematics, Vol. 323, Springer-Verlag, Berlin, 1973. [11] B. Miara and E. Sanchez-Palencia, Asymptotic analysis of linearly elastic shells, Asymptotic Analysis, 12 (1996), 41-54. [12] D. Percivale, Thin elastic beams: The variational approach to St. Venant's problem, Asymptotic Analysis, 20 (1999), 39-59. [13] J. Sanchez-Hubert and E. Sanchez-Palencia, Introduction aux Méthodes Asymptotiques et à L'homogéneisation, Masson, Paris, 1992. [14] I. S. Sokolnikoff, Mathematical Theory of Elasticity, McGraw-Hill, 1956. [15] S. P. Timoshenko, De la stabilité à la flexion plane d ne poutre en double té, Nouvelles de l'Institut Polytechnique de Saint-Pétesbourg, T. Ⅳ-Ⅴ (1905-1906). [16] L. Trabucho and J. M. Viaño, Mathematical modelling of rods, In Handbook of Numerical Analysis, Volume IV, Elsevier, Amsterdam, 1996, 487-974. [17] B. Z. Vlassov, Pièces Longues en Voiles Minces, Éditions Eyrolles, Paris, 1962. French translation of the Russian original book.
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