Linear Models for Composite Thin-Walled Beams by $\Gamma$-Convergence. Part II: Closed Cross-Sections

Abstract

Abstract. We consider a beam whose cross section is a tubular neighborhood of a simple closed curve γ. We assume that the wall thickness, i.e., the size of the neighborhood, scales with a parameter δε while the length of γ scales with ε. We characterize a thin-walled beam by assuming that δε goes to zero faster than ε. Starting from the three-dimensional linear theory of elasticity, by letting ε go to zero, we derive a one-dimensional Γ-limit problem for the case in which the ratio between ε 2 and δε is bounded. The limit model is obtained for a fully anisotropic and inhomogeneous material, thus making the theory applicable for composite thin-walled beams. Our approach recovers in a systematic way, and gives account of, many features of the beam models in the theory of Vlasov.

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

“…The Γ-limit is consistent with the energy of the reduced model that is obtained in our paper. Hence, the analysis in [1] and [2] provides a convergence result which is a full and rigorous mathematical justification of the formal analysis performed in this paper. The particular case of isotropic homogeneous elasticity is not specifically considered in references [1] and [2].…”

“…Hence, the analysis in [1] and [2] provides a convergence result which is a full and rigorous mathematical justification of the formal analysis performed in this paper. The particular case of isotropic homogeneous elasticity is not specifically considered in references [1] and [2]. However, the analysis provided in our paper shows that, in that particular case of isotropic homogeneous elasticity, the Γ-limit obtained in [1] reduces to the classical Vlassov model for thin-walled beams with open profile and the Γ-limit obtained in [2] reduces to the classical Navier-Bernoulli model.…”

“…The particular case of isotropic homogeneous elasticity is not specifically considered in references [1] and [2]. However, the analysis provided in our paper shows that, in that particular case of isotropic homogeneous elasticity, the Γ-limit obtained in [1] reduces to the classical Vlassov model for thin-walled beams with open profile and the Γ-limit obtained in [2] reduces to the classical Navier-Bernoulli model. Hence, the classical linear Vlassov model is now mathematically fully justified.…”

“…Addendum. We have discovered references [1] and [2] after that this article was accepted for publication, and we were therefore unaware of this important contribution at the time of writing the present paper. In these references, three-dimensional linear elastic equilibrium problems in heterogeneous anisotropic thin-walled beams with the same geometry as in our sections 3.4 and 3.6 are considered.…”

Enhanced models for the nonlinear bending of planar rods: localization phenomena and multistability

“…In [23,24], the authors consider a rod whose cross section is a tubular neighbourhood, with thickness scaling with a parameter δ ε , of a simple curve whose length scales with ε; to model a thin-walled rods they assume that δ ε goes to zero faster than ε, and they measure the rate of convergence by a slenderness parameter. The approach recovers in a systematic way, and gives account of, many features of the rod models in the theory of Vlasov.…”