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

Abstract

Abstract. We consider a beam whose cross section is a tubular neighborhood, with thickness scaling with a parameter δε, of a simple curve γ whose length scales with ε. To model a thin-walled beam we assume that δε goes to zero faster than ε, and we measure the rate of convergence by a slenderness parameter s which is the ratio between ε 2 and δε. In this Part I of the work we focus on the case where the curve is open. Under the assumption that the beam has a linearly elastic behavior, for s ∈ {0, 1} we derive two one-dimensional Γ-limit problems by letting ε go to zero. The limit models are obtained for a fully anisotropic and inhomogeneous material, thus making the theory applicable for composite thin-walled beams. The approach recovers in a systematic way, and gives account of, many features of the beam models in the theory of Vlasov. 1. Introduction. Composite, i.e., anisotropic and inhomogeneous, thin-walled beams have been extensively studied by the engineering community. We refrain from citing the abundant literature on the subject; we quote, instead, the opening lines of the abstract of [12]: "There is no lack of composite beam theories. Quite to the contrary, there might be too many of them. Different approaches, notation, etc., are used by authors of those theories, so it is not always straightforward to compare the assumptions made and to assess the quantitative consequences of those assumptions." This excerpt well describes the status of the research on composite thin-walled beams. To shed some light on the huge variety of models present in the literature, it is necessary to take a rigorous approach, possibly free of assumptions. The aim of this paper is to derive mechanical models for composite thin-walled beams by Γ-convergence (see [2]), starting from the three-dimensional theory of linear elasticity.This line of research essentially started in [6], where a mechanical model was obtained for an isotropic homogeneous and linearly elastic thin-walled beam with rectangular cross section. In that paper the long side of the rectangle scaled with a small parameter ε > 0 and the other with ε 2 . This "double" scaling was chosen to model a thin-walled beam. One of the main results was a compactness theorem in which different orders of convergence for the various components of the displacement were established. Namely, a sequence of displacements with equi-bounded energy is such that

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

“…We adopt the same notation used in [1], which we recall hereafter for convenience of the reader. In the proofs, for brevity, we neglect to specify which identities hold only "almost everywhere," while we are explicit in the statements of the theorems.…”

“…Clearly, κ cannot be identically equal to zero because the curve is closed. As in [1], we assume that the cross section is thin in the sense established by…”

“…In that article, however, the analysis of Γ-convergence is carried on in terms of strains rather than displacements, as is usually done. In [1] we have studied inhomogeneous anisotropic thin-walled beams with an open curved cross section within the framework of linear elasticity. In particular, when applied to homogeneous and isotropic materials, our results validate Vlasov's theory.…”

“…The requirement that the displacements fulfill periodicity conditions yields that additional information can be obtained from the compactness theorem. Accordingly, the kinematical analysis of [1] can be significantly enhanced. In particular, it is found that the twist used for an open cross section, hereafter denoted by ϑ, is identically equal to zero.…”

“…In sections 2 and 3 we introduce the problem and recall some useful formulas from [1]. After recalling the compactness results, in section 4 we prove that the limit twist ϑ, as we defined it for the open cross sections, vanishes and we introduce a new measure of twist Θ ε .…”