Thin-walled Beams: A Derivation of Vlassov Theory via Γ-Convergence

Freddi, Lorenzo; Morassi, Antonino; Paroni, Roberto

doi:10.1007/s10659-006-9094-9

Cited by 23 publications

(25 citation statements)

References 8 publications

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“…Thin-walled beams with a multirectangular cross section were studied in [7]. Each rectangle composing the cross section had sides that scaled with ε and ε 2 , respectively.…”

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confidence: 99%

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

Davini¹,

Freddi²,

Paroni³

2014

SIAM J. Math. Anal.

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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

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“…Thin-walled beams with a multirectangular cross section were studied in [7]. Each rectangle composing the cross section had sides that scaled with ε and ε 2 , respectively.…”

mentioning

confidence: 99%

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

Davini¹,

Freddi²,

Paroni³

2014

SIAM J. Math. Anal.

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“…Remark 2 The inertia tensors Θ 0 α that appear in the right-hand sides of the equations of motion (39) are expressed by (5). We observe that the relation Θ 0 1 = 0 holds true if and only if Σ ρ * xμ dxdy = Σ ρ * yμ dxdy = 0.…”

Section: Linear Theory Of Porous Thermoelastic Rodsmentioning

confidence: 92%

“…From relations (5) and (6), we can prove that Θ 1 is antisymmetric and Θ 2 is symmetric, and we can prove that the tensor Θ 2 − Θ T 1 · Θ 1 is symmetric and positive definite. Let us denote by l the length of the curve C 0 .…”

Section: Remarkmentioning

confidence: 94%

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Theory of thin thermoelastic rods made of porous materials

Bîrsan

Altenbach

2010

Arch Appl Mech

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In this paper, we consider thin rods modeled by the direct approach, in which the rod-like body is regarded as a one-dimensional continuum (i.e., a deformable curve) with a triad of rigidly rotating orthonormal vectors attached to each material point. In this context, we present a model for porous thermoelastic curved rods, having natural twisting and arbitrary shape of cross-section. To describe the porosity, we employ the theory of elastic materials with voids. The basic laws of thermodynamics are applied directly to the one-dimensional continuum, and the nonlinear governing equations are established. We formulate the constitutive equations and determine the structure of constitutive tensors. We prove the uniqueness of solution to the boundaryinitial-value problem associated with the deformation of porous thermoelastic rods in the framework of linear theory. Then, we show the decoupling of the bending-shear and extension-torsion problems for straight porous rods. Using a comparison with three-dimensional equations, we identify and give interpretations to the relevant fields introduced in the direct approach. Finally, we consider the case of orthotropic materials and determine the constitutive coefficients for deformable curves in terms of three-dimensional constitutive constants by means of comparison between simple solutions obtained in the two approaches for porous thermoelastic rods.

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“…Inspired by works of Freddi, Morassi and Paroni (see [6,7]), of Anzellotti, Baldo and Percivale (see [2]) and of Trabucho and Viano (see [14]), we derive, by means of Γ -convergence (see [4]), a variational model for a thin-walled beam with residual stress, which, to our knowledge, is new.…”

Section: Introductionmentioning

confidence: 99%