Nonlinear Thin-Walled Beams With a Rectangular Cross-Section — Part I

Freddi, Lorenzo; Mora, Maria Giovanna; Paroni, Roberto

doi:10.1142/s0218202511500163

Cited by 27 publications

(29 citation statements)

References 14 publications

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“…In [4,5] a hierarchy of models for homogeneous anisotropic thin-walled beams with rectangular cross sections have been deduced starting from the three-dimensional theory of nonlinear elasticity. In the nonlinear setting the scaling of the energy determines the limit model: for "very small" energy a linear model is obtained, while for "large" energies different nonlinear limit models are deduced.…”

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confidence: 99%

“…In the nonlinear setting the scaling of the energy determines the limit model: for "very small" energy a linear model is obtained, while for "large" energies different nonlinear limit models are deduced. Some of the compactness results used in the present paper were inspired by the nonlinear counterpart studied in [4,5].…”

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confidence: 99%

“…The curve γ can be either open or closed but not completely straight. As in [4,5], the thinness of the "wall" is characterized by the assumption that lim ε→0 δ ε ε = 0.…”

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confidence: 99%

“…Recently Davoli [3] generalized the works [4,5] by studying thin-walled beams with both open and curved cross section, starting from the three-dimensional theory of nonlinear elasticity. Her impressive work does not contain a compactness theorem as detailed as ours and thus the Γ-convergence analysis is carried out in terms of strains and not, as customary, in terms of displacements.…”

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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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confidence: 99%

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confidence: 99%

“…The curve γ can be either open or closed but not completely straight. As in [4,5], the thinness of the "wall" is characterized by the assumption that lim ε→0 δ ε ε = 0.…”

mentioning

confidence: 99%

mentioning

confidence: 99%

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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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“…In particular, in [7] the authors consider the case of an inhomogeneous anisotropic rectangular cross section, and in [3,4] the analysis has been extended to the nonlinear context. Recently, Davoli [2] considered the case of homogeneous anisotropic beams with curved open cross section within the framework of finite deformations.…”

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confidence: 99%

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

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A third gradient elastic material resulting from the homogenization of a von Kármán ribbon‐reinforced composite

Jarroudi

2018

Z Angew Math Mech

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We study the homogenization of an elastic material reinforced with thin periodic parallel elastic ribbons. We assume perfect adhesion along the common interface between the matrix and the ribbons. We assume that the ribbons are isotropic von Kármán plates of highly contrasting rigidity with respect to that of the matrix. The main result is the derivation of an effective stored energy involving new degrees of freedom implying second and third gradient of displacements.

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Nonlinear Thin-Walled Beams With a Rectangular Cross-Section — Part I

Cited by 27 publications

References 14 publications

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

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

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

A third gradient elastic material resulting from the homogenization of a von Kármán ribbon‐reinforced composite

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