Calculus of Variations and Geometric Measure Theory

C. Davini - L. Freddi - R. Paroni

Linear composite thin-walled beams by $\Gamma$-convergence. Part I: open cross-sections

created by freddi on 03 Jan 2014
modified on 01 Sep 2015


Published Paper

Inserted: 3 jan 2014
Last Updated: 1 sep 2015

Journal: Siam Journal on Mathematical Analysis
Volume: 46
Number: 5
Pages: 3296–3331
Year: 2014


We consider a beam whose cross-section is a tubular neighborhood, with thickness scaling with a parameter $\delta_\varepsilon$, of a simple curve $\gamma$ whose length scales with $\varepsilon$. To model a thin-walled beam we assume that $\delta_\varepsilon$ goes to zero faster than $\varepsilon$, and we measure the rate of convergence by a slenderness parameter $ \mathfrak{s}$ which is the ratio between $\varepsilon^2$ and $\delta_\varepsilon$. 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 $ \mathfrak{s}\in \{0, 1\}$ we derive two one-dimensional $\Gamma$-limit problems by letting $\varepsilon$ 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.

Keywords: Gamma-convergence, dimension reduction, linear elasticity, Thin-walled beams, curved cross-section, open cross-section