Calculus of Variations and Geometric Measure Theory

C. Davini - L. Freddi - R. Paroni

Linear models for composite thin-walled beams by $\Gamma$-convergence. Part II: closed cross-sections

created by freddi on 29 Jul 2014
modified on 11 Jan 2017


Published Paper

Inserted: 29 jul 2014
Last Updated: 11 jan 2017

Journal: Siam Journal on Mathematical Analysis
Volume: 46
Number: 5
Pages: 3332–3360
Year: 2014


We consider a beam whose cross-section is a tubular neighborhood of a simple closed curve $\gamma$. We assume that the wall thickness, i.e., the size of the neighborhood, scales with a parameter $\delta_\varepsilon$ while the length of $\gamma$ scales with $\varepsilon$. We characterize a thin-walled beam by assuming that $\delta_\varepsilon$ goes to zero faster than $\varepsilon$. Starting from the three dimensional linear theory of elasticity, by letting $\varepsilon$ go to zero, we derive a one-dimensional $\Gamma$--limit problem for the case in which the ratio between $\varepsilon^2$ and $\delta_\varepsilon$ 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.