Linear Models for Composite Thin-Walled Beams by $\Gamma$-Convergence. Part I: Open Cross Sections
Cesare Davini, Lorenzo Freddi, Roberto Paroni · SIAM Journal on Mathematical Analysis · 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.