Inserted: 5 nov 2010
Last Updated: 16 jul 2012
Journal: Mathematical Models and Methods in Applied Sciences (M3AS)
Our aim is to rigorously derive a hierarchy of one-dimensional models for thin-walled beams with a rectangular cross-section, starting from three-dimensional nonlinear elasticity. The different limit models are distinguished by the different scaling of the elastic energy and of the ratio between the sides of the cross-section. In this paper we report the first part of our results. More precisely, denoting by $h$ and $\delta_h$ the length of the sides of the cross section and by $\varepsilon_h$ the scaling factor of the bulk elastic energy, we analyse the cases in which $\delta_h/\varepsilon_h\to0$ (subcritical) and $\delta_h/\varepsilon_h\to1$ (critical).
Keywords: $\Gamma$-convergence, dimension reduction, nonlinear elasticity, thin-walled cross-section beams