Asymptotic analysis of Poisson's equation in a thin domain and its application to thin-walled elastic beams and tubes

Mathematical justification of Kelvin–Voigt beam models by asymptotic methods

“…Later, the method was successfully applied by Trabucho and Viaño [29,30] for mathematically derive more complete models as the Saint-Venant, Timoshenko and Vlassov theories for thick and thin profile cross section, isotropic and anisotropic materials and curved beams (see Rodríguez-Seijo and Viaño [25,26], Alvarez-Dios and Viaño [1][2][3], Irago and Viaño [13], Irago, Kerdid and Viaño [14]). The nonlinear case was studied by Cimetière et al [8].…”

Coupled local/nonlocal models in thin domains

“…Concerning references for equations in thin domains we refer to [2,3,4,27,5,33] that develop some techniques and methods to understand the effects of the geometry of the thin domain on the solutions of elliptic and parabolic singular problems. We can find some applications in elastic beam theories (as torsion and warping functions) [28], lubrification [12], fluid flows as ocean dynamics, geophysical fluid dynamics, and fluid flows in cell membranes, see for instance [25].…”

An Asymptotic Approach to the Torsion Problem in Thin Walled Beams

“…We shall now study the limit behaviour of the variational problem defined in (16) and (17). Preliminarily, let us state some auxiliary results.…”

“…In 1970, Popov [10] extended the result to generally curved simply connected profiles. More recently, the most comprehensive treatment to date of De Saint Venant's torsion using classical asymptotic techniques is due to Rodriguez, Trabucho and Viaño [12][13][14][15][16]19]. They recover the limit solution for all possible typologies including domains with junctions and multiply connected ones.…”