Asymptotic Justification of Equations for von Kármán Membrane Shells

The objective of this work is to study the asymptotic justification of the two- dimensional equations for membrane shells with boundary conditions of von Kármán's type. More precisely, we consider a three-dimensional model for a nonlinearly elastic membrane shell of Saint Venant–Kirchhoff material, where only a portion of the lateral face is subjected to boundary conditions of von Kármán's type. Using technics from formal asymptotic analysis with the thickness of the shell as a small parameter, we show that the scaled three-dimensional solution still leads to the so-called two-dimensional equations of von Kármán membrane shell.