A justification of the Donnell–Mushtari–Vlasov model by the asymptotic expansion method

Abstract

We apply here the asymptotic expansion method to the nonlinear three-dimensional equations for the equilibrium of a thin elastic shell, following the work of Destuynder (1980) for the linear case. We consider two small parameters: ε that is the half-thickness of the shell, and ρ that is the ratio of ε to the lower bound of the radius of curvature of the middle surface of the shell. We show that the leading term of the asymptotic expansion is the solution of the Donnell–Mushtari–Vlasov model (cf. Sanders [1963]), if ρ is of the order ε². We also study the cases ρ=O(ε^2+r), r > 0 and ρ= constant as ε→0.