Shallow-shell models by Γ-convergence

Abstract

In this paper we derive, by means of Γ-convergence, the shallow-shell models starting from non-linear three-dimensional elasticity. We use the approach analogous to the one for shells and plates. We start from the minimization formulation of the general three-dimensional elastic body, which is subjected to normal volume forces and free boundary conditions and do not presuppose any constitutional behavior. To derive the model we need to propose how the order of magnitude of the external loads is related to the thickness of the body h, as well as to the order of the ‘geometry’ of the shallow shell. We analyze the situation when the external normal forces are of order hα, where α > 2. For α = 3 we obtain the Marguerre–von Kármán model and for α > 3 the linearized Marguerre–von Kármán model. For α ∈ (2,3) we are able to obtain only the lower bound for the Γ-limit. This is analogous to recent results for the ordinary shell models.