Shallow shell models by Gamma convergence

TL;DR

This paper derives shallow shell models from 3D nonlinear elasticity using Gamma convergence, identifying different models based on the order of external forces relative to shell thickness.

Contribution

It provides a Gamma convergence derivation of shallow shell models without assuming specific constitutive behavior, considering various load magnitudes.

Findings

Derives Marguerre-von Kármán model for specific load scaling.

Obtains linearized Marguerre-von Kármán model for higher load scaling.

Establishes lower bounds for intermediate load scaling.

Abstract

In this paper we derive, by means of $\Gamma$-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 is the order of magnitudes of the external loads related to the thickness of the body $h$ as well as the order of the "geometry" of the shallow shell. We analyze the situation when the external normal forces are of order $h^\alpha$, where $\alpha>2$. For $\alpha=3$ we obtain the Marguerre-von K\'{a}rm\'{a}n model and for $\alpha>3$ the linearized Marguerre-von K\'{a}rm\'{a}n model. For $\alpha \in (2,3)$ we are able to obtain only the lower bound for the $\Gamma$-limit. This is analogous to the recent results for the ordinary shell models.