Existence and uniqueness results for a nonlinear Budiansky-Sanders shell model

TL;DR

This paper studies a nonlinear extension of the Budiansky-Sanders shell model introduced by Destuynder, formulating it as a variational problem with energy $J_{BS}$ that combines a modified membrane and flexural energy ($W_{BS}$) and the work of applied forces. It proves the existence of a minimizer on a general class of shell geometries under controlled force magnitudes, and shows uniqueness of the minimizer when the applied forces are sufficiently small, thereby strengthening and generalizing prior results. The analysis hinges on a careful a priori control of nonlinear strain measures, weak lower semicontinuity of the energy, and a coercivity framework built around a special force class $\mathcal{A}$. Overall, the results guarantee well-posedness of the nonlinear shell model for broad geometries and small-to-moderate loading regimes, with implications for the mathematical foundation of nonlinear shell theories.

Abstract

A nonlinear shell model is studied in this paper. This is a nonlinear variant of the Budiansky-Sanders linear shell model. Under some suitable assumptions on the magnitude of the applied force, we will prove the existence of a minimizer for this shell model. In addition, we will also show that our existence result can be applied to all kinds of geometries of the middle surface of the shell. We will also show that the minimizer found in this fashion is unique, provided the applied forces are small enough. Our result hence extends the one given by Destuynder in [1].

Theorems & Definitions (12)

• Theorem 1.1
• Theorem 1.2
• Definition 3.1
• Proposition 3.2
• proof
• Remark 3.3
• Theorem 3.4
• Lemma 3.5
• proof
• proof : Proof of Theorem \ref{['s4-t1']}