On variational dimension reduction in structure mechanics

TL;DR

This paper reviews and reformulates variational dimension reduction techniques in structure mechanics, emphasizing rigorous justification of classical low-dimensional models for thin structures.

Contribution

It provides a unified overview of variational dimension reduction methods and reformulates key ideas using explicit problems as examples.

Findings

Clarifies the main ideas behind dimension reduction techniques

Reformulates existing methods for better understanding

Uses explicit problems to illustrate the concepts

Abstract

The classical low-dimensional models of thin structures are based on certain a priori assumptions on the three-dimensional deformation and/or stress fields, diverse in nature but all motivated by the smallness of certain dimensions with respect to others. In recent years, a considerable amount of work has been done in order to rigorously justify these a priori assumptions; in particular, several techniques have been introduced to make dimension re- duction rigorous. We here review, and to some extent reformulate, the main ideas common to these techniques, using some explicit dimension-reduction problems to exemplify the points we want to make.