On the Kirchhoff-Love hypothesis (revised and vindicated)

TL;DR

This paper revises and validates the Kirchhoff-Love hypothesis, demonstrating its usefulness in deriving accurate two-dimensional models for elastic plates, including effects of stretching, and resolving previous conflicts with $ extGamma$-convergence methods.

Contribution

It provides a revised version of the Kirchhoff-Love hypothesis that effectively derives 2D elastic plate models from 3D energy functionals, incorporating stretching effects.

Findings

Revised hypothesis successfully derives 2D models from 3D energy.

Bending energies include measures of stretching of the mid surface.

Compatibility with $ extGamma$-convergence is established in certain cases.

Abstract

The Kirchhoff-Love hypothesis expresses a kinematic constraint that is assumed to be valid for the deformations of a three-dimensional body when one of its dimensions is much smaller than the other two, as is the case for plates. This hypothesis has a long history checkered with the vicissitudes of life: even its attribution has been questioned, and recent rigorous dimension-reduction tools (based on $\Gamma$-convergence) have proven to be incompatible with it. We find that an appropriately revised version of the Kirchhoff-Love hypothesis is a valuable means to derive a two-dimensional variational model for elastic plates from a three-dimensional nonlinear free-energy functional. The bending energies thus obtained for a number of materials also show to contain measures of stretching of the plate's mid surface (alongside the expected measures of bending). The incompatibility with $\Gamma$-convergence also appears to be removed in the cases where contact with that method and ours can be made.