Shell energies derived from three-dimensional isotropic strain-gradient elasticity

TL;DR

This paper derives shell energies for thin elastic bodies incorporating strain-gradient effects, revealing that consistent asymptotic reductions start at cubic order-in-thickness and connect to classical shell models as length scales vanish.

Contribution

It introduces a systematic derivation of two-dimensional shell energies from three-dimensional strain-gradient elasticity, including new models with intrinsic length scales.

Findings

Shell energies include kinetic and stored surface energies at cubic order-in-thickness.

The theory reduces to Koiter's classical shell energy as length scales tend to zero.

Explicit computations for deformations demonstrate the model's applicability.

Abstract

We derive a class of two-dimensional shell energies for thin elastic bodies exhibiting small-length scale effects modeled via strain-gradient elasticity. Building on the final author's earlier work on plate models, the kinetic and stored surface energies arise as the leading cubic order-in-thickness expressions for three-dimensional kinetic energies with velocity-gradient effects and a broad class of isotropic stored energies, each possessing an intrinsic length scale $\ell$. These include both classical Toupin-Mindlin and more recent dilatational strain-gradient elastic stored energies. A key insight of this work is that consistent asymptotic reductions of strain-gradient theories necessarily begin at cubic order-in-thickness due to the natural scaling assumption $\ell = O(h)$ where $h$ is the thickness of the body. In the limit as the intrinsic length scales vanish, the theory reduces to Koiter's classical shell energy. We illustrate the theory using the shell energy derived from dilatational strain-gradient elasticity, computing the body force, edge tractions and edge double force densities required to support a variety of finite deformations.