Generalized Hooke's law for isotropic second gradient materials

TL;DR

This paper derives a comprehensive generalized Hooke's law for isotropic second gradient materials, identifying seven elastic moduli and establishing conditions for positive energy definiteness, with a focus on torsion measurement methods.

Contribution

It extends previous models by formulating a general linear isotropic constitutive relation for second gradient materials with seven elastic moduli.

Findings

Seven elastic moduli characterize isotropic second gradient materials.

Necessary and sufficient conditions for positive energy are established.

A measurement procedure for one elastic modulus via torsion is proposed.

Abstract

In the spirit of Germain the most general objective stored elastic energy for a second gradient material is deduced using a literature result of Fortun\'e & Vall\'ee. Linear isotropic constitutive relations for stress and hyperstress in terms of strain and strain-gradient are then obtained proving that these materials are characterized by seven elastic moduli and generalizing previous studies by Toupin, Mindlin and Sokolowski. Using a suitable decomposition of the strain-gradient, it is found a necessary and sufficient condition, to be verified by the elastic moduli, assuring positive definiteness of the stored elastic energy. The problem of warping in linear torsion of a prismatic second gradient cylinder is formulated, thus obtaining a possible measurement procedure for one of the second gradient elastic moduli.