Generalized Hooke's law for isotropic second gradient materials

Francesco Dell'Isola (1, 2) , Giulio Sciarra (3) , Stefano Vidoli (1)

1 Dipartimento di Ingegneria Strutturale e Geotecnica
2 Laboratorio Strutture e Materiali Intelligenti - Fondazione Tullio Levi-Civita
3 Dipartimento Ingegneria Chimica Materiali Ambiente

In the spirit of Germain the most general objective stored elastic energy for a second gradient material is deduced using a literature result of Fortuné & Vallée. Linear isotropic constitutive relations for stress and hyperstress in terms of strain and straingradient 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.

Domaines: Mécanique [physics.med-ph]

Francesco Dell'Isola, Giulio Sciarra, Stefano Vidoli. Generalized Hooke's law for isotropic second gradient materials. Royal Society of London, 2009, pp.20. ⟨hal-00499570⟩