  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.
