The strain energy density for a material undergoing small strain
For linear elastic deformations and small strains,
If the strain energy density is given by equation (1), then
(for linear elastic materials) the stress and strain can be related
We will show that equation (2) is equivalent to equation (3).
We start off with equation (2) and work backward.
For linear elastic materials,
Substituting equation (5) into equation (4), we get,
Recall that, for a second order tensor ,
and that for a fourth order rensor (substitution rule),
Differentiating equation (6) with respect to ,
Using the symmetry of the stiffness tensor, we have,
which is the same as equation (3). Hence shown.