The complementary strain energy density is given by
For linear elastic materials
Let be the set of all statically admissible states of stress.
Let the complementary energy functional be
Then the Principle of Stationary Complementary Energy states that:
The Principle of Minimum Complementary Energy states that:
For linear elastic materials, the complementary energy functional is rendered an absolute minimum by the actual stress field.
Note that the complementary energy corresponding to the actual stress field is the negative of the potential energy corresponding to the actual displacement field.
Let be a solution of a the mixed boundary value problem of linear elasticity.
Let .
Define
Then satisfies equilibrium and the traction BCs, i.e.,
Since
and
we have,
We can also show that
Therefore,
Now,
Hence,
Therefore,
From equations (1), (2), and (3), we have,
Since on , we have,
Hence, proved.