  In this article, we develop duality principles applicable to primal
variational formulations found in the non-linear elasticity theory. As a first
application, we establish the concerning results in details for one and
three-dimensional models. We emphasize such duality principles are applicable
to a larger class of variational optimization problems, such as non-linear
models of plates and shells and other models in elasticity. Finally, we
formally prove there is no duality gap between the primal and dual
formulations, in a local extremal context.
