  Two mathematical models are developed within the theoretical framework of
large strain elasticity for the determination of upper and lower bounds on the
total strain energy of a finitely deformed hyperelastic body in unilateral
contact with a rigid surface or with an elastic substrate. The model problems
take the form of two continuous optimization problems with inequality
constraints, the solutions of which are used to provide an enclosure on the
uniform external load acting on the body's surface away from the contact zone.
