In large-strain elasticity, the stationary potential energy and the complementary-type energy of a finitely deformed hyperelastic body in frictionless unilateral contact bracket the minimum potential energy from above and below. Casting the contact boundary value problem as two continuous optimization problems with inequality constraints — one over kinematically admissible displacements, one over statically admissible deformation gradients — turns that bracket into guaranteed upper and lower bounds on the uniform external load applied away from the contact zone. The construction uses trial deformation gradients rather than stresses, so the non-linear constitutive relation never has to be inverted, and it extends to two bodies in mutual non-penetrative contact and to internally constrained (e.g. incompressible) materials via a Lagrange multiplier.
This paper develops mathematical models within large strain elasticity theory to determine rigorous upper and lower bounds on the total strain energy of finitely deformed hyperelastic bodies in unila…