Mixed-form extremum problem statements for small-deformation elastostatics

In situations outside those identified with routine elastic structural analysis, there is often a need for formulation in mixed form. Small-deformation elastostatics, expressed in terms of stress, strain, and displacement, is described here in the form of either of two complementary constrained-extremum problems. The set of governing equations and boundary conditions of elastostatics are obtained by an interpretation of the generalized “necessary conditions” for each of these fully mixed variational formulations. While the objectives in the problem statements are bilinear and therefore nonconvex, a simple proof is available to confirm that the solution to these conditions is an extremizer. Extensions of the basic formulation, obtained by the introduction of constraints or optimal relaxations, simulate constitutively nonlinear systems. The mixed formulations also provide a convenient representation of the mechanics requirements in connection with structural optimization.