Any problem that has a characterization in the form of a “minimum energy” principle usually has a dual characterization in the form of a “complementary energy” principle. Substitution of trial fields into the “energy” and “complementary energy” functionals provides bounds for the energy associated with the actual solution.
Inhomogeneous media such as composites have a complex structure; correspondingly, suitable trial fields for substitution into the “classical” principles for such media are hard to find and some alternative principles, which make use of a simpler “comparison medium”, have proved useful. These alternative principles were developed for linear elasticity and similar linear problems by Z. Hashin and S. Shtrikman. The major purpose of the present work is to provide generalizations of these principles to non-linear problems. The presentation is abstract so that the essential structure is exposed. From a primal problem (corresponding to the minimum energy principle), a dual problem (corresponding to complementary energy) is derived and then some new problems are generated which, when specialized to linear systems, reproduce the Hashin-Shtrikman formalism. Relationships between the various problems (and the bounds that they generate) are discussed. Linear elasticity, non-linear electrostatics and a non-linear diffusion problem are considered as examples.