ABSTRACT
This chapter details the odd- and even-order variational principles to multiphase composites to derive the first-, second-, third-, and fourth-order bounds. Based on these bounds, a phase transition is particularly discussed from a mechanical point of view, which predicts a peculiar phenomenon of fluid-antifluid annihilation in terms of elastic rigidities. The formulae to compare the first four orders of bounds on conductivity are summarized in a table. When a solid is completely mixed with a fluid, for example, a mixture of sand and water with surface force neglected, there are four cases of mixing results that are theoretically exact as presented in another table. With increase of solid content, in all the four cases, the mixture goes through a mechanical phase transition process from a fluid state to a mud-like state, and finally to a solid state. This phenomenon is well observed in daily experience when mixing powder with water.
