A variational approach to the overall sink strength of a nonlinear lossy composite medium

An extension of the Hashin-Shtrikman variational structure to nonlinear problems, initiated by J. R. Willis (J. appl. Mech. 50, 1202-1209 (1983)) and developed generally by D. R. S. Talbot & J. R. Willis (IMA J. appl. Math.35, 39-54 (1985)) is applied to the problem of homogenization of the semilinear steady-state diffusion equation ∇2c- (∂W/∂c) (c, x) +K(x)= 0, where the functionWis convex incbut varies randomly with positionx. A strict definition for an ‘overall’ functionŴwhich defines the homogenized medium is given and its properties are discussed. Bounds forŴare then developed from variational principles. Classical energy principles yield bounds analogous to the Voigt and Reuss bounds of elasticity theory while the new ‘Hashin-Shtrikman’ structure yields tighter bounds, which are sensitive to the two-point statistics of the medium. Explicit results are presented for a particular two-phase com­posite medium for which one phase is linear (so thatWis quadratic) whereas the second has a quadratic nonlinearity (so thatWis cubic).