Bounds for nonlinear composite conductors via the translation method

Hashin-Shtrikman type bounds are proposed for nonlinear isotropic composite conductors in two dimensions. Those bounds are obtained by combining the translation method with the idea of embedding the original two-dimensional problem in an extended problem of dimension 6. Invariance properties allow the evaluation of the bounds to be dramatically simplified. Explicit results are obtained for the problem of dielectric breakdown. Numerical results are given for two-phase composites governed by power-law energy functions. The obtained bounds are shown to improve on the linear comparison bounds of the Hashin-Shtrikman type that are delivered by the Talbot-Willis (1985) approach and the Ponte Castañeda (1991) variational method.