Bounds for nonlinear composite conductors via the translation method

B E Peigney (1) , Michaël Peigney (2)

1 DAM/DIF - DAM Île-de-France
2 navier umr 8205 - Laboratoire Navier

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.

Mots clés: composite materials; nonlinear homogenization; conductivity; bounds; translation method

Domaines: Physique [physics]; Mécanique [physics]; Mécanique des solides [physics.class-ph]

B E Peigney, Michaël Peigney. Bounds for nonlinear composite conductors via the translation method. Journal of the Mechanics and Physics of Solids, 2017, 101, pp.93 - 117. ⟨10.1016/j.jmps.2017.01.017⟩. ⟨hal-01477382⟩