Electrical properties of random checkerboards at finite scales

Abstract

Under investigation is the scale dependent electrical conductivity (and resistivity) of two-phase random checkerboards at arbitrary volume fractions and phase contrasts. Using variational principles, rigorous mesoscale bounds are obtained on the electrical properties at finite scales by imposing a boundary condition that is either uniform electric potential or uniform current density. We demonstrate the convergence of these bounds to the effective properties with increasing length scales. This convergence gives rise to the notion of a scalar-valued scaling function that accounts for the statistical nature of the mesoscale responses. A semi-analytical closed form solution for the scaling function is obtained as a function of phase contrast, volume fraction and the mesoscale. Finally, a material scaling diagram is constructed with which the convergence to the effective properties can be assessed for any random checkerboard with arbitrary phase contrast and volume fraction.

Electrostatic and magnetostatic properties of random materials

“…For an anisotropic medium, the constitutive relations cannot be expressed in a simple form as in Eq. (17). Instead, they have to be expressed as…”

“…Such a scale-dependent homogenization technique has already been employed in many different settings, e.g. thermal conductivity [14], [15], [16], linear or finite elasticity or thermo-elasticity, elasto-plasticity, viscoelasticity, electrical conductivity [17] and permeability [8]. All the linear continuum mechanics/physics problems have been found to share common trends in terms of a scaling function, leading to a question whether they will carry over to electromagnetism.…”