Homogenized Maxwell's equations; A model for ceramic varistors

Varistor ceramics are very heterogeneous nonlinear conductors, used in devices to protect electrical equipment against voltage surges in power lines. The fine structure in the material induces highly oscillating coefficients in the elliptic electrostatic equation as well as in the Maxwell equations. We suggest how the properties of ceramic varistors can be simulated by solving the homogenized problems, i.e. the corresponding homogenized elliptic problem and the homogenized Maxwell equations. The fine scales in the model yield local equations coupled with the global homogenized equations. Lower and upper bounds are also given for the overall electric conductivity of varistor ceramics. These two bounds are associated with two types of failures in varistor ceramics. The upper bound corresponds to thermal heating and the puncture failure due to localization of strong currents. The lower bound corresponds to fracturing of the varistor, due to charge build up at the grain boundaries resulting in stress caused by the piezoelectric property of the varistor.