Bounds on effective magnetic permeability of three-phase composites with coated spherical inclusions

An effective model is developed to bound the effective magnetic permeability of three-phase composites with coated spherical inclusions. In the present model, the trial magnetic potential for the upper bound and the trial magnetic induction field for the lower bound are constructed to satisfy continuity interface conditions. According to the variational principle, the upper and lower bounds on the effective magnetic permeability of three-phase composites with coated spherical inclusions are derived. In this paper, trial magnetic potentials with different function forms are taken and the optimal upper bound is obtained for the trail magnetic potential corresponding to the third-order function. When the three-phase model degenerates into the composite spheres assemblage model [1], it is interesting that the optimal upper and lower bounds are the same. The effects of the volume fraction of coated spherical inclusions and the thickness and magnetic permeability of coated layers between the matrix and spherical inclusions on the effective magnetic permeability of composites are analyzed. The upper and lower bounds are finite non-zero values when the magnetic permeability of spherical inclusions tends to ∞ and 0, respectively.