On existence of energy minimizing configurations for mixtures of two imperfectly bonded conductors
We consider a domain filled with a suspension of heat conducting spheres of conductivity σp embedded in a matrix of lesser conductivity σm. It is assumed that there exists a thermal contact resistance at the sphere - matrix interface. The contact resistance is characterized by a scalar β, which has dimensions of conductivity per unit length. A current flux is prescribed on the domain boundary and we seek the energy minimizing configuration among all suspensions satisfying a resource constraint on the total volume of spheres. We establish the existence of an energy minimizing configuration within the class of polydisperse suspensions of spheres. The optimal suspension depends upon the size of the domain and consists of spheres of radii greater than or equal to Rcr = β-1(σ-1m - σ-1p)-1 or no spheres at all. Here Rcr is the ratio between the interfacial resistance and the mismatch between the resistivity of each phase.
Publication Source (Journal or Book title)
Control and Cybernetics
Lipton, R. (1998). On existence of energy minimizing configurations for mixtures of two imperfectly bonded conductors. Control and Cybernetics, 27 (2), 216-234. Retrieved from https://digitalcommons.lsu.edu/mathematics_pubs/772