Effective behavior of clusters of microscopic cracks inside a homogeneous conductor

We study the effective behavior of a periodic array of microscopic cracks inside a homogenous conductor. Special emphasis is placed on a rigorous study of the case in which the corresponding effective conductivity becomes nearly singular - due to the fact that adjacent cracks nearly touch. It is heuristically shown how thin clusters of such extremely close cracks may macroscopically appear a single crack. These results have implications for our earlier work on impedance imaging.