Effective behavior of clusters of microscopic cracks inside a homogeneous conductor

Article type: Research Article

Authors: Bryan, Kurt | Vogelius, Michael

Affiliations: Department of Mathematics, Rose‐Hulman Institute of Technology, Terre Haute, IN 47803, USA | Departement de Mathematiques, Ecole Polytechnique Federale de Lausanne, CH‐1015 Lausanne, Switzerland, and Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, USA

Abstract: We study the effective behaviour of a periodic array of microscopic cracks inside a homogeneous 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 as a single crack. These results have implications for our earlier work on impedance imaging.

Journal: Asymptotic Analysis, vol. 16, no. 2, pp. 141-179, 1998