Asymptotic bounds on overall moduli of cracked bodies

The interactions among multiple parallel microcracks in an elastic body are examined asymptotically in an explicit and quantitative manner in order to reveal fully their so-called shielding and magni®cation e€ects on the overall moduli of the body. Based upon this asymptotic analysis, analytical upper and lower bounds are proposed for the overall moduli of bodies containing randomly distributed multiple parallel unbridged/bridged microcracks. The bounds are used to assess the accuracy of the approximate methods Ðthe dilute distribution approximation, the di€erential scheme and the self-consistent model Ð that are commonly used to determine the overall moduli of bodies containing multiple microcracks.