Effective Spring Constant for Planar Arrays of Circular Cracks

A doubly-periodic array of coplanar circular cracks is used as a micromechanical model for crack bridging due to unbroken ligaments between the crack faces and for a localized deformation band. Two complementary approaches are described for determining the effective spring constant for the unbroken ligaments and the root-mean-square average of the stress intensity factor for the microcracks. The first approach exploits the periodicity to formulate a computationally efficient numerical solution which can be considered to provide exact results for the specific configuration of a doubly-periodic array. The second approach is similar in spirit to the self-consistent procedure for deriving the effective moduli of two-phase composites and leads to explicit analytical approximations. The results obtained for the spring constant by these two approaches are in good agreement over the range of cracked area fraction for which the micromechanical model is most reliable; a consistent difference, but a similar trend, is noted in the results obtained for the stress intensity factor. The relative merits and limitations of the two approaches are briefly discussed.