The cement sheath around the oil wellbore casing is a potential site for the formation of annular cracks along the casing-cement and/or cement-rock interfaces. The lack of cement integrity may endanger environment, which makes important the prediction and assessment of damages at the cement interface with rock and casing. We use the theory of eigenstrains to analyze the effect of the presence of n partial interfacial cracks on the mechanical response of the casing-cement-rock system. Partial annular cracks are considered to experience leaking fluid pressure p, and rock formation is considered to be under the effect of in situ tectonic stresses. The theory of eigenstrains enables us to consider partial cracks as inclusions and cement sheath as an inhomogeneity embedded within the rock matrix. Moreover, using this methodology we can study the interacting effect of partial cracks with different central angles randomly distributed along the casing-cement and/or cement-rock interfaces. In contrast to the previous studies which are mostly restricted to one or two symmetric interfacial arc cracks in bi-material systems (a circular inclusion embedded in a matrix), the present study considers n randomly distributed partial cracks around the interfaces of an inclusion-inhomogeneity-matrix system.