An interface of a circular arc formed by two isotropic, homogeneous elastic materials is investigated. It is shown that L integral satisfies the conservation law for the interface if it is perfectly bonded, in frictionless contact or separated such as in a crack with the origin of the coordinate system being located at the center of the circular arc. The property of path independence of the L integral is applied to an interfacial crack problem, to obtain the stress intensity factors, where the interfacial crack is located along the arc of the circular inclusion embedded in infinite matrix. It is assumed here that the contact zone exist as in the model proposed by Comninou, thus removing the overlapping of the materials along the interface. Another example is shown for case of a circular interfacial crack in the matrix of finite size, where the stress intensity factors are determined by computing a value of the L integral numerically along the path far from the crack tip.