The paper deals with an equilibrium problem of two bodies adhesively bonded to each other along the part of interface between them. There is a crack on the rest part of the interface. The bonding between the bodies is described by “spring type” condition modeling a soft and thin material layer. We also impose non-penetration conditions and Tresca's friction conditions on the interface including both the adhesive layer and the crack. The non-penetration condition excludes mutual penetration of bodies. A formula for the derivative of the energy functional with respect to the crack length is obtained. It is shown that the derivative can be represented as a path-independent integral (J-integral). Moreover, a non-overlapping domain decomposition method for the bonded structure is proposed and its convergence is studied theoretically and numerically. The numerical study shows the efficiency of the proposed method and the importance of the non-penetration condition.