We show that the mixed-mode fracture/adhesion energy of an interface with periodically varying cohesive interactions generally depends on the size of the cohesive zone near the tip of a crack along the interface: it is equal to the average cohesive energy of the interface, if the cohesive zone size is much larger than the period of cohesive interaction but becomes the peak value of the local cohesive energy when the opposite is true. It is also interesting that the cohesive zone size can be strongly influenced by the geometry and velocity of the crack. As an example of geometry-constrained cohesive zone, we consider peeling of a thin film on substrate and show that the cohesive zone size under 90° peeling scales with the bending stiffness of the film, while that under 0° peeling scales with the tension stiffness of the film. As an example of a velocity-constrained cohesive zone, we consider crack propagation along an interfacial layer of weak molecular bonds joining two elastic media and show that the cohesive zone size can be altered by an order of magnitude over feasible regimes of crack velocity. These results suggest possible strategies to control fracture/adhesion strength of interfaces in both engineering and biological systems.