The interface behavior of a thin film bonded imperfectly to a finite thickness gradient substrate

The interface behavior of a thin film bonded imperfectly to a finite thickness gradient substrate with or without an adhesive interlayer is investigated in the present paper. The governing integro-differential equations for such an interface problem are formulated based on the equilibrium and compatibility conditions. The distribution of the interfacial shear stress and the transverse stress in the thin film is comprehensively studied. The stress singularity near the bonded edges is also focused on. It is found that a more durable film and a more reliable interface could be achieved if Young's modulus of the substrate decreases in its thickness direction. The effect of the adhesive interlayer is to change the stress field, which acts like a "rubber gasket in mechanical engineering" to reduce the stress concentration at the interface. The result should be very useful for the design of various film/substrate systems in flexible and stretchable electronic devices.