First-order and second-order wrinkling of thin elastic film laminated on a graded substrate

Wrinkling of the film-on-substrate structure is widely used in stretchable electronics, and the literature has shown that the substrate with gradient modulus has the advantage of maintaining the structure's integrity. Hence, it is necessary to have a comprehensive understanding of the evolution of the wrinkles in the film-on-elastic-graded-substrate structure. Firstly, a unified theoretical framework is established. Secondly, by using the energy method and the principle of minimum potential energy, the wrinkling behaviour of this structure is investigated, and the analytical expressions of the critical buckling strain, wavelength and wrinkling amplitude of this structure are obtained. Then, based on the energy increment method and Maxwell stability criterion, the post-wrinkling behaviour of this structure is studied, and the theoretical formulations for the critical buckling strain, wavelength and wrinkling amplitude are obtained. Finally, numerical examples are carried out to verify the correctness of the proposed formulations. Through the numerical results, it is found that by modulating the applied strain and decay rate of the exponential substrate, the wrinkling pattern of the film-on-exponential-substrate structure can be transferred from a first-order wrinkling pattern to a second-order wrinkling pattern. Moreover, by tuning the applied strain and characteristic length of the bilayer substrate, the wrinkling pattern of the film-on-bilayer-substrate structure can be determined. These findings are useful for guiding the design of stretchable electronics based on the film-on-elastic-graded-substrate structure.