A displacement-based approach to geometric instabilities of a film on a substrate

When a thin film adhered to a compliant substrate is growing, it will eventually buckle in order to release the compressive stresses accumulated within the film due to growth. Such geometric instabilities caused by compressive stresses prevail among all living systems in nature and their outcomes range from highly beneficial to destructive. Therefore, understanding compression induced instabilities is of crucial importance. Note that the origin of the “compression” need not necessarily be differential growth, as it may be due to pre-stretch or thermal expansion. A commonly accepted solution strategy for instabilities in bilayer structures dates back to the seminal work of Allen and employs the Airy stress functions. Owing to its reliance on a stress-based approach, the Allen solution is limited to linear two-dimensional problems and its success depends entirely on choosing an appropriate Airy function. The main objective of this contribution is to circumvent these limitations via a displacement-based approach formally suitable for three-dimensional problems, anisotropic materials, and even applicable to finite deformations. Furthermore, the Allen solution in its original form is valid for the plane-stress condition but often it is mistakenly compared with the numerical simulations corresponding to the plane-strain condition. We analyze the subtle difference between the solutions associated with the plane-strain and plane-stress conditions. Next, the analytical solution is compared against the computational results using the finite element method via eigenvalue analysis. Finally, it is briefly explained how the current approach can be utilized beyond the classical bilayer systems.