A multi-scale modeling framework for instabilities of film/substrate systems

•A multi-scale modeling framework for film/substrate systems is first developed.•The reduced-order model provides an efficient way to simulate large-scale instability problems with numerous undulations.•The proposed macroscopic model can significantly reduce DOF and CPU time by 90% and 98%, respectively.•Bifurcation portrait and post-buckling evolution are investigated from a quantitative standpoint.•A new bifurcation scenario with alternating packets of large and small undulations has been found numerically.

Spatial pattern formation in stiff thin films on soft substrates is investigated from a multi-scale point of view based on a technique of slowly varying Fourier coefficients. A general macroscopic modeling framework is developed and then a simplified macroscopic model is derived. The model incorporates Asymptotic Numerical Method (ANM) as a robust path-following technique to trace the post-buckling evolution path and to predict secondary bifurcations. The proposed multi-scale finite element framework allows sinusoidal and square checkerboard patterns as well as their bifurcation portraits to be described from a quantitative standpoint. Moreover, it provides an efficient way to compute large-scale instability problems with a significant reduction of computational cost compared to full models.

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