Shaping stress: how curvature governs the mechanics of film-substrate systems undergoing volumetric expansion.

Composite structures with conformal coatings on porous backbones are widely employed in energy storage, flexible electronics, and biomedical devices. However, expansion-induced stresses can lead to mechanical degradation of the coatings, thereby limiting their performance. In this study, we use finite element simulations to evaluate how substrate morphology - including curvature, shape, and coating configuration - governs the mechanical response of expanding thin-film coatings, using lithiation of silicon anodes as a model case of extreme expansion. The peak stress and strain energy density of the expanding film are used as indicators of failure, and empirical relationships are introduced to predict their scaling with curvature. We find that films on shell-backbones consistently exhibit higher tensile stress but lower strain energy density than those on solid-backbones, reflecting a trade-off between cracking and delamination risks. In all studied configurations, substrates with positive Gaussian curvature amplify the in-plane stresses of the film and increase the propensity for mechanical degradation, whereas substrates with negative Gaussian curvature effectively redistribute stresses and enhance the mechanical resilience. This work highlights the advantages of shell-backbone saddle substrates for expanding thin-film systems and provides general guidelines for the design of mechanically robust architected composites and shell-based metamaterials.