Surface Phase-Field-Crystal-Helfrich model for out-of-plane deformations in thin crystalline sheets with lattice mismatch
Thin, flexible crystalline sheets exhibit unique elastic properties due to their ability to undergo out-of-plane deformations. Understanding this behavior requires a description that couples in-plane elasticity, out-of-plane deformation, and their coupling, taking the crystalline structure and its defects into account. We develop a multiscale description for these systems by extending the surface Phase-Field-Crystal-Helfrich model. The extension permits a spatially varying equilibrium lattice spacing, enabling the representation of localized lattice eigenstrain to mimic lattice mismatch in heterostructures. We validate the extended model against analytical predictions from classical F\"oppl-von K\'arm\'an equations for uniaxial compression and from Eshelby's inclusion problem. Using this validated framework, we then show how locally induced compressive stresses drive out-of-plane deformation in the sheets.