Bending of a circular disk: from cylinder to ultrathin membrane

The article discusses methods of mathematical modeling of the stress-strain state of a circular disc at various ratios of its thickness to radius, ranging from $1$ to $10^{-3}$. For sufficiently thick plates, the solution of three-dimensional linear elasticity theory is used, for plates of medium thickness — the solution of linear bending equations within the Kirchhoff – Love hypotheses and nonlinear equations of Foppl – von Karman, and for ultrathin plates — the nonlinear equations of Adkins – Rivlin – Green. A comparative analysis of the solutions has been conducted, and ranges of relative thickness have been identified in which the considered solutions adequately describe the deformation process. This result enables the selection of a method for mathematical modeling of the stress-strain state of circular plates used in microelectromechanical systems that is most suitable for their relative size.