Nonlinear mechanics of doubly curved shallow microshells

This paper performs a thorough investigation on the nonlinear size-dependent bending characteristics and natural frequencies of doubly curved shallow microshells. A nonlinear continuous model for a general doubly curved microshell is developed on the basis of Donnell's nonlinear shell theory and in the framework of the modified couple-stress strain gradient theory. In particular, the doubly curved microshell equations of motion of partial differential type are derived while accounting for geometric nonlinearities and small-scale effects. The continuous model is transformed into a discretised set of equations via application of the two-dimensional Galerkin technique. A large number of modes are retained in both linear and nonlinear investigations of microshells to ensure converged and reliable results. The linear natural frequencies are reported for various microshells of rectangular and square bases. The nonlinear static deflection curves for both in-plane and out-of-plane displacements are constructed and the effects of different parameters, such as the radius of curvature, sign of the radius of the curvature, and the length-scale parameter are examined.