Nonlinear mechanical behaviour of microshells

Abstract

This study examines the nonlinear large-amplitude static and dynamic responses of a doubly curved shallow microshell in the framework of the modified couple stress (MCS) theory. To this end, the expressions for the classical and higher-order stresses and strains are consistently derived in an orthogonal curvilinear coordinate system employing the Novozhilov shell theory. The strain energy of the system is then consistently derived utilising the Novozhilov shell formulations in the framework of the MCS theory. The kinetic energy of the microshell is obtained while accounting for all out-of-plane and in-plane displacements. Furthermore, the work of the distributed out-of-plane load is accounted for and the energy dissipation is taken into account via the Rayleigh energy dissipation function. An assumed-mode technique is utilised to expand the out-of-plane and in-plane displacements via series expansions. The Lagrange equations are then utilised to derive the discretised equations of motion in the form of a set of nonlinearly coupled ordinary differential equations (ODEs). This set of nonlinear ODEs is solved making use of a continuation technique (for the nonlinear static and dynamic analyses) as well as an eigenvalue extraction method (for the linear natural frequency analysis). Extensive numerical simulations are carried out for both static and dynamic cases and the effects of different parameters, such as the radius of curvature, the magnitude and direction of the applied distributed load, and the small-scale parameter are investigated. The numerical results are constructed in the form of nonlinear static deflection curves, nonlinear dynamic frequency-amplitude diagrams, time traces, and phase-plane portraits.