Strain-gradient theory for shear deformation free-form microshells: Governing equations of motion and general boundary conditions

Abstract

A first-order shear deformation (FOSD) free-form microshell model described in general curvilinear coordinates was developed within the complete framework of Mindlin’s form II linear isotropic strain-gradient theory (SGT), considering both strain-gradient and micro-inertia effects. The high-order governing equations of motion and consistent boundary conditions were simultaneously obtained through a variational formulation based on Hamilton’s principle. The established microshell model contains five strain-gradient material constants and one length parameter resulting from micro-inertia effects in addition to two classical Lamé constants, thereby capturing microstructure-induced size-dependent phenomena in both static and dynamic analyses. The constructed model within the general SGT can be flexibly reduced to those based on the modified strain-gradient, modified couple stress, and simplified strain-gradient theories. Moreover, by enforcing geometric restrictions, the general formulation for free-form microshells can be specialized for doubly curved microshells. Finally, the forced vibration results of a simply supported closed cylindrical microshell confirmed the reliability and accuracy of the proposed model and strain-gradient and high-order inertia effects on the size-dependent dynamic characteristics of microshells. Altogether, the presented mathematical formulations can provide a foundation for the in-depth understanding of microscale shell structures.