Strain-gradient theory for shear deformation free-form microshells: Governing equations of motion and general boundary conditions. (1st July 2022) - Le, Toan Minh; Vo, Duy; Rungamornrat, Jaroon; Bui, Tinh Quoc
Graphical abstract: Highlights: Equations of motion and boundary conditions are derived for free-form microshells. First-order shear deformation and complete form-II strain gradient theories are used. Specializations to variants of SGT and doubly curved microshells are presented. Forced vibration responses of closed cylindrical microshells are investigated. Roles of strain gradients and micro-inertia on predicted response are elucidated. 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.