Large deformations of gradient elastic shells

Based on Budiansky’s nonlinear shell theory, a consistent mechanical formulation for a fully nonlinear gradient shell theory in the large deformation regime is developed. This work extends the Piola micro-macro identification procedure, which is grounded in the correspondence between micro and macro kinematical quantities within the framework of the Kirchhoff–Love hypothesis. The field equations are derived in their weak forms using a kinematically exact model, expressed in terms of the shell mid-surface membrane and bending strains along with their covariant derivatives. This formulation retains higher-order terms in the Lagrangian strain tensor, enabling a comprehensive representation of the deformation behavior. An extension of the Saint-Venant constitutive equation is introduced to incorporate the influence of an internal material length scale. A suitable finite element is developed to accommodate the additional degrees of freedom introduced by the strain gradient theory. To demonstrate the capability of the proposed formulation in capturing large deformations, numerical examples involving gradient microplates are presented. The results show that the predicted deflections of micro-scale plates are significantly smaller than those predicted by classical shell theory in both small and large deformation regimes.