A shell theory with scale effects and higher order gradients

This works follows a generalised continuum framework developed by Sansour (1998) to derive a strain gradient formulation suitable to address scale effects of structures where one dimension is very small (e.g. thin films, nano tubes etc.). Whereas a previous strain gradient approach by Sansour et al. (2009) considered the fully three-dimensional setting, the approach here proposes a shell theory which aims to run computations of thin structures more efficiently and to include scale effects. The theory features a generalised deformation description, new strain and stress measures. As consequence of these new quantities a corresponding generalised variational principle is formulated. The approach is completed by Dirichlet boundary conditions for the displacement field and its derivatives. A numerical example is presented based on a meshfree formulation which provides the necessary C1C1 continuity.