On the shell theory on the nanoscale with surface stresses

Below we discuss the derivation of the governing nonlinear shell equations taking into account the surface stresses. The surface effects are significant for the modeling of some structures as nanofilms, nanoporous materials and other nano-size structures. In particular, the surface stresses are responsible for the size effect, i.e. dependence of the material properties on the specimen size. The theory of elasticity with surface stresses is applied to the modeling of shells with nano-scaled thickness. It will be shown that the resultant stress and couple stress tensors can be represented as a sum of two terms. The first term in the sum depends on the stress distribution in the bulk material while the second one relates to the surface stresses. Hence, the resultant stress and couple stress tensors are linear functions with respect to the surface stresses. As an example the effective stiffness properties of a linear elastic Cosserat shells taking into account the surface stresses are presented.